## Information Measurement with SQL Server, Part 2.2: The Rényi Entropy and Its Kin

**By Steve Bolton**

…………I kicked off this far-ranging series on using SQL Server to quantify information by discussing two of the earliest and most important measures, the Hartley function and Shannon’s Entropy. These foundations of information theory are intimately related to a more general measure, Rényi Entropy, which is a bit more complex but nonetheless worthwhile to discuss, since its unites many different information measures under one umbrella. The underlying math formula isn’t much more difficult than the one for Shannon’s Entropy I posted last time around, but its alpha parameter (α-parameter) enables it to give rise to a wider range of results. This assortment of entropy types can be adapted to solving a wider range of problems.

…………A general-purpose means of parameterizing the concept of entropy was the explicit goal of Alfréd Rényi (1920-1971), a mathematician who overcame anti-Semitic persecution at the hands of Hungary’s World War II regime, which was allied with the Nazis and passed stringent laws against their own Jewish minority. Incidentally, his coffee addiction was the inspiration for the colorful saying, “A mathematician is a device for turning coffee into theorems”, which apparently predates the saying, “A programmer is a machine that turns coffee into code.”[1] The formula that bears his name has proven useful in many diverse fields and industries, from quantum mechanics to fractals to ecology and statistics, where it is useful in deriving indices of diversity.[2] When the α-parameter is set to 2, it becomes Collision or Quadratic Entropy (or sometimes just “the Rényi Entropy”), which “has been used in physics, in signal processing and in economics” and is attractive to statisticians because they “have found an easy way to estimate it directly from samples.”[3] Given that it has multifarious uses and is both easy to explain and compute in SQL Server, it makes sense to add it to our toolbelt, in order to derive our own DIY data mining methods. One thing I’ve come to realize since writing my initial A Rickety Stairway to SQL Server Data Mining tutorial series awhile back is that the data mining marketplace is decades behind the research in some respects, to the point where all of the available software taken together is probably several orders of magnitude behind the formulas available in the math books. If we encounter situations where the particular algorithms that would work best for our use cases aren’t yet implemented, waiting might not be an option, and if we’re going to learn to build our own, Rényi Entropy is bound to come in handy at some point.

**Coding the Range of α-Parameter Values**

The key to understanding Rényi Entropy is grasping how it intersects with its cousins. The α-parameter can be any non-negative value except for 1, which would result in a divide-by-zero error since one of the terms is 1 / (1 – α). This result is multiplied by a logarithm taken on the sum of the probabilities taken to the power of α; this differs from Shannon’s famous equation in that his log operation is performed on each probability and then multiplied by that probability, then summed afterwards and multiplied by -1. This logic is all implemented in the INSERT operation in Figure 1, in which most of the code is devoted to deriving actual probability values from the proportions of a single float column, taken from the same Higgs Boson dataset[4] I’ve been using for practice purposes for the past few tutorial series. I omitted any display of the intermediate proportions stored in the @EntropyTable variable, since they’re identical to the sample results posted in last week’s article. Evidently, it’s economical to compute these entropies, given that this code takes less than 2 seconds to run on my ancient abacus of a development machine (I might as well be using quipus) and most of that was devoted to deriving the probabilities for all 11 million rows; if we already had probability values derived from some other source like estimates, sampling or deductive methods, it would run practically instantaneously. I also hard-coded the results we already calculated over the last two articles for the Hartley and Shannon Entropies, given that they are also unchanged. The final SELECT in Figure 2 provides the same information from another vantage point, by plugging a range of α-values rather than a single parameter into the Rényi formula.

…………Note that as the values approach 1, they converge towards the Shannon Entropy, although they can never quite reach it because of the divide-by-zero error that would occur if plugged in the forbidden parameter value of 1.Likewise, the return values approach the Hartley measure as the parameter values approach 0; this acts as a cap on the permissible values, hence the frequent use of the synonym Max Entropy. Because SQL Server’s calculation powers are impressive yet finite, we can’t plug in values approaching infinity (if such a thing were possible). In fact, we can’t even go much higher than α-values of about 50 (which are rarely used in practice anyways) on this particular dataset without getting floating point errors. Figure 3 is sufficient to nevertheless sufficient to illustrate how the Rényi results approach an information measure known as the Min Entropy, which acts as a cap on the other end. References to it as the Chebyshev Entropy[5] seem to be few and far between outside the realm of quantum physics, but I’ve run into a few here and there. It is nonetheless easy to calculate, given that all we have to do is find the maximum probability value and take the negative log. As Wikipedia points out quite succinctly, “The min entropy is never greater than the ordinary or Shannon entropy (which measures the average unpredictability of the outcomes) and that in turn is never greater than the Hartley or max entropy, defined as the logarithm of the number of outcomes.”[6] It is also useful as the “most conservative way of measuring the unpredictability of a set of outcomes,” a property that makes it useful in determining randomness and quantum cryptography, in a way that the Shannon Entropy can’t handle.[7] Each Rényi value in between the fences set by the Min, Hartley and Shannon Entropies has its own distinct uses in ascertaining “the unpredictability of a nonuniform distribution in different ways.”[8] We can thus choose whatever α-parameter is ideal for dealing with the particular distribution of our data. As an amateur, I’m lacking in the experience needed to determine which α-values are best-suited to specific problems, but this code at least provides a launching pad for exploring such questions.

** Figure 1: The Rényi Entropy in T-SQL**DECLARE @LogarithmBase decimal(38,36) = 2 –2.7182818284590452353602874713526624977 — 10

DECLARE @AlphaParameter decimal(38,35) = 2

DECLARE @Count bigint, @DistinctValueCount bigint, @RenyiEntropy float, @MinOrChebyshevEntropy float

SELECT @Count = Count(*)

FROM Physics.HiggsBosonTable

WHERE Column1 IS NOT NULL

DECLARE @EntropyTable table

(Value decimal(33,29),

ValueCount bigint,

Proportion float,

SummationInput float

)

INSERT INTO @EntropyTable

(Value, ValueCount, Proportion, SummationInput)

SELECT Value, ValueCount, Proportion, Power(Proportion, @AlphaParameter) AS SummationInput

FROM (SELECT Value, ValueCount, ValueCount / CAST(@Count AS float) AS Proportion

FROM (SELECT Column1 AS Value, Count(*) AS ValueCount

FROM Physics.HiggsBosonTable

WHERE Column1 IS NOT NULL

GROUP BY Column1) AS T1) AS T2

SELECT @RenyiEntropy = 1 / (1 – @AlphaParameter) * Log(SUM(SummationInput), @LogarithmBase), @MinOrChebyshevEntropy = –1 * Log(Max(Proportion), @LogarithmBase)

FROM @EntropyTable

SELECT @RenyiEntropy AS RenyiEntropy, @MinOrChebyshevEntropy AS MinEntropy, 13.2879304339032 AS ShannonsEntropy, 14.772263018717 AS HartleyOrMaxEntropy

** Figure 2: A Range of Sample T-SQL**–WHOLE RANGE OF RENYI ENTROPY

SELECT AlphaParameter, 1 / (1 – AlphaParameter) * Log(SUM(Power(Proportion, AlphaParameter)), @LogarithmBase) AS ResultingRenyiEntropy

FROM (VALUES (0.01), (0.5),(1.01),(2),(3),(4),(5),(10),(50)) AS T1(AlphaParameter)

LEFT JOIN @EntropyTable AS T2

ON 1 = 1

GROUP BY AlphaParameter

ORDER BY AlphaParameter

__Figure 3: Results from the Higgs Boson Dataset
__

…………Jose C. Principe, a neural net expert who teaches at the University of Florida, has written a handy 35-page guide to using the Rényi Entropy for tasks even further beyond my level of inexpertise, such as deriving measures like Information Potential based on probability distribution function (PDF) estimates. This quickly leads into the bleeding edge topic of information geometry, which I won’t touch with a ten-foot pole for a few more years, at least until I have a better understanding of such mind-blowing concepts as statistical Riemann manifolds and multidimensional spaces. The introductory pages are nevertheless useful in illustrating how Rényi Entropy works on simpler expressions of information. This is particularly true of the excellent diagram on the page 4, which shows how the various α-values measure the distance of a PDF to the origin point, thereby filling in the available space for information content in different proportions. As we shall see later in this series of self-tutorials, plugging different α-values into a related concept known as the Rényi Divergence also allows us to measure the distances between two PDFs in various ways. This is intimately related to the Kullback–Leibler Divergence in much the same way as the Rényi Entropy is related to the Shannon Entropy. The KL-Divergence is among the most important measures in information theory and related fields so I’ll be writing quite a bit on the subject later on in the series, as I did with the Shannon Entropy in the last article. It also goes by the name of Relative Entropy, but I’ll have to put off that discussion for a later time, when I can set aside a separate segment of the series for distance and divergence measures between probability distributions. Over the next four articles I’ll stick to the topic of entropy measures on single distributions, including Leaf and Root Entropy, which are inextricably related to the SSDM algorithm I covered in A Rickety Stairway to SQL Server Data Mining, Algorithm 3: Decision Trees.

[1] See the __Wikipedia__ page “Alfréd Rényi” at http://en.wikipedia.org/wiki/Alfr%C3%A9d_R%C3%A9nyi

[2] See the __Wikipedia__ article “Rényi Entropy” at http://en.wikipedia.org/wiki/R%C3%A9nyi_entropy

[3] pp. 6-7, Principe, Jose C., 2009, “Rényi Entropy,” course notes posted at the __University of Florida__ webpage http://www.cnel.ufl.edu/courses/EEL6814/renyis_entropy.pdf

[4] I downloaded this publicly available dataset from University of California at Irvine’s Machine Learning Repository__ and while back and converted it to a SQL Server table, which now takes up about 5 gigabytes in a sham DataMiningProjects database.__

[5] One source I found it mentioned in was Bengtsson, Ingemar, 2008, __Geometry of Quantum States: An Introduction to Quantum Entanglement__. Cambridge University Press: New York.

[6] See the __Wikipedia__ article “Min Entropy” at http://en.wikipedia.org/wiki/Min_entropy

[7] *IBID*.

[8] *IBID*.

## Information Measurement with SQL Server, Part 2.1: The Uses and Abuses of Shannon’s Entropy

**By Steve Bolton**

…………In the first installment of this wide-ranging series of amateur tutorials, I noted that the Hartley function indeed returns “information,” but of a specific kind that could be described as “newsworthiness.” This week’s measure also quantifies how much we add to our existing knowledge from each new fact, but through explicitly stochastic methods, whereas the Hartley function is more closely related to counts of state descriptions than probabilities.[1] Shannon’s Entropy is of greater renown than its predecessor, but isn’t much more difficult to code in T-SQL. In fact, it should be a breeze given that I already posted a more advanced version of it tailor-made for Implementing Fuzzy Sets in SQL Server, Part 10.2: Measuring Uncertainty in Evidence Theory, just as I had already coded a modified version of Hartley’s measure for Implementing Fuzzy Sets in SQL Server, Part 9: Measuring Nonspecificity with the Hartley Function.

…………Historically, the real difficulty with the metrics of information theory is with their interpretation, largely due to the broad and fuzzy meaning of the imprecise term, “information.” As we shall see, many brilliant theorists far smarter than we have made the mistake of overextending it beyond its original domain, just as some thinkers have gotten carried with the hype surrounding fuzzy sets and chaos theory. Like those cutting-edge topics, Shannon’s Entropy and its relatives within information theory are indeed powerful, but can go badly wrong when misapplied outside its specific use cases. It has really subtle yet mind-blowing implications for all of the different classes of information measurement I hope to cover in this wide-ranging and open-ended series, not all of which are fully understood. My purposes in this series is to illuminate datasets from every possible direction, using a whole smorgasbord of measures of meaning (semantic information), randomness, complexity, order, redundancy and sensitivity to initial conditions, among others; since many of these metrics serve as the foundation of many mining algorithms, we can use them in a SQL Server environment to devise DIY data mining algorithms. Shannon’s Entropy intersects many of these disparate class of information in multifarious ways, such as the fact that it is equal to the Hartley measure in the case of uniform distributions (i.e. when all values are equally likely). It is not a measure of what is known, but of how much can be learned by adding to it; entropy measures rise in tandem with uncertainty precisely we because we can learn more from new facts when we don’t already know everything. It can thus be considered complementary to measures of existing knowledge, such as Bayes Factors. I’ve read more on information theory than many other fields I’ve commented on over the past couple of years, when I started writing self-tutorials on data mining, but that doesn’t mean I understand all of these complex interrelationships well. In fact, I’m writing this series in part because it helps me absorb the material a lot faster, and posting it publicly in the hopes that it can at least help readers avoid my inevitable mistakes. I lamented often in A Rickety Stairway to SQL Server Data Mining that SSDM was woefully under-utilized in comparison to its potential benefits, but the same can also be said of the algorithms that underpin it.

**Channel Capacity as a Godsend**

Few of these information metrics are as well-known as the measure that Claude E. Shannon (1916-2001), a key American cryptographic specialist during the Second World War, introduced in 1948 in a two-part journal article titled “A Mathematical Theory of Communication.”[2] Most of it consists of math theorems and formulas that only got thicker as researchers built on his theory over the years, but it is noteworthy to mention that he credits earlier work by electrical engineers Ralph Hartley[3] (1888-1970) and Harry Nyquist (1889-1976) in the opening paragraphs.[4] I’ll skip over most of the proofs and equations involved – even the ones I understand – for the usual reasons: as I’ve pointed out in past articles, users of mining tools shouldn’t be burdened with these details, for the same reason that commuters don’t need a degree in automotive engineering in order to drive their cars. Suffice it to say that the original purpose was to determine the shortest possible codes in the case of noisy transmission lines. In this effort to extend coding theory, Shannon gave birth to the whole wider field of information theory. Masud Mansuripur, the chair of the Optical Data Storage Department at the University of Arizona, sums up Shannon’s surprising mathematical discovery best: “In the past, communications engineers believed that the rate of information transmission over a noisy channel had to decline to zero, if we require the error probability to approach zero. Shannon was the first to show that the information-transmission rate can be kept constant for an arbitrarily small probability of error.”[5] It is also possible to determine the information-carrying capacity of the channel at the same time.[6]

…………The idea is still somewhat startling to this day, as many writers in the field of information theory have pointed out ever since; it almost seems too good to be true. Basically, the idea is to engineer a “decrease of uncertainty as to what message might have been enciphered,” [7] in part by leveraging combinatorics to derive probabilities. For example, the Law of Large Numbers states that rolling a pair of dice is far more likely to result in a number like eight, since there are multiple ways of adding up both dice to get that result, whereas there’s only one combination apiece for snake eyes or boxcars. The edges of the resulting dataset are thus far less likely than the values in the middle, just as we’d find in a Gaussian or “normal” distribution, i.e. the ubiquitous bell curve. This gives rise to a dizzying array of axioms and lemmas and a whole set of procedures for determining codes, all of which I’ll skip over. For perhaps the first time in the history of this blog, however, I have a sound reason for posting the underlying equation: *H* = -Σ p_{i} log_{b} p_{i}. It’s not a household name like E = mc^{2} or 2 + 2 = 4, but it’s still one of the most famous formulas of all-time. It relatives can be spotted in the literature by the telltale combination of a negative summation operator with a logarithm operation. Aside from the fact that it possesses many ideal mathematical properties[8], “it was proven in numerous ways, from several well-justified axiomatic characterizations, that this function is the only sensible measure of uncertainty in probability theory.”[9]

**Translating H into Code**

As with Hartley’s measure, a base 10 logarithm results in units known as hartleys or bans (as famed cryptographer Alan Turing called them). With Euler’s Number (as in the natural logarithm) the units are known as nats and with base 2, they’re referred to as shannons or more commonly, bits. The code in Figure 1 uses bits, but users can uncomment the numbers that follow to use one of the other units. It is also worth noting that physicist Léon Brillouin used a “negentropy” measure that is basically the inverse of *H*, to compensate for the fact that Shannon’s Entropy measures information in terms of uncertainty, which is a bit counter-intuitive; it never really caught on though.[10]

…………My sample code derives some fake probabilities by taking the known proportions of values for a column in the same Higgs Boson dataset[11] I’ve been using for practice purposes for several tutorial series. The probabilities are then multiplied by the logs for those probabilities, then summed across the dataset and multiplied by -1. It’s actually quite simpler than it looks; the table variable and INSERT statement could be done away with altogether if we already had probability figures calculated through some other means, such as some sort of statistical sampling method or even reasoning out the probabilities from the underlying process, as writers on information theory often do with dice and cards to illustrate the concepts. In fact, if we know the underlying probability distribution in advance, we can sometimes calculate the entropy in reverse by using special formulas specific to certain distributions, such as the normal. Moreover, all of the code pertaining the @UniformProportion is included solely for the purpose of validating the results, which equal the results we received in the last article for the DISTINCT version of the Hartley function, as expected. If need be, users can also validate their results using Lukasz Kozlowski’s Shannon Entropy Calculator. All of that can code can also be removed at will. Despite all of this extraneous baggage, the code ran in just 1.9 seconds according to the Client Statistics, on 11 million rows of float values on my beat-up, wheezing semblance of a development machine. The execution plan was uneventful and consisted mainly of seeks on the nonclustered index I put on the column a couple of tutorial series ago.

** Figure 1: Shannon’s Entropy in T-SQL**DECLARE @LogarithmBase decimal(38,36) = 2 –2.7182818284590452353602874713526624977 — 10

DECLARE @Count bigint, @DistinctValueCount bigint, @ShannonsEntropy float, @EntropyForUniformDistribution float,

@MaxProbability float, @ChebyshevEntropy float, @MetricEntropy float, @UniformProportion float

SELECT @Count = Count(*), @DistinctValueCount = Count(DISTINCT Column1)

FROM Physics.HiggsBosonTable

WHERE Column1 IS NOT NULL

SELECT @UniformProportion = 1 / CAST(@DistinctValueCount as float)

FROM Physics.HiggsBosonTable

WHERE Column1 IS NOT NULL

DECLARE @EntropyTable table

(Value decimal(33,29),

ValueCount bigint,

Proportion float,

SelfInformation float

)

INSERT INTO @EntropyTable

(Value, ValueCount, Proportion, SelfInformation)

SELECT Value, ValueCount, Proportion, –1 * Proportion * Log(Proportion, @LogarithmBase) AS SelfInformation

FROM (SELECT Value, ValueCount, ValueCount / CAST(@Count AS float) AS Proportion

FROM (SELECT Column1 AS Value, Count(*) AS ValueCount

FROM Physics.HiggsBosonTable

WHERE Column1 IS NOT NULL

GROUP BY Column1) AS T1) AS T2

SELECT * FROM @EntropyTable

SELECT @ShannonsEntropy = SUM(SelfInformation), @EntropyForUniformDistribution= SUM (@UniformProportion * Log(@UniformProportion, @LogarithmBase)) * –1, @MaxProbability = Max(Proportion)

FROM @EntropyTable

— ====================================

— SOME RELATED ENTROPIES

— ====================================

SELECT @ChebyshevEntropy = –1 * Log(@MaxProbability, @LogarithmBase)

SELECT @MetricEntropy = @ShannonsEntropy / CAST(@Count as float)

SELECT @ShannonsEntropy AS ShannonsEntropy, @MetricEntropy AS MetricEntropy, @ChebyshevEntropy AS ChebyshevEntropy, @EntropyForUniformDistribution as EntropyForUniformDistribution

__Figure 2: Calculating Shannon’s Entropy on the Higgs Boson Dataset
__

…………Note that in the last section of the code, I tacked on the Metric and Chebyshev Entropies, which are trivial to calculate once we have building blocks like Shannon’s Entropy. References to the Chebyshev Entropy[12], a.k.a the Min Entropy because it represents the minimum amount a variable can exhibit, seem to be few and far between outside the realm of quantum physics. Metric Entropy, on the other hand, can serve as a simple measure of randomness[13], which is more likely to be useful to SQL Server users. These are among a couple dozen extensions and relatives of Shannon’s Entropy, a few of which I’ve already dealt with, like the Hartley Entropy (i.e. the Max Entropy). I won’t discuss Differential Entropy, the extensions to continuous variables using various methods of calculus, because SQL Server data types like float, decimal and numeric are actually discrete in the strict sense; they can’t actually represent infinitesimal grades in between values, any more than the data types of any other software can on finite computers. Nor will I delve into the myriad applications that have been developed from Shannon’s Entropy for signal transmission and coding theory, since these are off-topic.

**Detection of “Randomness” and Other Uses**

Its relationship to cryptography might be a more appropriate subject, but for the sake of brevity I’ll limit my discussion to pointing out how it sheds light on the nature of “randomness.” Like “information,” it’s a broad term that raises the danger of “definition drift,” even within one’s own mind; I’ve seen big names in certain scientific fields unconsciously shift their usage of the term from “unintentioned” to “uncaused” to “indeterminate” and back again, without discerning the subtle differences between them all. In data mining, we’re basically trying to uncover information that is badly concealed, whereas in cryptography it’s deliberately concealed by mixing it in with apparently random information; the patterns generated by cryptographers are the opposite of random in one specific sense, that it takes great intelligence and hard work to deliberately create these false patterns. Cryptanalysis involves detecting these deliberate patterns, in order to remove the noise obscuring the original message. Of course, “noise” is a subjective matter determined entirely by the questions we choose to ask of the data (although the answers to those questions are entirely objective). Randomness can thus be viewed as excess information of the wrong kind, not an absence of pattern or lack of causation. For example, if we’re looking for evidence of solar flares, then the static on shortwave radios might tell us a lot, but if we’re trying to listen to a broadcast from some foreign land (like the mysterious “numbers stations,” which I picked up when I was younger) then it degrades our information. Consider the case of paleontologists trying to follow dinosaur tracks: each rain drop that has fallen upon them over the eons has interfered with that pattern, but the rain itself follows a pattern, albeit the wrong kind. It is only random for our chosen purposes and pattern recognition goals, but perhaps not to some prehistoric weathermen to whom the pattern of rain might’ve been of keen interest. The relationship between randomness and information is thus quite deep; this brief introduction is merely a wade into the kiddie pool. Later on in this series we may have to dive in, if the measures of randomness I hope to discuss call for it.

…………Shannon’s Entropy and related concepts from information theory have all sorts of far-flung implications, some of which are understood in great detail (particularly when they relate directly to designing codes with minimum information loss) and others which are really intellectually challenging. One of the simplest yet trickiest rules is that the greater the uncertainty, the greater the amount of probabilistic information that can be conveyed with each new record. [14] Some of the corollaries of information theory can be put to good use in reasoning from the data we’ve mined, which is after all, its raison d’etre. The principle of maximum entropy, for example, states that the ideal probability density functions (PDFs) maximize Shannon’s Entropy when producing the expected values.[15] I’d imagine that this could be leveraged for such good purposes as mining model selection. There are all kinds of hidden relationships to other forms of information I hope to cover eventually, such as redundancy, which is a big topic within coding theory. Someday I hope to tackle measures of semantic information, which can be extremely difficult to grasp because they quantify the actual meaning of data, which can be quite slippery. Sometimes the absence of any data at all can tell us everything we need to know, which many mystery writers and crime scene investigation shows have put to good use. More often, people often differ not only between but within themselves as the meaning they choose to assign to terms and numbers, without thinking it through clearly. Perhaps this would make it an ideal field to apply fuzzy sets to, since their chief use cases include modeling imprecision in natural language, as I discussed ad nauseum in the last tutorial series.

**Semantic Misinterpretation, Cybernetics, “Disorder” and Other Subtle Stumbling Blocks**

Shannon’s Entropy is probably further away from semantic information than fuzzy sets, but that hasn’t stopped many theorists from mistakenly conflating stochastic information with meaning. Shannon himself warned against this kind of bandwagon-jumping.[16] In fact, that’s probably the most common stumbling block of all. Perhaps the clearest cautionary tale is the whole philosophy developed out of information theory by bombastic mathematician Norbert Wiener (1894-1964), who asserted explicitly and deliberately” the very same connection to semantic meaning that Shannon and others cautioned against.[17] Another great mathematician who took Shannon’s Entropy a little too far into the realm of semantic meaning was Shannon’s colleague, Warren Weaver (1894-1978). Wiener’s philosophy is known by the cool name of “cybernetics,” which has a certain intriguing flair to it – just like the loaded terms “fuzzy sets” and “chaos theory”, which are often used in a manner precisely opposite to their meaning and purpose. Nobody is really quite certain what “cybernetics” is, but that hasn’t stopped academics from establishing a journal by that name; some of its contributors are among the leading names in analytics I most respect, but I find other commentators on the field as disturbing as some of the frenetic theologians who followed Pierre Teilhard de Chardin. To put it simply, information theory has been put to even more bad uses in the last seven decades than other hot mathematical ideas, like chaos theory and fuzzy sets – some of which border on madness.

…………I’m not convinced that even a lot of the authors who specialize this sort of thing really grasp all of these intricacies yet, although most at least have the common sense not to idolize these ideas and blow them up into full-blown crackpot philosophies. For example, there seems to be a lot of misunderstanding about its relationship to measures of complexity, structure and order, which are alike but not precisely the same. To complicate matters further, they all overlap thermodynamic entropy, in which not the same thing as probabilistic entropy. They both ultimately proceed from mathematical relationships like the Law of the Large Numbers, but measure different things which are not always connected; Shannon noticed the resemblance between the two off the bat, as did Leo Szilard (1898-1964), hence the name “entropy.” This kind of entropy is not really “disorder”; it merely removes the energy a system one would need to move from a disordered to an ordered state and vice-versa. Essentially, it freezes a system into its current state, no matter what order it exhibits. Likewise, zero probabilistic entropy always signifies complete certainty, but not necessarily lack of structure; we could for example, be quite certain that our data is disorganized. It is perhaps true that “…*H* can also be considered as a measure of disorganization…The more organized a system, the lower the value of *H*”[18] but only in a really broad sense. Furthermore, entropy prevents new complexity from arising in a system; the range of possible states that are reachable from a system’s current state are determined by its energy, so that the greater the entropy, the greater the number of unreachable states. Without an input of fresh energy, this set of reachable states cannot increase. This means that entropy *tends* towards simplicity, which can nevertheless exhibit order, while still not ruling out complexity. The same is true of probabilistic entropy, which intersects with algorithmic complexity at certain points.

…………The latter measures the shortest possible description of a program, which Shannon also investigated in terms of the minimum length of codes. He “further proclaimed that random sources – such as speech, music, or image signals – possess an irreducible complexity beyond which they cannot be compressed distortion-free. He called this complexity the source entropy (see the discussion in Chapter 5). He went on to assert that if a source has an entropy that is less than the capacity of a communication channel, the asymptotically error-free transmission of the source over the channel can be achieved.”[19] It is not wise to conflate these two powerful techniques, but when probabilistic and algorithmic information intersect, we can leverage the properties of both to shed further light on our data from both angles at once. In essence, we can borrow the axioms of both at the same time to discover more new knowledge, with greater certainty. One of the axioms of algorithmic complexity is that a program cannot contain a program more sophisticated than itself, for essentially the same reasons that a smaller box cannot contain a larger one in the physical realm. This is related to a principle called the Conservation of Information, which operates like the Second Law of Thermodynamics, in that absent or lost information cannot be added to a system from itself; since its violation would be a logical contradiction, actually more solid than its thermodynamic counterpart, which is based merely on empirical observation. It is essentially a violation of various No Free Lunch theorems. This has profound implications for fields like artificial intelligence and concepts like “self-organization” that are deeply intertwined with data mining. Since the thermodynamic entropies aren’t as closely related to data mining and information theory, I’ll only spend a little bit of time on them a couple of articles from now, where I’ll also dispense with extraneous topics like quantum entropies. There are many other directions we could take this discussion in by factoring in things like applications without replacement, compound events, multiple possible outcomes, mutually exclusive events, unordered pairs and sources with memory (non-Markov models). Instead, I’ll concentrate on using this article as a springboard to more complex forms of probabilistic entropy that might be of more use to SQL Server data miners, like leaf and root entropies, binary entropies like the conditional and joint, the information and entropy rates and I’ll gradually build up towards more complex cases like Mutual, Lautum and Shared Information at the tail end of this segment of the tutorial series, whereas the Cross Entropy will be saved for a future segment on an important distance measure called the Kullback-Leibler Divergence. The next logical step is to discuss the Rényi Entropy, which subsumes the Shannon and Hartley Entropies with other relatives in a single formula.

[1] I lost my original citation for this, but believe it is buried somewhere in Klir, George J., 2006, __Uncertainty and Information: Foundations of Generalized Information Theory__, Wiley-Interscience: Hoboken, N.J.

[2] See the __Wikipedia__ page “Claude Shannon” at http://en.wikipedia.org/wiki/Claude_Shannon

[3] See the __Wikipedia__ articles “Hartley Function” and “Ralph Hartley” at http://en.wikipedia.org/wiki/Hartley_function and http://en.wikipedia.org/wiki/Ralph_Hartley respectively.

[4] pp. 5-6, Shannon, C.E., 1974, “A Mathematical Theory of Communication,” pp. 5-18 in __Key Papers in the Development of Information Theory__, Slepian, David Slepian ed. IEEE Press: New York.

[5] p. xv, Mansuripur, Masud, 1987, __Introduction to Information Theory.__ Prentice-Hall: Englewood Cliffs, N.J.

[6] “It is nevertheless quite remarkable that, as originally shown by Shannon, one can show that, by proper encoding into long signals, one can attain the maximum possible language transmisssion capacity of a system while at the same time obtaining a vanishingly small percentage of errors.” p. 172, Goldman, Stanford, 1953, Information Theory. Prentice-Hall: New York.

[7] p. 272, Pierce, John Robinson, 1980, __An Introduction to Information Theory: Symbols, Signals & Noise__. Dover Publications: New York. Also see Pierce, John Robinson, 1961, __Symbols, Signals and Noise: The Nature and Process of Communication__. Harper: New York

[8] p. 31, Ritchie, L. David., 1991, Information. Sage Publications: Newbury Park, Calif.

[9] p. 259, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J.

[10] p. 12, Brillouin, Léon, 1964, __Science, Uncertainty and Information__. Academic Press: New York. .

[11] Which I originally downloaded from the University of California at Irvine’s Machine Learning Repository__.__

[12] One source I found it mentioned in was Bengtsson, Ingemar, 2008, __Geometry of Quantum States: An Introduction to Quantum Entanglement__. Cambridge University Press: New York.

[13] See the Wikipedia page “Entropy (Information Theory)” at http://en.wikipedia.org/wiki/Entropy_(information_theory)

[14] p. 87, Wright, Robert, 1988,__ Three Scientists and Their Gods: Looking For Meaning in an Age of Information__. Times Books: New York.

[15] See the Wikipedia page “Minimum Fisher Information” at http://en.wikipedia.org/wiki/Minimum_Fisher_information

[16] p. xvi, Mansuripur and p. 61, Ritchie, 1991.

[17] p. 289, Bar-Hillel, Yehoshua, 1964, __Language and Information: Selected Essays On Their Theory and Application__. Addison-Wesley Pub. Co.: Reading, Mass.

[18] p. 5, Ritchie.

[19] p. 143, Moser, Stefan M. and Po-Ning, Chen, 2012, __A Student’s Guide to Coding and Information Theory__. Cambridge University Press: New York

## Information Measurement with SQL Server, Part 1: A Quick Review of the Hartley Function

**By Steve Bolton**

…………This long-delayed series of amateur self-tutorials has been in the works ever since I began writing my A Rickety Stairway to SQL Server Data Mining series, which made it clear to me that I didn’t know enough about what was going on under the hood in SSDM. I still don’t know enough about the reasoning behind the various data mining algorithms implemented by SQL Server and other tools, but I am certain of one thing: I never will know enough, even if I actually became competent in these topics. These fields are just too detailed, broad and rooted in poorly understood corners of pure reason for anyone to master, let alone myself. Like my series on SSDM, this foray into coding all of the basic measures of information theory and related fields may well exemplify University of Connecticut statistician Daniel T. Larose’s witticism that “data mining is easy to do badly.”[1] My purpose in the Rickety series was merely to demonstrate that useful results can still be derived from SSDM, even when it is badly mishandled. In this series, I will try to explain how the metrics used in many data mining algorithms can be used to answer a whole cornucopia of questions about our datasets, such as: How much information might there be, as measured in terms of possible state descriptions and probabilities? How much meaning (i.e. semantic information) might it have? How many state descriptions does it rule out? How much is already known? How random, aperiodic, redundant, complex or ordered is it? Another interesting challenge is determining the shortest possible specifications of a structure.

…………There are literally dozens upon dozens of measures available to answer all of these questions, taken from such diverse fields as information theory, chaos theory, algorithmic complexity and many others which provide the basic building blocks of most data mining algorithms. The techniques provided by these fields are powerful, yet contain many logical stumbling blocks that far smarter people than ourselves have tripped over, often without even knowing it; these range from instances of subtle “definition drift” in the meaning of terms like “randomness” over the course of textbooks, to the development of full-blown crackpot theories by scholars overexcited by the potential of systems like chaos and information theory. I am bound to make mistakes along the way, given that I’m an amateur at this, so by all means take care when implementing my code and trusting my analysis, which is sometimes just plain wrong. On the other hand, these techniques are so powerful and so under-utilized that there is a crying need for some explanation of how they can be applied in a SQL Server environment, even a poor one. I know a little bit more of certain areas of philosophy – especially historical instances of when it has gone very wrong – so I can occasionally make a contribution by commenting on how to avoid fallacious reasoning, which is even more of a problem once these sophisticated topics are clouded over by excess jargon and complex math formulas.

**DIY Data Mining and the Scope of the Series**

One thing I’ve learned while trying to force-feed myself the underlying math is that the analysis marketplace is decades behind the research in some ways; there is no way that any single company is ever going to be able to code all of the extant mining algorithms, assuming it is even possible to tally them all up. This means that it may be beneficial in the years to come to have the skills to build DIY solutions. Throughout this series I will provide T-SQL code so that SQL Server DBAs and data miners can implement some of these techniques on their own, without waiting for developers of mining software to code the particular algorithms that fit their use cases. Why T-SQL? I could make a strong case that the gradual accretion of features like windowing functions is slowly making set-based languages ideal for this purpose, although they are rarely thought of in that way; it boils down to the fact that most of the problems covered in these interrelated fields are much easier to express in terms of sets. Furthermore, the sheer size of “Big Data” (which has steadily gotten “bigger” ever since the first records were kept; like “globalization”, it has only accelerated in recent years) requires taking computing to a new level of abstraction in order to simplify things, which is a purpose that set-based languages can fulfill. This series will be a learning experience for me as well, in which I hope to at least help others avoid my mistakes, by teaching through misadventure; writing a series like this aids me in absorbing the material a lot faster, while also getting valuable practice in translating the difficult underlying math formulas into code. Since I don’t know what I’ll discover along the way, I suspect that at some point I may resort to using some of my other favorite languages, like Multidimensional Expressions (MDX) and Visual Basic .Net, possibly in conjunction with Common Language Runtime (CLR) stored procedures. One of the benefits I hope to provide is to take most of the math out of the picture, so that readers don’t get bogged down in it; the jargon and formulas are absolutely necessary for the researchers to communicate with each other, but there is no need for data miners to be writing formal mathematical proofs, just as commuters shouldn’t have to give a dissertation on automotive engineering in order to drive their cars. I’ve sometimes received comments to the effect that there’s too much text in these articles, but that’s because they don’t see the hundreds of pages of math formulas that gave rise to them; rather than stringing together some meaningless screenshots or rehashing MSDN tutorials from AdventureWorks, I aim to show how these techniques might be coded and how they can be used appropriately.

…………The goal in this series is to corner the uncertainty in our datasets by shining lights on it from every possible direction, so that organizations can make better decisions that result in more effective actions. To that end, everything from measures used in Bayesian inference to Solomonoff Algorithmic Probability to the calculation of periodicities to the Lyapunov Exponent will be fair game. These metrics vary quite widely in terms of sophistication, so at points we’ll cross the imprecise boundaries separating them from full-blown data mining algorithms; the dividing lines separating statistics, data mining, machine learning, “soft computing,” predictive analytics and the like seem to boil down to degrees of intricacy, rather than differences in kind, especially since their algorithms are derived from the same metrics and building blocks. My last mistutorial series was designed from the ground up to build on existing concepts, one article at a time. This one will be inherently disorganized, since the scope is so wide and I don’t know what I will find along the way. I will have to skip around quite a bit across topics that may only be distantly related, or across levels of sophistication exhibited by the measures of information. It will also be an open-ended series, whereas the Rickety series was merely necessary to cover a specific set of easily denumerable features. We could delve into dozens of obscure metrics if the need arises, or even concoct our own special-purpose metrics, if a use case calls for it.

…………The series may explore a wide-ranging topics along paths are still somewhat unknown, but I can at least kick it off by introducing some of the primordial foundations of information theory. Perhaps the simplest is a function developed in 1928 by electronics pioneer Ralph Hartley[2], who applied it to signal transmission[3] a few decades before Claude Shannon did the same with his own renowned entropy measure. Although Hartley considered it a measure of information, his function did not lead to the use of the term of “information theory,”[4] which was coined when Shannon’s famous equation gave birth to the field. This introduction will be made even easier by the fact that I already discussed a more advanced version of Hartley’s measure in Implementing Fuzzy Sets in SQL Server, Part 9: Measuring Nonspecificity with the Hartley Function. The version for ordinary “crisp” sets of the kind DBAs and data miners are accustomed to working is actually quite a bit easier to code and interpret: all we have to do is count the records in a set and take the logarithm. The code in Figure 1 is actually longer than it has to be, given that I used three different logarithm bases to measure the same quantity, for the sake of completeness. When base 2 is used, the units are known as bits or “shannons.” When the default value of Euler’s Number is used, they’re known as “nats,” but with base 10 we’re measuring in hartleys or “bans” (a term coined by famed cryptographer Alan Turing). It would be trivial to turn this into a stored procedure with an option to select the desired units. It only took about a second to calculate the results in Figure 2 on the first float column of the Higgs Boson dataset I downloaded from University of California at Irvine’s Machine Learning Repository a few tutorial series ago, which I converted to a 5-gigabyte SQL Server table. This was calculated effortlessly across all 11 million rows only because we had to performs some simple counts, without traversing the whole table.

** Figure 1: Code for the Ordinary “Crisp” Version of the Hartley Function**DECLARE @HartleyEntropy float, @DistinctCount bigint, @Count bigint

SELECT @DistinctCount = Count(DISTINCT Column1), @Count = Count(*)

FROM Physics.HiggsBosonTable

SELECT Log(@DistinctCount, 2) AS BitsOrShannons, Log(@DistinctCount, 2.7182818284590452353602874713526624977) AS Nats, Log(@DistinctCount, 10) AS Hartleys,

Log(@Count, 2) AS MultisetBitsOrShannons, Log(@Count, 2.7182818284590452353602874713526624977) AS MultisetNats, Log(@Count, 10) AS MultisetHartleys

__ ____Figure 2: Results from the Higgs Boson Dataset
__

…………The main problem I ran into was a fundamental one: the formula calls for plugging in the cardinality of the set, but in ordinary set theory parlance, duplicate values are only counted in multisets. The set notation includes the symbols |A|, but the bars specify a cardinality measure rather than the use of the ABS function. The question is, which cardinality? I’ve included both versions in Figure 1, which differ solely by the fact that one uses a DISTINCT operator and the other takes a standard COUNT. This discrepancy could have its uses though. I have yet to see this issue raised in the information literature, where Hartley’s metric is often skipped over as a topic of mainly historical interest in comparison to Shannon’s, but it may be possible to derive a third metric based on the difference between the two. A simple subtraction might help us quantify the contribution of the repeated values to the uncertainty, which could have its uses in the kinds of uncertainty management programs I spoke of in the fuzzy set series. In essence, this difference could act as a crude measure of redundancy. If it reflects the information gained by using the DISTINCT operator, we could use this to assess its costs. We know for a fact that the DISTINCT version can’t exceed the multiset version, which acts as a cap on its range. At the other end of the scale, the measure reaches the limit of 0 when all records have the same value, therefore giving us perfect certainty. Another important issue is whether or not the DISTINCT clause adds information, by reducing the amount of uncertainty about how many different values a variable can take on.

…………Of course, the clause might not be necessary at all if we knew in advance precisely which values were permissible for a column, such as the range of a decimal type or a varchar column limited to a handful of known category values. On the other hand, this raises a subtle distinction between values that are permissible for a column, which can be determined by the data type, and the values actually found, which can only be counted through a DISTINCT operator. The issue becomes more sophisticated if we are able to determine the counts of each individual value; these measures of “multiplicity,” as they are known in multiset theory, further reduce the uncertainty associated with the dataset. It is easy enough to implement these internal counts using windowing functions and GROUP BY statements, but the issue of how to factor them in quickly complicates the discussion of the otherwise simple Hartley function. Thankfully, the order of the records is only an issue with tuples, not the kinds of sets or multisets we plug into it – except when we get to the end of the dataset and can determine the last records from the remaining counts, which is something I’ve not seen addressed in the literature. This brings applications without replacement (i.e., the kind of probabilities associated with decks of cards where no reshuffling takes place) into play, at least briefly.

**“Newsworthiness”: The Narrow Definition of “Information”**

Aside from all of these hidden subtleties, the information provided by both the DISTINCT and multiset versions can be summed up thus: how much am I learning each time I inspect a row and verify the actual value found there? This is equivalent to asking how much we learn from each slip of paper in a drawing, if we know the count of the jar in advance. In more advanced terms, we can think of this as increasing information by reducing the number of possible state descriptions the next record can take on; in such a context, whether or not repeated values are allowed makes a big difference. The same goes for their individual counts, if the answer is yes. Keep in mind that this type of “information” is practically the polar opposite of existing knowledge; basically, the higher the Hartley measure is, the less we don’t already know, so the more the next record can tell us. It is a highly specific type of information, which former journalists like myself might equate with “newsworthiness.” This is the dividing line between existing and new knowledge is precisely where measures of entropy (like the Hartley function) intersect with topics like Bayesian probability, which I will also address at some point in this series; as we shall see, many of these information measures are interrelated in complex ways. This highly specific definition of information is an important distinction that pertains to all of the other measures of entropy we’ll discuss in this series; interpretation is a critical stage in every field associated with data mining, particularly information theory, which should never be shortchanged in any workflow.

…………The Hartley function can be leveraged in a data mining workflow in various ways, such as calculating the reduction in uncertainty between two separate measures; this could be useful, for example, in specifying a numerical cut-off point in bits or bans, after which it’s not worthwhile to go on inspecting rows, for whatever end purpose that might be, such as sampling. The sample code in Figure 3 takes the Hartley measure after the 10 millionth row, or about 9 percent away from the end of the dataset, which is why remaining uncertainty in Figure 4 is so low; as we approach the last uninspected record, the remaining uncertainty would approach zero. This formula would be equivalent to counting the remaining records and plugging the results into the Hartley function. Another interesting question is whether or not we could we pair this with cardinality estimation, to get a ballpark figure of how much we can learn from each record we inspect, before we’ve even traversed a dataset. I don’t know much about cardinality estimation yet, but the possibility is tantalizing

** Figure 3: Calculating the Remaining Uncertainty with the Hartley Function**DECLARE @DistinctCountOfKnownValues bigint, @CountOfKnownValues bigint

SELECT @DistinctCountOfKnownValues = Count(DISTINCT Column1), @CountOfKnownValues = Count(*)

FROM Physics.HiggsBosonTable

WHERE ID BETWEEN 1 AND 10000000

SELECT Bits, KnownBits, Bits – KnownBits AS RemainingUncertaintyInBits, MultisetBits, KnownMultisetBits, MultisetBits – KnownMultisetBits AS RemainingMultisetUncertaintyInBits

FROM (SELECT Log(@DistinctCount, 2) AS Bits, Log(@DistinctCountOfKnownValues, 2) AS KnownBits, Log(@Count, 2) AS MultisetBits, Log(@CountOfKnownValues, 2) AS KnownMultisetBits) AS T1

__Figure 4: Uncertainty Reduction Results
__

…………The kinship between the Hartley function and the rest of information theory is evident in some of its alternative names, like the Hartley Entropy or Max Entropy. It is equivalent to the Rényi Entropy[5] with its alpha parameter (α) set to 0, as I’ll explain a few articles from now. It’s also identical to the Shannon Entropy in cases of the uniform distribution, i.e. when all values are equally likely.[6] I’ll be spending a couple of articles on various aspects of entropy early on this series, since it’s such an important concept in information theory. The math and logic can get thick pretty quickly in this field, so it is best to start off with the measure that started it all, Shannon’s infamous “H.” I recognized its signature combination of a negative summation operator and log operation when translating some of the equations used in Implementing Fuzzy Sets in SQL Server, Part 10.2: Measuring Uncertainty in Evidence Theory into T-SQL. As with the Hartley function, this previous exposure to more advanced fuzzy derivatives ought to make the material a little easier to swallow. The difficulty with Shannon’s Entropy, however, is not in its calculation, but in its proper interpretation. Yet as long as we have the rigor to avoid assigning unwarranted shades of meaning to the term “information,” it can be a powerful addition to our data mining toolbelts.

[1] p. xii, LaRose, Daniel T., 2005, __Discovering Knowledge in Data: An Introduction to Data Mining__. Wiley-Interscience: Hoboken, N.J.

[2] See the __Wikipedia__ articles “Hartley Function” and “Ralph Hartley” at http://en.wikipedia.org/wiki/Hartley_function and http://en.wikipedia.org/wiki/Ralph_Hartley respectively.

[3] p. 5 Ritchie, L. David., 1991, __Information__. Sage Publications: Newbury Park, Calif.

[4] p. 288, Bar-Hillel, Yehoshua, 1964, __Language and Information: Selected Essays On Their Theory and Application__. Addison-Wesley Pub. Co.: Reading, Mass.

[5] See the Wikipedia article “Rényi Entropy” at http://en.wikipedia.org/wiki/R%C3%A9nyi_entropy

[6] See the __Wikipedia__ article “Hartley Function” at http://en.wikipedia.org/wiki/Hartley_function

## Implementing Fuzzy Sets in SQL Server, Part 11: Fuzzy Addenda

**By Steve Bolton**

…………One of the key reasons I looked into the topic of fuzzy sets in the first place was my suspicion that T-SQL, as a set-based language, would be ideal for modeling them. That turned out to be an understatement of sorts: I definitely was not prepared to discover just how useful they can be for translating imprecise linguistic modifiers in Behavior-Driven Development (BDD) environments and user stories, nor did I realize how little information has percolated down from the mammoth amount of theoretical research done on fuzzy topics over the last 40 years. Hopefully this series of amateur mistutorials helped rectify that gap by giving fuzzy sets some badly needed free press, of the kind I tried to bring SSDM in my older A Rickety Stairway to SQL Server Data Mining series awhile back. I originally set aside this final article as a kitchen drawer of sorts, to dispense with some postscripts that would’ve interfered with the flow of the rest of the series, in which one concept was used as a building block onto the next. One leftover concept I thought might be worthy of significant attention was fuzzy orders, which sounds as if it would be right up SQL Server’s alley. After all, DBAs use the ORDER BY statement every day. The problem is that it turns out T-SQL, like most other set-based languages, is not ideal for modeling this kind of fuzzy object.

**Fuzzy Orders and the Limitations of Hierarchies in SQL**

In the literature, fuzzy set orders are created by applying continuous membership grades to a record’s position in a particular fuzzy set. Devices like Hesse diagrams and properties like “dominated” and “undominated” are useful in implementing them[1], but I won’t bother, for the simple reason that SQL Server lacks robust graph database capabilities. Modeling relationships of this kind is still notoriously difficult in the relational realm, even though they’ve been augmented by such useful tools as hierarchyid data type in recent years. I am rather fond of hierarchyid, but it is unable to model multiparent trees in an efficient way, let alone multidimensional directed graphs. Just try modeling a simple genealogical tree with it. Trees are instances of what are known in mathematical parlance as partial orders; when you really stop and think about it, they represent a form of order, except in more than one dimension, such as “my grandparents and I have a descendant-ancestor relationship, but not my cousins and I.”[2] As far as I can tell, directed graphs open up more possibilities by relaxing the rules of composition, in the same way the Riemann manifolds give us access to curved hyperspace. I for one would cast my vote for adding graph database capabilities similar to those found in Neo4j[3] to SQL Server, which would add a whole new dimension to the product in the same way that Analysis Services and Reporting Services do, without being a separate service.

…………Alas, until such capabilities are added to SQL Server, it wouldn’t be useful to model most forms of fuzzy orders in T-SQL, let alone Multidimensional Expressions (MDX) in SQL Server Analysis Server (SSAS) cubes, because they immediately require the flexibility of multiparent trees and directed graphs. These tasks could be accomplished in SQL Server 2014 as it stands, but in contrast to the other fuzzy objects I’ve introduced throughout this series, I doubt it can be done in an efficient way. It also doesn’t help matters at all that the Windows Presentation Foundation (WPF) tree control is a walking disaster – for years now, its shortcomings have been a thorn in the side of .Net developers of all skill levels. Microsoft simply didn’t build in such basic functionality as searching for specific members in a collapsed tree, and in fact made it virtually impossible for third-party developers to do it themselves. Needless to say, neither the WPF TreeView nor hierarchyid is well-suited to modeling directed graphs, which are simply a more flexible generalizations of trees. The kissing cousins of fuzzy orders, like fuzzy rankings[4] and fuzzy morphisms[5], aren’t really feasible either. George J. Klir and Bo Yuan’s *Fuzzy Sets and Fuzzy Logic: Theory and Applications*, my favorite go-to resource for fuzzy math formulas, provides a decent starting point for all three[6], but from my little experience, I wouldn’t even try to implement them unless I had access to a good third-party product like GoXAM’s directed graph control (which may be expensive, but would probably recoup its costs by saving weeks of wasted labor on the unworkable WPF TreeView). If it one day does become worthwhile to model fuzzy orders and ranks in some future edition of SQL Server (or I turn out to be wrong), they’ll probably require the use of a lot of CASE statements in ORDER BY clauses and windowing functions, respectively. Given that there’s a mountain of currently unsolved problems out there that other aspects of fuzzy sets could tackle right away, we’ll save this topic for a later date. It’ll be a long time before all the low-hanging fruit is used up and we’re to the point where struggling to model them will become worthwhile.

**Some Simple T-SQL for Fuzzy Medians**

Because I realized early on that fuzzy orders were an afterthought – at least by the present capabilities of SQL Server and other relational databases – I left the subject of fuzzy medians for this junk drawer of an article. After all, medians are inherently dependent on the order of data, given that the pick the one or two values that occur precisely in the middle of a set. Furthermore, I noticed that the formulas involved calculations on two sets rather than one, which would have cluttered Implementing Fuzzy Sets in SQL Server, Part 7: The Significance of Fuzzy Stats, where the sample code was all done on a single table. That should have been a clue, however, that the fuzzy medians in the literature are a separate subject, not just a fuzzified version of ordinary medians. That would be easy enough to implement, given the principles of fuzzy sets introduced throughout this series; for example, instead of selecting the one or two records at the dead center of the dataset, we could select a fuzzy range. The trapezoidal numbers discussed in Implementing Fuzzy Sets in SQL Server, Part 6: Fuzzy Numbers and Linguistic Modifiers might be ideal for this purpose. The type of fuzzy medians under discussion here instead belong in the taxonomic hierarchy of fuzzy objects I mentioned in the fuzzy stats article, like Ordered Weighted Averages (OWAs), Lambda Averages (λ-Averages), T-norms, T-conorms and the like. Compared to some of those operations, the logic of fuzzy medians is fairly simple: we take the maximum of the values of two sets at each corresponding row when both membership scores are between 0 and the @LambdaParameter, the minimum values when both are between the @LambdaParameter and 1 and just the @LambdaParameter (which must be set between 0 and 1) in all other cases.[7] Assuming I read the formulas correctly – which is not a given, since I’m a novice at this – then this should all be implemented in Figure 1. As usual, it looks a lot longer than it really is; everything through the second UPDATE statement is just the same sample code I’ve used this series to populate the membership functions for binary set relations. Keep in mind that we don’t need to use Z-Scores to assign membership values here; I’m just using them to illustrate how to assign memberships in a fuzzy set, using familiar code from older tutorials. The sky’s the limit as far as the number of functions you can use to assign such values; the key thing is to find the right match to the problem you’re trying to solve. This would be a good match if we were trying to rate outliers by two different forms of Z-Scores, for example. The only novel part is the last SELECT, which isn’t difficult at all. As always, the results in Figure 2 are derived from the Duchennes muscular dystrophy dataset I downloaded a few tutorial series ago from Vanderbilt University’s Department of Biostatistics and have been using for practice data ever since.

** Figure 1: Sample Code for a Simple Fuzzy Median**DECLARE @RescalingMax decimal(38,6), @RescalingMin decimal(38,6), @RescalingRange decimal(38,6)

DECLARE @ZScoreTable table

(PrimaryKey sql_variant,

Value decimal(38,6),

ZScore decimal(38,6),

ReversedZScore as CAST(1 as decimal(38,6)) – ABS(ZScore),

MembershipScore decimal(38,6),

GroupRank bigint

)

DECLARE @ModifiedZScoreTable table

(PrimaryKey sql_variant,

Value decimal(38,6),

ZScore decimal(38,6),

ReversedZScore as CAST(1 as decimal(38,6)) – ABS(ZScore),

MembershipScore decimal(38,6),

GroupRank bigint,

OutlierCandidate bit

)

INSERT INTO @ZScoreTable

(PrimaryKey, Value, ZScore, GroupRank)

EXEC Calculations.ZScoreSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’LactateDehydrogenase‘,

@PrimaryKeyName = N’ID’,

@DecimalPrecision = ’38,32′,

@OrderByCode = 8

— RESCALING

SELECT @RescalingMax = Max(ReversedZScore), @RescalingMin= Min(ReversedZScore)

FROM @ZScoreTable

SELECT @RescalingRange = @RescalingMax – @RescalingMin

UPDATE @ZScoreTable

SET MembershipScore = (ReversedZScore – @RescalingMin) / @RescalingRange

INSERT INTO @ModifiedZScoreTable

(PrimaryKey, Value, ZScore, GroupRank, OutlierCandidate)

EXEC Calculations.ModifiedZScoreSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’LactateDehydrogenase‘,

@PrimaryKeyName = N’ID’

@OrderByCode = 8,

@DecimalPrecision = ’38,32′

— RESCALING

SELECT @RescalingMax = Max(ReversedZScore), @RescalingMin= Min(ReversedZScore)

FROM @ModifiedZScoreTable

SELECT @RescalingRange = @RescalingMax – @RescalingMin

UPDATE @ModifiedZScoreTable

SET MembershipScore = (ReversedZScore – @RescalingMin) / @RescalingRange

DECLARE @LambdaParameter float = 0.43

SELECT T1.PrimaryKey, T1.Value, T1.MembershipScore, T2.MembershipScore,

CASE WHEN (T1.MembershipScore BETWEEN 0 AND @LambdaParameter) AND (T2.MembershipScore BETWEEN 0 AND @LambdaParameter) THEN (SELECT MAX(Value) FROM (VALUES (T1.MembershipScore), (T2.MembershipScore) ) AS T1(Value))

WHEN (T1.MembershipScore BETWEEN @LambdaParameter AND 1) AND (T2.MembershipScore BETWEEN @LambdaParameter AND 1) THEN (SELECT MIN(Value) FROM (VALUES (T1.MembershipScore), (T2.MembershipScore) ) AS T1(Value))

ELSE @LambdaParameter END AS FuzzyMedian

FROM @ZScoreTable AS T1

INNER JOIN @ModifiedZScoreTable AS T2

ON T1.PrimaryKey = T2.PrimaryKey AND T1.Value IS NOT NULL AND T2.Value IS NOT NULL

__Figure 2: Results from the Duchennes Dataset
__

…………I barely began to scratch the surface of fuzzy objects like fuzzy medians, λ-Averages, T-norms, T-conorms and OWAs in this series. In fact, there’s an entire sea of ripe research out there on all topics fuzzy that could be quite useful to relational DBAs and decision support specialists, but which has gone unpicked. There are many different directions this topic can be taken in, so I may revisit this series and tack some additional articles onto it in the future. I didn’t get a chance to mention the extension principle[8] at all and glossed over important applications of fuzzy techniques in Decision Theory, way back in Implementing Fuzzy Sets in SQL Server, Part 4: From Fuzzy Unions to Fuzzy Logic. I might provide more detail on the use cases for particular T-norms and T-conorms (if I can ever get my hands on the relevant academic journal articles, which are expensive), model more linguistic states and get into indexing considerations, other brands of fuzzy aggregates and other types of fuzzy partitions besides alpha cuts (α-cuts), among other things. Yet I’d rather branch off into “soft computing,” which is a grab-bag and hodge-podge of cutting edge fields that are quite hard, which make its name something of an oxymoron. Fuzzy logic is merely one of the buzz words associated with it, like chaos theory, neural nets, support vector machines (SVMs) and genetic algorithms. What they all have in common is that they’re useful in situations where inexact solutions are acceptable, including NP-Complete problems.[9] The same hype and intellectual intoxication I spoke of in Implementing Fuzzy Sets in SQL Server, Part 1: Membership Functions and the Fuzzy Taxonomy also surrounds certain aspects of soft computing, which seems to make some theoreticians go soft in the head; I guarantee there will still be useful innovations occurring in these fields a century from now, assuming the human race lasts that long, but these incredible tools aren’t cure-alls. There are some things they just can’t do and I’d wager that certain brands of artificial intelligence and machine learning are among them; I love science fiction but it’s not wise to confuse it with cold, hard reality.

…………That’s a discussion I’ll take up by dribs and drabs in my next, long-delayed mistutorial series, Information Measurement with SQL Server, which may serve as stepping stone to my favorite topic, neural nets. Both topics dovetail nicely with fuzzy sets and many of the tangential topics we’ve covered in this series, like Shannon’s Entropy and the Hartley function. These are among dozens of metrics which can be coded in T-SQL and Multidimensional Expressions (MDX) and put to good use for data mining purposes, as I will demonstrate over the course of this long and possibly nomadic series. I aim to familiarize myself with semantic information, measures of order, measures of sensitivity to initial conditions (like the Lyapunov Exponent used in chaos theory), various means of quantifying algorithmic complexity – anything that will reduce uncertainty and glean whatever unused information is left in our datasets, by quantifying it in some way. Some of these metrics can be plugged into the formulas I introduced in this series for measuring fuzziness in terms of set complements, such as the Küllback-Leibler Divergence and Bhattacharyya Distance. We’ve already gotten our toes wet by introducing fuzzy stats and metrics for quantifying nonspecificity and fuzziness; now it’s time to jump in. Some of the topics will be quite shallow and easy to follow, while others may be incredibly deep. It’s largely unexplored territory for me as well, so I may have to skip around from topic to topic in an unsystematic way, instead of deliberately building up to more complex concepts as I did towards Dempster-Shafer Evidence Theory in this series. At a minimum, readers should at least benefit from learning from my mistakes, which don’t require a fancy fuzzy expert system to tell us that they’re inevitable; like death and taxes, they’re one of the few pieces of information that come with any certainty in predictive analytics and data mining.

[1] pp . 137-141, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J. On this particular page, they’re extending the meaning of the term even further, to complex network topologies.

[2] For more information, see the article “Partially Ordered Set” at the __Wikipedia__ web address http://en.wikipedia.org/wiki/Partially_ordered_set

[3] Which I have yet to try; I’m only speaking here of what’ve read about Neo4j casually.

[4] pp. 405-408, Klir and Yuan.

[5] *IBID*., pp. 141-144.

[6] *IBID*., pp. 137-144,

[7] *IBID*., p. 94.

[8] *IBID*., pp. 44-45.

[9] See the __Wikipedia__ article “Soft Computing” at http://en.wikipedia.org/wiki/Soft_computing__.__

## Implementing Fuzzy Sets in SQL Server, Part 10.2: Measuring Uncertainty in Evidence Theory

**By Steve Bolton **

…………To avoid overloading readers with too many concepts at once, I split my discussion of Dempster-Shafer Evidence Theory into two parts, with the bulk of the data modeling aspects and theory occurring in the last article. This time around, I’ll cover how fuzzy measures can be applied to it to quantify such forms of uncertainty as nonspecificity and imprecision (i.e., “fuzziness”) that were introduced in prior articles. Since the Plausibility, Belief and probability mass assignment figures work together to assign degrees of truth, they also introduce the potential for contradictory evidence, which leads to a few other measures of uncertainty: Strife, Discord and Conflict, which aren’t as relevant to possibility distributions and ordinary fuzzy sets. In addition, the probability mass for a universal hypothesis can be interpreted as a form of uncertainty left over after all of the probabilities for the subsets have been partitioned out. For example, in Figure 1, this crude type of uncertainty would be associated with the 0.0334928229665072 value for row 6. For the sake of brevity, I won’t rehash how I derived the ordinal LactateDehydrogenaseState category and the first three fuzzy measures associated with it, since the numbers are identical to those in the last tutorial. For the sake of convenience I added three columns with nearly identical names and calculated some sham data for them (based on the frequencies of some CreatineKinase data in the original table) so that we have some Conflicting data to work with. Ordinarily, such comparisons would be made using joins against an external view or table with its own separate ProbabilityMassAssignment, BeliefScore and PlausibilityScore columns, or a query that calculated them on the fly.

__Figure 1: Some Sample Evidence Theory Data from the Last Tutorial
__

…………In Figure 2, I translated some of the most popular formulas for evidence theory measures into T-SQL, such as Strife, Discord and Conflict.[1] For these, I used a simpler version of the equations that performed calculations on differences in set values rather than fuzzy intersections and unions.[2] Despite the fact the two measures only differ by the divisor and order of the difference operation, Discord is apparently not used as often as Strife on the grounds that it does not capture as much information. These subtle differences occur only in the alternate measures of Conflict they’re based on; since the one related to Strife is more important, I only included that one in Figure 3, where it’s represented by a score of 0.286225667126791. Versions of Strife and Discord are available for possibility distributions, but I omitted these because the fact that possibility theory is “almost conflict-free” signifies that they’re of “negligible” benefit.[3] I also coded the evidence theory version of nonspecificity and essentially rehashed the crude fuzziness measure I used in Implementing Fuzzy Sets in SQL Server, Part 2: Measuring Imprecision with Fuzzy Complements, except with the YagerComplement parameter arbitrarily set to 0.55 and the probability mass used in place of the membership function results. Both of these are unary fuzzy measures that apply only to the set defined by the first three float columns, whereas Strife, Discord and Conflict are binary measures that are calculated on the differences between the two sets encoded in the Health.DuchennesEvidenceTheoryTable. We can also add the Strife and fuzziness figures together to derive a measure of total uncertainty, plus interpret the height of a fuzzy set – i.e., the count of records with the maximum MembershipScore of 1 – as a sort of credibility measure. Keep in mind that I’m not only a novice at this, but am consulting mathematical resources that generally don’t have the kind of step-by-step examples with sample data used in the literature on statistics. This means I wasn’t able to validate my implementation of these formulas well at all, so it would be wise to recheck them before putting them to use in a production environments where accuracy is an issue. I’m most concerned by the possibility that I may be incorrectly aggregating the individual focal elements for evidentiary fuzziness and nonspecificity, each of which should be weighted by the corresponding probability mass.

** Figure 2: Several Evidence Theory Measures Implemented in T-SQL**DECLARE @Conflict float, @ConflictForDiscord float

SELECT @Conflict = SUM(CASE WHEN BeliefScore2 = 0 THEN ProbabilityMassAssignment2 * ABS(BeliefScore – BeliefScore2)

ELSE ProbabilityMassAssignment2 * ABS(BeliefScore – BeliefScore2) / ABS(CAST(BeliefScore AS float))

END),

@ConflictForDiscord = SUM(CASE WHEN BeliefScore2 = 0 THEN ProbabilityMassAssignment2 * ABS(BeliefScore2 – BeliefScore)

ELSE ProbabilityMassAssignment2 * ABS(BeliefScore2 –BeliefScore) / ABS(CAST(BeliefScore2 AS float))

END)

FROM Health.DuchennesEvidenceTheoryTable

— FUZZINESS

DECLARE @Count bigint, @SimpleMeasureOfFuzziness float

DECLARE @OmegaParameter float = 0.55 — ω

SELECT @Count=Count(*)

FROM Health.DuchennesEvidenceTheoryTable

SELECT @SimpleMeasureOfFuzziness = SUM(ABS(ProbabilityMassAssignment – YagerComplement)) /@Count

FROM (SELECT ProbabilityMassAssignment, Power(1 – Power(ProbabilityMassAssignment, @OmegaParameter), 1 / CAST(@OmegaParameter AS float)) AS YagerComplement

FROM Health.DuchennesEvidenceTheoryTable) AS T1

— NONSPECIFICITY

DECLARE @EvidenceTheoryNonspecificityInBits float

SELECT @EvidenceTheoryNonspecificityInBits = SUM(ProbabilityMassAssignment * Log(@Count, 2))

FROM Health.DuchennesEvidenceTheoryTable

SELECT Strife, Discord, Conflict, EvidenceTheoryNonspecificityInBits,SimpleMeasureOfFuzziness, Strife + EvidenceTheoryNonspecificityInBits

AS TotaUncertainty, (SELECT ProbabilityMassAssignment

FROM Health.DuchennesEvidenceTheoryTable

WHERE LactateDehydrogenaseState = ‘Any’) AS ProbabilityMassRemainderUncertainty

FROM (SELECT –1 * SUM(ProbabilityMassAssignment * Log((1 – @Conflict), 2)) AS Strife,

–1 * SUM(ProbabilityMassAssignment * Log((1 – @ConflictForDiscord), 2)) AS Discord, @Conflict AS Conflict, @EvidenceTheoryNonspecificityInBits AS EvidenceTheoryNonspecificityInBits, @SimpleMeasureOfFuzziness AS SimpleMeasureOfFuzziness

FROM Health.DuchennesEvidenceTheoryTable) AS T1

__Figure 3: Sample Results from the Duchennes Evidence Theory Table
__

…………The nonspecificity measure in evidence theory is merely the Hartley function weighted by the probability mass assignments. On paper, the equation for Strife ought to appear awfully familiar to data miners who have worked with Shannon’s Entropy before. The evidence theory version incorporates some additional terms so that a comparison can be performed over two sets, but the negative summation operator and logarithm operation are immediately reminiscent of its more famous forerunner, which measures probabilistic uncertainty due to a lack of stochastic information. Evidentiary nonspecificity trumps entropy in many situations because it is measured linearly, therefore avoiding computationally difficult nonlinear math (my paraphrase), but sometimes doesn’t produce unique solutions, in which case Klir and Yuan recommend using measures of Strife to quantify uncertainty.[4] Nevertheless, when interpreted correctly and used judiciously, they can be used in conjunction with axioms like the principles of minimum uncertainty, maximum uncertainty[5] and uncertainty invariance[6] to perform ampliative reasoning[7] and draw useful inferences about datasets:

“Once uncertainty (and information) measures become well justified, they can very effectively be utilized for managing uncertainty and the associated information. For example, they can be utilized for extrapolating evidence, assessing the strength of relationship on between given groups of variables, assessing the influence of given input variables on given output variables, measuring the loss of information when a system is simplified, and the like. In many problem situations, the relevant measures of uncertainty are applicable only in their conditional or relative terms.”[8]

…………That often requires some really deep thinking in order to avoid various pitfalls in analysis; in essence, they all involve honing the use of pure reason, which I now see the benefits of, but could definitely use a lot more practice in. For example, Dempster-Shafer Theory has well-known issues with counter-intuitive results at the highest and lowest Conflict values, which may require mental discipline to ferret out; perhaps high values of Strife can act as a safeguard against this, by alerting analysts that inspection for these logical conundrums is warranted.[9] Critics like Judea Pearl have apparently elaborated at length on various other fallacies that can arise from “confusing probabilities of truth with probabilities of provability,” all of which need to be taken into account when modeling evidentiary uncertainty.[10] Keep in mind as well that Belief or Plausibility scores of 1 do not necessarily signify total certainty; as we saw a few articles ago, Possibility values of 1 only signify a state of complete surprise when an event does not occur rather than assurance that it will happen.

…………The issue with evidence theory is even deeper in a certain sense, especially if those figures are derived from subjective ratings. Nevertheless, even perfectly objective and accurate observations can be quibbled with, for reasons that basically boil down to Bill W.’s adage “Denial ain’t just a river in Egypt.” One of the banes of the human condition is our propensity to squeeze our eyes shut to evidence we don’t like, which can only be overcome by honesty, not education; more schooling may even make things worse, by enabling people to lie to themselves with bigger words than before. In that case, they may end up getting tenure for developing entirely preposterous philosophies, like solipsism, or doubting their own existence. As G.K. Chesterton warned more than a century ago, nothing can stop a man from piling doubt on top of doubt, perhaps by reaching for such desperate excuses as “perhaps all we know is just a dream.” He provided a litmus test for recognizing bad chains of logic, which can indeed go on forever, but can be judged on whether or not they tend to drive men into lunatic asylums. Cutting edge topics like fuzzy sets, chaos theory and information theory inevitably give birth to extravagant half-baked philosophies, born of the precisely the kind of obsession and intellectual intoxication that Chesterton speaks of in his chapter on The Suicide of Thought[11] and his colleague Arnold Lunn’s addresses in *The Flight from Reason*.[12] These are powerful techniques, but only when kept within the proper bounds; problems like “definition drift” and subtle, unwitting changes in the meanings assigned to fuzzy measures can easily lead to unwarranted, fallacious or even self-deceptive conclusions. As we shall see in the next series, information theory overlays some of its own interpretability issues on top of this, which means we must trend even more carefully when integrating it with evidence theory.

…………Fuzzy measures and information theory mesh so well together than George J. Klir and Bo Yuan included an entire chapter on the topic of “Uncertainty-Based Information” in my favorite resource for fuzzy formulas, *Fuzzy Sets and Fuzzy Logic: Theory and Applications*.[13] The field of uncertainty management is still in its infancy, but scholars now recognize that uncertainty is often “the result of some information deficiency. Information…may be incomplete, imprecise, fragmentary, not fully reliable, vague, contradictory, or deficient in some other way. In general, these various information deficiencies may result in different types of uncertainty.”[14] Information in this context is interpreted as uncertainty reduction[15]; the more information we have, the more certain we become. Methods to ascertain how the reduction of fuzziness (i.e how imprecise the boundaries of fuzzy sets are) contributes to information gain were not fully worked out two decades ago when most of the literature I consulted for this series was written, but I have the impression that still holds today. When we adapt the Hartley function to measure the nonspecificity of evidence, possibility distributions and fuzzy sets, all we’re doing is taking a count of how many states a dataset might take on. With Shannon’s Entropy, we’re performing a related calculation that incorporates the probabilities associated with those states. Given their status as the foundations of information theory, I’ll kick off my long-delayed tutorial series Information Measurement with SQL Server by discussing both from different vantage points. I hope to tackle a whole smorgasbord of various ways in which the amount of information associated with a dataset can be quantified, thereby helping to cut down further on uncertainty. Algorithmic complexity, the Lyapunov exponent, various measures of order and semantic information metrics can all be used to partition uncertainty and preserve the information content of our data, so that organizations can make more accurate decisions in the tangible world of the here and now.

[1] pp. 259, 262-263, 267, 269, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J. The formulas are widely available, but I adopted this as my go-to resource whenever the math got thick.

[2] *IBID*., p. 263.

[3] *IBID*., pp. 262-265.

[4] *IBID*., p. 274

[5] *IBID*., pp. 271-272. Klir and Yuan’s explanation of how to use maximum uncertainty for ampliative reasoning almost sounds a sort of reverse parsimony: “use all information available, but make sure that no additional information is unwittingly added…the principle requires that conclusions resulting from any ampliative inference maximize the relevant uncertainty within the constraints representing the premises. The principle guarantees that our ignorance be fully recognized when we try to enlarge our claims beyond the given premises and, as the same time, that all information contained in the premises be fully utilized. In other words, the principle guarantees that our conclusions are maximally noncommittal with regard to information not contained in the premises.”

[6] *IBID*., p. 275.

[7] *IBID*., p. 271.

[8] *IBID*., p. 269.

[9] See the __Wikipedia__ webpage “Dempster Shafer Theory” at http://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory

[10] *IBID*.

[11] See Chesterton, G.K., 2001, __Orthodoxy.__ Image Books: London. Available online at the __G. K. Chesterton’s Works on the Web__ address http://www.cse.dmu.ac.uk/~mward/gkc/books/

[12] Lunn, Arnold, 1931, __The Flight from Reason__. Longmans, Green and Co.: New York.

[13] pp. 245-276, Klir and Yuan.

[14] *IBID*.

[15] *IBID*., p. 245.

## Implementing Fuzzy Sets in SQL Server, Part 10.1: A Crude Introduction to Dempster-Shafer Evidence Theory

**By Steve Bolton**

…………Early on in this series, we learned how the imprecision in natural language statements like “the weather is hot” can be modeled using fuzzy sets. Ordinarily, the membership grades assigned to fuzzy sets are not to be interpreted as probabilities, even though they’re both implemented on continuous scales between 0 and 1; the exception to this rule is when a probabilistic meaning is consciously assigned to the type of fuzziness. A couple of articles ago we saw how membership scores can be interpreted as assessing the logical possibility of the associated statements; the possibility distributions this nuance gives rise to quantifies whether or not an event *can* occur, whereas a probability distribution assesses whether it *will *actually occur. The two scales are independent except at the maximum and minimum values, where possibility values acts as caps on probabilities, since an event must be possible if it is to have a non-zero probability. The possibility and necessity measures that factor into possibility distributions are actually special cases of the plausibility and belief measures used in Dempster-Shafer Evidence Theory, which has a related shade of meaning: instead of gauging whether or not an event can or will happen, plausibility and belief work together to grade the credibility of the associated evidence. If we were sifting through user stories in a Behavior-Driven Development (BDD) process, we wouldn’t use evidence theory for fuzzy terms like “the weather is hot,” or questions like “the weather could be cold” or “the weather is probably mild,”[1] which might be candidates for possibilistic or stochastic modeling. “As far as I can tell, the weather will be hot,” might be fair game, since the subject of the sentence is the trustworthiness of the associated statement. The clearest example I’ve yet run across in the literature occurs in George J. Klir and Bo Yuan’s *Fuzzy Sets and Fuzzy Logic: Theory and Applications*, which I’ve used as my go-to resource throughout this series for the heavy math formulas:

“Consider, however, the jury members for a criminal trial who are uncertain about the guilt or innocence of the defendant. The uncertainty in this situation seems to be of a different type; the set of people who are guilty of the crime and the set of innocent people are assumed to have very distinct boundaries. The concern, therefore, is not with the degree to which the defendant is guilty, but with the degree to which the evidence proves his membership in either the crisp set of guilty people or the crisp set of innocent people.”[2]

In the last article, I gave a monologue on how organizations can benefit from uncertainty management programs, which begins with partitioning uncertainty into various types, like probabilities, nonspecificity, fuzziness and conflicting information; these in turn stretch across five mathematical subtopics, information theory, stochastics, possibility theory, fuzzy sets and evidence theory. The last of these has its own corresponding formulas for measures like nonspecificity, but is particularly useful for quantifying the degree of conflict between pieces of information. For this reason, it is widely used to aggregate disparate sources of information, which in turn integrates seamlessly with Decision Theory; for example, one of its most common implementations is sensor fusion.[3] Klir and Yuan also provide a concise list of possible use cases in various fields:

“For instance, suppose we are trying to diagnose an ill patient. In simplified terms, we may be trying to determine whether the patient belongs to the set of people with, say, pneumonia, bronchitis, emphysema, or a common cold. A physical examination may provide us with helpful yet inconclusive evidence. For example, we might assign a high value, say 0.75, to our best guess, bronchitis, and a lower value to the other possibilities, such as 0.45 for the set consisting of pneumonia and emphysema and 0 for a common cold. These values reflect the degree to which the patient’s symptoms provide evidence for the individual diseases or sets of diseases; their collection constitutes a fuzzy measure representing the uncertainty associated with several well-defined alternatives; It is important to realize that this type of uncertainty, which results from information deficiency, is fundamentally different from fuzziness, which arises from the lack of sharp boundaries.”[4]

…………Thankfully, a sturdy mathematical scaffolding to model these types of evidence-based uncertainty already exists, although it isn’t being tested much these days in the relational database, data warehousing and data mining fields. The modeling process is akin to the one I introduced a few weeks ago for possibility distributions, but a tad more complicated. A continuous data type like float, numeric or decimal is required for probability values, but possibility theory also calls for the addition of a bit column, which is often assigned to the Necessity measure. In the theory developed independently by statisticians Glenn Shafer and Arthur Dempster, we need three measures: a Probability Mass Assignment (often denoted by a lower case m) that tells us the strength of the evidence that a record belongs just to one set; a Belief measure that measures the same, plus the evidence for belonging to its subsets; and a Plausibility measure, which covers both of those, as well as “the additional evidence or belief associated with sets that overlap with A.”[5] The easy part is that all three are measured on scale of 0 to 1, the same as fuzzy sets, probabilities, possibilities and the like; the complexity arises from the fact that they measure evidence at different levels. This leads to nested bodies of evidence, which alpha cuts (α-cuts) are ideal for modeling, as explained a couple of articles ago; I saved this topic for the next-to-last article precisely because it unites many of the concepts introduced throughout the series, like α-cuts, fuzzy unions, intersections and complements.

…………These relationships also give rise to various mathematical properties, some of which are similar to those used in possibility distributions. For example, just as Necessity is equal to 1 minus the complement of Possibility, so too is Plausibility equal to 1 minus the complement of the Belief measure. Plausibility must be greater than or equal to the Belief, since it models evidence at a higher scope. These “fuzzy measures” have weakened forms of properties like monotonicity, continuity and additivity than probabilities do.[6] Belief measures are superadditive, which means that if you sum them together across the subsets, the result must be greater than or equal to the Belief function for the whole set. For example, the Belief function for the whole set can be a figure less than 1, say 0.97, but the individual measures of each subset can be assigned degrees of belief like 0.5, 0.87, 0.3, etc. which together sum to 1.67, which is valid because it’s greater than 0.97. In contrast, probabilities must always sum to 1 across a dataset, including the probability mass assignments used in evidence theory. Plausibility is subadditive, which signifies the opposite relationship, so that the measures taken across the subsets must sum to the at least the Plausibility for the whole set. In short, they act as maximums rather than sums. This all sounds weird, but it’s a necessary logical consequence of the nesting of evidence. As explained in the discussion on α-cuts a couple of articles ago, this signifies that records can belong to multiple hierarchical partitions of a set, which is an unfamiliar situation in the relational world (despite the fact that it is easily modeled using set-theoretic relational technology). The good news is that this web of interrelationships makes the three evidence theory measures reconstructible from each other; this makes it possible to validate the values using queries like the samples in Figure 2.

**Two Common Illustrations of Dempster-Shafer Evidence Theory in Action**

The Wikipedia article on Dempster-Shafer Theory has comprehensible examples of how these three measures work together, beginning with a sensor that detects whether a cat concealed in a box is in a Dead or Alive state. The value for Either obviously reaches the maximum value of 1 for both Belief and Plausibility, since it must be one of the two by logical necessity (that is, unless our cat happens to belong to Erwin Schröedinger or was buried in Pet Sematary). It is thus an instance of a “universal hypothesis,” which encompasses the whole dataset. Yet the probability mass assignment for the Either state is only 0.3, which signifies the fact that we don’t have solid information on its status; the probability figure for the whole dataset still sums to 1 though, once the stats for Alive and Dead are factored in. The probability value for the universal hypothesis thus constitutes a measure of the uncertainty remaining in the data, once the probability, Belief and Plausibility measures have partitioned it off. Since Dead and Alive are discrete states without fuzzy intervals, the Wikipedia example assigns them Belief figures equal to their probability masses – which when added to the value of 1 for the Either state, means that the total Belief for the whole dataset is greater than 1, unlike the probability mass. The Plausibility can then be reconstructed using the inverse of the complement of the Belief.

…………The tricky part is that the Belief measures must sum to 1 for each subset, which calls for looking at our data in an unfamiliar way. I initially thought that the existence of these subsets meant that we could simply model this by applying the appropriate normal form, but that’s not the case. The second example in the Wikipedia article has examples of states like Red, Yellow, Green which are mutually exclusive, as well as some that carry a bit of measurement uncertainty, like “Red or Yellow” and “Red or Green.” In this situation, the Belief figures for Red, Yellow and “Red or Yellow” must sum to 1, as must the Belief figures for Red, Green and “Red or Green,” since there are two overlapping subsets. Red, Yellow and Green are all members of more than one subset, but not the same ones. This leads to an odd predicament where each state is discrete and thus difficult to denormalize, yet the associated column still represents subsets; this is one situation where the presence of logical OR statements is not a hint that the design requires normalization. Since we can’t be certain how many other state descriptions a child could be related to, a single self-referencing ParentID column won’t do the job either. The next best thing is an interleaved solution, in which a separate table with two foreign keys pointing to the primary key of the table holding the Belief measures to keep track of which subsets each record belongs to. To aggregate the Belief figures for each subset in the parent table, we just inspect the interleaved table for all of the categories a record can belong to.

**Server States: A SQL Server-Specific Example**

Let me give an example that might be more intuitive and relevant to SQL Server users: the state_desc column of sys.databases will assign one of seven mutually exclusive states to each database: Online, Offline, Restoring, Recovering; Recovery Pending, Suspect and Emergency. As far as I know, these states do not rule out which user modes a database can be, which range from SINGLE_USER to RESTRICTED_USER to MULTI_USER. Nevertheless, many combinations would be improbable, so each unique pair of descriptions requires a probability assignment that will probably differ from other pairs of state_desc and user mode values. Now let’s pretend we have a sensor that guesses which of pair of states a server is in at any given moment, perhaps based on I/O data or network bandwidth usage. If it can tell us the user mode plus whether we’re in one of the three recovery states, but can’t differentiate between them accurately, then we’re dealing with a fuzzy interval-valued set. From the point of view of the sensor, “Restoring | Recovering | Recovery Pending” is a discrete state and ought to be recorded as such in the database table. Nevertheless, to derive the Belief we must sum together all of the probabilities for the subsets it gives rise to, while the Plausibility equals one minus the sum of the probability assignments in the subsets it does *not *participate in. We could create a separate category like “Unknown” for situations where the sensor went offline or was otherwise unable to return accurate data – or better yet, establish a universal hypothesis like “Any State” with the Belief and Plausibility both set to 1 and we add all of its possible subsets. Subtracting the sum of the probabilities of all known states from that of the universal hypothesis would allow us to measure one type of uncertainty associated with the table. In order to measure the uncertainty inherent in the interval-valued fuzzy subsets that the Belief and Plausibility measures are attached to, we’d have to use a measure of fuzziness tailored to evidence theory. In the same vein, the count of possible state descriptions could be used to derive a measure of nonspecificity, albeit through a different formula than the ones introduced in the last article. In addition, we can define measures of uncertainty based on how much

…………It is easier to illustrate all of this with T-SQL code samples, beginning with the easiest part, a simple snapshot of a table with probability mass, Belief and Plausibility measures defined on it. Degrees of Belief are usually derived from some kind of input method, akin to fuzzy set membership functions – except that subjective ratings tend to be more common in evidence theory. It is no surprise that Bayesian methods are often applied in deriving Belief functions, given that they actually represent a more specific subset of evidence theory measures. Instead of complicating the topic any further, I’ve derived the values in Figure 1 by creating an artificial category in the Duchennes muscular dystrophy data I’ve been using for practice data for the last few tutorial series[7], then simply assigned probability mass assignments based on the frequency of the values for the LactateDehydrogenase column. From there, I derived the Belief measures, then constructed the Plausibility measures from those. I used the float data type for all three of the columns that associate measured with the LactateDehydrogenaseState column, an ordinal category; this represents yet another use of fuzzy sets to model ordinals on continuous scales, except at a more advanced level where three columns are required.

__Figure 1: Simple Evidence Theory Measures Defined on the LactateDehydrogenase Column
__

** Figure 2: Sample Validation Code for the Relationships Between the Three Evidence Theory Measures**— verifying the Belief via the ProbabilityMassAssignment mass assignment

SELECT ID, LactateDehydrogenaseState, ProbabilityMassAssignment, BeliefScore, PlausibilityScore,

CASE WHEN IntervalProbabilityMassAssignmentSum IS NOT NULL THEN IntervalProbabilityMassAssignmentSum ELSE ProbabilityMassAssignment END

AS BeliefReconstructedFromProbabilityMass

FROM Health.DuchennesEvidenceTheoryTable AS T3

LEFT JOIN (SELECT ParentID, SUM(ProbabilityMassAssignment) AS IntervalProbabilityMassAssignmentSum

FROM Health.DuchennesEvidenceTheoryTable AS T1

INNER JOIN Health.DuchennesEvidenceTheoryIntervalTable AS T2

ON T1.ID = T2.BeliefSubsetID

GROUP BY ParentID) AS T4

ON T3.ID = T4.ParentID

SELECT ID, LactateDehydrogenaseState, BeliefScore, ProbabilityMassAssignment, ProbabilityMassAssignmentBySum,

CASE WHEN ProbabilityMassAssignmentBySum IS NULL THEN 1 ELSE ABS(1 – (ProbabilityMassAssignment+ ProbabilityMassAssignmentBySum)) END AS PlausibilityScoreReconstructedFromProbability

FROM (SELECT ID, LactateDehydrogenaseState, BeliefScore, ProbabilityMassAssignment

FROM Health.DuchennesEvidenceTheoryTable) AS T5

LEFT JOIN (SELECT BeliefSubsetID, SUM(ProbabilityMassAssignment) AS ProbabilityMassAssignmentBySum

FROM (SELECT DISTINCT T1.BeliefSubsetID, T2.ParentID

FROM Health.DuchennesEvidenceTheoryIntervalTable AS T1

INNER JOIN Health.DuchennesEvidenceTheoryIntervalTable AS T2

ON T1.ParentID = T2.BeliefSubsetID AND T1.BeliefSubsetID != T2.BeliefSubsetID) AS T4

INNER JOIN Health.DuchennesEvidenceTheoryTable AS T3

ON T4.ParentID = T3.ID

GROUP BY BeliefSubsetID) AS T6

ON T5.ID = T6.BeliefSubsetID

…………Note how the Belief is equal to the ProbabilityMassAssignment for Low, Medium and High, which is reflective of the fact that they have no substates; Medium or Low and High or Medium have BeliefScore values higher than their masses, precisely because we have to tack the values for Low, Medium and High onto them. The PlausibilityScore is in each case determined by adding together all of the ProbabilityMassAssignment values for the columns that aren’t among a record’s subsets, then taking an inverse, which is equivalent to subtracting the complement of the BeliefScore from 1. The second image depicts the Health.DuchennesEvidenceTheoryIntervalTable, in which the ParentID and BeliefSubsetID determine the linkages between subsets. For example, the records with ParentIDs of 4 tie together the Medium | Low, Medium and High | Medium values, so that we can aggregate the ProbabilityAssignments to derive the BeliefScore. The PlausibilityScore can be determined using the same table. Code similar to what I provided in Figure 2 can be used to validate the relationships between these fuzzy measures, with your own particular column and table names plugged in of course. The IS NULL condition is due to a bizarre problem in which setting the first condition in the CASE to BeliefScore = 1 THEN 1, or using NullIf, both led to NULL values. It is also possible to derive the ProbabilityMassAssignment values in reverse, but I’ll omit validation code for that scenario in the interest of brevity. To avoid pummeling readers with too much information all at once, I’ll also put off discussion of how to derive uncertainty measures like Strife and Discord from this crude example. In the next article, I’ll also mention some principles for interpreting the results that can in turn provide an important bridge to Information Theory. Among other things, the first table tells us that, “the belief that the Lactate Dehydrogenase values are Medium or Low is higher than that for Low alone, by a margin of 0.679425837320574 to 0.349282296650718. It is more plausible that the value is High than Low, by a margin of 0.822966507177033.” Once we define measures of fuzziness, nonspecificity and the like on top of them and apply some principles of inference drawn from Information Theory, we can partition the uncertainty further in order to glean additional valuable insights.

[1] Here in Western New York the natural language term “mild” has interesting shades of meaning (at least among local weathermen) which would be a challenge to model in terms of a fuzzy set. As winter approaches, “mild” means warmer than normal, but as the peak of summer comes, it means cooler than expected, so the meaning is inverted depending on the season. If we were to use an interval-valued set, we’d need a range ofvalues somewhere between 30 and 70 degrees – which is so imprecise that it borders on meaningless.

[2] p. 177, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J.

[3] See the __Wikipedia__ article “Dempster Shafer Theory” at http://en.wikipedia.org/wiki/Dempster%E2%80%93Shafer_theory

[4] p. 179, Klir and Yuan.

[5] *IBID*., p. 181-182.

[6] *IBID*., p. 179-181.

[7] Which I downloaded from the Vanderbilt University’s Department of Biostatistics and converted into a SQL Server table in my sham DataMiningProjects database.

## Implementing Fuzzy Sets in SQL Server, Part 9: Measuring Nonspecificity with the Hartley Function

**By Steve Bolton**

…………Imagine how empowering it would be to quantify what you don’t know. Even an inaccurate measure might be helpful in making better decisions in any area of life, but particularly in the business world, where change is the only certainty. This is where a program of “uncertainty management” can come in handy and fuzzy set techniques find one of their most useful applications. Fuzzy sets don’t introduce new information, but they do conserve and put to good use some information left over after ordinary “crisp” sets are defined – particularly when it would be helpful to model ordinal categories on continuous number scales. As I pointed out at the beginning of this series, uncertainty reduction is akin to Stephen King’s adage that monsters are less fearsome once some scale of measurement can be applied to them; knowing that a bug is 10 feet tall is at least reassuring, in the sense that we now know that it is not 100 or 1,000 feet tall.[1] Uncertainty reduction can also be put to obvious uses in data mining activities like prediction and clustering. Another potential use is in simplification of data, so that information loss is minimized.[2] In today’s article I’ll shine a little light on the Hartley function, a tried and true method of quantifying one particular category of uncertainty that has been used since 1928 to simplify and demystify datasets of all kinds and could easily be extended to SQL Server data.

George J. Klir and Bo Yuan, the authors of my favorite resource for fuzzy set equations, note that data models must take uncertainty into account, along with complexity and credibility. Later in the book, they go onto subdivide uncertainty into three types that sprawl across possibility theory, stochastics, information theory, fuzzy sets and Dempster-Shafer Evidence Theory:

“The relationship is not as yet fully understood…Although usually (but not always) undesirable when considered alone, uncertainty becomes very valuable when considered in connection to the other characteristics of systems models; in general, allowing more uncertainty tends to reduce complexity and increase credibility of the resulting model. Our challenge in systems modelling is to develop methods by which an optimal level of allowable uncertainty can be estimated for each modelling problem…”[3]

“…Three types of uncertainty are now recognized in the five theories, in which measurement of uncertainty is currently well established. These three uncertainty types are: nonspecificity (or imprecision), which is connected with sizes (cardinalities) of relevant sets of alternatives; fuzziness (or vagueness), which results from imprecise boundaries of fuzzy sets; and strife (or discord), which expresses conflicts among the various sets of alternatives.

“It is conceivable that other types of uncertainty will be discovered when the investigation of uncertainty extends to additional theories of uncertainty.”[4]

…………Some authors also include “ambiguity (lack of information),”[5] which Klir and Yuan define as a parent class of both discord and nonspecificity in an excellent diagram I wish I could reprint.[6] Probabilities probably also ought to be included as well.[7]As soon as I introduced to the concept of uncertainty partitioning, I was intrigued by the possibility of defining human free will as an alternative form of uncertainty, but that raises many thorny philosophical questions. Among them is the contention that it doesn’t even exist, which is a disturbing tenet of many popular philosophies, like materialistic determinism and certain forms of theological predestination. I’d dispute that with evidence that would be hard to debunk and raise the possibility that it may not be possible to quantify it at all, by definition; the ability to assign values to it would certainly be helpful in academic fields like economics and psychology, where human behavior is the crux of the matter. This topic integrates quite nicely with the contention of authors like Lofti A. Zadeh, the father of fuzzy set theory, that it might be helpful to apply fuzzy techniques in these fields to model “humanistic systems.”[8] Other controversial candidates for new categories of uncertainty include the notion that reality is somewhat subjective (which I would argue is fraught with risk, since it is a key component of many forms of madness) and the contention that some events (particularly at the quantum level) can be truly random, in the sense of being indeterminate or “uncaused.” Albert Einstein drove home the point that uncertainty is deeply rooted in all we see in his famous quote from a lecture at the Prussian Academy of Sciences in 1921, in which he seemed to extend it right into the heart of mathematics itself: “…as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”[9]

**Partitioning Uncertainty**

The first step is to develop a habit of explicitly recognizing which type of uncertainty is under discussion, then partitioning it off using the appropriate type of fuzzy set. For example, whenever we need to cram continuous scales into finite data types like float, decimal and numeric, we end up creating measurement uncertainty about whatever values come after the precision we’ve chosen.[10] Like other types of measurement uncertainty, this is best addressed by fuzzy sets without any special probabilistic, possibilistic or evidence theory connotations attached to them. Incidentally, some theoreticians say that if we’re trying to quantify the uncertainty of a measurement, membership functions based on the normal distribution (i.e. the bell curve) are usually the best choice (based on empirical evidence from the aerospace industry).[11] If we were uncertain about the likelihood of an event occurring, we’d assign a probability value instead; if we were unsure of the logical necessity of an event, we’d use a possibility distribution, as explained in the last installment of this series. In the next installment, I’ll explain how Dempster-Shafer Theory can be used to judge the certainty and credibility of evidence, by assigning grades of membership in the set of true statements.

…………Once the appropriate method of uncertainty modeling has been selected, we can then apply its associated formulas to compute figures for nonspecificity, imprecision, discord and the like. The good news is that we already dispensed with the main means of computing fuzziness, back in Implementing Fuzzy Sets in SQL Server, Part 2: Measuring Imprecision with Fuzzy Complements. In the remainder of this article, I’ll provide sample T-SQL for implementing two of the three main methods for calculating the “U-Uncertainty,” a.k.a. the nonspecificity. Like many other authors I consulted for this series, Klir and Yuan stress that nonspecificity and fuzziness are completely independent stats, since they measure two distinct and unrelated types of uncertainty.[12] The former is dictated by the number of possible distinct states that a set can take on, whereas the latter quantifies imprecision in class boundaries.[13] A set can have many possible arrangements, yet still be entirely crisp; there’s no mistaking what a Lego or Lincoln Log is, but there’s apparently no end to the crazy things that can be built with either one. Sets with few arrangements but really fuzzy boundaries are also possible. That is why fuzzy sets sans any additional meaning like probability, possibility and credibility scores have both fuzziness and nonspecificity measures attached to them.

…………Possibility theory, the topic of the last blog post in this amateur series of self-tutorials, has a form of nonspecificity that is easier to specify (pun intended) than the ordinary fuzzy set version, so I’ll introduce that first. The SELECT in Figure 1 is performed on a column of muscular dystrophy data I downloaded from the Vanderbilt University’s Department of Biostatistics and added to a sham DataMiningProjects database a few tutorial series ago. The PossibilityScore was assigned by a random number generator in the last article and tacked onto the table definition, for the sake of convenience. It’s time for my usual disclaimer: I’m writing this in order to learn this topic, not because I know it well, so it is a good idea to check over my T-SQL samples before putting them to serious use. This is especially true of this SELECT, where I may be applying a Lead where there should be a Lag; in contrast to the topics I post on in previous series, examples with sample data are few and far between in the fuzzy set literature, which makes validation difficult. Furthermore, there is apparently a more compact version available for specific situations, but I’ll omit it for now because I’m still unclear on what mathematical prerequisites are needed.[14]

** Figure 1: Possibilistic Nonspecificity for the LactateDehydrogenase Column**SELECT SUM(PossiblityDifference * Log(RN, 2)) AS PossibilisticUUncertainty

FROM (SELECT ROW_NUMBER() OVER (ORDER BY ID) AS RN, PossibilityScore – Lead(PossibilityScore, 1, 0) OVER (ORDER BY ID) AS PossiblityDifference

FROM Health.DuchennesTable) AS T1

…………The SELECT returns a single value of 4.28638426128113, which measures that amount of uncertainty in bits; the greater the number of possible state descriptions, the higher the U-Uncertainty will be. The same relationship applies to the procedure below, which returns a value of 7.30278910848746 bits; the difference is that one measures uncertainty about the number of possible values the LactateDehydrogenase column can have, while the other measures lack of certainty about the number of membership function scores a row can be assigned. Figure 2 is practically identical to the sample code I’ve posted throughout this series, at least as far as the UPDATE; all I’m doing is running the stored procedure from Outlier Detection with SQL Server, part 2.1: Z-Scores__ on the DuchennesTable and storing the results in a table variable, then transforming them to a scale of 0 to 1 using the @Rescaling variables and ReversedZScores column. The GroupRank column can be safely ignored, as usual. __The first SELECT with the AlphaCutLeftBound and AlphaCutRightBound columns is only provided to illustrate the how the nonspecificity figure is arrived at in the last SELECT. What we’re basically doing is partitioning the dataset into nested levels, using the alpha cut (α-cut) technique I introduced in the last article, then applying a Base-2 LOG and summing the results across the hierarchy.[15] The tricky part is that with α-cuts, records can belong to more than one subset, as I pontificated on in my last post; the levels are widest at the bottom of the dataset, but narrowest at the top, where the MembershipScore values approach the maximum of 1.This calls for thinking about the data in an odd way, given that in most relational operations records are assigned to only a single subset.

** Figure 2: Code for Hartley Nonspecificity**DECLARE @RescalingMax decimal(38,6), @RescalingMin decimal(38,6), @RescalingRange decimal(38,6)

DECLARE @ZScoreTable table

(PrimaryKey sql_variant,

Value decimal(38,6),

ZScore decimal(38,6),

ReversedZScore as CAST(1 as decimal(38,6)) – ABS(ZScore),

MembershipScore decimal(38,6),

GroupRank bigint

)

INSERT INTO @ZScoreTable

(PrimaryKey, Value, ZScore, GroupRank)

EXEC Calculations.ZScoreSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’LactateDehydrogenase‘,

@PrimaryKeyName = N’ID’,

@DecimalPrecision = ’38,32′,

@OrderByCode = 8

— RESCALING

SELECT @RescalingMax = Max(ReversedZScore), @RescalingMin= Min(ReversedZScore) FROM @ZScoreTable

SELECT @RescalingRange = @RescalingMax – @RescalingMin

UPDATE @ZScoreTable

SET MembershipScore = (ReversedZScore – @RescalingMin) / @RescalingRange

SELECT AlphaCutBound AS AlphaCutLeftBound, Lag(AlphaCutBound, 1, 0) OVER (ORDER BY AlphaCutBound) AS AlphaCutRightBound,

AlphaCutBound – Lag(AlphaCutBound, 1, 0) OVER (ORDER BY AlphaCutBound) AS AlphaCutBoundaryChange, Log(AlphaCutCount, 2) AS IndividualLogValue

FROM (SELECT Count(*) AS AlphaCutCount, AlphaCutBound

FROM @ZScoreTable AS T1

INNER JOIN (SELECT DISTINCT MembershipScore AS AlphaCutBound

FROM @ZScoreTable) AS T2

ON MembershipScore >= AlphaCutBound

GROUP BY AlphaCutBound) AS T3

SELECT SUM(AlphaCutBoundaryChange * Log(AlphaCutCount, 2)) AS FuzzySetNonspecificityInBits

FROM (SELECT AlphaCutCount, AlphaCutBound – Lag(AlphaCutBound, 1, 0) OVER (ORDER BY AlphaCutBound) AS AlphaCutBoundaryChange

FROM (SELECT Count(*) AS AlphaCutCount, AlphaCutBound

FROM @ZScoreTable AS T1

INNER JOIN (SELECT DISTINCT MembershipScore AS AlphaCutBound

FROM @ZScoreTable) AS T2

ON MembershipScore >= AlphaCutBound

GROUP BY AlphaCutBound) AS T3) AS T4

__Figure 3: Results for the Hartley Nonspecificity Example
__

…………The point of using the α-cuts is to chop the dataset up into combinations of possible state descriptions, which is problematic with fuzzy sets because the boundaries between states are less clear. The interpretation depends entirely on the meaning of the fuzzy attribute; as Klir and Yuan note, it can reflect an “an unsettled historical question” in the case of retrodiction, possible future states in the case of prediction, prescriptive uncertainty in the case of policies, diagnostic uncertainty in the case of medical information and so forth.[16] In the same vein, we can interpret my sample above as measuring 7.30278910848746 bits of uncertainty about a record’s place within the range of Z-Scores, which can in turn be used as a form of outlier detection. The smaller the range of possible values, the smaller the number of possible state descriptions becomes, which means that the cardinality of the α-cuts and the value of the final statistic decline as well.

…………This is an adaptation of a function developed way back in 1928 by electronic pioneer Ralph Hartley[17]; since it serves as one of the foundations of information theory I’ll put off discussion of the crisp version until my long-delayed monster of a series, Information Measurement with SQL Server. We’ve got at least two more articles in the fuzzy set series to dispense with first, including an examination of Dempster-Shafer Theory in the next installment. Evidence theory also has its own brand of nonspecificity measure, also based on the Hartley function.[18] Measures like strife and discord are more relevant to that topic, since they deal with conflicts in evidence. Possibility theory has counterparts for both, but I’ll leave them out, given that Klir and Yuan counsel that “We may say that possibility theory is almost conflict-free. For large bodies of evidence, at least, these measures can be considered negligible when compared with the other type of uncertainty, nonspecificity. Neglecting strife (or discord), when justifiable, may substantially reduce computation complexity in dealing with large possibilistic bodies of evidence.”[19] Possibility theory is a useful springboard into the topic though, given that Belief and Plausibility measures are modeled in much the same way. In fact, Possibility and Necessity measures are just special cases of Belief and Plausibility, which should serve to decomplicate my introduction to Dempster-Shafer Theory a little.

[1] p. 114, King, Stephen, 1981, __Stephen King’s Danse Macabre__. Everest House: New York. I’m paraphrasing King, who in turn paraphrased an idea expressed to him by author William F. Nolan at the 1979 World Fantasy Convention.

[2] p. 269, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J.

[3] *IBID*., p. 3.

[4] *IBID*., p. 246.

[5] p. 2, Hinde, Chris .J. and Yang, Yingjie., 2009, “A New Extension of Fuzzy Sets Using Rough Sets: R-Fuzzy Sets,” pp. 354-365 in __Information Sciences__, Vol. 180, No. 3. Available online at the Loughborough University Institutional Repository web address https://dspace.lboro.ac.uk/dspace-jspui/bitstream/2134/13244/3/rough_m13.pdf

[6] p. 268, Klir and Yuan.

[7] *IBID*., p. 3.

[8] *IBID*., p. 451.

[9] Cited from the __Common Mistakes in Using Statistics__ web address https://www.ma.utexas.edu/users/mks/statmistakes/uncertaintyquotes.html

[10] *IBID*., pp. 327-328.

[11] Kreinovich, Vladik; Quintana, Chris and Reznik, L.,1992, “Gaussian Membership Functions are Most Adequate in Representing Uncertainty in Measurements,” pp. 618-624 in Proceedings of the North American Fuzzy Information Processing Society Conference, Vol. 2. NASA Johnson Space Center: Houston. Available online at the University of Texas at El Paso web address www.cs.utep.edu/vladik/2014/tr14-30.pdf

[12] p. 258, Klir and Yuan.

[13] p. 2, Hinde and Yang.

[14] pp. 253, 269, Klir and Yuan.

[15] *IBID*., pp. 248-251.

[16] *IBID*., p. 247.

[17] See the Wikipedia articles “Hartley Function” and “Ralph Hartley” at http://en.wikipedia.org/wiki/Hartley_function and http://en.wikipedia.org/wiki/Ralph_Hartley respectively.

[18] pp. 259, Klir and Yuan.

[19] *IBID*., p. 264.

## Implementing Fuzzy Sets in SQL Server, Part 8: Possibility Theory and Alpha Cuts

**By Steve Bolton**

…………To get the point across that fuzzy sets require membership grades of some sort, throughout this series I’ve borrowed the stored procedure I coded for Outlier Detection with SQL Server, part 2.1: Z-Scores and rescaled the results on the customary range of 0 to 1. The literature on fuzzy sets contains frequent warnings against automatically interpreting membership scores as probabilities, but I deliberately introduced a tie-in to stochastics by using Z-Scores, which are inherently probabilistic. Other shades of meaning may be assigned which are unfamiliar to modelers of ordinary “crisp” sets, which is why I pointed out early on in this series of amateur self-tutorials that interpretability is a more prominent issue with fuzzy sets. For example, membership functions can be viewed as assigning scores to the accuracy of the associated values, which is similar to the way in which we used fuzzy numbers two articles ago to code such linguistic concepts like “about” and “near.” If we add the subtle distinction that the membership scores may mean “cannot be near” or “can be around” a certain value, we’re stepping into the realm of Possibility Theory, which has important uses in fuzzy logic.[i]

…………Approximate reasoning and related concepts are more relevant to topics like expert systems that are beyond the purview of this series, but Possibility Theory can serve as a useful springboard into Evidence Theory, which is useful in developing programs of uncertainty management. Possibility distributions are in one sense a more restricted brand of probability distributions, while also acting as more restrictive versions of Evidence Theory measures; it may therefore be easier to use them as bridge from one relatively familiar topic to a lesser-known one. I originally thought the topic would be quite difficult to grasp, but it’s actually a good deal easier that stochastics. Perhaps the most difficult aspect is that possibility distributions can be modeled using alpha cuts (α-cuts), a method of partitioning fuzzy sets that will prove useful in the next two articles to come.

**From ‘Can’ and ‘Must’ to Surprise**

In fact, I’ll lighten the load further by dispensing with many of the details of Possibility Theory, since its simplicity can quickly give way to complexity, same as with any other fuzzy set topic. For example, stochastic concepts like conditional and marginal probabilities have their counterparts in Possibility Theory, all of which is too far afield for our purposes. For those have a need for the corresponding formulas and don’t mind wading through the thick math, I recommend consulting the seventh chapter of my favorite resource, George J. Klir and Bo Yuan’s *Fuzzy Sets and Fuzzy Logic: Theory and Applications*. I’m not even going to get into a discussion of how possibility scores are assigned; for the sake of argument, let’s assume any figures used in my examples are derived from subjective ratings by end users. The important thing to keep in mind is that we need* two* numbers to specify a possibility distribution, not just the single probability figure used in stochastics. One of these is known as the Possibility measure and the other as a measure of Necessity, which is the inverse of Possibility’s complement.

…………The two measures can be combined by adding them together and subtracting one, but the fact that this results in a non-standard range of -1 to 1 limits its usefulness.[ii] The simplest way to model this relationship is to use a bit column, in conjunction with the float, numeric or decimal columns normally used to represent fuzzy sets on a continuous scale between 0 and 1.[iii] The tricky thing is that an event *must* occur when Necessity equals 1, whereas a Possibility score of 0 means that it cannot; on the other hand, a Possibility score of 1 does not imply certainty, only a state of total surprise if it did; apparently this in analogous to a measure of “surprise” developed in the mid-20^{th} Century by economist G. L. S. Shackle,[iv] which has since been further developed by such household names in the fuzzy set field as like Henri Prade and Ronald R. Yager.[v] As Lofti A. Zadeh, the father of fuzzy set theory, explains it:

“Consider a numerical age, say u = 28, whose grade of membership in the fuzzy set ‘young’ is approximately 0.7. First we interpret 0.7 as the degree of compatibility of 28 with the concept labelled young. Then we postulate that the proposition ‘Peter is young’ converts the meaning of 0.7 from the degree of compatibility of 28 with young to the degree of possibility that Peter is 28 given the proposition ‘Peter is young.’ In short, the compatibility of a value of u given ‘Peter is young.’”[vi]

…………This lack of symmetry is comparable to the way possibilities and probabilities differ. A Necessity measure of 1 leads inevitably to a probability score of 1, since what *must* happen is entirely probable; conversely, a Possibility measure of 0 leads to a probability score of 0, since what cannot happen is entirely improbable. Apart from these extremes, however, the two theories diverge. A Necessity or Possibility score of 0.5 has no effect on the probability, since whether or not a thing is logically conceivable is not equivalent to whether it is likely to happen; it is entirely *possible* that we may win the lottery tomorrow, but I wouldn’t bet on it. This is the core difference between the two theories: one expresses confidence in our information about whether a thing can happen, while the other reflects confidence in information about whether it *will*.

…………Because of this relationship, a possibility distribution acts as a cap on the associated probability distribution; this has many mathematical consequences[vii], the most important of which is that the two distribution types intersect at their minimum and maximum values. This in turn leads to the interesting property that possibility scores do not have to sum to 1 across a set of records, unlike probabilities; the only restriction is that the maximum value per record is 1.[viii] This in turn means that to assess whether or not we’ve reached a certain threshold of possibility values, all of the records with scores greater than the threshold must be taken into account. In other words, if we want to know if an event has a possibility of 0.3, we must examine all of the records with scores higher than that to come to a verdict. Every record in a set will qualify for the lowest partition, where a possibility score of 0 is all it takes to qualify, but the number of records continually shrinks as we move up the dataset towards the perfect score of 1.

**Nested Sets and α-cuts**

This creates a nested set of evidence in which records can belong to multiple partitions, which can be easily implemented in T-SQL despite the fact that it calls for thinking about sets in unusual ways. We’re doing something uncommon here by cutting a set up hierarchically, so that a row belongs to more and more sets as we approach the maximum value of the membership function, rather than a single subset as we see in most relational joins. Klir and Yuan include a couple of handy illustrations which could get across the meaning of nested sets of evidence in a heartbeat, but I haven’t had a chance to seek permission to reprint them and don’t have the ability to draw my own.[ix] In turns out that the fuzzy set partitioning method known as α-cuts are an ideal tool for implementing these relationships[x] (not to mention many others that are beyond the scope of this series, like fuzzy equivalence relations[xi]). In plain English, this means that we have to use >= comparison operators to chop up a dataset into nested subsets, or > operators in the case of strong α-cuts.

…………I’m trying to keep the jargon to a minimum, but since the terms “cutworthy” and “strong cutworthy” occur frequently in the literature, it may be helpful to know that they refer to mathematical properties of fuzzy sets which are preserved in their α-cuts. [xii] Another important property is reconstructibility, which means that a fuzzy set can be rebuilt from its partitions. The manner in which possibility distributions establish maximum values for their associated probability distributions is essentially one and the same as the min/max types of unions and intersections we dealt with in previous articles, while the possibilities themselves are defined by their α-cuts.[xiii]

…………The first SELECT statement in Figure 1 illustrates how a simple GROUP BY and SUM with a ROWS UNBOUNDED PRECEDING clause can be used to partition a SQL Server table in this unconventional manner. I also have an alternate version of these SELECTs in which partitioning is done by deciles (or any other arbitrary percentile value) rather than DISTINCT MembershipScores, which I omitted to keep things simple; if anyone needs it though, I’d be happy to post it. As usual, the sample data comes from a dataset on the Duchennes form of muscular dystrophy I downloaded from Vanderbilt University’s Department of Biostatistics a few tutorial series ago, which now resides in a sham DataMiningProjects database. The code from the beginning to the UPDATE statement is basically identical to the T-SQL samples I’ve posted throughout this series, which always begins with plugging the results of the aforementioned Z-Score procedure into a table variable. The GroupRank column is only included because it was part of the original procedure and can’t be omitted from the INSERT EXEC, but it can be safely ignored. The @Rescaling variables and the ReversedZScore column are then used to adjust the Z-Scores to the 0 to 1 range used in almost all fuzzy sets. There are only 202 records in the DuchennesTable where LactateDehydrogenase where is NOT NULL, which is exactly equal to the count of values in Figure 2 where the MembershipScore is zero. The counts for each α-cut continually decline after that, till they reach the perfect score of 1, which is equivalent to the Height measure mentioned in last week’s article on fuzzy stats. I left out the middle values for the sake of brevity.

__Figure 1: An Example of α-Cut Partitioning__

DECLARE @RescalingMax decimal(38,6), @RescalingMin decimal(38,6), @RescalingRange decimal(38,6)

DECLARE @ZScoreTable table

(PrimaryKey sql_variant,

Value decimal(38,6),

ZScore decimal(38,6),

ReversedZScore as CAST(1 as decimal(38,6)) – ABS(ZScore),

MembershipScore decimal(38,6),

GroupRank bigint

)

INSERT INTO @ZScoreTable

(PrimaryKey, Value, ZScore, GroupRank)

EXEC Calculations.ZScoreSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’LactateDehydrogenase‘,

@PrimaryKeyName = N’ID’,

@DecimalPrecision = ’38,32′,

@OrderByCode = 8

— RESCALING

SELECT @RescalingMax = Max(ReversedZScore), @RescalingMin= Min(ReversedZScore) FROM @ZScoreTable

SELECT @RescalingRange = @RescalingMax – @RescalingMin

UPDATE @ZScoreTable

SET MembershipScore = (ReversedZScore – @RescalingMin) / @RescalingRange

— ALPHA CUTS BY DISTINCT VALUES

— =======================================

SELECT MembershipScore, SUM(DistinctCount) OVER (ORDER BY MembershipScore

DESC ROWS UNBOUNDED PRECEDING) AS AlphaCutCount

FROM (SELECT MembershipScore, Count(*) AS DistinctCount

FROM @ZScoreTable

WHERE MembershipScore IS NOT NULL

GROUP BY MembershipScore) AS T1

— MEASURE OF SURPRISE

— =======================================

SELECT ID, LactateDehydrogenase, NecessityMeasure, PossibilityScore, 1 – PossibilityScore AS SimpleMeasureOfSurprise

FROM Health.DuchennesTable

WHERE LactateDehydrogenase IS NOT NULL

__Figure 2: Sample α-Cut Values from the Beginning and End of the Duchennes Dataset__

__Figure 3: Possibility Scores and the Measurement of Surprise__

…………The second SELECT merely returns some fake PossibilityScore values I randomly generated and tacked onto the DuchennesTable, with a simple inverse calculation to illustrate the most basic measure of Surprise.[xiv] Authors like Prade and Yager have extended the measure to address more sophisticated use cases, but Figure 2 is sufficient to get the point across for our purposes. The interpretation of any Surprise measure is straightforward: the higher the value, the greater our bewilderment will be if the associated event occurs. In this context, the Surprise would be attached to the possibility of observing the corresponding LactateDehyrogenase value; of course, these are actual values taken from a muscular dystrophy in the 1980s, so if we weren’t using this for practice purposes we’d have to assign Necessity values of 1. These measurements of qualities like Surprise are of course not perfect, but they do allow us to attach some sort of ballpark figure to our expectations. As we shall see in the next two articles, one of the primary uses of fuzzy sets is to measure uncertainty, which can be valuable even when those measures are themselves uncertain. Two articles from now we’ll see how possibility theory is useful not merely in measuring surprise or in deriving interval-valued probabilities[xv], but also as a bridge to Dempster-Shafer Evidence Theory, which is useful in reckoning subtypes of uncertainty like Strife, Discord and Conflict. In the next installment, I’ll explain how both possibility distributions and α-cuts can measure nonspecificity, which is one of several types of uncertainty we can quantify with the aid of fuzzy sets.

[i] p. 200, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J.

[ii] *IBID*., p. 198.

[iii] I got this idea from the __Wikipedia__ article “Possiblity Theory” at http://en.wikipedia.org/wiki/Possibility_theory.

[iv] *IBID*.

[v] Prade, Henri and Yager, Ronald R., 1994, “Estimations of Expectedness and Potential Surprise in Possibility Theory,” pp. 417-428 in __International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems__, December 1994. Vol. 2, No. 4. Available online at the National Aeronautics and Space Administration (NASA) web address http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930020329.pdf

[vi] Posted by Kornelia Brutoczki July 4, 2001 at the __Fuzzy Logic Home Page__ address http://mazsola.iit.uni-miskolc.hu/DATA/diploma/brutoczki_kornelia/fu_gz_02.html. The original source is not given.

[vii] pp. 206-207, Klir and Yuan.

[viii] *IBID*., p. 204.

[ix] *IBID*., pp. 24, 195.

[x] *IBID*., pp. 19-21, 35.

[xi] *IBID*., p. 133.

[xii] *IBID*., p. 23, 25, 36.

[xiii] *IBID*., pp. 187-188, 198.

[xiv] See Prade and Yager, 1994.

[xv] p. 205, Klir and Yuan.

## Implementing Fuzzy Sets in SQL Server, Part 7: The Significance of Fuzzy Stats

**By Steve Bolton**

…………In the world of fuzzy sets and imprecision modeling, the concept of cardinality takes on new shades of meaning that are not applicable to ordinary “crisp” sets, i.e. those without membership grades. In the last article in this series of amateur-self-tutorials, I mentioned one type of “fuzzy cardinality,” based on triangular, trapezoidal and other fuzzy numbers that are quite useful in modeling many vague statements found in everyday speech. Of course, another means of expressing cardinality is through ordinary numbers that are defined by a single value, rather than intervals and other such fuzzy set types. This raises some interesting questions, because one of the logical implications of graded set membership is that records with a score of 0 shouldn’t be included in the count. By extension, the values that are non-zero should only be counted in proportion to the score assigned by their membership function; since these are almost always on a scale of 0 to 1, the count of a fuzzy set never exceeds that of its crisp count, but may be much lower. Moreover, since membership scores are represented by fractional values, we’d normally use SQL Server data types like float, decimal and numeric to represent them, rather than members of the int family as we would with ordinary counts.

…………This apparently gives rise to many different possible calculations for fuzzy counts, but the most common one in the literature is the sigma count, in which we simply add up all of the membership scores for an entire set. Another stat seen occasionally in the literature is Support, which is defined as a crisp count of all the non-zero members of a fuzzy set; it thus always results in an integer somewhere between the sigma count and the ordinary crisp count. The Height refers to the crisp count, the Bandwidth to the number of records with scores greater than 0.5 and the Core to those with the maximum score of 1; these concepts might be useful in such applications as fuzzy clustering, but I see the sigma count used far more often in connection with today’s topic, fuzzy stats, which come into play whenever we want to calculate aggregates on fuzzy sets in platforms like SQL Server.[1]

**Partial Credit for Partial Set Membership**

The trick with this topic is “Think partial credit!” to borrow a phrase from University of Minnesota Prof. Glen Medeen.[2] Even if we restrict ourselves to the sigma count definition of fuzzy counts, the concept carries many interesting implications for all of the statistics that are derived from it. Averages, standard deviations, variances and all of the more advanced statistics derived from them must be recalculated, given that they’re derived from fundamental measures like values and counts that no longer apply. The logic inherent in partial set membership demands this fundamental rethinking of basic statistics. The crisp versions of some of these stats are precalculated by SQL Server, so by switching to the fuzzy set versions we’ll incur some performance costs by computing them on the fly instead, with the aid of T-SQL aggregates and windowing functions. Thankfully, some of these fuzzy stats are worth the extra computations, because they can shed light on our data in unusual ways. Perhaps the most obvious example is the difference between the crisp count and sigma count, which might be used as an alternative measure of fuzziness in the place of fuzzy complements, which as we saw early on in this series, are normally ideal for that use case.

…………Figure 1 provides a simple example of how to code this possible alternate measure of imprecision, by subtracting the sigma count from the height, i.e. the crisp count. I also demonstrate how easy it is to derive the bandwidth, support and core stats, even though these are only used infrequently. As usual, most of the initial code involves assigning the membership scores, by drafting the procedure I wrote for Outlier Detection with SQL Server, part 2.1: Z-Scores for double duty as my membership function. The calculations are performed on the Duchennes muscular dystrophy data I downloaded a few tutorial series ago from Vanderbilt University’s Department of Biostatistics, which now resides in a dummy DataMiningProjects database; afterwards, they’re stored in a @ZScoreTable table variable, that can be operated on as needed. For the sake of consistency, I’ve stuck to the same format I’ve used throughout this series by using the three @Rescaling variables and ReversedZScore column to transform the ZScores in a membership score on the traditional 0 to 1 range.

**New Means, Medians and Modes**

Once we’ve derived the sigma count from these grades, I then calculate the standard fuzzy mean, which may be the simplest, most intuitive form of a “fuzzy absolute center.”[3] Another alternate measure of centrality is of course the mode, which I’ve thrown in because it’s so easy to calculate; to derive the fuzzy version, we just have to multiply each value’s count by its membership grade. This is one of the few fuzzy stats where the value is not affected by its score. In Figure 2 we can see that both versions of the mode return the same value of 198, which is within the general rule that both modes and their fuzzy counterparts will only return actual crisp values from their datasets. Since medians are dependent on orders, I’ll take up that topic when I address the fuzzification of ranks in a wrap-up of the whole series.

…………Instead, I’ve incorporated a higher class of averages known as Generalized Means, which can be used to derive a whole family of means between the minimum and maximum values, including the fuzzy arithmetic mean mentioned above, along with the harmonic and geometric means.[4] We basically plug in an @AlphaParameter bounded between 0 and 1, which allows us to cover the whole range, in much the same fashion that the various T-norm and T-conorm parameters empowered us to derive myriad types of fuzzy intersections and unions in previous articles. Note that in Figure 2, we see that the parameter value I arbitrarily chose led to a far different value for the GeneralizedMean than the one derived for the ordinary FuzzyMean.

** Figure 1: Sample Code for Fuzzy Counts and Means**DECLARE @RescalingMax decimal(38,6), @RescalingMin decimal(38,6), @RescalingRange decimal(38,6)

DECLARE @ZScoreTable table

(PrimaryKey sql_variant,

Value decimal(38,6),

ZScore decimal(38,6),

ReversedZScore as CAST(1 as decimal(38,6)) – ABS(ZScore),

MembershipScore decimal(38,6),

GroupRank bigint

)

(PrimaryKey, Value, ZScore, GroupRank)

EXEC Calculations.ZScoreSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’LactateDehydrogenase‘,

@PrimaryKeyName = N’ID’,

@DecimalPrecision = ’38,32′,

@OrderByCode = 8

SELECT @RescalingMax = Max(ReversedZScore), @RescalingMin= Min(ReversedZScore) FROM @ZScoreTable

SELECT @RescalingRange = @RescalingMax – @RescalingMin

UPDATE @ZScoreTable

SET MembershipScore = (ReversedZScore – @RescalingMin) / @RescalingRange

DECLARE @Count bigint, @SigmaCount float, @Support float, @Bandwidth float, @Core float,

@Mean float, @FuzzyMean float, @GeneralizedMean float, @Mode float, @FuzzyMode float

— COUNTS

SELECT @SigmaCount = SUM(MembershipScore), @Count = Count(*)

FROM @ZScoreTable

WHERE ZScore IS NOT NULL

SELECT @Support = Count(*)

FROM @ZScoreTable

WHERE ZScore IS NOT NULL AND MembershipScore > 0

SELECT @Bandwidth = Count(*)

FROM @ZScoreTable

WHERE ZScore IS NOT NULL AND MembershipScore > 0.5

SELECT @Core = Count(*)

FROM @ZScoreTable

WHERE ZScore IS NOT NULL AND MembershipScore = 1

— MODES

SELECT @Mode = Value

FROM (SELECT TOP 1 ValueCount, Value

FROM (SELECT Count(*) AS ValueCount, Value

FROM @ZScoreTable

WHERE ZScore IS NOT NULL

GROUP BY Value) AS T1

ORDER BY ValueCount DESC) AS T2

SELECT @FuzzyMode = Value

FROM (SELECT TOP 1 ValueCount, Value

FROM (SELECT Count(*) * MembershipScore AS ValueCount, Value

FROM @ZScoreTable

WHERE ZScore IS NOT NULL

GROUP BY Value,MembershipScore) AS T1

ORDER BY ValueCount DESC) AS T2

— AVERAGES

DECLARE @AlphaParameter float

SELECT @AlphaParameter = 0.3

SELECT @FuzzyMean = SUM(MembershipScore * Value) / @SigmaCount , @Mean = Avg(Value)

FROM @ZScoreTable

WHERE ZScore IS NOT NULL

SELECT @GeneralizedMean = Power(SUM(Power(Value, @AlphaParameter)) / CAST(@SigmaCount AS float), 1 / @AlphaParameter)

FROM @ZScoreTable

WHERE ZScore IS NOT NULL

SELECT @Count AS RegularCount, @SigmaCount AS SigmaCount,

@Count – @SigmaCount AS AlternativeMeasureOfFuzziness,

@Support AS Support, @Bandwidth As Bandwidth, @Core as Core,

@Mode AS Mode, @FuzzyMode as FuzzyMode,

@Mean AS Mean, @FuzzyMean AS FuzzyMean, @GeneralizedMean AS GeneralizedMean

__Figure 2: Sample Results from the Duchennes Table
__

…………Generalized means occupy a space in the set of norm operations in between T-norms and T-conorms, along with Ordered Weighted Averages.[5] Basically, each record in an OWA is multiplied by a weight which globally equals one, but the choice of weights is so broad that I won’t bother with them; I’ll merely point out that this obviously overlaps the topic of neural net weights, at least to anyone who has coded them before. I’ll also omit my sample code for Lambda Averages (i.e. λ-Averages), because it’s simply too long in comparison to its usefulness. This class of norm operations is derived from binary set relations, which means we first have to create a second table variable, fill it and adjust the scores, as we did in previous articles with T-norms and T-conorms. We’d then apply a CASE statement to select the MIN value of the @LambdaParameter and the outcome of the union, when the both records were between 0 and the @LambdaParameter; take the MAX of the @LambdaParameter and the outcome of the fuzzy intersection when the outcomes were greater than the @LambdaParameter; then use the @LambdaParameter value in the ELSE statement.[6]

…………As with fuzzy complements, unions and intersections, the applications are determined by the selection of appropriate parameter values.[7] One method of accomplishing this is of course parameter estimation.[8] A good starting point for fuzzy parameter estimation may be Seyed Mahmoud Taheri’s select bibliography of recent developments in fuzzy stats, which also lists many resources for extending fuzzy stats to standard statistical and data mining topics like Bayesian priors, fuzzy regression and hypothesis testing.[9] A couple of the sources also connect fuzzy sets to information theory, which I will also begin doing in my next tutorial.

**Fuzzy Variance: A Fresh Take on an Staid Stat**

Taheri also mentions published research on fuzzy sets and maximum likelihood, which makes me wonder if there is also some connection to Fisher Information. The same is true of the different types of fuzzy variance, given that variance is interpreted in Fisher Information as a form of uncertainty. This may be a more worthwhile topic to cover than λ-Averages and OWAs since the formulas are less broad and have clearer applications. First of all, it makes intuitive sense to calculate variance differently on fuzzy sets, for precisely the same reasons as fuzzy means: the crisp version of the statistic is dependent on counts, which ought to be replaced with alternative measures like the sigma count when possible. If a record has zero membership, for example, its value shouldn’t count at all in the computation, because it’s no longer part of the set. It thus follows that a value with partial membership should only be taken into consideration in proportion to its score, just as with fuzzy means; it stands to reason that the same principles would apply if we went beyond the first and second statistical moments, mean and variance, to the third and fourth, skewness and kurtosis.

…………This leads to some interesting questions over how we can interpret the differences in the crisp and fuzzy variances. Given that the difference between crisp and sigma counts reflects a measure of fuzziness – albeit not clearly as fuzzy complements – perhaps we can interpret this as a measure of how dispersed the fuzziness is. This might come in quite handy in many data mining applications. I haven’t seen it used that way in the literature – but it’s good to keep in mind that my exposure to the whole topic of fuzzy sets is limited, given that I can’t afford the hefty fees for many of the academic journals, which render them inaccessible to me. Nor have I seen trapezoidal numbers combined with variance, but a construction like the CASE statement for TrapezoidalRangeOnTheCrispVariance in Figure 3 might be useful in expressing natural language slang about dispersion like, “all over the place.” The TrapezoidalRangeOnTheFuzzyVariance expresses the same concept, except that it represents a fuzzy number on a fuzzy set, rather than a fuzzy number on a crisp set; it thus amounts to saying, “this graded set is all over the place.” I set the range boundaries arbitrarily so that both would have fractional scores in Figure 4, which serves as a better illustration of partial membership in a fuzzy number. Using the square root and power techniques mentioned in the last article, we could add superlatives to it like “really” or “somewhat.” If we were using Z-Scores in the context of a normal distribution, we might set graded boundaries based on the “68–95–99.7 Rule” I covered in Goodness-of-Fit Testing with SQL Server, part 1: The Simplest Methods, which involve the number of expected records between the first, second and third standard deviations. I left out the more complicated case of the superlative in sample code below, just to illustrate how a SQL Server user might write T-SQL for simple cases of the these two fuzzy variance types:

** Figure 3: Some Possible Measures of Fuzzy Variance**DECLARE @StDev float, @FuzzyStDev float, @Var float, @FuzzyVar float

SELECT @FuzzyVar = Sum(Power((Value * MembershipScore) – @FuzzyMean, 2)) / @SigmaCount, @Var

= Var(Value), @StDev = StDev(Value)

FROM @ZScoreTable

WHERE ZScore IS NOT NULL

SELECT @FuzzyStDev = Power(@FuzzyVar, 0.5)

DECLARE @LowerBound float, @UpperBound float

SELECT @LowerBound = 4000, @UpperBound = 5000

SELECT @StDev AS StDev, @FuzzyStDev AS FuzzyStDev,

@Var AS Var, @FuzzyVar AS FuzzyVar,

@Var – @FuzzyVar AS PossiblyTheVarianceOfTheFuzziness,

CASE WHEN @Var BETWEEN @LowerBound AND @UpperBound THEN 1

WHEN @Var < @LowerBound THEN ((@Var – @LowerBound)) / @Var + 1

WHEN @Var > @UpperBound THEN ((@UpperBound – @Var)) / @Var + 1

ELSE NULL END AS TrapezoidalRangeOnTheCrispVariance,

CASE WHEN @FuzzyVar BETWEEN @LowerBound AND @UpperBound THEN 1

WHEN @FuzzyVar < @LowerBound THEN ((@FuzzyVar – @LowerBound)) / @FuzzyVar + 1

WHEN @FuzzyVar > @UpperBound THEN ((@UpperBound – @FuzzyVar)) / @FuzzyVar + 1

ELSE NULL END AS TrapezoidalRangeOnTheFuzzyVariance

__Figure 4: Fuzzy Variance Result for the LactageDehydrogenase Column
__

…………Fuzzy variance may serve as a bridge to Fisher Information, a topic I want to cover in my long-delayed series, Information Measurement with SQL Server. Early on in this series we saw how fuzzy complements serve as one important measure of a different type of information, fuzziness, which quantifies the imprecision of a dataset in a different manner than variance. The difference between the sigma and crisp counts might serve the same purposes, although I’ve seen the various types of complements used more often for this purpose. One of the coolest things about fuzzy sets is that they give rise to several useful statistics that quantify different types of imprecision, which can be used to derive a program of “uncertainty management” for an organization. In the next installment we’ll see how we can use some of the fuzzy stats defined here to pin down a different brand of imprecision known as nonspecificity. This will involve discussion of the Hartley function and possibly Shannon’s Entropy, the latter of which is a fundamental concept in many data mining algorithms. Since entropy is among the foundations of information theory, this introduction to its applications in nonspecificity will serve as a bridge to my future Information Measurement series.

[1] pp. 25-28, Bonissone, Piero P., 1998, “Fuzzy Sets & Expert Systems in Computer Eng. (1).” Available online at http://homepages.rpi.edu/~bonisp/fuzzy-course/99/L1/mot-conc2.pdf. Bonissone’s material is reprinted at least in part from slides produced by artificial intelligence researchers Roger Jang and Enrique Ruspini.

[2] p. 5, Medeen, Glen, 2015, Two Examples of the Use of Fuzzy Set Theory in Statistics,” published online at the __University of Minnesota__ web address http://users.stat.umn.edu/~gmeeden/talks/fuzznov09.pdf

[3] p. 435, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J. On this particular page, they’re extending the meaning of the term even further, to complex network topologies.

[4] *IBID.*, p. 90.

[5] *IBID.*, pp. 92-93.

[6] *IBID.*, p. 94.

[7] *IBID.*, p. 93.

[8] *IBID.*, p. 94.

[9] p. 240, Taheri, Seyed Mahmoud, 2003, “Trends in Fuzzy Statistics,” pp. 239-257 in Austrian Journal of Statistics, Vol. 32, No. 3. Available online at __Vienna University of Technology__ web address http://www.statistik.tuwien.ac.at/oezstat/ausg033/papers/taheri.pdf

## Implementing Fuzzy Sets in SQL Server, Part 6: Fuzzy Numbers and Linguistic Modifiers

**By Steve Bolton**

…………I’ve written several amateur tutorial series on this blog in order to more quickly absorb difficult data mining, statistical and machine learning topics, while hopefully helping other SQL Server users avoid some of my inevitable mistakes. Since I don’t know what I’m talking about, I’m occasionally surprised at how useful some of the material turns out to be – particularly in the case of this week’s topic, fuzzy numbers, which I originally thought were a curiosity. In recent years I’ve made moderate progress in recapturing some of the advanced math skills I had as a kid, but some of the resources I’m consulting still wear me out with densely packed, arcane symbols; that was also true of George J. Klir and Bo Yuan’s *Fuzzy Sets and Fuzzy Logic: Theory and Applications* and several other sources I’ve used for this series on using T-SQL to implement fuzzy sets. Much of the literature on the subject is written by mathematicians who need thick equations and highly precise terminology to communicate with each other effectively, which can be taxing on non-specialists not accustomed to the jargon of the field.

…………That communication gap may be responsible for the enormous lag between the refinement of the theory and its adoption; this comes despite all of the empirical evidence that these data modeling techniques are insanely useful in solving certain classes of real-world problems, in a wide variety of industries. Adoption has also lagged in the relational database and data mining markets, in spite of the fact that set-based languages like T-SQL and Multidimensional Expressions (MDX) are ideal for implementing fuzzy sets. I’m trying to put my drop in the bucket to remedy that, but my lack of adequate equation translation skills almost caused me to skip over fuzzy numbers, which are actually one of the simplest and most useful aspects of fuzzy set theory.

**Modeling Everyday Speech **

…………One rule of thumb I’ve learned along the way is that whenever confusion arises over fuzzy sets, it is best to return to basics and state the problem in terms of natural language. After all, the main overarching use case for fuzzy set theory is to model imprecision that can be expressed linguistically, particularly when it would be useful to use continuous scales to ordinal categories. If we were using a Behavior-Driven Development (BDD) methodology, we might want to flag any qualifiers we encounter in user stories like “about,” “around,” “near,” “approximately,” “most,” “few” and the like as candidates for fuzzy numbers. These require several numbers and calculations of various intensity in order to assign a range of values to a single number – but since ordinary numbers require none of this extra computation, the obvious question is, why bother? It turns out that these particular instances of fuzzy sets are useful in modeling these types of natural language qualifiers, which express uncertainty about how close a value is to a definite target value.

…………“About half” is a clear and simple example from everyday speech. In fuzzy set parlance, this would be modeled as a “triangular number,” which is a fancy way of saying that we’d assign a perfect membership grade of 1 to values that were exactly equal to 0.5, but descending on either side of 0.5 in proportion to how far away the value was from that target. The term “triangular” is used because if we use a line chart to depict a membership function of this kind, it peaks at the target value and descends to 0 as the values decrease or increase away from it. Trapezoidal numbers are a mouthful, but really aren’t much more difficult; they have basically have the same shape as triangular numbers, except that the peak is flattened in order to express an interval of some kind.[1] University of Minnesota Prof. Glen Meeden’s natural language example of a trapezoidal number is the best I’ve yet run across in the literature: “What is the average yearly snowfall in the Twin Cities? You might answer somewhere between 20 and 50 inches.”[2]

…………Functions that only increase or decrease are useful in capturing related distinctions, like “a large number of” or “a small number of.”[3] Although it is wise to obey a few restrictions that lead to certain useful mathematical properties – particularly the industry-standard requirement of a boundary between 0 and 1 – it is not necessary for the membership functions to be symmetric. In fact, assigning a lopsided peak to a triangular number can be useful in modeling statements like “almost all,” which would be closer to the right edge of a line graph than a function that implements the term “most.”[4] They can even be bell-shaped.[5]

…………At a higher level of sophistication, fuzzy numbers can be used to model terms like “very,” which fall under the rubric of linguistic hedges and fuzzy modifiers – including statements like “very true,” which can be useful in fuzzy logic.[6] Klir and Yuan suggest applying powers and square roots to model such distinctions, since they don’t follow a linear scale. For example, they say that if we use a score of 0.8 to model the term “John is young,” squaring it would could model the phrase “very young” and using a square root could lead be used for “fairly young.” This is because the result of the first is 0.64, which strongly modifies the original term, while the second returns 0.89, which modifies it slightly.[7] Of course, the exact boundaries of all of these terms have to be set in light of domain knowledge of some kind. “Half” is a definite term but the modifier “about” can mean different things to different people, which may require aggregating the viewpoints of users in some way, perhaps using Decision Theory methods that integrate seamlessly with fuzzy sets. Neural nets likewise work well together with fuzzy sets because they are highly useful for encoding unknown functions, which means they can be put to use to derive such boundaries if greater precision is required. For many use cases, an informal guesstimate may suffice.

**Translating Fuzzy Numbers into T-SQL**

…………My sample code below implements four types of fuzzy numbers, using quite simple and arbitrary criteria that is merely designed to illustrate the concepts. For the sake of consistency, I’m once again using the procedure I wrote for Outlier Detection with SQL Server, part 2.1: Z-Scores to my derive membership grades and storing the results in a table variable (which has an extraneous column named GroupRank that can be safely ignored). In this case we’re calculating Z-Scores on the LactateDehydrogenase column of the Duchennes muscular dystrophy dataset I’ve been using as practice data for the last several tutorial series, which I downloaded ages ago from Vanderbilt University’s Department of Biostatistics and converted to a SQL Server table in a dummy DataMiningProjects database. Some of the key differences from previous articles include the absence of the ReversedZScore column and associated @Rescaling variables. This is because we don’t need to perform rescaling of any kind, since we’re measuring nearness to a few Z-Score values in all four examples, not calculating a relative score on a range of 0 to 1. The MembershipScore column is missing for the same reason. In its place, we have four computed columns, two of which measure the closeness of each Z-Score in the dataset to a target value or range.

…………To put it simply, the triangular number assigns a grade to the natural language statement, “around a Z-Score of 0.960526,” while the trapezoidal expresses the concept of “somewhere between 0.450526 and 1.360526.” The other two columns define increasing and decreasing functions that model how close the values are to either the top or bottom of the dataset, which can be interpreted as “few” or “most.” The hard-coded numbers in the UPDATE and the declarations above it are picked out of thin air merely to illustrate the point, not because they express any domain knowledge. There are probably more efficient ways of coding this, but the point is to get the concepts across.

** Figure 1: Code for the Fuzzy Number Sample**DECLARE @ZScoreTable table

(PrimaryKey sql_variant,

Value decimal(38,6),

ZScore decimal(38,6),

TriangularNearnessScore decimal(38,6),

TrapezoidalNearnessScore decimal(38,6),

FewScore decimal(38,6),

MostScore decimal(38,6),

GroupRank bigint

)

(PrimaryKey, Value, ZScore, GroupRank)

EXEC Calculations.ZScoreSP

@DatabaseName = N’DataMiningProjects‘,

@SchemaName = N’Health‘,

@TableName = N’DuchennesTable‘,

@ColumnName = N’LactateDehydrogenase‘,

@PrimaryKeyName = N’ID’,

@DecimalPrecision = ’38,32′,

@OrderByCode = 8

DECLARE @ComparisonPoint float = 0.960526

DECLARE @LowerBound float = @ComparisonPoint – 0.5, — we could of course make it lopsided if that would model our data better

@UpperBound float = @ComparisonPoint + 0.5

UPDATE @ZScoreTable

SET TriangularNearnessScore = CASE WHEN ZScore NOT BETWEEN @LowerBound and @UpperBound THEN 0

WHEN ZScore = @ComparisonPoint THEN 1

WHEN ZScore < @ComparisonPoint THEN (ZScore – @ComparisonPoint) + 1

WHEN ZScore > @ComparisonPoint THEN (@ComparisonPoint – ZScore) + 1

ELSE NULL END,

TrapezoidalNearnessScore = CASE WHEN ZScore NOT BETWEEN @LowerBound and @UpperBound THEN 0

WHEN ZScore BETWEEN @LowerBound + 0.1 and @UpperBound – 0.1 THEN 1

WHEN ZScore < @LowerBound + 0.1 THEN (ZScore – @ComparisonPoint) + 1

WHEN ZScore > @UpperBound – 0.1 THEN (@ComparisonPoint – ZScore) + 1 ELSE NULL END,

FewScore = CASE WHEN ZScore BETWEEN 0.2 AND 0.7 THEN 1 ELSE 1 – (ZScore * 0.3) END,

MostScore = CASE WHEN ZScore BETWEEN 1 AND 2 THEN 1 ELSE ZScore / 1.2 END

SELECT PrimaryKey, ZScore, TriangularNearnessScore, TrapezoidalNearnessScore, FewScore, MostScore

FROM @ZScoreTable

WHERE ZScore BETWEEN 0.2 AND 2 AND TriangularNearnessScore != 0 AND TrapezoidalNearnessScore != 0

ORDER BY ZScore DESC

__Figure 2: Sample Results from the Duchennes Practice Dataset
__

…………Note how the TriangularNearnessScore peaks at a single value, while the TrapezoidalNearnessScore is contained within a particular range. The FewScore and MostScore values peak at opposite ends of the dataset. Of course, a picture is worth a thousand words: if the values above aren’t clear, then the Reporting Services line graphs below ought to clear up any confusion. The TriangularNearnessScore doesn’t precisely follow a triangular shape, but it does come to single peak, which is good enough. The trapezoidal example reaches the same peak, but encompasses a range of values represented in the flat line. The FewScore and MostScore also have flat peaks, but these occur at the far edges of the membership grades. The shapes aren’t as neat as those in the literature, in part because I’m using real-world data from the Duchennes dataset together with some arbitrary range values, but I’m sure that readers will get the gist of it.

** Figure 3: Reporting Services Line Graphs for 4 Fuzzy Number Samples**…………Asking questions like “what is about half of A plus almost all of B” is probably a rare use case, one that is more likely to come up in data mining than in relational situations. If the need for this kind of comparison does arise, be aware that at least some of the math has been worked out, so there’s no need to reinvent the entire wheel. I say “some” because, at least at the time Klir and Yuan wrote, mathematicians were still struggling with some of the strange properties and enigmatic logic associated with these kinds of comparisons.[8] Just imagine several of the trapezoids and triangles above overlaid and you can see how quickly the topics of fuzzy arithmetic, fuzzy set relations and fuzzy matrix math can become. Some basic procedures for solving them are available[9], as well as neural net techniques for solving fuzzy equations[10], but they lead to certain logical difficulties that I believe still aren’t fully understood, such as the fact that many of the approximate solutions may not be unique.[11] Thankfully, we don’t encounter statements like “between 3 and 5 of A minus a little of B” often in ordinary speech, so I imagine that SQL Server users are unlikely to encounter it. In contrast, qualifiers like “around” or “near” are so common that I guarantee these fuzzy numbers will prove valuable to many users in the long run.

…………Instead of taking fuzzy numbers in directions of this kind that lack real-world applications, I’ll instead use them as a stepping stone towards quantifying imprecision, which can be helpful in programs of uncertainty management. Many of the linguistic qualifiers mentioned here are actually instances of what is known as “fuzzy cardinality,” which enable modeling of phrases like “about a quarter” or “near.”[12] In the next article, I’ll delve into the realm of fuzzy statistics, where the implications of relative membership in a set leads automatically to a range of different types of cardinality, not just the single type of Count used in T-SQL. I have yet to see this done in the literature (I simply can’t afford access to most of the research published on certain advanced topics like fuzzy stats), but I’ll provide an example of how trapezoidal numbers might be implemented in order to create fuzzy analogues of standard deviation and variance. Given that the ordinary “crisp” versions of these aggregates are determined in part by counts, the fuzzification of counts plays into that as well. It may be useful to probe for connections to Fisher’s Information, a metric I hope to code for a long-delayed series titled Information Measurement with SQL Server, given that it apparently uses variance to model uncertainty. Using increasing and decreasing functions to model statements like “almost all” as we have done here can be seen as fuzzy instances of MIN and MAX aggregates, which essentially express the same sentiment as “near the bottom” or “near the top.” These can also be used to create new methods of fuzzy outlier detection, as I’ve essentially been doing throughout this series by using Z-Scores in my sample code; fuzzy set grades don’t have anything to do with stochastics or outlier detection unless such meanings are deliberately assigned.

…………In the next article I will probably continue to use Z-Scores in my sample code mainly for the sake of consistency and to reuse old, familiar concepts, but also to kill two birds with one stone and investigate possible uses in outlier detection. As we shall see, membership functions can also be interpreted in the light of Evidence Theory, in which each grade indicates the levels of credibility and truthfulness of a statement. In such cases, we’re speaking of fuzzy measures[13], which are also useful in partitioning and quantifying different types of uncertainty. As we already saw in Implementing Fuzzy Sets in SQL Server, Part 2: Measuring Imprecision with Fuzzy Complements, fuzzy complements can be used to measure one type of uncertainty, in quantifying just how imprecise the boundaries of fuzzy sets are; in essence, they become measures of fuzziness. There are other types of uncertainty, however, which require entirely different modeling techniques, which is where fuzzy stats and measures come in handy.

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[1] Most fuzzy set references discuss triangulars and trapezoidals. One source I found to be helpful was pp. 12-13, Alavala, Chennakesava R., 2008, __Fuzzy Logic and Neural Networks: Basic Concepts and Applications__. New Age International Pvt. Ltd.: New Delhi.

[2] p. 3, Medeen, Glen, 2015, Two Examples of the Use of Fuzzy Set Theory in Statistics,” published online at the __University of Minnesota__ web address http://users.stat.umn.edu/~gmeeden/talks/fuzznov09.pdf

[3] pp. 96-98, Klir, George J. and Yuan, Bo, 1995, __Fuzzy Sets and Fuzzy Logic: Theory and Applications__. Prentice Hall: Upper Saddle River, N.J.

[4] *IBID.*, p. 228.

[5] *IBID.*, p. 99.

[6] *IBID.*, p. 222.

[7] *IBID.*, p. 230.

[8] *IBID.*, p. 115.

[9] *IBID.*, pp. 160-162.

[10] *IBID.*, pp. 171-173.

[11] *IBID.*, pp. 153-157, 166-167.

[12] *IBID.*, p. 98.

[13] *IBID.*, p. 177.