RED BOX RULES ARE FOR PROOF STAGE ONLY DELETE BEFORE FINAL PRINTING VIERTL Statistical Methods for Fuzzy Data Reinhard Viertl, Vienna University of Technology, Austria Statistical analysis methods have to be adapted for the analysis of fuzzy data In this book the foundations of the description of fuzzy data are explained, including methods on how to obtain the characterizing function of fuzzy measurement results Furthermore, statistical methods are then generalized to the analysis of fuzzy data and fuzzy a-priori information Key features: • Provides basic methods for the mathematical description of fuzzy data, as well as statistical methods that can be used to analyze fuzzy data • Describes methods of increasing importance with applications in areas such as environmental statistics and social science • Complements the theory with exercises and solutions and is illustrated throughout with diagrams and examples • Explores areas such as quantitative description of data uncertainty and mathematical description of fuzzy data This book is aimed at statisticians working with fuzzy logic, engineering statisticians, finance researchers, and environmental statisticians The book is written for readers who are familiar with elementary stochastic models and basic statistical methods Statistical Methods for Fuzzy Data Statistical data are not always precise numbers, or vectors, or categories Real data are frequently what is called fuzzy Examples where this fuzziness is obvious are quality of life data, environmental, biological, medical, sociological and economics data Also the results of measurements can be best described by using fuzzy numbers and fuzzy vectors respectively Statistical Methods for Fuzzy Data Reinhard Viertl P1: SBT fm JWST020-Viertl November 30, 2010 13:58 Printer Name: Yet to Come P1: SBT fm JWST020-Viertl November 30, 2010 13:58 Printer Name: Yet to Come Statistical Methods for Fuzzy Data P1: SBT fm JWST020-Viertl November 30, 2010 13:58 Printer Name: Yet to Come P1: SBT c01 JWST020-Viertl November 12, 2010 16:50 Printer Name: Yet to Come Part I FUZZY INFORMATION Fuzzy information is a special kind of information and information is an omnipresent word in our society But in general there is no precise definition of information However, in the context of statistics which is connected to uncertainty, a possible definition of information is the following: Information is everything which has influence on the assessment of uncertainty by an analyst This uncertainty can be of different types: data uncertainty, nondeterministic quantities, model uncertainty, and uncertainty of a priori information Measurement results and observational data are special forms of information Such data are frequently not precise numbers but more or less nonprecise, also called fuzzy Such data will be considered in the first chapter Another kind of information is probabilities Standard probability theory is considering probabilities to be numbers Often this is not realistic, and in a more general approach probabilities are considered to be so-called fuzzy numbers The idea of generalized sets was originally published in Menger (1951) and the term ‘fuzzy set’ was coined in Zadeh (1965) P1: SBT c01 JWST020-Viertl November 12, 2010 16:50 Printer Name: Yet to Come P1: SBT c04 JWST020-Viertl November 18, 2010 14:17 Printer Name: Yet to Come Part II DESCRIPTIVE STATISTICS FOR FUZZY DATA In standard statistics – objectivist and Bayesian analysis – observations are assumed to be numbers, vectors, or classical functions By the fuzziness of real data of continuous quantities it is necessary to adapt descriptive statistical methods to the situation of fuzzy data This is done for the most frequently used descriptive methods in this part P1: SBT c04 JWST020-Viertl November 18, 2010 14:17 Printer Name: Yet to Come P1: SBT c08 JWST020-Viertl October 26, 2010 19:21 Printer Name: Yet to Come Part III FOUNDATIONS OF STATISTICAL INFERENCE WITH FUZZY DATA Statistical inference is based on stochastic models like probability distributions, parametric families of probability distributions, cumulative distribution functions, expectations, dependence structures, and others In this part the basic mathematical concepts for statistical inference in the case of fuzzy data as well as fuzzy probabilities are explained In standard statistical inference the combination of observations of a classical random variable to form an element of the sample space is trivial Different from that for fuzzy data the combination is nontrivial because a vector of fuzzy numbers is not a fuzzy vector Therefore it is important to distinguish between observation space and sample space There are survey papers on different concepts concerned with statistical inference for fuzzy data by Gebhardt et al (1997) and Taheri (2003) P1: SBT c08 JWST020-Viertl October 26, 2010 19:21 Printer Name: Yet to Come P1: SBT app02 JWST020-Viertl 232 November 18, 2010 14:19 Printer Name: Yet to Come STATISTICAL METHODS FOR FUZZY DATA Therefore ξ (x , y1 , , xn , yn ) is an indicator function From that it follows that also the characterizing function ψr ∗ (·) of the generalized correlation coefficient r ∗ is an indicator function (b) In this situation the vector-characterizing function of (x, y)i∗ , i = 1(1)n is a function ξi (x, y) of the following form: The δ-cuts of (x, y)i∗ are given by ◦ Cδ (x, y)i∗ = {x i } × Cδ [ξi (·)] where ξi (·) is the characterizing function of the fuzzy y-component yi∗ of (x, y)i∗ The δ-cuts of the fuzzy combined sample (x1 , y2 , , x n , yn )∗ are given by n C δ (x1 , y1 , , xn , yn )∗ = X Cδ (x, y)i∗ i=1 n ◦ = X {x i } × Cδ [ξi (·)] i=1 The vector-characterizing function ξ (x1 , y1 , , x n , yn ) can be obtained by the construction lemma for membership functions P1: SBT app02 JWST020-Viertl November 18, 2010 14:19 Printer Name: Yet to Come SOLUTIONS TO THE PROBLEMS 233 Chapter (a) Let the fuzzy values f ∗ (x) be LR-fuzzy numbers (cf Section 2.1) Then the δ-cuts are given by Cδ f ∗ (x) = m x − sx − l x · L −1 (δ); m x + sx + r x · R −1 (δ) Therefore the upper and lower δ-level curves f δ (·) and f δ (·) are ⎫ f δ (x) = m(x) + s(x) + r (x) · R −1 (δ)⎬ and ⎭ f δ (x) = m(x) − s(x) − l(x) · L −1 (δ) The fuzzy integral J ∗ = ∀δ ∈ (0; 1] f ∗ (x)d x is defined by its δ-cuts Cδ (J ∗ ) which are given by Cδ (J ∗ ) = 1 f δ (x)d x; f δ (x)d x For a fuzzy uniform density we can assume m(x) ≡ ∀x ∈ [0; 1] Now we obtain 1 + s(x) + r (x) · R −1 (δ)d x f δ (x)d x = 0 and 1 − s(x) − l(x) · L −1 (δ)d x f δ (x)d x = 0 Assuming the functions s(·) and l(·) to be integrable we obtain 1 s(x)d x + R −1 (δ) f δ (x)d x = + r (x)d x and 1 f δ (x)d x = − s(x)d x − L −1 (δ) l(x)d x From this the δ-cuts Cδ (J ∗ ) are obtained, and by the representation lemma 2.1 the characterizing function of J ∗ is determined P1: SBT app02 JWST020-Viertl 234 November 18, 2010 14:19 Printer Name: Yet to Come STATISTICAL METHODS FOR FUZZY DATA (b) The fuzzy probability P ∗ [0; 12 ] based on the fuzzy density f ∗ (·) of (a) is obtained via its δ-cuts [P δ ; P δ ] ⎧ 1/2 ⎨ ⎫⎫ ⎬⎪ ⎪ ⎪ P δ = sup f (x)d x : f (·) ∈ Sδ ⎪ ⎪ ⎩ ⎭⎪ ⎪ ⎬ ⎫⎪ ⎧ 1/2 ⎬⎪ ⎨ ⎪ ⎪ ⎪ P δ = inf f (x)d x : f (·) ∈ Sδ ⎪ ⎭ ⎭⎪ ⎩ ∀δ ∈ (0; 1] Depending on the functions f δ (·) and f δ (·) the values P δ and P δ are calculated For the special case of constant fuzzy value f ∗ (x) with trapezoidal characterizing function t ∗ (m, s, l, r ) the calculation of P δ and P δ is relatively simple Chapter (a) If all δ-cuts of ψ(·) are finite unions of compact intervals then the convex hull of ψ(·) is ψ(·) itself If some δ-cuts of ψ(·) are not compact or not finite unions of compact intervals then the function ξ (·) is defined by ξ (x) := sup {min{ψ(x ), ψ(x )} : αx + (1 − α)x2 = x} A concrete example is the following: The 13 -cut of ψ(·) is (a; b] which is not closed Therefore the convex hull ξ (·) is the following P1: SBT app02 JWST020-Viertl November 18, 2010 14:19 Printer Name: Yet to Come SOLUTIONS TO THE PROBLEMS 235 (b) From β < δ it follows Cβ (x ∗ ) ⊇ Cδ (x ∗ ) by definition of δ-cuts Therefore we obtain Cβ (x ∗ ) ⊇ Cδ (x ∗ ) β δ Then ξx ∗ (x) ≥ β for all β < δ Assuming ξx ∗ (x) < δ yields a contradiction Therefore Cβ (x ∗ ) ⊆ Cδ (x ∗ ) and the equality follows β − α P1: SBT app02 JWST020-Viertl 240 November 18, 2010 14:19 Printer Name: Yet to Come STATISTICAL METHODS FOR FUZZY DATA ◦ Let Then H,1−α be the classical HPD-region based on the precise sample x ◦ π(θ | x) dθ = − α and ◦ π (θ | x) ≥ C ∀θ ∈ H,1−α , H,1−α where C is the maximal constant such that the foregoing equation is fulfilled ◦ For θ ∈ H,1−α it follows π (θ | x)dθ ≤ − α and therefore ψ(θ ) = ◦ B(θ ;x) For θ ∈ / H,1−α ◦ ◦ π (θ | x)dθ > − α and therefore ψ(θ ) = we have B(θ ;x ) Chapter 17 (a) For discrete quantity X ∼ p(x|θ ), θ ∈ with finite observation space M = {x1 , , x m } and continuous parameter space , for fuzzy a posteriori density π ∗ (·|D ∗ ) the fuzzy numbers p ∗ (x j |D ∗ ) = − p(x j |θ ) · π ∗ (θ |D ∗ )dθ are fulfilling the following Since p(·|θ ) is a classical discrete probability distribution and π ∗ (·|D ∗ ) a fuzzy probability density, the fuzzy values p∗ (x j |D ∗ ) = − p(x j |θ ) · π ∗ (θ |D ∗ )dθ have δ-cuts Cδ [ p ∗ (x j |D ∗ )] = [ p δ (x j ); p δ (x j )] with p δ (x j ) = ⎫ p(x j |θ ) · π δ (θ |D ∗ )dθ ⎪ ⎪ ⎬ and p δ (x j ) = ⎪ p(x j |θ ) · π δ (θ |D ∗ )dθ ⎪ ⎭ ∀δ ∈ (0; 1] m The fuzzy sum ⊕ p ∗ (x j |D ∗ ) is a fuzzy interval whose characterizing j=1 function has δ-cuts m m j=1 ⎡ C1 m ⊕ p (x j |D ) = ⎣ ∗ ∗ j=1 m By p δ (x j ); m j=1 p δ (x j ) , and ∗ m p(x j |θ ) · π (θ |D )dθ ; j=1 p(x j |θ ) = ∀θ ∈ ⎤ p(x j |θ ) · π (θ |D )dθ ⎦ ∗ j=1 and π ∗ (·|D ∗ ) is a fuzzy probability density on j=1 , ∈ C1 m ⊕ p ∗ (x j |D ∗ j=1 discrete distribution on M and by Section 14.1 p ∗ (·|D ∗ ) generates a fuzzy P1: SBT app02 JWST020-Viertl November 18, 2010 14:19 Printer Name: Yet to Come SOLUTIONS TO THE PROBLEMS 241 (b) In order to prove that f ∗ (·|D ∗ ) is a fuzzy probability density we have to prove the existence of a classical probability density g(·) on M X which fulfills f (x|D ∗ ) ≤ g(x) ≤ f (x|D ∗ ) − f (x|θ ) ∀ x ∈ MX π ∗ (θ |D ∗ )dθ is calculated using δ-level functions f (x|θ ) · π δ (θ |D ∗ ) and f (x|θ ) · π δ (θ |D ∗ ), i.e f δ (x|D ∗ ) = f (x|θ )π δ (θ |D ∗ ) dθ f δ (x|D ∗ ) = f (x|θ )π δ (θ |D ∗ ) dθ and For δ = we obtain f (x|D ∗ ) = f (x|θ )π (θ |D ∗ ) dθ f (x|D ∗ ) = f (x|θ )π (θ |D ∗ ) dθ and Since f (·|θ ) is a classical probability density and π ∗ (·|D ∗ ) is a fuzzy probability density, there exists a probability density h(·) on with π (θ |D ∗ ) ≤ h(θ ) ≤ π (θ |D ∗ ) ∀θ ∈ Therefore f (x|θ ) · h(θ ) is a two-dimensional density on M X × , and f (x|θ )h(θ ) dθ as marginal density is a probability density on M X for which f (x|D ∗ ) ≤ f (x|θ )h(θ ) dθ ≤ f (x|D ∗ ) ∀ x ∈ M X This proves the assertion P1: SBT app02 JWST020-Viertl 242 November 18, 2010 14:19 Printer Name: Yet to Come STATISTICAL METHODS FOR FUZZY DATA Chapter 18 (a) The classical situation is characterized by the indicator function IU (θ,d) (·) of U (θ, d) and a classical probability distribution P on Therefore the δ-level functions U δ (·, d) and U δ (·, d) are all identical and equal to U (·, d) The characterizing function of the generalized expected utility is the indicator function IE P U (θ,d) (·) (b) If there exists a decision d B such that the support of the characterizing function of E P ∗ U ∗ (θ˜ , d B ) has empty intersection with the supports of the characterizing functions χd (·) of the fuzzy expected utilities of all other decisions d, and the left end of supp[χd B (·)] is greater than all values in all supports of all other decisions, then d B is the uniquely determined Bayesian decision Chapter 19 (a) Polynomial regression functions ψ (x, θ0 , , θk ) = k θ j x j are linear with j=0 respect to the parameters θ0 , , θk Taking x j = x j this kind of regression models can be written in the following form: ◦ x = 1, x, x , , x k , θ = (θ0 , θ1 , , θk ) ◦T ψ (x, θ0 , , θk ) = θ x This is a special form of the linear regression model (19.4) (b) The classical linear regression function can be written as general linear model based on the following functions f j (·): ⎞ x1 ⎜ ⎟ f (·) = ⎝ ⎠ , ⎛ xk ⎛ ⎞ ⎜ x1 ⎟ ◦ ⎜ ⎟ f (·) = ⎜ ⎟ , ⎝ ⎠ xk θ = (θ0 , θ1 , , θk ) (c) Assumption (3) in Theorem 19.1 implies that Yxi , i = (1) n are independent Chapter 20 (a) First the fuzzy quantities x1∗ , x 2∗ , y1∗ , and y∗2 have to be combined into a fuzzy vector with vector-characterizing function τ (x1 , x2 , y1 , y2 ) = {ξ1 (x ) , ξ2 (x2 ), η1 (y1 ) , η2 (y2 )} ∀ (x , x2 , y1 , y2 ) ∈ R4 P1: SBT app02 JWST020-Viertl November 18, 2010 14:19 Printer Name: Yet to Come SOLUTIONS TO THE PROBLEMS 243 Based on this fuzzy combined vector the estimatiors θˆ0 = x12 + x 22 (y1 + y2 ) − (x1 + x ) (x1 y1 + x y2 ) x12 + x22 − (x1 + x )2 (x1 y1 + x y2 ) − (x1 + x2 ) (y1 + y2 ) θˆ1 = x 12 + x22 − (x1 + x )2 are generalized to the fuzzy estimatiors θˆ0∗ and θˆ1∗ by applying the extension principle For details compare the characterizing functions φ j (·) in Remark 20.3 ad (d) (b) Here the fuzzy elements θˆ0∗ , θˆ1∗ , and x ∗ with corresponding membership functions have to be combined into a fuzzy 3-dimensional vector z ∗ Then the extension principle has to be applied to the function g (x) = θ0 + θ1 · x Chapter 21 (a) If the density is given by g ( | ψ (θ, xi )) the likelihood function for observed data (x1 , y1 ) , , (xn , yn ) is given by n n f x i (yi | θ ) = l (θ0 , , θk ; (x1 , y1 ) , , (xn , yn )) = i=1 g (yi | ψ(θ , xi )) i=1 Bayes’ theorem obtains the following form: π (θ0 , , θk | (x1 , yi ) , , (xn , yn )) ∝ n g (yi | ψ (θ0 , , θk , xi )) π (θ0 , , θk ) i=1 n (b) f xi (yi | τ = θ0 + θ1 x) l (θ0 , θ1 , (x1 , y1 ) , , (xn , yn )) = i=1 n = i=1 yi exp − θ0 + θ1 xi θ0 + θ x i n = exp − i=1 yi θ0 + θ x i n i=1 Bayes’ theorem in this case reads I(0,∞) (yi ) I(0,∞) (yi ) θ0 + θ x i P1: SBT app02 JWST020-Viertl 244 November 18, 2010 14:19 Printer Name: Yet to Come STATISTICAL METHODS FOR FUZZY DATA n π (θ0 , θ1 | (x1 , y1 ) , , (xn , yn )) ∝ π (θ0 , θ1 ) exp − i=1 n × i=1 yi θ0 + θ x i I(0,∞) (yi ) θ0 + θ x i Chapter 22 (a) The fuzzy valued function − π ∗ (x , , xr ) d x d xs−1 d xs+1 d xr has δ-level functions Rr −1 π¯ δ (x , , xr ) d x d xs−1 d xs+1 d xr ≥ f¯δ (x s ) = Rr −1 By assumption the integral R Therefore also f¯δ (xs ) d xs exists and is finite for all δ ∈ (0; 1] f δ (xs ) d x s < ∞ ∀δ ∈ (0; 1] 0≤ R Now the fuzzy marginal density f ∗ (·) is defined by its δ-level functions f δ (·) and f¯δ (·) respectively Furthermore by f ∗ (x) d x = R ∗ Rr ∗ π (x1 , , xr ) d x , d xr it follows ∈ supp − f (x) d x Therefore R f ∗ (·) is a fuzzy density (b) Let f (·) be a classical probability density with existing expectation, i.e µ = x f (x) d x < ∞ The δ-level functions of this density are all equal to R f (·) Using the definition of Section 22.1 for the expectation we obtain µ∗ = − x f (x) d x = µ R The characterizing function ψ(·) of µ∗ is given by the family of single point sets Aδ = ψ(·) = I{µ} (·) R x f (x) d x = {µ} ∀ δ ∈ (0; 1] Therefore we obtain P1: SBT app03 JWST020-Viertl October 26, 2010 16:31 Printer Name: Yet to Come A3 Glossary Bayes’ formula: Bayesian inference: Bayes’ theorem: Characterizing function: Complete sample: Fuzzy Bayesian inference: Fuzzy combined sample: Fuzzy confidence region: Fuzzy data: Fuzzy estimate: Fuzzy histogram: Fuzzy HPD-regions: Statistical Methods for Fuzzy Data © 2011 John Wiley & Sons, Ltd Rule for the calculation of probabilities of hypotheses conditional on observed events Statistical inference method where all unknown quantities are modeled by stochastic quantities Rule for the calculation of probability densities of continuous parameters in stochastic models conditional on observed data of continuous variables Mathematical description of a fuzzy number If n copies of a stochastic quantity are observed and all n observations are obtained then the sample is called complete The generalization of Bayesian inference to the situation of fuzzy a priori information and fuzzy data Combination of a sample of fuzzy numbers to form a fuzzy element of the sample space Generalization of the concept of confidence region for the situation of fuzzy data Data which are not precise numbers or classical vectors Generalized statistical estimation technique for the situation of fuzzy data A generalized histogram based on fuzzy data Generalization of highest probability density regions for parameters in Bayesian inference Reinhard Viertl 245 P1: SBT app03 JWST020-Viertl 246 October 26, 2010 16:31 Printer Name: Yet to Come STATISTICAL METHODS FOR FUZZY DATA Fuzzy information: Information which is not given in the form of precise numbers, precise vectors, precise functions or precisely defined models Fuzzy number: Mathematical description of a fuzzy one-dimensional quantity Fuzzy probability density: Generalization of a probability density whose values are fuzzy intervals Fuzzy probability distribution: Generalized probability distributions assigning fuzzy intervals to events Fuzzy sample: Data in the form of fuzzy numbers or fuzzy vectors Fuzzy statistics: Statistical analysis methods for the situation of fuzzy information Fuzzy utility: Generalized utility functions assuming fuzzy intervals as values Fuzzy valued function: Generalized real function whose values are fuzzy numbers Fuzzy vector: Mathematical description of a nonprecise (fuzzy) vector quantity Gamma function: Extension of the factorial function to real arguments x > Hybrid data analysis methods: Data analysis methods using methods from statistics and fuzzy models Nonprecise data: Synonym for fuzzy data Triangular norm: Mathematical operation to combine fuzzy numbers into a fuzzy vector Vector-characterizing function: Mathematical description of a fuzzy vector ... 2.1 Statistical Methods for Fuzzy Data © 2011 John Wiley & Sons, Ltd Reinhard Viertl P1: SBT c02 JWST020-Viertl October 29, 2010 20:52 Printer Name: Yet to Come STATISTICAL METHODS FOR FUZZY DATA. .. one-dimensional fuzzy data are data in the form of intervals [a; b] ⊆ R Such data are generated by digital measurement equipment, because they have only a finite number of digits Statistical Methods for Fuzzy. .. typical examples of fuzzy information in the sense of fuzzy a priori distributions Both kinds of fuzzy information, fuzzy data as well as fuzzy probability distributions for the quantification