In Many Practical . . . First Problem: How to . . . What Are the . . . How to Estimate the . . . Optimizing Computer Representation and Estimates for U - . . . Computer Processing of Epistemic Need to Take into . . . Uncertainty for Risk-Informed Decision Estimating the Heavy- . . . Making: Finances etc. Conclusion Acknowledgments Vladik Kreinovich 1 , Nitaya Buntao 2 , and Home Page Olga Kosheleva 1 Title Page 1 University of Texas at El Paso ◭◭ ◮◮ El Paso, TX 79968, USA ◭ ◮ vladik@utep.edu, olgak@utep.edu Page 1 of 28 2 Department of Applied Statistics King Mongkut’s University of Technology Go Back North Bangkok Bangkok 10800 Thailand Full Screen taltanot@hotmail.com Close Quit
In Many Practical . . . First Problem: How to . . . 1. Formulation of the Problem What Are the . . . • Traditionally, most statistical techniques assume that How to Estimate the . . . the random variables are normally distributed. Estimates for U - . . . Need to Take into . . . • For such distributions: Estimating the Heavy- . . . – a natural characteristic of the “average” value is Conclusion the mean, and Acknowledgments – a natural characteristic of the deviation from the Home Page average is the variance. Title Page • In practice, we encounter heavy-tailed distributions, with ◭◭ ◮◮ infinite variance; what are analogs of: ◭ ◮ – “average” and deviation from average? Page 2 of 28 – correlation? Go Back – how to take into account interval uncertainty? Full Screen Close Quit
In Many Practical . . . First Problem: How to . . . 2. Normal Distributions Are Most Widely Used What Are the . . . • Most statistical techniques assume that the random How to Estimate the . . . variables are normally distributed: Estimates for U - . . . � � Need to Take into . . . − ( x − m ) 2 1 ρ ( x ) = √ · exp . Estimating the Heavy- . . . 2 V 2 π · V Conclusion • For such distributions: Acknowledgments Home Page – a natural characteristic of the “average” value is def Title Page the mean m = E [ x ], and – a natural characteristic of the deviation from the ◭◭ ◮◮ def = E [( x − m ) 2 ]. average is the variance V ◭ ◮ • It is known that a normal distribution is uniquely de- Page 3 of 28 termined by m and V . Go Back • Thus, each characteristic (mode, median, etc.) is uniquely Full Screen determined by m and V . Close Quit
In Many Practical . . . First Problem: How to . . . 3. Estimating the Values of the Characteristics: What Are the . . . Case of Normal Distributions How to Estimate the . . . • We have a sample consisting of the values x 1 , . . . , x n . Estimates for U - . . . Need to Take into . . . • We can use the Maximum Likelihood Method: m and Estimating the Heavy- . . . V maximizing � � Conclusion � n − ( x i − m ) 2 1 √ Acknowledgments L = ρ ( x 1 ) · . . . · ρ ( x n ) = · exp . 2 V 2 π · V Home Page i =1 Title Page • Maximizing L is equivalent to minimizing � 1 � ◭◭ ◮◮ n � 2 · ln(2 π · V ) + ( x i − m ) 2 def = − ln( L ) = ψ . ◭ ◮ 2 V i =1 Page 4 of 28 • Equating derivatives to 0, we get: Go Back n n � � m = 1 V = 1 m ) 2 . � � n · x i ; n · ( x i − � Full Screen i =1 i =1 Close Quit
In Many Practical . . . First Problem: How to . . . 4. In Many Practical Situations, We Encounter What Are the . . . Heavy-Tailed Distributions How to Estimate the . . . • In the 1960s, Benoit Mandelbrot empirically studied Estimates for U - . . . fluctuations. Need to Take into . . . Estimating the Heavy- . . . • He showed that larger-scale fluctuations follow the power- law distribution ρ ( x ) = A · x − α , with α ≈ 2 . 7. Conclusion Acknowledgments • For this distribution, variance is infinite. Home Page • Such distributions are called heavy-tailed . Title Page • Similar heavy-tailed laws were empirically discovered ◭◭ ◮◮ in other application areas. ◭ ◮ • These result led to the formulation of fractal theory . Page 5 of 28 • Since then, similar heavy-tailed distributions have been Go Back empirically found: Full Screen – in other financial situations and – in many other application areas. Close Quit
In Many Practical . . . First Problem: How to . . . 5. First Problem: How to Characterize Such Dis- What Are the . . . tributions? How to Estimate the . . . • Usually, variance is used to describe deviation from the Estimates for U - . . . average. Need to Take into . . . Estimating the Heavy- . . . • For heavy-tailed distributions, variance is infinite . Conclusion • So, we cannot use variance to describe the deviation Acknowledgments from the “average”. Home Page • Thus, we need to come up with other characteristics Title Page for describing this deviation. ◭◭ ◮◮ • We will describe such characteristics in the first part of ◭ ◮ this talk. Page 6 of 28 • We will also describe how we can estimate these char- Go Back acteristics. Full Screen Close Quit
In Many Practical . . . First Problem: How to . . . 6. How to Describe Deviation from the “Average” What Are the . . . for Heavy-Tailed Distributions: Analysis How to Estimate the . . . • A standard way to describe preferences of a decision Estimates for U - . . . maker is to use the notion of utility u . Need to Take into . . . Estimating the Heavy- . . . • According to decision theory, a user prefers an alter- Conclusion native for which the expected value � Acknowledgments ρ ( x ) · u ( x ) dx → max . Home Page Title Page � • Alternative, the expected value ρ ( x ) · U ( x ) dx of the ◭◭ ◮◮ def disutility U = − u is the smallest possible. ◭ ◮ • If we replace x → m ≈ x , there is disutility U ( x − m ). Page 7 of 28 � • So, we choose m s.t. ρ ( x ) · U ( x − m ) dx → min . Go Back • The resulting minimum describes the deviation of the Full Screen values from this “average”. Close Quit
In Many Practical . . . First Problem: How to . . . 7. Resulting Definitions What Are the . . . • Let U : I R → I R 0 be a function for which: How to Estimate the . . . Estimates for U - . . . • U (0) = 0, Need to Take into . . . • U ( d ) is (non-strictly) increasing for d ≥ 0, and Estimating the Heavy- . . . • U ( d ) is (non-strictly) decreasing for d ≤ 0. Conclusion • For a distribution ρ ( x ), by a U -mean , we mean the Acknowledgments � Home Page value m U that minimizes ρ ( x ) · U ( x − m ) dx . Title Page • By a U -deviation , we mean � ◭◭ ◮◮ def V U = ρ ( x ) · U ( x − m U ) dx. ◭ ◮ Page 8 of 28 • When U ( x ) = x 2 , m U is mean, and V U is variance. Go Back • When U ( x ) = | x | , m U is median, and V U is average � absolute deviation V U = ρ ( x ) · | x − m U | dx. Full Screen Close Quit
In Many Practical . . . First Problem: How to . . . 8. What Are the Reasonable Measures of Depen- What Are the . . . dence for Heavy-Tailed Distributions? How to Estimate the . . . • In the traditional statistics, a reasonable measure of Estimates for U - . . . dependence is the correlation Need to Take into . . . Estimating the Heavy- . . . ρ xy = E [( x − E ( x )) · ( y − E ( y ))] � . Conclusion V x · V y Acknowledgments • For heavy-tailed distributions, variances are infinite, so Home Page this formula cannot be applied. Title Page • Possibility: Kendall’s tau, the proportion of pairs ( x, y ) ◭◭ ◮◮ and ( x ′ , y ′ ) s.t. x and y change in the same direction: ◭ ◮ either ( x ≤ x ′ & y ≤ y ′ ) or ( x ′ ≤ x & y ′ ≤ y ) . Page 9 of 28 Go Back • Remaining problem: what if we are interested only in linear dependencies? Full Screen Close Quit
In Many Practical . . . First Problem: How to . . . 9. Proposed Definition What Are the . . . • Idea: c describes how much disutility decreases when How to Estimate the . . . we use x to help predict y : Estimates for U - . . . Need to Take into . . . = V U ( y ) − V U, F ( y | x ) def c , Estimating the Heavy- . . . V U ( y ) Conclusion where � Acknowledgments def Home Page V U ( y ) = min ρ ( x, y ) · U ( y − m ) dx dy m Title Page and � ◭◭ ◮◮ def V U, F ( y | x ) = min ρ ( x, y ) · U ( y − f ( x )) dx dy. ◭ ◮ f ∈F Page 10 of 28 • The function f at which the minimum is attained is called F -regression . Go Back • When U ( d ) = d 2 and F is the class of all linear func- Full Screen tions, c = ρ 2 . Close Quit
In Many Practical . . . First Problem: How to . . . 10. Discussion What Are the . . . • For normal distributions and linear functions, correla- How to Estimate the . . . tion is symmetric: Estimates for U - . . . Need to Take into . . . – if we can reconstruct y from x , Estimating the Heavy- . . . – then we can reconstruct x from y . Conclusion • Our definition is, in general, not symmetric. Acknowledgments Home Page • This asymmetry make perfect sense. Title Page • For example, suppose that y = x 2 : ◭◭ ◮◮ – then, if we know x , then we can uniquely recon- ◭ ◮ struct y ; Page 11 of 28 – however, if we know y , we can only reconstruct x modulo sign. Go Back Full Screen Close Quit
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