Adding Constraints to Situations When, Illustration: . . . In - PowerPoint PPT Presentation

Adding Interval . . . Limitations of the . . . How to Describe . . . Kolmogorov-Smirnov . . . Illustration: . . . Adding Constraints to Situations When, Illustration: . . . In Addition to Intervals, We Also Have Computing V Computing V Partial Information about Probabilities Computational . . . How to Handle . . . Martine Ceberio 1 , Scott Ferson 2 , Cliff Joslyn 3 , Gauging Amount of . . . Vladik Kreinovich 1 , and Gang Xiang 1 Case of a Continuous . . . Case of a p-Box 1 Department of Computer Science Acknowledgments University of Texas, El Paso, TX 79968, USA Shannon’s Derivation: . . . 2 Applied Biomathematics Shannon’s Derivation . . . 3 Los Alamos National Laboratory Title Page mceberio@utep.edu, scott@ramas.com, joslyn@lanl.gov ◭◭ ◮◮ vladik@utep.edu, gxiang@utep.edu ◭ ◮ Page 1 of 19 Go Back Full Screen Close

Adding Interval . . . Limitations of the . . . 1. Statistical Analysis Is Important How to Describe . . . Kolmogorov-Smirnov . . . • Fact: statistical analysis of measurement and observation results Illustration: . . . is an important part of data processing and data analysis. Illustration: . . . • Specifics: Computing V Computing V – when faced with new data, Computational . . . – engineers and scientists usually start with estimating stan- How to Handle . . . dard statistical characteristics such as: Gauging Amount of . . . ∗ the mean E , Case of a Continuous . . . ∗ the variance V , Case of a p-Box ∗ the probability distribution function (pdf) F ( x ) of each Acknowledgments variable, and Shannon’s Derivation: . . . ∗ the covariance and correlation between different variables. Shannon’s Derivation . . . Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 19 Go Back Full Screen Close

Adding Interval . . . Limitations of the . . . 2. Limitations of Traditional Statistical Techniques and How to Describe . . . the Need to Consider Interval Uncertainty Kolmogorov-Smirnov . . . Illustration: . . . • Main assumption: traditional statistical techniques assume that Illustration: . . . the measured values � x 1 , . . . � x n coincide with the actual values Computing V x 1 , . . . , x n of the measured quantities. Computing V Computational . . . • This assumption is often true: if the variability of each variable How to Handle . . . def is much higher than the measurement errors ∆ x i = � x i − x i . Gauging Amount of . . . • Example: the accuracy of measuring a person’s height ( ≈ 1 cm) Case of a Continuous . . . is ≪ variability in height. Case of a p-Box Acknowledgments • Sometimes, this assumption is not true: when the measurement Shannon’s Derivation: . . . errors ∆ x i are of the same order of magnitude. Shannon’s Derivation . . . • Conclusion: ∆ x i cannot be ignored in statistical analysis. Title Page • Frequent situation: the only information about ∆ x i is the upper ◭◭ ◮◮ bound ∆ i : | ∆ x i | ≤ ∆ i . ◭ ◮ • Interval uncertainty: the only information about x i is that x i ∈ def = [ � x i − ∆ i , � x i + ∆ i ]. x i Page 3 of 19 Go Back Full Screen Close

Adding Interval . . . Limitations of the . . . 3. Adding Interval Uncertainty to Statistical Tech- How to Describe . . . niques: What Is Known Kolmogorov-Smirnov . . . Illustration: . . . • We start with: a statistic C ( x 1 , . . . , x n ), such as: Illustration: . . . Computing V � n – population mean E = 1 n · x i ; Computing V i =1 Computational . . . � n – population variance V = 1 How to Handle . . . ( x i − E ) 2 ; n · Gauging Amount of . . . i =1 Case of a Continuous . . . – histogram pdf F n ( x ) = # i : x i ≤ x ; Case of a p-Box n n Acknowledgments � – population covariance C x,y = 1 n · ( x i − E x ) · ( y i − E y ). Shannon’s Derivation: . . . i =1 Shannon’s Derivation . . . • Interval extension: find the range Title Page def C = C ( x 1 , . . . , x n ) = { C ( x 1 , . . . , x n ) : x 1 ∈ x 1 , . . . , x n ∈ x n } . ◭◭ ◮◮ • General case: the general problem is NP-hard, even for V . ◭ ◮ • Conclusion: in general, we can only compute an enclosure. Page 4 of 19 • Specific cases: efficient algorithms are possible: for E , for V , for Go Back V when [ x i , x i ] �⊆ ( x j , x j ), etc. Full Screen Close

Adding Interval . . . Limitations of the . . . 4. Limitations of the Existing Approach How to Describe . . . Kolmogorov-Smirnov . . . • Currently used idea: Illustration: . . . – we start with a statistic C ( x 1 , . . . , x n ) for estimating a given Illustration: . . . characteristic S ; Computing V Computing V – we evaluate this statistic under interval uncertainty, resulting Computational . . . in C = C ( x 1 , . . . , x n ). How to Handle . . . • First limitation of this idea: Gauging Amount of . . . – we know that C ( x 1 , . . . , x n ) ≈ S ; Case of a Continuous . . . – sometimes, the estimation error C ( x 1 , . . . , x n ) − S � = 0 is not Case of a p-Box always taken into account when estimating C . Acknowledgments Shannon’s Derivation: . . . • Solution: instead of the original statistic C , we consider the bounds C − and C + of the confidence interval. Shannon’s Derivation . . . � + � Title Page • Good news: the interval C − , C is an enclosure for S (with ◭◭ ◮◮ appropriate certainty). • Remaining limitation: excess width. ◭ ◮ • New idea: find S = { S ( F ) : F is possible } . Page 5 of 19 • Related problem: how to describe class F of possible probability Go Back distributions F . Full Screen Close

Adding Interval . . . Limitations of the . . . 5. How to Describe Possible Probability Distributions: How to Describe . . . p-Boxes Kolmogorov-Smirnov . . . Illustration: . . . • General situation: Illustration: . . . Computing V – we do not know the probability distribution of the actual Computing V values x i ; Computational . . . – we want to determine this distribution. How to Handle . . . • Question: which characteristics of this distribution are practically Gauging Amount of . . . useful? Case of a Continuous . . . • Practical example: Case of a p-Box Acknowledgments – there is a critical threshold x 0 after which a chip delays too Shannon’s Derivation: . . . much, a panel cracks, etc.; Shannon’s Derivation . . . – we want to make sure that the probability of exceeding x 0 is Title Page small. ◭◭ ◮◮ • Resulting characteristic: Prob( x i ≤ x 0 ), i.e., cdf F ( x 0 ). • p-box: ◭ ◮ – we cannot determine the exact values of F ( x ); Page 6 of 19 – thus, we should look for bounds F ( x ) = [ F ( x ) , F ( x )]; Go Back – the function x → F ( x ) is called a p-box . Full Screen Close

Adding Interval . . . Limitations of the . . . 6. Kolmogorov-Smirnov (KS) p-Box How to Describe . . . Kolmogorov-Smirnov . . . • New idea (reminder): Illustration: . . . Illustration: . . . – transform observations x 1 , . . . , x n into a p-box; Computing V – estimate a characteristic S based on the p-box. Computing V • How to transform: use KS inequalities. Computational . . . How to Handle . . . • Main idea behind KS: for each x 0 , we have Gauging Amount of . . . – actual (unknown) probability p = F ( x 0 ) that x ≤ x 0 , and Case of a Continuous . . . – frequency F n ( x 0 ) = # i : x i ≤ x 0 Case of a p-Box . n Acknowledgments Shannon’s Derivation: . . . • Known: for large n , F n ( x 0 ) ≈ normal, and with given certainty � Shannon’s Derivation . . . p · (1 − p ) α , we have p − k · σ ≤ F n ( x 0 ) ≤ p + k · σ , where σ = Title Page n and k = k ( α ). ◭◭ ◮◮ • Conclusion: with certainty α , we get bounds on p = F ( x 0 ) in ◭ ◮ terms of F n ( x 0 ). Page 7 of 19 • We use these bounds for x 0 = x i and use monotonicity to get bounds [ F n ( x ) − ε, F n ( x ) + ε ] for all x ∈ [ x i , x i +1 ]. Go Back Full Screen Close

Adding Interval . . . Limitations of the . . . 7. Illustration: Histogram Pdf How to Describe . . . Kolmogorov-Smirnov . . . Illustration: . . . ✻ F n ( x ) Illustration: . . . Computing V Computing V Computational . . . How to Handle . . . Gauging Amount of . . . Case of a Continuous . . . Case of a p-Box Acknowledgments Shannon’s Derivation: . . . Shannon’s Derivation . . . Title Page ✲ x 1 x 2 x 3 ◭◭ ◮◮ ◭ ◮ Page 8 of 19 Go Back Full Screen Close

Adding Interval . . . Limitations of the . . . 8. Illustration: Kolmogorov-Smirnov p-Box How to Describe . . . Kolmogorov-Smirnov . . . Illustration: . . . ✻ F Illustration: . . . Computing V Computing V Computational . . . How to Handle . . . Gauging Amount of . . . Case of a Continuous . . . Case of a p-Box Acknowledgments Shannon’s Derivation: . . . Shannon’s Derivation . . . Title Page ✲ x 1 x 2 x 3 ◭◭ ◮◮ ◭ ◮ Page 9 of 19 Go Back Full Screen Close

Adding Interval . . . Limitations of the . . . 9. Computing V How to Describe . . . Kolmogorov-Smirnov . . . Illustration: . . . ✻ F Illustration: . . . Computing V Computing V Computational . . . How to Handle . . . Gauging Amount of . . . Case of a Continuous . . . Case of a p-Box Acknowledgments Shannon’s Derivation: . . . Shannon’s Derivation . . . Title Page ✲ x 1 x 2 x 3 ◭◭ ◮◮ ◭ ◮ Page 10 of 19 Go Back Full Screen Close