A Practical Problem: . . . How This Problem is . . . A Standard Way to . . . Selecting the Parameter k Computing Case of Interval . . . Standard-Deviation-to-Mean What is Known What We Do in This Talk and Theorem 1 Theorem 2 Variance-to-Mean Ratios Home Page under Interval Uncertainty is Title Page NP-Hard ◭◭ ◮◮ ◭ ◮ Sio-Long Lo Page 1 of 31 Faculty of Information Technology Macau University of Science and Technology (MUST) Go Back Avenida Wai Long, Taipa, Macau SAR, China Email: akennetha@gmail.com Full Screen Close Quit
A Practical Problem: . . . How This Problem is . . . 1. A Practical Problem: Checking Whether an A Standard Way to . . . Object Belongs to a Class Selecting the Parameter k • In many practical situations, we want to check whether Case of Interval . . . a new object belongs to a given class. What is Known What We Do in This Talk • In such situations, we usually have a sample of objects Theorem 1 which are known to belong to this class. Theorem 2 • Example: Home Page – a biologist who is studying bats has observed sev- Title Page eral bats from a local species; ◭◭ ◮◮ – the question is ◭ ◮ ∗ whether a newly observed bat belongs to the Page 2 of 31 same species – ∗ or whether the newly observed bat belongs to a Go Back different bat species. Full Screen Close Quit
A Practical Problem: . . . How This Problem is . . . 2. How This Problem is Usually Solved A Standard Way to . . . • Problem: checking whether an object belongs to a class. Selecting the Parameter k Case of Interval . . . • To solve this problem: we usually What is Known – measure one or more quantities for the objects from What We Do in This Talk this class and for the new object, and Theorem 1 – compare the resulting values. Theorem 2 Home Page • Simplest case of a single quantity. In this case, we have: Title Page ◭◭ ◮◮ – a collection of values x 1 , . . . , x n corresponding to objects from the known class, and ◭ ◮ – a value x corresponding to the new object. Page 3 of 31 Go Back Full Screen Close Quit
A Practical Problem: . . . How This Problem is . . . 3. A Standard Way to Decide Whether an Object A Standard Way to . . . Belongs to a Class Selecting the Parameter k • Problem (reminder): to decide whether Case of Interval . . . What is Known • a new object with the value x What We Do in This Talk • belongs to the class characterized by the values Theorem 1 x 1 , . . . , x n . Theorem 2 • Usual solution: check whether the value x belongs to Home Page the “ k sigma” interval [ E − k · σ, E + k · σ ], where: Title Page n � = 1 def ◭◭ ◮◮ • E n · x i is the sample mean, i =1 ◭ ◮ n � √ = 1 ( x i − E ) 2 is the sample def • σ = V , where V n · Page 4 of 31 i =1 Go Back variance, and Full Screen • the parameter k is determined by the degree of con- fidence with which we want to make the decision. Close Quit
A Practical Problem: . . . How This Problem is . . . 4. Selecting the Parameter k A Standard Way to . . . • Problem (reminder): to decide whether Selecting the Parameter k Case of Interval . . . • a new object with the value x What is Known • belongs to the class characterized by the values What We Do in This Talk x 1 , . . . , x n . Theorem 1 • Solution (reminder): check whether the value x belongs Theorem 2 to the “ k sigma” interval [ E − k · σ, E + k · σ ] Home Page • Usually, we take: Title Page ◭◭ ◮◮ • k = 2 (corresponding to confidence 0.9), • k = 3 (corresponding to 0.999), or ◭ ◮ • k = 6 (corresponding to 1 − 10 − 8 ). Page 5 of 31 Go Back Full Screen Close Quit
A Practical Problem: . . . How This Problem is . . . 5. Formulation of the Problem A Standard Way to . . . • Problem: checking whether an object belongs to a class. Selecting the Parameter k Case of Interval . . . • Standard approach: an object belongs to the class if What is Known the value x belongs to the “ k sigma” interval What We Do in This Talk [ E − k · σ, E + k · σ ] , where: Theorem 1 n � Theorem 2 = 1 def • E n · x i is the sample mean, and Home Page i =1 Title Page n � √ = 1 ( x i − E ) 2 is the sample def • σ = V , where V n · ◭◭ ◮◮ i =1 variance ◭ ◮ Page 6 of 31 • How confident are we about the decision • depends on the smallest values k − for which Go Back x ≥ E − k − · σ , and Full Screen • on the smallest value k + for which x ≤ E + k + · σ . Close Quit
A Practical Problem: . . . How This Problem is . . . 6. How to Compute the Parameters Describing A Standard Way to . . . Confidence? Selecting the Parameter k • The inequality x ≥ E − k − · σ is equivalent to Case of Interval . . . k − · σ ≥ E − x and k − ≥ E − x What is Known . σ What We Do in This Talk • Thus, when x < E , the corresponding smallest value Theorem 1 is equal to k − = E − x . Theorem 2 σ Home Page • Similarly, the inequality x ≤ E + k + · σ is equivalent to k + · σ ≥ x − E and k + ≥ x − E Title Page . σ ◭◭ ◮◮ • Thus, when x > E , the corresponding smallest value is equal to k + = x − E ◭ ◮ . σ Page 7 of 31 • So, to determine the parameter describing confidence, Go Back we must compute one of the ratios Full Screen = E − x = x − E k − def or k + def . σ σ Close Quit
A Practical Problem: . . . How This Problem is . . . 7. How to Compute the Parameters Describing A Standard Way to . . . Confidence (cont-d) Selecting the Parameter k • Reminder: to determine the parameter describing con- Case of Interval . . . fidence, we must compute one of the ratios What is Known = E − x = x − E What We Do in This Talk k − def or k + def . Theorem 1 σ σ Theorem 2 • Often, reciprocal ratio are used: Home Page = 1 σ = 1 σ r − def E − x and r + def k − = k + = x − E. Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 31 Go Back Full Screen Close Quit
A Practical Problem: . . . How This Problem is . . . 8. Case of Interval Uncertainty A Standard Way to . . . • Simplifying assumption: we know the exact values Selecting the Parameter k x 1 , . . . , x n of the corresponding quantity. Case of Interval . . . What is Known • In practice: these values come from measurement, and What We Do in This Talk measurements are never absolutely accurate. Theorem 1 • Specifics: the measurement results � x 1 , . . . , � x n are, in Theorem 2 general, different from the actual (unknown) values x i . Home Page • Traditional engineering techniques assume that we know Title Page def the probabilities of different values of ∆ x i = � x i − x i . ◭◭ ◮◮ • In many practical situations: we only know the upper ◭ ◮ bound ∆ i : | ∆ x i | ≤ ∆ i . Page 9 of 31 • In this case: we only know that the actual (unknown) Go Back def value x i belongs to the interval x i = [ � x i − ∆ i , � x i + ∆ i ]. Full Screen Close Quit
A Practical Problem: . . . How This Problem is . . . 9. Case of Interval Uncertainty (cont-d) A Standard Way to . . . • Reminder: we only know that the actual (unknown) Selecting the Parameter k def Case of Interval . . . value x i belongs to the interval x i = [ � x i − ∆ i , � x i + ∆ i ]. What is Known • Different possible values x i ∈ x i lead, in general, to dif- What We Do in This Talk ferent values of the corresponding ratios r ( x 1 , . . . , x n ). Theorem 1 • Thus, it is desirable to compute the range of possible Theorem 2 values of this ratio: Home Page def r = [ r, r ] = { r ( x 1 , . . . , x n ) | x 1 ∈ x 1 , . . . , x n ∈ x n } . Title Page ◭◭ ◮◮ • This problem is a particular case of the main problem ◭ ◮ of interval computation : Page 10 of 31 • given: an algorithm f ( x 1 , . . . , x n ) and intervals x i , Go Back • compute: the range Full Screen def y = [ y, y ] = { f ( x 1 , . . . , x n ) | x 1 ∈ x 1 , . . . , x n ∈ x n } . Close Quit
A Practical Problem: . . . How This Problem is . . . 10. What is Known A Standard Way to . . . • What was analyzed earlier: Selecting the Parameter k Case of Interval . . . – the problem of computing the range of r , and What is Known – similar problems of computing ranges for the thresh- What We Do in This Talk olds E − k · σ and E + k · σ for a given k . Theorem 1 • Feasible algorithms were described: Theorem 2 – for computing the upper bounds for E − k · σ , and Home Page σ – for computing the lower bounds for E + k · σ , E − x , Title Page σ and x − E . ◭◭ ◮◮ ◭ ◮ • For other bounds, feasible algorithms are known under certain conditions on the intervals: Page 11 of 31 – for computing the lower bounds for E − k · σ , and Go Back σ – for computing the upper bounds for E + k · σ , E − x , Full Screen σ and x − E . Close Quit
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