Estimating Variance under Hierarchical Case: . . . Interval and - PowerPoint PPT Presentation

Case of Measurement . . . Case of Expert . . . Estimating Mean . . . Estimating Variance . . . Estimating Variance under Hierarchical Case: . . . Interval and Fuzzy Hierarchical Case: . . . Formulation of the . . . Uncertainty: Case of Analysis of the Problem Efficient Algorithm for . . . Hierarchical Estimation Number of . . . Gang Xiang and Vladik Kreinovich Acknowledgments Title Page Department of Computer Science ◭◭ ◮◮ University of Texas at El Paso El Paso, Texas 79968, USA ◭ ◮ email vladik@utep.edu Page 1 of 13 http://www.cs.utep.edu/vladik Go Back http://www.cs.utep.edu/interval-comp Full Screen Close Quit

Case of Measurement . . . Case of Expert . . . 1. Estimating Variance under Uncertainty Estimating Mean . . . • Computing statistics is important: traditional data pro- Estimating Variance . . . cessing starts with computing population mean and Hierarchical Case: . . . population variance: Hierarchical Case: . . . n n � � E = 1 V = 1 Formulation of the . . . ( x i − E ) 2 . n · x i , n · Analysis of the Problem i =1 i =1 Efficient Algorithm for . . . • Traditional approach: assumes that we know the exact Number of . . . values x i . Acknowledgments • In practice: these values come either from measure- Title Page ments or from expert estimates. ◭◭ ◮◮ • Uncertainty: in both case, we get only approximations ◭ ◮ � x i to the actual (unknown) values x i . Page 2 of 13 • Result: we only get approximate valued � E and � V . Go Back • Question: what is the accuracy of these approxima- Full Screen tions? Close Quit

Case of Measurement . . . Case of Expert . . . 2. Case of Measurement Uncertainty Estimating Mean . . . • The result � x of the measurement is, in general, different Estimating Variance . . . def from the (unknown) actual value x : ∆ x = � x − x � = 0. Hierarchical Case: . . . Hierarchical Case: . . . • Upper bound ∆ is usually supplied by the manufac- Formulation of the . . . turer: | ∆ x | ≤ ∆. Analysis of the Problem • Interval uncertainty: x ∈ [ � x − ∆ , � x + ∆]. Efficient Algorithm for . . . • Probabilistic approach: often, we know probabilities of Number of . . . different values of ∆ x . Acknowledgments Title Page • How these probabilities are determined: by comparing with standard measuring instrument (SMI). ◭◭ ◮◮ ◭ ◮ • Cases when we do not know probabilities: Page 3 of 13 – cutting-edge measurements; – manufacturing. Go Back Full Screen • Resulting problem: find the ranges E and V of E and V . Close Quit

Case of Measurement . . . Case of Expert . . . 3. Case of Expert Uncertainty Estimating Mean . . . • Situation: an expert use natural language. Estimating Variance . . . Hierarchical Case: . . . • Example: “most probably, the value of the quantity is Hierarchical Case: . . . between 6 and 7, but it is somewhat possible to have Formulation of the . . . values between 5 and 8”. Analysis of the Problem • Natural formalization: for every i , a fuzzy set µ i ( x i ). Efficient Algorithm for . . . • Resulting problem: given fuzzy numbers x i , find the Number of . . . fuzzy numbers for E and V . Acknowledgments Title Page • Reduction to interval case: the α -cut for C ( x 1 , . . . , x n ) is equal to the range of C when x i care in the corre- ◭◭ ◮◮ sponding α -cuts: x i ∈ x i ( α ). ◭ ◮ • Conclusion: for each characteristic C ( x 1 , . . . , x n ), it is Page 4 of 13 sufficient to be able to compute the range Go Back def C ( x 1 , . . . , x n ) = { C ( x 1 , . . . , x n ) | x 1 ∈ x 1 , . . . , x n ∈ x n } . Full Screen Close Quit

Case of Measurement . . . Case of Expert . . . 4. Estimating Mean under Interval Uncertainty: What Is Known Estimating Mean . . . Estimating Variance . . . • Fact: the arithmetic average E ( x 1 , . . . , x n ) is an in- Hierarchical Case: . . . creasing function of x 1 , . . . , x n . Hierarchical Case: . . . Formulation of the . . . • Conclusions: Analysis of the Problem – the smallest possible value E of E is attained when Efficient Algorithm for . . . each value x i is the smallest possible ( x i = x i ); Number of . . . – the largest possible value E of E is attained when Acknowledgments x i = x i for all i . Title Page • Resulting formulas: the range E of E is equal to ◭◭ ◮◮ [ E ( x 1 , . . . , x n ) , E ( x 1 , . . . , x n )] , ◭ ◮ i.e., to Page 5 of 13 � 1 � n · ( x 1 + . . . + x n ) , 1 Go Back E = [ E, E ] = n · ( x 1 + . . . + x n ) . Full Screen Close Quit

Case of Measurement . . . Case of Expert . . . 5. Estimating Variance under Interval Uncertainty: What is Known Estimating Mean . . . Estimating Variance . . . • Problem: compute the range V = [ V , V ] of the vari- Hierarchical Case: . . . ance V over interval data x i ∈ [ � x i − ∆ i , � x i + ∆ i ]. Hierarchical Case: . . . • Known: there is a polynomial-time algorithm for com- Formulation of the . . . puting V . Analysis of the Problem Efficient Algorithm for . . . • In general: computing V is NP-hard. Number of . . . • In many practical situations: there are efficient algo- Acknowledgments rithms for computing V . Title Page • Example: consider narrowed intervals [ x − i , x + i ], where ◭◭ ◮◮ x i − ∆ i x i + ∆ i def def x − n and x + = � = � n . ◭ ◮ i i • Case: no two narrowed intervals are proper subsets of Page 6 of 13 one another, i.e., [ x − i , x + i ] �⊆ ( x − j , x + j ) for all i and j . Go Back • For this case: here exists an O ( n · log( n )) time algo- Full Screen rithm for computing V . Close Quit

Case of Measurement . . . Case of Expert . . . 6. Hierarchical Case: Formulation of the Problem Estimating Mean . . . • Situation: often, Estimating Variance . . . Hierarchical Case: . . . – we do not know the individual values of the obser- Hierarchical Case: . . . vations x i , Formulation of the . . . – we only have average values corresponding to sev- Analysis of the Problem eral ( m < n ) groups I 1 , . . . , I m of observations. Efficient Algorithm for . . . • Typically: for each group j , we know Number of . . . – the frequency p j of this group (i.e., the probabil- Acknowledgments ity that a randomly selected observation belongs Title Page to this group), ◭◭ ◮◮ – the average E j over this group, and ◭ ◮ – the standard deviation σ j within j -th group. Page 7 of 13 � m • Formulas: E = p j · E j and V = V E + V σ , where Go Back j =1 � � m m def def def = M E − E 2 , M E p j · E 2 p j · σ 2 Full Screen V E = j , and V σ = j . j =1 j =1 Close Quit

Case of Measurement . . . Case of Expert . . . 7. Hierarchical Case: Interval Uncertainty Estimating Mean . . . • Practical situation: we only know the intervals E j = Estimating Variance . . . [ E j , E j ] and [ σ j , σ j ] that contain E j and σ j . Hierarchical Case: . . . Hierarchical Case: . . . • Mean E is monotonic in E j , hence � m � Formulation of the . . . m � � Analysis of the Problem E = [ E, E ] = p j · E j , p j · E j . Efficient Algorithm for . . . j =1 j =1 Number of . . . • Variance: the terms V E and V σ in the expression for V Acknowledgments depend on different variables. Title Page • Conclusion: the range V = [ V , V ] of the population ◭◭ ◮◮ variance V is equal to the sum of the ranges V E = ◭ ◮ [ V E , V E ] and V σ = [ V σ , V σ ]. � � Page 8 of 13 � m � m p j · ( σ j ) 2 , p j · ( σ j ) 2 • Due to monotonicity, V σ = . Go Back j =1 j =1 Full Screen • Thus, it is sufficient to compute V E . Close Quit

Case of Measurement . . . Case of Expert . . . 8. Formulation of the Problem in Precise Terms Estimating Mean . . . GIVEN: Estimating Variance . . . Hierarchical Case: . . . • an integer m ≥ 1; Hierarchical Case: . . . � m • m numbers p j > 0 for which p j = 1; and Formulation of the . . . j =1 Analysis of the Problem • m intervals E j = [ E j , E j ]. Efficient Algorithm for . . . Number of . . . COMPUTE the range Acknowledgments V E = { V E ( E 1 , . . . , E m ) | E 1 ∈ E 1 , . . . , E m ∈ E m } , Title Page where ◭◭ ◮◮ m m � � ◭ ◮ def def p j · E 2 j − E 2 ; V E = E = p j · E j . Page 9 of 13 j =1 j =1 Go Back Full Screen Close Quit

Case of Measurement . . . Case of Expert . . . 9. Analysis of the Problem Estimating Mean . . . • Fact: the function V E is convex. Estimating Variance . . . Hierarchical Case: . . . • Fact: the box E 1 × . . . × E m is convex. Hierarchical Case: . . . • Known: a polynomial-time algorithm for computing Formulation of the . . . minima of convex functions on convex sets. Analysis of the Problem • Conclusion: we can compute V E in polynomial time. Efficient Algorithm for . . . Number of . . . • Computing V E : in general, NP-hard. Acknowledgments • Proof of NP-hardness: Title Page – for p 1 = . . . = p m = 1 m , the expression V E becomes ◭◭ ◮◮ a standard formula for the sample variance V ; ◭ ◮ – so, in this case, V E = V ; Page 10 of 13 – computing V under interval uncertainty is NP-hard; Go Back – thus, the more general problem of computing V E Full Screen is also NP-hard. Close Quit