Estimating Variance under Estimating Mean . . . Interval and Fuzzy - PowerPoint PPT Presentation

Estimating Variance . . . Case of Measurement . . . Case of Expert . . . Estimating Mean . . . Estimating Variance under Estimating Mean . . . Interval and Fuzzy Estimating Variance . . . Estimating Variance . . . Uncertainty: Estimating Variance . . . Estimating Variance . . . Parallel Algorithms Acknowledgments Title Page Karen Villaverde ◭◭ ◮◮ Department of Computer Science New Mexico State University ◭ ◮ Las Cruces, NM 88003, USA email kvillave@cs.nmsu.edu Page 1 of 11 Gang Xiang Go Back Philips Healthcare Informatics Full Screen El Paso, Texas email gxiang@acm.org Close Quit

Estimating Variance . . . Case of Measurement . . . 1. Estimating Variance under Uncertainty Case of Expert . . . • Computing statistics is important: traditional data pro- Estimating Mean . . . cessing starts with computing population mean and Estimating Mean . . . population variance: Estimating Variance . . . n n � � E = 1 V = 1 Estimating Variance . . . ( x i − E ) 2 . n · x i , n · Estimating Variance . . . i =1 i =1 Estimating Variance . . . • Traditional approach: assumes that we know the exact Acknowledgments values x i . Title Page • In practice: these values come either from measure- ◭◭ ◮◮ ments or from expert estimates. ◭ ◮ • Uncertainty: in both cases, we get only approximations Page 2 of 11 � x i to the actual (unknown) values x i . Go Back • Result: we only get approximate valued � E and � V . Full Screen • Question: what is the accuracy of these approxima- Close tions? Quit

Estimating Variance . . . Case of Measurement . . . 2. Case of Measurement Uncertainty Case of Expert . . . • The result � x of the measurement is, in general, different Estimating Mean . . . def from the (unknown) actual value x : ∆ x = � x − x � = 0. Estimating Mean . . . Estimating Variance . . . • Upper bound ∆ is usually supplied by the manufac- Estimating Variance . . . turer: | ∆ x | ≤ ∆. Estimating Variance . . . • Interval uncertainty: x ∈ [ � x − ∆ , � x + ∆]. Estimating Variance . . . • Probabilistic approach: often, we know probabilities of Acknowledgments different values of ∆ x . Title Page ◭◭ ◮◮ • How these probabilities are determined: by comparing with standard measuring instrument (SMI). ◭ ◮ • Cases when we do not know probabilities: Page 3 of 11 – cutting-edge measurements; Go Back – manufacturing. Full Screen • Resulting problem: find the ranges E and V of E and V . Close Quit

Estimating Variance . . . Case of Measurement . . . 3. Case of Expert Uncertainty Case of Expert . . . • Situation: an expert use natural language. Estimating Mean . . . Estimating Mean . . . • Example: “most probably, the value of the quantity is Estimating Variance . . . between 6 and 7, but it is somewhat possible to have Estimating Variance . . . values between 5 and 8”. Estimating Variance . . . • Natural formalization: for every i , a fuzzy set µ i ( x i ). Estimating Variance . . . • Resulting problem: given fuzzy numbers x i , find the Acknowledgments fuzzy numbers for E and V . 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 are in the corre- ◭ ◮ sponding α -cuts: x i ∈ x i ( α ). Page 4 of 11 • Conclusion: for each characteristic C ( x 1 , . . . , x n ), it is Go Back sufficient to be able to compute the range Full Screen def C ( x 1 , . . . , x n ) = { C ( x 1 , . . . , x n ) | x 1 ∈ x 1 , . . . , x n ∈ x n } . Close Quit

Estimating Variance . . . Case of Measurement . . . 4. Estimating Mean under Interval Uncertainty: What Is Known Case of Expert . . . Estimating Mean . . . • Fact: the arithmetic average E ( x 1 , . . . , x n ) is an in- Estimating Mean . . . creasing function of x 1 , . . . , x n . Estimating Variance . . . Estimating Variance . . . • Conclusions: Estimating Variance . . . – the smallest possible value E of E is attained when Estimating Variance . . . each value x i is the smallest possible ( x i = x i ); Acknowledgments – the largest possible value E of E is attained when Title Page x i = x i for all i . ◭◭ ◮◮ • Resulting formulas: the range E of E is equal to ◭ ◮ [ E ( x 1 , . . . , x n ) , E ( x 1 , . . . , x n )] , Page 5 of 11 i.e., to Go Back � 1 � n · ( x 1 + . . . + x n ) , 1 Full Screen E = [ E, E ] = n · ( x 1 + . . . + x n ) . Close Quit

Estimating Variance . . . Case of Measurement . . . 5. Estimating Mean under Interval Uncertainty: Par- allelization Case of Expert . . . Estimating Mean . . . • General problem: for large n , the corresponding algo- Estimating Mean . . . rithm may require a large computation time. Estimating Variance . . . • Possible solution: if we have several ( p ) processors, we Estimating Variance . . . may speed up computations by parallelization. Estimating Variance . . . Estimating Variance . . . • Case of the mean – reminder: � 1 � Acknowledgments n · ( x 1 + . . . + x n ) , 1 E = [ E, E ] = n · ( x 1 + . . . + x n ) . Title Page ◭◭ ◮◮ • Parallelization: ◭ ◮ – divide n elements into p groups of n/p elements; Page 6 of 11 – each of p processors computes the sum of all the ele- Go Back ments from the corresponding group in time O ( n/p ); – then, we add p subsums. Full Screen • Resulting computation time: O ( n/p + log( p )). Close Quit

Estimating Variance . . . Case of Measurement . . . 6. Estimating Variance under Interval Uncertainty: What is Known Case of Expert . . . Estimating Mean . . . • Problem: compute the range V = [ V , V ] of the vari- Estimating Mean . . . ance V over interval data x i ∈ [ � x i − ∆ i , � x i + ∆ i ]. Estimating Variance . . . • Known: there is a polynomial-time algorithm for com- Estimating Variance . . . puting V . Estimating Variance . . . Estimating Variance . . . • In general: computing V is NP-hard. Acknowledgments • In many practical situations: there are efficient algo- Title Page rithms for computing V . ◭◭ ◮◮ • Example: consider narrowed intervals [ x − i , x + i ], where ◭ ◮ x i − ∆ i x i + ∆ i def def x − n and x + = � = � n . i i Page 7 of 11 • Case: no two narrowed intervals are proper subsets of Go Back one another, i.e., [ x − i , x + i ] �⊆ ( x − j , x + j ) for all i and j . Full Screen • For this case: there exists an O ( n · log( n )) time algo- Close rithm for computing V . Quit

Estimating Variance . . . Case of Measurement . . . 7. Estimating Variance Under Interval Uncertainty: Main Idea Case of Expert . . . Estimating Mean . . . � n • Reminder: V = M − E 2 , where M = 1 x 2 Estimating Mean . . . n · i and Estimating Variance . . . i =1 � n E = 1 Estimating Variance . . . n · x i . Estimating Variance . . . i =1 Estimating Variance . . . • Main lemma: if we sort the narrowed intervals in lexi- Acknowledgments cographic order, then V is attained at one of the vectors Title Page x = ( x 1 , . . . , x k , x k +1 , . . . , x n ) . ◭◭ ◮◮ • Conclusion: for some k , we have V = M k − E 2 k , where ◭ ◮ k � M k = M k + M k , E k = E k + E k , M k = 1 Page 8 of 11 ( x i ) 2 , n · Go Back i =1 n k n � � � M k = 1 ( x i ) 2 , E k = 1 x i , E k = 1 Full Screen n · n · n · x i . i = k +1 i =1 i = k +1 Close Quit

Estimating Variance . . . Case of Measurement . . . 8. Estimating Variance Under Interval Uncertainty: Resulting Algorithm Case of Expert . . . Estimating Mean . . . • First, we sort the intervals; this takes time O ( n · log( n )). Estimating Mean . . . Estimating Variance . . . • Then, for every k , we compute M k , M k , E k , E k , and Estimating Variance . . . V k = ( M k + M k ) − ( E k + E k ) 2 : Estimating Variance . . . Estimating Variance . . . – computing the values for k = 0 takes linear time O ( n ); Acknowledgments Title Page – then, we update in O (1) steps for each k , e.g., M k +1 = M k + 1 ◭◭ ◮◮ n · ( x k +1 ) 2 . ◭ ◮ • Finally, we find the largest of the values V 0 , . . . , V n +1 ; Page 9 of 11 this takes O ( n ) time. Go Back • Total time O ( n · log( n )) + O ( n ) + n · O (1) + O ( n ) = O ( n · log( n )). Full Screen Close Quit

Estimating Variance . . . Case of Measurement . . . 9. Estimating Variance Under Interval Uncertainty: Parallel Algorithm Case of Expert . . . Estimating Mean . . . If we have p < n processors, then we can: Estimating Mean . . . � n · log( n ) � Estimating Variance . . . • on Stage 1, sort n values in time O + log( n ) ; p Estimating Variance . . . Estimating Variance . . . • on Stage 2, compute all n + 1 values M k , M k , E k , E k Estimating Variance . . . (and hence V k ) in time in O ( n/p + log( p )); Acknowledgments • on Stage 3, compute the maximum of V 0 , . . . , V n in time Title Page O ( n/p + log( p )). ◭◭ ◮◮ Overall, we thus need time (since p ≤ n ): ◭ ◮ � n · log( n ) � � n � � n � Page 10 of 11 O + log( n ) + O p + log( p ) + O p + log( p ) = p Go Back � n · log( n ) � O + log( n ) . Full Screen p Close Quit