Analyzing a Sample Need to Estimate . . . Computing the Range . . . Estimating Mean under Variance Constraints Cases When This . . . Interval Uncertainty and Main Result: A . . . Variance Constraint Computation Time of . . . Toy Example Ali Jalal-Kamali 1 , Luc Longpr´ e 1 , and Proof: Main Lemmas Misha Koshelev 2 Home Page 1 Department of Computer Science Title Page University of Texas at El Paso El Paso, TX 79968, USA ◭◭ ◮◮ ajalalkamali@miners.utep.edu ◭ ◮ longpre@utep.edu Page 1 of 15 2 Human Neuroimaging Lab Division of Neuroscience Go Back Baylor College of Medicine Houston, TX 77030, USA Full Screen misha680hnl@gmail.com Close Quit
Analyzing a Sample Need to Estimate . . . 1. Analyzing a Sample Computing the Range . . . • Often, we have a sample of values x 1 , . . . , x n corre- Variance Constraints sponding to objects of a certain type. Cases When This . . . Main Result: A . . . • In this case, a standard way to describe the correspond- Computation Time of . . . ing population is to estimate its mean and variance: Toy Example n n � � E = 1 V = 1 ( x i − E ) 2 . n · x i ; n · Proof: Main Lemmas Home Page i =1 i =1 Title Page • In practice, the values x i come from measurements, and measurements are never absolutely accurate. ◭◭ ◮◮ • Often, the only information we have is an upper bound ◭ ◮ ∆ i on the measurement error: | ∆ x i | ≤ ∆ i . Page 2 of 15 • In this case, based on the measured value � x i , we con- Go Back clude that the actual value x i is in the interval Full Screen x i = [ x i , x i ] = [ � x i − ∆ i , � x i + ∆ i ] . Close Quit
Analyzing a Sample Need to Estimate . . . 2. Need to Estimate Mean and Variance under Computing the Range . . . Interval Uncertainty Variance Constraints • In general, different values x i ∈ x i lead to different Cases When This . . . values of E and V . Main Result: A . . . Computation Time of . . . • It is therefore desirable to describe the range of possible values of mean and variance when x i ∈ x i . Toy Example Proof: Main Lemmas • This is a particular case of a general problem of interval Home Page computation : computing the range Title Page def y = [ y, y ] = { f ( x 1 , . . . , x n ) | x 1 ∈ x 1 , . . . , x n ∈ x n } . ◭◭ ◮◮ • Sometimes, we have fuzzy values X 1 , . . . , X n , and we ◭ ◮ want to find Y = f ( X 1 , . . . , X n ). Page 3 of 15 • It is known that for α -cuts X i ( α ), we have Go Back Y ( α ) = { f ( x 1 , . . . , x n ) | x 1 ∈ X 1 ( α ) , . . . , x n ∈ X n ( α ) } . Full Screen • In view of this reduction, we will concentrate on algo- Close rithms for interval uncertainty. Quit
Analyzing a Sample Need to Estimate . . . 3. Computing the Ranges of the Mean and Vari- Computing the Range . . . ance: What Is Known Variance Constraints • The mean E is an increasing function of each x i ; thus: Cases When This . . . Main Result: A . . . – the smallest value E is attained when each x i is the Computation Time of . . . smallest x i = x i , and Toy Example – the largest value E is attained when each x i is the Proof: Main Lemmas largest x i = x i : Home Page n n � � E = 1 E = 1 n · x i ; n · x i . Title Page i =1 i =1 ◭◭ ◮◮ • Variance V is, in general, not monotonic, so its range ◭ ◮ is more difficult to compute: Page 4 of 15 – the lower endpoint V is computable in linear time, Go Back – but computing V is, in general, NP-hard. Full Screen • There are also efficient algorithms for computing V in some cases. Close Quit
Analyzing a Sample Need to Estimate . . . 4. Variance Constraints Computing the Range . . . • In the previous expressions, we assume that there is no Variance Constraints a priori information about the values of E and V . Cases When This . . . Main Result: A . . . • In some cases, we have a priori constraint on the vari- Computation Time of . . . ance: V ≤ V 0 , for a given V 0 . Toy Example • For example, we know that within a species, there can Proof: Main Lemmas be ≤ 0 . 1 variation of a certain characteristic. Home Page • Thus, we arrive at the following problem: Title Page – given: n intervals x i = [ x i , x i ] and a number V 0 ≥ 0; ◭◭ ◮◮ – compute: the range ◭ ◮ [ E, E ] = { E ( x 1 , . . . , x n ) : x i ∈ x i & V ( x 1 , . . . , x n ) ≤ V 0 } ; Page 5 of 15 – under the assumption that there exist values x i ∈ x i Go Back for which V ( x 1 , . . . , x n ) ≤ V 0 . Full Screen • This is the problem that we will solve in this paper. Close Quit
Analyzing a Sample Need to Estimate . . . 5. Cases Where This Problem Is (Relatively) Easy Computing the Range . . . to Solve Variance Constraints • First case: V 0 is ≥ the largest possible value V of the Cases When This . . . variance corresponding to the given sample. Main Result: A . . . Computation Time of . . . • In this case, the constraint V ≤ V 0 is always satisfied. Toy Example • Thus, in this case, the desired range simply coincides Proof: Main Lemmas with the range of all possible values of E . Home Page • Second case: V 0 = 0. Title Page • In this case, the constraint V ≤ V 0 means that the ◭◭ ◮◮ variance V should be equal to 0, i.e., x 1 = . . . = x n . ◭ ◮ • In this case, we know that this common value x i be- Page 6 of 15 longs to each of n intervals x i . Go Back • So, the set of all possible values E is the intersection: Full Screen E = x 1 ∩ . . . ∩ x n . Close Quit
Analyzing a Sample Need to Estimate . . . 6. Main Result: A Feasible Algorithm that Com- Computing the Range . . . putes [ E, E ] under Interval Uncertainty and Vari- Variance Constraints ance Constraint Cases When This . . . • First, we compute the values Main Result: A . . . n n � � Computation Time of . . . = 1 = 1 E − def x i and V − def ( x i − E − ) 2 ; n · n · Toy Example i =1 i =1 Proof: Main Lemmas n n � � = 1 = 1 Home Page E + def x i and V + def ( x i − E + ) 2 . n · n · Title Page i =1 i =1 • If V − ≤ V 0 , then we return E = E − . ◭◭ ◮◮ • If V + ≤ V 0 , then we return E = E + . ◭ ◮ • If V 0 < V − or V 0 < V + , we sort the all 2 n endpoints Page 7 of 15 x i and x i into a non-decreasing sequence Go Back z 1 ≤ z 2 ≤ . . . ≤ z 2 n Full Screen and consider 2 n − 1 zones [ z k , z k +1 ]. Close Quit
Analyzing a Sample Need to Estimate . . . 7. Algorithm (cont-d) Computing the Range . . . • For each zone [ z k , z k +1 ], we take: Variance Constraints Cases When This . . . – for every i for which x i ≤ z k , we take x i = x i ; Main Result: A . . . – for every i for which z k +1 ≤ x i , we take x i = x i ; Computation Time of . . . – for every other i , we take x i = α ; let us denote the Toy Example number of such i ’s by n k . Proof: Main Lemmas Home Page • The value α is determined from the condition that for the selected vector x , we have V ( x ) = V 0 : Title Page ◭◭ ◮◮ � � 1 ( x i ) 2 + ( x i ) 2 + n k · α 2 − n · ◭ ◮ i : x i ≤ z k i : z k +1 ≤ x i Page 8 of 15 2 � � 1 Go Back n 2 · x i + x i + n k · α = V 0 . Full Screen i : x i ≤ z k i : z k +1 ≤ x i Close Quit
Analyzing a Sample Need to Estimate . . . 8. Algorithm: Last Part Computing the Range . . . • If none of the two roots of the above quadratic equation Variance Constraints belongs to the zone, this zone is dismissed. Cases When This . . . Main Result: A . . . • If one or more roots belong to the zone, then for each Computation Time of . . . of these roots α , we compute the value Toy Example � � E k ( α ) = 1 . Proof: Main Lemmas n · x i + x i + n k · α Home Page i : x i ≤ z k i : z k +1 ≤ x i Title Page • After that: ◭◭ ◮◮ – if V 0 < V − , we return the smallest of the values E k ( α ) as E : ◭ ◮ E = min k,α E k ( α ); Page 9 of 15 – if V 0 < V + , we return the largest of the values Go Back E k ( α ) as E : Full Screen E = max k,α E k ( α ) . Close Quit
Analyzing a Sample Need to Estimate . . . 9. Computation Time of the Algorithm Computing the Range . . . • Sorting 2 n numbers requires time O ( n · log( n )). Variance Constraints Cases When This . . . • Once the values are sorted, we can then go zone-by- Main Result: A . . . zone, and perform the corresponding computations: Computation Time of . . . – for each of 2 n zones, Toy Example – we compute several sums of n numbers. Proof: Main Lemmas Home Page • The sum for the first zone requires linear time. Title Page • Once we have the sums for one zone, computing the ◭◭ ◮◮ sums for the next zone requires changing a few terms. ◭ ◮ • Each value x i changes status once, so overall, to com- pute all these sums, we need linear time O ( n ). Page 10 of 15 • So, the total time is: Go Back Full Screen O ( n · log( n )) + O ( n ) = O ( n · log( n )) . Close Quit
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