Data Processing in the A Computational . . . Priority Approach to . - PowerPoint PPT Presentation

Design and Control . . . How to Describe . . . Constraint . . . What Are Soft . . . Priority Approach to . . . Data Processing in the A Computational . . . Priority Approach to . . . When Is a Method . . . Presence of Interval Main Result Proof (cont-d) Uncertainty and Erroneous Acknowledgments What If Constraints . . . Measurements: Practical Proof of NP-hardness Constraint . . . New Idea Problems, Results, New Algorithm Title Page Challenges ◭◭ ◮◮ M. Ceberio, O. Kosheleva, V. Kreinovich, G. R. Keller, ◭ ◮ R. Araiza, M. Averill, and G. Xiang Page 1 of 34 University of Texas at El Paso, 500 W. University El Paso, TX 79968, USA, olgak@utep.edu Go Back Full Screen Close Quit

Design and Control . . . How to Describe . . . 1. Formulation of the Problem Constraint . . . What Are Soft . . . • There are two main reasons why measurement results differ from the actual Priority Approach to . . . values of the measured quantities: A Computational . . . • There is a small difference caused by the inaccuracy of the measuring instru- Priority Approach to . . . ment. When Is a Method . . . Main Result • This inaccuracy is characterized by probabilistic or interval uncertainty. Proof (cont-d) • Sometimes, due to an instrument malfunction or a human error, we get an Acknowledgments erroneous measurement result (outlier) that is drastically different from the What If Constraints . . . actual value. Proof of NP-hardness Constraint . . . • This uncertainty is usually characterized by a proportion of measurement New Idea results that could be erroneous (e.g., ≤ 1%). New Algorithm • Situation: most data processing algorithms based on interval computations Title Page only take into account the first type of uncertainty. ◭◭ ◮◮ • Problem: take the presence of erroneous measurements into account as well. ◭ ◮ Page 2 of 34 Go Back Full Screen Close Quit

Design and Control . . . How to Describe . . . 2. Sometimes, It Is Relatively Easy to Detect Outliers Constraint . . . What Are Soft . . . • In some cases, when the data is smooth, we can (rather easily) detect the Priority Approach to . . . corresponding outliers. A Computational . . . • Traditional engineering approach: a new measurement result x is classified Priority Approach to . . . as an outlier if x �∈ [ L, U ], where When Is a Method . . . Main Result def def L = E − k 0 · σ, U = E + k 0 · σ, Proof (cont-d) Acknowledgments and k 0 > 1 is pre-selected (most frequently, k 0 = 2, 3, or 6. What If Constraints . . . • Minor problem: in some practical situations, we only have intervals x i = Proof of NP-hardness [ x i , x i ]. Constraint . . . New Idea • For different values x i ∈ x i , we get different k 0 -sigma intervals [ L, U ]. New Algorithm • A value x is a guaranteed outlier if x �∈ [ L, U ]. Title Page • Conclusion: to detect outliers, we must know the ranges of L = E − k 0 · σ ◭◭ ◮◮ and U = E + k 0 · σ . • Good news: there exist algorithm for computing these ranges. ◭ ◮ • Not so good news: in many practical situations, e.g., in non-destructive test- Page 3 of 34 ing (NDT) of aeroplanes and roads, and in geophysical analysis, we are ac- tually interested in unusual non-smooth data points. Go Back • Problem: separating correct but unusual measurement results from the erro- Full Screen neous measurement results is a challenge. Close Quit

Design and Control . . . How to Describe . . . 3. Presence of Erroneous Measurements Make Prob- Constraint . . . lems Computationally Difficult What Are Soft . . . Priority Approach to . . . • Known fact: the presence of outliers turns easy-to-solve interval problems A Computational . . . into difficult-to-solve (NP-hard) ones. Priority Approach to . . . When Is a Method . . . • New result: this difficulty may appear even without interval uncertainty. Main Result • Situation: we know how the measured quantity y is related to the desired Proof (cont-d) parameters x j . Acknowledgments What If Constraints . . . � n • Simplest case: linear dependence, i.e., a ij · x j = y i , where y i is the result Proof of NP-hardness j =1 Constraint . . . of i -th measurement, and a ij are (known) parameters corresponding to i -th New Idea measurement. New Algorithm • Problem: given a ij , y i , and ε ∈ (0 , 1), and constraints Title Page n � ◭◭ ◮◮ a ij · x j = y i , i = 1 , . . . , N j =1 ◭ ◮ check whether we can select a consistent set of N · (1 − ε ) constraints. Page 4 of 34 Go Back Full Screen Close Quit

Design and Control . . . How to Describe . . . 4. Result: The Problem Is NP-hard Even for the Lin- Constraint . . . ear Case What Are Soft . . . Priority Approach to . . . • Idea: reduce to a known NP-hard problem. A Computational . . . Priority Approach to . . . • Subset sum: given positive integers s 1 , . . . , s n , and s , check whether s = When Is a Method . . . n � x i · s i for some x i ∈ { 0 , 1 } . Main Result i =1 Proof (cont-d) Acknowledgments • Reduction: N = n/ε constraints: What If Constraints . . . • 2 n constraints x 1 = 0, x 1 = 1 . . . , x n = 0, x n = 1; Proof of NP-hardness � • N − 2 n identical constraints s i · x i = s. Constraint . . . New Idea • Since 0 � = 1, at most N − n are satisfied. New Algorithm • If the subset problem has a solution, then: Title Page � • all N − 2 n constraints s i · x i = s are satisfied, ◭◭ ◮◮ • and for each i , x i = 0 or x i = 1, ◭ ◮ to the total of N − n = N · (1 − ε ). Page 5 of 34 • If N − n constraints are satisfied, then for every i , x i ∈ { 0 , 1 } – a solution to the subset problem. Go Back Full Screen Close Quit

Design and Control . . . How to Describe . . . 5. Constraint Propagation Techniques (Semenov, Nu- Constraint . . . merica, Jaulin, etc): Reminder What Are Soft . . . Priority Approach to . . . • Constraint propagation – traditional technique for solving constraint satis- A Computational . . . faction problems. Priority Approach to . . . When Is a Method . . . • We start with the intervals [ x 1 , x 1 ] , . . . , [ x n , x n ] containing the actual values Main Result of the unknowns x 1 , . . . , x n . Proof (cont-d) • On each iteration: Acknowledgments What If Constraints . . . – select i and a constraint f j ( x 1 , . . . , x n ) = 0, Proof of NP-hardness def – replace [ x i , x i ] with new interval x ( j ) = [ x ( j ) i , x ( j ) i ] = Constraint . . . i New Idea { x i : x i ∈ [ x i , x i ] & f j ( x 1 , . . . , x i − 1 , x i , x i +1 , . . . , x n ) = 0 New Algorithm for some x k ∈ [ x k , x k ] } . Title Page • If the process stalls, we bisect the interval for one the variables into two and ◭◭ ◮◮ try to decrease both resulting half-boxes. ◭ ◮ • Problem: cannot use it if not all constraints are valid. Page 6 of 34 Go Back Full Screen Close Quit

Design and Control . . . How to Describe . . . 6. Traditional Interval-Related Constraint Propagation Constraint . . . Techniques: Example What Are Soft . . . Priority Approach to . . . • Toy problem: find x ∈ [ − 5 , 5] for which x − x 2 = 0. A Computational . . . Priority Approach to . . . • Pre-processing: parse the expression: When Is a Method . . . r = x 2 ; x − r = 0 . Main Result Proof (cont-d) • Originally: X = [ − 5 , 5], R = [ −∞ , ∞ ]. Acknowledgments • Use the first constraint: x ∈ [ − 5 , 5] implies r ∈ [0 , 25], so for r , the new What If Constraints . . . interval is [ −∞ , ∞ ] ∩ [0 , 25] = [0 , 25]: Proof of NP-hardness X = [ − 5 , 5] , R = [0 , 25] . Constraint . . . New Idea • Use the second constraint: for x , we have [ − 5 , 5] ∩ [0 , 25] = [0 , 5], and similarly New Algorithm for r , so X = [0 , 5] , R = [0 , 5] . Title Page • Use the first constraint: x = √ r , hence ◭◭ ◮◮ X = [0 , 2 . 24] , R = [0 , 5] . ◭ ◮ • Use the second constraint: Page 7 of 34 X = [0 , 2 . 24] , R = [0 , 2 . 24] . Go Back • After a while, we stall at X = R ≈ [0 , 1], so we bisect X to [0 , 1 / 2] and [1 / 2 , 1]. Full Screen • Then, we converge to x = 0 and x = 1. Close Quit

Design and Control . . . How to Describe . . . 7. New Idea Constraint . . . What Are Soft . . . • On each iteration, we still select a variable x i , but: Priority Approach to . . . – instead of selecting a single constraint, A Computational . . . Priority Approach to . . . – we try all N constraints, and get N resulting intervals [ x ( j ) i , x ( j ) i ]. When Is a Method . . . • We know that ≥ N · (1 − ε ) constraints are satisfied. Main Result Proof (cont-d) • Hence x i ≤ x ( j ) for ≥ N · (1 − ε ) different values j . i Acknowledgments • Let us sort all N upper endpoints x ( j ) What If Constraints . . . (1 ≤ j ≤ N ) into an increasing i Proof of NP-hardness sequence u 1 ≤ u 2 ≤ . . . ≤ u N , Constraint . . . • Then we can guarantee that x i is smaller than (or equal to) at least N · (1 − ε ) New Idea terms in this sequence. New Algorithm • So, x i ≤ u N · ε . Title Page • Similarly, if we sort the lower endpoints x ( j ) into a decreasing sequence l 1 ≥ ◭◭ ◮◮ i . . . ≥ l N , then x i ≥ l N · ε . ◭ ◮ Page 8 of 34 Go Back Full Screen Close Quit