Need to Combine . . . General Case: What . . . Taking Into Account . . . Need to Combine Interval and How to Take into . . . Probabilistic Uncertainty: Physical and . . . What Needs to Be Computed, Relation with . . . What Can Be Computed, Proofs What Can Be Feasibly Computed, Home Page and How Physics Can Help Title Page ◭◭ ◮◮ Vladik Kreinovich Department of Computer Science ◭ ◮ University of Texas at El Paso Page 1 of 85 500 W. University El Paso, Texas 79968, USA Go Back vladik@utep.edu Full Screen http://www.cs.utep.edu/vladik Close Quit
Need to Combine . . . Part I General Case: What . . . Need to Combine Interval and Taking Into Account . . . Probabilistic Uncertainty: How to Take into . . . Linearized Case Physical and . . . Relation with . . . Proofs Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 85 Go Back Full Screen Close Quit
Need to Combine . . . 1. Need to Take Uncertainty Into Account When General Case: What . . . Processing Data Taking Into Account . . . • In practice, we are often interested in a quantity y How to Take into . . . which is difficult to measure directly. Physical and . . . Relation with . . . • Examples: distance to a star, amount of oil in the well, Proofs tomorrow’s weather. Home Page • Solution: find easier-to-measure quantities x 1 , . . . , x n Title Page related to y by a known dependence y = f ( x 1 , . . . , x n ). ◭◭ ◮◮ • Then, we measure x i and use measurement results � x i to compute an estimate � y = f ( � x 1 , . . . , � x n ). ◭ ◮ Page 3 of 85 • Measurements are never absolutely accurate, so even if def x i � = x i leads to ∆ y y − y � = 0. the model f is exact, � = � Go Back • It is important to use information about measurement Full Screen def x i − x i to estimate the accuracy ∆ y . errors ∆ x i = � Close Quit
Need to Combine . . . 2. We Often Have Imprecise Probabilities General Case: What . . . • Usual assumption: we know the probabilities for ∆ x i . Taking Into Account . . . How to Take into . . . • To find them, we measure the same quantities: Physical and . . . – with our measuring instrument (MI) and Relation with . . . x st i ≈ x i . – with a much more accurate MI, with � Proofs • In two important cases, this does not work: Home Page – state-of-the art-measurements, and Title Page – measurements on the shop floor. ◭◭ ◮◮ • Then, we have partial information about probabilities. ◭ ◮ • Often, all we know is an upper bound | ∆ x i | ≤ ∆ i . Page 4 of 85 • Then, we only know that x i ∈ [ � x i − ∆ i , � Go Back x i + ∆ i ] and def Full Screen y ∈ [ y, y ] = { f ( x 1 , . . . , x n ) : x i ∈ [ � x i − ∆ i , � x i + ∆ i ] } . Close • Computing [ y, y ] is known as interval computation. Quit
Need to Combine . . . 3. Data Processing: Example General Case: What . . . • Example: Taking Into Account . . . How to Take into . . . – we want to measure coordinates X j of an object; Physical and . . . – we measure the distance Y i between this object and Relation with . . . objects with accurately known coordinates X ( i ) j : Proofs � � Home Page 3 � � � ( X j − X ( i ) j ) 2 . Y i = Title Page j =1 ◭◭ ◮◮ • General case: ◭ ◮ – we know the results � Y i of measuring Y i ; Page 5 of 85 – we want to estimate the desired quantities X j . Go Back Full Screen Close Quit
Need to Combine . . . 4. Usually Linearization Is Possible General Case: What . . . • In most practical situations, we know the approximate Taking Into Account . . . values X (0) of the desired quantities X j . How to Take into . . . j Physical and . . . • These approximation are usually reasonably good, in Relation with . . . def = X j − X (0) the sense that the difference x j are small. j Proofs • In terms of x j , we have Home Page Y i = f ( X (0) + x 1 , . . . , X (0) + x n ) . Title Page 1 n • We can safely ignore terms quadratic in x j . ◭◭ ◮◮ • Indeed, even if the estimation accuracy is 10% (0.1), ◭ ◮ its square is 1% ≪ 10%. Page 6 of 85 • We can thus expand the dependence of Y i on x j in Go Back Taylor series and keep only linear terms: Full Screen n � = ∂f i Y i = Y (0) a ij · x j , Y (0) def = f i ( X (0) def 1 , . . . , X (0) + n ) , a ij . Close i i ∂X j j =1 Quit
Need to Combine . . . 5. Least Squares General Case: What . . . • Thus, to find the unknowns x j , we need to solve a Taking Into Account . . . system of approximate linear equations How to Take into . . . Physical and . . . � n def Y i − Y (0) = � a ij · x i ≈ y i , where y i . Relation with . . . i j =1 Proofs Home Page • Usually, it is assumed that each measurement error is: Title Page – normally distributed ◭◭ ◮◮ – with 0 mean (and known st. dev. σ i ). ◭ ◮ • The distribution is indeed often normal: Page 7 of 85 – the measurement error is a joint result of many in- dependent factors, Go Back – and the distribution of the sum of many small in- Full Screen dependent errors is close to Gaussian; Close – this is known as the Central Limit Theorem. Quit
Need to Combine . . . 6. Least Squares (cont-d) General Case: What . . . Taking Into Account . . . • 0 mean also makes sense: How to Take into . . . – we calibrate the measuring instrument by compar- Physical and . . . ing it with a more accurate, Relation with . . . – so if there was a bias (non-zero mean), we delete it Proofs by re-calibrating the scale. Home Page • It is also assumed that measurement errors of different Title Page measurements are independent. ◭◭ ◮◮ • In this case, for each possible combination x = ◭ ◮ ( x 1 , . . . , x n ), the probability of observing y 1 , . . . , y m is: � � 2 Page 8 of 85 � n y i − a ij · x j Go Back m � 1 j =1 √ · exp − . 2 σ 2 Full Screen 2 π · σ i i i =1 Close Quit
Need to Combine . . . 7. Least Squares (final) General Case: What . . . Taking Into Account . . . • It is reasonable to select x j for which this probability is the largest, i.e., equivalently, for which How to Take into . . . � � 2 Physical and . . . � n Relation with . . . y i − a ij · x j n � j =1 Proofs → min . σ 2 Home Page i i =1 Title Page • The set S γ of all possible combinations x is: � � 2 ◭◭ ◮◮ � n y i − a ij · x j ◭ ◮ n � j =1 ≤ χ 2 S γ = x : . Page 9 of 85 m − n,γ σ 2 i i =1 Go Back Full Screen • If S = ∅ , this means that some measurements are out- Close liers. Quit
Need to Combine . . . 8. Need to Take into Account Systematic Error General Case: What . . . • In the traditional approach, we assume that y i = Taking Into Account . . . � n How to Take into . . . a ij · x j + e i , where the meas. error e i has 0 mean. Physical and . . . j =1 • Sometimes: Relation with . . . Proofs def – in addition to the random error e r = e i − E [ e i ] with i Home Page 0 mean, def Title Page – we also have a systematic error e s = E [ e i ]: i ◭◭ ◮◮ n � a ij · x j + e r i + e s y i = i . ◭ ◮ j =1 Page 10 of 85 • Sometimes, we know the upper bound ∆ i : | e s i | ≤ ∆ i . Go Back • In other cases, we have different bounds ∆ i ( p ) corre- Full Screen sponding to different degree of confidence p . Close • What can we then say about x j ? Quit
Need to Combine . . . 9. Combining Probabilistic and Interval Uncer- General Case: What . . . tainty: Main Idea Taking Into Account . . . • If we knew the values e s i , then we would conclude that How to Take into . . . � n Physical and . . . for e r a ij · x j − e s i = y i − i , we have Relation with . . . j =1 � � 2 Proofs � n a ij · x j − e s y i − Home Page i m m � � i ) 2 ( e r j =1 ≤ χ 2 = m − n,γ . Title Page σ 2 σ 2 i i i =1 i =1 ◭◭ ◮◮ • In practice, we do not know the values e s i , we only know ◭ ◮ that these values are in the interval [ − ∆ i , ∆ i ]. Page 11 of 85 • Thus, we know that the above inequality holds for some Go Back e s i ∈ [ − ∆ i , ∆ i ] . Full Screen Close Quit
Need to Combine . . . 10. Main Idea (cont-d) General Case: What . . . • The above condition is equivalent to v ( x ) ≤ χ 2 Taking Into Account . . . m − n,γ , where How to Take into . . . � � 2 Physical and . . . � n a ij · x j − e s y i − Relation with . . . i m � j =1 def Proofs v ( x ) = min . σ 2 e s i ∈ [ − ∆ i , ∆ i ] i Home Page i =1 • So, the set S γ of all combinations X = ( x 1 , . . . , x n ) Title Page which are possible with confidence 1 − γ is: ◭◭ ◮◮ S γ = { x : v ( x ) ≤ χ 2 m − n,γ } . ◭ ◮ • The range of possible values of x j can be obtained by Page 12 of 85 maximizing and minimizing x j under the constraint Go Back v ( x ) ≤ χ 2 m − n,γ . Full Screen • In the fuzzy case, we have to repeat the computations Close for every p . Quit
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