Need for Data Processing Need to Take . . . Case of Interval . . . Case of Fuzzy Uncertainty When Is Data Processing How to Describe the . . . Under Interval and Fuzzy Formulation of the . . . Answer to the First . . . Uncertainty Feasible: What Analysis of the . . . Answering the Second . . . If Few Inputs Interact? Does Home Page Feasibility Depend on How Title Page We Describe Interaction? ◭◭ ◮◮ ◭ ◮ ık 1 , Michal ˇ y 2 , and Vladik Kreinovich 3 Milan Hlad´ Cern´ Page 1 of 35 1 Department of Applied Mathematics, Charles University Prague, Czech Republic, milan.hladik@matfyz.cz Go Back 2 Department of Econometrics, University of Economics Prague, Czech Republic, cernym@vse.cz Full Screen 3 Department of Computer Science, University of Texas at El Paso Close El Paso, Texas 79968, USA, vladik@utep.edu Quit
Need for Data Processing Need to Take . . . 1. Need for Data Processing Case of Interval . . . • In many practical situations: Case of Fuzzy Uncertainty How to Describe the . . . – we are interested in the value of a quantity y Formulation of the . . . – which is difficult or even impossible to measure di- Answer to the First . . . rectly. Analysis of the . . . • For example, we may be interested in a distance to a Answering the Second . . . faraway star, or in tomorrow’s temperature. Home Page • Since we cannot measure y directly, we measure it in- Title Page directly: namely, ◭◭ ◮◮ – we find easier-to-estimate quantities x 1 , . . . , x n re- ◭ ◮ lated to y by a known dependence Page 2 of 35 y = f ( x 1 , . . . , x n ) , Go Back – and we use the results � x i of measuring or estimating Full Screen the quantities x i to estimate y as � y = f ( � x n ) . x 1 , . . . , � Close Quit
Need for Data Processing Need to Take . . . 2. Need to Take Uncertainty into Account Case of Interval . . . • Measurements are never absolutely accurate. Case of Fuzzy Uncertainty How to Describe the . . . • As a result, the measurement results � x i are, in general, Formulation of the . . . different from the actual (unknown) values x i . Answer to the First . . . • Thus, even if the relation y = f ( x 1 , . . . , x n ) is precise, Analysis of the . . . – the result � y of applying the algorithm f to the mea- Answering the Second . . . Home Page surement results – is, in general, different from the actual value y . Title Page ◭◭ ◮◮ • How accurate is the estimate � y ? ◭ ◮ • In other words, what can we conclude about the mea- def surement errors ∆ y = � y − y. Page 3 of 35 Go Back Full Screen Close Quit
Need for Data Processing Need to Take . . . 3. Taking Uncertainty into Account (cont-d) Case of Interval . . . • This is definitely important: Case of Fuzzy Uncertainty How to Describe the . . . y = − 2 ◦ C, – if we predict tomorrow’s temperature as � Formulation of the . . . and the accuracy of this prediction is ± 1 ◦ , Answer to the First . . . – then we know that tomorrow will be freezing, with Analysis of the . . . the possibility of ice on the road, Answering the Second . . . – so we need to send a warning to the public, put Home Page sand (or salt) on the roads, etc. Title Page • On the other hand, if the accuracy is ± 10 degrees: ◭◭ ◮◮ – we may still alert the public, ◭ ◮ – but we better wait for more accurate information Page 4 of 35 before placing sand (or salt). Go Back Full Screen Close Quit
Need for Data Processing Need to Take . . . 4. Taking Uncertainty into Account (cont-d) Case of Interval . . . • This is even more important for a spaceship sent to Case of Fuzzy Uncertainty Mars; we want to make sure that: How to Describe the . . . Formulation of the . . . – with all the uncertainty taken into account, Answer to the First . . . – the spaceship will land in the desired Martian re- Analysis of the . . . gion. Answering the Second . . . Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 35 Go Back Full Screen Close Quit
Need for Data Processing Need to Take . . . 5. Case of Interval Uncertainty Case of Interval . . . • In many practical situations: Case of Fuzzy Uncertainty How to Describe the . . . – the only information that we have about the mea- Formulation of the . . . def surement errors ∆ x i = � x i − x i is Answer to the First . . . – the upper bound ∆ i on its absolute value: Analysis of the . . . | ∆ x i | ≤ ∆ i . Answering the Second . . . Home Page • In this case, once we know the measurement result � x i : Title Page – the only information that we have about the actual ◭◭ ◮◮ (unknown) values x i is ◭ ◮ – that x i belongs to the interval Page 6 of 35 def [ x i , x i ] = [ � x i − ∆ i , � x i + ∆ i ] . Go Back Full Screen Close Quit
Need for Data Processing Need to Take . . . 6. Case of Interval Uncertainty (cont-d) Case of Interval . . . • Different values x i from the corresponding intervals Case of Fuzzy Uncertainty lead, in general, to different values y = f ( x 1 , . . . , x n ). How to Describe the . . . Formulation of the . . . • In this case, we would like to find the range of all pos- Answer to the First . . . def sible values of y : [ y, y ] = f ([ x 1 , x 1 ] , . . . , [ x n , x n ]) = Analysis of the . . . { f ( x 1 , . . . , x n ) : x 1 ∈ [ x 1 , x 1 ] , . . . , x n ∈ [ x n , x n ] } . Answering the Second . . . Home Page • The problem of computing this range is known as the Title Page problem of interval computations . ◭◭ ◮◮ ◭ ◮ Page 7 of 35 Go Back Full Screen Close Quit
Need for Data Processing Need to Take . . . 7. Already for Interval Uncertainty, the Corre- Case of Interval . . . sponding Problem Is NP-Hard Case of Fuzzy Uncertainty • Sometimes, the function f ( x 1 , . . . , x n ) is linear: How to Describe the . . . n � Formulation of the . . . f ( x 1 , . . . , x n ) = a 0 + a i · x i . Answer to the First . . . i =1 Analysis of the . . . • Then, we have explicit formulas for the range: Answering the Second . . . Home Page n � y = � y − ∆ , y = � y + ∆ , where ∆ = | a i | · ∆ i . Title Page i =1 ◭◭ ◮◮ • However, already for quadratic functions f ( x 1 , . . . , x n ), ◭ ◮ the problem of computing the range [ y, y ] is NP-hard. Page 8 of 35 • This means that, if P � =NP (as most computer scientists believe) then: Go Back – no feasible algorithms is possible Full Screen – that would solve all particular cases of this problem. Close Quit
Need for Data Processing Need to Take . . . 8. Case of Fuzzy Uncertainty Case of Interval . . . • In many practical situations, in addition to the upper Case of Fuzzy Uncertainty bounds ∆ i on the measurement error: How to Describe the . . . Formulation of the . . . – experts also tell us which values from the corre- Answer to the First . . . sponding interval [ − ∆ i , ∆ i ] are more probable and Analysis of the . . . – which values are less probable. Answering the Second . . . • This information is usually given in terms of imprecise Home Page (“fuzzy”) words form natural langauge, such as: Title Page – “somewhat probable”, ◭◭ ◮◮ – “very probable”, etc. ◭ ◮ • We need to describe such knowledge in precise Page 9 of 35 computer-understandable terms. Go Back • For this purpose, Zadeh invented the technique of fuzzy Full Screen logic . Close Quit
Need for Data Processing Need to Take . . . 9. Case of Fuzzy Uncertainty (cont-d) Case of Interval . . . • In this technique: Case of Fuzzy Uncertainty How to Describe the . . . – to describe each imprecise property like “somewhat Formulation of the . . . probable”, Answer to the First . . . – we ask the expert to mark, on a scale from 0 to 10, Analysis of the . . . to what extent the corresponding value is possible. Answering the Second . . . • If an expert marks 7, we take 7/10 as the degree to Home Page which the corresponding value is possible. Title Page • As a result, in addition to the interval [ − ∆ i , ∆ i ], we ◭◭ ◮◮ also have: ◭ ◮ – for each value ∆ x i from this interval, Page 10 of 35 – a degree µ i (∆ x i ) to which this value is possible. Go Back • The function that assigns, to each value ∆ x i , the corre- Full Screen sponding degree, is known as the membership function . Close Quit
Need for Data Processing Need to Take . . . 10. Data Processing Under Fuzzy Uncertainty Case of Interval . . . • A value y is possible if y = f ( x 1 , . . . , x n ) for some tu- Case of Fuzzy Uncertainty ples for which: How to Describe the . . . Formulation of the . . . – x 1 is a possible value of the first input and Answer to the First . . . – x 2 is a possible value of the second inputs, etc. Analysis of the . . . • We know the degrees µ i ( x i ) to which each x i is a pos- Answering the Second . . . sible value of the i -th input. Home Page • We need to estimate the degree to which x 1 is possible Title Page and x 2 is possible, etc. ◭◭ ◮◮ • It is reasonable to use a corresponding “and”-operation ◭ ◮ & ( a, b ) (t-norm) of fuzzy logic, resulting in Page 11 of 35 f & ( µ 1 ( x 1 ) , . . . , µ n ( x n )) . Go Back • The simplest such operation is f & ( a, b ) = min( a, b ) , in Full Screen which case the corresponding inputs has the form Close min( µ 1 ( x 1 ) , . . . , µ n ( x n )) . Quit
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