Need for Data . . . Need for Interval . . . Need for Interval . . . Sometimes, We do not . . . Practical Need for Algebraic Situation When We . . . (Equality-Type) Solutions of Sometimes, the Values . . . How to Find the Set A ? Interval Equations and for Independence What If There Is No . . . Extended-Zero Solutions Home Page Ludmila Dymova 1 , Pavel Sevastjanov 1 , Andrzej Pownuk 2 , and Title Page Vladik Kreinovich 2 ◭◭ ◮◮ 1 Institute of Computer and Information Science Czestochowa University of Technology ◭ ◮ Dabrowskiego 73, 42-200 Czestochowa, Poland sevast@icis.pcz.pl Page 1 of 28 2 Computational Science Program, University of Texas at El Paso Go Back El Paso, TX 79968, USA, ampownuk@utep.edu, vladik@utep.edu Full Screen Close Quit
Need for Data . . . Need for Interval . . . 1. Need for Data Processing Need for Interval . . . • We are often interested in the values of quantities Sometimes, We do not . . . y 1 , . . . , y m which are difficult to measure directly. Situation When We . . . Sometimes, the Values . . . • Examples: distance to a faraway star, tomorrow’s tem- How to Find the Set A ? perature at a certain location. Independence • Since we cannot measure these quantities directly, to What If There Is No . . . estimate these quantities we must: Home Page – find easier-to-measure quantities x 1 , . . . , x n which Title Page are related to y i by known formulas y i = ◭◭ ◮◮ f i ( x 1 , . . . , x n ), – measure these quantities x j , and ◭ ◮ – use the results � x j of measuring the quantities x j to Page 2 of 28 compute the estimates for y i : Go Back y i = f ( � x n ) . � x 1 , . . . , � Full Screen • Computation of these estimates is called indirect mea- Close surement or data processing. Quit
Need for Data . . . Need for Interval . . . 2. Need for Data Processing under Uncertainty Need for Interval . . . • Measurements are never 100% accurate. Sometimes, We do not . . . Situation When We . . . • Hence, the measurement result � x j is, in general, differ- Sometimes, the Values . . . ent from the actual (unknown) value x j . How to Find the Set A ? def • In other words, the measurement errors ∆ x j = � x j − x j Independence are, in general, different from 0. What If There Is No . . . Home Page • Because of this, the estimates � y i are, in general, differ- ent from the desired values y i . Title Page ◭◭ ◮◮ • It is therefore desirable to know how accurate are the resulting estimates. ◭ ◮ Page 3 of 28 Go Back Full Screen Close Quit
Need for Data . . . Need for Interval . . . 3. Need for Interval Uncertainty Need for Interval . . . • The manufacturer of the measuring instrument usually Sometimes, We do not . . . provides a bound ∆ j on the measurement error: Situation When We . . . Sometimes, the Values . . . | ∆ x j | ≤ ∆ j . How to Find the Set A ? • If no such bound is known, this is not a measuring Independence instrument, but a wild-guess-generator. What If There Is No . . . Home Page • Sometimes, we also know the probabilities of different values ∆ x j within this interval. Title Page • However, in many practical situations, the upper ◭◭ ◮◮ bound is the only information that we have; then: ◭ ◮ – after we know the result � x j of measuring x j , Page 4 of 28 – the only information that we have about the actual Go Back (unknown) value x j is that x j ∈ [ x j , x j ] , where: Full Screen def def x j = � x j − ∆ j and x j = � x j + ∆ j . Close Quit
Need for Data . . . Need for Interval . . . 4. Need for Interval Computations Need for Interval . . . • In this case, all we know about each x i is that Sometimes, We do not . . . Situation When We . . . x i ∈ [ x i , x i ] . Sometimes, the Values . . . How to Find the Set A ? • We also know that y i = f i ( x 1 , . . . , x n ). Independence • Then, the only thing that we can say about each value What If There Is No . . . y i = f i ( x 1 , . . . , x n ) is that y i is in the range Home Page { f i ( x 1 , . . . , x n ) : x 1 ∈ [ x 1 , x 1 ] , . . . , x n ∈ [ x n , x n ] } . Title Page ◭◭ ◮◮ • Computation of this range is one of the main problems of interval computations . ◭ ◮ Page 5 of 28 Go Back Full Screen Close Quit
Need for Data . . . Need for Interval . . . 5. Sometimes, We do not Know the Exact Depen- Need for Interval . . . dence Sometimes, We do not . . . • So far, we assumed that when we know the exact de- Situation When We . . . pendence y i = f i ( x 1 , . . . , x n ) between y i and x j . Sometimes, the Values . . . How to Find the Set A ? • In practice, often, we do not know the exact depen- dence. Independence What If There Is No . . . • Instead, we know that the dependence belongs to a Home Page finite-parametric family of dependencies, i.e., that Title Page y i = f i ( x 1 , . . . , x n , a 1 , . . . , a k ) for some parameters a 1 , . . . , a k . ◭◭ ◮◮ • Example: y i is a linear function of x j , i.e., � n ◭ ◮ y i = c i + c ij · x j for some c i and c ij . j =1 Page 6 of 28 • The presence of these parameters complicates the cor- Go Back responding data processing problem. Full Screen • Depending on what we know about the parameters, we Close have different situations. Quit
Need for Data . . . Need for Interval . . . 6. Specific Case: Control Solution Need for Interval . . . • Sometimes, we can control the values a ℓ , by setting Sometimes, We do not . . . them to any values within certain intervals [ a ℓ , a ℓ ]. Situation When We . . . Sometimes, the Values . . . • By setting the appropriate values of the parameters, How to Find the Set A ? we can change the values y i . Independence • We would like the values y i to be within some given What If There Is No . . . ranges [ y i , y i ]. Home Page • For example, we would like the temperature to be Title Page within a comfort zone. ◭◭ ◮◮ • So, we need to find x j for which, by applying controls ◭ ◮ a i ∈ [ a ℓ , a ℓ ], we can place each y i within [ y i , y i ]: Page 7 of 28 X = { x : ∃ a ℓ ∈ [ a ℓ , a ℓ ] ∀ i f i ( x 1 , . . . , x n , a 1 , . . . , a k ) ∈ [ y i , y i ] } . Go Back • This set is known as the control solution to the corre- Full Screen sponding interval system of equations f ( x, a ) = y . Close Quit
Need for Data . . . Need for Interval . . . 7. Situation When We Need to Find the Parame- Need for Interval . . . ters from the Data Sometimes, We do not . . . • Sometimes, we do not know these values a ℓ , we must Situation When We . . . determine these values from the measurements. Sometimes, the Values . . . How to Find the Set A ? • After each cycle c of measurements, we conclude that: Independence – the actual (unknown) value of x ( c ) is in the interval j What If There Is No . . . [ x ( c ) j , x ( c ) j ] and Home Page – the actual value of y ( c ) i , y ( c ) is in the interval [ y ( c ) i ]. Title Page i ◭◭ ◮◮ • We want to find the set A of all the values a for which y ( c ) = f ( x ( c ) , a ) for some x ( c ) and y ( c ) : ◭ ◮ A = { a : ∀ c ∃ x ( c ) ∈ [ x ( c ) j , x ( c ) j ] ∃ y ( c ) i , y ( c ) ∈ [ y ( c ) i ] ( f ( x ( c ) , a ) = y ( c ) ) } . Page 8 of 28 j i Go Back • This set A is known as the united solution to the inter- val system of equations. Full Screen Close Quit
Need for Data . . . Need for Interval . . . 8. Comment About Notations Need for Interval . . . • In general, in our description: Sometimes, We do not . . . Situation When We . . . • y denotes the desired quantities, Sometimes, the Values . . . • x denote easier-to-measure quantities, and How to Find the Set A ? • a denote parameters of the dependence between Independence these quantities. What If There Is No . . . Home Page • In some cases, we have some information about a , and we need to know x – case of the control solution. Title Page • In other cases, we have some information about x , and ◭◭ ◮◮ we need to know a – case of the united solution. ◭ ◮ • As a result, sometimes x ’s are the unknowns, and some- Page 9 of 28 times a ’s are the unknowns. Go Back Full Screen Close Quit
Need for Data . . . Need for Interval . . . 9. What Can We Do Once We Have Found the Need for Interval . . . Range of Possible Values of a Sometimes, We do not . . . • Once we have found the set A of possible values of a , Situation When We . . . we can find the range of possible values of y i : Sometimes, the Values . . . How to Find the Set A ? { f i ( x 1 , . . . , x n , a ) : x j ∈ [ x j , x j ] and a ∈ A } . Independence • This is a particular case of the main problem of interval What If There Is No . . . computations. Home Page • Often, we want to make sure that each value y i lies Title Page within the given bounds [ y i , y i ]. ◭◭ ◮◮ • Then we must find the set X of possible values of x for ◭ ◮ which f i ( x, a ) ∈ [ y i , y i ] for all a ∈ A : Page 10 of 28 X = { x : ∀ a ∈ A ∀ i ( f i ( x, a ) ∈ [ y i , y i ]) } . Go Back • This set is known as the tolerance solution to the in- Full Screen terval system of equations. Close Quit
Recommend
More recommend
