Which Distributions (or Families of Continuous . . . Distributions) - PowerPoint PPT Presentation

Interval Uncertainty Is . . . Data Processing . . . Interval Data . . . Family of Distributions . . . Which Distributions (or Families of Continuous . . . Distributions) Best Represent Example: Estimating . . . Interval Uncertainty: Case of Maximum Entropy . . . Permutation-Invariant Criteria Beyond the Uniform . . . Let Us Use Symmetries Michael Beer 1 , Julio Urenda 2 Home Page Olga Kosheleva 2 , and Vladik Kreinovich 2 Title Page 1 Leibniz University Hannover ◭◭ ◮◮ 30167 Hannover, Germany beer@irz.uni-hannover.de ◭ ◮ 2 University of Texas at El Paso 500 W. University Page 1 of 36 El Paso, Texas 79968, USA Go Back jcurenda@utep.edu, olgak@utep.edu, vladik@utep.edu Full Screen Close Quit

Interval Uncertainty Is . . . Data Processing . . . 1. Interval Uncertainty Is Ubiquitous Interval Data . . . • An engineering designs comes with numerical values of Family of Distributions . . . the corresponding quantities, be it: Continuous . . . Example: Estimating . . . – the height of ceiling in civil engineering or Maximum Entropy . . . – the resistance of a certain resistor in electrical en- Beyond the Uniform . . . gineering. Let Us Use Symmetries • Of course, in practice, it is not realistic to maintain the Home Page exact values of all these quantities. Title Page • We can only maintain them with some tolerance. ◭◭ ◮◮ • As a result, the engineers: ◭ ◮ – not only produce the desired (“nominal”) value x Page 2 of 36 of the corresponding quantity, Go Back – they also provide positive and negative tolerances Full Screen ε + > 0 and ε − > 0. Close Quit

Interval Uncertainty Is . . . Data Processing . . . 2. Interval Uncertainty Is Ubiquitous (cont-d) Interval Data . . . • The actual value must be in the interval x = [ x, x ], Family of Distributions . . . def def Continuous . . . where x = x − ε − and x = x + ε + . Example: Estimating . . . • All the manufacturers need to do is to follow these Maximum Entropy . . . interval recommendations. Beyond the Uniform . . . • There is no special restriction on probabilities of dif- Let Us Use Symmetries ferent values within these intervals. Home Page • These probabilities depends on the manufacturer. Title Page ◭◭ ◮◮ • Even for the same manufacturer, they may change when the manufacturing process changes. ◭ ◮ Page 3 of 36 Go Back Full Screen Close Quit

Interval Uncertainty Is . . . Data Processing . . . 3. Data Processing Under Interval Uncertainty Is Interval Data . . . Often Difficult Family of Distributions . . . • Interval uncertainty is ubiquitous. Continuous . . . Example: Estimating . . . • So, many researchers have considered different data Maximum Entropy . . . processing problems under this uncertainty. Beyond the Uniform . . . • This research area is known as interval computations . Let Us Use Symmetries Home Page • The problem is that the corresponding computational problems are often very complex. Title Page • They are much more complex than solving similar prob- ◭◭ ◮◮ lems under probabilistic uncertainty: ◭ ◮ – when we know the probabilities of different values Page 4 of 36 within the corresponding intervals, Go Back – we can use Monte-Carlo simulations to gauge the Full Screen uncertainty of data processing results. Close Quit

Interval Uncertainty Is . . . Data Processing . . . 4. Interval Data Processing Is Difficult (cont-d) Interval Data . . . • A similar problem for interval uncertainty: Family of Distributions . . . Continuous . . . – is NP-hard already for the simplest nonlinear case Example: Estimating . . . – when the whole data processing means computing Maximum Entropy . . . the value of a quadratic function. Beyond the Uniform . . . • It is even NP-hard to find the range of variance when Let Us Use Symmetries inputs are known with interval uncertainty. Home Page • This complexity is easy to understand. Title Page • Interval uncertainty means that we may have different ◭◭ ◮◮ probability distributions on the given interval. ◭ ◮ • So, to get guaranteed estimates, we need, in effect, to Page 5 of 36 consider all possible distributions. Go Back • And this leads to very time-consuming computations. Full Screen • For some problems, this time can be sped up, but in general, the problems remain difficult. Close Quit

Interval Uncertainty Is . . . Data Processing . . . 5. It Is Desirable to Have a Family of Distribu- Interval Data . . . tions Representing Interval Uncertainty Family of Distributions . . . • Interval computation problems are NP-hard. Continuous . . . Example: Estimating . . . • In practical terms, this means that the corresponding Maximum Entropy . . . computations will take forever. Beyond the Uniform . . . • So, we cannot consider all possible distributions on the Let Us Use Symmetries interval. Home Page • A natural idea is to consider some typical distributions. Title Page • This can be a finite-dimensional family of distributions. ◭◭ ◮◮ • This can be even a finite set of distributions – or even ◭ ◮ a single distribution. Page 6 of 36 • For example, in measurements, practitioners often use Go Back uniform distributions on the corresponding interval. Full Screen • This selection is even incorporated in some interna- tional standards for processing measurement results. Close Quit

Interval Uncertainty Is . . . Data Processing . . . 6. Family of Distributions (cont-d) Interval Data . . . • Of course, we need to be very careful which family we Family of Distributions . . . choose. Continuous . . . Example: Estimating . . . • By limiting the class of possible distributions, we in- Maximum Entropy . . . troduce an artificial “knowledge”. Beyond the Uniform . . . • Thus, we modify the data processing results. Let Us Use Symmetries Home Page • So, we should select the family depending on what characteristic we want to estimate. Title Page • We need to beware that: ◭◭ ◮◮ – a family that works perfectly well for one charac- ◭ ◮ teristic Page 7 of 36 – may produce a completely misleading result when Go Back applied to some other desired characteristic. Full Screen • Examples of such misleading results are well known. Close Quit

Interval Uncertainty Is . . . Data Processing . . . 7. Continuous Vs. Discrete Distributions Interval Data . . . • Usually, in statistics and in measurement theory: Family of Distributions . . . Continuous . . . – when we say that the actual value x belongs to the Example: Estimating . . . interval [ a, b ], Maximum Entropy . . . – we assume that x can take any real value between Beyond the Uniform . . . a and b . Let Us Use Symmetries • However, in practice: Home Page – even with the best possible measuring instruments, Title Page – we can only measure the value of the physical quan- ◭◭ ◮◮ tity x with some uncertainty h . ◭ ◮ • Thus, from the practical viewpoint, it does not make Page 8 of 36 any sense to distinguish between a and a + h . Go Back • Even with the best measuring instruments, we will not Full Screen be able to detect this difference. Close Quit

Interval Uncertainty Is . . . Data Processing . . . 8. Continuous Vs. Discrete (cont-d) Interval Data . . . • From the practical viewpoint, it makes sense to divide Family of Distributions . . . the interval [ a, b ] into small subintervals Continuous . . . Example: Estimating . . . [ a, a + h ] , [ a + h, a + 2 h ] , . . . Maximum Entropy . . . • Within each of them the values of x are practically Beyond the Uniform . . . indistinguishable. Let Us Use Symmetries • It is sufficient to find the probabilities p 1 , p 2 , . . . , p n Home Page that the actual value x is in one of the subintervals: Title Page – the probability p 1 that x is in the first small subin- ◭◭ ◮◮ terval [ a, a + h ]; ◭ ◮ – the probability p 2 that x is in the first small subin- terval [ a + h, a + 2 h ]; etc. Page 9 of 36 • These probabilities should, of course, add up to 1: Go Back n Full Screen � p i = 1 . Close i =1 Quit

Interval Uncertainty Is . . . Data Processing . . . 9. Continuous Vs. Discrete (cont-d) Interval Data . . . • In the ideal case, we get more and more accurate mea- Family of Distributions . . . suring instruments – i.e., h → 0. Continuous . . . Example: Estimating . . . • Then, the corresponding discrete probability distribu- Maximum Entropy . . . tions will tend to continuous ones. Beyond the Uniform . . . • So, from this viewpoint: Let Us Use Symmetries Home Page – selecting a probability distribution means selecting a tuple of values p = ( p 1 , . . . , p n ), and Title Page – selecting a family of probability distributions means ◭◭ ◮◮ selecting a family of such tuples. ◭ ◮ Page 10 of 36 Go Back Full Screen Close Quit