Towards A Neural-Based Understanding of the Cauchy Deviate Method for Processing Interval and Fuzzy Uncertainty

One of the most efficient techniques for processing interval and fuzzy data is a Monte-Carlo type technique of Cauchy deviates that uses Cauchy distributions. This technique is mathematically valid, but somewhat counterintuitive. In this paper, following the ideas of Paul Werbos, we provide a natural neural network explanation for this technique. Keywords— Cauchy deviate method, fuzzy uncertainty, interval uncertainty, Monte-Carlo simulations, neural networks 1 Formulation of the Problem: Cauchy Deviate Method and Need for Intuitive Explanation 1.1 Practical Need for Uncertainty Propagation In many practical situations, we are interested in the value of a quantity y which is difficult or even impossible to measure directly. To estimate this difficult-to-measure quantity y, we measure or estimate related easier-to-measure quantities x1, . . . , xn which are related to the desired quantity y by a known relation y = f(x1, . . . , xn). Then, we apply the relation f to the estimates x̃1, . . . , x̃n for xi and produce an estimate ỹ = f(x̃1, . . . , x̃n) for the desired quantity y. In the simplest cases, the relation f(x1, . . . , xn) may be an explicit expression: e.g., if we know the current x1 and the resistance x2, then we can measure the voltage y by using Ohm’s law y = x1 · x2. In many practical situations, the relation between xi and y is much more complicated: the corresponding algorithm f(x1, . . . , xn) is not an explicit expression, but a complex algorithm for solving an appropriate non-linear equation (or system of equations). Estimates are never absolutely accurate: • measurements are never absolutely precise, and • expert estimates can only provide approximate values of the directly measured quantities x1, . . . , xn. In both cases, the resulting estimates x̃i are, in general, different from the actual (unknown) values xi. Due to these estimation errors ∆xi def = x̃i − xi, even if the relation f(x1, . . . , xn) is exact, the estimate ỹ = f(x̃1, . . . , x̃n) is different from the actual value y = f(x1, . . . , xn): ∆y def = ỹ − y 6= 0. (In many situations, when the relation f(x1, . . . , xn) is only known approximately, there is an additional source of the approximation error in y caused by the uncertainty in knowing this relation.) It is therefore desirable to find out how the uncertainty ∆xi in estimating xi affects the uncertainty ∆y in the desired quantity, i.e., how the uncertainties ∆xi propagate via the algorithm f(x1, . . . , xn). 1.2 Propagation of Probabilistic Uncertainty Often, we know the probabilities of different values of ∆xi. For example, in many cases, we know that the approximation errors ∆xi are independent normally distributed with zero mean and known standard deviations σi; see, e.g., [16]. In this case, we can use known statistical techniques to estimate the resulting uncertainty ∆y in y. For example, since we know the probability distributions, we can simulate them in the computer, i.e., use the Monte-Carlo simulation techniques to get a sample population ∆y, . . . , ∆y of the corresponding errors ∆y. Based on this sample, we can then estimate the desired statistical characteristics of the desired approximation error ∆y. 1.3 Propagation of Interval Uncertainty In many other practical situations, we do not know these probabilities, we only know the upper bounds ∆i on the (absolute values of) the corresponding measurement errors ∆xi: |∆xi| ≤ ∆. In this case, based on the known approximation x̃i, we can conclude that the actual (unknown) value of i-th auxiliary quantity xi can take any value from the interval xi = [x̃i −∆i, x̃i + ∆i]. (1) To find the resulting uncertainty in y, we must therefore find the range y = [y, y] of possible values of y when xi ∈ xi: y = f(x1, . . . ,xn) def = {f(x1, . . . , xn) |x1 ∈ x1, . . . , xn ∈ xn}. (2) Computations of this range under interval uncertainty is called interval computations; see, e.g., [4, 5]. The corresponding computational problems are, in general, NP-hard [9]. Crudely speaking, this means that, in general, such problems require a large amount of computation time – and that therefore faster methods are needed. 1.4 Propagation of Fuzzy Uncertainty In many practical situations, the estimates x̃i come from experts. Experts often describe the inaccuracy of their estimates in terms of imprecise words from natural language, such as “approximately 0.1”, etc. A natural way to formalize such words is to use special techniques developed for formalizing this type of estimates – specifically, the technique of fuzzy logic; see, e.g., [6, 15]. In this technique, for each possible value of xi ∈ xi, we describe the degree μi(xi) to which this value is possible. For each degree of certainty α, we can determine the set of values of xi that are possible with at least this degree of certainty – the α-cut xi(α) = {x |μ(x) ≥ α} of the original fuzzy set. Vice versa, if we know α-cuts for every α, then, for each object x, we can determine the degree of possibility that x belongs to the original fuzzy set [3, 6, 12, 13, 15]. A fuzzy set can be thus viewed as a nested family of its (interval) α-cuts. We already know how to propagate interval uncertainty. Thus, to propagate this fuzzy uncertainty, we can therefore consider, for each α, the fuzzy set y with the α-cuts y(α) = f(x1(α), . . . ,x1(α)); (3) see, e.g., [3, 6, 12, 13, 15]. So, from the computational viewpoint, the problem of propagating fuzzy uncertainty can be reduced to several interval propagation problems. 1.5 Need for Faster Algorithms for Uncertainty Propagation Summarizing the above analysis, we can conclude that in principle, we need to consider three basic types of uncertainty propagation: situations when we propagate probabilistic, interval, and fuzzy uncertainty. It is also possible that some quantities are represented by fuzzy sets, while others may be represented by probabilities. For probabilistic uncertainty, there exist reasonable efficient uncertainty propagation algorithms such as Monte-Carlo simulations. In contrast, the problems of propagating interval and fuzzy uncertainty are, in general, computationally difficult. It is therefore desirable to design faster algorithms for propagating interval and fuzzy uncertainty. Once such methods are developed, we can then use these methods to propagate interval and fuzzy uncertainty components, and Monte-Carlo simulations to propagate the probabilistic uncertainty. The computational problem of propagating fuzzy uncertainty can be naturally reduced to the problem of propagating interval uncertainty. Because of this reduction, in the following text, we will mainly concentrate on faster algorithms for propagating interval uncertainty. 1.6 Linearization Situations: Description Due to the approximation errors ∆xi = x̃i− xi, the unknown (actual) values xi = x̃i−∆xi of the input quantities xi are, in general, different from the approximate estimates x̃i. In many practical situations, the approximation errors ∆xi are small – e.g., when the approximations are obtained by reasonably accurate measurements. In such situations, we can ignore terms which are quadratic (and of higher order) in ∆xi. 1.7 Linearization Situations: Analysis In the above situations, we can expand the expression for ∆y = ỹ − y = f(x̃1, . . . , x̃n)− f(x1, . . . , xn) = f(x̃1, . . . , x̃n)− f(x̃1 −∆x1, . . . , x̃n −∆xn) (4) in Taylor series in ∆xi and keep only the linear terms in this expansion. In this case, we get ∆y = c1 ·∆x1 + . . . + cn ·∆xn, (5)