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Mathematical Structures and Modeling 2022. N. 1 (61). PP. 115-123
UDC 519.254 DOI 10.24147/2222-8772.2022.1 .115-123
TOWARDS OPTIMAL TECHNIQUES INTERMEDIATE BETWEEN INTERVAL AND AFFINE, AFFINE AND TAYLOR
M. Ceberio
Student, e-mail: mceberio@utep.edu O. Kosheleva
Ph.D. (Phys.-Math.), Associate Professor, e-mail: olgak@utep.edu
V. Kreinovich Ph.D. (Phys.-Math.), Professor, e-mail: vladik@utep.edu
Abstract. In data processing, it is important to gauge how input uncertainty affects the results of data processing. Several techniques have been proposed for this gauging, from interval to affine to Taylor techniques. Some of these techniques result in more accurate estimates but require longer computation time, others' results are less accurate but can be obtained faster. Sometimes, we do not have enough time to use more accurate (but more time-consuming) techniques, but we have more time than needed for less accurate ones. In such cases, it is desirable to come up with intermediate techniques that would utilize the available additional time to get somewhat more accurate estimates. In this paper, we formulate the problem of selecting the best intermediate techniques, and provide a solution to this optimization problem.
Keywords: data processing, interval method, optimal method, Taylor method.
1. Formulation of the Problem
Interval, affine, and Taylor techniques: reminder. In many practical problems, we need to estimate the value of a quantity y based on the values of the quantities x\,...,xn on which y depends in a known way, as y = f (xi,... ,xn) for a known algorithm f (xx,..., xn).
The problem is that we do not know the exact values of the quantities Xi, all we know are the results x^ of measuring x^ and these results are, in general different from the actual values of the corresponding quantities: there is usually a non-zero measurement error Ax; oc^csee, e.g., [7]. Often, the only information that we have about each of these measurement error is the upper bound A, on its absolute value: |Ax,| ^ A,. In this case, the only information that we have about the actual (unknown) value Xi is that this value belongs to the interval [xi — Ai,Xi + Aj], In such situations, it is desirable not only to compute the value y = f (xi,..., xn), but also to find the range of possible values of y:
University of Texas at El Paso, El Paso, USA
{f (xi,... ,xn) : Xi E [xí - Ai,Xi + Aj ]}.
One of the natural ideas for computing this range is to take into account that computing y consists of several computational steps. So, on each of these steps, we do not only compute the corresponding intermediate result z, but we also keep some information about the dependence of this result on Xi, information that will eventually help us to find the desired range. There exist several implementations of this idea.
• In interval computations (see, e.g., [4-6]), for each intermediate result z, we keep an interval of possible values of z.
• In affine arithmetic (see, e.g., [2,3]), for each intermediate result z, we represent Az = z — z as the expression
Az = ^^ ai • Axi + 8z,
i=l
in which we know the coefficients ai and the upper bound A^ on the absolute value of the remaining term 8z: |£z| ^ A^.
• In the more general Taylor arithmetic (see, e.g., [1]), instead of a generic linear expression, we keep a generic polynomial expression of a given order k:
n n n
Az ah • Axh + ... + ...Yl
+ sz,
¿1 = 1 ¿1 = 1 ¿fc = 1
in which we know the coefficients ail...ij and the upper bound A^ on the absolute value of the remaining term 8z.
Then, for each elementary computational step - addition, subtraction, multiplication, etc. - we use expressions for this step's inputs to come up with a similar expression for the output of this step. For example, if we know that
n n
Az = ^^ ai • Axi + 8z and At = ^^ bi • Axi + 8t,
i=1 i=1
with |£z| ^ A^ and |ii| ^ At, then for s = z +1, we get
As = ^2(ai + bi) • Axi + 8s,
i=\
where |5s| ^ A^ + As.
Need for intermediate techniques. The more terms we keep in the dependence of Az on Axi, the more accurately we represent this dependence - after all, any continuous function on a bounded domain can be approximated by polynomials as accurately as possible, but the more accuracy we want, the more terms we need. On the other hand, the more terms we keep and process for each intermediate
result, the more memory we need and the more computation time we need - and both memory and computation time are often limited.
As of now, the usual choice is either go with interval computations, or use affine arithmetic, or use quadratic Taylor series, or use cubic Taylor series, etc. But what if we do not have enough time to use affine techniques but we still have extra time left when using intervals? In this case, it is desirable to use this extra time to come up with computations which are less time consuming that affine arithmetic, but more accurate than interval computations. Similarly, if we cannot afford quadratic Taylor series but we still have extra time left when using affine arithmetic, it is desirable to come up with computations which are less time consuming that quadratic Taylor technique, but more accurate than affine arithmetic.
Which intermediate techniques should we choose? There are many possible intermediate techniques. We can choose some monomials and only use their linear combinations. Alternatively, we can select some other basis in the linear space of all polynomials of given order, and use linear combinations of some elements of this basis.
In this paper, we show that the optimal choice is selecting monomials.
2. Analysis of the Problem
What we want. If we can only afford to have a limited number L of coefficients at each computation stage, then we need to represent the difference Az corresponding to each intermediate result as
where fl(Axi,..., Axn) are pre-selected analytical functions, and we know the coefficients aa and a bound A^ of the absolute value of the remainder 5z.
In this approach, we approximate each dependence of Az on Axi by a linear combination of the functions fl(Axi,..., Axn), i.e., by an element of the corresponding L-dimensional space
So, selecting an intermediate method means selecting an L-dimensional linear (sub)space in the linear space of all analytical functions.
What we mean by optimal. We want to select a subspace which is, in some reasonable sense, optimal. In some cases, optimal means attaining the largest or the smallest value of some objective function. However, optimality criteria can be more general. For example, if we select average approximation error as the objective function, we may end up with several different spaces with the same smallest possible value of this objective function. In this case, it is reasonable to select,
among them, the space that requires the smallest possible average computation time. This is equivalent to selecting an optimality criterion which is more complex than numerical: according to this criterion, a space A is better than a family A' if:
• either A has a smaller average approximation error,
• or they have the same average approximation error, but A' has a smaller average computation time.
We can have even more complex criteria. In general, what all these criteria do is for some pairs of alternatives A and A' that A is better - we will denote it by A < N - or that they are of equal quality with respect to this criterion; this we denote by A ~ A'. It is also possible that for some pairs, the criterion does not tell us which alternative is worse. Of course, these conclusions should be consistent: e.g., if A is better than A', and N is better than A", then A should be better than A".
What is important is that there should be exactly one alternative which is, according to this criterion, better than or of equal quality than all others. Indeed, as we have mentioned, if there are several optimal alternatives, this would mean that we can use the corresponding non-uniqueness to optimize something else -and thus, that the original optimality criterion is not final.
Scale-invariance. We process the values of physical quantities, but the numerical values of these quantities depend on the choice of a measuring unit. If we replace meters with centimeters, the lengths remain the same, but the numerical values of all the lengths become multiplied by c = 100. In general, if we select a different measuring unit for the quantity Xi, then its numerical value (and thus, the numerical value of the difference Axi = Xi — Xi) gets multiplied by the corresponding factor Ci > 0:
It is reasonable to assume that the relative quality of different approximation families do not depend on the choice of units. Indeed, it would be very strange if one family is better for meters and kilograms, and another is better for centimeters and grams.
Now, we are ready to formulate our main result.
3. Definition and the Main Result Definition 1. Let A be a set; its elements will be called alternatives.
• By an optimality criterion on the set S, we mean a pair of relations (<, that satisfy the following properties:
- if A < N and A' < A", then A < A!';
- if A < N and A' ~ A", then A < A!';
- if A ~ A' and N < A", then A < A!';
- if A ~ A' and N ~ A!', then A ~ A";
- always A — A; and
- if A < N then A — A' and N < A.
• We say that an alternative A is optimal for every A' e A, we have A < A' or A — A'.
• We say that an optimality criterion is final if there is exactly one optimal alternative.
Definition 2. Let A be the set of all L-dimensional linear subspaces of the linear space of all analytical functions. We say that the optimality criterion is scale-invariant if for all tuples c = (cl,...,cn) of positive numbers, we have A<A' & SC(A) < SC(A') and A — A' & SC(A) — SC(A'), where
Proposition. For every scale-invariant final optimality criterion, the optimal linear space is the set of all linear combinations of given L monomials.
Comment. In other words, the optimal method between interval and affine means selecting L <n variables il}... ,il, and considering expressions
The optimal method between affine and quadratic Taylor methods means selecting L — n pairs (ii,ji), and considering expressions
Proof of the Proposition. Let us first prove that the optimal space A0pt is itself scale-invariant, i.e., that Tc(Aopt) = Aopt for all c. Indeed, by definition of optimality, for every A', we have Aopt < A' or ^opt —
A'. This is true for all A', in
particular, for A' = Tc-i(A), where (cl,... ,cn)-1 =f (c-l,... ,c-1). By using scale-invariance, from Aopt < Tc-i(A), we conclude that Tc(Aopt) < Tc(Tc-i(A)) = A, and from Aopt — Tc-i(A), we conclude that Tc(Aopt) — Tc(Tc-i(A)) = A. Thus, for each alternative A, we have either Tc(Aopt) < A or Tc(Aopt) — A. By definition of an optimal alternative, this means that the alternative Tc(Aopt) is optimal. But our optimality criterion is final, which means that there is only one optimal alternative, and therefore, Tc(Aopt) = Aopt.
Each function from the basis of the optimal family is an analytical function, i.e., a sum - finite or infinite - of monomials, i.e., of the expressions of the type
SC(A) = [f (Cl • Axu ...,cn • Axn) : f (Axu..., Axn) e A}.
etc.
(Ax1)kl • ... • (Axn)kn. Let m1 be the smallest possible value of k1 in all L basic functions. Then, the function containing a non-zero term with x™1 has the form
fe (Ax1, A2,..., Axn) = (Ax1)m1 • P1 (Ax2 ,..., Axn) + (Ax1)mi+1 • P2 (Ax2 ,..., Axn) +
where Pk are polynomials and P1 is not identically 0. Due to scale-invariance, for each c1, the function
fe(c1 • Ax1, A2,..., Axn) =
c? • (Ax1)mi • P1(Ax2, ..., Axn) + c?+1 • (Ax1)mi+1 • P2(Ax2,..., Axn) + ... also belongs to the space Aopt, and thus, the function
c-m • f£(c1 • AX1, A2,..., Axn) = (Ax1)mi • P1(Ax2, ..., Axn) + C1 • (Ax1)mi+1 • P2(Ax2,..., Axn) + ...
A finite-dimensional linear space is closed, i.e., contains all its limits. In particular, in the limit c1 ^ 0, we conclude that the space L contains the function
(Ax1)mi • P1(Ax2,..., Axn).
Similarly, by considering the smallest possible power of Ax2 in this expression and using scale-invariance, we conclude that the optimal linear space contains a function (Ax1)mi • (Ax2)m2 • Q1(Ax3,..., Axn), etc., and in the end, that the optimal linear space contains a monomial (Ax1)mi • (Ax2)m2 •... • (Axn)mn.
By subtracting terms proportional to this monomial from all the basic functions, we thus get a new basis, in which we can also select a monomial, etc. At the end, we indeed get a representation of the optimal linear space as the set of all linear combinations of L monomials.
The proposition is proven.
How we can implement this idea. In the case of techniques intermediate between interval and affine, we can select the variables Xi for which the initial uncertainty is the largest.
Alternatively, at each step like computing s = z +1, we can first combine all 2L terms from both expressions for z and for t, and then keep L of them with the largest uncertainty - i.e., the largest values of the corresponding term |a»| • Aj.
Acknowledgments
This work was supported in part by the National Science Foundation grants 1623190 (A Model of Change for Preparing a New Generation for Professional Practice in Computer Science), and HRD-1834620 and HRD-2034030 (CAHSI Includes), and by the AT&T Fellowship in Information Technology.
It was also supported by the program of the development of the Scientific-Educational Mathematical Center of Volga Federal District No. 075-02-2020-1478, and by a grant from the Hungarian National Research, Development and Innovation Office (NRDI).
The authors are thankful to Christoph Lauter for valuable discussions.
References
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2. de Figueiredo L.H. and Stolfi J. Self-Validated Numerical Methods and Applications. IMPA, Rio de Janeiro, 1997.
3. Hansen E.R. A generalized interval arithmetic. In: Nickel K. (ed.), Interval Mathematics, Springer Lecture Notes in Computer Science, 1975, vol. 29, pp. 7-18.
4. Jaulin L., Kiefer M., Didrit O., and Walter E. Applied Interval Analysis, with Examples in Parameter and State Estimation. Robust Control, and Robotics, Springer, London, 2001.
5. Mayer G. Interval Analysis and Automatic Result Verification. de Gruyter, Berlin, 2017.
6. Moore R.E., Kearfott R.B., and Cloud M.J. Introduction to Interval Analysis. SIAM, Philadelphia, 2009.
7. Rabinovich S.G. Measurement Errors and Uncertainty: Theory and Practice. Springer Verlag, New York, 2005.
НА ПУТИ К ОПТИМАЛЬНЫМ МЕТОДАМ, ПРОМЕЖУТОЧНЫМ МЕЖДУ ИНТЕРВАЛЬНЫМ И АФФИННЫМ, АФФИННЫМ И ТЕЙЛОРОВСКИМ
М. Цеберио
студент, e-mail: mceberio@utep.edu O. Кошелева к.ф.-м.н., доцент, e-mail: olgak@utep.edu В. ^ейнович
к.ф.-м.н., профессор, e-mail: vladik@utep.edu
Техасский университет в Эль-Пасо, Эль-Пасо, США
Аннотация. При обработке данных важно оценить, как неопределенность ввода влияет на результаты обработки данных. Для этой калибровки было предложено несколько методов, от интервальных до аффинных и методов Тейлора. Некоторые из этих методов дают более точные оценки, но требуют больше времени для вычислений, другие результаты менее точны, но могут быть получены быстрее. Иногда у нас не хватает времени, чтобы использовать более точные (но более трудоёмкие) методы, но у нас есть больше времени, чем нужно, для менее точных. В таких случаях желательно разработать промежуточные методы, которые позволили бы использовать имеющееся дополнительное время для получения несколько более точных оценок. В данной статье мы формулируем задачу выбора наилучших промежуточных методов и даём решение этой задачи оптимизации.
Ключевые слова: обработка данных, интервальный метод, оптимальный метод, метод Тейлора.
Дата поступления в редакцию: 14.02.2022
