How interval measurement uncertainty affects the results of Data processing: a calculus-based approach to computing the range of a box Текст научной статьи по специальности «Компьютерные и информационные науки»

Mathematical Structures and Modeling 2018. N. 2(46). PP. 118-124

UDC 53.08 DOI: 10.25513/2222-8772.2018.2.118-124

HOW INTERVAL MEASUREMENT UNCERTAINTY AFFECTS THE RESULTS OF DATA PROCESSING: A CALCULUS-BASED APPROACH TO COMPUTING THE RANGE OF A BOX

Andrew Pownuk

Ph.D. (Phys.-Math.), e-mail: ampownuk@utep.edu Vladik Kreinovich

Ph.D. (Phys.-Math.), Professor, e-mail: vladik@utep.edu

University of Texas at El Paso, El Paso, Texas 79968, USA

Abstract. In many practical applications, we are interested in the values of the quantities yi,...,ym which are difficult (or even impossible) to measure directly. A natural idea to estimate these values is to find easier-to-measure related quantities x1,...,xn and to use the known relation to estimate the desired values yj. Measurements come with uncertainty, and often, the only thing we know about the actual value of each auxiliary quantity Xi is that it belongs to the interval [xi,xi] = \x% — Ai,Xi+Aj], where Xi is the measurement result, and Aj is the upper bound on the absolute value of the measurement error Xi — Xi. In such situations, instead of a single value of a tuple y = (y1,... ,ym), we have a range of possible values. In this paper, we provide calculus-based algorithms for computing this range.

Keywords: Data processing, interval uncertainty, indirect measurements, calculus.

1. Formulation of the Problem

Need for indirect measurements. In many practical situations, we are interested in the values of the quantities y1,... ,ym which are difficult — or even impossible — to measure directly. Since we cannot measure these quantities directly, a natural idea is to measure them indirectly (see, e.g., [6]), i.e.:

• to measure related quantities xi,...,xn which are related to the desired quantities yj by known relations, and

to use appropriate algorithms to find the values of the desired quantities:

yi = fi (xi,...,xn);

V2 = /2(xi,... ,xn);

Comment. In the real world, the relations are usually smooth; see, e.g., [1,7].

Need to take into account measurement uncertainty. If we knew the exact values xi,... ,xn of all the auxiliary quantities, then, by using the relations (1), we would be able to find the exact values of all the desired quantities yi,... ,ym.

In practice, however, measurements are never absolutely precise. The measurement result Xi is, in general, different from the actual (unknown) values of the corresponding quantity. When we plus in values cci = xi into the formula (1), we, in general, get the values yj = f)(xi,... ,xn) which are, in general, different from the desired values yj. How can we gauge the resulting uncertainty in yj ?

Case of interval measurement uncertainty. In many practical situations,the only information that we have about the measurement error Axi — 00i Xi is the upper bound Ai provided by the manufacturer of the corresponding measuring instrument. (If the manufacturer provide no such bound, then it is not a measuring instrument, it is a device for producing wild guesses.)

In this case, once we know the measurement result xi, the only information we have about the actual value xi is that it is somewhere on the interval [xi,xi], where xi =f — Ai and xi =f + Ai; see, e.g., [2,4-6].

There is no a priori known relation between the values x,, so the set of all possible values of Xi should not depend on the values of all other quantities Xj, j = i. Thus, the set of all possible values of the tuple x = (xi,... ,xn) is the box

[xi, Xi] x ... x [x_n, xn]. (2)

Resulting problem. Once we know that x belongs to the box (2), what are the possible values of the tuple y = (yx,... ,ym)? In mathematical terms, what is the range of the box (2) under the mapping (1)?

In this paper, we describe calculus-based techniques for solving this problem.

2. Analysis of the Problem and the Resulting Algorithms

Simplest case when we have only one desired quantity yx: analysis of the problem. Let us start with the simplest case, when we have only one desired quantity yi. In this case, we are interested in the range of the function fi(xi,... ,xn) when each Xi is in the corresponding interval [xi,xi]. For smooth (even for continuous) functions, this range is connected and is, thus, an interval [yi,yi], where:

• y is the smallest possible value of the function fi(xi,... ,xn) on the given box, and

• yi is the largest possible value of the function fi(xi,... ,xn) on the given box.

For each variable x^ the maximum (or minimum) of the expression yi = fi(x,..., xn) is attained:

either at one of the endpoints of this interval, i.e., for

• or inside the corresponding interval (xi,xi).

According to calculus, if the maximum or minimum is attained inside an interval,

d f

then the corresponding derivative is equal to 0. So, for each i, it is sufficient

OXi

to consider three possible cases:

• the case when xi =

• the case when xi = xi, and

d f

• the case when —1 = 0.

OXi _

Thus, to find the minimum y and the maximum yi of the function yi = fi(xi,...,xn) over the box, it is sufficient to consider all possible combinations of these 3 cases.

In other words, we arrive at the following algorithm.

Case when we have only one desired quantity yi: algorithm. Consider all systems of equations, in which, for each i, we have one of the three alternatives:

d f

Xi = Xi, Xi = Xi, and = 0. There are 3n such systems.

OXi

For each of these systems, we find the corresponding values x = (xi,...,xn) and compute the corresponding value yi = f (xi}... ,xn). The largest of thus computed values is yi, the smallest is y

Comment. This algorithm requires solving an exponential number of systems and thus takes exponential time. This is, however, unavoidable, since it is known that already for quadratic functions fi(xi,... ,xn), the problem of computing the bounds y and y is NP-hard; see, e.g., [3]. This means that, unless P=NP (which most computer scientists believe to be impossible), super-polynomial (e.g., exponential) computation time is unavoidable — at least for some inputs.

Exponential time does not mean that the algorithm is not practical — for reasonably small n, solving 3n system is quite reasonable. For example, for n = 10, we need to solve less than 60,000 systems, it is a large number, but it is quite doable. For n = 15, we need to solve about 5 million systems — still possible.

What we plan to do next. In the following subsections, we show how we can extend this calculus-based approach to the general case, and thus reduce the difficult-to-solve problem of finding the range to more well-studied problems of solving systems of equations.

Case when the number m of desired quantities is equal to the number n of auxiliary ones: analysis of the problem. To find the range means to find its border. At almost all points on the border, there is — locally — at least one tangent plane. A plane in an m-dimensional space has the form

m

• yi = c°.

3 = i

Thus, at this border point y = (yi,... ,ym), the linear expression

m m

y =Y1 Ci • yi = f (xi' ...,Xn) d= YI Ci • h (xi'...' Xn) j=i j=i

attains its local maximum or local minimum.

Similarly to the previous case, this may mean that one of the inputs x, attains its largest possible value Xi or its smallest possible value Xi = x,. In this case, the corresponding condition ^ci ^cï or ^ci ^ci determines the (n — 1)-dimensional set — which could be part of the border.

It may also means that the maximum or minimum of the linear function is attained when all the values x, are inside the corresponding intervals. In this case, we get

K

dxi

0

for all i, i.e., we get

3=1

dfl dxi

for all i.

In algebraic terms, the existence of non-always-zero values Cj that satisfy the above equality for all i means that m = n gradient vectors

(—1

\dxi '' ' ' ' dxn)

,dxi

that correspond to different j are linearly dependent. According to linear algebra

df-

this is equivalent to requiring that the determinant of the Jacobian matrix j

is equal to 0:

dxi

det

dfi

dxi

0.

(3)

So, we arrive at the following algorithm.

Case when the number m of desired quantities is equal to the number n of auxiliary ones: algorithm. To find the border of the desired range, for each i from 1 to m = n, we form two systems of equations:

• the system (1) in which we substitute Xi = x^ and

• the system (1) in which we substitute Xi = Xi.

Each of these systems provides a set of co-dimension 1 that could potentially serve as part of the border of the desired set.

To these possible border sets, we add the set corresponding to the equation (3). This equation defined a set of co-dimension 1, and plugging this set into (1), we can a y-set of co-dimension one - which can also be part of the border.

We know that the actual border can contain only segments of the above type, so once we have computed all these segments, we can reconstruct the border.

General case: analysis of the problem. We have already considered the case when m = n. There are two remaining cases: when n < m and when m < n.

When n < m, the set of all possible values of the tuple y is of of smaller dimension than the m, so this set is its own boundary.

0

j

Let us now consider the case when m < n. In this case, also, some linear combination

attains its maximum or its minimum. Let v denote the number of inputs xi for which at this maximum-or-minimum point, we have xi = xi or xi = xi. For each of the remaining n — v variables xi; we then have the equation

This equality (4) must hold for all (n — v) values of i, so we must have (n — v) equations.

We can select one of the values Cj equal to 1, then the other m — 1 values of Cj can be determining if we consider the first m — 1 conditions (4) as a system of linear equations with m — 1 unknowns. Substituting these values for Cj into the remaining n — v — (m — 1) equalities (4), we thus get n — v — (m — 1) equalities that relate n — v unknowns.

In general, each additional equality imposed on elements of a set decreases its dimension by 1. For example, in the 3-D space:

the set of all the points that satisfy a certain equality is usually a 2-D surface, the set of points that satisfy two independent equalities is a 1-D line, etc.

In our case, the dimension of the set of all the (n — w)-dimensional tuples x that satisfy all n — v — (m — 1) equalities is equal to the difference

The image of this (m — 1)-dimensional set under the transformation (1) is also (m— 1)-dimensional, so it forms a surface in the m-dimensional space of all possible tuples y = (yi,...,ym).

As a result, we get the following algorithm.

General case: algorithm. We consider all possible subsets I of the set {1,... ,n] of all indices of the inputs xi. For each such subset I of size v, we consider all 2V possible combinations of values xi and xi.

For each such combination, we consider the system of equations (4) for all i I. We can set up one of the values Cj to 1 and the first m — 1 equations (4) to describe Cj as a function of xi,...,xm. Substituting the resulting expressions for Cj in terms of xi into the remaining n — v — (m — 1) equalities (4), we get a (m — 1)-dimensional set of tuples x. Substituting this set of tuples into the formula (1), we get a (m — 1)-dimensional set of y-tuples.

We thus get several (m — 1)-dimensional sets, and we know that the actual border can only consist of the above fragments.

m

3 = 1

(4)

(n — v) — (n — v — (m — 1)) = m — 1.

Acknowledgments

This work was supported in part by the National Science Foundation grant HRD-1242122 (Cyber-ShARE Center of Excellence).

The authors are thankful to all the participants of the 2017 UTEP/NMSU Workshop on Mathematics, Computer Science, and Computation Science (El Paso, Texas, November 4, 2017) for valuable discussions.

References

1. Feynman R., Leighton R., Sands M. The Feynman Lectures on Physics. Addison Wesley, Boston, Massachusetts, 2005.

2. Jaulin L., Kiefer M., Didrit O., Walter E. Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control, and Robotics. Springer, London, 2001.

3. Kreinovich V., Lakeyev A., Rohn J., Kahl P. Computational Complexity and Feasibility of Data Processing and Interval Computations. Kluwer, Dordrecht, 1998.

4. Mayer G. Interval Analysis and Automatic Result Verification. De Gruyter, Berlin, 2017.

5. Moore R.E., Kearfott R.B., Cloud M.J. Introduction to Interval Analysis. SIAM, Philadelphia, 2009.

6. Rabinovich S.G. Measurement Errors and Uncertainty: Theory and Practice. Springer Verlag, Berlin, 2005.

7. Thorne K.S., Blandford R.D. Modern Classical Physics: Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics. Princeptn University Press, Princeton, New Jersey, 2017

КАК НЕОПРЕДЕЛЁННОСТЬ ИЗМЕРЕНИЯ ИНТЕРВАЛА ВЛИЯЕТ НА РЕЗУЛЬТАТЫ ОБРАБОТКИ ДАННЫХ: ВЫЧИСЛИТЕЛЬНЫЙ ПОДХОД ДЛЯ РАСЧЁТА ОБЛАСТИ ЗНАЧЕНИЙ ПРЯМОУГОЛЬНИКА

А. ^внук

к.ф.-м.н., ст. преподаватель, e-mail: ampownuk@utep.edu В. Крейнович

к.ф.-м.н., профессор, e-mail: vladik@utep.edu

Техасский университет в Эль Пасо, США

Аннотация. Во многих практических приложениях нас интересуют значения величин ух,..., ут, которые трудно (или даже невозможно) измерить непосредственно. Естественная идея оценить эти значения — найти более лёгкие для оценки величины хх,...,хп и использовать известное отношение для оценки желаемых значений yj. Измерения проходят с неопределённостью, и часто единственное, что мы знаем о фактическом значении каждой вспомогательной величины xit — это то, что оно принадлежит интервалу [xi,xi] = [х^ — Ai,Xi + Aj], где x^ — результат измерения, а Aj — верхняя граница по абсолютной величине ошибки измерения

а^ — Хг. В таких ситуациях вместо одного значения кортежа у = (у1,... ,ут) мы имеем диапазон возможных значений. В этой статье мы предлагаем вычислительные алгоритмы для расчёта этого диапазона.

Ключевые слова: обработка данных, интервальная неопределённость, косвенные измерения, вычисления.

Дата поступления в редакцию: 25.04.2018