How to Represent Measurement Errors ?

What is the set of possible values of a measurement error? In the majority of practical applications, an error is caused not by a single cause; it is caused by a large number of independent causes, each of which adds a small component to the total error. As a result, we get a geometric description of the area of possible values of error. In this paper, we formulate the known result (for 1-dimensional case, when this area is known to be an interval), and a hypothesis for multi-dimensional case (where it is conjectured to be a convex set). 1. FORMULATION OF A REAL-LIFE PROBLEM Suppose that we have a measuring device that measures the values of one or several physical quantities x1, x2, ..., xk (e.g., current, voltage, frequency, etc). A measurement cannot be absolutely accurate, there are always some sources of error. As a result, the measured values x̃j differ from the actual ones xj , and the errors ej = x̃j − xj are different from 0. For each measuring instrument, possible values of the error vector e⃗ = (e1, ..., ek) form an area in k−dimensional space. In 1-dimensional case, it is usually an interval. In multidimensional case, it is usually either a parallelotope, or an ellipsoid (see, e.g., [Fuller 1987] and [Rabinovich 1993]). The main question that will be addressed in this paper is as follows: What other areas can be thus represented? 1 This work was partially supported by an NSF grant No. CDA9015006, NASA Grant No. 9–482, and Grant No. PF90–018 from the General Services Administration (GSA), administered by the Materials Research Institute and the Institute of Manufacturing and Materials Management. 2. HOW TO FORMULATE THIS PROBLEM IN GEOMETRIC TERMS? To find an answer to the above-formulated question, we can take into consideration the fact that an error is usually caused by a large number of different factors. Therefore, this error e⃗ is a sum of the large number n of small independent component vectors: e⃗ = e⃗1 + ...+ e⃗n. What do we mean by “small”, and what do we mean by “independent”? (Let’s recall that we are talking about geometry here, not about statistics, so independence has to be defined). Let us denote the set of all possible values of a component e⃗i by Ei. • It is natural to understand “small” as follows: some number δ > 0 is fixed, and a component is small (or, to be more precise, δ−small) if all its possible values do not exceed δ, i.e., if ∥e⃗i∥ ≤ δ for all e⃗i ∈ Ei. • In order to understand independence, let us first give an example when two error components are e⃗i and e⃗j are not independent. For example, they may be mainly caused by the same factor and must therefore be α−close for some small α. Then, for a given value of e⃗i, the corresponding set of possible values of e⃗j is the ball of radius δ with a center in e⃗i, and is thus different for different e⃗i. So, it is natural to call the components e⃗i and e⃗j independent if the set of possible values of e⃗j does not depend on the value of e⃗i. In other words, this means that all pairs (e⃗i, e⃗j), where e⃗i ∈ Ei and e⃗j ∈ Ej , are possible. Therefore, the set of all possible values of the sum e⃗i + e⃗j coincides with the set {e⃗i + e⃗j : e⃗i ∈ Ei, e⃗j ∈ Ej}, i.e., with the sum Ei + Ej of the two sets Ei and Ej . Before we turn to formal definitions, let’s show that if the set of all possible values of an error is not closed, we will never be able to find that out. Indeed, suppose that E is not closed. This means that there exists a value e⃗ that belongs to the closure of E, but does not belong to E itself. In every test measurement, we measure error with some accuracy δ. Since e⃗ belongs to the closure of E, there exists a value e⃗ ′ ∈ E such that ∥e⃗ ′ − e⃗∥ ≤ δ. So, if the actual error is e⃗ ′ (and e⃗ ′ ∈ E, and is thus a possible value of an error), we can get e⃗ as a result of measuring that error. So, no matter how precisely we measure errors, e⃗ is always possible. Therefore, we will never be able to experimentally distinguish between the cases when e⃗ is possible and when it is not. In view of that, to add the limit point e⃗ to E or not to add is purely a matter of convenience. Usually, the limit values are added. For example, in 1-D case, we usually consider closed intervals [−ε, ε] as sets of possible values of error [Fuller 1993], [Rabinovich 1993]. In view of that, we will assume that the sets E and Ei are closed. Now, we are ready for formal definitions. 3. FORMAL DEFINITIONS AND WHAT IS CURRENTLY KNOWN By a sum A+B of two sets A,B ⊆ R, we understand the set {a⃗+ b⃗ : a⃗ ∈ A, b⃗ ∈ B}. For a given δ > 0, a set A is called δ−small if ∥a⃗∥ ≤ δ for all a⃗ ∈ A. By a distance ρ(A,B) between sets A and B, we will understand Hausdorff distance (so, for sets, terms like “δ−close” will mean δ−close in the sense of ρ). Comment. For reader’s convenience, let us reproduce the definition of Hausdorff distance: ρ(A,B) is the smallest real number δ for which the following two statements are true: • for every a⃗ ∈ A, there exists a b⃗ ∈ B such that ∥a⃗− b⃗∥ ≤ δ; • for every b⃗ ∈ B, there exists an a⃗ ∈ A such that ∥a⃗− b⃗∥ ≤ δ. We are interested in describing sets that can be represented as E = E1 + ...+En for δ−small Ei. Such sets are fully described for 1-dimensional case (k = 1). Namely, in that case, the following results are true (see, e.g., [Kreinovich 1993]): PROPOSITION 1. If E = E1 + ... + En is a sum of δ−small closed sets from R, then E is δ−close to an interval. PROPOSITION 2. If E ⊆ R is a bounded set, and for every δ > 0, E can be represented as a finite sum of δ−small closed sets, then E is an interval. For reader’s convenience (and in the hope that they will help to solve the open problems), the proofs are included in the Appendix (they are very simple).