Corrections and Generalizations of the Absolute and Relative Errors

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

RUAG Aerospace Services GmbH, Germany

Mathematical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

6 (2006), 1

By estimating any inexact data, for the simplest formal (correct or not) equality a =? b with two real numbers, the absolute error |a - b| [1] alone offers no sufficient quality estimation giving, e.g., the same result 1 for the acceptable formal equality 1000 =? 999 and for the inadmissible one 1 =? 0. This error is not invariant by equivalent transformations of a problem because, for instance, when multiplying a formal equality by a nonzero number, the absolute error is multiplied by the modulus (absolute value) of that number. For the relative error [1], there are at once two propositions, namely to use either |a - b|/|a| or |a - b|/|b| as an estimating fraction, which is a generally inadmissible uncertainty that could be acceptable only if |a/b| is close to 1. Further the relative error should always belong to the segment [0, 1]. But for 1 =? 0 by choosing 0 as the denominator, the result is +∞, for 1 =? -1 by each denominator choice the result is 2. Hence, the relative error has a restricted range of applicability. By more complicated formal equalities with at least three elements, e.g., by 100 - 99 =? 0 or 1 - 2 + 3 - 4 =? -1, the choice of a denominator seems to be vague at all.

For general problem setting, let Z ⊆ X × Y be any given subset of the direct product of two sets X and Y and have a projection Z/X on X consisting of all x ∈ X really represented in Z, i.e., of all such x that for each of them there is a y ∈ Y such that (x, y) ∈ Z. Let further { y = F(x) } (x ∈ X, y ∈ Y) be a certain class of functions defined on X with range in Y. Then the graph of such a function is a curve in X × Y. The problem consists in finding (in this class) functions with graphs nearest to Z in a certain reasonable sense. To exactly fit this with a specific function y = F(x), the set Z has to be included in the graph of this function: Z ⊆ { (x, F(x)) | x ∈ X }, or, equivalently, F(x) = y for each x ∈ Z/X. But this inclusion (or equality) does not necessarily hold in the general case. Then it seems to be reasonable to estimate the error E( F(x) =? y | x ∈ Z/X ) of the formal equality (true or not true) F(x) =? y on this set Z/X via a certain error function E defined at least on Z/X. To suitably construct such a function, it seems to be reasonable to first consider two stages of its building: 1) defining local error functions to estimate errors at separate points x; 2) defining global error functions using the values of local error functions to estimate errors on the whole set Z/X. Possibly the simplest and most straightforward approach includes the following steps:

1) defining on Y × Y certain nonnegative functions r_yy’(y, y’) generally individual for different y, y’ and, e.g., similar to a distance [1] between any two elements y, y’ of Y (but not necessarily with holding the distance axioms [1]),

2) defining certain nonnegative functions R_x(r(F(x), y)) generally individual for different x,

summing (possibly including integrating) their values on Z/X, and

3) using this sum (possibly including integrals) as a nearness measure.

Unimathematics [2] proposes a unierror irreproachably correcting the relative error and generalizing it possibly for any conceivable range of applicability. For a =? b, the linear estimating fraction is δ_{a =? b} = |a - b|/(|a| + |b|) by |a| + |b| > 0, which should simply vanish by a = b = 0. Introduce extended division: a//b = a/b by a ≠ 0 and a//b = 0 by a = 0 independently of the existence and value of b. Then δ_{a =? b} = |a - b|//(|a| + |b|). The quadratic estimating fraction is ²δ_{a =? b} = |a - b|//[2(a² + b²)]^1/2. The outputs (return values) of such unierrors always belong to [0, 1]. By the principle of tolerable simplicity [2], it is reasonable to use the linear estimating fraction alone if it suffices. For a formal vector equality Σ_ω∈Ω z_ω=? 0,

δ(Σ_ω∈Ωz_ω=? 0) = ||Σ_ω∈Ωz_ω||//Σ_ω∈Ω||z_ω||, ²δ(Σ_ω∈Ω z_ω =? 0)= ||Σ_ω∈Ω z_ω||//((Q(Ω)Σ_ω∈Ω ||z_ω||²)^1/2

whose denominators contain all elements that have been initially in the equality, i.e., before any transformations. If all the vectors are replaced with numbers, the norms can be replaced with the moduli (absolute values). Examples:

δ_{100 - 99 =? 0} = 1/199 = δ_{100 =? 99}; δ_{1 - 2 + 3 - 4 =? -1}= |1 - 2 + 3 - 4 + 1|/(1 + 2 + 3 + 4 + 1) = 1/11.

The absolute error, the relative error, and the unierror of any exact object or model always vanish. It is often reasonable to additionally discriminate exact objects or models by the confidence in their exactness reliability. For example, both x₁ = 1 + 10^-10 and x₂ = 1 + 10¹⁰ are exact solutions to the inequation x > 1, x₁ practically unreliable and x₂ guaranteed. Their discrimination is especially important by any inexact data. Classical mathematics [1] cannot provide this at all. Elastic mathematics [2] proposes for this purpose the basic concept of the reserve which is quite new in mathematics and extends the unierror in the following sense. The values of a unierror H belong to the segment [0, 1], those of a reserve R to [-1, 1]. For each inexact object I, H(I) > 0 and we can take R(I) = -H(I). For each exact object E, H(E) = 0 and R(E) ≥ 0. A proposition to determine the reserve of an inexact object as its unierror with the opposite sign is at once evident. For an exact object, it seems to be reasonable, to first define a suitable mapping of the object with respect to its exactness boundary and to further take the unierror of the mapped object. It is exact if and only if the object itself precisely lies on its exactness boundary where the reserve vanishes. Otherwise, the mapped object is inexact and the object itself has a positive reserve. For inequalities, such a mapping can be replaced with negating inequality relations and conserving equality ones. In our example, we have

R_{x > 1} (x₁) = R_{x > 1} (1 + 10 ^-10) = H_{x <? 1} (1 + 10 ^-10) = 10^-10/(2 + 10 ^-10),

R_{x > 1} (x₂) = R_{x > 1} (1 + 10¹⁰) = H_{x <? 1} (1 + 10¹⁰) = 10¹⁰/(2 + 10¹⁰).

[1]         Encyclopaedia of Mathematics. Ed. M. Hazewinkel. Volumes 1 to 10. Kluwer Academic Publ., Dordrecht, 1988-1994

[2]         Lev Gelimson. Elastic Mathematics. General Strength Theory. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2004