Correction and Generalization of the Least Square Method

by

© Ph. D. & Dr. Sc. Lev Gelimson

Academic Institute for Creating Fundamental Sciences (Munich, Germany)

Mathematical Journal

of the "Collegium" All World Academy of Sciences

Munich (Germany)

8 (2008), 1

For any data processing, the least square method [1] by Legendre and Gauss is practically the unique known one applicable to contradictory problems. But this method is based on the absolute error not invariant by equivalent transformations of a problem and ignores the possibly noncoinciding physical dimensions (units) of relations in a problem. The method does not correlate the deviations of the desired approximation from the approximated objects with these objects themselves and simply mixes those deviations without their adequate weighing. The method considers equal changes of the squares of those deviations with relatively less and greater moduli (absolute values) as equivalent ones. It foresees no iterating, is based on a fixed algorithm accepting no a priori flexibility, and provides no own a posteriori adapting. These defects in the essence of the least square method result in many fundamental shortcomings in its applicability. The method loses any sense and is not applicable at all to problems simulated by a set of equations with different physical dimensions (units), e.g., to a mechanics problem with one equation via the impulse conservation law and another equation via the energy conservation law. There is no result invariance by equivalent transformations of a problem and no objective sense of the result along with possibly ignoring some part of a problem and paradoxical approximation. And the method presents each result as the highest truth without any justification.

For extending the least square method via setting a general problem, let Z ⊆ X × Y be a point set to be approximated by a function (curve) in X × Y with an expansion y = Σ_k=0 ^m a_k f_k(x). Here x ∈ X, y ∈ Y, a_k ∈ R, m ∈ N = {0, 1, 2, ...} with using certain given ”standard” functions f_k(x) defined on X with ranges in Y and taken with any real factors (coefficients) to be chosen. The problem consists in finding (in this class) functions with graphs nearest to Z in a certain reasonable sense. To exactly fit this, Z ⊆ { (x, y) | x ∈ X } should hold. Otherwise,

(²S = Σ_x∈Z/XR_x (r_x (Σ_k=0  ^m a_k f_k(x), y) ), ²S = Σ_x∈Z/Xr²(Σ_k=0 ^m a_k f_k(x), y)

where R_x and r_x (which can be similar to distances in Y × Y) are some nonnegative functions generally individual for different x ∈ Z/X; R and r are (independently from x) the second power function and the distance, respectively. ²S has to be minimized as a measure of nearness. Let Z/X be a finite point set

{_i=0  ⁿ  (x_i , y_i) } = { (x₀ , y₀),  (x₁ , y₁), (x₂ , y₂), ... , (x_n , y_n) }.

To provide belonging all the n + 1 points to the above graph, the set of n + 1 equations

Σ_k=0  ^m a_k f_k(x_i) = a₀ f₀(x_i) + a₁ f₁(x_i) + a₂f₂(x_i) + ... + a_m f_m(x_i)  = y_i , i ∈ {0, 1, 2, ... , n}

with m +1 unknown factors (coefficients)a₀ , a₁ , a₂ , ... , a_m to be found, has to hold. Keeping X arbitrary, consider Y multidimensional and receive the set of m + 1 linear equations

Y ⊆ R^p , y_i = (y_i⁽¹⁾, y_i⁽²⁾, ... , y_i^(p)), f_k(x_i) = (f_k⁽¹⁾(x_i), f_k⁽²⁾(x_i), ... , f_k^(p)(x_i)), p ∈ N⁺ = {1, 2, ... },

Σ_k=0  ^m a_kΣ_i=0  ⁿ Σ_l=1  ^p f_k^(l)(x_i) f_s^(l)(x_i) = Σ_i=0  ⁿ Σ_l=1  ^p f_s^(l)(x_i) y_i^(l), s ∈ {0, 1, 2, ... , m}.

By the proposed iteration method of the least normed powers [2], instead of absolute errors, unierrors are minimized, and both the power and the unierror parameters may be freely chosen. The known formulae in the least square method can be used also here by m = 2 to determine all the currently unknown variables in the next approximation in the numerators after replacing all the variables in the prior approximation with their values already known in the denominators. Replacing the unierrors with the reserves that naturally have to be maximized brings a further method generalization also applicable to determining the supersolution [2] to a set of relations.

By a unierror equalizing iteration method [2], order the quantiset of the unierrors δ_λ^(k) of each (λth) equation (λ∈Λ). The left-hand sides of two equations with the greatest absolute value of the difference of their unierrors δ_λ^(k) with corresponding indexes λ are setting equal to each other. Repeat this for the remaining equations, and so on. By every step of designing such an equalizing quantiset of equations, each initial equation is used at most once if possible. If this quantiset contains all the possible independent new equations and nevertheless has more than one solution, the required amount of additional new equations that express equalizing the greatest unierrors δ_λ^(k) to zero must be also designed. Stabilizing the equalizing quantiset, its quasisolution approximation, and the unierrors of the separate initial equations and of their quantiset can be a criterion for finishing the iterations.

A direct-solution method [2] proposes the approximation (Q the multiquantity)

x= Q(A’)//Q(A)(... + αa/||a||ⁿ+ ... + βb/||b||ⁿ+ …)//(… + α/||a||ⁿ+ … + β/||b||ⁿ+ …)

to any quantiset A using also A’ after excluding exactly all quantielements with zero elements:

A= {... , _αa, ... , _βb, ... , _γ0, ...}°,  A’ = {... , _αa, ... , _βb, ... }°

whose elements ... , a, ... , b, ... , 0, ... belong to a common normed vector space and their quantities ... , α, ... , β, ... , γ, ... are nonnegative real numbers.

These methods are very effective by solving problems simulated by overdetermined quantisets of equations, which is always the case by best approximating experimental data.

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

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