Distance Quadrat Theories for Three Dimensions in Fundamental Sciences of Estimation, Approximation, Data Modeling and Processing
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)
11 (2011), 19
By estimation, approximation, and data processing, the least square method (LSM) [1] by Legendre and Gauss only usually applies to contradictory (e.g. overdetermined) problems, by methods of finite elements, points, etc. Overmathematics [2, 3] and fundamental sciences of estimation [4], approximation [5], and data processing [6] have discovered a lot of principal shortcomings [7] of the least square method. Additionally consider its simplest approach which is typical. Minimizing the sum of the squared differences of the alone preselected coordinates (e.g., ordinates in a two-dimensional problem) of the graph of the desired approximation function and of everyone among the given data depends on this preselection, ignores the remaining coordinates, and provides no coordinate system rotation invariance and hence no objective sense of the result. Moreover, the method is correct by constant approximation or no data scatter only and gives systematic errors increasing together with data scatter and the deviation (namely declination) of an approximation from a constant.
In fundamental sciences of estimation [4], approximation [5], and data processing [6], distance quadrat theories (DQT) are valid in the case of coordinate system rotation invariance. By coordinate system translation invariance of the given data, centralize them by subtracting every coordinate of the data center from the corresponding coordinate of every data point.
Additionally to showing the essence of least squared distance theories in the simplest but most important linear approximation in the two-dimensional case in which it is possible to explicitly algebraically provide the unique best quasisolution [2, 3] to the corresponding problem setting, now consider the two-dimensional case, too.
Given n (n ∈ N+ = {1, 2, ...}, n > 3) points [j=1n (x'j , y'j , z'j)] = {(x'1 , y'1 , z'1), (x'2 , y'2 , z'2), ... , (x'n , y'n , z'n)] with any real coordinates. Use clearly invariant centralization transformation x = x' - Σj=1n x'j / n , y = y' - Σj=1n y'j / n , z = z' - Σj=1n z'j / n to provide coordinate system Oxyz central for the given data and further work in this system with points [j=1n (xj , yj , zj)] to be approximated with a plane ax + by + cz = 0 containing origin O(0, 0, 0).
Case c = 0 bringing clear simplification should be considered separately. Now regard the general case c ≠ 0. It is possible to equivalently transform plane equation ax + by + cz = 0 via simultaneously multiplicating (or dividing) all the three factors a , b , and c by any nonzero number, in particular - 1/c (or - c , respectively). Therefore, we can consider case c = -1 and the plane equation z = ax + by only.
The distance between this plane and the jth data point (xj , yj , zj) and further the sum of the squared distances between this plane and everyone of the n data points [j=1n (xj , yj , zj)] are, respectively,
dj = |axj + byj - zj|/(1 + a2 + b2)1/2,
2S(a , b) = Σj=1n dj2 = Σj=1n(axj + byj - zj)2/(1 + a2 + b2).
This nonnegative differentiable function takes its minimum at a point with vanishing the both first order derivatives of this function by a and b:
2S'a = - 2a/(1 + a2 + b2)2 Σj=1n (xj2a2 + yj2b2 + zj2 + 2xjyjab - 2xjzja - 2yjzjb) + 2/(1 + a2 + b2) Σj=1n (xj2a + xjyjb - xjzj) = 0,
2S'b = - 2b/(1 + a2 + b2)2 Σj=1n (xj2a2 + yj2b2 + zj2 + 2xjyjab - 2xjzja - 2yjzjb) + 2/(1 + a2 + b2) Σj=1n (xjyja + yj2b - yjzj) = 0;
- Σj=1n xjyja2b +Σj=1n xjzja2+ (Σj=1n xj2 - Σj=1n yj2)ab2 + 2Σj=1n yjzjab+ (Σj=1n xj2 - Σj=1n zj2)a + Σj=1n xjyjb3 - Σj=1n xjzjb2 + Σj=1n xjyjb - Σj=1n xjzj = 0,
Σj=1n xjyja3+ (Σj=1n yj2 - Σj=1n xj2)a2b - Σj=1n xjyjab2 - Σj=1n yjzja2+ 2Σj=1n xjzjab + Σj=1n yjzjb2 + Σj=1n xjyja + (Σj=1n yj2 - Σj=1n zj2)b - Σj=1n yjzj = 0;
(Σj=1n xj2 - Σj=1n zj2)a + Σj=1n xjyjb = Σj=1n xjzj - Σj=1n xjzja2- 2Σj=1n yjzjab + Σj=1n xjzjb2+ Σj=1n xjyja2b + (Σj=1n yj2 - Σj=1n xj2)ab2 - Σj=1n xjyjb3 ,
Σj=1n xjyja + (Σj=1n yj2 - Σj=1n zj2)b = Σj=1n yjzj + Σj=1n yjzja2- 2Σj=1n xjzjab - Σj=1n yjzjb2 - Σj=1n xjyja3+ (Σj=1n xj2 - Σj=1n yj2)a2b + Σj=1n xjyjab2 .
Nota bene: The last two equations in a and b (namely in these forms) are very suitable for using iteration approach. Let us use the next approximations ak+1 and bk+1 (k = 0, 1, 2, ...) to a and b , respectively, instead of a and b themselves in the left-hand parts of these equations, as well as the previous approximations ak and bk to a and b , respectively, instead of a and b themselves in the right-hand parts of these equations:
(Σj=1n xj2 - Σj=1n zj2)ak+1 + Σj=1n xjyjbk+1 = Σj=1n xjzj - Σj=1n xjzjak2- 2Σj=1n yjzjakbk + Σj=1n xjzjbk2+ Σj=1n xjyjak2bk + (Σj=1n yj2 - Σj=1n xj2)akbk2 - Σj=1n xjyjbk3 ,
Σj=1n xjyjak+1 + (Σj=1n yj2 - Σj=1n zj2)bk+1 = Σj=1n yjzj + Σj=1n yjzjak2- 2Σj=1n xjzjakbk - Σj=1n yjzjbk2- Σj=1n xjyjak3+ (Σj=1n xj2 - Σj=1n yj2)ak2bk + Σj=1n xjyjakbk2.
The last two formulae provide simply determining every next approximations ak+1 and bk+1 (k = 0, 1, 2, ...) to a and b , respectively, via the previous approximations ak and bk to a and b , respectively.
To begin with, there are many reasonable possibilities for the initial approximations a0 and b0 to a and b , respectively, e.g.:
1) zero approximations a0 = 0 and b0 = 0;
2) the solution via the least square method (LSM) [1]
2S(a , b) = Σj=1n(axj + byj - zj)2;
2S'a = 2Σj=1n (xj2a + xjyjb - xjzj) = 0,
2S'b = 2Σj=1n (xjyja + yj2b - yjzj) = 0
via solving equation set
Σj=1n xj2 a + Σj=1n xjyjb = Σj=1n xjzj ,
Σj=1n xjyja + Σj=1n yj2 b = Σj=1n yjzj
in a and b .
Nota bene: Zero approximations a0 = 0 and b0 = 0 lead to similar but another equation set in a and b for determining a1 and b1 :
(Σj=1n xj2 - Σj=1n zj2)a + Σj=1n xjyjb = Σj=1n xjzj ,
Σj=1n xjyja + (Σj=1n yj2 - Σj=1n zj2)b = Σj=1n yjzj .
By linear approximation in the two-dimensional case in which it is possible to explicitly algebraically provide the unique best quasisolution [2, 3] to the corresponding problem setting but iteration approach is also possible, the corresponding two sets of equations coincide. Namely, for straight line z = ax , the least square method (LSM) [1] gives
2S(a , b) = Σj=1n(axj - zj)2;
2S'a = 2Σj=1n (xj2a - xjzj) = 0
and equation set consisting of one equation
Σj=1n xj2 a = Σj=1n xjzj
in a . Distance quadrat theories (DQT) give
2S'a = - 2a/(1 + a2)2 Σj=1n (xj2a2+ zj2 - 2xjzja) + 2/(1 + a2) Σj=1n (xj2a - xjzj) = 0
and for zero approximation a0 = 0, the same equation set consisting of one equation
Σj=1n xj2 a = Σj=1n xjzj
in a .
This is an interesting phenomenon of adding the third dimension.
Now we also see that it is possible to introduce another iteration approach than above. Namely, at once substitute every next approximations ak+1 and bk+1 (k = 0, 1, 2, ...) to a and b , respectively, for a and b in expressions
xj2a + xjyjb - xjzj ,
xjyja + yj2b - yjzj
only of the initial formulae for 2S'a and 2S'b because these two expressions only correspond to ignoring the plane declinations, e.g. via the least square method (LSM) [1]. At all the remaining occurences of a and b in the initial formulae for 2S'a and 2S'b , substitute every previous approximations ak and bk to a and b , respectively. Then we obtain:
(2S'a =) - 2ak/(1 + ak2 + bk2)2 Σj=1n (xj2ak2 + yj2bk2 + zj2 + 2xjyjakbk - 2xjzjak - 2yjzjbk) + 2/(1 + ak2 + bk2) Σj=1n (xj2ak+1 + xjyjbk+1 - xjzj) = 0,
(2S'b =) - 2bk/(1 + ak2 + bk2)2 Σj=1n (xj2ak2 + yj2bk2 + zj2 + 2xjyjakbk - 2xjzjak - 2yjzjbk) + 2/(1 + ak2 + bk2) Σj=1n (xjyjak+1 + yj2bk+1 - yjzj) = 0;
Σj=1n (xj2ak+1 + xjyjbk+1 - xjzj) = ak/(1 + ak2 + bk2) Σj=1n (xj2ak2 + yj2bk2 + zj2 + 2xjyjakbk - 2xjzjak - 2yjzjbk),
Σj=1n (xjyjak+1 + yj2bk+1 - yjzj)= bk/(1 + ak2 + bk2) Σj=1n (xj2ak2 + yj2bk2 + zj2 + 2xjyjakbk - 2xjzjak - 2yjzjbk);
Σj=1n xj2ak+1 + Σj=1n xjyj bk+1 = Σj=1n xjzj + ak/(1 + ak2 + bk2) (Σj=1n xj2ak2 + Σj=1n yj2bk2 + Σj=1n zj2 + 2Σj=1n xjyj akbk - 2Σj=1n xjzj ak - 2Σj=1nyjzj bk),
Σj=1n xjyj ak+1 + Σj=1n yj2bk+1 = Σj=1n yjzj + bk/(1 + ak2 + bk2) (Σj=1n xj2ak2 + Σj=1n yj2bk2 + Σj=1n zj2 + 2Σj=1n xjyjakbk - 2Σj=1n xjzj ak - 2Σj=1n yjzjbk).
In this second approach, zero approximations a0 = 0 and b0 = 0 lead to the same equation set in a and b for determining a1 and b1 as the least square method (LSM) [1] does:
(Σj=1n xj2 - Σj=1n zj2)a + Σj=1n xjyjb = Σj=1n xjzj ,
Σj=1n xjyja + (Σj=1n yj2 - Σj=1n zj2)b = Σj=1n yjzj .
Therefore, this second, so-called essential, approach seems to be less formal and more natural than the first, so-called formal, approach with formally substituting every next approximations ak+1 and bk+1 (k = 0, 1, 2, ...) to a and b , respectively, for a and b in all the linear terms, the powers of a and b being added, whereasevery previous approximations ak and bk to a and b , respectively, are substituted for a and b , respectively, in all the remaining (higher, nonlinear) terms, which provides simple solvability of the equation set expressing every next approximations ak+1 and bk+1 (k = 0, 1, 2, ...) to a and b , respectively, via every previous approximations ak and bk to a and b , respectively.
Due to intuition, we can expect that the essential approach can give results in less iteration steps than the formal approach does.
Nota bene: Both the essential and the formal approaches principally give the same results which are solutions to their common initial equation setsproviding vanishing the both derivatives of the first order before considering iterations via substituting approximations to a and b .
Distance quadrat theories (DQT) are very efficient in data estimation, approximation, and processing by coordinate system rotation invariance.
Acknowledgements to Anatolij Gelimson for our constructive discussions on coordinate system transformation invariances and his very useful remarks.
References
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