Quadratic Mean Theories for 1D Bisectors of 3D Data 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), 17

By multidimensional data estimation, approximation, modeling and processing of [1], it is reasonable to use approximation (to such data) or, equivalently, bisectors (of such data) with less numbers of dimensions then it is the case by data.

In particular, by 3D data, it is possible to use not only 2D but also 1D approximation (to such data) or, equivalently, bisectors (of such data).

In overmathematics [2, 3] and fundamental sciences of estimation [4], approximation [5], data modeling [6] and processing [7], quadratic mean theories are valid by coordinate system linear transformation invariance of the given data.

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 coordinate system centralization transformation x = x' - Σ_j=1ⁿ x'_j / n , y = y' - Σ_j=1ⁿ y'_j / n , z = z' - Σ_j=1ⁿ z'_j / n to provide coordinate system Oxyz central for the given data and further work in this system with points [_j=1ⁿ (x_j , y_j , z_j)] to be approximated with a straight line

x/a = y/b = z/c

containing origin O(0, 0, 0).

Nota bene: Consider cases a = 0, or b = 0, or c = 0, separately. Now regard nonzero values of a , b , and c only.

After centralization, additionally introduce coordinate system normalization transformation

X = x/(Σ_j=1ⁿ x_j²)^1/2 ,

Y = y/(Σ_j=1ⁿ y_j²)^1/2 ,

Z = z/(Σ_j=1ⁿ z_j²)^1/2

to provide coordinate system OXYZ which is central normalized for the given data and further work in this system with points [_j=1ⁿ (X_j , Y_j , Z_j)] to be approximated with a straight line

X/A = Y/B = Z/C

containing origin O(0, 0, 0).

Nota bene: For these points, we have

Σ_j=1ⁿ X_j² = 1,

Σ_j=1ⁿ Y_j² = 1,

Σ_j=1ⁿ Z_j² = 1,

(Σ_j=1ⁿ X_j²)^1/2 = 1,

(Σ_j=1ⁿ Y_j²)^1/2 = 1,

(Σ_j=1ⁿ Z_j²)^1/2 = 1.

Therefore, similarly to 1D approximation (to 2D data) or, equivalently, 1D bisectors (of such data), additionally using the possibility to simultaneously multiplicate or divide real numbers A , B , and C by any nonzero real number, we can consider

|A| = |B| = |C| = 1.

All the more, we can freely select the sign of one of these numbers A , B , and C , e.g. consider A positive. Then, similarly to 1D approximation (to 2D data) or, equivalently, 1D bisectors (of such data), we have

A = 1,

B = sign(Σ_j=1ⁿ X_jY_j) = sign(Σ_j=1ⁿ x_jy_j),

C = sign(Σ_j=1ⁿ X_jZ_j) = sign(Σ_j=1ⁿ x_jz_j).

Check that necessary condition

sign(Σ_j=1ⁿ x_jy_j) sign(Σ_j=1ⁿ y_jz_j) = sign(Σ_j=1ⁿ x_jz_j)

is valid.

Then the equations of the desired straight line are

X = Y sign(Σ_j=1ⁿ X_jY_j) = Z sign(Σ_j=1ⁿ X_jZ_j)

in coordinate system OXYZ ,

x/(Σ_j=1ⁿ x_j²)^1/2 = y/(Σ_j=1ⁿ y_j²)^1/2 sign(Σ_j=1ⁿ x_jy_j) = z/(Σ_j=1ⁿ z_j²)^1/2 sign(Σ_j=1ⁿ x_jz_j)

in coordinate system Oxyz , and

(x' - Σ_j=1ⁿ x'_j / n)/[Σ_j=1ⁿ (x'_j - Σ_j=1ⁿ x'_j / n)²]^1/2 =

(y' - Σ_j=1ⁿ y'_j / n)/[Σ_j=1ⁿ (y'_j - Σ_j=1ⁿ y'_j / n)²]^1/2 sign[Σ_j=1ⁿ (x'_j - Σ_j=1ⁿ x'_j / n)(y'_j - Σ_j=1ⁿ y'_j / n)] =

(z' - Σ_j=1ⁿ z'_j / n)/[Σ_j=1ⁿ (z'_j - Σ_j=1ⁿ z'_j / n)²]^1/2 sign[Σ_j=1ⁿ (x'_j - Σ_j=1ⁿ x'_j / n)(z'_j - Σ_j=1ⁿ z'_j / n)]

in coordinate system O'x'y'z' .

To introduce data scatter and trend measures via distances in coordinate system OXYZ , first consider a distance between any point (x'_j , y'_j , z'_j) and any straight line

(x' - x'')/a = (y' - y'')/b = (z' - z'')/c

containing any point (x'' , y'' , z''). Planes which are orthogonal to this straight line have equations of type

ax' + by' + cz' = d .

In particular, the plane which is orthogonal to this straight line and contains point (x'_j , y'_j , z'_j) has equation

a(x' - x'_j) + b(y' - y'_j) + c(z' - z'_j) = 0.

To determine the projection of point (x'_j , y'_j , z'_j) onto this straight line, or the intersection of this straight line and this plane, consider a parametric equation of this straight line via introducing a real parameter

t = (x' - x'')/a = (y' - y'')/b = (z' - z'')/c :

x' = x'' + at ,

y' = y'' + bt ,

z' = z'' + ct .

Substituting these formulae for x' , y' , and z' into the equation of this plane, we obtain

t = [a(x'_j - x'') + b(y'_j - y'') + c(z'_j - z'')]/(a² + b² + c²)

and the intersection coordinates

x'_nj = x'' + a[a(x'_j - x'') + b(y'_j - y'') + c(z'_j - z'')]/(a² + b² + c²),

y'_nj = y'' + b[a(x'_j - x'') + b(y'_j - y'') + c(z'_j - z'')]/(a² + b² + c²),

z'_nj = z'' + c[a(x'_j - x'') + b(y'_j - y'') + c(z'_j - z'')]/(a² + b² + c²).

The Euclidean distance between point (x'_j , y'_j , z'_j) and this straight line, or, equivalently, between point (x'_j , y'_j , z'_j) and its projection (x'_nj , y'_nj , z'_nj) on this straight line, is

²d_j = [(x'_j - x'_nj)² + (y'_j - y'_nj)² + (z'_j - z'_nj)²]^1/2 .

The sum of the squared Euclidean distances between all the points (x'_j , y'_j , z'_j) and this straight line, or, equivalently, between all the points (x'_j , y'_j , z'_j) and their projections (x'_nj , y'_nj , z'_nj) on this straight line, is

^1|2|2S = Σ_j=1^{n 2}d_j² = Σ_j=1ⁿ [(x'_j - x'_nj)² + (y'_j - y'_nj)² + (z'_j - z'_nj)²].

In our particular case in coordinate system OXYZ , we have

x'' = y'' = z'' = 0,

a = A = 1,

b = B = sign(Σ_j=1ⁿ X_jY_j) = sign(Σ_j=1ⁿ x_jy_j),

c = C = sign(Σ_j=1ⁿ X_jZ_j) = sign(Σ_j=1ⁿ x_jz_j).

The projections of point (X_j , Y_j , Z_j) onto this straight line

X = Y sign(Σ_j=1ⁿ X_jY_j) = Z sign(Σ_j=1ⁿ X_jZ_j)

are

X_nj = [X_j + Y_j sign(Σ_j=1ⁿ X_jY_j) + Z_j sign(Σ_j=1ⁿ X_jZ_j)]/3,

Y_nj = [X_j sign(Σ_j=1ⁿ X_jY_j) + Y_j + Z_j sign(Σ_j=1ⁿ Y_jZ_j)]/3,

Z_nj = [X_j sign(Σ_j=1ⁿ X_jZ_j) + Y_j sign(Σ_j=1ⁿ Y_jZ_j) + Z_j]/3.

Now we obtain

^1|2|2S_min = Σ_j=1^{n 2}d_j² = Σ_j=1ⁿ [(X_j - X_nj)² + (Y_j - Y_nj)² + (Z_j - Z_nj)²].

This is the desired least value of ^1|2|2S by different straight lines. To check this and to additionally determine the greatest value ^1|2|2S_max of ^1|2|2S by different straight lines, consider all the 4 sign combinations in A = 1, B = ± 1, and C = ± 1, i.e. all the 4 straight lines

X/A = Y/B = Z/C

with

|A| = |B| = |C| = 1.

Remember, we can freely select the sign of one of these numbers A , B , and C , e.g. consider A positive.

Then for each central normalized data point (X_j , Y_j , Z_j) and for each of the 4 combinations of the signs of B and C separately, determine 4 projections

X_nj = (X_j + BY_j + CZ_j)/3,

Y_nj = (BX_j + Y_j + BCZ_j)/3,

Z_nj = (CX_j + BCY_j + Z_j)/3

of this point (X_j , Y_j , Z_j) onto these 4 straight lines.

Then determine 4 sums

^1|2|2S = Σ_j=1^{n 2}d_j² = Σ_j=1ⁿ [(X_j - X_nj)² + (Y_j - Y_nj)² + (Z_j - Z_nj)²]

for each of the 4 combinations of the signs of B and C separately, too.

Now determine ²S_min and ²S_max among those 4 values of ²S and, finally, data scatter measure

^1|2|2S = (^1|2|2S_min / ^1|2|2S_max)^1/2

and data trend measure

^1|2|2T = 1 - ^1|2|2S = 1 - (^1|2|2S_min / ^1|2|2S_max)^1/2.

Quadratic mean theories are very efficient in data estimation, approximation, and processing and reliable even by great data scatter.

Acknowledgements to Anatolij Gelimson for our constructive discussions on coordinate system transformation invariances and his very useful remarks.

References

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

[2] Lev Gelimson. Providing Helicopter Fatigue Strength: Flight Conditions. In: Structural Integrity of Advanced Aircraft and Life Extension for Current Fleets – Lessons Learned in 50 Years After the Comet Accidents, Proceedings of the 23rd ICAF Symposium, Dalle Donne, C. (Ed.), 2005, Hamburg, Vol. II, 405-416

[3] Lev Gelimson. Overmathematics: Fundamental Principles, Theories, Methods, and Laws of Science. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[4] Lev Gelimson. Fundamental Science of Estimation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[5] Lev Gelimson. Fundamental Science of Approximation. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[6] Lev Gelimson. Fundamental Science of Data Modeling. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010

[7] Lev Gelimson. Fundamental Science of Data Processing. The ”Collegium” All World Academy of Sciences Publishers, Munich, 2010