Θ Theta Criteria

Multivariate Analysis

By Steve Halitsky and Edward Halitsky

Intro

ΘTheta Criteria

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Existing

ΘFormal

ΘNumerical

ΘApplications

III. Theta Criteria Numerical Study

3.1. Numerical Experiments Details
3.2. Sequence Matrices
3.3. Sequence Matrices
3.4. Sequence Matrices
Table 1
3.5. The matrices block linkage and are variable
Figure 4
Figure 5
Figure 6
Table 2
Conclusions for Numerical Experiments
References

3.1. Numerical Experiments Details

Let be two positively defined matrices. Let construct on the sequence of symmetrical positively defined matrices:

, with

Numerical results have been obtained on MATLAB Version 6.5 for matrices Accuracies of algorithms have been verified by methods from [6] - [8]. The matrix block linkage and sequence cardinality are :,. We assumed that and distinction on sequence can be represented by criteria

The adequate criteria compared with and criteria in logarithmic scale.

3.2. Sequence of matrices

3.2.1.

The positively defined matrices and are:

Figure 3.2.1.The results for , and .

3.3. Sequence of Matrices

3.3.1:

The positively defined matrices and are:

Figure 3.3.1.The results for , and .

3.3.2:

The positively defined matrices and are:

Figure 3.3.2.The results for , and

3.4. Sequence Matrices

The initial 10 x 10 positively defined matrix is:

Formally, 10 x 10 matrix is presented below:
3.4.1:

Let will be 10-dimensional correlation matrix:

Figure 3.4.1.The results for , and criteria.
3.4.2:

Let will be 10-dimensional correlation matrix:

Figure 3.4.2.The results for

and

criteria.

3.4.3:

Let will be 10-dimensional correlation matrix:

Figure 3.4.3.The results for , and .

3.4.4.

Let

will be 10-dimensional correlation matrix:

Figure 3.4.4.The results for , and .

3.4.5.

Let will be 10-dimensional correlation matrix:

Figure 3.4.5. The results for , and .

3.4.6.

Let will be 10-dimensional correlation matrix:

Figure 3.4.6. The results for , and .

3.4.7:

Let will be 10-dimensional correlation matrix:

Figure 3.4.7. The results for , and .

3.4.8.

Let will be 10-dimensional correlation matrix:

Figure 3.4.8. The results for , and .

3.4.9:

Let will be 10-dimensional correlation matrix:

Figure 3.4.9. The results for , and .

Table 1.

	Criteria Type	Interval	Criteria graphical description
3.2. Sequence of matrices
3.2.1.
		(1, 100)	(1, 100)- Convex decreasing curve;
		(1, 100)	(1, 100)- Convex decreasing curve;
		(1, 12)	(1, 12) - Segment of Convex decreasing curve; (12, 100) - Segment of Horizontal line
		(1, 12)	(1, 12) - Segment of Convex decreasing curve; (12, 100) - Segment of Horizontal line
3.3. Sequence Matrices
3.3.1.
		[1,20], [25,100]	2 segments of convex decreasing curve with small horizontal plateau (20,25) in the interval
		[1,23], [25,100]	2 segments of convex decreasing curve with small horizontal plateau (23,25) in the interval
		(1, 12)	(1, 12) - Segment of Convex decreasing curve; (12, 100) - Segment of Horizontal line
		(1, 12)	(1, 12) - Segment of Convex decreasing curve; (12, 100) - Segment of Horizontal line
*3.3.2:*
		(1, 100)	(1, 100) - Convex decreasing curve
		(1,20)	(1, 20) - Segment of Convex decreasing curve; (20, 100) - Segment of Horizontal line
		(1, 12)	(1, 12) - Segment of Convex decreasing curve; (12, 100) - Segment of Horizontal line
		(1, 12)	(1, 12) - Segment of Convex decreasing curve; (12, 100) - Segment of Horizontal line

3.4. Sequence Matrices
3.4.1:
	(1,50)	(1,50) - Convex decreasing curve; (50, 100) -Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
3.4.2:
	(1,50)	(1,50) - Convex decreasing curve; (50, 100)-Horizontal line
	(6,20), (50, 100)	(1,5) - Segment of horizontal line; (6,20) - Segment of Convex decreasing curve; (20,50) - Segment of horizontal line; (50,100) - Segment of Convex decreasing curve
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
3.4.3:
	(1,40)	(1,40) - Convex decreasing curve with 2 small segments of horizontal line; (40, 100) -Horizontal line
	(20, 100)	(1, 20) - Segment of horizontal line; (20,100) - Convex decreasing curve with 2 small segments of horizontal line;
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
3.4.4:
	(10, 100)	(10,100) - Segment of Convex decreasing curve with 4 small segments of horizontal line; (40, 100) -Horizontal line
	(1,50)	(1,50) - Segment of Convex decreasing curve; (50, 100) - Segment of horizontal line;
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line

3.4.5:
	(1,100)	(1,100) - Segment of Convex decreasing curve with 4 small segments of horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
3.4.6:
	(1,55)	(1,55) - Segment of Convex decreasing curve with 2 small segments of horizontal line; (55,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (25,100) - Segment of Horizontal line
3.4.7:
	(1,50)	(1,50) - Segment of Convex decreasing curve with 2 small segments of horizontal line; (50,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (1,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (1,100) - Segment of Horizontal line
3.4.8:
	(20,45), (60,100)	(1,20) - Segment of Horizontal line; (20,45) - Segment of Convex decreasing curve; (45,60) Segment of horizontal line; (60,100) - Segment of Convex decreasing curve
	(1,50)	(1,50) - Segment of Convex decreasing curve with 3 "wild points"; (50,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (1,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (1,100) - Segment of Horizontal line
3.4.9
	(15,100)	(1,15) - Segment of Horizontal line; (15,100) - Segment of Convex decreasing curve
	(1,25)	(1,25) - Segment of Convex decreasing curve; (1,100) - Segment of Horizontal line
	(1,25)	(1,25) - Segment of Convex decreasing curve; (1,100) - Segment of Horizontal line

3.5. The matrices block linkage and are variable

Let's construct two sequences of matrices:

= with , if , ,

then

and

where

R _ik = ,

with and , if , .

Let construct the response functions F_{Θ ,} and on surface (, ) for , and .

F_Θ = F (, , ), Î {10^-16, 10⁰}, Î {10⁷, 10¹³}.

The presented as contours on Fig. 4, - on Fig. 5 and - on Fig. 6.

Figure 4:

Figure 5:

Figure 6:

Table 2.

Criteria Type	Correct Region	Det R Correct Region	Incorrect Region for	Det R Incorrect Region	Criteria Slope
	(-14, -3)	(6, 13)	None	None	1
	(-14, -1.5)	(6, 13)	None	None	2
	(-10, -1.5)	(8, 13)	(-13, -10)	(6.5, 10)	2

Conclusions for Numerical Experiments

For det R₁ = const and = Var:

Criteria correctly represented matrices linkage decrease process for all iterations for 3.2.1, 3.3.1, Cases.
Criteria correctly represented matrices linkage decrease process for all iterations for 3.3.1, 3.3.2 Cases.
Criteria , adequately represented process only for 10 first iterations for all 3.4. Cases.
There are several Theta Criteria, applicable for 3.4. Cases. They are better in accuracy than , .
In all of our experiments exists at least onecriteria with superior accuracy to and for identification of very weak linkages between matrices.

For det R₁ = Var, = Var :

The criteria is correct for the whole domain with constant slope m = 1.
The criteria is very steep and can be used for the whole domain.
The criteria was incorrect for small .
The criteria have a tremendous potential for improving accuracy in analysis of multi-dimensional objects and systems.

References:

G. Golub, C. Van Loan. Matrix Computations. The John Hopkins University Press, 1996.
C. De Boor. A Practical Guide to Splines. Springer-Verlag, 1978.