Θ Theta Criteria

Multivariate Analysis

By Steve Halitsky and Edward Halitsky
Theta Criteria Slide: Numerical Study

III. Theta Criteria Numerical Study

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.

Theta Criteria: Figure 3.2.1 Formula

The positively defined matrices and  are:

Theta Criteria: Figure 3.2.1 Formula
Theta Criteria: Figure 3.2.1

Figure 3.2.1.The results for , and .

3.3.              Sequence of Matrices

3.3.1:

Theta Criteria: Figure 3.3.1

The positively defined matrices and  are:

Theta Criteria: Figure 3.3.1

Figure 3.3.1.The results for , and .

3.3.2:

Theta Criteria: Figure 3.3.2 Formula

The positively defined matrices and  are:

Theta Criteria: Figure 3.3.2

Figure 3.3.2.The results for , and

3.4.    Sequence Matrices

The initial 10 x 10 positively defined matrix is:

Theta Criteria: Figure 3.4.1 Formula
Theta Criteria: Figure 3.4.1

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

Let will be 10-dimensional correlation matrix:

Theta Criteria: Figure 3.4.1 Formula
Theta Criteria: Figure 3.4.1

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

Let will be 10-dimensional correlation matrix:

Theta Criteria: Figure 3.4.2 Formula
Theta Criteria: Figure 3.4.2
Figure 3.4.2.The results for , and  criteria.

3.4.3:

Let will be 10-dimensional correlation matrix:

Theta Criteria: Figure 3.4.3 Formula
Theta Criteria: Figure 3.4.3

Figure 3.4.3.The results for , and .

3.4.4.  

Theta Criteria: Figure 3.4.4 Formula
Let will be 10-dimensional correlation matrix:

Theta Criteria: Figure 3.4.4

Figure 3.4.4.The results for , and .

3.4.5.  

 

Let will be 10-dimensional correlation matrix:

Theta Criteria: Figure 3.4.5 Formula
Theta Criteria: Figure 3.4.5

Figure 3.4.5. The results for , and .

3.4.6.  

Let will be 10-dimensional correlation matrix:

Theta Criteria: Figure 3.4.6 Formula
Theta Criteria: Figure 3.4.6

Figure 3.4.6. The results for , and .

3.4.7:

Theta Criteria: Figure 3.4.7

Let will be 10-dimensional correlation matrix:

Theta Criteria: Figure 3.4.7 Formula
Theta Criteria: Figure 3.4.7

Figure 3.4.7. The results for , and .

3.4.8.  

Theta Criteria: Figure 3.4.8 Formula
Theta Criteria: Figure 3.4.8

Let will be 10-dimensional correlation matrix:

Theta Criteria: Figure 3.4.8

Figure 3.4.8. The results for , and .

3.4.9:  

Theta Criteria: Figure 3.4.9 Formula

Let will be 10-dimensional correlation matrix:

Theta Criteria: Figure 3.4.9 Formula
Theta Criteria: Figure 3.4.9

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 FΘ ,  and  on surface (, ) for ,  and .

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

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

Figure 4:

Theta Criteria: Figure 4
Theta Criteria: Figure 4 Formula

Figure 5:

Theta Criteria: Figure 5
Theta Criteria: Figure 5 Formula

Figure 6:

Theta Criteria: Figure 6
Theta Criteria: Figure 6 Formula


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

  1. For det R1  = const and = Var:
  1. For det R1  = Var, = Var :

References:

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