[vtkusers] [EXTERNAL] vtkMultiCorrelativeStatistics

[David Thompson]

Hi Darshan,

> I was just goings through the Testing Cxx for  
> vtkMultiCorrelativeteStatistics class . ... tables below ...
> I am trying to understand the result in the Derived Statistics  
> part . In the Derived Statistics 0 the Cholesky value for M1,M1 ,  
> which is the result in the last column , last row , the output is  
> 1.11199 , If I understand correctly , the answer should be  
> sqrt(7.54) = 2.74 , isn't it ? Just wondering about the result.


I believe the values in the table are correct, if perhaps a little  
difficult to parse. The covariance matrix -- of which only the upper  
triangle of the symmetric matrix is presented in the table below -- is

    5.9829   6.1452
    6.1452   7.5484

The Cholesky factorization of this matrix is

    2.44599   2.51234
    0.00000   1.11199

whose transpose is presented below the covariance matrix as

    2.44599     --
    2.51234   1.11199

(where I've put "--" in place of the covariance entry reported by VTK).

When I multiply the above by its transpose, I get the original,  
symmetric covariance matrix. The actual relationship between the  
covariance entry and its Cholesky decomposition is 7.54839 =  
2.51234**2 + 1.11199**2 because the transpose leaves both terms in the  
Cholesky matrix row-column product non-zero.

> Also in the Primary Statistics , what are those values after row 4 ...

Those are the sums of centered moments used to derive the covariance  
matrices. Dividing by (Cardinality - 1) will get you the covariance  
matrix entry. For example, take the M0,M0 entry of row 5: 185.469 /  
(32-1) == 5.98286.

	David

> Primary Statistics
> +-----------------+-----------------+------------------+
> | Column1         | Column2         | Entries          |
> +-----------------+-----------------+------------------+
> | Cardinality     |                 | 32               |
> | M0              |                 | 49.2188          |
> | M1              |                 | 49.5             |
> | M2              |                 | -1.00003         |
> | M0              | M0              | 185.469          |
> | M0              | M1              | 190.5            |
> | M0              | M2              | -0.00278125      |
> | M1              | M1              | 234              |
> | M1              | M2              | -0.0045          |
> | M2              | M2              | 9.6875e-007      |
> +-----------------+-----------------+------------------+
> Derived Statistics 0
> +-----------------+-----------------+----------------- 
> +------------------+
> | Column          | Mean            | M0              |  
> M1               |
> +-----------------+-----------------+----------------- 
> +------------------+
> | M0              | 49.2188         | 5.98286         |  
> 6.14516          |
> | M1              | 49.5            | 2.44599         |  
> 7.54839          |
> | Cholesky        | 32              | 2.51234         |  
> 1.11199          |
> +-----------------+-----------------+----------------- 
> +------------------+
> Derived Statistics 1
> +-----------------+-----------------+----------------- 
> +-----------------+-------
> -----------+
> | Column          | Mean            | M0              |  
> M1              | M2
>            |
> +-----------------+-----------------+----------------- 
> +-----------------+-------
> -----------+
> | M0              | 49.2188         | 5.98286         |  
> 6.14516         | -8.971
> 77e-005    |
> | M1              | 49.5            | 2.44599         |  
> 7.54839         | -0.000
> 145161     |
> | M2              | -1.00003        | 2.51234         |  
> 1.11199         | 3.125e
> -008       |
> | Cholesky        | 32              | -3.66795e-005   |  
> -4.7671e-005    | 0.0001
> 66229      |
> +-----------------+-----------------+----------------- 
> +-----------------+-------
>