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TRANSFORM>NORMALIZE  

PURPOSE Normalize the values in a matrix.

DESCRIPTION This routine normalizes using a variety of techniques. 

Each technique can be applied to either the whole matrix or just the rows or columns.  In addition an iterative facility is provided to Normalize both rows and columns simultaneously.  These operate on the matrix as follows:

Marginal:  normalizes the sum to be 100.  This is achieved by dividing by the current sum of the rows, columns or matrix and multiplying by 100.

Mean:  normalizes the mean to be zero.  This is achieved by subtracting from every row, column, or matrix element the current mean.

Standard Deviation:  normalizes the standard deviation to be one.  This is achieved by dividing the rows, columns or matrix by the current  standard deviation. 

Z-Score:  standardizes the mean to be zero and the standard deviation to be one.  This is achieved by subtracting from every row, column or matrix element the current mean and then dividing the rows, columns or matrix by the current standard deviation. 

Euclidean:  standardizes the Euclidean norm to be one.  This is achieved by dividing the rows, columns or matrix by the current Euclidean norm. 

Maximum:  standardizes the rows, columns or matrix to each have a maximum value of 100.  This is achieved by dividing the matrix or each row or column by the current maximum and multiplying by 100.

The routine also allows each of these options to be applied to the rows and columns simultaneously.  This involves an iterative procedure in which the technique is first applied to the rows and then the columns and then the rows etc.  It is terminated when (and if) there is convergence.

PARAMETERS
Input dataset
Name of file containing matrix to be standardized. Data type: Matrix.

Which dimension(s) to standardize: (Default = Columns).
Choices are:

Rows - Normalization is applied to the rows of the matrix independently.
Columns - Normalization is applied to the columns of the matrix independently.
Matrix - Normalization is applied to the entire matrix.
Both - Normalization is applied to the rows, then the columns, then the rows etc iteratively until convergence.

Standardizing criterion: (Default = Marginal).
Choices are:

Marginal - Forces the sum of elements to be 100.  By row, column, matrix or row and column.

Mean - Forces the mean of elements to be zero.  By row, column, matrix, or row and column.

Std-Dev - Forces the standard deviation to be one.  By row, column, matrix or row and column.  If standard deviation is initially zero then elements of matrix are treated as missing.

Z-Score - Forces the mean of the elements to be zero and the standard deviation to be 1.  By row, column, matrix or row and column.  If standard deviation is initially zero then elements of matrix are treated as missing.

Euclidean - Forces the Euclidean norm,  to be one.  By row, column, matrix or row and column.

Maximum - Forces the maximum of the elements to be 100.  By row, column or row and column.  Forces the maximum element to be one for the whole matrix.

Constant to replace zeros with (Default =0.0)
Zeros can cause this procedure to crash and this can be overcome by replacing them with a relatively small value. 

(Sq. matrices only) Include diagonal values?  (Default = Yes).
Yes includes diagonals.  No treats diagonal values as missing.
  
(For iterative norm.) Convergence tolerance (Default=0.001)
When both is selected the routine iterates to convergence the tolerance specifies a point at which when the values change by less than the tolerance the routine has converged.

(For iterative norm.)  Max # of iterations (Default=100)
When both is selected the routine iterates to convergence. Convergence will be deemed to have failed if the tolerance has not been achieved before the maximum number of iterations has taken place.     
 
Output dataset: (Default = 'Normalize').
Name of file which contains normalized matrix.


LOG FILE Normalized matrix.

TIMING O(N^2).

COMMENTS None.

REFERENCES None.