The module "cumvar" of the Mastrave modelling library

Copyright and license notice of the function cumvar

Copyright © 2006,2007,2008,2009,2010,2011 Daniele de Rigo

The file cumvar.m is part of Mastrave.

Mastrave is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

Mastrave is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Mastrave. If not, see http://www.gnu.org/licenses/.

Function declaration

answer = cumvar( values      ,
                 dim    = [] ,
                 mode   = 0  )

Description

Utility to extend the function var(.) providing cumulative variances of the elements of the array values along a given dimension dim .

To mitigate unwanted numeric cancellations, all cumulative variances are first approximated by centering values elements with their corresponding non-cumulative means and then refined by correcting the centering with the correct cumulative means. This way, if N is the total number of elements of values and n is the size of the dimension dim of values , then the computation remains O( n * N ).

Input arguments

 values            ::numeric::
                   Vector, matrix or multi-dimensional array of numbers.

 dim               ::scalar_index|empty::
                   Scalar positive integer representing the dimension along
                   which the cumulative variances have to be computed.       
                   If  dim  is an empty array  [] , the dimension is the 
                   first non-singleton dimension.  In case  values  is a
                   vector, this definition means that the default
                   dimension is the one along which the elements of the
                   vector  values  are aligned.
                   If omitted, the default value is  [].

 mode              ::numstring::
                   Boolean or string declaring the kind of variance to be
                   computed.
                   If omitted, its default value is: 0.
                   Valid modes are:

                        mode        │      meaning
                   ─────────────────┼───────────────────────────────────
                     0              │ Compute the unbiased sample 
                     '--sample'     │ variance by normalizing the i-th
                                    │ variance with  (i-1) .  Variances
                                    │ of the first elements of  values  
                                    │ along  dim  are therefore NaN.
                   ─────────────────┼───────────────────────────────────
                     1              │ Compute the population variance
                     '--population' │ (or biased sample variance) by
                                    │ normalizing the i-th variance 
                                    │ with  i .  Variances of the first
                                    │ elements of  values  along  dim  
                                    │ are therefore zeros.

Example of usage

   % Correctness check function
   check = @(computed,expected)assert(                          ...
      abs( (computed - expected)./expected ) < 10*eps           ...
   )

   % Basic usage

   % Vectors:
   v    = ceil( rand(1,7)* 100 )
   cv   = cumvar( v )
   check( cv(end), var(v) );

   % Matrices:
   v    = ceil( rand(5,7)* 100 )
   cv   = cumvar( v )
   check( cv(end,:), var(v) );
   

   % Passing a custom dimension
   cv   = cumvar( v , 2 )
   check( cv(:,end), var(v,0,2) );

   % Dealing with multi-dimensional arrays
   v    = ceil( rand(5,7,3)* 100 )
   cv   = cumvar( v )
   cv   = cumvar( v , [] )
   check( cv(end,:,:), var(v) );

   cv   = cumvar( v , 3 )
   check( cv(:,:,end), var(v,0,3) );
   

   % Passing the kind of variance to be computed
   % Unbiased sample variance
   cv   = cumvar( v , [], '--sample' )
   cv   = cumvar( v , [], 0 )
   check( cv(end,:,:), var(v) );

   % Population variance (biased sample variance)
   cv   = cumvar( v , [], '--population' )
   cv   = cumvar( v , [], 1 )
   check( cv(end,:,:), var(v,1,1) );
   cv   = cumvar( v , 2 , 1 )
   check( cv(:,end,:), var(v,1,2) );
   cv   = cumvar( v , 3 , 1 )
   check( cv(:,:,end), var(v,1,3) );


   % Complex-valued elements of  values 
   v    = ceil( rand(5,7)* 100 ) +1i * ceil( rand(5,7)* 100 )
   cv   = cumvar( v )
   check( cv(end,:), var(v) );
   cv   = cumvar( v , 2 )
   check( cv(:,end), var(v,0,2) );  

Memory requirements:
   O( numel(  values  ) )



See also:
   cummean, cumstd, cumsumsq, groupfun



Keywords:
   cumulative operators, scan



Version: 0.3.3

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