## The module "cumvar" of the Mastrave modelling library

Daniele de Rigo

#### Copyright and license notice of the function cumvar

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  ) )

cummean, cumstd, cumsumsq, groupfun

Keywords:
cumulative operators, scan

Version: 0.3.3

#### Support

The Mastrave modelling library is committed to provide reusable and general - but also robust and scalable - modules for research modellers dealing with computational science.  You can help the Mastrave project by providing feedbacks on unexpected behaviours of this module.  Despite all efforts, all of us - either developers or users - (should) know that errors are unavoidable.  However, the free software paradigm successfully highlights that scientific knowledge freedom also implies an impressive opportunity for collectively evolve the tools and ideas upon which our daily work is based.  Reporting a problem that you found using Mastrave may help the developer team to find a possible bug.  Please, be aware that Mastrave is entirely based on voluntary efforts: in order for your help to be as effective as possible, please read carefully the section on reporting problems.  Thank you for your collaboration.

Copyright (C) 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016 Daniele de Rigo