# Matlab winsorized mean over all dimension

This is a function I wrote back in 2014. I think it illustrates an advanced functionality in matlab that I hadn’t found written about before.

### The problem:

Calculate the winsorized mean of a multidimensional matrix over an arbitrary dimension.

### Winsorized Mean

The benefits of the winsorized mean can be seen here:

We replace the top 10% and bottom 10% by the remaining most extreme value before calculating the mean (left panel). The right panel shows how the mean is influenced by a single outlier, but the winsorized mean is not (ignore the “yuen”-box”)

### Current Implementation

I adapted an implementation from the LIMO toolbox based on Original Code from Prof. patrick J Bennett, McMaster University. In this code the dimension is fixed at dim = 3, the third dimension.

They solve it in three steps:

- sort the matrix along dimension 3 [matlab] xsort=sort(x,3); [/matlab]
- replace the upper and lower 10% by the remaining extreme value [matlab] % number of items to winsorize and trim
- calculate the mean over the sorted matrix [matlab]wvarx=var(wx,0,3);[/matlab]

g=floor((percent/100)*n);

wx(:,:,1:g+1)=repmat(xsort(:,:,g+1),[1 1 g+1]);

wx(:,:,n-g:end)=repmat(xsort(:,:,n-g),[1 1 g+1]);

[/matlab]

### Generalisation

To generalize this to any dimension I have seen two previous solution that feels unsatisfied:

– Implement it for up to X dimension hardcoded and then use a switch-case to get the solution for the case.

– use permute to reorder the array and then go for the first dimension (which can be slow depending on the array)

Let’s solve it for X = 20 x 10 x 5 x 2 over the third dimension

[matlab]

function [x] = winMean(x,dim,percent)

% x = matrix of arbitrary dimension

% dim = dimension to calculate the winsorized mean over

% percent = default 20, how strong to winsorize

% How long is the matrix in our required dimension

n=size(x,dim);

% number of items to winsorize and trim

g=floor((percent/100)*n);

x=sort(x,dim);

We are now using the subsasgn command and subsref

We need to generate a structure that mimics the syntax of

[matlab] x(:,:,1:g+1,:) = y [/matlab] for arbitrary dimensions and we need to construct

*y*[matlab] % Prepare Structs

Srep.type = ‘()’;

S.type = ‘()’;

% replace the left hand side

nDim = length(size(x));

beforeColons = num2cell(repmat(‘:’,dim-1,1));

afterColons = num2cell(repmat(‘:’,nDim-dim,1));

Srep.subs = {beforeColons{:} [g+1] afterColons{:}};

S.subs = {beforeColons{:} [1:g+1] afterColons{:}};

x = subsasgn(x,S,repmat(subsref(x,Srep),[ones(1,dim-1) g+1 ones(1,nDim-dim)])); % general case

[/matlab]
The output of *Srep* is:

Srep =

type: ‘()’

subs: {‘:’ ‘:’ [2] ‘:’ }

thus subsref(x,Srep) outputs what x(:,:,2,:) would output. And then we need to repmat it, to fit the number of elements we replace by the winsorizing method.

This is put into subsasgn, where the S here is :

Srep =

type: ‘()’

subs: {‘:’ ‘:’ [1 2] ‘:’ }

Thus equivalent to x(:,:,[1 2],:).

The evaluated structure then is:

[matlab] x(:,:,[1:2]) = repmat(x[:,:,1],[1 1 2 1]) [/matlab]

The upper percentile is replaced analogous:

[matlab]
% replace the right hand side

Srep.subs = {beforeColons{:} [n-g] afterColons{:}};

S.subs = {beforeColons{:} [n-g:size(x,dim)] afterColons{:}};

x = subsasgn(x,S,repmat(subsref(x,Srep),[ones(1,dim-1) g+1 ones(1,nDim-dim)])); % general case

[/matlab]And in the end we can take the mean, var, nanmean or whatever we need:

[matlab]
x = squeeze(nanmean(x,dim));

[/matlab]

That finishes the implementation.

### Timing

But how about speed? I thus generated a random matrix of 200 x 10000 x 5 and measured the timing (n=100 runs) of the original limo implementation and mine:

algorithm |
timing (95% bootstraped CI of mean) |
---|---|

limo_winmean | 185 – 188 ms |

my_winmean | 202 – 203ms |

limo_winmean otherDimension than 3 | 218 – 228 ms |

For the last comparison I permuted the array prior to calculating the winsorized mean, thus the overhead. In my experience, the overhead is greater the larger the arrays are (I’m talking about 5-10GB matrices here).

## Conclusion

My generalization seems to work fine. As expected it is slower than the hardcoded version. But it is faster than permuting the whole array.