zscoreFS

zscoreFS computes (robust) standardized z scores

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Syntax

Description

X can be a vector of length(n) or data matrix containing n observations on v variables or 3D array of size n-by-v-by-r.

Z = zscoreFS(X) returns a centered, scaled version of X, with the same size as X. For vector input X, Z is the vector of z-scores (X-median(X)) ./ (1.4826* mad(X)).

Z=zscoreFS(X,loc,scale) returns a centered, scaled version of X, the same size as X using location and scale are specified in input parameters 'loc' and 'scale'. For vector input X, Z is the vector of z-scores (X-location(X)) ./ scale(X).

where scaled(X) is the corrected estimator of scale (corrected in the sense that it is multiplied by a coefficient to achieve consistency for normally distributed data).

Z=zscoreFS(X,loc,scale) computes robust standardized zscores using the estimates of location and scale specified in loc and scale strings. If X is a 2D matrix, zscores are computed using loc and scale along each column of X. If X is a 3D array zscores are computed using the location and scale along the first non-singleton dimension. For example if X is n-by-v-by-r (with n>1) and loc='median'; n-by-r medians are computed for each of the n rows of X and each third dimension r.

Z=zscoreFS(X,loc) computes standardized zscores using the estimates of location specified in loc and the mad as measure of dispersion.

[Z,mu,sigma] = zscoreFS(X) also returns median(X) in mu and mad in sigma.

[Z,mu,sigma] = zscoreFS(X,loc,scale) also returns the estimates of location in mu and of scale in sigma as specified in loc and scale strings.

Z=zscoreFS(X,loc,scale,dim) computes robust standardized zscores along the dimension dim of X using the estimates of location and scale specified in loc and scale strings. dim standardizes X by working along the dimension dim of X. For example if X is a two dimensional matrix dim=2 (default) standardizes the columns of X else if dim=1 standardizes the rows. If X is a three dimensional dim = 1 standardizes the columns, dim =2 standardizes the rows and dim =3 standardizes the third dimension.

zscoreFS is an extension of function zscore of statistic toolbox because it enables to specify alternative measures of location and scale.

example

Z =zscoreFS(X) Scale using medians and mads.

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Z =zscoreFS(X, loc) Scale using mean and mads.

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Z =zscoreFS(X, loc, scale) Remove the medians and divide by Qn.

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Z =zscoreFS(X, loc, scale, dim) Examples with 3D arrays.

example

[Z, mu] =zscoreFS(___) Report also location and scale measures which have have been used.

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[Z, mu, sigma] =zscoreFS(___) 3D arrays with dim=1, dim=2 and dim=3.

Examples

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  • Scale using medians and mads.

    • zscoreFS with all default options (that is remove the medians and divide by mads) 

    n=200;
v=3;
randn('state', 123456);
Y=randn(n,v);
% Contaminated data
Ycont=Y;
Ycont(1:5,:)=Ycont(1:5,:)+10;
[out]=zscoreFS(Ycont);

  • Scale using mean and mads.

    • Computes standardized zscores using mean and mads estimates of location the medians and the measure of dispersion specified in scale 

    loc='mean'
X=randn(10,2);
Z=zscoreFS(X,loc,'mad');

  • Remove the medians and divide by Qn.

    • n=200;
v=1;
randn('state', 123456);
Y=randn(n,v);
% Contaminated data
Ycont=Y;
Ycont(1:5,:)=Ycont(1:5,:)+10;
[out]=zscoreFS(Ycont,[],'Qn');
% Alternatively it is possible to use the following sintax
[out]=zscoreFS(Ycont,'median','Qn');

  • Examples with 3D arrays.

    • n=200;
v=3;
q=5;
randn('state', 123456);
Y=randn(n,v,q);
% Contaminated data
Ycont=Y;
Ycont(1:5,:,:)=Ycont(1:5,:,:)+10;
[out1,Mu,Sigma]=zscoreFS(Ycont,[],'Sn',1);
% [out,Mu1,Sigma1]=zscoreFS(Ycont,[],'Sn',1);

  • Report also location and scale measures which have have been used.

    • zscoreFS produces the same output as function zscore of statistics toolbox if centroid is arithmetic mean and scale measure is the standard deviation 

    X=randn(10,3,6);
[Z,mu,sig]=zscoreFS(X,'mean','std',3);
[Z1,mu1,sig1]=zscore(X,[],3);
if isequal(Z,Z1) + isequal(mu,mu1) + isequal(sig,sig) ==3
disp('Everything is equal')
else
disp('Equality not reached')
end

  • 3D arrays with dim=1, dim=2 and dim=3.

    • n=200;
v=3;
q=5;
randn('state', 123456);
Y=randn(n,v,q);
% Contaminated data
Ycont=Y;
Ycont(1:5,:,:)=Ycont(1:5,:,:)+10;
scale='Qn';
loc='mean';
dim=2; % work along rows
[Z,Mu1,Sigma1]=zscoreFS(Ycont,loc,scale,dim);
isequal(Z(3,:,2)',zscoreFS(Ycont(3,:,2),loc,scale))
scale='Qn';
loc='median';
dim=1; % work along columns
[Z,Mu1,Sigma1]=zscoreFS(Ycont,loc,scale,dim);
isequal(Z(:,2,4),zscoreFS(Ycont(:,2,4),loc,scale))
scale='Sn';
loc='median';
dim=3; % work along third dimension
[Z,Mu1,Sigma1]=zscoreFS(Ycont,loc,scale,dim);
isequal(squeeze(Z(7,2,:)),zscoreFS(squeeze(Ycont(7,2,:)),loc,scale))

    Related Examples

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  • Example of use of modmad as a scale measure.

    • p=3;
X=randn(100,p);
loc='median';
scale=['modmad' num2str(p)];
% Project the data using v vectors   
v=10;
proj=randn(p,v);
Y=X*proj;
% Standardize the n projected points using median and modified MAD
% Note that Y has v columns but the original matrix X has p columns
[Z,Mu1,Sigma1]=zscoreFS(Y,loc,scale);

    Input Arguments

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    X — Input data. Vector or Matrix or 3D array.

    Vector of length n or data matrix containing n observations on v variables or 3D array of size n-by-v-by-r.

    Missing values (NaN's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

    Data Types: single|double

    Optional Arguments

    loc — location measure to use. 'median' (default) or 'mean'.

    String which specifies the location measure to use. The default value is 'median'.

    Example: 'median'

    Data Types: character

    scale — scale measure to use. 'mad' (default) or 'Qn' or 'Sn' or 'std' or moddmadp'.

    String which specifies the dispersion measure to use 'mad' is the default. Traditional (corrected) mad is $Me(|x_i-Me(X)|)/norminv(3/4)$;

    'Qn' first quartile of interpoint distances $|x_i-x_j|$ corrected by the consistency factor. See function Qn.m;

    'Sn' robust Gini's average difference index corrected by the consistency factor. See function Sn.m;

    'std' Unbiased standard deviations. See function std.m;

    'modmadp'. Modified mad where the last letter(s) p of string modmap is (are) a number converted to string necessary to compute the modified MAD.

    Modified MAD = (order statistic $ceil((n+p-1)/2)$ of $|x_i-Me(X)|$ + order statistic $floor((n+p-1)/2+1)$ of $|x_i-Me(X)|)$ / $(2 \sigma)$ where $\sigma= norminv(0.5*((n+p-1)/(2*n)+1))$.

    Note that $p$ is different from $v$ (columns of X if X is a matrix) and must be supplied by the user.

    For example if p=5 then the user can supply the string 'modmad5' as follows. p=5; modmadp=['modmap' num2str(p)];

    Example: 'mad'

    Data Types: character

    dim — Dimension to operate along. Positive integer scalar.

    Dimension to operate along, specified as a positive integer scalar. If no value is specified, then the default is the first array dimension whose size does not equal 1.

    Example: 2

    Data Types: ingle | double | int8 | int16 | int32 | int64 |uint8 | uint16 | uint32 | uint64

    Output Arguments

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    Z —centered, scaled version of X. Array with the same dimension as input X

    Array with the same size as X using location and scale are specified in input parameters 'loc' and 'scale'. For vector input X, Z is the vector of z-scores (X-location(X)) ./ scale(X).

    mu —location estimate. Scalar, vector or matrix depending on the size of input matrix X

    Estimates of location specified in loc input string.

    sigma —scale estimate. Scalar, vector or matrix depending on the size of input matrix X

    Estimates of scale specified in scale input string.

    References

    See Also

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