Smult

Smult computes S estimators in multivariate analysis

Syntax

• out =Smult(Y)example
• out =Smult(Y,Name,Value)example
• [out , varargout]=Smult(___)example

Description

 out =Smult(Y) Smult with all default options.

 out =Smult(Y, Name, Value) Smult with optional arguments.

 [out , varargout] =Smult(___) Smult with exctracted subsamples.

Examples

expand all

Smult with all default options.

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

Smult with optional arguments.

n=200;
v=3;
randn('state', 123456);
Y=randn(n,v);
% Contaminated data
Ycont=Y;
Ycont(1:5,:)=Ycont(1:5,:)+3;
[out]=Smult(Ycont,'plots',1);
Warning: Using 'state' to set RANDN's internal state causes RAND, RANDI, and
RANDN to use legacy random number generators.  This syntax is not recommended.
See <a href="matlab:helpview([docroot '\techdoc\math\math.map'],'update_random_number_generator')">Replace Discouraged Syntaxes of rand and randn</a> to use RNG to replace the
old syntax.
Total estimated time to complete S estimator:  1.80 seconds


Smult with exctracted subsamples.

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

Input Arguments

Y — Input data. Matrix.

n x v data matrix; n observations and v variables. Rows of Y represent observations, and columns represent variables.

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

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as  Name1,Value1,...,NameN,ValueN.

Example:  'bdp',0.4 , 'nsamp',1000 , 'refsteps',0 , 'reftol',1e-8 , 'refstepsbestr',10 , 'reftolbestr',1e-10 , 'minsctol',1e-7 , 'bestr',10 , 'conflev',0.99 , 'nocheck',1 , 'plots',0 , 'msg',0 , 'ysave',1 

bdp —breakdown point.scalar.

It measures the fraction of outliers the algorithm should resist. In this case any value greater than 0 but smaller or equal than 0.5 will do fine (default=0.5).

Note that given bdp nominal efficiency is automatically determined.

Example:  'bdp',0.4 

Data Types: double

nsamp —Number of subsamples which will be extracted to find the robust estimator.scalar.

If nsamp=0 all subsets will be extracted.

They will be (n choose p).

If the number of all possible subset is <1000 the default is to extract all subsets otherwise just 1000.

Example:  'nsamp',1000 

Data Types: single | double

refsteps —Number of refining iterations.scalar.

Number of refining iterationsin each subsample (default = 3).

refsteps = 0 means "raw-subsampling" without iterations.

Example:  'refsteps',0 

Data Types: single | double

reftol —scalar.default value of tolerance for the refining steps.

The default value is 1e-6;

Example:  'reftol',1e-8 

Data Types: single | double

refstepsbestr —number of refining iterations for each best subset.scalar.

Scalar defining number of refining iterations for each best subset (default = 50).

Example:  'refstepsbestr',10 

Data Types: single | double

reftolbestr —Tolerance for the refining steps.scalar.

Tolerance for the refining steps for each of the best subsets The default value is 1e-8;

Example:  'reftolbestr',1e-10 

Data Types: single | double

minsctol —tolerance for the iterative procedure for finding the minimum value of the scale.scalar.

Value of tolerance for the iterative procedure for finding the minimum value of the scale for each subset and each of the best subsets (It is used by subroutine minscale.m) The default value is 1e-7;

Example:  'minsctol',1e-7 

Data Types: single | double

bestr —number of "best betas" to remember.scalar.

Scalar defining number of "best betas" to remember from the subsamples.

These will be later iterated until convergence (default=5)

Example:  'bestr',10 

Data Types: single | double

conflev —Confidence level which is used to declare units as outliers.scalar.

Usually conflev=0.95, 0.975 0.99 (individual alpha) or 1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha).

Default value is 0.975

Example:  'conflev',0.99 

Data Types: double

nocheck —Check input arguments.scalar.

If nocheck is equal to 1 no check is performed on matrix Y. As default nocheck=0.

Example:  'nocheck',1 

Data Types: double

plots —Plot on the screen.scalar | structure.

If plots is a structure or scalar equal to 1, generates: (1) a plot of Mahalanobis distances against index number. The confidence level used to draw the confidence bands for the MD is given by the input option conflev. If conflev is not specified a nominal 0.975 confidence interval will be used.

(2) a scatter plot matrix with the outliers highlighted.

If plots is a structure it may contain the following fields

Value Description
labeladd

if this option is '1', the outliers in the spm are labelled with their unit row index. The default value is labeladd='', i.e. no label is added.

nameY

cell array of strings containing the labels of the variables. As default value, the labels which are added are Y1, ...Yv.

Example:  'plots',0 

Data Types: single | double

msg —Level of output to display.scalar.

If msg==1 (default) messages are displayed on the screen about estimated time to compute the final estimator else no message is displayed on the screen

Example:  'msg',0 

Data Types: single | double

ysave —save input matrix Y.scalar.

Scalar that is set to 1 to request that the data matrix Y is saved into the output structure out. This feature is meant at simplifying the use of function malindexplot.

Default is 0, i.e. no saving is done.

Example:  'ysave',1 

Data Types: double

Output Arguments

out — description Structure

A structure containing the following fields

Value Description
loc

1 x v vector containing S estimate of location

shape

v x v matrix containing robust estimate of the shape matrix. Remark det|shape|=1

scale

scalar, robust estimate of the scale

cov

out.scale^2 * out.shape = robust estimate of covariance matrix

bs

(v+1) x 1 vector containing the units forming best subset associated with S estimate of location.

md

n x 1 vector containing the estimates of the robust Mahalanobis distances (in squared units)

outliers

A vector containing the list of the units declared as outliers using confidence level specified in input scalar conflev

conflev

Confidence level that was used to declare outliers

singsub

Number of subsets without full rank. Notice that out.singsub > 0.1*(number of subsamples) produces a warning

weights

n x 1 vector containing the estimates of the weights

Y

Data matrix Y. The field is present only if option ysave is set to 1.

class

'Smult'

References

Maronna, R.A., Martin D. and Yohai V.J. (2006), "Robust Statistics, Theory and Methods", Wiley, New York.

Acknowledgements

This function follows the lines of MATLAB/R code developed during the years by many authors.

For more details see the R library robustbase http://robustbase.r-forge.r-project.org/ The core of these routines, e.g. the resampling approach, however, has been completely redesigned, with considerable increase of the computational performance.