# MMmulteda

MMmulteda computes MM estimators in multivariate analysis for a series of values of eff

## Syntax

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

## Description

 out =MMmulteda(Y) MMmult with all default options.

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

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

## Examples

expand all

### MMmult with all default options.

    load('swiss_banknotes');
Y=swiss_banknotes.data;
Y=Y(1:100,:);
[outMM]=MMmulteda(Y);
malfwdplot(outMM);

Total estimated time to complete S estimator:  0.65 seconds


### MMmult with optional arguments.

    Y = load('geyser.txt');
[out1]=MMmulteda(Y,'conflev',0.99,'plots',1);

Total estimated time to complete S estimator:  0.77 seconds


### MMmulteda with exctracted subsamples.

    load('swiss_banknotes');
Y=swiss_banknotes.data;
Y=Y(1:100,:);
[outMM,C]=MMmulteda(Y);


## 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:  'InitialEst',[] , 'Snsamp',1000 , 'Sbdp',0.4 , 'Sbestr',10 , 'Sminsctol',1e-7 , 'Snsamp',1000 , 'Srefsteps',0 , 'Sreftol',1e-8 , 'Srefstepsbestr',10 , 'Sreftolbestr',1e-10 , 'eff',[0.85 0.90 0.95 0.99] , 'effshape',1 , 'refsteps',10 , 'tol',1e-10 , 'conflev',0.99 , 'plots',0 , 'nocheck',1 

### InitialEst —starting values of the MM-estimator.[] (default) | structure.

InitialEst must contain the following fields:

Value Description
loc0

1 x v vector (estimate of the centroid)

shape0

v x v matrix (estimate of the shape matrix)

auxscale

scalar (estimate of the scale parameter).

If InitialEst is empty (default) program uses S estimators. In this last case it is possible to specify the options given in function Smult.

Example:  'InitialEst',[] 

Data Types: struct

### Soptions —options if initial estimator is S and InitialEst is empty.srhofunc,Snsamp,Srefsteps, Sreftol, Srefstepsbestr, Sreftolbestr, Sminsctol, Sbestr.

See function Smult for more details on these options.

It is necessary to add to the S options the letter S at the beginning. For example, if you want to use the optimal rho function the supplied option is 'Srhofunc','optimal'. For example, if you want to use 3000 subsets, the supplied option is 'Snsamp',3000

Example:  'Snsamp',1000 

Data Types: single | double

### Sbdp —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:  'Sbdp',0.4 

Data Types: double

### Sbestr —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:  'Sbestr',10 

Data Types: single | double

### Sminsctol —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:  'Sminsctol',1e-7 

Data Types: single | double

### Snsamp —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:  'Snsamp',1000 

Data Types: single | double

### Srefsteps —Number of refining iterations.scalar.

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

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

Example:  'Srefsteps',0 

Data Types: single | double

### Sreftol —scalar.default value of tolerance for the refining steps.

The default value is 1e-6;

Example:  'Sreftol',1e-8 

Data Types: single | double

### Srefstepsbestr —number of refining iterations for each best subset.scalar.

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

Example:  'Srefstepsbestr',10 

Data Types: single | double

### Sreftolbestr —Tolerance for the refining steps.scalar.

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

Example:  'Sreftolbestr',1e-10 

Data Types: single | double MM options

### eff —nominal efficiency.scalar | vector.

Vector defining nominal efficiency (i.e. a series of numbers between 0.5 and 0.99). The default value is the sequence 0.5:0.01:0.99 Asymptotic nominal efficiency is:

$(\int \psi' d\Phi)^2 / (\psi^2 d\Phi)$

Example:  'eff',[0.85 0.90 0.95 0.99] 

Data Types: double

### effshape —location or scale effiicency.dummy scalar.

If effshape=1 efficiency refers to shape efficiency else (default) efficiency refers to location

Example:  'effshape',1 

Data Types: double

### refsteps —Maximum iterations.scalar.

Scalar defining maximum number of iterations in the MM loop. Default value is 100.

Example:  'refsteps',10 

Data Types: double

### tol —Tolerance.scalar.

Scalar controlling tolerance in the MM loop.

Default value is 1e-7

Example:  'tol',1e-10 

Data Types: 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

### 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

### 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

## Output Arguments

### out — description Structure

A structure containing the following fields

Value Description
Loc

length(bdp)-by-v matrix containing MM estimate of location for each value of eff

Shape

v-by-v-by-length(eff) 3D array containing robust estimate of the shape for each value of eff. Remark det|shape|=1

Scale

length(eff) vector. Robust estimate of the scale for each value of eff

Cov

v-by-v-by-length(eff) 3D array containing robust estimate of Note that out.scale(i)^2 * out.shape(:,:,i) = robust estimate of covariance matrix

bs

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

MAL

n x length(eff) matrix containing the estimates of the robust Mahalanobis distances (in squared units) for each value of eff

Outliers

n-by-length(bdp) matrix containing true for the outliers.

It is a Boolean matrix containing the list of the units declared as outliers for each value of bdp using confidence level specified in input scalar conflev

Weights

n x length(eff) matrix containing the weights for each value of eff

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

eff

vector which contains the values of efficiency which have been used

Y

Data matrix Y.

class

'MMmulteda'

## References

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