MMregeda

MMregeda computes MM estimator in linear regression for a series of values of efficiency

Syntax

Description

example

out =MMregeda(y, X) MMregeda with all default options.

example

out =MMregeda(y, X, Name, Value) MMregeda with optional input arguments.

example

[out , varargout] =MMregeda(___) Comparing the output of different MMreg runs.

Examples

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  • MMregeda with all default options.
  • n=200;
    p=3;
    randn('state', 123456);
    X=randn(n,p);
    % Uncontaminated data
    y=randn(n,1);
    % Contaminated data
    ycont=y;
    ycont(1:5)=ycont(1:5)+6;
    [out]=MMregeda(ycont,X);

  • MMregeda with optional input arguments.
  • MMregeda using the optimal rho function

    n=200;
    p=3;
    randn('state', 123456);
    X=randn(n,p);
    % Uncontaminated data
    y=randn(n,1);
    % Contaminated data
    ycont=y;
    ycont(1:5)=ycont(1:5)+6;
    [out]=MMregeda(ycont,X,'Srhofunc','optimal','rhofunc','optimal');

  • Comparing the output of different MMreg runs.
  • state=100;
    randn('state', state);
    n=100;
    X=randn(n,3);
    bet=[3;4;5];
    y=3*randn(n,1)+X*bet;
    y(1:20)=y(1:20)+13;
    %For outlier detection we consider both the nominal individual 1%
    %significance level and the simultaneous Bonferroni confidence level.
    % Define nominal confidence level
    conflev=[0.99,1-0.01/length(y)];
    % Define number of subsets
    nsamp=3000;
    % MM  estimators
    [out]=MMregeda(y,X,'conflev',conflev(1));
    laby='Scaled MM residuals';
    resfwdplot(out)
    Total estimated time to complete S estimate:  0.05 seconds 
    
    Click here for the graphical output of this example (link to Ro.S.A. website). Graphical output could not be included in the installation file because toolboxes cannot be greater than 20MB. To load locally the image files, download zip file http://rosa.unipr.it/fsda/images.zip and unzip it to <tt>(docroot)/FSDA/images</tt> or simply run routine <tt>downloadGraphicalOutput.m</tt>

    Input Arguments

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    y — Response variable. Vector.

    A vector with n elements that contains the response variable. y can be either a row or a column vector.

    Data Types: single| double

    X — Data matrix of explanatory variables (also called 'regressors') of dimension (n x p-1). Rows of X 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: 'conflev',0.99 , 'eff',[0.85 0.90 0.95 0.99] , 'effshape',1 , 'InitialEst',[] , 'intercept',false , 'nocheck',true , 'refsteps',10 , 'rhofunc','optimal' , 'rhofuncparam',5 , 'Snsamp',1000 , 'tol',1e-10 , 'plots',0

    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

    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

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

    InitialEst must contain the following fields

    Value Description
    beta

    v x 1 vector (estimate of the initial regression coefficients)

    scale

    scalar (estimate of the scale parameter).

    If InitialEst is empty (default) or InitialEst.beta continas NaN values, program uses S estimators. In this last case it is possible to specify the options given in function Sreg.

    Example: 'InitialEst',[]

    Data Types: struct or empty value

    intercept —Indicator for constant term.true (default) | false.

    Indicator for the constant term (intercept) in the fit, specified as the comma-separated pair consisting of 'Intercept' and either true to include or false to remove the constant term from the model.

    Example: 'intercept',false

    Data Types: boolean

    nocheck —Check input arguments.boolean.

    If nocheck is equal to true no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added.

    As default nocheck=false.

    Example: 'nocheck',true

    Data Types: boolean

    refsteps —Maximum iterations.scalar.

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

    Example: 'refsteps',10

    Data Types: double

    rhofunc —rho function.string.

    String which specifies the rho function which must be used to weight the residuals in MM step.

    Possible values are 'bisquare';

    'optimal';

    'hyperbolic';

    'hampel';

    'mdpd'.

    'bisquare' uses Tukey's $\rho$ and $\psi$ functions.

    See TBrho and TBpsi.

    'optimal' uses optimal $\rho$ and $\psi$ functions.

    See OPTrho and OPTpsi.

    'hyperbolic' uses hyperbolic $\rho$ and $\psi$ functions.

    See HYPrho and HYPpsi.

    'hampel' uses Hampel $\rho$ and $\psi$ functions.

    See HArho and HApsi.

    'mdpd' uses Minimum Density Power Divergence $\rho$ and $\psi$ functions.

    See PDrho and PDpsi.

    The default is bisquare

    Example: 'rhofunc','optimal'

    Data Types: char

    rhofuncparam —Additional parameters for the specified rho function in the MM step.scalar | vector.

    For hyperbolic rho function it is possible to set up the value of k = sup CVC (the default value of k is 4.5).

    For Hampel rho function it is possible to define parameters a, b and c (the default values are a=2, b=4, c=8)

    Example: 'rhofuncparam',5

    Data Types: single | double

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

    See function Sreg 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.

    Note that the rho function which is used in the MMstep is the same as the one used in the S step.

    Example: 'Snsamp',1000

    Data Types: single | double

    tol —Tolerance.scalar.

    Scalar controlling tolerance in the MM loop.

    Default value is 1e-7

    Example: 'tol',1e-10

    Data Types: double

    plots —Plot on the screen.scalar | structure.

    If plots = 1, generates a plot with the robust residuals against index number. The confidence level used to draw the confidence bands for the residuals is given by the input option conflev. If conflev is not specified a nominal 0.975 confidence interval will be used.

    Example: 'plots',0

    Data Types: single | double

    Output Arguments

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    out — description Structure

    A structure containing the following fields

    Value Description
    auxscale

    scalar, S estimate of the scale (or supplied external estimate of scale, if option InitialEst is not empty)

    Beta

    p x length(eff) matrix containing MM estimate of regression coefficients for each value of eff

    RES

    n x length(eff) matrix containing scaled MM residuals for each value of eff out.RES(:,jj)=(y-X*out.Beta(:,jj))/out.auxscale

    Weights

    n x length(eff) matrix. Weights assigned to each observation for each value of eff out.Outliers : n x length(eff) Boolean matrix containing the outliers which have been found for each value of eff.

    out.Sbeta : p x 1 vector containing S estimate of regression coefficients (or supplied initial external estimate of regression coefficients, if option InitialEst is not empty)

    Ssingsub

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

    conflev

    Confidence level that was used to declare outliers

    rhofunc

    string identifying the rho function which has been used

    rhofuncparam

    vector which contains the additional parameters for the specified rho function which have been used. For hyperbolic rho function the value of k =sup CVC. For Hampel rho function the parameters a, b and c

    Outliers

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

    eff

    vector containing the value of eff which have been used.

    Sbeta

    vector. S initial estimate of regression coefficients.

    y

    response vector y.

    X

    data matrix X.

    class

    'MMregeda'.

    References

    Riani, M., Cerioli, A., Atkinson, A.C. and Perrotta, D. (2014), Monitoring Robust Regression, "Electronic Journal of Statistics", Vol. 8, pp. 646-677.

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

    Acknowledgements

    See Also

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