Sregeda

Sregeda computes S estimators in linear regression for a series of values of bdp

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Syntax

Description

example

out =Sregeda(y, X) Sregeda with msg=0.

example

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

example

[out , varargout] =Sregeda(___) Sregeda with hyperbolic rho function.

Examples

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  • Sregeda with msg=0.

    • Run this code to see the output shown in the help file 

    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]=Sregeda(ycont,X,'msg',0);
resfwdplot(out)
ylabel('Scaled S residuals');
    Click here for the graphical output of this example (link to Ro.S.A. website).

  • Sreg with optional input arguments.

    • Sreg with optimal rho function. Run this code to see the output shown in the help file 

    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]=Sregeda(ycont,X,'rhofunc','optimal');

  • Sregeda with hyperbolic rho function.

    • Run this code to see the output shown in the help file 

    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]=Sregeda(ycont,X,'rhofunc','hyperbolic');

    Related Examples

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  • Sreg on Stars data.

    • Run this code to see the Figure 2 of the article in the References 

    load('stars');
X=stars{:,1};
y=stars{:,2};
[out]=Sregeda(y,X,'rhofunc','bisquare');
standard.Color={'b'}
standard.xvalues=size(out.RES,1)-size(out.RES,2)+1:size(out.RES,1)
fground.Color={'r'}
resfwdplot(out,'standard',standard,'fground',fground)
ylabel('Scaled S residuals');
RHO = [];
for i=1:49
RHO(i,1) = corr(out.RES(:,i),out.RES(:,i+1),'type','Spearman');
RHO(i,2) = corr(out.RES(:,i),out.RES(:,i+1),'type','Kendall');
RHO(i,3) = corr(out.RES(:,i),out.RES(:,i+1),'type','Pearson');
end
minc = min(RHO);
maxc = max(RHO);
ylimits = [min(minc)*0.8,max(maxc)*1.1];
figure;
subplot(3,1,1);
plot(out.bdp(1:49),RHO(:,1)');
if strcmp(out.class,'Sregeda')
set(gca,'XDir','reverse','ylim',ylimits);
title('Spearman');
end
subplot(3,1,2);
plot(out.bdp(1:49),RHO(:,2)');
if strcmp(out.class,'Sregeda')
set(gca,'XDir','reverse','ylim',ylimits);
title('Kendall');
end
subplot(3,1,3);
plot(out.bdp(1:49),RHO(:,3)');
if strcmp(out.class,'Sregeda')
set(gca,'XDir','reverse','ylim',ylimits);
title('Pearson');
end
    Total estimated time to complete S estimate:  0.33 seconds 
Total estimated time to complete S estimate:  0.20 seconds 
Total estimated time to complete S estimate:  0.21 seconds 
Total estimated time to complete S estimate:  0.17 seconds 
Total estimated time to complete S estimate:  0.18 seconds 
Total estimated time to complete S estimate:  0.17 seconds 
Total estimated time to complete S estimate:  0.17 seconds 
Total estimated time to complete S estimate:  0.19 seconds 
Total estimated time to complete S estimate:  0.16 seconds 
Total estimated time to complete S estimate:  0.16 seconds 
Total estimated time to complete S estimate:  0.17 seconds 
Total estimated time to complete S estimate:  0.16 seconds 
Total estimated time to complete S estimate:  0.16 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.18 seconds 
Total estimated time to complete S estimate:  0.16 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.16 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.16 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.16 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.16 seconds 
Total estimated time to complete S estimate:  0.16 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.13 seconds 
Total estimated time to complete S estimate:  0.20 seconds 
Total estimated time to complete S estimate:  0.15 seconds 
Total estimated time to complete S estimate:  0.10 seconds 
Total estimated time to complete S estimate:  0.09 seconds 
Total estimated time to complete S estimate:  0.09 seconds 
Total estimated time to complete S estimate:  0.09 seconds 
Total estimated time to complete S estimate:  0.09 seconds 
Total estimated time to complete S estimate:  0.09 seconds 
Total estimated time to complete S estimate:  0.09 seconds 
Total estimated time to complete S estimate:  0.08 seconds 
Total estimated time to complete S estimate:  0.08 seconds 
Total estimated time to complete S estimate:  0.10 seconds 
Total estimated time to complete S estimate:  0.08 seconds 
Total estimated time to complete S estimate:  0.08 seconds 
Total estimated time to complete S estimate:  0.08 seconds 

standard = 

  struct with fields:

    Color: {'b'}


standard = 

  struct with fields:

      Color: {'b'}
    xvalues: [-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 … ] (1×50 double)


fground = 

  struct with fields:

    Color: {'r'}
    Click here for the graphical output of this example (link to Ro.S.A. website)

    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: 'bdp',[0.5 0.4 0.3 0.2 0.1] , 'bestr',10 , 'conflev',0.99 , 'tstattype',3 , 'intercept',false , 'minsctol',1e-7 , 'msg',0 , 'nocheck',true , 'nsamp',1000 , 'refsteps',10 , 'refstepsbestr',10 , 'reftol',1e-05 , 'reftolbestr',1e-10 , 'rhofunc','optimal' , 'rhofuncparam',5 , 'plots',0

    bdp —breakdown point.scalar | vector.

    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.

    The default for bdp is a sequence from 0.5 to 0.01 with step -0.01. The sequence is forced to be monotonically decreasing.

    Example: 'bdp',[0.5 0.4 0.3 0.2 0.1]

    Data Types: 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

    covrob —scalar.a number in the set 0, 1, .

    .., 5 which specifies the type of covariance matrix of robust beta coefficients.

    These numbers correspond to estimators covrob, covrob1, covrob2, covrob4, covrob4 and covrobc detailed inside file RobCov. The default value is 5 (i.e. estimator covrobc).

    Example: 'tstattype',3

    Data Types: single | double

    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

    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

    msg —Level of output to display.scalar.

    It controls whether to display or not messages on the screen.

    If msg==1 (default) messages are displayed on the screen about estimated time to compute the estimator and the proportion of singular elemental subsets, if this proportion exceeds 0.1.

    Note that the warnings about: 'MATLAB:rankDeficientMatrix', 'MATLAB:singularMatrix' and 'MATLAB:nearlySingularMatrix' are always set to off else no message is displayed on the screen

    Example: 'msg',0

    Data Types: single | double

    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

    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',10

    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

    reftol —tolerance for the refining steps.scalar.

    The default value is 1e-6;

    Example: 'reftol',1e-05

    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

    rhofunc —rho function.string.

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

    Possible values are 'bisquare' 'optimal' 'hyperbolic' 'hampel' 'mdpd'.

    'AS'.

    '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.

    'AS' uses Andrews' sine $\rho$ and $\psi$ functions.

    See ASrho and ASpsi.

    The default is bisquare

    Example: 'rhofunc','optimal'

    Data Types: character

    rhofuncparam —Additional parameters for the specified rho function.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

    plots —Plot on the screen.scalar.

    If plots = 1, generates a plot with the robust residuals for each value of bdp. 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
    Beta

    matrix of size length(bdp)-by-(p+1) containing the S estimator of regression coefficients for each value of bdp.

    The first column contains the value of bdp.
    tStat

    matrix of size length(bdp)-by-(p+1) containing the S estimator of t statistics for each value of bdp. The first column contains the value of bdp. The type of t stat which is monitored depends on option tstattype.
    Scale

    vector containing the estimate of the scale (sigma) for each value of bdp. This is the value of the objective function
    BS

    p x 1 vector containing the units forming best subset associated with S estimate of regression coefficient.
    RES

    n x length(bdp) matrix containing the robust scaled residuals for each value of bdp
    Weights

    n x length(bdp) vector containing the estimates of the weights for each value of bdp
    Outliers

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

    confidence level which has been used to declare outliers.
    Singsub

    Number of subsets wihtout full rank. Notice that out.singsub(bdp(jj)) > 0.1*(number of subsamples) produces a warning
    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. This field is present only if input argument 'rhofunc' is 'hyperbolic' or 'hampel'.
    bdp

    vector which contains the values of bdp which have been used
    y

    response vector y.
    X

    Data matrix of explanatory variables which has been used (it also contains the column of ones if input option intercept was missing or equal to 1)
    class

    'Sregeda'

    varargout —Indices of the subsamples extracted for computing the estimate (the so called elemental sets). C : matrix

    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.

    See Also

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