RhoPsiWei

RhoPsiWei finds rho, psi, psi', w functions given bdp, or eff or tuning constant c

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Syntax

Description

example

out =RhoPsiWei(u, v) Example of call to RhoPsiWei with 2 input arguments.

example

out =RhoPsiWei(u, v, Name, Value) Example of call to RhoPsiWei with bdp prespecified.

Examples

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  • Example of call to RhoPsiWei with 2 input arguments.

    • In this case TB link is invoved with bdp=0.5; 

    n=20;
u=randn(n,1);
out=RhoPsiWei(u,1);

  • Example of call to RhoPsiWei with bdp prespecified.

    • n=20;
u=randn(n,1);
out=RhoPsiWei(u,1,'bdp',0.1);

    Related Examples

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  • Example of call to RhoPsiWei with bdp and rhofunc prespecified.

    • n=20;
u=randn(n,1);
rhofunc='PD';
out=RhoPsiWei(u,1,'bdp',0.1,'rhofunc',rhofunc);

  • Example of call to RhoPsiWei with eff and rhofunc prespecified.

    • n=20;
u=randn(n,1);
rhofunc='OPT';
eff=0.9;
out=RhoPsiWei(u,1,'eff',eff,'rhofunc',rhofunc);

  • Example of call to RhoPsiWei with c prespecified.

    • n=20;
u=randn(n,1);
rhofunc='HA';
c=1.5;
out=RhoPsiWei(u,1,'c',c,'rhofunc',rhofunc);

  • Example of call to RhoPsiWei with c prespecified, and personalized input parameters for HA link.

    • n=20;
u=randn(n,1);
rhofunc='HA';
rhofuncparam=[2 3 8];
eff=.95;
c=0.5;
out=RhoPsiWei(u,1,'eff',eff,'rhofunc',rhofunc,'rhofuncparam',rhofuncparam);

  • Example of call to RhoPsiWei with eff prespecified, and personalized input parameters fpr HYP link.

    • n=20;
u=-4:0.01:4;
rhofunc='HYP';
k=4.2;
eff=0.9;
out=RhoPsiWei(u,1,'eff',eff,'rhofunc',rhofunc,'rhofuncparam',k);
subplot(2,3,1)
plot(u,out.rho)
title('HYP: rho function')
subplot(2,3,2)
plot(u,out.psi)
title('HYP: psi function')
subplot(2,3,3)
plot(u,out.psider)
title('HYP: derivative of  psi function')
subplot(2,3,4)
plot(u,out.psider)
title('HYP: psi(x)*x function')
subplot(2,3,5)
plot(u,out.wei)
title('HYP: weight function')
    Click here for the graphical output of this example (link to Ro.S.A. website)

  • Example of calling RhoPsiWei with all parameters related to the constant: c, k, A, B and d First call RhoPsiWei with efficiency fixed and then use the calculated constant for a second call - it will be much faster n=20; u=-4:0.

    • n=20;
u=-4:0.01:4;
rhofunc='HYP';
k=4.2;
eff=0.9;
out=RhoPsiWei(u,1,'eff',eff,'rhofunc',rhofunc,'rhofuncparam',k);
out1=RhoPsiWei(u,1,'c',out.c1(1),'rhofunc',rhofunc,'rhofuncparam',out.c1(2:5));
isequal(out.rho, out1.rho)
isequal(out.psi, out1.psi)
[bdp1, eff1]=HYPc(out.c1(1), 1, 'k', out.c1(2), 'param', out.c1(3:5));
isequal(out.bdp, bdp1)
isequal(out.eff, eff1)
    ans =

  logical

   1


ans =

  logical

   1


ans =

  logical

   1


ans =

  logical

   0

  • Example of call to RhoPsiWei with both bdp and eff.

    • In this case, given that it is possible to supply just one between bdp, eff and c the call throws an error 

    n=20;
u=-4:0.01:4;
rhofunc='HYP';
k=4.2;
eff=0.85;
try
out=RhoPsiWei(u,1,'bdp',0.2,'eff',eff,'rhofunc',rhofunc,'rhofuncparam',k);
catch
disp('Only a value among bdp, eff and c must be supplied')
end

  • Example of call to RhoPsiWei with u empty In this case fiels psi, psider, psix and wei are empty n=20; u=randn(n,1); rhofunc='PD'; eff=0.

    • n=20;
u=randn(n,1);
rhofunc='PD';
eff=0.9;
out=RhoPsiWei([],1,'eff',eff,'rhofunc',rhofunc);

    Input Arguments

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    u — scaled residuals or Mahalanobis distances. Vector.

    Vector containing residuals or Mahalanobis distances for the n units of the sample. If u is empty fields rho, wei, psi, psix and psider of output structure out will be empty.

    Data Types: single|double

    v — number of response variables. Scalar.

    e.g. in regression v=1. So far v can be set to a value greater than 1 just for TB and OPT functions

    Data Types: single|double|int32|int64

    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',0.2 , 'c',3 , 'eff',0.9 , 'rhofunc','optimal' , 'rhofuncparam',5

    bdp —breakdown point.scalar.

    Scalar defining breakdown point (i.e a number between 0 and 0.5). If none of the optional input arguments c, bdp and eff, are not specified bdp is set to 0.5. Note that if c or eff are specified bdp is automatically found.

    Example: 'bdp',0.2

    Data Types: single|double

    c —tuning constant.scalar.

    Scalar defining the tuning constant which determines bdp or eff. c corresponds to alpha for mdpd link. Note that if bdp or eff are specified c is automatically found.

    Example: 'c',3

    Data Types: single|double

    eff —asymptotic efficiency.scalar.

    Scalar defining requested asymptotic efficiency of the estimate (in general a number in the range 0.5 0.99.

    Typical values of eff are 0.85, 0.90 or 0.95.

    Note that if bdp or c are specified eff is automatically found.

    Example: 'eff',0.9

    Data Types: single|double

    rhofunc —link function.string.

    String which specifies the rho function which must be used.

    Possible values are 'TB' or ''biweight', 'OPT' or 'optimal', 'HYP' or 'hyperbolic';

    'HA' or 'hampel', 'PD' or 'mdpd', 'HU' or 'huber', 'AS'.

    'biweight' 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.

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

    See HArho and HApsi.

    'AS' uses Andrew's 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). For Hyperbolic link function it is possible to pass either one parameter, k (default is k=4.5) or four parameters: k, A, B and d.

    Example: 'rhofuncparam',5

    Data Types: single | double

    Output Arguments

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    out — description Structure

    Structure which contains the following fields

    Value Description
    rho

    vector of length n which contains the rho function associated to the residuals or Mahalanobis distances for the n units of the sample. This field is empty if input u is empty.
    psi

    vector of length n which contains the psi function associated to the residuals or Mahalanobis distances for the n units of the sample. This field is empty if input u is empty.
    psider

    vector of length n which contains the derivative of the psi function associated to the residuals or Mahalanobis distances for the n units of the sample. This field is empty if input u is empty.
    wei

    vector of length n which contains the weights associated to the residuals or Mahalanobis distances for the n units of the sample. This field is empty if input u is empty.
    psix

    vector of length n which contains the psi function mutiplied by u associated to the residuals or Mahalanobis distances for the n units of the sample. This field is empty if input u is empty.
    class

    Character which specifies the link function which has be used.

    Possible values are 'bisquare', 'optimal', 'hyperbolic';

    'hampel', 'huber' or 'mdpd'.
    bdp

    scalar which contains the bdp which has been used.
    eff

    scalar which contains the eff which has been used.
    c1

    consistency factor (and other parameters) associated to required breakdown point or nominal efficiency.

    More precisely, out.c1(1) contains consistency factor c associated to required breakdown point or nominal efficiency out.c1(2:end) contain other parameters associated with the rho (psi) function.

    For example, if rhofunc='hampel', c1(2:4) must contain parameters (a, b and c) of Hampel rho function.

    If rhofunc is hyperbolic out.c1(1) specifies the value of the tuning constant c (scalar greater than 0 which controls the robustness/efficiency of the estimator) out.c1(2) contains the prefixed value k (sup of the change-of-variance sensitivity) and out.c1(3:5) contain parameters A, B and d.
    kc1

    Expectation of rho associated with out.c1(1) to get a consistent estimator at the model distribution kc1=E(rho)=sup(rho)*bdp

    References

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

    See Also

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