RKpsi

RKpsi computes psi function for Rocke (translated Tukey's) biweight

Syntax

Description

example

psiRK =RKpsi(u, c, M) Plot of psi function.

Examples

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  • Plot of psi function.
  • close all
    % Find the values of c and M given bdp=0.4 and v=5 for ARP=0.01
    x=0:0.01:5;
    bdp=0.4;
    v=5;
    ARP=0.01;
    [c,M]=RKbdp(bdp,v,ARP);
    psiRK=RKpsi(x,c,M);
    % psiRK=psiRK/max(psiRK);
    plot(x,psiRK,'LineWidth',2)
    xlabel('$u$','Interpreter','Latex')
    ylabel('$\psi (u,c,M)$','Interpreter','Latex')
    title('$\psi (u,c,M)$','Interpreter','Latex')
    hold('on')
    stem(M,M,'LineStyle',':','LineWidth',1)
    text(M,0,'M')
    stem(M+c,0,'LineStyle',':','LineWidth',1)
    text(M+c,0,'M+c')
    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>

    Related Examples

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  • Compare Rocke psi function for 3 different values of bdp.
  • close all
    x=0:0.01:6;
    % Number of variables v is fixed to 5
    v=5;
    % ARP is fixed to 0.01
    ARP=0.01;
    lwd=2;
    hold('on')
    % Use bdp=0.3
    bdp=0.3;
    [c,M]=RKbdp(bdp,v,ARP);
    psiRK030=RKpsi(x,c,M);
    psiRK030=psiRK030/max(psiRK030);
    plot(x,psiRK030,'LineStyle','-','LineWidth',lwd)
    % Use bdp=0.4
    bdp=0.4;
    [c,M]=RKbdp(bdp,v,ARP);
    psiRK040=RKpsi(x,c,M);
    psiRK040=psiRK040/max(psiRK040);
    plot(x,psiRK040,'LineStyle','-.','LineWidth',lwd)
    % Use bdp=0.5
    bdp=0.5;
    [c,M]=RKbdp(bdp,v,ARP);
    psiRK050=RKpsi(x,c,M);
    psiRK050=psiRK050/max(psiRK050);
    plot(x,psiRK050,'LineStyle','--','LineWidth',lwd)
    xlabel('$x$','Interpreter','Latex','FontSize',16)
    ylabel('RK. Normalized $\psi(x,c,M)$','Interpreter','Latex','FontSize',20)
    legend({'$bdp=0.3$', '$bdp=0.4$' '$bdp=0.5$'},'Interpreter','Latex','Location','SouthEast','FontSize',16)
    Click here for the graphical output of this example (link to Ro.S.A. website)

    Input Arguments

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    u — scaled residuals or Mahalanobis distances. Vector.

    n x 1 vector containing residuals or Mahalanobis distances for the n units of the sample

    Data Types: single| double

    c — tuning parameter. Scalar.

    Non negative scalar which (together with the other optional parameter M) controls the robustness/efficiency of the estimator (beta in regression or mu in the location case ...).

    Data Types: single| double

    M — tuning parameter. Scalar.

    Scalar greater than 0 which (together with the other optional parameter c) controls the robustness/efficiency of the estimator (beta in regression or mu in the location case ...).

    Data Types: single| double

    More About

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    Additional Details

    function RKpsi transforms vector u as follows \[ RKpsi(u)= \left\{ \begin{array}{cc} u & 0 \leq u \leq M \\ u \left(1-\left( \frac{u-M}{c} \right)^2 \right)^2 & M < u \leq M+c \\ 0 & u > M+c \\ \end{array} \right. \]

    See equation (2.19) p. 1332 of Rocke (1996).

    References

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

    Rocke D.M. (1996), Robustness properties of S-estimators of multivariate location and shape in high dimension, "The Annals of Statistics", Vol. 24, pp. 1327-1345.

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