RKpsider

RKpsider computes derivative of psi function (second derivative of rho function) for Rocke (translated Tukey's) biweight

Syntax

Description

example

psiderRK =RKpsider(u, c, M) Plot of psider function.

Examples

expand all

  • Plot of psider 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);
        psiderRK=RKpsider(x,c,M);
        % psiderRK=psiderRK/max(psiderRK);
        plot(x,psiderRK,'LineWidth',2)
        xlabel('$u$','Interpreter','Latex')
        ylabel('$\psi'' (u,c,M)$','Interpreter','Latex')
        title('$\psi'' (u,c,M)$','Interpreter','Latex')
        hold('on')
        stem(M,1,'LineStyle',':','LineWidth',1)
        text(M,0,'M')
        stem(M+c,0,'LineStyle',':','LineWidth',1)
        text(M+c,0,'M+c')
    

    Related Examples

  • Compare Rocke psider 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);
        psiderRK030=RKpsider(x,c,M);
        plot(x,psiderRK030,'LineStyle','-','LineWidth',lwd)
    
        % Use bdp=0.4
        bdp=0.4;
        [c,M]=RKbdp(bdp,v,ARP);
        psiderRK040=RKpsider(x,c,M);
        plot(x,psiderRK040,'LineStyle','-.','LineWidth',lwd)
    
        % Use bdp=0.5
        bdp=0.5;
        [c,M]=RKbdp(bdp,v,ARP);
        psiderRK050=RKpsider(x,c,M);
        plot(x,psiderRK050,'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)
    

    Input Arguments

    expand all

    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

    expand all

    Additional Details

    function RKpsider transforms vector u as follows:

    \[ RKpsider(u)= \left\{ \begin{array}{cc} 1 & 1 \leq |u| \leq M \\ \left(1-\left( \frac{u-M}{c} \right)^2 \right)\left(1-\left( \frac{u-M}{c} \right)^2 -4u \frac{u-M}{c^2} \right) & M \leq u \leq M+c \\ 0 & u > M+c \\ \end{array} \right. \]

    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.

    This page has been automatically generated by our routine publishFS