RKrho

RKrho computes rho function for Rocke (translated Tukey's) biweight

Syntax

Description

example

rhoRK =RKrho(u, c, M) Plot of rho function.

Examples

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  • Plot of rho 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);
        rhoRK=RKrho(x,c,M);
        % rhoRK=rhoRK/max(rhoRK);
        plot(x,rhoRK,'LineWidth',2)
        xlabel('$u$','Interpreter','Latex')
        ylabel('$\rho (u,c,M)$','Interpreter','Latex')
        title('$\rho (u,c,M)$','Interpreter','Latex')
        hold('on')
        stem(M,M^2/2,'LineStyle',':','LineWidth',1)
        text(M,0,'M')
        stem(M+c,M^2/2+c*(5*c+16*M)/30,'LineStyle',':','LineWidth',1)
            text(M+c,0,'M+c')
    
    

    Related Examples

  • Compare Rocke rho functions 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);
        rhoRK030=RKrho(x,c,M);
        rhoRK030=rhoRK030/max(rhoRK030);
        plot(x,rhoRK030,'LineStyle','-','LineWidth',lwd)
    
        % Use bdp=0.4
        bdp=0.4;
        [c,M]=RKbdp(bdp,v,ARP);
        rhoRK040=RKrho(x,c,M);
        rhoRK040=rhoRK040/max(rhoRK040);
        plot(x,rhoRK040,'LineStyle','-.','LineWidth',lwd)
    
        % Use bdp=0.5
        bdp=0.5;
        [c,M]=RKbdp(bdp,v,ARP);
        rhoRK050=RKrho(x,c,M);
        rhoRK050=rhoRK050/max(rhoRK050);
        plot(x,rhoRK050,'LineStyle','--','LineWidth',lwd)
        
        xlabel('$x$','Interpreter','Latex','FontSize',16)
        ylabel('RK. Normalized $\rho(x,c,M)$','Interpreter','Latex','FontSize',20)
        legend({'$bdp=0.3$', '$bdp=0.4$' '$bdp=0.5$'},'Interpreter','Latex','Location','SouthEast','FontSize',16)
    

    Input Arguments

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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 RKrho transforms vector u as follows \[ TBrho(u)= \left\{ \begin{array}{cc} \frac{u^2}{2} & 0\leq |u| \leq M \\ \frac{M^2}{2} -M^2\frac{M^4-5 M^2 c^2 + 15c^4}{30c^4} + u^2 \left( 0.5+ \frac{M^4}{2c^4} -\frac{M^2}{c^2} \right) \\ +u^3 \left( \frac{4M}{3c^2} -\frac{4 M^3}{3c^4} \right) +u^4 \left( \frac{3M^2}{2c^4}- \frac{1}{2c^2} \right) \\ -4M \frac{u^5}{5c^4} + \frac{u^6}{6c^4} & M \leq u \leq M+c \\ \frac{M^2}{2} + \frac{c(5c+ 16M)}{30} & u > M+c \\ \end{array} \right. \]

    See equation (2.20) p. 1333 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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