RKrho

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

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Syntax

Description

example

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

Examples

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  • Plot of rho function.

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% 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')
    Click here for the graphical output of this example (link to Ro.S.A. website).

    Related Examples

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  • Compare Rocke rho functions for 3 different values of bdp.

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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)
    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

    Output Arguments

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    rhoRK —Rocke rho (translated Tukey's biweight) rho associated to the residuals or Mahalanobis distances for the n units of the sample. n -by- 1 vector

    More About

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    Additional Details

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

    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.

    See Also

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