On some multivariate descriptive statistics based on multivariate signs and ranks

ON SOME MULTIVARIATE DESCRIPTIVE STATISTICS BASED ON MULTIVARIATE SIGNS AND RANKS NELUKA DEVPURA NATIONAL UNIVERSITY OF SINGAPORE 2004 ON SOME MULTIVARIATE DESCRIPTIVE STATISTICS BASED ON MULTIVARIATE SIGNS AND RANKS NELUKA DEVPURA (B.Sc.(Statistics) University of Colombo) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2004 i Acknowledgments I was helped by many people professionally and personally to complete and perfect the thesis Where words alone will not suffice to express my heartiest gratitude to those wonderful people, who assisted me and encouraged me to achieve the objectives of this thesis First of all, I owe an immense depth of gratitude to my supervisor Dr Biman Chakraborty who had provided me much needed support and unending encouragement throughout the thesis I truly appreciate all the time and effort he has spent in helping me and for the valuable comments and suggestions I wish to thank the staffs of my department for providing me very much support during my study and special thanks goes to my colleagues and friends for their generous help given to me during preparation of the thesis I would like to take this opportunity to thank my father Dharmasena Devpura and mother Lakshmi for looking after my daughter for last two years They have been supporting me all the way upto now by taking most of my burden onto them and thanks to them only, I have come so far in my life Finally, I would like to thank my husband and loving sisters for their support given I wish to contribute the completion of my thesis to my dearest family ii Contents Acknowledgment Summary xiv Introduction 1.1 Outline of the thesis Multivariate Medians 2.1 2.2 i 1 Notions of Multivariate Symmetry 2.1.1 Spherical Symmetry 2.1.2 Elliptical Symmetry 2.1.3 Central and Sign Symmetry 2.1.4 Angular and Halfspace Symmetry Notions of Multivariate Medians 2.2.1 Co-ordinatewise Median 10 CONTENTS 2.3 2.4 iii 2.2.2 Spatial Median 11 2.2.3 Convex Hull Peeling Median 12 2.2.4 Oja’s Simplex Volume Median 13 2.2.5 Liu’s Simplicial Median 15 2.2.6 Tukey’s Half-space Depth Median 16 Transformation Retransformation Based Approaches 16 2.3.1 Data Driven Co-ordinate System 17 2.3.2 Tyler’s Approach 18 Computing the TR Median Multivariate Quantiles, Signs and Ranks 3.1 3.2 23 3.1.1 Computing lp -Quantiles 26 3.1.2 Affine Equivariant lp-Quantiles 28 Multivariate Signs and Ranks 29 Quantile Contour Plot 32 Examples with Real Data Sets 40 Some Multivariate Descriptive Statistics 4.1 23 Multivariate lp-Quantiles 3.2.1 3.3 20 Scale Curves 45 45 CONTENTS 4.2 4.3 4.4 iv 4.1.1 Algorithm for Computation of Central Rank Regions 47 4.1.2 Affine Equivariant Scale Curve 48 4.1.3 Scale Curves for Real Data Sets 59 Bivariate Boxplots 64 4.2.1 Constructing Bivariate Boxplot 64 4.2.2 Affine Equivariant Boxplot 69 4.2.3 Examples with Real Data 72 Multivariate Kurtosis Curve 78 4.3.1 Applications 84 Multivariate Skewness Curve 91 4.4.1 99 Applications Multivariate Skew-Symmetric Distributions 106 5.1 Multivariate g-and-h Distribution 106 5.2 Conclusion 114 Appendix 117 Bibiliography 136 v List of Figures 3.1 Co-ordinatewise quantile contour plot for bivariate normal data 3.2 Co-ordinatewise quantile contour plot for bivariate Laplace distri- 33 bution 33 3.3 Co-ordinatewise quantile contour plot for t-distribution with d.f 34 3.4 Spatial quantile contour plot for bivariate normal data 35 3.5 Spatial quantile contour plot for bivariate Laplace distribution 35 3.6 Spatial quantile contour plot for t-distribution with d.f 36 3.7 Co-ordinatewise quantile contour plot for bivariate normal data with TR 3.8 Co-ordinatewise quantile contour plot for bivariate Laplace distribution with TR 3.9 37 37 Co-ordinatewise quantile contour plot for t-distribution with d.f using TR 38 3.10 Spatial quantile contour plot for bivariate normal with TR 38 LIST OF FIGURES vi 3.11 Spatial quantile contour plot for bivariate Laplace distribution with TR 39 3.12 Spatial quantile contour plot for t-distribution with d.f with TR 39 3.13 Quantile contour plots for the concentrations of cholesterol and triglycerides in the plasma of 320 patients 41 3.14 Quantile contour plots for the concentrations of PCB and thickness of shell data 42 3.15 Quantile contour plots for open book examination marks 43 3.16 Quantile contour plots for closed book examination marks 44 4.1 Scale curve for bivariate normal distribution with p = 49 4.2 Scale curve for bivariate normal distribution with p = using TR 50 4.3 Scale curve for bivariate Laplace distribution with p = 51 4.4 Scale curve for bivariate Laplace distribution with p = using TR 52 4.5 Scale curve for bivariate t-distribution with d.f with p = 52 4.6 Scale curve for bivariate t-distribution with d.f.and p = using TR 53 4.7 Scale curve for bivariate normal distribution with p = 53 4.8 Scale curve for bivariate normal distribution with p = using TR 54 4.9 Scale curve for bivariate Laplace distribution with p = 54 4.10 Scale curve for bivariate Laplace distribution with p = using TR 55 LIST OF FIGURES 4.11 Scale curve for bivariate t-distribution with d.f with p = vii 55 4.12 Scale curve for bivariate t-distribution with d.f with p = using TR 56 4.13 Scale curve for bivariate normal, bivariate Laplace and t4 with p = 57 4.14 Scale curve for bivariate normal, bivariate Laplace and t4 with p = 57 4.15 Scale curve for bivariate normal, Laplace and t4 with p = using TR 58 4.16 Scale curve for bivariate normal, Laplace and t4 with p = using TR 58 4.17 Scale curves with p = 1, with and without TR for the concentrations of cholesterol and triglycerides in the plasma of 320 patients 59 4.18 Scale curves with p = 2, with and without TR for the concentrations of cholesterol and triglycerides in the plasma of 320 Patients 60 4.19 Scale curves with p = 1, with and without TR for the concentrations of PCB and thickness of shell data 61 4.20 Scale curves with p = 2, with and without TR for the concentrations of PCB and thickness of shell data 61 4.21 Scale curves with p = 1, with and without TR for the open book examination marks 62 4.22 Scale curves with p = 2, with and without TR for the open book examination marks 62 LIST OF FIGURES viii 4.23 Scale curves with p = 1, with and without TR for the closed book examination marks 63 4.24 Scale curves with p = 2, with and without TR for the closed book examination marks 4.25 Boxplot with p = for bivariate normal data 63 66 4.26 Boxplot with p = for bivariate Laplace distribution 66 4.27 Boxplot with p = for t4 distribution 67 4.28 Boxplot with p = for bivariate normal distribution 68 4.29 Boxplot with p = for bivariate Laplace distribution 68 4.30 Boxplot with p = for t4 distribution 69 4.31 Boxplot with p = using TR for bivariate standard normal 70 4.32 Boxplot with p = using TR for bivariate Laplace distribution 71 4.33 Boxplot with p = using TR for t4 distribution 71 4.34 Boxplot with p = using TR for bivariate normal distribution 72 4.35 Boxplot with p = using TR for bivariate Laplace distribution 73 4.36 Boxlot with p = using TR for t4 distribution 73 4.37 Boxplots for the concentrations of cholesterol and triglycerides in the plasma of 320 patients 74 4.38 Boxplots for the concentrations of PCB and thickness of shell 75 APPENDIX if alpha ~= quant=[]; r=ysort(ceil(n*alpha)); u1=-r; for u2=-r:r/5:r u=[u1 u2]; qn=lpquantile(u,xdata,p) quant=[quant;qn]; end u2=r; for u1=-r:r/5:r u=[u1 u2]; qn=lpquantile(u,xdata,p); quant=[quant;qn]; end u1=r; for u2=r:-r/5:-r u=[u1 u2]; qn=lpquantile(u,xdata,p); quant=[quant;qn]; end u2=-r; for u1=r:-r/5:-r u=[u1 u2]; qn=lpquantile(u,xdata,p); quant=[quant;qn]; end if r~=0 v=[v; polyarea(quant(:,1),quant(:,2))]; end end end end if p~=1 for alpha=0.0:0.1:1.0 if alpha==0 v=0; end if alpha ~= quant=[]; r=ysort(ceil(n*alpha)); r1=-r:r/20:r; x1=real(((r^q)-abs(r1).^q).^(1/q)); u=[x1’ r1’;-x1’ -r1’]; for j=1:length(u) [qn, exitflag]=lpquantile(u(j,:),xdata,p); 125 APPENDIX 126 if exitflag == quant=[quant;qn]; end end v=[v; polyarea(quant(:,1),quant(:,2))]; end end end function v=volpoly_after(newdata,y2,p) % % This computes volume of a polygon when p>=1 after TR u=[0 0]; [n,d]=size(newdata) if p==1 q=inf; else q=p/(p-1); end for i=1:n y1=lprank(newdata(i,:),newdata,p); y(i,:)=norm(y1,q); end ysort=sort(y); v=[]; if p==1; for alpha=0.0:0.1:1.0 if alpha==0 v=0; end if alpha ~= quant=[]; r=ysort(ceil(n*alpha)); u1=-r; for u2=-r:r/5:r u=[u1 u2]; qn=lpquantile(u,newdata,p) quant=[quant;qn]; end u2=r; for u1=-r:r/5:r u=[u1 u2]; qn=lpquantile(u,newdata,p); quant=[quant;qn]; end APPENDIX u1=r; for u2=r:-r/5:-r u=[u1 u2]; qn=lpquantile(u,newdata,p); quant=[quant;qn]; end u2=-r; for u1=r:-r/5:-r u=[u1 u2]; qn=lpquantile(u,newdata,p); quant=[quant;qn]; end quant=quant*transpose(inv(y2)); if r~=0 v=[v; polyarea(quant(:,1),quant(:,2))]; end end end end if p~=1 for alpha=0.0:0.1:1.0 if alpha==0 v=0; end if alpha ~= quant=[]; r=ysort(ceil(n*alpha)); r1=-r:r/20:r; x1=real(((r^q)-abs(r1).^q).^(1/q)); u=[x1’ r1’;-x1’ -r1’]; for j=1:length(u) [qn, exitflag]=lpquantile(u(j,:),newdata,p); if exitflag == quant=[quant;qn]; end end quant=quant*transpose(inv(y2)); v=[v; polyarea(quant(:,1),quant(:,2))]; end end end 127 APPENDIX function []=boxplot(xdata,p); % u=zeros(1,2); if p==1 q=inf; else q=p/(p-1); end [n,d]=size(xdata); for i=1:n y1=lprank(xdata(i,:),xdata,p); y(i,:)=norm(y1,q) ; end ysort=sort(y); alpha=median(ysort); plot(xdata(:,1),xdata(:,2),’*’,’Markersize’,5); hold on alpha if p==1 quant=[]; r=ysort(ceil(n*alpha)) u1=-r; for u2=-r:r/5:r u=[u1 u2]; qn=lpquantile(u,xdata,p); quant=[quant;qn]; end u2=r; for u1=-r:r/5:r u=[u1 u2]; qn=lpquantile(u,xdata,p); quant=[quant;qn]; end u1=r; for u2=r:-r/5:-r u=[u1 u2]; qn=lpquantile(u,xdata,p); quant=[quant;qn]; end u2=-r; for u1=r:-r/5:-r u=[u1 u2]; qn=lpquantile(u,xdata,p); quant=[quant;qn]; end if r~=0 128 APPENDIX plot(quant(:,1),quant(:,2)); end hold on end if p~=1 quant=[]; r=ysort(ceil(n*alpha)) r1=-r:r/20:r; x1=real(((r^q)-abs(r1).^q).^(1/q)); u=[x1’ r1’;-x1’ -r1’]; for j=1:length(u) [qn, exitflag]=lpquantile(u(j,:),xdata,p); if exitflag == quant=[quant;qn]; end end plot(quant(:,1),quant(:,2)); end hold on end xx=median(xdata); % Plotting the median of the data as "M" plot(xx(:,1),xx(:,2)) text(xx(:,1),xx(:,2),’M’,’FontSize’,10) hold on quant; fence=[]; %Drawing the fence [s,t]=size(quant); for i=1:s l=xx+2*(quant(i,:)-xx); fence=[fence;l] end plot(fence(:,1),fence(:,2),’LineStyle’,’- -’) function []=affine_boxplot(xdata,y2,p); % %Computes affine equivariant boxplots % u=zeros(1,2); if p==1 q=inf; else q=p/(p-1); end [n,d]=size(xdata); newdata=xdata*transpose(y2); for i=1:n 129 APPENDIX y1=lprank(newdata(i,:),newdata,p); y(i,:)=norm(y1,q) ; end ysort=sort(y); alpha=median(ysort); plot(xdata(:,1),xdata(:,2),’*’,’Markersize’,5); hold on alpha if p==1 quant=[]; r=ysort(ceil(n*alpha)) u1=-r; for u2=-r:r/5:r u=[u1 u2]; qn=lpquantile(u,newdata,p); quant=[quant;qn]; end u2=r; for u1=-r:r/5:r u=[u1 u2]; qn=lpquantile(u,newdata,p); quant=[quant;qn]; end u1=r; for u2=r:-r/5:-r u=[u1 u2]; qn=lpquantile(u,newdata,p); quant=[quant;qn]; end u2=-r; for u1=r:-r/5:-r u=[u1 u2]; qn=lpquantile(u,newdata,p); quant=[quant;qn]; end quant=quant*transpose(inv(y2)); if r~=0 plot(quant(:,1),quant(:,2)); end hold on end if p~=1 quant=[]; r=ysort(ceil(n*alpha)) r1=-r:r/20:r; x1=real(((r^q)-abs(r1).^q).^(1/q)); 130 APPENDIX u=[x1’ r1’;-x1’ -r1’]; for j=1:length(u) [qn, exitflag]=lpquantile(u(j,:),newdata,p); if exitflag == quant=[quant;qn]; end end quant=quant*transpose(inv(y2)); plot(quant(:,1),quant(:,2)); end hold on end xx=median(xdata); % Plotting median of the xdata as "M" plot(xx(:,1),xx(:,2)) text(xx(:,1),xx(:,2),’M’,’FontSize’,10) hold on quant; fence=[]; %Plotting the fence [s,t]=size(quant); for i=1:s l=xx+2*(quant(i,:)-xx); fence=[fence;l] end plot(fence(:,1),fence(:,2),’LineStyle’,’- -’) function [v,qu]=lpscatter(xdata,alpha,p) % %LPSCATTER(XDATA,ALPHA,P) % % It computes a measure of scatter based on l_p rank of % the data XDATA corresponding to the probability ALPHA % % One must have =1, d is dimension % skew=[]; u=[0 0]; inc=1.0/(2*k); inck=1.0/k; alpha=0.1:inc:1 n=length(alpha); med=lpquantile(u,x,p) for i=1:k [vol,qu]=lpscatter(x,alpha(i),p); lr=mean(qu) ; sr=(lr-med); vf=(vol^(1/d)); ss=2*(sr/vf); skew=[skew;norm(ss)] end 134 APPENDIX return; function [skew,alpha]=affine_skewcurve(x,p,d,k) % % This computes affine skewness for x data % temp=[]; skew=[]; u=[0 0]; inc=1.0/(2*k); inck=1.0/k; alpha=0.1:inc:1 n=length(alpha); y2=transform(x) med=lpquantile(u,x,p) for i=1:k [vol,qu]=lpscatter_after(x,alpha(i),p,y2); lr=mean(qu) ; sr=(lr-med); vf=(vol^(1/d)); ss=2*(sr/vf); skew=[skew;norm(ss)] end return; 135 136 Bibliography [1] Arcones, M A., Chen, Z and Gine, E (1994) Estimators related to U-processes with applications to multivariate medians: asymptotic normality, The Annals 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Mathematicians, 2, 523-531 [30] Tyler, D E (1987) A distribution-free M estimator of multivariate scatter, The Annals of Statistics, 15, 234-251 [...]... scale Multivariate skewness and kurtosis curves are new tools in multivariate analysis We consider a generalization of the univariate g -and- h distribution to the multivariate situation Although there has been much attention to symmetrical distributions like multivariate normal, Laplace and t-distributions, researchers have also investigated non-symmetrical distributions such as multivariate g -and- h... non-normal univariate g -and- h distribution to multivariate case, that is multivariate g-andh distribution We plot some illustrations for bivariate box plots, scale curves, skewness and kurtosis by simulating data from multivariate g -and- h distribution This chapter ends with a conclusion 3 Chapter 2 Multivariate Medians 2.1 Notions of Multivariate Symmetry There has been a lot of attention to the univariate... discussed To illustrate some applications on lp-quantiles, we plot quantile contour plots for some simulated data sets namely, bivariate normal, bivariate Laplace and t-distribution with 4 d.f., on zero mean, unit variance and varying correlations ρ = 0, 0.5, 0.85 and 0.95 We illustrate these quantile contour plots with some real data as well Chapter 4 explores some multivariate descriptive statistics such... no proper ordering for multivariate set up One of our main interest is to construct lp-quantiles and lp -ranks and make use of these generalized lp -quantiles and lp -ranks as a basis in developing graphical representations such as quantile contour plots and bivariate boxplots Since lp - xv quantiles and lp -ranks are not affine equivariant, when there are high correlations among multivariate data, they... quantile contour plot for g -and- h distribution 108 5.2 Spatial quantile contour plot for g -and- h distribution 109 5.3 Coordinatewise quantile contour plot with TR for g -and- h distribution 109 5.4 Spatial quantile contour plot with TR for g -and- h distribution 110 5.5 Box plot with p = 1 for g -and- h distribution 5.6 Box plot with p = 2 for g -and- h distribution... are mostly multivariate by nature, this has led to gain much awareness from the researchers Many tools have been emerged as a consequence of the curiosity into the behaviour of multivariate data In this thesis, we discuss several multivariate descriptive statistics such as median, quantiles based on signs and ranks with some illustrations 1.1 Outline of the thesis In Chapter 2, notions of multivariate. .. of a multivariate data set are possible and these definitions have the common property of producing the usual definitions when applied to univariate data or a univariate distribution Some common ideas of equivariance and breakdown properties are discussed as well as with computational convenience for each definition Although univariate quantiles provide an order of the real line, an extension to multivariate. .. probability distribution and especially, to detect outliers We explore descriptive plots for analyzing multivariate distributional characteristics such as spread, skewness and kurtosis All graphs are two dimensional curves and can be easily visualized and interpreted The spread of a distribution can be plotted using scale curves based on lp -ranks If the scale is larger, then scale curve is consistently above... g -and- h distribution Since in reality, we may come across many natural phenomena that do not follow the normal law, thus multivariate non-normal distributions are needed to cope with such situations Finally, we illustrate these descriptive measures by applying them to some simulated and real data sets 1 Chapter 1 Introduction Multivariate descriptive measures have received considerable attention in the literature... with and without TR for g -and- h distribution 114 5.13 Skewness curve with p = 1 with and without TR for g -and- h distribution 115 5.14 Skewness curve with p = 2 with and without TR for g -and- h distribution 115 xiv Summary Extending the univariate concepts to multivariate setting has a long history in statistics .. .ON SOME MULTIVARIATE DESCRIPTIVE STATISTICS BASED ON MULTIVARIATE SIGNS AND RANKS NELUKA DEVPURA (B.Sc. (Statistics) University of Colombo) A THESIS... chapter and generalization of univariate signs and ranks to multivariate set up is also discussed To illustrate some applications on lp-quantiles, we plot quantile contour plots for some simulated... a consequence of the curiosity into the behaviour of multivariate data In this thesis, we discuss several multivariate descriptive statistics such as median, quantiles based on signs and ranks