R语言实现的多变量非参数分析：基于空间标志与排名的方法

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"《多变量非参数方法与R》是一本由Hannu Oja编写的统计学专著，属于Lecture Notes in Statistics系列。本书介绍了一种分析多变量数据的新方法，该方法基于多变量空间标志和秩，强调效率和稳健性，无需假设数据符合多元正态分布。书中构建了一个统一的理论框架，从简单的单样本位置问题到多样本位置问题，再到多元线性回归模型，最后扩展到集群依赖数据的分析。" 在《多变量非参数方法与R》中，作者提出了一种创新的数据分析策略，该策略不依赖于传统的多变量正态分布假设。传统方法通常假设数据服从正态分布，然后利用L2范数进行分析。然而，这本书提出用不同的L1范数来替代L2范数，同时将观测向量替换为它们的标准化和中心化的空间标志和秩。这种方法具有较强的鲁棒性，意味着它对异常值和非正态分布的数据有较好的适应性。书中详细阐述了如何利用R语言进行这种非参数分析。R是一种广泛应用于统计计算和图形绘制的开源软件，它拥有丰富的统计包和功能，使得数据分析变得更为便捷。通过R，读者可以实现这些新方法，并应用到实际的多变量数据集上。作者Hannu Oja逐步引导读者从基础的一样本问题开始，逐渐深入到更复杂的多样本问题，如方差分析（ANOVA）和多元线性回归。此外，他还探讨了如何处理集群依赖数据，这对于社会科学、生物统计学以及许多其他领域中的研究至关重要，因为这些领域的数据往往存在内在的结构或关联。书中的内容不仅涵盖了理论基础，还包括了许多实例和应用，使得读者能够理解和掌握这些方法的实际操作。书中可能包括了如何计算和解释空间标志和秩，如何建立和评估非参数模型，以及如何处理各种复杂的数据结构。《多变量非参数方法与R》为统计学家、数据科学家和研究人员提供了一种新的、强大的工具，以处理那些难以用传统正态分布假设分析的多变量数据。通过学习本书，读者将能够利用R语言实现这些方法，从而提升其在数据分析领域的专业能力。

Chapter 1

Introduction

Classical multivariate statistical inference methods (Hotelling’s T

, multivariate

analysis of variance, multivariate regression, tests for independence, canonical cor-

relation analysis, principal component analysis, etc.) are based on the use of the reg-

ular sample mean vector and covariance matrix. See, for example, the monographs

by Anderson (2003)andMardia et al. (1979). These standard moment-based mul-

tivariate techniques are optimal under the assumption of multivariate normality but

unfortunately poor in their efﬁciency for heavy-tailed distributions and are highly

sensitive to outlying observations. The book by Puri and Sen (1971)givesacom-

plete presentation of multivariate analysis methods based on marginal signs and

ranks. Pesarin (2001) considers permutation tests for multidimensional hypotheses.

In this book nonparametric and robust competitors to standard multivariate infer-

ence methods based on (multivariate) spatial signs and ranks are introduced and

discussed in detail. An R statistical software package MNM (Multivariate Nonpara-

metric Methods) to implement the procedures is freely available to users of spatial

sign and rank methods.

The univariate concepts of sign and rank are based on the ordering of the uni-

variate data y

,..., y

. The ordering is manifested with the univariate sign function

U(y) with values −1, 0, and 1 for y < 0, y = 0, and y > 0, respectively. The sign

and centered rank of the observation y

is then U(y

) and AV E

{U(y

−y

)}.Inthe

multivariate case there is no natural coordinate-free ordering of the data points; see

Barnett (1976) for a discussion on the problem. An approach utilizing L

objective

or criterion functions is therefore often used to extend these concepts to the multi-

variate case. Let Y =(y

,..., y

)



be an n × p data matrix with n observations and

p variables. The multivariate spatial sign U

, multivariate spatial (centered) rank R

and multivariate spatial signed-rank Q

, i = 1,...,n, may be implicitly deﬁned us-

ing the three L

criterion functions with Euclidean norm |·|. The sign, rank and

signed-rank are then deﬁned implicitly by

H. Oja, Multivariate Nonparametric Methods with R: An Approach Based on Spatial Signs 1

and Ranks, Lecture Notes in Statistics 199, DOI 10.1007/978-1-4419-0468-3

 Springer Science+Business Media, LLC 2010

2 1 Introduction

AVE {|y

|} = AV E {U



AVE {|y

−y

|} = AV E {R



}, and

AVE {|y

−y

|+ |y

+ y

|} = AV E {Q



See Hettmansperger and Aubuchon (1988). Note also that the sign, centered rank,

and signed-rank may be seen as scores T(y) corresponding to the three objective

functions. These score functions then are

U(y)=|y|

−1

R(y)=AVE {U(y −y

)}, and

Q(y)=

AVE {U(y −y

)+U(y + y

)}.

The identity score T(y

)=y

, i = 1,...,n, is the score corresponding to the regular

criterion AVE {|y

} = AV E {y



Multivariate spatial sign and spatial rank methods are thus based on the L

ob-

jective functions and corresponding score functions. The L

methods have a long

history in the univariate case but they are often regarded as computationally highly

demanding. See, however, Portnoy and Koenker (1997). The ﬁrst objective function

AVE {|y

|}, if applied to the residuals in the general linear regression model, is the

mean deviation of the residuals from the origin, and it is the basis for the so called

least absolute deviation (LAD) methods. It yields different median-type estimates

and sign tests in the one-sample, two-sample, several-sample and ﬁnally general

linear model settings. The second objective function AV E{|y

−y

|} is the mean

difference of the residuals which in fact measures how close together the residu-

als are. The second and third objective functions generate Hodges-Lehmann type

estimates and rank tests for different location problems.

The general strategy in the analysis of the multivariate data followed in this book

is ﬁrst to replace the original observations y

by some scores T

= T(y

) or, in more

complex designs, by centered and/or standardized scores

T(y

), i = 1,...,n.

The statistical tests are then based on the new data matrix

T =(T

,..., T

)



T =(

,...,

)



The spatial sign score U(y), the spatial rank score R(y), and the spatial signed-

rank score Q(y) thus correspond to the three L

criterion functions given above.

The tests are then rotation invariant but not afﬁne invariant. Inner centering and/or

standardization is used to attain the desired afﬁne invariance property of the tests.

The location estimates are chosen to minimize the selected criterion function; they

are also obtained if one applies inner centering with the corresponding score. Inner

standardization may be used to construct afﬁne equivariant versions of the estimates

1 Introduction 3

and as a side product one gets scatter matrix estimates for the inference on the

covariance structure of the data.

The tests and estimates for the multivariate location problem based on multivari-

ate signs and ranks have been widely discussed in the literature. See, for example,

M¨ott¨onen and Oja (1995), Choi and Marden (1997), Marden (1999a), and Oja and

Randles (2004). The scatter matrix estimates by Tyler (1987)andD¨umbgen (1998)

are often used in inner standardizations. The location tests and estimates are robust

and they have good efﬁciency properties even in the multivariate normal model.

M¨ott¨onen et al. (1997) calculated the asymptotic efﬁciencies e

(p,

) and e

(p,

)

of the multivariate spatial sign and rank methods, respectively, in the p-variate t

distribution case (t

∞,p

is the p-variate normal distribution). In the 3-variate case, for

example, the asymptotic efﬁciencies are

(3,3)=2.162, e

(3,10)=1.009, e

(3,∞)=0.849,

(3,3)=1.994, e

(3,10)=1.081, e

(3,∞)=0.973,

and in the 10-variate case one has even higher

(10,3)=2.422, e

(10,10)=1.131, e

(10,∞)=0.951,

(10,3)=2.093, e

(10,10)=1.103, e

(10,∞)=0.989.

The procedures based on spatial signs and ranks, however, yield only one

possible approach to multivariate nonparametric tests (sign test, rank test) and cor-

responding estimates (median, Hodges-Lehmann estimate). Randles (1989), for

example, developed an afﬁne invariant sign test based on interdirections. Interdi-

rections measure the angular distance between two observation vectors relative to

the rest of the data. Randles (1989) was followed by a series of papers introduc-

ing nonparametric sign and rank interdirection tests for different location problems.

These tests are typically asymptotically equivalent with afﬁne invariant versions of

the spatial sign and rank tests. The tests and estimates based on interdirections are,

unfortunately, computationally heavy.

The multivariate inference methods based on marginal signs and ranks are de-

scribed in detail in the monograph by Puri and Sen (1971). This ﬁrst extension of

the univariate sign and rank methods to the multivariate setting is based on the cri-

terion functions

AVE {|y

|+ ···+ |y

|} and AVE {|y

−y

|+ ···+ |y

−y

|},

respectively. Note that here the Euclidean norm, the L

-norm |y| =



+ ···+ y

is replaced by the L

-norm |y| = |y

| + ···+ |y

|. Unfortunately, the tests are

not afﬁne invariant and the estimates are not afﬁne equivariant in this approach.

Chakraborty and Chaudhuri (1996)andChakraborty and Chaudhuri (1998)usedthe

4 1 Introduction

so-called transformation retransformation technique (TR-technique) to ﬁnd afﬁne

invariant versions of the tests and afﬁne equivariant versions of the estimates.

Afﬁne equivariant multivariate signs and ranks are obtained if one uses the L

criterion functions

AVE {V(y

,..., y

,0)} and AV E{V(y

,..., y

p+1

)}

where the average is over all the p-tuples and (p+1)-tuples of observations, respec-

tively, and

V(y

,..., y

p+1

abs



det



1 ··· 1

··· y

p+1



is the volume of the p-variate simplex with vertices y

,..., y

p+1

. The one-sample

location estimate is known as the Oja median Oja (1983). The approach in the

one-sample and two-sample location cases is described in Oja (1999). In a paral-

lel approach Koshevoy and Mosler (1997a,b, 1998) used so-called zonotopes and

lift-zonotopes to illustrate and characterize a multivariate data cloud. The duality

relationship between these two approaches is analyzed in Koshevoy et al. (2004).

In all the approaches listed above the power of the rank tests may often be in-

creased, if some further transformations are applied to the ranks. In a series of pa-

pers, Hallin and Paindaveine constructed optimal signed-rank tests for the location

and scatter problems in the elliptical model; see the series of papers starting with

Hallin and Paindaveine (2002). In their approach, the location tests were based on

the spatial signs and optimally transformed ranks of the Euclidean lengths of the

standardized observations. As the model of elliptically symmetric distributions, the

so called independent component (IC) model is also an extension of the multivariate

normal model. In the test construction in this model, one ﬁrst transforms the obser-

vations to the estimated independent coordinates, then calculates the values of the

marginal sign and rank test statistics, and ﬁnally combines the asymptotically inde-

pendent tests in the regular way. See Nordhausen et al. (2009)andOja et al. (2009)

for optimal rank tests in the IC model.

In this book the approach is thus based on the spatial signs and ranks. A uni-

ﬁed theory starting with the simple one-sample location problem and proceeding

through the several-sample location problems to the general multivariate linear re-

gression model and ﬁnally to the analysis of cluster-dependent data is presented.

The theory is often presented using a general score function and, for comparison

to classical methods, the classical normal-based approach (L

criterion and identity

score function) is carefully reviewed in each case. Also statistical inference on scat-

ter or shape matrices based on the spatial signs and ranks is discussed. The theory is

illustrated with several examples. For the computation of the statistical procedures

described in the book, R packages MNM and SpatialNP are available on CRAN.

The readers who are not so familiar with R are advised to learn more from Venables

et al. (2009), Dalgaard (2008), or Everitt (2005).

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