, Klaus Nordhausen
[cre, aut] , Hannu Oja
[aut], Sara Taskinen [aut]
| Title: | Multivariate Nonparametric Methods Based on Spatial Signs and Ranks |
|---|---|
| Description: | Test and estimates of location, tests of independence, tests of sphericity and several estimates of shape all based on spatial signs, symmetrized signs, ranks and signed ranks. For details, see Oja and Randles (2004) <doi:10.1214/088342304000000558> and Oja (2010) <doi:10.1007/978-1-4419-0468-3>. |
| Authors: | Seija Sirkia [aut], Jari Miettinen [aut]
, Klaus Nordhausen
[cre, aut] , Hannu Oja
[aut], Sara Taskinen [aut]
|
| Maintainer: | Klaus Nordhausen <[email protected]> |
| License: | GPL-2 |
| Version: | 1.1-5 |
| Built: | 2024-11-18 06:52:54 UTC |
| Source: | CRAN |
Test and estimates of location, tests of independence, tests of sphericity and several estimates of shape all based on spatial signs, symmetrized signs, ranks and signed ranks. For details, see Oja and Randles (2004) <doi:10.1214/088342304000000558> and Oja (2010) <doi:10.1007/978-1-4419-0468-3>.
| Package: | SpatialNP |
| Type: | Package |
| Version: | 1.1-5 |
| Date: | 2021-12-08 |
| License: | GPL (>= 2) |
There are three functions for inference,
sr.loc.test, sr.indep.test and
sr.sphere.test, for location, independence and
sphericity tests. The so called inner and
outer standardization matrices are also available as
well as the actual sign and rank score
functions, together with a utility function to.shape.
Seija Sirkia, Jari Miettinen, Klaus Nordhausen, Hannu Oja, Sara Taskinen
Maintainer: Klaus Nordhausen, [email protected]
Test of independence between two sets of variables. Inference is based on the spatial signs of the observations, symmetrized signs of the observations or spatial signed ranks of the observations.
sr.indep.test(X, Y = NULL, g = NULL, score = c("sign", "symmsign", "rank"), regexp = FALSE, cond = FALSE, cond.n = 1000, na.action = na.fail)sr.indep.test(X, Y = NULL, g = NULL, score = c("sign", "symmsign", "rank"), regexp = FALSE, cond = FALSE, cond.n = 1000, na.action = na.fail)
X |
a matrix or a data frame |
Y |
an optional matrix or a data frame |
g |
a factor giving the two sets of variables, or numeric vector or vector of column names giving the first set of variables. See details |
score |
a character string indicating which transformation of the observations should be used |
regexp |
logical. Is |
cond |
logical. Should the conditionally distribution free test be used? |
cond.n |
Number of permutations to use in the conditionally distribution free test |
na.action |
a function which indicates what should happen when the data contain 'NA's. Default is to fail. |
X should contain the first set of variables and
Y the second with matching rows. Alternatively, X should
contain both sets and g should be a factor of length equal to
number of columns of X, or, g should be a numeric or
character vector naming the variables in the first set. If g is
a character vector it is assumed to name all wanted columns exactly,
unless regexp is TRUE.
A list with class 'htest' containing the following components:
statistic |
the value of the statistic |
parameter |
the degrees of freedom for the statistic or the number of replications if conditionally distribution free p-value was used |
p.value |
the p-value for the test |
null.value |
the specified hypothesized value of the measure of dependence (always 0) |
alternative |
a character string with the value 'two.sided'. |
method |
a character string indicating what type of test was performed |
data.name |
a character string giving the name of the data (and grouping vector) |
Seija Sirkia, [email protected]
Taskinen, S., Oja, H., Randles R. (2004) Multivariate Nonparametric Tests of Independence. JASA, 100, 916-925
A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(3000),ncol=3)%*%t(A) Y<-cbind(X+runif(3000,-1,1),runif(1000)) sr.indep.test(X,Y) #alternative calls: Z<-cbind(X,Y) colnames(Z)<-c("a1","a2","a3","b1","b2","b3","b4") g<-factor(c(rep(1,3),rep(2,4))) sr.indep.test(Z,g=g) sr.indep.test(Z,g=c("b"),regexp=TRUE) sr.indep.test(Z,g=1:3)A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(3000),ncol=3)%*%t(A) Y<-cbind(X+runif(3000,-1,1),runif(1000)) sr.indep.test(X,Y) #alternative calls: Z<-cbind(X,Y) colnames(Z)<-c("a1","a2","a3","b1","b2","b3","b4") g<-factor(c(rep(1,3),rep(2,4))) sr.indep.test(Z,g=g) sr.indep.test(Z,g=c("b"),regexp=TRUE) sr.indep.test(Z,g=1:3)
Multivariate tests of location of one or more samples based on spatial signs and (signed) ranks. In case of one sample the null hypothesis about a given location is tested. In case of several samples the null hypothesis is that all samples have the same location.
sr.loc.test(X, Y = NULL, g = NULL, score = c("sign", "rank"), nullvalue = NULL, cond = FALSE, cond.n = 1000, na.action = na.fail,...)sr.loc.test(X, Y = NULL, g = NULL, score = c("sign", "rank"), nullvalue = NULL, cond = FALSE, cond.n = 1000, na.action = na.fail,...)
X |
a matrix or a data frame |
Y |
an optional matrix or a data frame |
g |
a factor giving the groups (may contain just one level) |
score |
a character string indicating which transformation of the observations should be used |
nullvalue |
location to be tested in the one sample case (ignored if there is more than one sample) |
cond |
logical. Should the conditionally distribution free test be used? (Ignored if |
cond.n |
number of permutations to use in the conditionally distribution free test |
na.action |
a function which indicates what should happen when the data contain 'NA's. Default is to fail. |
... |
further arguments to be passed to other functions |
X should contain the the whole data set and g should describe the groups, or, if there is only one group, g may be missing. Alternatively, if there are two samples X may contain only the first sample while the second sample is given in Y and g is ignored. Note that in the one sample case when rank is chosen as score the function in fact uses signed ranks.
Note that the conditionally distribution free p-value is only provided for the sign based version of the test.
A list with class 'htest' containing the following components:
statistic |
the value of the statistic |
parameter |
the degrees of freedom for the statistic or the number of replications if conditionally distribution free p-value was used |
p.value |
the p-value for the test |
null.value |
the specified hypothesized value of the (common) location |
alternative |
a character string with the value 'two.sided'. |
method |
a character string indicating what type of test was performed |
data.name |
a character string giving the name of the data (and grouping vector) |
Seija Sirkia, [email protected]
Oja, H., Randles R. (2004) Multivariate Nonparametric Tests. Statistical Science 19, 598-605.
A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-rbind(matrix(rnorm(1500),ncol=3),matrix(rnorm(750)+1,ncol=3))%*%t(A) sr.loc.test(X,cond=TRUE) X[1:250,]<-X[1:250,]+1 g<-factor(rep(c(1,2,3),each=250)) sr.loc.test(X,g=g,score="rank")A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-rbind(matrix(rnorm(1500),ncol=3),matrix(rnorm(750)+1,ncol=3))%*%t(A) sr.loc.test(X,cond=TRUE) X[1:250,]<-X[1:250,]+1 g<-factor(rep(c(1,2,3),each=250)) sr.loc.test(X,g=g,score="rank")
Iterative algorithms to estimate M-estimators of location and scatter as well as symmetrized M-estimator using Huber's weight functions.
mvhuberM(X, qg = 0.9, fixed.loc = FALSE, location = NULL, init = NULL, steps = Inf, eps = 1e-06, maxiter = 100, na.action = na.fail) symmhuber(X, qg = 0.9, init = NULL, steps = Inf, eps = 1e-6, maxiter = 100, na.action = na.fail) symmhuber.inc(X, qg=0.9, m=10, init=NULL, steps=Inf, permute=TRUE, eps=1e-6, maxiter=100, na.action = na.fail)mvhuberM(X, qg = 0.9, fixed.loc = FALSE, location = NULL, init = NULL, steps = Inf, eps = 1e-06, maxiter = 100, na.action = na.fail) symmhuber(X, qg = 0.9, init = NULL, steps = Inf, eps = 1e-6, maxiter = 100, na.action = na.fail) symmhuber.inc(X, qg=0.9, m=10, init=NULL, steps=Inf, permute=TRUE, eps=1e-6, maxiter=100, na.action = na.fail)
X |
a matrix or a data frame |
qg |
a tuning parameter. The default is 0.9, see details |
fixed.loc |
a logical, see details |
location |
an optional vector giving the location of the data or the initial value for the location if it is estimated |
init |
an optional starting value for scatter |
steps |
fixed number of iteration steps to take, if |
m |
a parameter in |
permute |
logical in |
eps |
tolerance for convergence |
maxiter |
maximum number of iteration steps. Ignored if |
na.action |
a function which indicates what should happen when the data contain 'NA's. Default is to fail. |
mvhuberM computes multivariate M-estimators of location and scatter
using Huber's weight functions. The tuning parameter qg defines cutoff-point c for weight functions so that , where F is the cdf of -distribution with p degrees of freedom. The estimators with maximal breakdown point are obtained with the choice qg=F(p+1). If fixed.loc is set TRUE, scatter estimator is computed with fixed location given by
location (default is column means).
symmhuber computes Huber's M-estimator of scatter using pairwise
differences of the data therefore avoiding location estimation.
symmhuber.inc is a computationally lighter estimator to approximate symmetrized Huber's M-estimator of scatter. Only a subset of the pairwise
differences are used in the computation in the incomplete case. The magnitude of the subset used is controlled by the argument m which is half
of the number of how many differences each observation is part of. Differences of successive observations are used, and therefore random permutation
of the rows of X is suggested and is the default choice in the function. For details see Miettinen et al., 2016.
mvhuberM returns a list with components
location |
a vector |
scatter |
a matrix |
symmhuber returns a matrix.
symmhuber.inc returns a matrix.
Klaus Nordhausen, [email protected],
Jari Miettinen, [email protected]
Huber, P.J. (1981), Robust Statistics, Wiley, New York.
Lopuhaa, H.P. (1989). On the relation between S-estimators and M-estimators of multivariate location and covariance. Annals of Statistics, 17, 1662-1683.
Sirkia, S., Taskinen, S., Oja, H. (2007) Symmetrised M-estimators of scatter. Journal of Multivariate Analysis, 98, 1611-1629.
Miettinen, J., Nordhausen, K., Taskinen, S., Tyler, D.E. (2016) On the computation of symmetrized M-estimators of scatter. In Agostinelli, C. Basu, A., Filzmoser, P. and Mukherje, D. (editors) ”Recent Advances in Robust Statistics: Theory and Application”, 131-149, Springer India, New Delhi.
A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(1500),ncol=3)%*%t(A) mvhuberM(X) symmhuber(X) symmhuber.inc(X, m=5) symm.mvtmle.inc(X, m=5)A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(1500),ncol=3)%*%t(A) mvhuberM(X) symmhuber(X) symmhuber.inc(X, m=5) symm.mvtmle.inc(X, m=5)
Iterative algorithms to find shape matrices based on spatial signs and ranks and the k-step versions of these.
spatial.shape(X, score = c("sign", "symmsign", "rank", "signrank"), fixed.loc = FALSE, location = NULL, init = NULL, steps = Inf, eps = 1e-06, maxiter = 100, na.action = na.fail) signs.shape(X, fixed.loc = FALSE, location = NULL, init = NULL, steps = Inf, eps = 1e-6, maxiter = 100, na.action = na.fail) symmsign.shape(X, init = NULL, steps = Inf, eps = 1e-6, maxiter = 100, na.action = na.fail) symmsign.shape.inc(X, m=10, init=NULL, steps=Inf, permute=TRUE, eps=1e-6, maxiter=100, na.action=na.fail) rank.shape(X, init = NULL, steps = Inf, eps = 1e-06, maxiter = 100, na.action = na.fail) signrank.shape(X, fixed.loc = FALSE, location = NULL, init = NULL, steps = Inf, eps = 1e-06, maxiter = 100, na.action = na.fail)spatial.shape(X, score = c("sign", "symmsign", "rank", "signrank"), fixed.loc = FALSE, location = NULL, init = NULL, steps = Inf, eps = 1e-06, maxiter = 100, na.action = na.fail) signs.shape(X, fixed.loc = FALSE, location = NULL, init = NULL, steps = Inf, eps = 1e-6, maxiter = 100, na.action = na.fail) symmsign.shape(X, init = NULL, steps = Inf, eps = 1e-6, maxiter = 100, na.action = na.fail) symmsign.shape.inc(X, m=10, init=NULL, steps=Inf, permute=TRUE, eps=1e-6, maxiter=100, na.action=na.fail) rank.shape(X, init = NULL, steps = Inf, eps = 1e-06, maxiter = 100, na.action = na.fail) signrank.shape(X, fixed.loc = FALSE, location = NULL, init = NULL, steps = Inf, eps = 1e-06, maxiter = 100, na.action = na.fail)
X |
a matrix or a data frame |
score |
a character string indicating which transformation of the observations should be used |
fixed.loc |
a logical, see details |
location |
an optional vector giving the location of the data or the initial value for the location if it is estimated |
init |
an optional starting value for the iteration |
steps |
fixed number of iteration steps to take, if |
m |
a parameter in |
permute |
logical in |
eps |
tolerance for convergence |
maxiter |
maximum number of iteration steps. Ignored if |
na.action |
a function which indicates what should happen when the data contain 'NA's. Default is to fail. |
sign.shape is Tyler's shape matrix and symmsign.shape is Duembgen's shape matrix. Function
symmsign.shape.inc is for a computationally lighter estimator to approximate Duembgen's shape matrix. Only a subset of the pairwise differences are used
in the computation in the incomplete case. The magnitude of the subset used is controlled by the argument m which is half of the number of how many
differences each observation is part of. Differences of successive observations are used, and therefore random permutation of the rows of X is suggested
and is the default choice in the function. For details see Miettinen et al., 2016.
rank.shape and signrank.shape are the so called inner standardization matrices of location etc. tests based on spatial signs and ranks. When data is standardized using these matrices the corresponding sign or rank scores will appear “uncorrelated”: the corresponding outer standardization matrices will be proportional to the identity matrix, see examples.
spatial.shape is a wrapper function for a unified access to all
four shape estimates (not including symmsign.shape.inc). The choice of estimate is done via score:
"sign" for signs.shape
"symmsign" for symmsign.shape
"rank" for rank.shape
"signrank" for signrank.shape
signrank.shape and sign.shape include options to compute the shape matrix either with respect to fixed location (fixed.loc = TRUE) or so that the location and the shape are estimated simultaneously (fixed.loc = FALSE).
The estimate matrix with the (final estimate of or given) location vector
as attribute "location".
Seija Sirkia, [email protected], Jari Miettinen, [email protected]
Oja, H., Randles R. (2004) Multivariate Nonparametric Tests. Statistical Science 19, 598-605.
Sirkia et al. (2009) Tests and estimates of shape based on spatial signs and ranks. Journal of Nonparametric Statistics, 21, 155-176.
Sirkia, S., Taskinen, S., Oja, H. (2007) Symmetrised M-estimators of scatter. Journal of Multivariate Analysis, 98, 1611-1629.
Miettinen, J., Nordhausen, K., Taskinen, S., Tyler, D.E. (2016) On the computation of symmetrized M-estimators of scatter. In Agostinelli, C. Basu, A., Filzmoser, P. and Mukherje, D. (editors) ”Recent Advances in Robust Statistics: Theory and Application”, 131-149, Springer India, New Delhi.
tyler.shape, duembgen.shape, also spatial sign and rank covariance matrices and spatial signs and ranks
A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(1500),ncol=3)%*%t(A) symmsign.shape(X) to.shape(symmsign.shape(X),trace=3) spatial.shape(X,score="sign") spatial.shape(X,score="sign",fixed.loc=TRUE) to.shape(A%*%t(A)) # one-step shape estimate based on spatial ranks and covariance matrix: spatial.shape(X,score="rank",init=cov(X),steps=1) symmsign.shape.inc(X, m=5)A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(1500),ncol=3)%*%t(A) symmsign.shape(X) to.shape(symmsign.shape(X),trace=3) spatial.shape(X,score="sign") spatial.shape(X,score="sign",fixed.loc=TRUE) to.shape(A%*%t(A)) # one-step shape estimate based on spatial ranks and covariance matrix: spatial.shape(X,score="rank",init=cov(X),steps=1) symmsign.shape.inc(X, m=5)
Iterative algorithms to find spatial median, multivariate Hodges-Lehmann estimate of location, their affine equivariant versions and k-step versions of these.
spatial.location(X, score = c("sign", "signrank"), init = NULL, shape = TRUE, steps = Inf, maxiter = 500, eps = 1e-6, na.action = na.fail) ae.spatial.median(X, init = NULL, shape = TRUE, steps = Inf, maxiter = 500, eps = 1e-6, na.action = na.fail) ae.hl.estimate(X, init = NULL, shape = TRUE, steps = Inf, maxiter = 500, eps = 1e-06, na.action = na.fail)spatial.location(X, score = c("sign", "signrank"), init = NULL, shape = TRUE, steps = Inf, maxiter = 500, eps = 1e-6, na.action = na.fail) ae.spatial.median(X, init = NULL, shape = TRUE, steps = Inf, maxiter = 500, eps = 1e-6, na.action = na.fail) ae.hl.estimate(X, init = NULL, shape = TRUE, steps = Inf, maxiter = 500, eps = 1e-06, na.action = na.fail)
X |
a matrix or a data frame |
score |
a character string indicating which transformation of the observations should be used |
init |
an optional vector giving the initial point of the iteration |
shape |
logical, or a matrix. See details |
steps |
fixed number of iteration steps to take, if |
eps |
tolerance for convergence |
maxiter |
maximum number of iteration steps |
na.action |
a function which indicates what should happen when the data contain 'NA's. Default is to fail. |
Spatial median and Hodges-Lehmann estimator (spatial median of the pairwise differences) are not affine equivariant. Affine
equivariance can be achieved by simultaneously estimating the
corresponding shape, as proposed for the spatial median by
Hettmansperger and Randles (2002). For spatial median the corresponding
shape is signs.shape and for the Hodges-Lehmann estimate it
is signrank.shape.
spatial.location is a wrapper function for a unified access to
both location estimates. The choice of estimate is done via
score:
"sign" for spatial median
"signrank" for Hodges-Lehmann estimate
If a matrix (must be symmetric and positive definite, but this is not
checked) is given as shape the location estimate is found with
respect to that shape and no further shape estimation is done. If a
logical TRUE is given as shape the shape is estimated
and consequently the affine equivariant version of the location
estimate is found. If shape is FALSE then shape
estimation is not done and the non affine equivariant versions of the
location estimate, that is the spatial median and the Hodges-Lehmann estimate are found.
The estimate vector with the (final estimate of or given) shape matrix
as attribute "shape".
Seija Sirkia, [email protected], Jari Miettinen, [email protected]
Hettmansperger, T. and Randles, R. (2002) A Practical Affine Equivariant Multivariate Median, Biometrika, 89, pp. 851-860
spatial.median, signrank.shape
A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(3000),ncol=3)%*%t(A) spatial.location(X,score="signrank") spatial.location(X,score="sign") #compare with: colMeans(X) ae.hl.estimate(X,shape=A%*%t(A)) ae.hl.estimate(X,shape=FALSE)A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(3000),ncol=3)%*%t(A) spatial.location(X,score="signrank") spatial.location(X,score="sign") #compare with: colMeans(X) ae.hl.estimate(X,shape=A%*%t(A)) ae.hl.estimate(X,shape=FALSE)
Functions to compute spatial sign, spatial symmetrized sign, spatial rank and spatial signed rank covariance matrices
SCov(X, location = NULL, na.action = na.fail) SSCov(X, na.action = na.fail) RCov(X, na.action = na.fail) SRCov(X, location = NULL, na.action = na.fail)SCov(X, location = NULL, na.action = na.fail) SSCov(X, na.action = na.fail) RCov(X, na.action = na.fail) SRCov(X, location = NULL, na.action = na.fail)
X |
matrix or a data frame |
location |
numeric vector (may be missing) |
na.action |
a function which indicates what should happen when the data contain 'NA's. Default is to fail. |
These functions compute the matrices of the form
where are the appropriate scores of the data:
spatial signs, spatial symmetrized signs, spatial ranks or spatial
signed ranks. These are the so called outer standardization matrices
of location etc. tests based on spatial signs and ranks. They are
not affine equivariant.
SCov and SRCov require a location vector with respect
to which they are computed. If none is provided, SCov uses
spatial median and SRCov uses Hodges-Lehmann estimator.
Seija Sirkia, [email protected]
Visuri, S., Koivunen, V. and Oja, H. (2000). Sign and rank covariance matrices. J. Statistical Planning and Inference, 91, 557-575.
spatial signs and ranks, corresponding shape matrices (inner standardization matrices)
A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rt(150,1),ncol=3)%*%t(A) SCov(X) SSCov(X) RCov(X) SRCov(X) to.shape(A%*%t(A),trace=1)A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rt(150,1),ncol=3)%*%t(A) SCov(X) SSCov(X) RCov(X) SRCov(X) to.shape(A%*%t(A),trace=1)
Functions to compute spatial signs, symmetrized signs, ranks and signed ranks.
spatial.signs(X, center = TRUE, shape = TRUE, na.action = na.fail,...) spatial.symmsign(X, shape = TRUE, na.action = na.fail, ...) spatial.rank(X, shape = TRUE, na.action = na.fail, ...) spatial.signrank(X, center = TRUE, shape = TRUE, na.action = na.fail,...)spatial.signs(X, center = TRUE, shape = TRUE, na.action = na.fail,...) spatial.symmsign(X, shape = TRUE, na.action = na.fail, ...) spatial.rank(X, shape = TRUE, na.action = na.fail, ...) spatial.signrank(X, center = TRUE, shape = TRUE, na.action = na.fail,...)
X |
a matrix or a data frame |
center |
a vector or a logical, see details |
shape |
a matrix or a logical, see details |
... |
arguments that can be passed on to function used for the estimation of shape. |
na.action |
a function which indicates what should happen when the data contain 'NA's. Default is to fail. |
The spatial signs of an observed vector is simply the
vector, possibly affinely transformed first, multiplied by its
Euclidian length. See spatial.sign for a precise
definition. Symmetrized spatial signs are the spatial signs of the pairwise
differences of the data
(there are n over 2 of these). Spatial
rank of an observation is the average of the signs of the differences
of that observation and the others:
Spatial signed rank of an observation is defined as
If a numerical value is given for shape and/or center
these are used to transform the data before the computation of signs
or ranks. A logical TRUE indicates that the shape or center should be
estimated. In this case an affine transformation that makes the
resulting signs or ranks have a covariance matrix equal or
proportional to the identity matrix and centerd on the origin is
found. A logical FALSE indicates that the null value, that is, the
identity matrix or the origin, should be used. Note that only signed
ranks depend on a center.
The value of shape and/or location used are returned as attributes.
Seija Sirkia, [email protected]
Visuri, S., Koivunen, V. and Oja, H. (2000). Sign and rank covariance matrices. J. Statistical Planning and Inference, 91, 557-575.
spatial.sign for the signs, spatial sign and rank covariance matrices and spatial.shape for the standardizing transformations
A<-matrix(c(1,2,-3,4),ncol=2) X<-matrix(rnorm(100),ncol=2)%*%t(A) def.par<-par(no.readonly=TRUE) # for resetting layout(matrix(1:4,ncol=2,nrow=2,byrow=TRUE)) plot(X,col=c(2,rep(1,19))) plot(spatial.symmsign(X),col=c(2,rep(1,19)),xlim=c(-1,1),ylim=c(-1,1)) theta<-seq(0,2*pi,length=1000) lines(sin(theta),cos(theta)) plot(spatial.rank(X),col=c(2,rep(1,19)),xlim=c(-1,1),ylim=c(-1,1)) lines(sin(theta),cos(theta)) plot(spatial.signrank(X),col=c(2,rep(1,19)),xlim=c(-1,1),ylim=c(-1,1)) lines(sin(theta),cos(theta)) par(def.par)A<-matrix(c(1,2,-3,4),ncol=2) X<-matrix(rnorm(100),ncol=2)%*%t(A) def.par<-par(no.readonly=TRUE) # for resetting layout(matrix(1:4,ncol=2,nrow=2,byrow=TRUE)) plot(X,col=c(2,rep(1,19))) plot(spatial.symmsign(X),col=c(2,rep(1,19)),xlim=c(-1,1),ylim=c(-1,1)) theta<-seq(0,2*pi,length=1000) lines(sin(theta),cos(theta)) plot(spatial.rank(X),col=c(2,rep(1,19)),xlim=c(-1,1),ylim=c(-1,1)) lines(sin(theta),cos(theta)) plot(spatial.signrank(X),col=c(2,rep(1,19)),xlim=c(-1,1),ylim=c(-1,1)) lines(sin(theta),cos(theta)) par(def.par)
Tests of sphericity based on spatial signs and spatial signs of pairwise differences.
sr.sphere.test(X, score = c("sign", "symmsign"), shape = NULL, na.action = na.fail)sr.sphere.test(X, score = c("sign", "symmsign"), shape = NULL, na.action = na.fail)
X |
a matrix or a data frame |
score |
a character string indicating which transformation of the observations should be used |
shape |
a matrix with which the data should be standardized before the sphericity test |
na.action |
a function which indicates what should happen when the data contain 'NA's. Default is to fail. |
The test is for a null hypothesis of the form “true
shape matrix is equal to the identity matrix”. Effectively, giving a
matrix as shape will produce a test of whether the true shape
is equal (in fact, proportional, since the scale of shape will
have no effect) to it. In that case the test will still be for
sphericity but the data is standardized beforehand.
A list with class 'htest' containing the following components:
statistic |
the value of the statistic |
parameter |
the degrees of freedom for the statistic |
p.value |
the p-value for the test |
null.value |
the specified hypothesized value of the shape (always |
alternative |
a character string with the value 'two.sided'. |
method |
a character string indicating what type of test was performed |
data.name |
a character string giving the name of the data |
Seija Sirkia, [email protected]
Sirkia et al. (2009) Tests and estimates of shape based on spatial signs and ranks. Journal of Nonparametric Statistics, 21, 155-176.
sign and rank covariance matrices
A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(600),ncol=3)%*%t(A) sr.sphere.test(X,score="sign")A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(600),ncol=3)%*%t(A) sr.sphere.test(X,score="sign")
Iterative algorithms to estimate symmetrized M-estimators of scatter using weights of the t-distribution.
symm.mvtmle(X, nu=1, init=NULL, steps=Inf, eps=1e-6, maxiter=100, na.action = na.fail) symm.mvtmle.inc(X, nu=1, m=10, init=NULL, steps=Inf, permute=TRUE, eps=1e-6, maxiter=100, na.action = na.fail)symm.mvtmle(X, nu=1, init=NULL, steps=Inf, eps=1e-6, maxiter=100, na.action = na.fail) symm.mvtmle.inc(X, nu=1, m=10, init=NULL, steps=Inf, permute=TRUE, eps=1e-6, maxiter=100, na.action = na.fail)
X |
a matrix or a data frame |
nu |
the degrees of freedom of the t-distribution. The default is 1. Must be larger than 0. |
init |
an optional starting value for scatter |
steps |
fixed number of iteration steps to take, if |
m |
a parameter in |
permute |
logical in |
eps |
tolerance for convergence |
maxiter |
maximum number of iteration steps. Ignored if |
na.action |
a function which indicates what should happen when the data contain 'NA's. Default is to fail. |
symm.mvtmle computes M-estimator of scatter using weights of the t-distribution and pairwise
differences of the data. Hence, location estimation is not needed.
symm.mvtmle.inc is a computationally lighter estimator to approximate symmetrized M-estimator of scatter which uses weights of the t-distribution.
Only a subset of the pairwise differences are used in the computation in the incomplete case. The magnitude of the subset used is controlled
by the argument m which is half of the number of how many differences each observation is part of. Differences of successive observations are used,
and therefore random permutation of the rows of X is suggested and is the default choice in the function. For details see Miettinen et al., 2016.
symm.mvtmle returns a matrix.
symm.mvtmle.inc returns a matrix.
Jari Miettinen, [email protected],
Klaus Nordhausen, [email protected]
Huber, P.J. (1981), Robust Statistics, Wiley, New York.
Sirkia, S., Taskinen, S., Oja, H. (2007) Symmetrised M-estimators of scatter. Journal of Multivariate Analysis, 98, 1611-1629.
Duembgen, L., Pauly, M., Schweizer, T. (2015) M-Functionals of multivariate scatter. Statistics Surveys 9, 32-105.
Miettinen, J., Nordhausen, K., Taskinen, S., Tyler, D.E. (2016) On the computation of symmetrized M-estimators of scatter. In Agostinelli, C. Basu, A., Filzmoser, P. and Mukherje, D. (editors) ”Recent Advances in Robust Statistics: Theory and Application”, 131-149, Springer India, New Delhi.
A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(1500),ncol=3)%*%t(A) symm.mvtmle(X, nu=2) symm.mvtmle.inc(X, nu=2, m=20)A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3) X<-matrix(rnorm(1500),ncol=3)%*%t(A) symm.mvtmle(X, nu=2) symm.mvtmle.inc(X, nu=2, m=20)
This function rescales a given matrix such that its determinant, trace or the value of the first diagonal element meets a criteria.
to.shape(M, determ, trace, first)to.shape(M, determ, trace, first)
M |
a matrix to be scaled |
determ |
required value for determinant |
trace |
required value for trace |
first |
required value of the first diagonal element |
If determ, trace or first is given
M is scaled such that its determinant, trace or first diagonal
element, respectively, equals that value. If none of the three is
given M is scaled such that its determinant equals one. If more
than one criteria is given the first of them is used and the others
silently ignored.
The rescaled matrix
A shape matrix is symmetric and positive definite square matrix. In order for the result to be such the argument matrix M should also be symmetric and positive definite square matrix. However, the function does not check for this. Expect to see errors if M is of inappropriate type.
Seija Sirkia, [email protected]
Paindaveine D. (2008) A Canonical Definition of Shape. Statistics and Probability Letters 78, 2240-2247
(A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3)) to.shape(A%*%t(A)) to.shape(A%*%t(A),trace=3) to.shape(A%*%t(A),first=1)(A<-matrix(c(1,2,-3,4,3,-2,-1,0,4),ncol=3)) to.shape(A%*%t(A)) to.shape(A%*%t(A),trace=3) to.shape(A%*%t(A),first=1)