[R] boundary check
Dear R, I have a covariates matrix with 10 observations, e.g. X - matrix(rnorm(50), 10, 5) X [,1][,2][,3][,4] [,5] [1,] 0.24857135 0.30880745 -1.44118657 1.10229027 1.0526010 [2,] 1.24316806 0.36275370 -0.40096866 -0.24387888 -1.5324384 [3,] -0.33504014 0.42996246 0.03902479 -0.84778875 -2.4754644 [4,] 0.06710229 1.01950917 -0.09325091 -0.03222811 0.4127816 [5,] -0.13619141 1.33143821 -0.79958805 2.08274102 0.6901768 [6,] -0.45060357 0.19348831 -1.23793647 -0.72440163 0.5057326 [7,] -1.20740516 0.20231086 1.15584485 0.8170 -1.2719855 [8,] -1.81166284 -0.07913113 -0.91080581 -0.34774436 0.9552182 [9,] 0.19131383 0.14980569 -0.37458224 -0.09371273 -1.7667203 [10,] -0.85159276 -0.66679528 1.63019340 0.56920196 -2.4049600 And I define a boundary of X: The smallest ball that nests all the observations of X. I wish to check if a particular point x_i x_i - matrix(rnorm(5), 1, 5) x_i [,1] [,2] [,3] [,4] [,5] [1,] -0.1525543 0.4606419 -0.1011011 -1.557225 -1.035694 is inside the boundary of X or not. I know it's easy to do it with 1-D or 2-D, but I don't knot how to manage it when the dimension is large. Can someone give a hint? Thanks in advance! Feng -- Feng Li Department of Statistics Stockholm University 106 91 Stockholm, Sweden http://feng.li/ [[alternative HTML version deleted]] __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] boundary check
Hello
convex hulls in large numbers of dimensions are hard.
For your problem, though, one can tell whether a given
point is inside or outside by using linear programming:
X - matrix(rnorm(50), 10, 5)
x_i - matrix(rnorm(5), 1, 5)
isin.chull
function(candidate,p,plot=FALSE,give.answers=FALSE,
...){
if(plot){
plot(p,...)
p(candidate[1],candidate[2], pch=16)
}
n - nrow(p) # number of points
d - ncol(p) # number of dimensions
p - t(sweep(p,2,candidate))
jj - simplex(a=rep(1,n),A3=rbind(p,1),b3=c(0*candidate,1))
if(give.answers){
return(jj)
} else {
return((jj$solved = 0) all(jj$soln1))
}
}
isin.chull(x_i,X)
[1] FALSE
(we can discuss offline; I'll summarize)
HTH
rksh
On 24/09/10 10:44, Feng Li wrote:
Dear R,
I have a covariates matrix with 10 observations, e.g.
X - matrix(rnorm(50), 10, 5)
X
[,1][,2][,3][,4] [,5]
[1,] 0.24857135 0.30880745 -1.44118657 1.10229027 1.0526010
[2,] 1.24316806 0.36275370 -0.40096866 -0.24387888 -1.5324384
[3,] -0.33504014 0.42996246 0.03902479 -0.84778875 -2.4754644
[4,] 0.06710229 1.01950917 -0.09325091 -0.03222811 0.4127816
[5,] -0.13619141 1.33143821 -0.79958805 2.08274102 0.6901768
[6,] -0.45060357 0.19348831 -1.23793647 -0.72440163 0.5057326
[7,] -1.20740516 0.20231086 1.15584485 0.8170 -1.2719855
[8,] -1.81166284 -0.07913113 -0.91080581 -0.34774436 0.9552182
[9,] 0.19131383 0.14980569 -0.37458224 -0.09371273 -1.7667203
[10,] -0.85159276 -0.66679528 1.63019340 0.56920196 -2.4049600
And I define a boundary of X: The smallest ball that nests all the
observations of X. I wish to check if a particular point x_i
x_i - matrix(rnorm(5), 1, 5)
x_i
[,1] [,2] [,3] [,4] [,5]
[1,] -0.1525543 0.4606419 -0.1011011 -1.557225 -1.035694
is inside the boundary of X or not. I know it's easy to do it with 1-D or
2-D, but I don't knot how to manage it when the dimension is large.
Can someone give a hint? Thanks in advance!
Feng
--
Robin K. S. Hankin
Uncertainty Analyst
University of Cambridge
19 Silver Street
Cambridge CB3 9EP
01223-764877
__
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] boundary check
Hello, If an N-dimensional convex hull fits your idea of a smallest ball then you could try the convhulln function in the geometry package. For testing if a new point is inside a previously derived hull, one brute force approach is to rbind the new point to your data, generate a new hull and see if it is the same as the previous one. I've only used convhulln in low dimensions so I don't know how efficient it is when N is large. Hope this helps. Michael On 24 September 2010 19:44, Feng Li feng...@stat.su.se wrote: Dear R, I have a covariates matrix with 10 observations, e.g. X - matrix(rnorm(50), 10, 5) X [,1] [,2] [,3] [,4] [,5] [1,] 0.24857135 0.30880745 -1.44118657 1.10229027 1.0526010 [2,] 1.24316806 0.36275370 -0.40096866 -0.24387888 -1.5324384 [3,] -0.33504014 0.42996246 0.03902479 -0.84778875 -2.4754644 [4,] 0.06710229 1.01950917 -0.09325091 -0.03222811 0.4127816 [5,] -0.13619141 1.33143821 -0.79958805 2.08274102 0.6901768 [6,] -0.45060357 0.19348831 -1.23793647 -0.72440163 0.5057326 [7,] -1.20740516 0.20231086 1.15584485 0.8170 -1.2719855 [8,] -1.81166284 -0.07913113 -0.91080581 -0.34774436 0.9552182 [9,] 0.19131383 0.14980569 -0.37458224 -0.09371273 -1.7667203 [10,] -0.85159276 -0.66679528 1.63019340 0.56920196 -2.4049600 And I define a boundary of X: The smallest ball that nests all the observations of X. I wish to check if a particular point x_i x_i - matrix(rnorm(5), 1, 5) x_i [,1] [,2] [,3] [,4] [,5] [1,] -0.1525543 0.4606419 -0.1011011 -1.557225 -1.035694 is inside the boundary of X or not. I know it's easy to do it with 1-D or 2-D, but I don't knot how to manage it when the dimension is large. Can someone give a hint? Thanks in advance! Feng -- Feng Li Department of Statistics Stockholm University 106 91 Stockholm, Sweden http://feng.li/ [[alternative HTML version deleted]] __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] boundary check
Hi, I remember a discussion we had on this list a few months ago for a better way to decide if a point is inside a convex hull. It eventually lead to a R function in this post, http://tolstoy.newcastle.edu.au/R/e8/help/09/12/8784.html I don't know if it was included in the geometry package in the end. HTH, baptiste On 24 September 2010 12:44, Michael Bedward michael.bedw...@gmail.com wrote: Hello, If an N-dimensional convex hull fits your idea of a smallest ball then you could try the convhulln function in the geometry package. For testing if a new point is inside a previously derived hull, one brute force approach is to rbind the new point to your data, generate a new hull and see if it is the same as the previous one. I've only used convhulln in low dimensions so I don't know how efficient it is when N is large. Hope this helps. Michael On 24 September 2010 19:44, Feng Li feng...@stat.su.se wrote: Dear R, I have a covariates matrix with 10 observations, e.g. X - matrix(rnorm(50), 10, 5) X [,1] [,2] [,3] [,4] [,5] [1,] 0.24857135 0.30880745 -1.44118657 1.10229027 1.0526010 [2,] 1.24316806 0.36275370 -0.40096866 -0.24387888 -1.5324384 [3,] -0.33504014 0.42996246 0.03902479 -0.84778875 -2.4754644 [4,] 0.06710229 1.01950917 -0.09325091 -0.03222811 0.4127816 [5,] -0.13619141 1.33143821 -0.79958805 2.08274102 0.6901768 [6,] -0.45060357 0.19348831 -1.23793647 -0.72440163 0.5057326 [7,] -1.20740516 0.20231086 1.15584485 0.8170 -1.2719855 [8,] -1.81166284 -0.07913113 -0.91080581 -0.34774436 0.9552182 [9,] 0.19131383 0.14980569 -0.37458224 -0.09371273 -1.7667203 [10,] -0.85159276 -0.66679528 1.63019340 0.56920196 -2.4049600 And I define a boundary of X: The smallest ball that nests all the observations of X. I wish to check if a particular point x_i x_i - matrix(rnorm(5), 1, 5) x_i [,1] [,2] [,3] [,4] [,5] [1,] -0.1525543 0.4606419 -0.1011011 -1.557225 -1.035694 is inside the boundary of X or not. I know it's easy to do it with 1-D or 2-D, but I don't knot how to manage it when the dimension is large. Can someone give a hint? Thanks in advance! Feng -- Feng Li Department of Statistics Stockholm University 106 91 Stockholm, Sweden http://feng.li/ [[alternative HTML version deleted]] __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] boundary check
Thanks. I agree with you that the speed and memory issues might be
(actually is) a big problem for big dimensions. It is interesting to
know to solve this by using linear programming. Buy the way, it seems
a potential bug in your function if you try this
X - matrix(rnorm(50), 10, 5)
x_i-X[1,,drop=FALSE]
isin.chull(x_i,X)
Error in A.out[, basic] - iden(M) : subscript out of bounds
Or I must be wrong somewhere.
Feng
On Sep 24, 12:39 pm, Robin Hankin rk...@cam.ac.uk wrote:
Hello
convex hulls in large numbers of dimensions are hard.
For your problem, though, one can tell whether a given
point is inside or outside by using linear programming:
X - matrix(rnorm(50), 10, 5)
x_i - matrix(rnorm(5), 1, 5)
isin.chull
function(candidate,p,plot=FALSE,give.answers=FALSE,
...){
if(plot){
plot(p,...)
p(candidate[1],candidate[2], pch=16)
}
n - nrow(p) # number of points
d - ncol(p) # number of dimensions
p - t(sweep(p,2,candidate))
jj - simplex(a=rep(1,n),A3=rbind(p,1),b3=c(0*candidate,1))
if(give.answers){
return(jj)
} else {
return((jj$solved = 0) all(jj$soln1))
}
}
isin.chull(x_i,X)
[1] FALSE
(we can discuss offline; I'll summarize)
HTH
rksh
On 24/09/10 10:44, Feng Li wrote:
Dear R,
I have a covariates matrix with 10 observations, e.g.
X - matrix(rnorm(50), 10, 5)
X
[,1] [,2] [,3] [,4] [,5]
[1,] 0.24857135 0.30880745 -1.44118657 1.10229027 1.0526010
[2,] 1.24316806 0.36275370 -0.40096866 -0.24387888 -1.5324384
[3,] -0.33504014 0.42996246 0.03902479 -0.84778875 -2.4754644
[4,] 0.06710229 1.01950917 -0.09325091 -0.03222811 0.4127816
[5,] -0.13619141 1.33143821 -0.79958805 2.08274102 0.6901768
[6,] -0.45060357 0.19348831 -1.23793647 -0.72440163 0.5057326
[7,] -1.20740516 0.20231086 1.15584485 0.8170 -1.2719855
[8,] -1.81166284 -0.07913113 -0.91080581 -0.34774436 0.9552182
[9,] 0.19131383 0.14980569 -0.37458224 -0.09371273 -1.7667203
[10,] -0.85159276 -0.66679528 1.63019340 0.56920196 -2.4049600
And I define a boundary of X: The smallest ball that nests all the
observations of X. I wish to check if a particular point x_i
x_i - matrix(rnorm(5), 1, 5)
x_i
[,1] [,2] [,3] [,4] [,5]
[1,] -0.1525543 0.4606419 -0.1011011 -1.557225 -1.035694
is inside the boundary of X or not. I know it's easy to do it with 1-D or
2-D, but I don't knot how to manage it when the dimension is large.
Can someone give a hint? Thanks in advance!
Feng
--
Robin K. S. Hankin
Uncertainty Analyst
University of Cambridge
19 Silver Street
Cambridge CB3 9EP
01223-764877
__
r-h...@r-project.org mailing listhttps://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guidehttp://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
__
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] boundary check
Hi,
Below Baptiste's message I attach the R code and the .Rd documentation I
treated as 'final', it may be slightly different from that in the Dec 2009
post.
I did submit if for inclusion in the geometry package, but last time I
checked it wasn't there.
I have found (and others have reported via private e-mail) that high
dimensional data sets cause crashes (R exits with no warnings or messages).
I tentatively believe this is a 'bug' in convhulln, but tracking it takes me
to a complicated package and compiled code. It isn't a problem for me, so I
can't make time to follow it up.
hth.
Keith J
baptiste auguie baptiste.aug...@googlemail.com wrote in message
news:aanlktikf3+higwyhwyxeu2brwqs0x9mtywz9xzyq_...@mail.gmail.com...
Hi,
I remember a discussion we had on this list a few months ago for a
better way to decide if a point is inside a convex hull. It eventually
lead to a R function in this post,
http://tolstoy.newcastle.edu.au/R/e8/help/09/12/8784.html
I don't know if it was included in the geometry package in the end.
HTH,
baptiste
-
inhull - function(testpts, calpts, hull=convhulln(calpts),
tol=mean(mean(abs(calpts)))*sqrt(.Machine$double.eps)) {
#
# R implementation of the Matlab code by John D'Errico 04 Mar 2006 (Updated
30 Oct 2006)
# downloaded from
http://www.mathworks.com/matlabcentral/fileexchange/10226-inhull
# with some modifications and simplifications
#
# Efficient test for points inside a convex hull in n dimensions
#
#% arguments: (input)
#% testpts - nxp array to test, n data points, in p dimensions
#% If you have many points to test, it is most efficient to
#% call this function once with the entire set.
#%
#% calpts - mxp array of vertices of the convex hull, as used by
#% convhulln.
#%
#% hull - (OPTIONAL) tessellation (or triangulation) generated by convhulln
#% If hull is left empty or not supplied, then it will be
#% generated.
#%
#% tol - (OPTIONAL) tolerance on the tests for inclusion in the
#% convex hull. You can think of tol as the distance a point
#% may possibly lie outside the hull, and still be perceived
#% as on the surface of the hull. Because of numerical slop
#% nothing can ever be done exactly here. I might guess a
#% semi-intelligent value of tol to be
#%
#% tol = 1.e-13*mean(abs(calpts(:)))
#%
#% In higher dimensions, the numerical issues of floating
#% point arithmetic will probably suggest a larger value
#% of tol.
#%
# In this R implementation default
tol=mean(mean(abs(calpts)))*sqrt(.Machine$double.eps)
#
# VALUE: Matlab returns a vector of TRUE (inside/on) or FALSE (outside)
# This R implementation returns an integer vector of length n
# 1 = inside hull
# -1 = inside hull
# 0 = on hull (to precision indicated by tol)
#
# ensure arguments are matrices (not data frames) and get sizes
calpts - as.matrix(calpts)
testpts - as.matrix(testpts)
p - ncol(calpts) # columns in calpts
cx - nrow(testpts) # rows in testpts
nt - nrow(hull)# number of simplexes in hull
# find normal vectors to each simplex
nrmls - matrix(NA, nt, p) # predefine each nrml as NA,
degenerate
degenflag - matrix(TRUE, nt, 1)
for (i in 1:nt) {
nullsp - t(Null(t(calpts[hull[i,-1],] -
matrix(calpts[hull[i,1],],p-1,p, byrow=TRUE
if (dim(nullsp)[1] == 1) { nrmls[i,] - nullsp
degenflag[i] - FALSE}}
# Warn of degenerate faces, and remove corresponding normals
if(sum(degenflag) 0) warning(sum(degenflag), degenerate faces in
convex hull)
nrmls - nrmls[!degenflag,]
nt - nrow(nrmls)
# find center point in hull, and any (1st) point in the plane of each
simplex
center = colMeans(calpts)
a - calpts[hull[!degenflag,1],]
# scale normal vectors to unit length and ensure pointing inwards
nrmls - nrmls/matrix(apply(nrmls, 1, function(x) sqrt(sum(x^2))), nt, p)
dp - sign(apply((matrix(center, nt, p, byrow=TRUE)-a) * nrmls, 1, sum))
nrmls - nrmls*matrix(dp, nt, p)
# if min across all faces of dot((x - a),nrml) is
# +ve then x is inside hull
# 0 then x is on hull
# -ve then x is outside hull
# Instead of dot((x - a),nrml) use dot(x,nrml) - dot(a, nrml)
aN - diag(a %*% t(nrmls))
val - apply(testpts %*% t(nrmls) - matrix(aN, cx, nt, byrow=TRUE),
1,min)
# code values inside 'tol' to zero, return sign as integer
val[abs(val) tol] - 0
as.integer(sign(val))
}
\name{inhull}
\alias{inhull}
%- Also NEED an '\alias' for EACH other topic documented here.
\title{
Test if one set of N-D points is inside the convex hull defined by another
set
}
\description{
R implementation of the Matlab code by John D'Errico 04 Mar 2006 (Updated 30
Oct 2006)
downloaded from
Re: [R] boundary check
Feng
thanks for this. The problem you report is
reproducible; it originates from simplex()
of the boot packge.
It is ultimately due to
the fact that x_i is precisely *on* the convex hull,
which is evidently causing problems. I'll
investigate it.
In the short term, you can break the degeneracy:
isin.chull(X[1,]+1e-10,X)
[1] FALSE
but I don't know if that is acceptable to you.
best
rksh
On 24/09/10 13:17, Feng Li wrote:
Thanks. I agree with you that the speed and memory issues might be
(actually is) a big problem for big dimensions. It is interesting to
know to solve this by using linear programming. Buy the way, it seems
a potential bug in your function if you try this
X - matrix(rnorm(50), 10, 5)
x_i-X[1,,drop=FALSE]
isin.chull(x_i,X)
Error in A.out[, basic] - iden(M) : subscript out of bounds
Or I must be wrong somewhere.
Feng
On Sep 24, 12:39 pm, Robin Hankin rk...@cam.ac.uk wrote:
Hello
convex hulls in large numbers of dimensions are hard.
For your problem, though, one can tell whether a given
point is inside or outside by using linear programming:
X - matrix(rnorm(50), 10, 5)
x_i - matrix(rnorm(5), 1, 5)
isin.chull
function(candidate,p,plot=FALSE,give.answers=FALSE,
...){
if(plot){
plot(p,...)
p(candidate[1],candidate[2], pch=16)
}
n - nrow(p) # number of points
d - ncol(p) # number of dimensions
p - t(sweep(p,2,candidate))
jj - simplex(a=rep(1,n),A3=rbind(p,1),b3=c(0*candidate,1))
if(give.answers){
return(jj)
} else {
return((jj$solved = 0) all(jj$soln1))
}
}
isin.chull(x_i,X)
[1] FALSE
(we can discuss offline; I'll summarize)
HTH
rksh
On 24/09/10 10:44, Feng Li wrote:
Dear R,
I have a covariates matrix with 10 observations, e.g.
X - matrix(rnorm(50), 10, 5)
X
[,1][,2][,3][,4] [,5]
[1,] 0.24857135 0.30880745 -1.44118657 1.10229027 1.0526010
[2,] 1.24316806 0.36275370 -0.40096866 -0.24387888 -1.5324384
[3,] -0.33504014 0.42996246 0.03902479 -0.84778875 -2.4754644
[4,] 0.06710229 1.01950917 -0.09325091 -0.03222811 0.4127816
[5,] -0.13619141 1.33143821 -0.79958805 2.08274102 0.6901768
[6,] -0.45060357 0.19348831 -1.23793647 -0.72440163 0.5057326
[7,] -1.20740516 0.20231086 1.15584485 0.8170 -1.2719855
[8,] -1.81166284 -0.07913113 -0.91080581 -0.34774436 0.9552182
[9,] 0.19131383 0.14980569 -0.37458224 -0.09371273 -1.7667203
[10,] -0.85159276 -0.66679528 1.63019340 0.56920196 -2.4049600
And I define a boundary of X: The smallest ball that nests all the
observations of X. I wish to check if a particular point x_i
x_i - matrix(rnorm(5), 1, 5)
x_i
[,1] [,2] [,3] [,4] [,5]
[1,] -0.1525543 0.4606419 -0.1011011 -1.557225 -1.035694
is inside the boundary of X or not. I know it's easy to do it with 1-D or
2-D, but I don't knot how to manage it when the dimension is large.
Can someone give a hint? Thanks in advance!
Feng
--
Robin K. S. Hankin
Uncertainty Analyst
University of Cambridge
19 Silver Street
Cambridge CB3 9EP
01223-764877
__
r-h...@r-project.org mailing listhttps://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guidehttp://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
--
Robin K. S. Hankin
Uncertainty Analyst
University of Cambridge
19 Silver Street
Cambridge CB3 9EP
01223-764877
__
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] boundary check
You did not originally define ball, the other respondents have discussed using a convex hull, but here is another approach: Use ball to mean sphere (or technically hypersphere) and find the sphere with the smallest radius that contains all the points, optim or other optimizers could be programmed to do this (or an approximation that may be good enough is to use the means as the center and the distance to the furthest point as the radius). Then finding if a new point is within the sphere is just a matter of computing the Euclidean distance from the new point to the center and comparing that to the radius. -- Gregory (Greg) L. Snow Ph.D. Statistical Data Center Intermountain Healthcare greg.s...@imail.org 801.408.8111 -Original Message- From: r-help-boun...@r-project.org [mailto:r-help-boun...@r- project.org] On Behalf Of Feng Li Sent: Friday, September 24, 2010 3:44 AM To: r-help@r-project.org Subject: [R] boundary check Dear R, I have a covariates matrix with 10 observations, e.g. X - matrix(rnorm(50), 10, 5) X [,1][,2][,3][,4] [,5] [1,] 0.24857135 0.30880745 -1.44118657 1.10229027 1.0526010 [2,] 1.24316806 0.36275370 -0.40096866 -0.24387888 -1.5324384 [3,] -0.33504014 0.42996246 0.03902479 -0.84778875 -2.4754644 [4,] 0.06710229 1.01950917 -0.09325091 -0.03222811 0.4127816 [5,] -0.13619141 1.33143821 -0.79958805 2.08274102 0.6901768 [6,] -0.45060357 0.19348831 -1.23793647 -0.72440163 0.5057326 [7,] -1.20740516 0.20231086 1.15584485 0.8170 -1.2719855 [8,] -1.81166284 -0.07913113 -0.91080581 -0.34774436 0.9552182 [9,] 0.19131383 0.14980569 -0.37458224 -0.09371273 -1.7667203 [10,] -0.85159276 -0.66679528 1.63019340 0.56920196 -2.4049600 And I define a boundary of X: The smallest ball that nests all the observations of X. I wish to check if a particular point x_i x_i - matrix(rnorm(5), 1, 5) x_i [,1] [,2] [,3] [,4] [,5] [1,] -0.1525543 0.4606419 -0.1011011 -1.557225 -1.035694 is inside the boundary of X or not. I know it's easy to do it with 1-D or 2-D, but I don't knot how to manage it when the dimension is large. Can someone give a hint? Thanks in advance! Feng -- Feng Li Department of Statistics Stockholm University 106 91 Stockholm, Sweden http://feng.li/ [[alternative HTML version deleted]] __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting- guide.html and provide commented, minimal, self-contained, reproducible code. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
