Nonparametric tests of independence between random vectors

Description

This function can be used for the following two problems: 1) testing mutual independence between many numerical random vectors, and 2) testing for serial independence of a multivariate stationary numerical time series. The proposed test does not assume continuous marginals. It is valid for any probability distribution. It is also invariant with respect to the affine general linear group of transformations on the vectors. This test is based on a characterization of mutual independence defined from probabilities of half-spaces in a combinatorial formula of Mobius. As such, it is a natural generalization of tests of independence between univariate random variables using the empirical distribution function. Without the assumption that each vector is one-dimensional with a continuous cumulative distribution function, any test of independence can not be distribution free. The critical values of the proposed test are thus computed with the bootstrap which was shown to be consistent in this context.

Usage

dependogram(X,vecd.ou.p,N=10,B=2000,alpha=0.05,display=TRUE,graphics=TRUE,nbclus=1)

Arguments

Value

A list with the following components:

In the mutual independence case:

In the serial case:

Author(s)

M. Bilodeau, P. Lafaye de Micheaux

Examples

n <- 100
W1 <- rpois(n,1)
W3 <- rpois(n,1)
W4 <- rpois(n,1)
W6 <- rpois(n,1)
W2 <- rpois(n,3)
W5 <- rpois(n,3)
X1 <- W1 + W2
X2 <- W2 + W3
X3 <- W4 + W5
X4 <- W5 + W6
X <- cbind(X1,X2,X3,X4)
dependogram(X,vecd.ou.p=c(1,1,1,1),N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)


n <- 50
Sigma <- matrix(c(1, 0, 0, 0, 0, 0,
                  0, 1, 0, 0, 0, 0,
                  0, 0, 1, 0,.4,.5,
                  0, 0, 0, 1,.1,.2,
                  0, 0,.4,.1, 1, 0,
                  0, 0,.5,.2, 0, 1),nrow=6,ncol=6,byrow=TRUE)
W <- chol(Sigma) 
X1 <- cbind(W[1,],W[2,])
X2 <- cbind(W[3,],W[4,])
X3 <- cbind(W[5,],W[6,])
X <- cbind(X1,X2,X3)

dependogram(X,vecd.ou.p=c(2,2,2),N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)



n <- 100
W <- sample(x=1:8,size=n,TRUE)
X1 <- W 
X2 <- W 
X3 <- W 
X4 <- W 
X <- cbind(X1,X2,X3,X4)
dependogram(X,vecd.ou.p=c(1,1,1,1),N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)


n <- 100
W <- rbinom(n,1,0.8)
Y <- W[1:(n-3)]*W[4:n]
dependogram(W,vecd.ou.p=4,N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)
dependogram(Y,vecd.ou.p=4,N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)


n <- 75
U <- matrix(rnorm(2*n),nrow=n,ncol=2)
W <- U[1:(n-1),] + sqrt(2)*U[2:n,]
Y <- W/apply(W,MARGIN=1,FUN=function(x) sqrt(x[1]^2+x[2]^2))

dependogram(Y,vecd.ou.p=3,N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.

R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.

Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.

> library(IndependenceTests)
Loading required package: xtable
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/IndependenceTests/dependogram.Rd_%03d_medium.png", width=480, height=480)
> ### Name: dependogram
> ### Title: Nonparametric tests of independence between random vectors
> ### Aliases: dependogram
> 
> ### ** Examples
> 
> 
> 
> n <- 100
> W1 <- rpois(n,1)
> W3 <- rpois(n,1)
> W4 <- rpois(n,1)
> W6 <- rpois(n,1)
> W2 <- rpois(n,3)
> W5 <- rpois(n,3)
> X1 <- W1 + W2
> X2 <- W2 + W3
> X3 <- W4 + W5
> X4 <- W5 + W6
> X <- cbind(X1,X2,X3,X4)
> dependogram(X,vecd.ou.p=c(1,1,1,1),N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)
1 : A= 1 2 : ||RnA||= 1.604 
2 : A= 1 3 : ||RnA||= 0.456 
3 : A= 1 4 : ||RnA||= 0.2 
4 : A= 2 3 : ||RnA||= 0.264 
5 : A= 2 4 : ||RnA||= 0.212 
6 : A= 3 4 : ||RnA||= 1.692 
7 : A= 1 2 3 : ||RnA||= 0.191 
8 : A= 1 2 4 : ||RnA||= 0.18 
9 : A= 1 3 4 : ||RnA||= 0.173 
10 : A= 2 3 4 : ||RnA||= 0.13 
11 : A= 1 2 3 4 : ||RnA||= 0.285 
$norm.RnA
 [1] 1.6040000 0.4560000 0.2000000 0.2640000 0.2120000 1.6920000 0.1910000
 [8] 0.1803200 0.1730000 0.1298400 0.2853096

$Rn
[1] 1.692

$rA
 [1] 0.724000 0.628000 0.720000 0.578000 0.568000 0.633000 0.391200 0.360200
 [9] 0.392000 0.314000 0.169152

$r
[1] 0.72

$RnAsstar
           [,1]      [,2]     [,3]      [,4]     [,5]     [,6]    [,7]     [,8]
 [1,] 0.3020000 0.7240000 0.386000 0.5450000 0.292000 0.478000 0.64700 0.283000
 [2,] 0.4880000 0.1680000 0.376000 0.4520000 0.264000 0.274000 0.22200 0.449000
 [3,] 0.4120000 0.3890000 0.320000 0.4930000 0.290000 0.221000 0.44000 0.340000
 [4,] 0.4740000 0.3160000 0.302000 0.2820000 0.308000 0.236000 0.27500 0.252000
 [5,] 0.4260000 0.4240000 0.552000 0.1980000 0.306000 0.422000 0.26200 0.284000
 [6,] 0.3680000 0.3130000 0.360000 0.3010000 0.444000 0.450000 0.41400 0.419000
 [7,] 0.2544000 0.2859600 0.209560 0.2986800 0.279000 0.127000 0.25030 0.391200
 [8,] 0.2870000 0.1826800 0.217120 0.3602000 0.210800 0.240620 0.30600 0.166920
 [9,] 0.2084800 0.2639600 0.391800 0.1777600 0.241800 0.265600 0.22160 0.349560
[10,] 0.2388000 0.2628800 0.184080 0.2105200 0.194080 0.184200 0.20720 0.125240
[11,] 0.1459736 0.1337112 0.169152 0.0907445 0.116984 0.106128 0.10853 0.136016
           [,9]     [,10]     [,11]   [,12]     [,13]   [,14]   [,15]     [,16]
 [1,] 0.3360000 0.3400000 0.3040000 0.33500 0.2540000 0.18000 0.42000 0.2940000
 [2,] 0.4380000 0.4140000 0.4000000 0.33200 0.3970000 0.42800 0.32800 0.3730000
 [3,] 0.3200000 0.4560000 0.4440000 0.30200 0.3010000 0.34000 0.58000 0.3970000
 [4,] 0.4560000 0.4070000 0.2600000 0.43600 0.3530000 0.30800 0.34000 0.5430000
 [5,] 0.3760000 0.4680000 0.5680000 0.32500 0.3060000 0.24700 0.27000 0.3620000
 [6,] 0.3600000 0.3580000 0.6000000 0.46000 0.2780000 0.63300 0.42400 0.5800000
 [7,] 0.1851200 0.1973600 0.1864000 0.23480 0.3187400 0.21528 0.20480 0.2340000
 [8,] 0.2419200 0.1976400 0.2239600 0.24080 0.2610000 0.24400 0.29600 0.1777200
 [9,] 0.2260000 0.2304400 0.2682000 0.17916 0.2441800 0.25232 0.19920 0.1853600
[10,] 0.2528400 0.1715600 0.2008800 0.22980 0.2065000 0.18912 0.31400 0.1495200
[11,] 0.1509684 0.1285292 0.1052422 0.10178 0.1520925 0.12208 0.13361 0.1090282
        [,17]     [,18]    [,19]    [,20]
 [1,] 0.28000 0.3060000 0.276000 0.460000
 [2,] 0.50200 0.6280000 0.360000 0.324000
 [3,] 0.56000 0.4120000 0.378000 0.720000
 [4,] 0.35000 0.3000000 0.578000 0.420000
 [5,] 0.27700 0.3400000 0.404000 0.480000
 [6,] 0.35000 0.3560000 0.416000 0.366000
 [7,] 0.18528 0.3162400 0.234400 0.205920
 [8,] 0.18004 0.2130000 0.176800 0.252000
 [9,] 0.25200 0.3920000 0.177800 0.195360
[10,] 0.20412 0.2692000 0.254800 0.222480
[11,] 0.12533 0.1315728 0.130632 0.121866

> 
> n <- 50
> Sigma <- matrix(c(1, 0, 0, 0, 0, 0,
+                   0, 1, 0, 0, 0, 0,
+                   0, 0, 1, 0,.4,.5,
+                   0, 0, 0, 1,.1,.2,
+                   0, 0,.4,.1, 1, 0,
+                   0, 0,.5,.2, 0, 1),nrow=6,ncol=6,byrow=TRUE)
> W <- chol(Sigma) 
> X1 <- cbind(W[1,],W[2,])
> X2 <- cbind(W[3,],W[4,])
> X3 <- cbind(W[5,],W[6,])
> X <- cbind(X1,X2,X3)
> ## No test: 
> dependogram(X,vecd.ou.p=c(2,2,2),N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)
1 : A= 1 2 : ||RnA||= 0.544 
2 : A= 1 3 : ||RnA||= 0.272 
3 : A= 2 3 : ||RnA||= 0.408 
4 : A= 1 2 3 : ||RnA||= 0.181 
$norm.RnA
[1] 0.5443311 0.2721655 0.4082483 0.1814437

$Rn
[1] 0.5443311

$rA
[1] 0.6123724 0.5443311 0.6123724 0.2721655

$r
[1] 0.6123724

$RnAsstar
          [,1]       [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
[1,] 0.0000000 0.20412415 0.3402069 0.0000000 0.3402069 0.4082483 0.4082483
[2,] 0.0000000 0.34020691 0.1360828 0.2041241 0.2041241 0.2721655 0.4082483
[3,] 0.2721655 0.06804138 0.5443311 0.2721655 0.5443311 0.2721655 0.2041241
[4,] 0.0000000 0.04536092 0.1360828 0.1360828 0.2041241 0.2041241 0.2041241
          [,8]      [,9]     [,10]     [,11]     [,12]     [,13]     [,14]
[1,] 0.4082483 0.2721655 0.4082483 0.6123724 0.3402069 0.0000000 0.3402069
[2,] 0.4082483 0.4082483 0.5443311 0.4082483 0.2721655 0.0000000 0.1360828
[3,] 0.4082483 0.4082483 0.3402069 0.5443311 0.4082483 0.4082483 0.4082483
[4,] 0.2721655 0.1360828 0.2041241 0.1360828 0.1814437 0.0000000 0.1134023
         [,15]     [,16]      [,17]     [,18]      [,19]     [,20]
[1,] 0.2721655 0.5443311 0.20412415 0.2721655 0.34020691 0.5443311
[2,] 0.2041241 0.3402069 0.06804138 0.3402069 0.13608276 0.4082483
[3,] 0.5443311 0.6123724 0.20412415 0.2041241 0.27216553 0.5443311
[4,] 0.1360828 0.2721655 0.06804138 0.1360828 0.09072184 0.1360828

> ## End(No test)
> 
> n <- 100
> W <- sample(x=1:8,size=n,TRUE)
> X1 <- W 
> X2 <- W 
> X3 <- W 
> X4 <- W 
> X <- cbind(X1,X2,X3,X4)
> dependogram(X,vecd.ou.p=c(1,1,1,1),N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)
1 : A= 1 2 : ||RnA||= 2.484 
2 : A= 1 3 : ||RnA||= 2.484 
3 : A= 1 4 : ||RnA||= 2.484 
4 : A= 2 3 : ||RnA||= 2.484 
5 : A= 2 4 : ||RnA||= 2.484 
6 : A= 3 4 : ||RnA||= 2.484 
7 : A= 1 2 3 : ||RnA||= 0.961 
8 : A= 1 2 4 : ||RnA||= 0.961 
9 : A= 1 3 4 : ||RnA||= 0.961 
10 : A= 2 3 4 : ||RnA||= 0.961 
11 : A= 1 2 3 4 : ||RnA||= 0.833 
$norm.RnA
 [1] 2.4840000 2.4840000 2.4840000 2.4840000 2.4840000 2.4840000 0.9609600
 [8] 0.9609600 0.9609600 0.9609600 0.8326032

$Rn
[1] 2.484

$rA
 [1] 0.6010000 0.6520000 0.8000000 0.6720000 0.7780000 0.5600000 0.3205000
 [8] 0.3780000 0.3005600 0.3470000 0.1967264

$r
[1] 0.778

$RnAsstar
           [,1]      [,2]      [,3]     [,4]      [,5]     [,6]     [,7]
 [1,] 0.2920000 0.4040000 0.5520000 0.348000 0.4680000 0.402000 0.425000
 [2,] 0.2940000 0.2960000 0.4000000 0.450000 0.4200000 0.484000 0.343000
 [3,] 0.4610000 0.2620000 0.5840000 0.465000 0.5320000 0.466000 0.452000
 [4,] 0.5480000 0.3440000 0.6720000 0.480000 0.4840000 0.500000 0.350000
 [5,] 0.3360000 0.3630000 0.5040000 0.625000 0.4650000 0.354000 0.375000
 [6,] 0.4540000 0.4160000 0.5250000 0.560000 0.3510000 0.359000 0.375000
 [7,] 0.2423600 0.1652000 0.2250000 0.263700 0.2036400 0.310100 0.233000
 [8,] 0.2126000 0.2255200 0.2723200 0.225000 0.2014600 0.220600 0.317140
 [9,] 0.2332000 0.2176000 0.2557600 0.296140 0.2278400 0.195000 0.274800
[10,] 0.2496000 0.2460800 0.2608000 0.264800 0.2132400 0.275000 0.191600
[11,] 0.1331016 0.1927312 0.1193712 0.134613 0.1222408 0.113592 0.167546
           [,8]     [,9]   [,10]   [,11]    [,12]     [,13]    [,14]    [,15]
 [1,] 0.4760000 0.300000 0.35600 0.37000 0.270000 0.4210000 0.414000 0.424000
 [2,] 0.2930000 0.488000 0.46800 0.59500 0.420000 0.4510000 0.384000 0.248000
 [3,] 0.2080000 0.287000 0.23200 0.39000 0.383000 0.4820000 0.424000 0.800000
 [4,] 0.4720000 0.360000 0.30200 0.42000 0.556000 0.2700000 0.488000 0.415000
 [5,] 0.6580000 0.430000 0.36600 0.49000 0.689000 0.2860000 0.352000 0.505000
 [6,] 0.3920000 0.338000 0.24000 0.50600 0.321000 0.4340000 0.339000 0.512000
 [7,] 0.1878000 0.230480 0.28800 0.17892 0.314640 0.1889600 0.208400 0.176360
 [8,] 0.2573200 0.378000 0.21880 0.30630 0.167480 0.2459400 0.247800 0.213500
 [9,] 0.1805600 0.218640 0.30056 0.24100 0.290560 0.2330800 0.295420 0.227840
[10,] 0.2365200 0.226520 0.27872 0.34700 0.174400 0.2978000 0.207200 0.247000
[11,] 0.1535536 0.098688 0.13476 0.10727 0.135288 0.1115188 0.118072 0.138476
         [,16]    [,17]   [,18]     [,19]     [,20]
 [1,] 0.353000 0.288000 0.51000 0.3320000 0.6010000
 [2,] 0.652000 0.299000 0.44000 0.3580000 0.3520000
 [3,] 0.328000 0.600000 0.36400 0.5170000 0.6480000
 [4,] 0.238000 0.400000 0.34000 0.5350000 0.5160000
 [5,] 0.402000 0.460000 0.25500 0.4300000 0.7780000
 [6,] 0.315000 0.550000 0.41800 0.2780000 0.2840000
 [7,] 0.218920 0.320500 0.24376 0.1628000 0.2250000
 [8,] 0.244000 0.214840 0.23072 0.2486000 0.2707200
 [9,] 0.180800 0.249960 0.22120 0.2238000 0.1934400
[10,] 0.340100 0.234400 0.30840 0.2283000 0.2638400
[11,] 0.133261 0.132834 0.13008 0.0857008 0.1967264

> 
> n <- 100
> W <- rbinom(n,1,0.8)
> Y <- W[1:(n-3)]*W[4:n]
> dependogram(W,vecd.ou.p=4,N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)
1 : A= 1 2 : ||SnA||= 0.046 
2 : A= 1 3 : ||SnA||= 0.055 
3 : A= 1 4 : ||SnA||= 0.046 
4 : A= 1 2 3 : ||SnA||= 0.182 
5 : A= 1 2 4 : ||SnA||= 0.021 
6 : A= 1 3 4 : ||SnA||= 0.08 
7 : A= 1 2 3 4 : ||SnA||= 0.026 
$norm.SnA
[1] 0.04605694 0.05547768 0.04605694 0.18199397 0.02107526 0.08045936 0.02586523

$Sn
[1] 0.181994

$sA
[1] 0.4951121 0.1758861 0.1183767

$s
[1] 0.38311

$SnAsstar
            [,1]       [,2]        [,3]       [,4]        [,5]       [,6]
[1,] 0.016747978 0.20620948 0.339146554 0.28052863 0.024075218 0.05547768
[2,] 0.186321255 0.02407522 0.085833387 0.22714445 0.147591556 0.24912617
[3,] 0.084786639 0.17899401 0.339146554 0.02407522 0.249126172 0.13503057
[4,] 0.055876952 0.09281531 0.043510211 0.03449954 0.047060523 0.02831617
[5,] 0.072624930 0.09281531 0.050502923 0.10567845 0.009463903 0.08986930
[6,] 0.031003183 0.09086210 0.043510211 0.10044470 0.007273284 0.11005968
[7,] 0.008773375 0.05465065 0.005895345 0.05346728 0.036099867 0.01408744
            [,7]       [,8]        [,9]      [,10]      [,11]       [,12]
[1,] 0.028262213 0.05547768 0.495112099 0.04501019 0.03454270 0.003140246
[2,] 0.073272404 0.15701229 0.292042866 0.25959366 0.16852653 0.003140246
[3,] 0.231331446 0.04605694 0.088973633 0.08269314 0.13607732 0.033495956
[4,] 0.030798150 0.04192390 0.156958337 0.09228654 0.08138740 0.029460039
[5,] 0.008978297 0.03762899 0.139163611 0.16044390 0.03898869 0.081430568
[6,] 0.008978297 0.06390562 0.037628994 0.06204953 0.06254593 0.020104048
[7,] 0.029552265 0.02256578 0.007666663 0.07632565 0.01954635 0.118376712
            [,13]       [,14]       [,15]       [,16]       [,17]        [,18]
[1,] 1.465448e-01 0.093160628 0.132937075 0.095254125 0.102581365 0.0198882239
[2,] 5.652443e-02 0.109908606 0.006280492 0.107815108 0.100487868 0.0994411193
[3,] 1.727135e-01 0.008373989 0.095254125 0.006280492 0.001046749 0.2009757358
[4,] 6.107832e-03 0.055876952 0.007165372 0.175886142 0.138645632 0.0610675305
[5,] 9.938716e-02 0.026567991 0.143566430 0.049229559 0.087452071 0.0059243814
[6,] 5.395611e-05 0.026567991 0.052305058 0.052305058 0.014082546 0.0007338032
[7,] 1.876126e-02 0.025056997 0.027025895 0.017741994 0.014643356 0.0192642242
           [,19]      [,20]
[1,] 0.002093497 0.38311000
[2,] 0.099441119 0.15701229
[3,] 0.099441119 0.18946150
[4,] 0.036323256 0.05760355
[5,] 0.018528530 0.09167144
[6,] 0.083006087 0.00644236
[7,] 0.001110161 0.05136166

> dependogram(Y,vecd.ou.p=4,N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)
1 : A= 1 2 : ||SnA||= 0.061 
2 : A= 1 3 : ||SnA||= 0.042 
3 : A= 1 4 : ||SnA||= 1.073 
4 : A= 1 2 3 : ||SnA||= 0.209 
5 : A= 1 2 4 : ||SnA||= 0.127 
6 : A= 1 3 4 : ||SnA||= 0.036 
7 : A= 1 2 3 4 : ||SnA||= 0.096 
$norm.SnA
[1] 0.06144637 0.04169575 1.07311700 0.20920249 0.12655805 0.03583594 0.09569975

$Sn
[1] 1.073117

$sA
[1] 0.6210473 0.2483535 0.1018472

$s
[1] 0.5738652

$SnAsstar
           [,1]       [,2]         [,3]       [,4]       [,5]        [,6]
[1,] 0.04608478 0.06144637 0.0449875224 0.13715708 0.47401487 0.092169558
[2,] 0.01206982 0.35112213 0.2644388514 0.22822938 0.24798000 0.229326639
[3,] 0.25236903 0.16458850 0.2918702676 0.24029921 0.01316708 0.092169558
[4,] 0.13097042 0.18910168 0.0068870364 0.08371835 0.15095450 0.003735342
[5,] 0.01893351 0.06546186 0.0004435718 0.04648166 0.04085530 0.027011190
[6,] 0.14413750 0.24835354 0.2100195910 0.14077569 0.10946886 0.003735342
[7,] 0.09139095 0.08517076 0.0265099987 0.06742540 0.01420076 0.036561893
           [,7]       [,8]        [,9]      [,10]       [,11]      [,12]
[1,] 0.01865336 0.13386531 0.311620887 0.07900248 0.573865225 0.20628425
[2,] 0.01865336 0.27431416 0.621047261 0.01755611 0.098753098 0.25785531
[3,] 0.18763089 0.10094761 0.002194513 0.18872814 0.004389027 0.00000000
[4,] 0.05278505 0.11395127 0.015408285 0.21366155 0.031983864 0.13057354
[5,] 0.23908522 0.01694911 0.139304902 0.11332093 0.028692094 0.05815460
[6,] 0.17348328 0.20133492 0.063687577 0.01797633 0.210416471 0.17226929
[7,] 0.01068335 0.04361061 0.101847173 0.02939942 0.030694128 0.04586299
          [,13]      [,14]      [,15]       [,16]        [,17]      [,18]
[1,] 0.08119699 0.16019947 0.10094761 0.228229382 0.2413964619 0.02413965
[2,] 0.33137151 0.14922690 0.20847876 0.182144603 0.3390523033 0.38842885
[3,] 0.27541142 0.41476301 0.26334159 0.070224425 0.5069325700 0.03291770
[4,] 0.06268370 0.01356396 0.08087015 0.226992050 0.1190873648 0.21543584
[5,] 0.05014696 0.02934578 0.09244971 0.005976547 0.0002334589 0.06824002
[6,] 0.05014696 0.07379635 0.01398419 0.070434538 0.0004202259 0.17973998
[7,] 0.04752949 0.04280816 0.03183584 0.040900254 0.0523827044 0.06841437
          [,19]        [,20]
[1,] 0.49815452 4.580430e-17
[2,] 0.29187027 3.609974e-01
[3,] 0.39501239 1.031421e-01
[4,] 0.06957074 2.359102e-01
[5,] 0.07197537 1.020449e-01
[6,] 0.02897224 2.413965e-02
[7,] 0.01119361 6.005729e-02

> 
> n <- 75
> U <- matrix(rnorm(2*n),nrow=n,ncol=2)
> W <- U[1:(n-1),] + sqrt(2)*U[2:n,]
> Y <- W/apply(W,MARGIN=1,FUN=function(x) sqrt(x[1]^2+x[2]^2))
> ## No test: 
> dependogram(Y,vecd.ou.p=3,N=10,B=20,alpha=0.05,display=TRUE,graphics=TRUE)
1 : A= 1 2 : ||SnA||= 1.267 
2 : A= 1 3 : ||SnA||= 0.694 
3 : A= 1 2 3 : ||SnA||= 0.442 
$norm.SnA
[1] 1.2668996 0.6940122 0.4424873

$Sn
[1] 1.2669

$sA
[1] 0.9542668 0.5614297

$s
[1] 0.9542668

$SnAsstar
          [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
[1,] 0.7922215 1.0541129 0.7660323 0.7660323 0.7725796 0.6383603 0.8838835
[2,] 0.9542668 0.5630665 0.6432708 0.8200474 0.7562114 0.8806098 0.6743704
[3,] 0.5614297 0.4182078 0.4190262 0.3875174 0.4484890 0.3756505 0.4550363
          [,8]      [,9]     [,10]     [,11]     [,12]     [,13]     [,14]
[1,] 0.8380525 0.7791269 0.7922215 0.7807637 0.8904308 0.6547285 0.8937044
[2,] 0.7332959 0.6678231 0.7332959 0.6776440 0.6760072 0.8740625 0.5827084
[3,] 0.4026580 0.3846530 0.4124790 0.3993844 0.4152070 0.4724048 0.4308477
         [,15]     [,16]     [,17]     [,18]     [,19]     [,20]
[1,] 0.8102265 0.6727335 0.7169277 0.6727335 0.8789730 0.8986149
[2,] 0.7332959 0.6841913 0.6334498 0.6154448 0.6416339 0.7954951
[3,] 0.4664941 0.4428966 0.4572187 0.4157526 0.3906547 0.4036583

> ## End(No test)
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>