Wishart: Wishart distribution

Description Usage Arguments Details Value Value for dWISHART Value for rWISHART Author(s) References See Also Examples

Description

Wishart distribution

Wishart(nu, S),

where nu are degrees of freedom of the Wishart distribution and S is its scale matrix. The same parametrization as in Gelman (2004) is assumed, that is, if W~Wishart(nu,S) then

E(W) = nu*S.

Prior to version 3.4-1 of this package, functions dWISHART and rWISHART were called as dWishart and rWishart, respectively. The names were changed in order to avoid conflicts with rWishart from a standard package stats.

Usage

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dWISHART(W, df, S, log=FALSE)

rWISHART(n, df, S)

Arguments

W

Either a matrix with the same number of rows and columns as S (1 point sampled from the Wishart distribution) or a matrix with ncol equal to ncol*(ncol+1)/2 and n rows (n points sampled from the Wishart distribution for which only lower triangles are given in rows of the matrix W).

n

number of observations to be sampled.

df

degrees of freedom of the Wishart distribution.

S

scale matrix of the Wishart distribution.

log

logical; if TRUE, log-density is computed

Details

The density of the Wishart distribution is the following

f(W) = (2^{nu*p/2} * pi^{p*(p-1)/4} * prod[i=1]^p Gamma((nu + 1 - i)/2))^{-1} * |S|^{-nu/2} * |W|^{(nu - p - 1)/2} * exp(-0.5*tr(S^{-1}*W)),

where p is number of rows and columns of the matrix W.

In the univariate case, Wishart(nu,S) is the same as Gamma(nu/2, 1/(2*S)).

Generation of random numbers is performed by the algorithm described in Ripley (1987, pp. 99).

Value

Some objects.

Value for dWISHART

A numeric vector with evaluated (log-)density.

Value for rWISHART

If n equals 1 then a sampled symmetric matrix W is returned.

If n > 1 then a matrix with sampled points (lower triangles of W) in rows is returned.

Author(s)

Arno<c5><a1>t Kom<c3><a1>rek arnost.komarek[AT]mff.cuni.cz

References

Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (2004). Bayesian Data Analysis, Second edition. Boca Raton: Chapman and Hall/CRC.

Ripley, B. D. (1987). Stochastic Simulation. New York: John Wiley and Sons.

See Also

rWishart.

Examples

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set.seed(1977)
### The same as gamma(shape=df/2, rate=1/(2*S))
df <- 1
S  <- 3

w <- rWISHART(n=1000, df=df, S=S)
mean(w)    ## should be close to df*S
var(w)     ## should be close to 2*df*S^2

dWISHART(w[1], df=df, S=S)
dWISHART(w[1], df=df, S=S, log=TRUE)

dens.w <- dWISHART(w, df=df, S=S)
dens.wG <- dgamma(w, shape=df/2, rate=1/(2*S))
rbind(dens.w[1:10], dens.wG[1:10])

ldens.w <- dWISHART(w, df=df, S=S, log=TRUE)
ldens.wG <- dgamma(w, shape=df/2, rate=1/(2*S), log=TRUE)
rbind(ldens.w[1:10], ldens.wG[1:10])


### Bivariate Wishart
df <- 2
S <- matrix(c(1,3,3,13), nrow=2)

print(w2a <- rWISHART(n=1, df=df, S=S))
dWISHART(w2a, df=df, S=S)

w2 <- rWISHART(n=1000, df=df, S=S)
print(w2[1:10,])
apply(w2, 2, mean)                ## should be close to df*S
(df*S)[lower.tri(S, diag=TRUE)]

dens.w2 <- dWISHART(w2, df=df, S=S)
ldens.w2 <- dWISHART(w2, df=df, S=S, log=TRUE)
cbind(w2[1:10,], data.frame(Density=dens.w2[1:10], Log.Density=ldens.w2[1:10]))


### Trivariate Wishart
df <- 3.5
S <- matrix(c(1,2,3,2,20,26,3,26,70), nrow=3)

print(w3a <- rWISHART(n=1, df=df, S=S))
dWISHART(w3a, df=df, S=S)

w3 <- rWISHART(n=1000, df=df, S=S)
print(w3[1:10,])
apply(w3, 2, mean)                ## should be close to df*S
(df*S)[lower.tri(S, diag=TRUE)]

dens.w3 <- dWISHART(w3, df=df, S=S)
ldens.w3 <- dWISHART(w3, df=df, S=S, log=TRUE)
cbind(w3[1:10,], data.frame(Density=dens.w3[1:10], Log.Density=ldens.w3[1:10]))

Example output

Loading required package: colorspace
Loading required package: lme4
Loading required package: Matrix

### Mixture of methods including mixtures
### Arnost Komarek

### See citation("mixAK") or toBibtex(citation("mixAK")) for the best way to cite
### the package if you find it useful.


[1] 2.967794
[1] 19.43923
[1] 0.0588085
[1] -2.833469
             1         2         3        4         5         6         7
[1,] 0.0588085 0.1449037 0.3470858 3.054181 0.2388643 0.1074662 0.1479219
[2,] 0.0588085 0.1449037 0.3470858 3.054181 0.2388643 0.1074662 0.1479219
             8         9       10
[1,] 0.8135921 0.7172008 2.210538
[2,] 0.8135921 0.7172008 2.210538
             1         2         3        4        5         6         7
[1,] -2.833469 -1.931686 -1.058183 1.116511 -1.43186 -2.230579 -1.911071
[2,] -2.833469 -1.931686 -1.058183 1.116511 -1.43186 -2.230579 -1.911071
              8          9        10
[1,] -0.2062962 -0.3323995 0.7932361
[2,] -0.2062962 -0.3323995 0.7932361
         [,1]      [,2]
[1,] 0.341001  2.091936
[2,] 2.091936 17.844946
[1] 0.00451184
         (1.1)      (2.1)      (2.2)
1   3.46855249  7.9162578  35.985898
2   1.77611610  0.5697106   2.019978
3   7.71710352 26.4729448  91.249136
4   0.95528853  1.5859176   4.814772
5   0.92952209 -0.4209225  14.579903
6   0.47219029  0.4200592   4.743177
7   2.98564445 15.7761746  91.794062
8   2.36146714  5.6494757  13.518222
9   0.01211207  0.2777356  13.394315
10 12.24134582 34.2778524 104.200794
    (1.1)     (2.1)     (2.2) 
 2.057971  6.160985 26.490188 
[1]  2  6 26
         (1.1)      (2.1)      (2.2)      Density Log.Density
1   3.46855249  7.9162578  35.985898 3.793826e-05  -10.179551
2   1.77611610  0.5697106   2.019978 7.317531e-04   -7.220067
3   7.71710352 26.4729448  91.249136 1.813020e-04   -8.615347
4   0.95528853  1.5859176   4.814772 5.251182e-03   -5.249302
5   0.92952209 -0.4209225  14.579903 1.415763e-04   -8.862672
6   0.47219029  0.4200592   4.743177 4.870213e-03   -5.324618
7   2.98564445 15.7761746  91.794062 4.431984e-05  -10.024078
8   2.36146714  5.6494757  13.518222 6.920151e-02   -2.670733
9   0.01211207  0.2777356  13.394315 1.543682e-02   -4.171000
10 12.24134582 34.2778524 104.200794 1.467829e-06  -13.431726
          [,1]      [,2]      [,3]
[1,]  4.420618  14.92187  10.96118
[2,] 14.921869 101.44288 100.68469
[3,] 10.961177 100.68469 126.92317
[1] 7.86458e-11
       (1.1)      (2.1)     (3.1)       (2.2)      (3.2)     (3.3)
1   1.575966  0.1652046 12.906412  24.1248781  41.104651 261.35484
2   3.083901  6.9410265  1.669144  85.6338825  38.009242  19.75609
3   6.948168 12.9984221 14.240827  57.8778758  67.564201 274.41267
4   6.089056 10.1538141 18.463781  82.0256934 156.165259 430.56446
5   4.977367 13.2492356 25.247122 132.0954065 161.218745 339.45961
6   5.169670  9.9097134 12.311678  55.2284374  98.241472 202.10080
7  10.762091 22.7189887 49.352652  59.8707432  90.608935 296.57284
8   3.037555 17.9022981 11.649918 145.4420432  37.573954 185.20094
9   0.630428 -0.4079311 -1.953571   0.4067281   1.778779 167.44650
10  2.133667  5.1083065 11.612041  51.2147903 179.996712 669.65562
     (1.1)      (2.1)      (3.1)      (2.2)      (3.2)      (3.3) 
  3.452541   7.036285  10.523561  72.353371  96.256739 254.865072 
[1]   3.5   7.0  10.5  70.0  91.0 245.0
       (1.1)      (2.1)     (3.1)       (2.2)      (3.2)     (3.3)      Density
1   1.575966  0.1652046 12.906412  24.1248781  41.104651 261.35484 1.243326e-10
2   3.083901  6.9410265  1.669144  85.6338825  38.009242  19.75609 1.405562e-10
3   6.948168 12.9984221 14.240827  57.8778758  67.564201 274.41267 3.157100e-12
4   6.089056 10.1538141 18.463781  82.0256934 156.165259 430.56446 2.813189e-12
5   4.977367 13.2492356 25.247122 132.0954065 161.218745 339.45961 2.526182e-12
6   5.169670  9.9097134 12.311678  55.2284374  98.241472 202.10080 1.161716e-10
7  10.762091 22.7189887 49.352652  59.8707432  90.608935 296.57284 3.129837e-12
8   3.037555 17.9022981 11.649918 145.4420432  37.573954 185.20094 9.543639e-13
9   0.630428 -0.4079311 -1.953571   0.4067281   1.778779 167.44650 2.235773e-09
10  2.133667  5.1083065 11.612041  51.2147903 179.996712 669.65562 2.020762e-11
   Log.Density
1    -22.80806
2    -22.68541
3    -26.48137
4    -26.59670
5    -26.70431
6    -22.87595
7    -26.49004
8    -27.67773
9    -19.91868
10   -24.62496

mixAK documentation built on May 29, 2017, 9:20 p.m.

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