Sample a covariance Matrix from the Posterior Inverse Wishart distribution.

Note this Inverse wishart rescaled to match the original scale of the covariance matrix.

Usage

cvPost(
  nu,
  omega,
  n = 1L,
  omegaIsChol = FALSE,
  returnChol = FALSE,
  type = c("invWishart", "lkj", "separation"),
  diagXformType = c("log", "identity", "variance", "nlmixrSqrt", "nlmixrLog",
    "nlmixrIdentity")
)

Arguments

nu

Degrees of Freedom (Number of Observations) for covariance matrix simulation.

omega

Either the estimate of covariance matrix or the estimated standard deviations in matrix form each row forming the standard deviation simulated values

n

Number of Matrices to sample. By default this is 1. This is only useful when omega is a matrix. Otherwise it is determined by the number of rows in the input omega matrix of standard deviations

omegaIsChol

is an indicator of if the omega matrix is in the Cholesky decomposition. This is only used when type="invWishart"

returnChol

Return the Cholesky decomposition of the covariance matrix sample. This is only used when type="invWishart"

type

The type of covariance posterior that is being simulated. This can be:

invWishart The posterior is an inverse wishart; This allows for correlations between parameters to be modeled. All the uncertainty in the parameter is captured in the degrees of freedom parameter.
lkj The posterior separates the standard deviation estimates (modeled outside and provided in the omega argument) and the correlation estimates. The correlation estimate is simulated with the rLKJ1(). This simulation uses the relationship eta=(nu-1)/2. This is relationship based on the proof of the relationship between the restricted LKJ-distribution and inverse wishart distribution (XXXXXX). Once the correlation posterior is calculated, the estimated standard deviations are then combined with the simulated correlation matrix to create the covariance matrix.
separation Like the lkj option, this separates out the estimation of the correlation and standard deviation. Instead of using the LKJ distribution to simulate the correlation, it simulates the inverse wishart of the identity matrix and converts the result to a correlation matrix. This correlation matrix is then used with the standard deviation to calculate the simulated covariance matrix.

diagXformType

Diagonal transformation type. These could be:

log The standard deviations are log transformed, so the actual standard deviations are exp(omega)
identity The standard deviations are not transformed. The standard deviations are not transformed; They should be positive.
variance The variances are specified in the omega matrix; They are transformed into standard deviations.
nlmixrSqrt These standard deviations come from an nlmixr omega matrix where diag(chol(inv(omega))) = x^2
nlmixrLog These standard deviations come from a nlmixr omega matrix omega matrix where diag(chol(solve(omega))) = exp(x)
nlmixrIdentity These standard deviations come from a nlmixr omega matrix omega matrix where diag(chol(solve(omega))) = x

The nlmixr transformations only make sense when there is no off-diagonal correlations modeled.

Value

a matrix (n=1) or a list of matrices (n > 1)

Details

If your covariance matrix is a 1x1 matrix, this uses an scaled inverse chi-squared which is equivalent to the Inverse Wishart distribution in the uni-directional case.

In general, the separation strategy is preferred for diagonal matrices. If the dimension of the matrix is below 10, lkj is numerically faster than separation method. However, the lkj method has densities too close to zero (XXXX) when the dimension is above 10. In that case, though computationally more expensive separation method performs better.

For matrices with modeled covariances, the easiest method to use is the inverse Wishart which allows the simulation of correlation matrices (XXXX). This method is more well suited for well behaved matrices, that is the variance components are not too low or too high. When modeling nonlinear mixed effects modeling matrices with too high or low variances are considered sub-optimal in describing a system. With these rules in mind, it is reasonable to use the inverse Wishart.

References

Alvarez I, Niemi J and Simpson M. (2014) Bayesian Inference for a Covariance Matrix. Conference on Applied Statistics in Agriculture.

Wang1 Z, Wu Y, and Chu H. (2018) On Equivalence of the LKJ distribution and the restricted Wishart distribution. <doi:10.48550/arXiv.1809.047463

Author

Matthew L.Fidler & Wenping Wang

Examples


## Sample a single covariance.
draw1 <- cvPost(3, matrix(c(1, .3, .3, 1), 2, 2))

## Sample 3 covariances
set.seed(42)
draw3 <- cvPost(3, matrix(c(1, .3, .3, 1), 2, 2), n = 3)

## Sample 3 covariances, but return the cholesky decomposition
set.seed(42)
draw3c <- cvPost(3, matrix(c(1, .3, .3, 1), 2, 2), n = 3, returnChol = TRUE)

## Sample 3 covariances with lognormal standard deviations via LKJ
## correlation sample
cvPost(3, sapply(1:3, function(...) {
  rnorm(10)
}), type = "lkj")
#> [[1]]
#>           [,1]      [,2]      [,3]
#> [1,] 96.862339 0.7748630 -4.626203
#> [2,]  0.774863 0.0283645  0.119055
#> [3,] -4.626203 0.1190550  4.094623
#> 
#> [[2]]
#>             [,1]       [,2]       [,3]
#> [1,]  0.06218003 0.05815138 -0.5246680
#> [2,]  0.05815138 0.70904612  0.3915278
#> [3,] -0.52466798 0.39152780  7.9264641
#> 
#> [[3]]
#>            [,1]       [,2]       [,3]
#> [1,]  0.5725945 -0.7039115  0.3184467
#> [2,] -0.7039115 11.3514944 -0.9360070
#> [3,]  0.3184467 -0.9360070  0.2958648
#> 
#> [[4]]
#>            [,1]      [,2]       [,3]
#> [1,]  0.7659467 -2.803861  0.3011139
#> [2,] -2.8038607 44.273527 -9.4665546
#> [3,]  0.3011139 -9.466555  2.7453546
#> 
#> [[5]]
#>            [,1]        [,2]        [,3]
#> [1,] 3.56762745  0.07176028  0.13656513
#> [2,] 0.07176028  0.42276523 -0.02050365
#> [3,] 0.13656513 -0.02050365  0.03225709
#> 
#> [[6]]
#>           [,1]      [,2]      [,3]
#> [1,] 0.5663711 0.0340262 0.1454173
#> [2,] 0.0340262 0.5977762 0.3269646
#> [3,] 0.1454173 0.3269646 0.2082704
#> 
#> [[7]]
#>              [,1]         [,2]         [,3]
#> [1,]  0.004927562 -0.002843181  0.005520158
#> [2,] -0.002843181  0.029412775 -0.013463952
#> [3,]  0.005520158 -0.013463952  0.182352220
#> 
#> [[8]]
#>              [,1]        [,2]         [,3]
#> [1,]  0.007589923 -0.06216448  0.002859140
#> [2,] -0.062164482  2.50977902 -0.118417072
#> [3,]  0.002859140 -0.11841707  0.007999188
#> 
#> [[9]]
#>            [,1]       [,2]       [,3]
#> [1,] 14.0163806  1.2051675  0.8401261
#> [2,]  1.2051675  0.2780401 -0.1994978
#> [3,]  0.8401261 -0.1994978  1.0749189
#> 
#> [[10]]
#>            [,1]       [,2]       [,3]
#> [1,]  0.5415731 -0.7109454  0.4040513
#> [2,] -0.7109454  2.4865600 -0.4319643
#> [3,]  0.4040513 -0.4319643  1.5098302
#> 

## or return cholesky decomposition
cvPost(3, sapply(1:3, function(...) {
  rnorm(10)
}),
type = "lkj",
returnChol = TRUE
)
#> [[1]]
#>         [,1]      [,2]       [,3]
#> [1,] 1.86548 0.9644577 -1.1377549
#> [2,] 0.00000 0.5131372  0.7633548
#> [3,] 0.00000 0.0000000  1.3396488
#> 
#> [[2]]
#>           [,1]       [,2]      [,3]
#> [1,] 0.3853808 -0.2396288 0.2759888
#> [2,] 0.0000000  0.8531100 0.3904699
#> [3,] 0.0000000  0.0000000 3.9907672
#> 
#> [[3]]
#>           [,1]       [,2]        [,3]
#> [1,] 0.5811021 0.02348261 0.040235833
#> [2,] 0.0000000 0.30199556 0.006542798
#> [3,] 0.0000000 0.00000000 0.326766208
#> 
#> [[4]]
#>          [,1]       [,2]       [,3]
#> [1,] 1.787819 -0.5701632 0.02973627
#> [2,] 0.000000  1.7537550 0.16276115
#> [3,] 0.000000  0.0000000 0.38911063
#> 
#> [[5]]
#>          [,1]       [,2]        [,3]
#> [1,] 2.155836 -0.1976721 -0.27043101
#> [2,] 0.000000  0.7801646  0.06882305
#> [3,] 0.000000  0.0000000  0.16161284
#> 
#> [[6]]
#>          [,1]       [,2]        [,3]
#> [1,] 1.590053 0.07434835 -0.17317880
#> [2,] 0.000000 0.82964610  0.13403995
#> [3,] 0.000000 0.00000000  0.07785199
#> 
#> [[7]]
#>           [,1]      [,2]       [,3]
#> [1,] 0.4123939 0.2062854 -0.8475394
#> [2,] 0.0000000 2.5346241  0.4594744
#> [3,] 0.0000000 0.0000000  0.4939941
#> 
#> [[8]]
#>          [,1]       [,2]         [,3]
#> [1,] 0.332944 -0.6235242 -0.002281778
#> [2,] 0.000000  2.1873959  0.885067020
#> [3,] 0.000000  0.0000000  1.705736993
#> 
#> [[9]]
#>          [,1]      [,2]      [,3]
#> [1,] 4.539001 0.4690774 -2.300368
#> [2,] 0.000000 3.9959179 -1.030652
#> [3,] 0.000000 0.0000000  2.165767
#> 
#> [[10]]
#>          [,1]         [,2]      [,3]
#> [1,] 1.294237 2.545228e-05 -0.891757
#> [2,] 0.000000 6.211554e-01 -1.993399
#> [3,] 0.000000 0.000000e+00  1.819896
#> 

## Sample 3 covariances with lognormal standard deviations via separation
## strategy using inverse Wishart correlation sample
cvPost(3, sapply(1:3, function(...) {
  rnorm(10)
}), type = "separation")
#> [[1]]
#>            [,1]       [,2]        [,3]
#> [1,] 17.8091343 0.73461953 -1.39358522
#> [2,]  0.7346195 0.43703688  0.08246014
#> [3,] -1.3935852 0.08246014  0.39231482
#> 
#> [[2]]
#>             [,1]       [,2]        [,3]
#> [1,]  0.11144462  0.2079765 -0.08551474
#> [2,]  0.20797652  9.2698949 -0.16544961
#> [3,] -0.08551474 -0.1654496  0.08403645
#> 
#> [[3]]
#>            [,1]       [,2]       [,3]
#> [1,]  0.7908562 -0.5415399 -0.8590748
#> [2,] -0.5415399  0.3821333  0.5923616
#> [3,] -0.8590748  0.5923616  0.9845958
#> 
#> [[4]]
#>            [,1]        [,2]        [,3]
#> [1,] 11.0562602  0.33123520  1.27030624
#> [2,]  0.3312352  0.42048853 -0.01175715
#> [3,]  1.2703062 -0.01175715  0.20178261
#> 
#> [[5]]
#>           [,1]     [,2]      [,3]
#> [1,] 0.3908392 1.232200 0.3605851
#> [2,] 1.2321999 4.029834 1.1549621
#> [3,] 0.3605851 1.154962 0.3440444
#> 
#> [[6]]
#>           [,1]      [,2]       [,3]
#> [1,] 0.9003795 0.2147410  3.1855956
#> [2,] 0.2147410 0.1209066  0.7990146
#> [3,] 3.1855956 0.7990146 13.1359204
#> 
#> [[7]]
#>            [,1]       [,2]       [,3]
#> [1,]  0.8417985 -0.8216620 -0.7381474
#> [2,] -0.8216620  0.9218277  0.7890609
#> [3,] -0.7381474  0.7890609  0.7039473
#> 
#> [[8]]
#>             [,1]        [,2]        [,3]
#> [1,]  0.16942278  0.01244557 -0.01024524
#> [2,]  0.01244557  0.04491023 -0.07210998
#> [3,] -0.01024524 -0.07210998  0.11723618
#> 
#> [[9]]
#>            [,1]      [,2]       [,3]
#> [1,]  0.4109154 -1.842263 -0.7241764
#> [2,] -1.8422630 10.322635  3.4070393
#> [3,] -0.7241764  3.407039  1.3859887
#> 
#> [[10]]
#>            [,1]       [,2]       [,3]
#> [1,]  0.9428107 -0.7354081  0.6621306
#> [2,] -0.7354081  0.5785147 -0.5232715
#> [3,]  0.6621306 -0.5232715  0.4840937
#> 

## or returning the cholesky decomposition
cvPost(3, sapply(1:3, function(...) {
  rnorm(10)
}),
type = "separation",
returnChol = TRUE
)
#> [[1]]
#>           [,1]       [,2]       [,3]
#> [1,] 0.5361344 -0.5646490 -1.2157714
#> [2,] 0.0000000  0.2706145  0.7833376
#> [3,] 0.0000000  0.0000000  1.4143150
#> 
#> [[2]]
#>           [,1]       [,2]       [,3]
#> [1,] 0.4763222 -0.1686305 -0.2638639
#> [2,] 0.0000000  0.3953129  0.2370631
#> [3,] 0.0000000  0.0000000  0.2553825
#> 
#> [[3]]
#>          [,1]       [,2]        [,3]
#> [1,] 1.457362 -0.1565739 0.003957496
#> [2,] 0.000000  0.2969488 0.004259446
#> [3,] 0.000000  0.0000000 0.034623292
#> 
#> [[4]]
#>           [,1]      [,2]       [,3]
#> [1,] 0.2697727 0.1901255  0.1775689
#> [2,] 0.0000000 0.1863785 -0.4415488
#> [3,] 0.0000000 0.0000000  1.3256841
#> 
#> [[5]]
#>          [,1]       [,2]       [,3]
#> [1,] 1.081252 -0.2458460 0.05591077
#> [2,] 0.000000  0.4958417 0.11182166
#> [3,] 0.000000  0.0000000 0.07342814
#> 
#> [[6]]
#>           [,1]      [,2]       [,3]
#> [1,] 0.4513931 -1.555660  0.9059502
#> [2,] 0.0000000  1.430568 -0.3169197
#> [3,] 0.0000000  0.000000  0.7111382
#> 
#> [[7]]
#>          [,1]      [,2]       [,3]
#> [1,] 5.850312 0.3103629 -3.1410818
#> [2,] 0.000000 0.3115511  0.2156489
#> [3,] 0.000000 0.0000000  2.5678462
#> 
#> [[8]]
#>           [,1]        [,2]        [,3]
#> [1,] 0.3920281 -0.17091427  1.44657948
#> [2,] 0.0000000  0.01081568 -0.09688330
#> [3,] 0.0000000  0.00000000  0.08276372
#> 
#> [[9]]
#>          [,1]       [,2]       [,3]
#> [1,] 2.021659 -0.6427884  0.8126606
#> [2,] 0.000000  0.2158093 -0.3291584
#> [3,] 0.000000  0.0000000  0.3722985
#> 
#> [[10]]
#>           [,1]      [,2]      [,3]
#> [1,] 0.4696165 0.1467044 -1.908483
#> [2,] 0.0000000 0.4906213  3.952318
#> [3,] 0.0000000 0.0000000 10.894538
#>