Final Notes
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The goal of condTruncMVN is to find densities, probabilities, and samples from a conditional truncated multivariate normal distribution. Suppose that Z = (X,Y) is from a fully-joint multivariate normal distribution of dimension n with mean \(\boldsymbol\mu\) and covariance matrix sigma ( \(\Sigma\) ) truncated between lower and upper. Then, Z has the density \[ f_Z(\textbf{z}, \boldsymbol\mu, \Sigma, \textbf{lower}, \textbf{upper})= \frac{exp(-\frac{1}{2}*(\textbf{z}-\boldsymbol\mu)^T \Sigma^{-1} (\textbf{z}-\boldsymbol\mu))} {\int_{\textbf{lower}}^{\textbf{upper}} exp(-\frac{1}{2}*(\textbf{z}-\boldsymbol\mu)^T \Sigma^{-1} (\textbf{z}-\boldsymbol\mu)) d\textbf{z}} \] for all z in [lower, upper] in \(\mathbb{R^{n}}\).
This package computes the conditional truncated multivariate normal distribution of Y|X. The conditional distribution follows a truncated multivariate normal [1]. Specifically, the functions are arranged such that
\[ Y = Z[ , dependent.ind] \] \[ X = Z[ , given.ind] \] \[ Y|X = X.given \sim MVN(mean, sigma, lower, upper) \] The [d,p,r]cmvtnorm() functions create a list of parameters used in truncated conditional normal and then passes the parameters to the source function below.
| Function Name | Description | Source Function | Univariate Case | Additional Parameters |
|---|---|---|---|---|
| condtMVN | List of parameters used in truncated conditional normal. | condMVNorm:: condMVN() | ||
| dcmvtnorm | Calculates the density f(Y=y| X = X.given) up to a constant. The integral of truncated distribution is not computed. | tmvtnorm:: dtmvnorm() | truncnorm:: dtruncnorm() | y, log |
| pcmvtnorm | Calculates the probability that Y|X is between lowerY and upperY given the parameters. | tmvtnorm:: ptmvnorm() | truncnorm:: ptruncnorm() | lowerY, upperY, maxpts, abseps, releps |
| rcmvtnorm | Generate random sample. | tmvmixnorm:: rtmvn() | truncnorm:: rtruncnorm() | n, init, burn, thin |
You can install the released version of condTruncMVN from CRAN with install.packages("condTruncMVN"). You can load the package by:
And the development version from GitHub with:
Suppose \(X2,X3,X5|X1 = 1,X4 = -1 \sim N_3(1, Sigma, -10, 10)\). The following code finds the parameters of the distribution, calculates the density, probability, and finds random variates from this distribution.
> library(condTruncMVN)
> d <- 5
> rho <- 0.9
> Sigma <- matrix(0, nrow = d, ncol = d)
> Sigma <- rho^abs(row(Sigma) - col(Sigma))
> Sigma
#> [,1] [,2] [,3] [,4] [,5]
#> [1,] 1.0000 0.900 0.81 0.729 0.6561
#> [2,] 0.9000 1.000 0.90 0.810 0.7290
#> [3,] 0.8100 0.900 1.00 0.900 0.8100
#> [4,] 0.7290 0.810 0.90 1.000 0.9000
#> [5,] 0.6561 0.729 0.81 0.900 1.0000First, we find the conditional Truncated Normal Parameters.
> condtMVN(mean = rep(1, d), sigma = Sigma, lower = rep(-10, d), upper = rep(10, d), dependent.ind = c(2,
+ 3, 5), given.ind = c(1, 4), X.given = c(1, -1))
#> $condMean
#> [1] 0.3430923 -0.3211143 -0.8000000
#>
#> $condVar
#> [,1] [,2] [,3]
#> [1,] 1.394510e-01 6.934025e-02 0.00
#> [2,] 6.934025e-02 1.394510e-01 0.00
#> [3,] 1.110223e-16 1.110223e-16 0.19
#>
#> $condLower
#> [1] -10 -10 -10
#>
#> $condUpper
#> [1] 10 10 10
#>
#> $condInit
#> [1] 0 0 0Find the log-density when X2,X3,X5 all equal \(0\):
> dcmvtruncnorm(rep(0, 3), mean = rep(1, 5), sigma = Sigma, lower = rep(-10, 5), upper = rep(10,
+ d), dependent.ind = c(2, 3, 5), given.ind = c(1, 4), X.given = c(1, -1), log = TRUE)
#> [1] -3.07231Find \(P( -0.5 < X2,X3,X5 < 0 | X1 = 1,X4 = -1)\):
> pcmvtruncnorm(rep(-0.5, 3), rep(0, 3), mean = rep(1, d), sigma = Sigma, lower = rep(-10, d),
+ upper = rep(10, d), dependent.ind = c(2, 3, 5), given.ind = c(1, 4), X.given = c(1, -1))
#> [1] 0.01306111
#> attr(,"error")
#> [1] 7.108294e-07
#> attr(,"msg")
#> [1] "Normal Completion"Generate two random numbers from the distribution.
> set.seed(2342)
> rcmvtruncnorm(2, mean = rep(1, d), sigma = Sigma, lower = rep(-10, d), upper = rep(10, d),
+ dependent.ind = c(2, 3, 5), given.ind = c(1, 4), X.given = c(1, -1))
#> [,1] [,2] [,3]
#> [1,] 0.02238382 -0.1426882 -1.7914223
#> [2,] 0.32684386 -0.5239659 -0.1189072Another Example: To find the probability that \(X1|X2, X3, X4, X5 \sim N(**1**, Sigma, **-10**, **10**)\) is between -0.5 and 0:
> pcmvtruncnorm(-0.5, 0, mean = rep(1, d), sigma = Sigma, lower = rep(-10, d), upper = rep(10,
+ d), dependent.ind = 1, given.ind = 2:5, X.given = c(1, -1, 1, -1))
#> univariate CDF: using truncnorm::ptruncnorm
#> [1] 7.080093e-08If I want to generate 2 random variates from \(X1|X2, X3, X4, X5 \sim N(**1**, Sigma, **-10**, **10**)\):
This vignette is successfully processed using the following.
#> -- Session info ---------------------------------------------------
#> setting value
#> version R version 4.0.2 (2020-06-22)
#> os macOS High Sierra 10.13.6
#> system x86_64, darwin17.0
#> ui X11
#> language (EN)
#> collate C
#> ctype en_US.UTF-8
#> tz America/New_York
#> date 2020-09-10
#> -- Packages -------------------------------------------------------
#> package * version date lib source
#> condMVNorm 2020.1 2020-03-18 [3] CRAN (R 4.0.2)
#> matrixNormal 0.0.4 2020-08-26 [3] CRAN (R 4.0.2)
#> tmvmixnorm 1.1.0 2020-05-19 [3] CRAN (R 4.0.2)
#> tmvtnorm 1.4-10 2015-08-28 [3] CRAN (R 4.0.2)
#> truncnorm 1.0-8 2020-07-27 [3] Github (olafmersmann/truncnorm@eea186e)
#>
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#> [2] /private/var/folders/55/n1q58r751yd7x4vqzdc8zk_w0000gn/T/RtmpWVLW4l/temp_libpath1170138ff3692
#> [3] /Library/Frameworks/R.framework/Versions/4.0/Resources/library
1. Horrace, W.C. Some results on the multivariate truncated normal distribution. Journal of Multivariate Analysis 2005, 94, 209–221.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.