### Configure numerical integration algorithms

Source: https://context7.com/cran/mvtnorm/llms.txt

Illustrates the configuration and usage of GenzBretz, Miwa, and TVPACK algorithms for calculating multivariate probabilities. These algorithms allow users to balance computational speed with precision requirements in normal and t-distributions.

```r
# GenzBretz algorithm
algo_gb <- GenzBretz(maxpts = 50000, abseps = 0.0001, releps = 0)
prob_gb <- pmvnorm(lower = rep(-2, 5), upper = rep(2, 5), mean = rep(0, 5), corr = diag(5), algorithm = algo_gb)

# Miwa algorithm
algo_miwa <- Miwa(steps = 256, checkCorr = TRUE)
prob_miwa <- pmvnorm(lower = rep(-1, 3), upper = rep(1, 3), mean = rep(0, 3), corr = diag(3), algorithm = algo_miwa)

# TVPACK algorithm
algo_tv <- TVPACK(abseps = 0.001)
prob_tv <- pmvt(lower = rep(-1, 4), upper = rep(1, 4), df = 5, corr = diag(4), algorithm = algo_tv)
```

--------------------------------

### Simulate multivariate normal samples

Source: https://context7.com/cran/mvtnorm/llms.txt

Demonstrates how to generate random samples from a defined multivariate normal object using the simulate function, followed by verification of sample dimensions and mean vector convergence.

```r
simulated <- simulate(mvn, nsim = 100)
dim(simulated)  # 4 x 100
colMeans(simulated)  # Should be close to mean_vec
```

--------------------------------

### Analyze Marginal and Conditional Distributions

Source: https://context7.com/cran/mvtnorm/llms.txt

Shows how to create multivariate normal objects and compute marginal or conditional distributions. This is useful for slicing multidimensional distributions based on specific variable subsets or known values.

```R
mvn <- mvnorm(mean = mean_vec, chol = chol_mat)
mvn_marg <- margDist(mvn, which = c(1, 3))
mvn_cond <- condDist(mvn, which_given = 1, given = matrix(1.5, nrow = 1, ncol = 1))
ll <- logLik(mvn, obs = obs)
```

--------------------------------

### Generate Multivariate Normal Random Samples

Source: https://context7.com/cran/mvtnorm/llms.txt

Demonstrates how to generate random samples from a multivariate normal distribution using the rmvnorm function. It supports various decomposition methods such as eigen, cholesky, and svd to handle different covariance matrix structures.

```r
library(mvtnorm)

# Generate 1000 samples from 3-dimensional MVN
mean_vector <- c(1, 2, 3)
cov_matrix <- matrix(c(1, 0.5, 0.3,
                       0.5, 1, 0.4,
                       0.3, 0.4, 1), nrow = 3)

samples <- rmvnorm(n = 1000, mean = mean_vector, sigma = cov_matrix)

# Using different decomposition methods
samples_eigen <- rmvnorm(100, mean = c(0, 0), sigma = diag(2), method = "eigen")
samples_chol <- rmvnorm(100, mean = c(0, 0), sigma = diag(2), method = "chol")
samples_svd <- rmvnorm(100, mean = c(0, 0), sigma = diag(2), method = "svd")
```

--------------------------------

### Compute Likelihood for Mixed and Censored Data

Source: https://context7.com/cran/mvtnorm/llms.txt

Demonstrates the use of ldpmvnorm for calculating likelihoods with mixed exact and censored data points. It requires defining observation matrices and lower/upper bounds for censored intervals, returning the log-likelihood values.

```R
obs_mixed <- matrix(rnorm(2 * N), nrow = 2, ncol = N)
lower_mixed <- matrix(-0.5, nrow = 1, ncol = N)
upper_mixed <- matrix(0.5, nrow = 1, ncol = N)
L_full <- ltMatrices(diag(3), diag = TRUE, byrow = TRUE)
ll_mixed <- ldpmvnorm(obs = obs_mixed, lower = lower_mixed, upper = upper_mixed, mean = 0, chol = L_full, M = 500)
print(ll_mixed)
```

--------------------------------

### Convert Covariance and Correlation Parameterizations

Source: https://context7.com/cran/mvtnorm/llms.txt

Provides utility functions for converting between Cholesky factors, inverse Cholesky factors, covariance matrices, and correlation matrices. These transformations are essential for changing statistical model parameterizations.

```R
cov_from_chol <- chol2cov(L)
cor_from_chol <- chol2cor(L)
prec_from_invchol <- invchol2pre(invL)
L_std <- standardize(chol = L)
```

--------------------------------

### Find Multivariate Normal Quantiles

Source: https://context7.com/cran/mvtnorm/llms.txt

Determines equicoordinate quantiles for multivariate normal distributions using qmvnorm. It supports both lower-tail and two-tailed quantile evaluations for various probability thresholds.

```r
mean <- c(0, 0, 0)
corr <- diag(3)

# Find the quantile q such that P(|X_i| <= q for all i) = 0.95
quantile_result <- qmvnorm(p = 0.95, tail = "both.tails", mean = mean, corr = corr)

# Lower tail quantile
quantile_lower <- qmvnorm(p = 0.95, tail = "lower.tail", mean = c(0, 0), 
                          corr = matrix(c(1, 0.5, 0.5, 1), 2, 2))
```

--------------------------------

### Manipulate Lower Triangular Matrices

Source: https://context7.com/cran/mvtnorm/llms.txt

Illustrates the use of the ltMatrices class for representing multiple lower triangular matrices. Includes operations for matrix creation, array conversion, matrix multiplication, and solving linear systems.

```R
L <- ltMatrices(matrix(values, nrow = 1, ncol = 3), diag = FALSE, byrow = TRUE, names = c("A", "B"))
result <- Mult(L1, matrix(1:4, nrow = 2))
x <- solve(L1, matrix(c(1, 2), nrow = 2))
```

--------------------------------

### Generate Multivariate t Random Samples

Source: https://context7.com/cran/mvtnorm/llms.txt

Generate random samples from a multivariate t distribution using either shifted or Kshirsagar parameterizations. The df parameter controls tail thickness, with Inf representing a multivariate normal distribution.

```r
n <- 500
df <- 5
delta <- c(1, 2)
sigma <- matrix(c(1, 0.6, 0.6, 1), nrow = 2)

samples_shifted <- rmvt(n = n, sigma = sigma, df = df, delta = delta, type = "shifted")
samples_kshir <- rmvt(n = n, sigma = sigma, df = df, delta = delta, type = "Kshirsagar")
```

--------------------------------

### Log-likelihood for Censored Multivariate Normal Data

Source: https://context7.com/cran/mvtnorm/llms.txt

Compute log-likelihoods for exact or interval-censored observations in multivariate Gaussian models. Utilizes Cholesky factorization and supports Monte Carlo methods for interval integration.

```r
J <- 3
L <- ltMatrices(diag(J), diag = TRUE, byrow = TRUE)
ll_exact <- ldmvnorm(obs = matrix(rnorm(J * 5), nrow = J), mean = rep(0, J), chol = L)
ll_censored <- lpmvnorm(lower = matrix(-1, nrow = J, ncol = 5), upper = matrix(1, nrow = J, ncol = 5), mean = rep(0, J), chol = L, M = 1000)
```

--------------------------------

### Compute Multivariate t Density

Source: https://context7.com/cran/mvtnorm/llms.txt

Calculate the probability density function (PDF) or log-density for multivariate t distributions. Supports vectorization for multiple observations and comparison with normal distributions by setting df=Inf.

```r
x <- matrix(c(1, 2, 0, 1, 2, 3), nrow = 3, byrow = TRUE)
densities_t <- dmvt(x, delta = c(1, 2), sigma = diag(2), df = 5, log = FALSE)
log_densities_t <- dmvt(x, delta = c(1, 2), sigma = diag(2), df = 5, log = TRUE)
```

--------------------------------

### Calculate Multivariate t and Normal Quantiles

Source: https://context7.com/cran/mvtnorm/llms.txt

Compute upper and equicoordinate quantiles for multivariate distributions. These functions support both standard and Kshirsagar parameterizations for bivariate and higher-dimensional distributions.

```r
quantile_upper <- qmvnorm(p = 0.05, tail = "upper.tail", mean = c(0, 0), corr = matrix(c(1, 0.3, 0.3, 1), 2, 2))
print(quantile_upper$quantile)

quantile_t <- qmvt(p = 0.95, tail = "both.tails", df = 4, delta = c(0, 0), corr = matrix(c(1, 0.3, 0.3, 1), nrow = 2))
print(quantile_t$quantile)
```

--------------------------------

### Evaluate Multivariate Normal Density

Source: https://context7.com/cran/mvtnorm/llms.txt

Calculates the probability density function for multivariate normal distributions using dmvnorm. Users can compute standard densities or log-densities for improved numerical stability.

```r
x <- matrix(c(1, 2, 3,
              0, 1, 2,
              2, 3, 4), nrow = 3, byrow = TRUE)
mean_vec <- c(1, 2, 3)
cov_mat <- diag(3)

# Compute densities
densities <- dmvnorm(x, mean = mean_vec, sigma = cov_mat)

# Get log densities for numerical stability
log_densities <- dmvnorm(x, mean = mean_vec, sigma = cov_mat, log = TRUE)
```

--------------------------------

### Compute Multivariate Normal Cumulative Distribution

Source: https://context7.com/cran/mvtnorm/llms.txt

Uses pmvnorm to calculate probabilities over specific rectangular regions. It supports specifying either covariance or correlation matrices and allows fine-tuning of integration algorithms like GenzBretz.

```r
mean <- c(0, 0)
corr_matrix <- matrix(c(1, 0.5, 0.5, 1), nrow = 2)

# Compute P(X1 <= 1, X2 <= 1)
prob <- pmvnorm(lower = c(-Inf, -Inf), upper = c(1, 1), mean = mean, corr = corr_matrix)

# Using GenzBretz algorithm with custom parameters
prob_precise <- pmvnorm(lower = c(-1, -1), upper = c(1, 1), mean = c(0, 0), corr = corr_matrix,
                        algorithm = GenzBretz(maxpts = 50000, abseps = 0.0001, releps = 0))
```

--------------------------------

### Perform Score Function Gradient Estimation

Source: https://context7.com/cran/mvtnorm/llms.txt

Calculates score functions (gradients) for likelihood-based inference using sldmvnorm and slpmvnorm. These functions compute derivatives with respect to mean, Cholesky factors, and interval bounds for normal and censored distributions.

```R
scores <- sldmvnorm(obs = obs, mean = true_mean, chol = true_chol)
scores_censored <- slpmvnorm(lower = lower, upper = upper, mean = 0, chol = true_chol, M = 500)
print(scores$logLik)
print(scores_censored$mean)
```

--------------------------------

### Compute Multivariate t Cumulative Distribution

Source: https://context7.com/cran/mvtnorm/llms.txt

Calculate the cumulative distribution function (CDF) for multivariate t distributions to evaluate probabilities over hyper-rectangular regions. The result depends on the degrees of freedom and correlation matrix.

```r
prob_t <- pmvt(lower = c(-1, -1), upper = c(1, 1), delta = c(0, 0), df = 5, corr = matrix(c(1, 0.5, 0.5, 1), nrow = 2), type = "shifted")
print(prob_t)
```

=== COMPLETE CONTENT === This response contains all available snippets from this library. No additional content exists. Do not make further requests.