Bayesian Supervised Learning with Heteroskedasticity in StochTree

This vignette demonstrates how to use the bart() function for Bayesian supervised learning (Chipman, George, and McCulloch (2010)), with an additional “variance forest,” for modeling conditional variance (see Murray (2021)). To begin, we load the stochtree package.

library(stochtree)

Demo 1: Variance-Only Simulation (simple DGP)

Simulation

Here, we generate data with a constant (zero) mean and a relatively simple covariate-modified variance function.

$$\begin{equation*} \begin{aligned} y &= 0 + \sigma^2(X) \epsilon\\ \sigma^2(X) &= \begin{cases} 0.25 & X_1 \geq 0 \text{ and } X_1 < 0.25\\ 1 & X_1 \geq 0.25 \text{ and } X_1 < 0.5\\ 4 & X_1 \geq 0.5 \text{ and } X_1 < 0.75\\ 9 & X_1 \geq 0.75 \text{ and } X_1 < 1\\ \end{cases}\\ X_1,\dots,X_p &\sim \text{U}\left(0,1\right)\\ \epsilon &\sim \mathcal{N}\left(0,1\right) \end{aligned} \end{equation*}$$

# Generate the data
n <- 500
p_x <- 10
X <- matrix(runif(n*p_x), ncol = p_x)
f_XW <- 0
s_XW <- (
    ((0 <= X[,1]) & (0.25 > X[,1])) * (0.5) + 
    ((0.25 <= X[,1]) & (0.5 > X[,1])) * (1) + 
    ((0.5 <= X[,1]) & (0.75 > X[,1])) * (2) + 
    ((0.75 <= X[,1]) & (1 > X[,1])) * (3)
)
y <- f_XW + rnorm(n, 0, 1)*s_XW

# Split data into test and train sets
test_set_pct <- 0.2
n_test <- round(test_set_pct*n)
n_train <- n - n_test
test_inds <- sort(sample(1:n, n_test, replace = FALSE))
train_inds <- (1:n)[!((1:n) %in% test_inds)]
X_test <- as.data.frame(X[test_inds,])
X_train <- as.data.frame(X[train_inds,])
W_test <- NULL
W_train <- NULL
y_test <- y[test_inds]
y_train <- y[train_inds]
f_x_test <- f_XW[test_inds]
f_x_train <- f_XW[train_inds]
s_x_test <- s_XW[test_inds]
s_x_train <- s_XW[train_inds]

Sampling and Analysis

Warmstart

We first sample the $\sigma^2(X)$ ensemble using “warm-start” initialization (He and Hahn (2023)). This is the default in stochtree.

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 100
num_trees <- 20
a_0 <- 1.5
num_samples <- num_gfr + num_burnin + num_mcmc
bart_params <- list(num_trees_mean = 0, num_trees_variance = num_trees, 
                    sample_sigma_global = F, sample_sigma_leaf = F)
bart_model_warmstart <- stochtree::bart(
    X_train = X_train, y_train = y_train, X_test = X_test, 
    num_gfr = num_gfr, num_burnin = num_burnin, num_mcmc = num_mcmc, 
    params = bart_params
)

Inspect the MCMC samples

plot(rowMeans(bart_model_warmstart$sigma_x_hat_test), s_x_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "standard deviation function")
abline(0,1,col="red",lty=2,lwd=2.5)

MCMC

We now sample the $\sigma^2(X)$ ensemble using MCMC with root initialization (as in Chipman, George, and McCulloch (2010)).

num_gfr <- 0
num_burnin <- 1000
num_mcmc <- 100
num_samples <- num_gfr + num_burnin + num_mcmc
bart_params <- list(num_trees_mean = 0, num_trees_variance = 50, 
                    sample_sigma_global = F, sample_sigma_leaf = F)
bart_model_mcmc <- stochtree::bart(
    X_train = X_train, y_train = y_train, X_test = X_test, 
    num_gfr = num_gfr, num_burnin = num_burnin, num_mcmc = num_mcmc, 
    params = bart_params
)

Inspect the MCMC samples

plot(rowMeans(bart_model_mcmc$sigma_x_hat_test), s_x_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "standard deviation function")
abline(0,1,col="red",lty=2,lwd=2.5)

Demo 2: Variance-Only Simulation (complex DGP)

Simulation

Here, we generate data with a constant (zero) mean and a more complex covariate-modified variance function.

$$\begin{equation*} \begin{aligned} y &= 0 + \sigma^2(X) \epsilon\\ \sigma^2(X) &= \begin{cases} 0.25X_3^2 & X_1 \geq 0 \text{ and } X_1 < 0.25\\ 1X_3^2 & X_1 \geq 0.25 \text{ and } X_1 < 0.5\\ 4X_3^2 & X_1 \geq 0.5 \text{ and } X_1 < 0.75\\ 9X_3^2 & X_1 \geq 0.75 \text{ and } X_1 < 1\\ \end{cases}\\ X_1,\dots,X_p &\sim \text{U}\left(0,1\right)\\ \epsilon &\sim \mathcal{N}\left(0,1\right) \end{aligned} \end{equation*}$$

# Generate the data
n <- 500
p_x <- 10
X <- matrix(runif(n*p_x), ncol = p_x)
f_XW <- 0
s_XW <- (
    ((0 <= X[,1]) & (0.25 > X[,1])) * (0.5*X[,3]) + 
    ((0.25 <= X[,1]) & (0.5 > X[,1])) * (1*X[,3]) + 
    ((0.5 <= X[,1]) & (0.75 > X[,1])) * (2*X[,3]) + 
    ((0.75 <= X[,1]) & (1 > X[,1])) * (3*X[,3])
)
y <- f_XW + rnorm(n, 0, 1)*s_XW

# Split data into test and train sets
test_set_pct <- 0.2
n_test <- round(test_set_pct*n)
n_train <- n - n_test
test_inds <- sort(sample(1:n, n_test, replace = FALSE))
train_inds <- (1:n)[!((1:n) %in% test_inds)]
X_test <- as.data.frame(X[test_inds,])
X_train <- as.data.frame(X[train_inds,])
W_test <- NULL
W_train <- NULL
y_test <- y[test_inds]
y_train <- y[train_inds]
f_x_test <- f_XW[test_inds]
f_x_train <- f_XW[train_inds]
s_x_test <- s_XW[test_inds]
s_x_train <- s_XW[train_inds]

Sampling and Analysis

Warmstart

We first sample the $\sigma^2(X)$ ensemble using “warm-start” initialization (He and Hahn (2023)). This is the default in stochtree.

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 100
num_samples <- num_gfr + num_burnin + num_mcmc
bart_params <- list(num_trees_mean = 0, num_trees_variance = 50, 
                    alpha_mean = 0.95, beta_mean = 2, min_samples_leaf_mean = 5, 
                    alpha_variance = 0.95, beta_variance = 1.25, min_samples_leaf_variance = 1, 
                    sample_sigma_global = F, sample_sigma_leaf = F)
bart_model_warmstart <- stochtree::bart(
    X_train = X_train, y_train = y_train, X_test = X_test, 
    num_gfr = num_gfr, num_burnin = num_burnin, num_mcmc = num_mcmc, 
    params = bart_params
)

Inspect the MCMC samples

plot(rowMeans(bart_model_warmstart$sigma_x_hat_test), s_x_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "standard deviation function")
abline(0,1,col="red",lty=2,lwd=2.5)

MCMC

We now sample the $\sigma^2(X)$ ensemble using MCMC with root initialization (as in Chipman, George, and McCulloch (2010)).

num_gfr <- 0
num_burnin <- 1000
num_mcmc <- 100
num_samples <- num_gfr + num_burnin + num_mcmc
bart_params <- list(num_trees_mean = 0, num_trees_variance = 50, 
                    alpha_mean = 0.95, beta_mean = 2, min_samples_leaf_mean = 5, 
                    alpha_variance = 0.95, beta_variance = 1.25, min_samples_leaf_variance = 5, 
                    sample_sigma_global = F, sample_sigma_leaf = F)
bart_model_mcmc <- stochtree::bart(
    X_train = X_train, y_train = y_train, X_test = X_test, 
    num_gfr = num_gfr, num_burnin = num_burnin, num_mcmc = num_mcmc, 
    params = bart_params
)

Inspect the MCMC samples

plot(rowMeans(bart_model_mcmc$sigma_x_hat_test), s_x_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "standard deviation function")
abline(0,1,col="red",lty=2,lwd=2.5)

Demo 3: Mean and Variance Simulation (simple DGP)

Simulation

Here, we generate data with (relatively simple) covariate-modified mean and variance functions.

$$\begin{equation*} \begin{aligned} y &= f(X) + \sigma^2(X) \epsilon\\ f(X) &= \begin{cases} -6 & X_2 \geq 0 \text{ and } X_2 < 0.25\\ -2 & X_2 \geq 0.25 \text{ and } X_2 < 0.5\\ 2 & X_2 \geq 0.5 \text{ and } X_2 < 0.75\\ 6 & X_2 \geq 0.75 \text{ and } X_2 < 1\\ \end{cases}\\ \sigma^2(X) &= \begin{cases} 0.25 & X_1 \geq 0 \text{ and } X_1 < 0.25\\ 1 & X_1 \geq 0.25 \text{ and } X_1 < 0.5\\ 4 & X_1 \geq 0.5 \text{ and } X_1 < 0.75\\ 9 & X_1 \geq 0.75 \text{ and } X_1 < 1\\ \end{cases}\\ X_1,\dots,X_p &\sim \text{U}\left(0,1\right)\\ \epsilon &\sim \mathcal{N}\left(0,1\right) \end{aligned} \end{equation*}$$

# Generate the data
n <- 500
p_x <- 10
X <- matrix(runif(n*p_x), ncol = p_x)
f_XW <- (
    ((0 <= X[,2]) & (0.25 > X[,2])) * (-6) + 
    ((0.25 <= X[,2]) & (0.5 > X[,2])) * (-2) + 
    ((0.5 <= X[,2]) & (0.75 > X[,2])) * (2) + 
    ((0.75 <= X[,2]) & (1 > X[,2])) * (6)
)
s_XW <- (
    ((0 <= X[,1]) & (0.25 > X[,1])) * (0.5) + 
    ((0.25 <= X[,1]) & (0.5 > X[,1])) * (1) + 
    ((0.5 <= X[,1]) & (0.75 > X[,1])) * (2) + 
    ((0.75 <= X[,1]) & (1 > X[,1])) * (3)
)
y <- f_XW + rnorm(n, 0, 1)*s_XW

# Split data into test and train sets
test_set_pct <- 0.2
n_test <- round(test_set_pct*n)
n_train <- n - n_test
test_inds <- sort(sample(1:n, n_test, replace = FALSE))
train_inds <- (1:n)[!((1:n) %in% test_inds)]
X_test <- as.data.frame(X[test_inds,])
X_train <- as.data.frame(X[train_inds,])
W_test <- NULL
W_train <- NULL
y_test <- y[test_inds]
y_train <- y[train_inds]
f_x_test <- f_XW[test_inds]
f_x_train <- f_XW[train_inds]
s_x_test <- s_XW[test_inds]
s_x_train <- s_XW[train_inds]

Sampling and Analysis

Warmstart

We first sample the $\sigma^2(X)$ ensemble using “warm-start” initialization (He and Hahn (2023)). This is the default in stochtree.

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 100
num_samples <- num_gfr + num_burnin + num_mcmc
bart_params <- list(num_trees_mean = 50, num_trees_variance = 50, 
                    alpha_mean = 0.95, beta_mean = 2, min_samples_leaf_mean = 5, 
                    alpha_variance = 0.95, beta_variance = 1.25, min_samples_leaf_variance = 5, 
                    sample_sigma_global = F, sample_sigma_leaf = F)
bart_model_warmstart <- stochtree::bart(
    X_train = X_train, y_train = y_train, X_test = X_test, 
    num_gfr = num_gfr, num_burnin = num_burnin, num_mcmc = num_mcmc, 
    params = bart_params
)

Inspect the MCMC samples

plot(rowMeans(bart_model_warmstart$y_hat_test), y_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "mean function")
abline(0,1,col="red",lty=2,lwd=2.5)

plot(rowMeans(bart_model_warmstart$sigma_x_hat_test), s_x_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "standard deviation function")
abline(0,1,col="red",lty=2,lwd=2.5)

MCMC

We now sample the $\sigma^2(X)$ ensemble using MCMC with root initialization (as in Chipman, George, and McCulloch (2010)).

num_gfr <- 0
num_burnin <- 1000
num_mcmc <- 100
num_samples <- num_gfr + num_burnin + num_mcmc
bart_params <- list(num_trees_mean = 50, num_trees_variance = 50, 
                    alpha_mean = 0.95, beta_mean = 2, min_samples_leaf_mean = 5, 
                    alpha_variance = 0.95, beta_variance = 1.25, min_samples_leaf_variance = 5, 
                    sample_sigma_global = F, sample_sigma_leaf = F)
bart_model_mcmc <- stochtree::bart(
    X_train = X_train, y_train = y_train, X_test = X_test, 
    num_gfr = num_gfr, num_burnin = num_burnin, num_mcmc = num_mcmc, 
    params = bart_params
)

Inspect the MCMC samples

plot(rowMeans(bart_model_mcmc$y_hat_test), y_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "mean function")
abline(0,1,col="red",lty=2,lwd=2.5)

plot(rowMeans(bart_model_mcmc$sigma_x_hat_test), s_x_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "standard deviation function")
abline(0,1,col="red",lty=2,lwd=2.5)

Demo 4: Mean and Variance Simulation (complex DGP)

Simulation

Here, we generate data with more complex covariate-modified mean and variance functions.

$$\begin{equation*} \begin{aligned} y &= f(X) + \sigma^2(X) \epsilon\\ f(X) &= \begin{cases} -6X_4 & X_2 \geq 0 \text{ and } X_2 < 0.25\\ -2X_4 & X_2 \geq 0.25 \text{ and } X_2 < 0.5\\ 2X_4 & X_2 \geq 0.5 \text{ and } X_2 < 0.75\\ 6X_4 & X_2 \geq 0.75 \text{ and } X_2 < 1\\ \end{cases}\\ \sigma^2(X) &= \begin{cases} 0.25X_3^2 & X_1 \geq 0 \text{ and } X_1 < 0.25\\ 1X_3^2 & X_1 \geq 0.25 \text{ and } X_1 < 0.5\\ 4X_3^2 & X_1 \geq 0.5 \text{ and } X_1 < 0.75\\ 9X_3^2 & X_1 \geq 0.75 \text{ and } X_1 < 1\\ \end{cases}\\ X_1,\dots,X_p &\sim \text{U}\left(0,1\right)\\ \epsilon &\sim \mathcal{N}\left(0,1\right) \end{aligned} \end{equation*}$$

# Generate the data
n <- 500
p_x <- 10
X <- matrix(runif(n*p_x), ncol = p_x)
f_XW <- (
    ((0 <= X[,2]) & (0.25 > X[,2])) * (-6*X[,4]) + 
    ((0.25 <= X[,2]) & (0.5 > X[,2])) * (-2*X[,4]) + 
    ((0.5 <= X[,2]) & (0.75 > X[,2])) * (2*X[,4]) + 
    ((0.75 <= X[,2]) & (1 > X[,2])) * (6*X[,4])
)
s_XW <- (
    ((0 <= X[,1]) & (0.25 > X[,1])) * (0.5*X[,3]) + 
    ((0.25 <= X[,1]) & (0.5 > X[,1])) * (1*X[,3]) + 
    ((0.5 <= X[,1]) & (0.75 > X[,1])) * (2*X[,3]) + 
    ((0.75 <= X[,1]) & (1 > X[,1])) * (3*X[,3])
)
y <- f_XW + rnorm(n, 0, 1)*s_XW

# Split data into test and train sets
test_set_pct <- 0.2
n_test <- round(test_set_pct*n)
n_train <- n - n_test
test_inds <- sort(sample(1:n, n_test, replace = FALSE))
train_inds <- (1:n)[!((1:n) %in% test_inds)]
X_test <- as.data.frame(X[test_inds,])
X_train <- as.data.frame(X[train_inds,])
W_test <- NULL
W_train <- NULL
y_test <- y[test_inds]
y_train <- y[train_inds]
f_x_test <- f_XW[test_inds]
f_x_train <- f_XW[train_inds]
s_x_test <- s_XW[test_inds]
s_x_train <- s_XW[train_inds]

Sampling and Analysis

Warmstart

We first sample the $\sigma^2(X)$ ensemble using “warm-start” initialization (He and Hahn (2023)). This is the default in stochtree.

num_gfr <- 10
num_burnin <- 0
num_mcmc <- 100
num_samples <- num_gfr + num_burnin + num_mcmc
bart_params <- list(num_trees_mean = 50, num_trees_variance = 50, 
                    alpha_mean = 0.95, beta_mean = 2, min_samples_leaf_mean = 5, 
                    alpha_variance = 0.95, beta_variance = 1.25, min_samples_leaf_variance = 5, 
                    sample_sigma_global = F, sample_sigma_leaf = F)
bart_model_warmstart <- stochtree::bart(
    X_train = X_train, y_train = y_train, X_test = X_test, 
    num_gfr = num_gfr, num_burnin = num_burnin, num_mcmc = num_mcmc, 
    params = bart_params
)

Inspect the MCMC samples

plot(rowMeans(bart_model_warmstart$y_hat_test), y_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "mean function")
abline(0,1,col="red",lty=2,lwd=2.5)

plot(rowMeans(bart_model_warmstart$sigma_x_hat_test), s_x_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "standard deviation function")
abline(0,1,col="red",lty=2,lwd=2.5)

MCMC

We now sample the $\sigma^2(X)$ ensemble using MCMC with root initialization (as in Chipman, George, and McCulloch (2010)).

num_gfr <- 0
num_burnin <- 1000
num_mcmc <- 100
num_samples <- num_gfr + num_burnin + num_mcmc
bart_params <- list(num_trees_mean = 50, num_trees_variance = 50, 
                    alpha_mean = 0.95, beta_mean = 2, min_samples_leaf_mean = 5, 
                    alpha_variance = 0.95, beta_variance = 1.25, min_samples_leaf_variance = 5, 
                    sample_sigma_global = F, sample_sigma_leaf = F)
bart_model_mcmc <- stochtree::bart(
    X_train = X_train, y_train = y_train, X_test = X_test, 
    num_gfr = num_gfr, num_burnin = num_burnin, num_mcmc = num_mcmc, 
    params = bart_params
)

Inspect the MCMC samples

plot(rowMeans(bart_model_mcmc$y_hat_test), y_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "mean function")
abline(0,1,col="red",lty=2,lwd=2.5)

plot(rowMeans(bart_model_mcmc$sigma_x_hat_test), s_x_test, 
     pch=16, cex=0.75, xlab = "pred", ylab = "actual", main = "standard deviation function")
abline(0,1,col="red",lty=2,lwd=2.5)

References

Chipman, Hugh A., Edward I. George, and Robert E. McCulloch. 2010. “BART: Bayesian additive regression trees.” The Annals of Applied Statistics 4 (1): 266–98. https://doi.org/10.1214/09-AOAS285.

He, Jingyu, and P Richard Hahn. 2023. “Stochastic Tree Ensembles for Regularized Nonlinear Regression.” Journal of the American Statistical Association 118 (541): 551–70.

Murray, Jared S. 2021. “Log-Linear Bayesian Additive Regression Trees for Multinomial Logistic and Count Regression Models.” Journal of the American Statistical Association 116 (534): 756–69.