Package 'spBPS' reference manual

Title:	Bayesian Predictive Stacking for Scalable Geospatial Transfer Learning
Description:	Provides functions for Bayesian Predictive Stacking within the Bayesian transfer learning framework for geospatial artificial systems, as introduced in "Bayesian Transfer Learning for Artificially Intelligent Geospatial Systems: A Predictive Stacking Approach" (Presicce and Banerjee, 2024) <doi:10.48550/arXiv.2410.09504>. This methodology enables efficient Bayesian geostatistical modeling, utilizing predictive stacking to improve inference across spatial datasets. The core functions leverage 'C++' for high-performance computation, making the framework well-suited for large-scale spatial data analysis in parallel and distributed computing environments. Designed for scalability, it allows seamless application in computationally demanding scenarios.
Authors:	Luca Presicce [aut, cre] , Sudipto Banerjee [aut]
Maintainer:	Luca Presicce <[email protected]>
License:	GPL (>= 3)
Version:	0.0-4
Built:	2024-10-27 05:19:36 UTC
Source:	https://github.com/lucapresicce/spbps

Compute the Euclidean distance matrix

Description

Compute the Euclidean distance matrix

Usage

arma_dist(X)
arma_dist(X)

Arguments

`X`	matrix (tipically of $N$ coordindates on $\mathbb{R}^2$ )

Value

matrix distance matrix of the elements of $X$

Examples

## Compute the Distance matrix of dimension (n x n)
n <- 100
p <- 2
X <- matrix(runif(n*p), nrow = n, ncol = p)
distance.matrix <- arma_dist(X)

## Compute the Distance matrix of dimension (n x n)
n <- 100
p <- 2
X <- matrix(runif(n*p), nrow = n, ncol = p)
distance.matrix <- arma_dist(X)

Gibbs sampler for Conjugate Bayesian Multivariate Linear Models

Description

Gibbs sampler for Conjugate Bayesian Multivariate Linear Models

Usage

bayesMvLMconjugate(Y, X, mu_B, V_B, nu, Psi, n_iter = 1000, burn_in = 500)
bayesMvLMconjugate(Y, X, mu_B, V_B, nu, Psi, n_iter = 1000, burn_in = 500)

Arguments

`Y`	matrix $n \times q$ of response variables
`X`	matrix $n \times p$ of predictors
`mu_B`	matrix $p \times q$ prior mean for $\beta$
`V_B`	matrix $p \times p$ prior row covariance for $\beta$
`nu`	double prior parameter for $\Sigma$
`Psi`	matrix prior parameter for $\Sigma$
`n_iter`	integer iteration number for Gibbs sampler
`burn_in`	integer number of burn-in iteration

Value

B_samples array of posterior sample for $\beta$

Sigma_samples array of posterior samples for $\Sigma$

Examples

## Generate data
n <- 100
p <- 3
q <- 2
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
X <- matrix(rnorm(n*p), nrow = n, ncol = p)

## Prior parameters
mu_B <- matrix(0, p, q)
V_B <- diag(10, p)
nu <- 3
Psi <- diag(q)

## Samples from posteriors
n_iter <- 1000
burn_in <- 500
set.seed(1234)
samples <- spBPS::bayesMvLMconjugate(Y, X, mu_B, V_B, nu, Psi, n_iter, burn_in)

## Generate data
n <- 100
p <- 3
q <- 2
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
X <- matrix(rnorm(n*p), nrow = n, ncol = p)

## Prior parameters
mu_B <- matrix(0, p, q)
V_B <- diag(10, p)
nu <- 3
Psi <- diag(q)

## Samples from posteriors
n_iter <- 1000
burn_in <- 500
set.seed(1234)
samples <- spBPS::bayesMvLMconjugate(Y, X, mu_B, V_B, nu, Psi, n_iter, burn_in)

Combine subset models wiht BPS

Description

Combine subset models wiht BPS

Usage

BPS_combine(fit_list, K, rp)
BPS_combine(fit_list, K, rp)

Arguments

`fit_list`	list K fitted model outputs composed by two elements each: first named $epd$ , second named $W$
`K`	integer number of folds
`rp`	double percentage of observations to take into account for optimization (`default=1`)

Value

matrix posterior predictive density evaluations (each columns represent a different model)

Examples


## Generate subsets of data
n <- 100
p <- 3
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n), nrow = n, ncol = 1)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competitive set of values for hyperparameters
delta_seq <- c(0.1, 0.2, 0.3)
phi_seq <- c(3, 4, 5)

## Perform Bayesian Predictive Stacking within subsets
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
    Yi <- data_part$Y_list[[i]]
    Xi <- data_part$X_list[[i]]
    crd_i <- data_part$crd_list[[i]]
    p <- ncol(Xi)
    bps <- spBPS::BPS_weights(data = list(Y = Yi, X = Xi),
                               priors = list(mu_b = matrix(rep(0, p)),
                                             V_b = diag(10, p),
                                             a = 2,
                                             b = 2), coords = crd_i,
                                             hyperpar = list(delta = delta_seq,
                                                             phi = phi_seq),
                                             K = 5)
     w_hat <- bps$W
     epd <- bps$epd
     fit_list[[i]] <- list(epd, w_hat) }

## Combination weights between partitions using Bayesian Predictive Stacking
comb_bps <- BPS_combine(fit_list = fit_list, K = 10, rp = 1)


## Generate subsets of data
n <- 100
p <- 3
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n), nrow = n, ncol = 1)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competitive set of values for hyperparameters
delta_seq <- c(0.1, 0.2, 0.3)
phi_seq <- c(3, 4, 5)

## Perform Bayesian Predictive Stacking within subsets
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
    Yi <- data_part$Y_list[[i]]
    Xi <- data_part$X_list[[i]]
    crd_i <- data_part$crd_list[[i]]
    p <- ncol(Xi)
    bps <- spBPS::BPS_weights(data = list(Y = Yi, X = Xi),
                               priors = list(mu_b = matrix(rep(0, p)),
                                             V_b = diag(10, p),
                                             a = 2,
                                             b = 2), coords = crd_i,
                                             hyperpar = list(delta = delta_seq,
                                                             phi = phi_seq),
                                             K = 5)
     w_hat <- bps$W
     epd <- bps$epd
     fit_list[[i]] <- list(epd, w_hat) }

## Combination weights between partitions using Bayesian Predictive Stacking
comb_bps <- BPS_combine(fit_list = fit_list, K = 10, rp = 1)

Perform the BPS sampling from posterior and posterior predictive given a set of stacking weights

Description

Perform the BPS sampling from posterior and posterior predictive given a set of stacking weights

Usage

BPS_post(data, X_u, priors, coords, crd_u, hyperpar, W, R)
BPS_post(data, X_u, priors, coords, crd_u, hyperpar, W, R)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`priors`	list priors: named $\mu_b$ , $V_b$ , $a$ , $b$
`coords`	matrix sample coordinates for X and Y
`crd_u`	matrix unboserved instances coordinates
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`W`	matrix set of stacking weights
`R`	integer number of desired samples

Value

list BPS posterior predictive samples

Examples


## Generate subsets of data
n <- 100
p <- 3
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n), nrow = n, ncol = 1)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competetive set of values for hyperparameters
delta_seq <- c(0.1, 0.2, 0.3)
phi_seq <- c(3, 4, 5)
## Fit local models
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
  Yi <- data_part$Y_list[[i]]
  Xi <- data_part$X_list[[i]]
  crd_i <- data_part$crd_list[[i]]
  p <- ncol(Xi)
  bps <- spBPS::BPS_weights(data = list(Y = Yi, X = Xi),
                           priors = list(mu_b = matrix(rep(0, p)),
                                         V_b = diag(10, p),
                                         a = 2,
                                         b = 2), coords = crd_i,
                                         hyperpar = list(delta = delta_seq,
                                                         phi = phi_seq),
                                                         K = 5)
  w_hat <- bps$W
  epd <- bps$epd
  fit_list[[i]] <- list(epd, w_hat) }

## Model combination weights between partitions using Bayesian Predictive Stacking
comb_bps <- BPS_combine(fit_list = fit_list, K = 10, rp = 1)
Wbps <- comb_bps$W
W_list <- comb_bps$W_list

## Generate prediction points
m <- 100
X_new <- matrix(rnorm(m*p), nrow = m, ncol = p)
crd_new <- matrix(runif(m*2), nrow = m, ncol = 2)

## Perform posterior and posterior predictive sampling
R <- 250
subset_ind <- sample(1:10, R, TRUE, Wbps)
postsmp_and_pred <- vector(length = R, mode = "list")
for (r in 1:R) {
 ind_s <- subset_ind[r]
 Ys <- matrix(data_part$Y_list[[ind_s]])
 Xs <- data_part$X_list[[ind_s]]
 crds <- data_part$crd_list[[ind_s]]
 Ws <- W_list[[ind_s]]
 result <- spBPS::BPS_post(data = list(Y = Ys, X = Xs), coords = crds,
                           X_u = X_new, crd_u = crd_new,
                           priors = list(mu_b = matrix(rep(0, p)),
                                         V_b = diag(10, p),
                                         a = 2,
                                         b = 2),
                                         hyperpar = list(delta = delta_seq,
                                                         phi = phi_seq),
                                                         W = Ws, R = 1)
 postsmp_and_pred[[r]] <- result}



## Generate subsets of data
n <- 100
p <- 3
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n), nrow = n, ncol = 1)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competetive set of values for hyperparameters
delta_seq <- c(0.1, 0.2, 0.3)
phi_seq <- c(3, 4, 5)
## Fit local models
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
  Yi <- data_part$Y_list[[i]]
  Xi <- data_part$X_list[[i]]
  crd_i <- data_part$crd_list[[i]]
  p <- ncol(Xi)
  bps <- spBPS::BPS_weights(data = list(Y = Yi, X = Xi),
                           priors = list(mu_b = matrix(rep(0, p)),
                                         V_b = diag(10, p),
                                         a = 2,
                                         b = 2), coords = crd_i,
                                         hyperpar = list(delta = delta_seq,
                                                         phi = phi_seq),
                                                         K = 5)
  w_hat <- bps$W
  epd <- bps$epd
  fit_list[[i]] <- list(epd, w_hat) }

## Model combination weights between partitions using Bayesian Predictive Stacking
comb_bps <- BPS_combine(fit_list = fit_list, K = 10, rp = 1)
Wbps <- comb_bps$W
W_list <- comb_bps$W_list

## Generate prediction points
m <- 100
X_new <- matrix(rnorm(m*p), nrow = m, ncol = p)
crd_new <- matrix(runif(m*2), nrow = m, ncol = 2)

## Perform posterior and posterior predictive sampling
R <- 250
subset_ind <- sample(1:10, R, TRUE, Wbps)
postsmp_and_pred <- vector(length = R, mode = "list")
for (r in 1:R) {
 ind_s <- subset_ind[r]
 Ys <- matrix(data_part$Y_list[[ind_s]])
 Xs <- data_part$X_list[[ind_s]]
 crds <- data_part$crd_list[[ind_s]]
 Ws <- W_list[[ind_s]]
 result <- spBPS::BPS_post(data = list(Y = Ys, X = Xs), coords = crds,
                           X_u = X_new, crd_u = crd_new,
                           priors = list(mu_b = matrix(rep(0, p)),
                                         V_b = diag(10, p),
                                         a = 2,
                                         b = 2),
                                         hyperpar = list(delta = delta_seq,
                                                         phi = phi_seq),
                                                         W = Ws, R = 1)
 postsmp_and_pred[[r]] <- result}

Perform the BPS sampling from posterior and posterior predictive given a set of stacking weights

Description

Perform the BPS sampling from posterior and posterior predictive given a set of stacking weights

Usage

BPS_post_MvT(data, X_u, priors, coords, crd_u, hyperpar, W, R)
BPS_post_MvT(data, X_u, priors, coords, crd_u, hyperpar, W, R)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`priors`	list priors: named $\mu_B$ , $V_r$ , $\Psi$ , $\nu$
`coords`	matrix sample coordinates for X and Y
`crd_u`	matrix unboserved instances coordinates
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$
`W`	matrix set of stacking weights
`R`	integer number of desired samples

Value

list BPS posterior predictive samples

Examples


## Generate subsets of data
n <- 100
p <- 3
q <- 2
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competitive set of values for hyperparameters
alfa_seq <- c(0.7, 0.8, 0.9)
phi_seq <- c(3, 4, 5)

## Fit local models
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
    Yi <- data_part$Y_list[[i]]
    Xi <- data_part$X_list[[i]]
    crd_i <- data_part$crd_list[[i]]
    bps <- spBPS::BPS_weights_MvT(data = list(Y = Yi, X = Xi),
                              priors = list(mu_B = matrix(0, nrow = p, ncol = q),
                                            V_r = diag(10, p),
                                            Psi = diag(1, q),
                                            nu = 3), coords = crd_i,
                                            hyperpar = list(alpha = alfa_seq,
                                                            phi = phi_seq),
                                            K = 5)
     w_hat <- bps$W
     epd <- bps$epd
     fit_list[[i]] <- list(epd, w_hat) }

## Model combination weights between partitions using Bayesian Predictive Stacking
comb_bps <- BPS_combine(fit_list = fit_list, K = 10, rp = 1)
Wbps <- comb_bps$W
W_list <- comb_bps$W_list

## Generate prediction points
m <- 100
X_new <- matrix(rnorm(m*p), nrow = m, ncol = p)
crd_new <- matrix(runif(m*2), nrow = m, ncol = 2)

## Perform posterior and posterior predictive sampling
R <- 250
subset_ind <- sample(1:10, R, TRUE, Wbps)
postsmp_and_pred <- vector(length = R, mode = "list")
for (r in 1:R) {
  ind_s <- subset_ind[r]
  Ys <- data_part$Y_list[[ind_s]]
  Xs <- data_part$X_list[[ind_s]]
  crds <- data_part$crd_list[[ind_s]]
  Ws <- W_list[[ind_s]]
  result <- spBPS::BPS_post_MvT(data = list(Y = Ys, X = Xs), coords = crds,
                                X_u = X_new, crd_u = crd_new,
                                priors = list(mu_B = matrix(0, nrow = p, ncol = q),
                                              V_r = diag(10, p),
                                              Psi = diag(1, q),
                                              nu = 3),
                                              hyperpar = list(alpha = alfa_seq,
                                                              phi = phi_seq),
                                              W = Ws, R = 1)

  postsmp_and_pred[[r]] <- result}



## Generate subsets of data
n <- 100
p <- 3
q <- 2
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competitive set of values for hyperparameters
alfa_seq <- c(0.7, 0.8, 0.9)
phi_seq <- c(3, 4, 5)

## Fit local models
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
    Yi <- data_part$Y_list[[i]]
    Xi <- data_part$X_list[[i]]
    crd_i <- data_part$crd_list[[i]]
    bps <- spBPS::BPS_weights_MvT(data = list(Y = Yi, X = Xi),
                              priors = list(mu_B = matrix(0, nrow = p, ncol = q),
                                            V_r = diag(10, p),
                                            Psi = diag(1, q),
                                            nu = 3), coords = crd_i,
                                            hyperpar = list(alpha = alfa_seq,
                                                            phi = phi_seq),
                                            K = 5)
     w_hat <- bps$W
     epd <- bps$epd
     fit_list[[i]] <- list(epd, w_hat) }

## Model combination weights between partitions using Bayesian Predictive Stacking
comb_bps <- BPS_combine(fit_list = fit_list, K = 10, rp = 1)
Wbps <- comb_bps$W
W_list <- comb_bps$W_list

## Generate prediction points
m <- 100
X_new <- matrix(rnorm(m*p), nrow = m, ncol = p)
crd_new <- matrix(runif(m*2), nrow = m, ncol = 2)

## Perform posterior and posterior predictive sampling
R <- 250
subset_ind <- sample(1:10, R, TRUE, Wbps)
postsmp_and_pred <- vector(length = R, mode = "list")
for (r in 1:R) {
  ind_s <- subset_ind[r]
  Ys <- data_part$Y_list[[ind_s]]
  Xs <- data_part$X_list[[ind_s]]
  crds <- data_part$crd_list[[ind_s]]
  Ws <- W_list[[ind_s]]
  result <- spBPS::BPS_post_MvT(data = list(Y = Ys, X = Xs), coords = crds,
                                X_u = X_new, crd_u = crd_new,
                                priors = list(mu_B = matrix(0, nrow = p, ncol = q),
                                              V_r = diag(10, p),
                                              Psi = diag(1, q),
                                              nu = 3),
                                              hyperpar = list(alpha = alfa_seq,
                                                              phi = phi_seq),
                                              W = Ws, R = 1)

  postsmp_and_pred[[r]] <- result}

Compute the BPS posterior samples given a set of stacking weights

Description

Compute the BPS posterior samples given a set of stacking weights

Usage

BPS_postdraws(data, priors, coords, hyperpar, W, R)
BPS_postdraws(data, priors, coords, hyperpar, W, R)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_b$ , $V_b$ , $a$ , $b$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`W`	matrix set of stacking weights
`R`	integer number of desired samples

Value

matrix BPS posterior samples

Compute the BPS posterior samples given a set of stacking weights

Description

Compute the BPS posterior samples given a set of stacking weights

Usage

BPS_postdraws_MvT(data, priors, coords, hyperpar, W, R, par)
BPS_postdraws_MvT(data, priors, coords, hyperpar, W, R, par)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_B$ , $V_r$ , $\Psi$ , $\nu$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$
`W`	matrix set of stacking weights
`R`	integer number of desired samples
`par`	if `TRUE` only $\beta$ and $\Sigma$ are sampled ( $\omega$ is omitted)

Value

matrix BPS posterior samples

Compute the BPS spatial prediction given a set of stacking weights

Description

Compute the BPS spatial prediction given a set of stacking weights

Usage

BPS_pred(data, X_u, priors, coords, crd_u, hyperpar, W, R)
BPS_pred(data, X_u, priors, coords, crd_u, hyperpar, W, R)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`priors`	list priors: named $\mu_b$ , $V_b$ , $a$ , $b$
`coords`	matrix sample coordinates for X and Y
`crd_u`	matrix unboserved instances coordinates
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`W`	matrix set of stacking weights
`R`	integer number of desired samples

Value

list BPS posterior predictive samples

Examples


## Generate subsets of data
n <- 100
p <- 3
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n), nrow = n, ncol = 1)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competetive set of values for hyperparameters
delta_seq <- c(0.1, 0.2, 0.3)
phi_seq <- c(3, 4, 5)

## Fit local models
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
    Yi <- data_part$Y_list[[i]]
    Xi <- data_part$X_list[[i]]
    crd_i <- data_part$crd_list[[i]]
    p <- ncol(Xi)
    bps <- spBPS::BPS_weights(data = list(Y = Yi, X = Xi),
                               priors = list(mu_b = matrix(rep(0, p)),
                                             V_b = diag(10, p),
                                             a = 2,
                                             b = 2), coords = crd_i,
                                             hyperpar = list(delta = delta_seq,
                                                             phi = phi_seq),
                                             K = 5)
     w_hat <- bps$W
     epd <- bps$epd
     fit_list[[i]] <- list(epd, w_hat) }

## Model combination weights between partitions using Bayesian Predictive Stacking
comb_bps <- BPS_combine(fit_list = fit_list, K = 10, rp = 1)
Wbps <- comb_bps$W
W_list <- comb_bps$W_list

## Generate prediction points
m <- 50
X_new <- matrix(rnorm(m*p), nrow = m, ncol = p)
crd_new <- matrix(runif(m*2), nrow = m, ncol = 2)

## Perform posterior predictive sampling
R <- 250
subset_ind <- sample(1:10, R, TRUE, Wbps)
predictions <- vector(length = R, mode = "list")
for (r in 1:R) {
  ind_s <- subset_ind[r]
  Ys <- matrix(data_part$Y_list[[ind_s]])
  Xs <- data_part$X_list[[ind_s]]
  crds <- data_part$crd_list[[ind_s]]
  Ws <- W_list[[ind_s]]
  result <- spBPS::BPS_pred(data = list(Y = Ys, X = Xs), coords = crds,
                            X_u = X_new, crd_u = crd_new,
                            priors = list(mu_b = matrix(rep(0, p)),
                                          V_b = diag(10, p),
                                          a = 2,
                                          b = 2),
                                          hyperpar = list(delta = delta_seq,
                                                          phi = phi_seq),
                                          W = Ws, R = 1)

  predictions[[r]] <- result}



## Generate subsets of data
n <- 100
p <- 3
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n), nrow = n, ncol = 1)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competetive set of values for hyperparameters
delta_seq <- c(0.1, 0.2, 0.3)
phi_seq <- c(3, 4, 5)

## Fit local models
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
    Yi <- data_part$Y_list[[i]]
    Xi <- data_part$X_list[[i]]
    crd_i <- data_part$crd_list[[i]]
    p <- ncol(Xi)
    bps <- spBPS::BPS_weights(data = list(Y = Yi, X = Xi),
                               priors = list(mu_b = matrix(rep(0, p)),
                                             V_b = diag(10, p),
                                             a = 2,
                                             b = 2), coords = crd_i,
                                             hyperpar = list(delta = delta_seq,
                                                             phi = phi_seq),
                                             K = 5)
     w_hat <- bps$W
     epd <- bps$epd
     fit_list[[i]] <- list(epd, w_hat) }

## Model combination weights between partitions using Bayesian Predictive Stacking
comb_bps <- BPS_combine(fit_list = fit_list, K = 10, rp = 1)
Wbps <- comb_bps$W
W_list <- comb_bps$W_list

## Generate prediction points
m <- 50
X_new <- matrix(rnorm(m*p), nrow = m, ncol = p)
crd_new <- matrix(runif(m*2), nrow = m, ncol = 2)

## Perform posterior predictive sampling
R <- 250
subset_ind <- sample(1:10, R, TRUE, Wbps)
predictions <- vector(length = R, mode = "list")
for (r in 1:R) {
  ind_s <- subset_ind[r]
  Ys <- matrix(data_part$Y_list[[ind_s]])
  Xs <- data_part$X_list[[ind_s]]
  crds <- data_part$crd_list[[ind_s]]
  Ws <- W_list[[ind_s]]
  result <- spBPS::BPS_pred(data = list(Y = Ys, X = Xs), coords = crds,
                            X_u = X_new, crd_u = crd_new,
                            priors = list(mu_b = matrix(rep(0, p)),
                                          V_b = diag(10, p),
                                          a = 2,
                                          b = 2),
                                          hyperpar = list(delta = delta_seq,
                                                          phi = phi_seq),
                                          W = Ws, R = 1)

  predictions[[r]] <- result}

Compute the BPS spatial prediction given a set of stacking weights

Description

Compute the BPS spatial prediction given a set of stacking weights

Usage

BPS_pred_MvT(data, X_u, priors, coords, crd_u, hyperpar, W, R)
BPS_pred_MvT(data, X_u, priors, coords, crd_u, hyperpar, W, R)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`priors`	list priors: named $\mu_B$ , $V_r$ , $\Psi$ , $\nu$
`coords`	matrix sample coordinates for X and Y
`crd_u`	matrix unboserved instances coordinates
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$
`W`	matrix set of stacking weights
`R`	integer number of desired samples

Value

list BPS posterior predictive samples

Examples


## Generate subsets of data
n <- 100
p <- 3
q <- 2
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competitive set of values for hyperparameters
alfa_seq <- c(0.7, 0.8, 0.9)
phi_seq <- c(3, 4, 5)

## Fit local models
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
    Yi <- data_part$Y_list[[i]]
    Xi <- data_part$X_list[[i]]
    crd_i <- data_part$crd_list[[i]]
    bps <- spBPS::BPS_weights_MvT(data = list(Y = Yi, X = Xi),
                              priors = list(mu_B = matrix(0, nrow = p, ncol = q),
                                            V_r = diag(10, p),
                                            Psi = diag(1, q),
                                            nu = 3), coords = crd_i,
                                            hyperpar = list(alpha = alfa_seq,
                                                            phi = phi_seq),
                                            K = 5)
     w_hat <- bps$W
     epd <- bps$epd
     fit_list[[i]] <- list(epd, w_hat) }

## Model combination weights between partitions using Bayesian Predictive Stacking
comb_bps <- BPS_combine(fit_list = fit_list, K = 10, rp = 1)
Wbps <- comb_bps$W
W_list <- comb_bps$W_list

## Generate prediction points
m <- 100
X_new <- matrix(rnorm(m*p), nrow = m, ncol = p)
crd_new <- matrix(runif(m*2), nrow = m, ncol = 2)

## Perform posterior predictive sampling
R <- 250
subset_ind <- sample(1:10, R, TRUE, Wbps)
predictions <- vector(length = R, mode = "list")
for (r in 1:R) {
  ind_s <- subset_ind[r]
  Ys <- data_part$Y_list[[ind_s]]
  Xs <- data_part$X_list[[ind_s]]
  crds <- data_part$crd_list[[ind_s]]
  Ws <- W_list[[ind_s]]
  result <- spBPS::BPS_pred_MvT(data = list(Y = Ys, X = Xs), coords = crds,
                                X_u = X_new, crd_u = crd_new,
                                priors = list(mu_B = matrix(0, nrow = p, ncol = q),
                                              V_r = diag(10, p),
                                              Psi = diag(1, q),
                                              nu = 3),
                                              hyperpar = list(alpha = alfa_seq,
                                                              phi = phi_seq),
                                              W = Ws, R = 1)

  predictions[[r]] <- result}



## Generate subsets of data
n <- 100
p <- 3
q <- 2
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competitive set of values for hyperparameters
alfa_seq <- c(0.7, 0.8, 0.9)
phi_seq <- c(3, 4, 5)

## Fit local models
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
    Yi <- data_part$Y_list[[i]]
    Xi <- data_part$X_list[[i]]
    crd_i <- data_part$crd_list[[i]]
    bps <- spBPS::BPS_weights_MvT(data = list(Y = Yi, X = Xi),
                              priors = list(mu_B = matrix(0, nrow = p, ncol = q),
                                            V_r = diag(10, p),
                                            Psi = diag(1, q),
                                            nu = 3), coords = crd_i,
                                            hyperpar = list(alpha = alfa_seq,
                                                            phi = phi_seq),
                                            K = 5)
     w_hat <- bps$W
     epd <- bps$epd
     fit_list[[i]] <- list(epd, w_hat) }

## Model combination weights between partitions using Bayesian Predictive Stacking
comb_bps <- BPS_combine(fit_list = fit_list, K = 10, rp = 1)
Wbps <- comb_bps$W
W_list <- comb_bps$W_list

## Generate prediction points
m <- 100
X_new <- matrix(rnorm(m*p), nrow = m, ncol = p)
crd_new <- matrix(runif(m*2), nrow = m, ncol = 2)

## Perform posterior predictive sampling
R <- 250
subset_ind <- sample(1:10, R, TRUE, Wbps)
predictions <- vector(length = R, mode = "list")
for (r in 1:R) {
  ind_s <- subset_ind[r]
  Ys <- data_part$Y_list[[ind_s]]
  Xs <- data_part$X_list[[ind_s]]
  crds <- data_part$crd_list[[ind_s]]
  Ws <- W_list[[ind_s]]
  result <- spBPS::BPS_pred_MvT(data = list(Y = Ys, X = Xs), coords = crds,
                                X_u = X_new, crd_u = crd_new,
                                priors = list(mu_B = matrix(0, nrow = p, ncol = q),
                                              V_r = diag(10, p),
                                              Psi = diag(1, q),
                                              nu = 3),
                                              hyperpar = list(alpha = alfa_seq,
                                                              phi = phi_seq),
                                              W = Ws, R = 1)

  predictions[[r]] <- result}

Combine subset models wiht Pseudo-BMA

Description

Combine subset models wiht Pseudo-BMA

Usage

BPS_PseudoBMA(fit_list)
BPS_PseudoBMA(fit_list)

Arguments

fit_list

list K fitted model outputs composed by two elements each: first named $epd$ , second named $W$

Value

matrix posterior predictive density evaluations (each columns represent a different model)

Examples


## Generate subsets of data
n <- 100
p <- 3
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n), nrow = n, ncol = 1)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competitive set of values for hyperparameters
delta_seq <- c(0.1, 0.2, 0.3)
phi_seq <- c(3, 4, 5)

## Perform Bayesian Predictive Stacking within subsets
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
    Yi <- data_part$Y_list[[i]]
    Xi <- data_part$X_list[[i]]
    crd_i <- data_part$crd_list[[i]]
    p <- ncol(Xi)
    bps <- spBPS::BPS_weights(data = list(Y = Yi, X = Xi),
                               priors = list(mu_b = matrix(rep(0, p)),
                                             V_b = diag(10, p),
                                             a = 2,
                                             b = 2), coords = crd_i,
                                             hyperpar = list(delta = delta_seq,
                                                             phi = phi_seq),
                                             K = 5)
     w_hat <- bps$W
     epd <- bps$epd
     fit_list[[i]] <- list(epd, w_hat) }

## Combination weights between partitions using Pseudo Bayesian Model Averaging
comb_bps <- BPS_PseudoBMA(fit_list = fit_list)


## Generate subsets of data
n <- 100
p <- 3
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n), nrow = n, ncol = 1)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
data_part <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Select competitive set of values for hyperparameters
delta_seq <- c(0.1, 0.2, 0.3)
phi_seq <- c(3, 4, 5)

## Perform Bayesian Predictive Stacking within subsets
fit_list <- vector(length = 10, mode = "list")
for (i in 1:10) {
    Yi <- data_part$Y_list[[i]]
    Xi <- data_part$X_list[[i]]
    crd_i <- data_part$crd_list[[i]]
    p <- ncol(Xi)
    bps <- spBPS::BPS_weights(data = list(Y = Yi, X = Xi),
                               priors = list(mu_b = matrix(rep(0, p)),
                                             V_b = diag(10, p),
                                             a = 2,
                                             b = 2), coords = crd_i,
                                             hyperpar = list(delta = delta_seq,
                                                             phi = phi_seq),
                                             K = 5)
     w_hat <- bps$W
     epd <- bps$epd
     fit_list[[i]] <- list(epd, w_hat) }

## Combination weights between partitions using Pseudo Bayesian Model Averaging
comb_bps <- BPS_PseudoBMA(fit_list = fit_list)

Compute the BPS weights by convex optimization

Description

Compute the BPS weights by convex optimization

Usage

BPS_weights(data, priors, coords, hyperpar, K)
BPS_weights(data, priors, coords, hyperpar, K)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_b$ , $V_b$ , $a$ , $b$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`K`	integer number of folds

Value

matrix posterior predictive density evaluations (each columns represent a different model)

Examples

## Generate subsets of data
n <- 100
p <- 3
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n), nrow = n)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)

## Select competitive set of values for hyperparameters
delta_seq <- c(0.1, 0.2, 0.3)
phi_seq <- c(3, 4, 5)

## Perform Bayesian Predictive Stacking within subsets
bps <- spBPS::BPS_weights(data = list(Y = Y, X = X),
                               priors = list(mu_b = matrix(rep(0, p)),
                                             V_b = diag(10, p),
                                             a = 2,
                                             b = 2), coords = crd,
                                             hyperpar = list(delta = delta_seq,
                                                             phi = phi_seq),
                                             K = 5)

## Generate subsets of data
n <- 100
p <- 3
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n), nrow = n)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)

## Select competitive set of values for hyperparameters
delta_seq <- c(0.1, 0.2, 0.3)
phi_seq <- c(3, 4, 5)

## Perform Bayesian Predictive Stacking within subsets
bps <- spBPS::BPS_weights(data = list(Y = Y, X = X),
                               priors = list(mu_b = matrix(rep(0, p)),
                                             V_b = diag(10, p),
                                             a = 2,
                                             b = 2), coords = crd,
                                             hyperpar = list(delta = delta_seq,
                                                             phi = phi_seq),
                                             K = 5)

Compute the BPS weights by convex optimization

Description

Compute the BPS weights by convex optimization

Usage

BPS_weights_MvT(data, priors, coords, hyperpar, K)
BPS_weights_MvT(data, priors, coords, hyperpar, K)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_B$ , $V_r$ , $\Psi$ , $\nu$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$
`K`	integer number of folds

Value

matrix posterior predictive density evaluations (each columns represent a different model)

Examples


## Generate subsets of data
n <- 100
p <- 3
q <- 2
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)

## Select competitive set of values for hyperparameters
alfa_seq <- c(0.7, 0.8, 0.9)
phi_seq <- c(3, 4, 5)

## Perform Bayesian Predictive Stacking within subsets
bps <- spBPS::BPS_weights_MvT(data = list(Y = Y, X = X),
                              priors = list(mu_B = matrix(0, nrow = p, ncol = q),
                                            V_r = diag(10, p),
                                            Psi = diag(1, q),
                                            nu = 3), coords = crd,
                                            hyperpar = list(alpha = alfa_seq,
                                                            phi = phi_seq),
                                            K = 5)


## Generate subsets of data
n <- 100
p <- 3
q <- 2
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)

## Select competitive set of values for hyperparameters
alfa_seq <- c(0.7, 0.8, 0.9)
phi_seq <- c(3, 4, 5)

## Perform Bayesian Predictive Stacking within subsets
bps <- spBPS::BPS_weights_MvT(data = list(Y = Y, X = X),
                              priors = list(mu_B = matrix(0, nrow = p, ncol = q),
                                            V_r = diag(10, p),
                                            Psi = diag(1, q),
                                            nu = 3), coords = crd,
                                            hyperpar = list(alpha = alfa_seq,
                                                            phi = phi_seq),
                                            K = 5)

Solver for Bayesian Predictive Stacking of Predictive densities convex optimization problem

Description

Solver for Bayesian Predictive Stacking of Predictive densities convex optimization problem

Usage

conv_opt(scores)
conv_opt(scores)

Arguments

scores

matrix $N \times K$ of expected predictive density evaluations for the K models considered

Value

W matrix of Bayesian Predictive Stacking weights for the K models considered

Examples

## Generate (randomly) K predictive scores for n observations
n <- 50
K <- 5
scores <- matrix(runif(n*K), nrow = n, ncol = K)

## Find Bayesian Predictive Stacking weights
opt_weights <- conv_opt(scores)

## Generate (randomly) K predictive scores for n observations
n <- 50
K <- 5
scores <- matrix(runif(n*K), nrow = n, ncol = K)

## Find Bayesian Predictive Stacking weights
opt_weights <- conv_opt(scores)

Compute the BPS weights by convex optimization

Description

Compute the BPS weights by convex optimization

Usage

CVXR_opt(scores)
CVXR_opt(scores)

Arguments

scores

matrix $N \times K$ of expected predictive density evaluations for the K models considered

Value

conv_opt function to perform convex optimiazion with CVXR R package

Evaluate the density of a set of unobserved response with respect to the conditional posterior predictive

Description

Evaluate the density of a set of unobserved response with respect to the conditional posterior predictive

Usage

d_pred_cpp(data, X_u, Y_u, d_u, d_us, hyperpar, poster)
d_pred_cpp(data, X_u, Y_u, d_u, d_us, hyperpar, poster)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`Y_u`	matrix unobserved instances response matrix
`d_u`	matrix unobserved instances distance matrix
`d_us`	matrix cross-distance between unobserved and observed instances matrix
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`poster`	list output from `fit_cpp` function

Value

vector posterior predictive density evaluations

Evaluate the density of a set of unobserved response with respect to the conditional posterior predictive

Description

Evaluate the density of a set of unobserved response with respect to the conditional posterior predictive

Usage

d_pred_cpp_MvT(data, X_u, Y_u, d_u, d_us, hyperpar, poster)
d_pred_cpp_MvT(data, X_u, Y_u, d_u, d_us, hyperpar, poster)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`Y_u`	matrix unobserved instances response matrix
`d_u`	matrix unobserved instances distance matrix
`d_us`	matrix cross-distance between unobserved and observed instances matrix
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$
`poster`	list output from `fit_cpp` function

Value

double posterior predictive density evaluation

Compute the KCV of the density evaluations for fixed values of the hyperparameters

Description

Compute the KCV of the density evaluations for fixed values of the hyperparameters

Usage

dens_kcv(data, priors, coords, hyperpar, K)
dens_kcv(data, priors, coords, hyperpar, K)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_b$ , $V_b$ , $a$ , $b$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`K`	integer number of folds

Value

vector posterior predictive density evaluations

Compute the KCV of the density evaluations for fixed values of the hyperparameters

Description

Compute the KCV of the density evaluations for fixed values of the hyperparameters

Usage

dens_kcv_MvT(data, priors, coords, hyperpar, K)
dens_kcv_MvT(data, priors, coords, hyperpar, K)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_B$ , $V_r$ , $\Psi$ , $\nu$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$
`K`	integer number of folds

Value

vector posterior predictive density evaluations

Compute the LOOCV of the density evaluations for fixed values of the hyperparameters

Description

Compute the LOOCV of the density evaluations for fixed values of the hyperparameters

Usage

dens_loocv(data, priors, coords, hyperpar)
dens_loocv(data, priors, coords, hyperpar)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_b$ , $V_b$ , $a$ , $b$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$

Value

vector posterior predictive density evaluations

Compute the LOOCV of the density evaluations for fixed values of the hyperparameters

Description

Compute the LOOCV of the density evaluations for fixed values of the hyperparameters

Usage

dens_loocv_MvT(data, priors, coords, hyperpar)
dens_loocv_MvT(data, priors, coords, hyperpar)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_B$ , $V_r$ , $\Psi$ , $\nu$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$

Value

vector posterior predictive density evaluations

Build a grid from two vector (i.e. equivalent to `expand.grid()` in `R`)

Description

Build a grid from two vector (i.e. equivalent to expand.grid() in R)

Usage

expand_grid_cpp(x, y)
expand_grid_cpp(x, y)

Arguments

`x`	vector first vector of numeric elements
`y`	vector second vector of numeric elements

Value

matrix expanded grid of combinations

Examples

## Create a matrix from all combination of vectors
x <- seq(0, 10, length.out = 100)
y <- seq(-1, 1, length.out = 20)
grid <- expand_grid_cpp(x = x, y = y)

## Create a matrix from all combination of vectors
x <- seq(0, 10, length.out = 100)
y <- seq(-1, 1, length.out = 20)
grid <- expand_grid_cpp(x = x, y = y)

Compute the parameters for the posteriors distribution of $\beta$ and $\Sigma$ (i.e. updated parameters)

Description

Compute the parameters for the posteriors distribution of $\beta$ and $\Sigma$ (i.e. updated parameters)

Usage

fit_cpp(data, priors, coords, hyperpar)
fit_cpp(data, priors, coords, hyperpar)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_b$ , $V_b$ , $a$ , $b$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$

Value

list posterior update parameters

Compute the parameters for the posteriors distribution of $\beta$ and $\Sigma$ (i.e. updated parameters)

Description

Compute the parameters for the posteriors distribution of $\beta$ and $\Sigma$ (i.e. updated parameters)

Usage

fit_cpp_MvT(data, priors, coords, hyperpar)
fit_cpp_MvT(data, priors, coords, hyperpar)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_B$ , $V_r$ , $\Psi$ , $\nu$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$

Value

list posterior update parameters

Function to subset data for meta-analysis

Description

Function to subset data for meta-analysis

Usage

forceSymmetry_cpp(mat)
forceSymmetry_cpp(mat)

Arguments

mat

matrix not-symmetric matrix

Value

matrix symmetric matrix (lower triangular of mat is used)

Examples

## Force matrix to be symmetric (avoiding numerical problems)
n <- 4
X <- matrix(runif(n*n), nrow = n, ncol = n)
X <- forceSymmetry_cpp(mat = X)

## Force matrix to be symmetric (avoiding numerical problems)
n <- 4
X <- matrix(runif(n*n), nrow = n, ncol = n)
X <- forceSymmetry_cpp(mat = X)

Return the CV predictive density evaluations for all the model combinations

Description

Return the CV predictive density evaluations for all the model combinations

Usage

models_dens(data, priors, coords, hyperpar, useKCV, K)
models_dens(data, priors, coords, hyperpar, useKCV, K)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_b$ , $V_b$ , $a$ , $b$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`useKCV`	if `TRUE` K-fold cross validation is used instead of LOOCV (no `default`)
`K`	integer number of folds

Value

matrix posterior predictive density evaluations (each columns represent a different model)

Return the CV predictive density evaluations for all the model combinations

Description

Return the CV predictive density evaluations for all the model combinations

Usage

models_dens_MvT(data, priors, coords, hyperpar, useKCV, K)
models_dens_MvT(data, priors, coords, hyperpar, useKCV, K)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`priors`	list priors: named $\mu_B$ , $V_r$ , $\Psi$ , $\nu$
`coords`	matrix sample coordinates for X and Y
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$
`useKCV`	if `TRUE` K-fold cross validation is used instead of LOOCV (no `default`)
`K`	integer number of folds

Value

matrix posterior predictive density evaluations (each columns represent a different model)

Sample R draws from the posterior distributions

Description

Sample R draws from the posterior distributions

Usage

post_draws(poster, R, par, p)
post_draws(poster, R, par, p)

Arguments

`poster`	list output from `fit_cpp` function
`R`	integer number of posterior samples
`par`	if `TRUE` only $\beta$ and $\sigma^2$ are sampled ( $\omega$ is omitted)
`p`	integer if `par = TRUE`, it specifies the column number of $X$

Value

list posterior samples

Sample R draws from the posterior distributions

Description

Sample R draws from the posterior distributions

Usage

post_draws_MvT(poster, R, par, p)
post_draws_MvT(poster, R, par, p)

Arguments

`poster`	list output from `fit_cpp` function
`R`	integer number of posterior samples
`par`	if `TRUE` only $\beta$ and $\Sigma$ are sampled ( $\omega$ is omitted)
`p`	integer if `par = TRUE`, it specifies the column number of $X$

Value

list posterior samples

Predictive sampler for Conjugate Bayesian Multivariate Linear Models

Description

Predictive sampler for Conjugate Bayesian Multivariate Linear Models

Usage

pred_bayesMvLMconjugate(X_new, B_samples, Sigma_samples)
pred_bayesMvLMconjugate(X_new, B_samples, Sigma_samples)

Arguments

`X_new`	matrix $n_new \times p$ of predictors for new data points
`B_samples`	array of posterior sample for $\beta$
`Sigma_samples`	array of posterior samples for $\Sigma$

Value

Y_pred matrix of posterior mean for response matrix Y predictions

Y_pred_samples array of posterior predictive sample for response matrix Y

Examples

## Generate data
n <- 100
p <- 3
q <- 2
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
X <- matrix(rnorm(n*p), nrow = n, ncol = p)

## Prior parameters
mu_B <- matrix(0, p, q)
V_B <- diag(10, p)
nu <- 3
Psi <- diag(q)

## Samples from posteriors
n_iter <- 1000
burn_in <- 500
set.seed(1234)
samples <- spBPS::bayesMvLMconjugate(Y, X, mu_B, V_B, nu, Psi, n_iter, burn_in)

## Extract posterior samples
B_samples <- samples$B_samples
Sigma_samples <- samples$Sigma_samples

## Samples from predictive posterior (based posterior samples)
m <- 50
X_new <- matrix(rnorm(m*p), nrow = m, ncol = p)
pred <- spBPS::pred_bayesMvLMconjugate(X_new, B_samples, Sigma_samples)

## Generate data
n <- 100
p <- 3
q <- 2
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
X <- matrix(rnorm(n*p), nrow = n, ncol = p)

## Prior parameters
mu_B <- matrix(0, p, q)
V_B <- diag(10, p)
nu <- 3
Psi <- diag(q)

## Samples from posteriors
n_iter <- 1000
burn_in <- 500
set.seed(1234)
samples <- spBPS::bayesMvLMconjugate(Y, X, mu_B, V_B, nu, Psi, n_iter, burn_in)

## Extract posterior samples
B_samples <- samples$B_samples
Sigma_samples <- samples$Sigma_samples

## Samples from predictive posterior (based posterior samples)
m <- 50
X_new <- matrix(rnorm(m*p), nrow = m, ncol = p)
pred <- spBPS::pred_bayesMvLMconjugate(X_new, B_samples, Sigma_samples)

Draw from the conditional posterior predictive for a set of unobserved covariates

Description

Draw from the conditional posterior predictive for a set of unobserved covariates

Usage

r_pred_cond(data, X_u, d_u, d_us, hyperpar, poster, post)
r_pred_cond(data, X_u, d_u, d_us, hyperpar, poster, post)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`d_u`	matrix unobserved instances distance matrix
`d_us`	matrix cross-distance between unobserved and observed instances matrix
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`poster`	list output from `fit_cpp` function
`post`	list output from `post_draws` function

Value

list posterior predictive samples

Draw from the conditional posterior predictive for a set of unobserved covariates

Description

Draw from the conditional posterior predictive for a set of unobserved covariates

Usage

r_pred_cond_MvT(data, X_u, d_u, d_us, hyperpar, poster, post)
r_pred_cond_MvT(data, X_u, d_u, d_us, hyperpar, poster, post)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`d_u`	matrix unobserved instances distance matrix
`d_us`	matrix cross-distance between unobserved and observed instances matrix
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$
`poster`	list output from `fit_cpp_MvT` function
`post`	list output from `post_draws_MvT` function

Value

list posterior predictive samples

Draw from the joint posterior predictive for a set of unobserved covariates

Description

Draw from the joint posterior predictive for a set of unobserved covariates

Usage

r_pred_joint(data, X_u, d_u, d_us, hyperpar, poster, R)
r_pred_joint(data, X_u, d_u, d_us, hyperpar, poster, R)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`d_u`	matrix unobserved instances distance matrix
`d_us`	matrix cross-distance between unobserved and observed instances matrix
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`poster`	list output from `fit_cpp` function
`R`	integer number of posterior predictive samples

Value

list posterior predictive samples

Draw from the joint posterior predictive for a set of unobserved covariates

Description

Draw from the joint posterior predictive for a set of unobserved covariates

Usage

r_pred_joint_MvT(data, X_u, d_u, d_us, hyperpar, poster, R)
r_pred_joint_MvT(data, X_u, d_u, d_us, hyperpar, poster, R)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`d_u`	matrix unobserved instances distance matrix
`d_us`	matrix cross-distance between unobserved and observed instances matrix
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$
`poster`	list output from `fit_cpp` function
`R`	integer number of posterior predictive samples

Value

list posterior predictive samples

Draw from the marginals posterior predictive for a set of unobserved covariates

Description

Draw from the marginals posterior predictive for a set of unobserved covariates

Usage

r_pred_marg(data, X_u, d_u, d_us, hyperpar, poster, R)
r_pred_marg(data, X_u, d_u, d_us, hyperpar, poster, R)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`d_u`	matrix unobserved instances distance matrix
`d_us`	matrix cross-distance between unobserved and observed instances matrix
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`poster`	list output from `fit_cpp` function
`R`	integer number of posterior predictive samples

Value

list posterior predictive samples

Draw from the joint posterior predictive for a set of unobserved covariates

Description

Draw from the joint posterior predictive for a set of unobserved covariates

Usage

r_pred_marg_MvT(data, X_u, d_u, d_us, hyperpar, poster, R)
r_pred_marg_MvT(data, X_u, d_u, d_us, hyperpar, poster, R)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`d_u`	matrix unobserved instances distance matrix
`d_us`	matrix cross-distance between unobserved and observed instances matrix
`hyperpar`	list two elemets: first named $\alpha$ , second named $\phi$
`poster`	list output from `fit_cpp` function
`R`	integer number of posterior predictive samples

Value

list posterior predictive samples

Function to sample integers (index)

Description

Function to sample integers (index)

Usage

sample_index(size, length, p)
sample_index(size, length, p)

Arguments

`size`	integer dimension of the set to sample
`length`	integer number of elements to sample
`p`	vector sampling probabilities

Value

vector sample of integers

Perform prediction for BPS accelerated models - loop over prediction set

Description

Perform prediction for BPS accelerated models - loop over prediction set

Usage

spPredict_BPS(data, X_u, priors, coords, crd_u, hyperpar, W, R, J)
spPredict_BPS(data, X_u, priors, coords, crd_u, hyperpar, W, R, J)

Arguments

`data`	list two elements: first named $Y$ , second named $X$
`X_u`	matrix unobserved instances covariate matrix
`priors`	list priors: named $\mu_b$ , $V_b$ , $a$ , $b$
`coords`	matrix sample coordinates for X and Y
`crd_u`	matrix unboserved instances coordinates
`hyperpar`	list two elemets: first named $\delta$ , second named $\phi$
`W`	matrix set of stacking weights
`R`	integer number of desired samples
`J`	integer number of desired partition of prediction set

Value

list BPS posterior predictive samples

Function to subset data for meta-analysis

Description

Function to subset data for meta-analysis

Usage

subset_data(data, K)
subset_data(data, K)

Arguments

`data`	list three elements: first named $Y$ , second named $X$ , third named $crd$
`K`	integer number of desired subsets

Value

list subsets of data, and the set of indexes

Examples

## Create a list of K random subsets given a list with Y, X, and crd
n <- 100
p <- 3
q <- 2
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
subsets <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

## Create a list of K random subsets given a list with Y, X, and crd
n <- 100
p <- 3
q <- 2
X <- matrix(rnorm(n*p), nrow = n, ncol = p)
Y <- matrix(rnorm(n*q), nrow = n, ncol = q)
crd <- matrix(runif(n*2), nrow = n, ncol = 2)
subsets <- subset_data(data = list(Y = Y, X = X, crd = crd), K = 10)

Package 'spBPS'

Help Index

Compute the Euclidean distance matrix

Description

Usage

Arguments

Value

Examples

Gibbs sampler for Conjugate Bayesian Multivariate Linear Models

Description

Usage

Arguments

Value

Examples

Combine subset models wiht BPS

Description

Usage

Arguments

Value

Examples

Perform the BPS sampling from posterior and posterior predictive given a set of stacking weights

Description

Usage

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Value

Examples

Perform the BPS sampling from posterior and posterior predictive given a set of stacking weights

Description

Usage

Arguments

Value

Examples

Compute the BPS posterior samples given a set of stacking weights

Description

Usage

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Value

Compute the BPS posterior samples given a set of stacking weights

Description

Usage

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Value

Compute the BPS spatial prediction given a set of stacking weights

Description

Usage

Arguments

Value

Examples

Compute the BPS spatial prediction given a set of stacking weights

Description

Usage

Arguments

Value

Examples

Combine subset models wiht Pseudo-BMA

Description

Usage

Arguments

Value

Examples

Compute the BPS weights by convex optimization

Description

Usage

Arguments

Value

Examples

Compute the BPS weights by convex optimization

Description

Usage

Arguments

Value

Examples

Solver for Bayesian Predictive Stacking of Predictive densities convex optimization problem

Description

Usage

Arguments

Value

Examples

Compute the BPS weights by convex optimization

Description

Build a grid from two vector (i.e. equivalent to `expand.grid()` in `R`)

Compute the parameters for the posteriors distribution of $\beta$ and $\Sigma$ (i.e. updated parameters)

Compute the parameters for the posteriors distribution of $\beta$ and $\Sigma$ (i.e. updated parameters)