Package 'bravo'

Bayesian Iterated Screening (ultra-high, high or low dimensional).

Description

Perform Bayesian iterated screening in Gaussian regression models

Usage

bits(X, y, lam = 1, w = 0.5, pp = FALSE, max.var = nrow(X), verbose = TRUE)

Arguments

X	An $n\times p$ matrix. Sparse matrices are supported and every care is taken not to make copies of this (typically) giant matrix. No need to center or scale.
y	The response vector of length n.
lam	The slab precision parameter. Default: 1.
w	The prior inclusion probability of each variable. Default: 1/2.
pp	Boolean: If FALSE (default) the algorithm stops after including max.var many variables. If true, the posterior probability stopping rule is used.
max.var	The maximum number of variables to be included.
verbose	If TRUE (default) will show the variable index included in each iteration.

Value

A list with components

model.pp	An integer vector of the screened model.
postprobs	The sequence of posterior probabilities until the last included variable.
lam	The value of lam, the slab precision parameter.
w	The value of w, the prior inclusion probability.

References

Wang, R., Dutta, S., Roy, V. (2021) Bayesian iterative screening in ultra-high dimensional settings. https://arxiv.org/abs/2107.10175

Examples

n=50; p=100;
TrueBeta <- c(rep(5,3),rep(0,p-3))

rho <- 0.6
x1 <- matrix(rnorm(n*p), n, p)
X <- sqrt(1-rho)*x1 + sqrt(rho)*rnorm(n)
y <- 0.5 + X %*% TrueBeta + rnorm(n)
res<-bits(X,y, pp=TRUE)
res$model.pp # the vector of screened model
res$postprobs # the log (unnormalized) posterior probabilities corresponding to the model.pp.

---

Compute marginal inclusion probabilities from a fitted "sven" object.

Description

This function computes the marginal inclusion probabilities of all variables from a fitted "sven" object.

Usage

mip.sven(object, threshold = 0)

Arguments

object	A fitted "sven" object
threshold	marginal inclusion probabilities above this threshold are stored. Default 0.

Value

The object returned is a data frame if the sven was run with a single matrix, or a list of two data frames if sven was run with a list of two matrices. The first column are the variable names (or numbers if column names of were absent). Only the nonzero marginal inclusion probabilities are stored.

Author(s)

Somak Dutta
Maintainer: Somak Dutta <[email protected]>

Examples

n <- 50; p <- 100; nonzero <- 3
trueidx <- 1:3
truebeta <- c(4,5,6)
X <- matrix(rnorm(n*p), n, p) # n x p covariate matrix
y <- 0.5 + X[,trueidx] %*% truebeta + rnorm(n)
res <- sven(X=X, y=y)
res$model.map # the MAP model
mip.sven(res)

Z <- matrix(rnorm(n*p), n, p) # another covariate matrix
y2 = 0.5 + X[,trueidx] %*% truebeta  + Z[,1:2] %*% c(-2,-2) + rnorm(n)
res2 <- sven(X=list(X,Z), y=y2)
mip.sven(res2) # two data frames, one for X and another for Z

---

Make predictions from a fitted "sven" object.

Description

This function makes point predictions and computes prediction intervals from a fitted "sven" object.

Usage

## S3 method for class 'sven'
predict(
  object,
  newdata,
  model = c("WAM", "MAP"),
  interval = c("none", "MC", "Z"),
  return.draws = FALSE,
  Nsim = 10000,
  level = 0.95,
  alpha = 1 - level,
  ...
)

Arguments

object	A fitted "sven" object
newdata	Matrix of new values for X at which predictions are to be made. Must be a matrix; can be sparse as in Matrix package.
model	The model to be used to make predictions. Model "MAP" gives the predictions calculated using the MAP model; model "WAM" gives the predictions calculated using the WAM. Default: "WAM".
interval	Type of interval calculation. If interval = "none", only point predictions are returned; if interval = "MC", Monte Carlo prediction intervals are returned; if interval = "Z", Z prediction intervals are returned.
return.draws	only required if interval = "MC". if TRUE, the Monte Carlo samples are returned. Default: FALSE.
Nsim	only required if interval = "MC". The Monte Carlo sample size. Default: 10000.
level	Confidence level of the interval. Default: 0.95.
alpha	Type one error rate. Default: 1-level.
...	Further arguments passed to or from other methods.

Value

The object returned depends on "interval" argument. If interval = "none", the object is an $\code{ncol(newdata)}\times 1$ vector of the point predictions; otherwise, the object is an $\code{ncol(newdata)}\times 3$ matrix with the point predictions in the first column and the lower and upper bounds of prediction intervals in the second and third columns, respectively.

if return.draws is TRUE, a list with the following components is returned:

prediction	vector or matrix as above
mc.draws	an $\code{ncol(newdata)} \times \code{Nsim}$ matrix of the Monte Carlo samples

Author(s)

Dongjin Li and Somak Dutta
Maintainer: Dongjin Li <[email protected]>

References

Li, D., Dutta, S., Roy, V.(2020) Model Based Screening Embedded Bayesian Variable Selection for Ultra-high Dimensional Settings http://arxiv.org/abs/2006.07561

Examples

n = 80; p = 100; nonzero = 5
trueidx <- 1:5
nonzero.value <- c(0.50, 0.75, 1.00, 1.25, 1.50)
TrueBeta = numeric(p)
TrueBeta[trueidx] <- nonzero.value

X <- matrix(rnorm(n*p), n, p)
y <- 0.5 + X %*% TrueBeta + rnorm(n)
res <- sven(X=X, y=y)
newx <- matrix(rnorm(20*p), 20, p)
# predicted values at a new data matrix using MAP model
yhat <- predict(object = res, newdata = newx, model = "MAP", interval = "none")
# 95% Monte Carlo prediction interval using WAM
MC.interval <- predict(object = res, model = "WAM", newdata = newx, interval = "MC", level=0.95)
# 95% Z-prediction interval using MAP model
Z.interval <- predict(object = res, model = "MAP", newdata = newx, interval = "Z", level = 0.95)

---

Selection of variables with embedded screening using Bayesian methods (SVEN) in Gaussian linear models (ultra-high, high or low dimensional).

Description

SVEN is an approach to selecting variables with embedded screening using a Bayesian hierarchical model. It is also a variable selection method in the spirit of the stochastic shotgun search algorithm. However, by embedding a unique model based screening and using fast Cholesky updates, SVEN produces a highly scalable algorithm to explore gigantic model spaces and rapidly identify the regions of high posterior probabilities. It outputs the log (unnormalized) posterior probability of a set of best (highest probability) models. For more details, see Li et al. (2023, https://doi.org/10.1080/10618600.2022.2074428)

Usage

sven(
  X,
  y,
  w = NULL,
  lam = NULL,
  Ntemp = 10,
  Tmax = NULL,
  Miter = 50,
  wam.threshold = 0.5,
  log.eps = -16,
  L = 20,
  verbose = FALSE
)

Arguments

X	The $n\times p$ covariate matrix or list of two matrices without intercept. The following classes are supported: matrix and dgCMatrix. Every care is taken not to make copies of these (typically) giant matrices. No need to center or scale these matrices manually. Scaling is performed implicitly and regression coefficient are returned on the original scale. Typically, in a combined GWAS-TWAS type analysis, X[[1]] should be a sparse matrix and X[[2]] should be a dense matrix.
y	The response vector of length $n$. No need to center or scale.
w	The prior inclusion probability of each variable. Default: NULL, whence it is set as $\sqrt{n}/p$ if $X$ is a matrix. Or $(\sqrt{n}/p_1,\sqrt{n}/p_2)$ if $X$ is a list of two matrices with $p_1$ and $p_2$ columns.
lam	The slab precision parameter. Default: NULL, whence it is set as $n/p^2$ for as suggested by the theory of Li et al. (2023). Similarly, it's a vector of length two with values $\sqrt{n}/P_1^2$ and $\sqrt{n}/p_2^2$ when X is a list.
Ntemp	The number of temperatures. Default: 10.
Tmax	The maximum temperature. Default: $\log\log p+\log p$.
Miter	The number of iterations per temperature. Default: 50.
wam.threshold	The threshold probability to select the covariates for WAM. A covariate will be included in WAM if its corresponding marginal inclusion probability is greater than the threshold. Default: 0.5.
log.eps	The tolerance to choose the number of top models. See detail. Default: -16.
L	The minimum number of neighboring models screened. Default: 20.
verbose	If FALSE, the function prints the current temperature SVEN is at; the default is TRUE.

Details

SVEN is developed based on a hierarchical Gaussian linear model with priors placed on the regression coefficients as well as on the model space as follows:

$y | X, \beta_0,\beta,\gamma,\sigma^2,w,\lambda \sim N(\beta_01 + X_\gamma\beta_\gamma,\sigma^2I_n)$

$\beta_i|\beta_0,\gamma,\sigma^2,w,\lambda \stackrel{indep.}{\sim} N(0, \gamma_i\sigma^2/\lambda),~i=1,\ldots,p,$

$(\beta_0,\sigma^2)|\gamma,w,p \sim p(\beta_0,\sigma^2) \propto 1/\sigma^2$

$\gamma_i|w,\lambda \stackrel{iid}{\sim} Bernoulli(w)$

where $X_\gamma$ is the $n \times |\gamma|$ submatrix of $X$ consisting of those columns of $X$ for which $\gamma_i=1$ and similarly, $\beta_\gamma$ is the $|\gamma|$ subvector of $\beta$ corresponding to $\gamma$. Degenerate spike priors on inactive variables and Gaussian slab priors on active covariates makes the posterior probability (up to a normalizing constant) of a model $P(\gamma|y)$ available in explicit form (Li et al., 2020).

The variable selection starts from an empty model and updates the model according to the posterior probability of its neighboring models for some pre-specified number of iterations. In each iteration, the models with small probabilities are screened out in order to quickly identify the regions of high posterior probabilities. A temperature schedule is used to facilitate exploration of models separated by valleys in the posterior probability function, thus mitigate posterior multimodality associated with variable selection models. The default maximum temperature is guided by the asymptotic posterior model selection consistency results in Li et al. (2020).

SVEN provides the maximum a posteriori (MAP) model as well as the weighted average model (WAM). WAM is obtained in the following way: (1) keep the best (highest probability) $K$ distinct models $\gamma^{(1)},\ldots,\gamma^{(K)}$ with

$\log P\left(\gamma^{(1)}|y\right) \ge \cdots \ge \log P\left(\gamma^{(K)}|y\right)$

where $K$ is chosen so that $\log \left\{P\left(\gamma^{(K)}|y\right)/P\left(\gamma^{(1)}|y\right)\right\} > \code{log.eps}$; (2) assign the weights

$w_i = P(\gamma^{(i)}|y)/\sum_{k=1}^K P(\gamma^{(k)}|y)$

to the model $\gamma^{(i)}$; (3) define the approximate marginal inclusion probabilities for the $j$th variable as

$\hat\pi_j = \sum_{k=1}^K w_k I(\gamma^{(k)}_j = 1).$

Then, the WAM is defined as the model containing variables $j$ with $\hat\pi_j > \code{wam.threshold}$. SVEN also provides all the top $K$ models which are stored in an $p \times K$ sparse matrix, along with their corresponding log (unnormalized) posterior probabilities.

When X is a list with two matrices, say, W and Z, the above method is extended to ncol(W)+ncol(Z) dimensional regression. However, the hyperparameters lam and w are chosen separately for the two matrices, the default values being nrow(W)/ncol(W)^2 and nrow(Z)/ncol(Z)^2 for lam and sqrt(nrow(W))/ncol(W) and sqrt(nrow(Z))/ncol(Z) for w.

The marginal inclusion probabities can be extracted by using the function mip.

Value

A list with components

model.map	A vector of indices corresponding to the selected variables in the MAP model.
model.wam	A vector of indices corresponding to the selected variables in the WAM.
model.top	A sparse matrix storing the top models.
beta.map	The ridge estimator of regression coefficients in the MAP model.
beta.wam	The ridge estimator of regression coefficients in the WAM.
mip.map	The marginal inclusion probabilities of the variables in the MAP model.
mip.wam	The marginal inclusion probabilities of the variables in the WAM.
pprob.map	The log (unnormalized) posterior probability corresponding to the MAP model.
pprob.top	A vector of the log (unnormalized) posterior probabilities corresponding to the top models.
stats	Additional statistics.

Author(s)

Dongjin Li, Debarshi Chakraborty, and Somak Dutta
Maintainer: Dongjin Li <[email protected]>

References

Li, D., Dutta, S., and Roy, V. (2023). Model based screening embedded Bayesian variable selection for ultra-high dimensional settings. Journal of Computational and Graphical Statistics, 32(1), 61-73.

See Also

[mip.sven()] for marginal inclusion probabilities, [predict.sven()](via [predict()]) for prediction for .

Examples

n <- 50; p <- 100; nonzero <- 3
trueidx <- 1:3
truebeta <- c(4,5,6)
X <- matrix(rnorm(n*p), n, p) # n x p covariate matrix
y <- 0.5 + X[,trueidx] %*% truebeta + rnorm(n)
res <- sven(X=X, y=y)
res$model.map # the MAP model


Z <- matrix(rnorm(n*p), n, p) # another covariate matrix
y2 = 0.5 + X[,trueidx] %*% truebeta  + Z[,1:2] %*% c(-2,-2) + rnorm(n)
res2 <- sven(X=list(X,Z), y=y2)

---