binaryGP_fit: Binary Gaussian Process (with/without time-series)

Description Usage Arguments Details Value Author(s) See Also Examples

View source: R/binaryGP_fit.R

Description

The function fits Gaussian process models with binary response. The models can also include time-series term if a time-series binary response is observed. The estimation methods are based on PQL/PQPL and REML (for mean function and correlation parameter, respectively).

Usage

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binaryGP_fit(X, Y, R = 0, L = 0, corr = list(type = "exponential", power =
  2), nugget = 1e-10, constantMean = FALSE, orthogonalGP = FALSE,
  converge.tol = 1e-10, maxit = 15, maxit.PQPL = 20, maxit.REML = 100,
  xtol_rel = 1e-10, standardize = FALSE, verbose = TRUE)

Arguments

X

a design matrix with dimension n by d.

Y

a response matrix with dimension n by T. The values in the matrix are 0 or 1. If no time-series, T = 1. If time-series is included, i.e., T > 1, the first column is the response vector of time 1, and second column is the response vector of time 2, and so on.

R

a positive integer specifying the order of autoregression only if time-series is included. The default is 1.

L

a positive integer specifying the order of interaction between X and previous Y only if time-series is included. The value couldn't nbe larger than R. The default is 1.

corr

a list of parameters for the specifing the correlation to be used. Either exponential correlation function or Matern correlation function can be used. See corr_matrix for details.

nugget

a positive value to use for the nugget. If NULL, a nugget will be estimated. The default is 1e-10.

constantMean

logical. TRUE indicates that the Gaussian process will have a constant mean, plus time-series structure if R or T is greater than one; otherwise the mean function will be a linear regression in X, plus an intercept term and time-series structure.

orthogonalGP

logical. TRUE indicates that the orthogonal Gaussian process will be used. Only available when corr is list(type = "exponential", power = 2).

converge.tol

convergence tolerance. It converges when relative difference with respect to η_t is smaller than the tolerance. The default is 1e-10.

maxit

a positive integer specifying the maximum number of iterations for estimation to be performed before the estimates are convergent. The default is 15.

maxit.PQPL

a positive integer specifying the maximum number of iterations for PQL/PQPL estimation to be performed before the estimates are convergent. The default is 20.

maxit.REML

a positive integer specifying the maximum number of iterations in lbfgs for REML estimation to be performed before the estimates are convergent. The default is 100.

xtol_rel

a postive value specifying the convergence tolerance for lbfgs. The default is 1e-10.

standardize

logical. If TRUE, each column of X will be standardized into [0,1]. The default is FALSE.

verbose

logical. If TRUE, additional diagnostics are printed. The default is TRUE.

Details

Consider the model

logit(p_t(x))=η_t(x)=∑^R_{r=1}\varphi_ry_{t-r}(\mathbf{x})+α_0+\mathbf{x}'\boldsymbol{α}+∑^L_{l=1}\boldsymbol{γ}_l\textbf{x}y_{t-l}(\mathbf{x})+Z_t(\mathbf{x}),

where p_t(x)=Pr(y_t(x)=1) and Z_t(\cdot) is a Gaussian process with zero mean, variance σ^2, and correlation function R_{\boldsymbol{θ}}(\cdot,\cdot) with unknown correlation parameters \boldsymbol{θ}. The power exponential correlation function is used and defined by

R_{\boldsymbol{θ}}(\mathbf{x}_i,\mathbf{x}_j)=\exp\{-∑^{d}_{l=1}\frac{(x_{il}-x_{jl})^p}{θ_l} \},

where p is given by power. The parameters in the mean functions including \varphi_r,α_0,\boldsymbol{α},\boldsymbol{γ}_l are estimated by PQL/PQPL, depending on whether the mean function has the time-series structure. The parameters in the Gaussian process including \boldsymbol{θ} and σ^2 are estimated by REML. If orthogonalGP is TRUE, then orthogonal covariance function (Plumlee and Joseph, 2016) is employed. More details can be seen in Sung et al. (unpublished).

Value

alpha_hat

a vector of coefficient estimates for the linear relationship with X.

varphi_hat

a vector of autoregression coefficient estimates.

gamma_hat

a vector of the interaction effect estimates.

theta_hat

a vector of correlation parameters.

sigma_hat

a value of sigma estimate (standard deviation).

nugget_hat

if nugget is NULL, then return a nugget estimate, otherwise return nugget.

orthogonalGP

orthogonalGP.

X

data X.

Y

data Y.

R

order of autoregression.

L

order of interaction between X and previous Y.

corr

a list of parameters for the specifing the correlation to be used.

Model.mat

model matrix.

eta_hat

eta_hat.

standardize

standardize.

mean.x

mean of each column of X. Only available when standardize=TRUE, otherwise mean.x=NULL.

scale.x

max(x)-min(x) of each column of X. Only available when standardize=TRUE, otherwise scale.x=NULL.

Author(s)

Chih-Li Sung <[email protected]>

See Also

predict.binaryGP for prediction of the binary Gaussian process, print.binaryGP for the list of estimation results, and summary.binaryGP for summary of significance results.

Examples

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library(binaryGP)

#####      Testing function: cos(x1 + x2) * exp(x1*x2) with TT sequences      #####
#####   Thanks to Sonja Surjanovic and Derek Bingham, Simon Fraser University #####
test_function <- function(X, TT)
{
  x1 <- X[,1]
  x2 <- X[,2]

  eta_1 <- cos(x1 + x2) * exp(x1*x2)

  p_1 <- exp(eta_1)/(1+exp(eta_1))
  y_1 <- rep(NA, length(p_1))
  for(i in 1:length(p_1)) y_1[i] <- rbinom(1,1,p_1[i])
  Y <- y_1
  P <- p_1
  if(TT > 1){
    for(tt in 2:TT){
      eta_2 <- 0.3 * y_1 + eta_1
      p_2 <- exp(eta_2)/(1+exp(eta_2))
      y_2 <- rep(NA, length(p_2))
      for(i in 1:length(p_2)) y_2[i] <- rbinom(1,1,p_2[i])
      Y <- cbind(Y, y_2)
      P <- cbind(P, p_2)
      y_1 <- y_2
    }
  }

  return(list(Y = Y, P = P))
}

set.seed(1)
n <- 30
n.test <- 10
d <- 2
X <- matrix(runif(d * n), ncol = d)

##### without time-series #####
Y <- test_function(X, 1)$Y  ## Y is a vector

binaryGP.model <- binaryGP_fit(X = X, Y = Y)
print(binaryGP.model)   # print estimation results
summary(binaryGP.model) # significance results

##### with time-series, lag 1 #####
Y <- test_function(X, 10)$Y  ## Y is a matrix with 10 columns

binaryGP.model <- binaryGP_fit(X = X, Y = Y, R = 1)
print(binaryGP.model)   # print estimation results
summary(binaryGP.model) # significance results

binaryGP documentation built on Sept. 19, 2017, 9:02 a.m.