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AllFunctions.R
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AllFunctions.R
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### This file contains functions to
### - fit the GP regression model (fitGPR),
### - compute nearest neighbour distances (Dnear),
### - make predictions from a GPR model (predictGPR),
### - fit the forward model (fitFWD)
### - make predictions from the forward model (predictFWD)
### - extract full posterior predictive distributions (posteriorFWD)
### You will need to install Python and also the GPy library for Python
### (run '!pip install GPy' in Python's command line)
### both are free to install. This code is based on Python 3.6.
### point to your Python distribution
Sys.setenv(RETICULATE_PYTHON = YOUR_PYTHON_PATH)
###
require(reticulate)
GPy <- import("GPy")
np <- import("numpy")
### import weights and nodes for 500-point Gauss-Hermite quadrature
weightsAndNodes <- read.csv("ghWeightsNodes.csv")[,2:3]
### This is a simple wrapper around the GPy functions for fitting GP regression models
### It takes the modern data and returns an environment containing information about the
### fitted GP regression model
### load_previous allows one to load the already-optimised GP model object (so is faster)
fitGPR <- function(modern.data,modern.temperatures, load_existing = TRUE){
if(load_existing){
mb <- np$load("mb.npy")
} else{
KK <- GPy$kern$RBF(input_dim = ncol(modern.data),ARD = TRUE)
mb <- GPy$models$GPRegression(as.matrix(modern.data),as.matrix(modern.temperatures),KK)
mb$optimize()
}
return(mb)
}
### This function computes nearest neighbour distances (weighted by the lengthscales of
### the kernel of a fitted GPR object (obtained via fitGPR)
Dnear <- function(newX,model){
if(ncol(newX) != ncol(model$X)){
stop("newX has the wrong number of columns")
}
K <- model$kern$K(as.matrix(model$X),as.matrix(newX))
dists <- -log(K / as.numeric(model$kern$variance))
return(apply(dists,2,min))
}
### This function predicts mean temperatures and standard deviations of predictions
### for the points in newX given the model object (obtained via fitGPR)
predictGPR <- function(newX,model){
if(ncol(newX) != ncol(model$X)){
stop("newX has the wrong number of columns")
}
pred <- mb$predict(newX)
list(means = pred[[1]],sds = sqrt(pred[[2]]))
}
### This function fits the forward model. It takes the modern data and returns a model
### object for use with other functions. Specifying load_existing will load a previously
### trained model object:
### - load_existing = 1 loads an MOGP model based on GDGTs 0-3
### - load_existing = 2 loads an MOGP model based on GDGTs 0-5
### - load_existing = NULL fits the model using GPy (can take some time)
fitFWD <- function(modern.data,modern.temp,load_existing = c(1,2,NULL)){
if(load_existing == 1){
message('Loading MOGP model object based on GDGTs 0-3')
mf <- np$load('mf4.npy')
} else if(load_existing == 2){
message('Loading MOGP model object based on all 6 GDGTs')
mf <- np$load('mf6.npy')
} else{
message("Loading required package robCompositions")
require(robCompositions)
message("Imputing zeros")
modern.data[modern.data == 0] <- NA
modern.data <- impCoda(modern.data)$xImp
message("ilr transforming the data")
modern.data.ilr <- pivotCoord(modern.data)
message("Setting up Multi-Output GP model")
KK <- GPy$kern$Matern32(1)
icm <- GPy$util$multioutput$ICM(input_dim = 1,
num_outputs = ncol(modern.data.ilr),
kernel = KK)
temp.list <- lapply(1:ncol(modern.data.ilr),function(j) as.matrix(modern.temp))
ilr.list <- lapply(1:ncol(modern.data.ilr),function(j) as.matrix(modern.data.ilr[,j]))
mf <- GPy$models$GPCoregionalizedRegression(temp.list,ilr.list,kernel = icm)
### horrible hacky way to fix kernel variance parameter
mf$constraints$add('fixed',c(0L,1L,2L))
mf$constraints$remove('fixed',c(1L,2L))
message("Optimising hyperparameters; this might take some time (tens of minutes) depending on your machine")
mf$optimize()
}
return(mf)
}
### make predictions from the forward model
### inputs: newX :- a matrix or data.frame of GDGT values
### model :- a model object obtained via fitFWD()
### prior :- a 2-vector containing the mean and sd of the Gaussian prior on
### temperature. Defaults to (15,10)
### PofXgivenT :- a list containing means, invcovs, dets of p(X|T_j) for each
### Gauss-Hermite node T_j
### returnFullPosterior :- one of:
### - FALSE (default): only return means and variances
### - A vector of indices for which full posterior should be computed
### - TRUE: return full posterior for every new point
predictFWD <- function(newX,
model,
prior = c(15,10),
PofXgivenT = NULL,
returnFullPosterior = FALSE,
transformed = F){
dd <- max(model$Y_metadata[[1]]) + 1
npred <- nrow(newX)
if(ncol(newX) != (dd + 1)){
stop("newX has the wrong number of columns")
}
if(returnFullPosterior){
returnFullPosterior <- 1:npred
}
whichzerorows <- NULL
if(!transformed){
if(npred > 2*dd){
message("Loading required package robCompositions")
require(robCompositions)
message("Imputing zeros")
newX[newX == 0] <- NA
newX <- impCoda(newX)$xImp
} else{
message("Not enough data points to impute zeros; removing rows containing zeros")
whichzerorows <- which(apply(newX,1,function(x) any(x == 0)))
}
message("ilr transforming the data")
newX <- as.matrix(pivotCoord(newX))
}
## 500 node Gauss-Hermite quadrature (straightforward to use fastGHquad package to
## change this if desired)
n_nodes <- 500
xx <- sqrt(2) * prior[2] *weightsAndNodes$x + prior[1]
if(!is.null(returnFullPosterior)){
priorAtNodes <- dnorm(xx,prior[1],prior[2])
}
ww <- weightsAndNodes$w
if(is.null(PofXgivenT)){
warning("For speed on repeated runs, it is recommended to provide PofXgivenT,
which can be obtained via getPofXgivenT()")
inds <- as.integer(0:(dd-1))
noise_dict <- dict(list(output_index = matrix(inds,dd,1)))
message("Computing p(X|T) at each quadrature node...")
pb <- txtProgressBar(0,n_nodes)
means <- matrix(NA,n_nodes,dd)
invcovs <- array(NA,c(n_nodes,dd,dd))
dets <- rep(NA,n_nodes)
for(j in 1:n_nodes){
X <- rep(xx[j],5)
X <- cbind(X,inds)
tmpp <- model$predict(X,Y_metadata=noise_dict,full_cov = TRUE)
means[j,] <- tmpp[[1]]
cholInvCov <- chol(tmpp[[2]])
invtmp = chol2inv(cholInvCov)
invcovs[j,,] = invtmp
dets[j] = prod(diag(cholInvCov)^2)
setTxtProgressBar(pb,j)
}
message("DONE")
} else{
means = PofXgivenT$means
invcovs = PofXgivenT$invcovs
dets = PofXgivenT$dets
}
posterior_means <- rep(NA,npred)
posterior_vars <- rep(NA,npred)
full_posteriors <- list()
Zout <- rep(NA,npred)
message("Computing p(T|X) for new data...")
pb <- txtProgressBar(0,npred)
for(i in which(!((1:npred)%in%whichzerorows))){
ff <- rep(NA,n_nodes)
xi <- newX[i,]
for (j in 1:n_nodes){
### evaluate multivariate Gaussian density at i-th ######################
### composition, at j-th temperature node, p(x_i|T_j) ####################
##########################################################################
qf <- t(xi - means[j,])%*%invcovs[j,,]%*%(xi - means[j,]) ##############
ff[j] <- exp(-0.5 * qf) / sqrt((2 * pi)^dd * dets[j]) ##############
##########################################################################
}
## compute normalising factor, int p(t) dt, by Gauss-Hermite quadrature
Z <- t(ww)%*%ff
mu <- t(ww)%*%(ff*xx) / Z ## Gauss-Hermite quadrature again
posterior_means[i] <- mu
posterior_vars[i] <- t(ww)%*%(ff * (xx - rep(mu,n_nodes))^2) / Z
Zout[i] <- Z
if(i %in% returnFullPosterior){
full_posteriors[[i]] <- data.frame(xx = xx,posterior = (ff * priorAtNodes) / rep(Z,n_nodes))
}
setTxtProgressBar(pb,i)
}
message("DONE")
if(!is.null(whichzerorows)){
message(paste("Predictions not made for points",whichzerorows,
"because they contained zero entries"))
}
return(list(mean = posterior_means,
variance = posterior_vars,
full_posteriors = full_posteriors,
Z = Zout,
transformedData = newX))
}
#### Function to obtain densities p(X|T) at the quadrature nodes
getPofXgivenT <- function(model){
dd <- max(model$Y_metadata[[1]]) + 1
## 500 node Gauss-Hermite quadrature (straightforward to use fastGHquad package to
## change this if desired)
n_nodes <- 500
xx <- sqrt(2) * prior[2] *weightsAndNodes$x + prior[1]
ww <- weightsAndNodes$w
inds <- as.integer(0:(dd-1))
noise_dict <- dict(list(output_index = matrix(inds,dd,1)))
message("Computing p(X|T) at each quadrature node...")
pb <- txtProgressBar(0,n_nodes)
means <- matrix(NA,n_nodes,dd)
invcovs <- array(NA,c(n_nodes,dd,dd))
dets <- rep(NA,n_nodes)
for(j in 1:n_nodes){
X <- rep(xx[j],5)
X <- cbind(X,inds)
tmpp <- model$predict(X,Y_metadata=noise_dict,full_cov = TRUE)
means[j,] <- tmpp[[1]]
cholInvCov <- chol(tmpp[[2]])
invtmp = chol2inv(cholInvCov)
invcovs[j,,] = invtmp
dets[j] = prod(diag(cholInvCov)^2)
setTxtProgressBar(pb,j)
}
return(list(means = means,invcovs = invcovs,dets = dets))
}
#### Compute the (unnormalised) posterior predictive density at the specified
#### points for the data points in newX.
#### Z is a vector of normalising constants (which can be obtained via predictFWD()).
#### transformed is a logical input indicating whether newX contains transformed data.
posteriorFWD <- function(newX,model,points = seq(-10,60,len = 200),
prior = c(15,10),Z = NULL, transformed = FALSE){
dd <- max(model$Y_metadata[[1]]) + 1
npoints <- length(points)
npred <- nrow(newX)
priorAtPoints <- dnorm(points,prior[1],prior[2])
inds <- as.integer(0:(dd-1))
noise_dict <- dict(list(output_index = matrix(inds,dd,1)))
whichzerorows <- NULL
if(!transformed){
if(npred > 2*dd){
message("Loading required package robCompositions")
require(robCompositions)
message("Imputing zeros")
newX[newX == 0] <- NA
newX <- impCoda(newX)$xImp
} else{
whichzerorows <- which(apply(newX,1,function(x) any(x == 0)))
}
message("ilr transforming the data")
newX <- as.matrix(pivotCoord(newX))
}
means <- matrix(NA,npoints,dd)
invcovs <- array(NA,c(npoints,dd,dd))
dets <- rep(NA,npoints)
for(j in 1:npoints){
X <- rep(points[j],5)
X <- cbind(X,inds)
tmpp <- model$predict(X,Y_metadata=noise_dict,full_cov = TRUE)
means[j,] <- tmpp[[1]]
cholInvCov <- chol(tmpp[[2]])
invtmp = chol2inv(cholInvCov)
invcovs[j,,] = invtmp
dets[j] = prod(diag(cholInvCov)^2)
}
PPD <- matrix(NA,npoints,npred)
for(i in which(!((1:npred)%in%whichzerorows))){
ff <- rep(NA,npoints)
xi <- newX[i,]
for (j in 1:npoints){
### evaluate multivariate Gaussian density at i-th ######################
### composition, at j-th temperature node, p(x_i|T_j) ####################
##########################################################################
qf <- t(xi - means[j,])%*%invcovs[j,,]%*%(xi - means[j,]) ##############
ff[j] <- exp(-0.5 * qf) / sqrt((2 * pi)^dd * dets[j]) ##############
##########################################################################
}
PPD[i,] <- ff*priorAtPoints / (ifelse(!is.null(Z),Z[i],1))
}
if(!is.null(whichzerorows)){
message(paste("Predictions not made for points",whichzerorows,
"because they contained zero entries"))
}
return(PPD)
}