update.hetGP {hetGP}

Update "hetGP"-class model fit with new observations

Description

Fast update of existing hetGP model with new observations.

Usage

## S3 method for class 'hetGP'
update(
  object,
  Xnew,
  Znew,
  ginit = 0.01,
  lower = NULL,
  upper = NULL,
  noiseControl = NULL,
  settings = NULL,
  known = NULL,
  maxit = 100,
  method = "quick",
  ...
)

Arguments

object

previously fit "hetGP"-class model

Xnew

matrix of new design locations; ncol(Xnew) must match the input dimension encoded in object

Znew

vector new observations at those design locations, of length nrow(X). NAs can be passed, see Details

ginit

minimal value of the smoothing parameter (i.e., nugget of the noise process) for optimization initialization. It is compared to the g hyperparameter in the object.

lower, upper, noiseControl, settings, known

optional bounds for mle optimization, see mleHetGP. If not provided, they are extracted from the existing model

maxit

maximum number of iterations for the internal L-BFGS-B optimization method; see optim for more details

method

one of "quick", "mixed" see Details.

...

no other argument for this method.

Details

The update can be performed with or without re-estimating hyperparameter. In the first case, mleHetGP is called, based on previous values for initialization. The only missing values are the latent variables at the new points, that are initialized based on two possible update schemes in method:

• "quick" the new delta value is the predicted nugs value from the previous noise model;

• "mixed" new values are taken as the barycenter between prediction given by the noise process and empirical variance.

The subsequent number of MLE computations can be controlled with maxit.

In case hyperparameters need not be updated, maxit can be set to 0. In this case it is possible to pass NAs in Znew, then the model can still be used to provide updated variance predictions.

Examples

##------------------------------------------------------------
## Sequential update example
##------------------------------------------------------------
set.seed(42)

## Spatially varying noise function
noisefun <- function(x, coef = 1){
  return(coef * (0.05 + sqrt(abs(x)*20/(2*pi))/10))
}

## Initial data set
nvar <- 1
n <- 20
X <- matrix(seq(0, 2 * pi, length=n), ncol = 1)
mult <- sample(1:10, n, replace = TRUE)
X <- rep(X, mult)
Z <- sin(X) + rnorm(length(X), sd = noisefun(X))

## Initial fit
testpts <- matrix(seq(0, 2*pi, length = 10*n), ncol = 1)
model <- model_init <- mleHetGP(X = X, Z = Z, lower = rep(0.1, nvar), 
  upper = rep(50, nvar), maxit = 1000)

## Visualizing initial predictive surface
preds <- predict(x = testpts, model_init) 
plot(X, Z)
lines(testpts, preds$mean, col = "red")

## 10 fast update steps
nsteps <- 5
npersteps <- 10
for(i in 1:nsteps){
  newIds <- sort(sample(1:(10*n), npersteps))
  
  newX <- testpts[newIds, drop = FALSE] 
  newZ <- sin(newX) + rnorm(length(newX), sd = noisefun(newX))
  points(newX, newZ, col = "blue", pch = 20)
  model <- update(object = model, Xnew = newX, Znew = newZ)
  X <- c(X, newX)
  Z <- c(Z, newZ)
  plot(X, Z)
  print(model$nit_opt)
}

## Final predictions after 10 updates
p_fin <- predict(x=testpts, model) 

## Visualizing the result by augmenting earlier plot
lines(testpts, p_fin$mean, col = "blue")
lines(testpts, qnorm(0.05, p_fin$mean, sqrt(p_fin$sd2)), col = "blue", lty = 2)
lines(testpts, qnorm(0.95, p_fin$mean, sqrt(p_fin$sd2)), col = "blue", lty = 2)
lines(testpts, qnorm(0.05, p_fin$mean, sqrt(p_fin$sd2 + p_fin$nugs)), 
  col = "blue", lty = 3)
lines(testpts, qnorm(0.95, p_fin$mean, sqrt(p_fin$sd2 + p_fin$nugs)), 
  col = "blue", lty = 3)

## Now compare to what you would get if you did a full batch fit instead
model_direct <-  mleHetGP(X = X, Z = Z, maxit = 1000,
                          lower = rep(0.1, nvar), upper = rep(50, nvar),
                          init = list(theta = model_init$theta, k_theta_g = model_init$k_theta_g))
p_dir <- predict(x = testpts, model_direct)
print(model_direct$nit_opt)
lines(testpts, p_dir$mean, col = "green")
lines(testpts, qnorm(0.05, p_dir$mean, sqrt(p_dir$sd2)), col = "green", 
  lty = 2)
lines(testpts, qnorm(0.95, p_dir$mean, sqrt(p_dir$sd2)), col = "green", 
  lty = 2)
lines(testpts, qnorm(0.05, p_dir$mean, sqrt(p_dir$sd2 + p_dir$nugs)), 
  col = "green", lty = 3)
lines(testpts, qnorm(0.95, p_dir$mean, sqrt(p_dir$sd2 + p_dir$nugs)), 
  col = "green", lty = 3)
lines(testpts, sin(testpts), col = "red", lty = 2)

## Compare outputs
summary(model_init)
summary(model)
summary(model_direct)