NoiseKriging::predict
Description
Predict from a NoiseKriging
Model Object
Usage
Python
# k = NoiseKriging(...) k.predict(x, stdev = True, cov = False, deriv = False)
R
# k = NoiseKriging(...) k$predict(x, stdev = TRUE, cov = FALSE, deriv = FALSE)
Matlab/Octave
% k = NoiseKriging(...) k.predict(x, stdev = true, cov = false, deriv = false)
Arguments
Argument 
Description 


Input points where the prediction must be computed. 






Details
Given \(n^\star\) “new” input points \(\mathbf{x}^\star_{j}\), the method
compute the expectation, the standard deviation and (optionally) the
covariance of the estimated values of the “trend \(+\) GP” stochastic
process \(\mu(\mathbf{x}_j^\star) + \zeta(\mathbf{x}_j^\star)\) at the
“new” observations. The estimation is based on the distribution
conditional on the \(n\) noisy observations \(y_i\) made at the input
points \(\mathbf{x}_i\) as used when fitting the model. The \(n^\star\)
input vectors (with length \(d\)) are given as the rows of a
\(\mathbf{X}^\star\) corresponding to x
.
The computation of these quantities is often called Universal Kriging see here for more details.
Value
A list containing the element mean
and possibly stdev
and
cov
.
The expectation in
mean
is the estimate of the vector \(\textsf{E}[\boldsymbol{\mu}^\star + \boldsymbol{\zeta}^\star \, \vert \,\mathbf{y}]\) with length \(n^\star\) where \(\boldsymbol{\mu}^\star\) and \(\boldsymbol{\zeta}^\star\) are for “new” points and \(\mathbf{y}\) corresponds to the observations. Similarly the conditional standard deviation instdev
is a vector with length \(n^\star\) and the conditional covariance incov
is a \(n^\star \times n^\star\) matrix.The (optional) derivatives are two \(n^\star \times d\) matrices
pred_mean_deriv
andpred_sdtdev_deriv
with their row \(j\) containing the vector of derivatives w.r.t. to the new input point \(\mathbf{x}^\star\) evaluated at \(\mathbf{x}^\star = \mathbf{x}^\star_j\). So the row \(j\) ofpred_mean_deriv
contains the derivative \(\partial_{\mathbf{x}^\star} \mathbb{E}[y(\mathbf{x}^\star) \, \vert \,\mathbf{y}]\). evaluated at \(\mathbf{x}^\star = \mathbf{x}^\star_j\).
Note that for a NoiseKriging
object the prediction is actually a
smoothing. The socalled Kriging mean function \(\mathbf{x}^\star
\mapsto \mathbb{E}[y(\mathbf{x}^\star) \, \vert \, \mathbf{y}]\) is a
smooth function. Depending on the given noise variances \(\sigma^2_i\)
given in the fit step, the prediction at
\(\mathbf{x}^\star \approx \mathbf{x}_i\) will be more or less close to
the observed value \(y_i\). As opposed to the
NuggetKriging
model case, duplicated inputs can be
used in the design.
Examples
f < function(x) 1  1 / 2 * (sin(12 * x) / (1 + x) + 2 * cos(7 * x) * x^5 + 0.7)
plot(f)
set.seed(123)
X < as.matrix(runif(10))
y < f(X) + X/10 * rnorm(nrow(X))
points(X, y, col = "blue", pch = 16)
k < NoiseKriging(y, (X/10)^2, X, "matern3_2")
x <seq(from = 0, to = 1, length.out = 101)
p < k$predict(x)
lines(x, p$mean, col = "blue")
polygon(c(x, rev(x)), c(p$mean  2 * p$stdev, rev(p$mean + 2 * p$stdev)),
border = NA, col = rgb(0, 0, 1, 0.2))
Results