Description Usage Arguments Details Value Author(s) References See Also Examples
Estimate parameters in the composite Gaussian process (CGP) model using maximum likelihood methods. Calculate the root mean squared (leave-one-out) cross validation error for diagnosis, and export intermediate values to facilitate predict.CGP
function.
1 2 |
X |
The design matrix |
yobs |
The vector of response values, corresponding to the rows of |
nugget_l |
Optional, default is “0.001”. The lower bound for the nugget value (λ in the paper) |
num_starts |
Optional, default is “5”. Number of random starts for optimizing the likelihood function |
theta_l |
Optional, default is “0.0001”. The lower bound for all correlation parameters in the global GP (θ in the paper) |
alpha_l |
Optional. The lower bound for all correlation parameters in the local GP (α in the paper). It is also the upper bound for all correlation parameters in the global GP (the θ).
Default is |
kappa_u |
Optional. The upper bound for κ, where we define α_j=θ_j+κ for j=1,…,p. Default value is |
This function fits a composite Gaussian process (CGP) model based on the given design matrix X
and the observed responses yobs
. The fitted model consists of a smooth GP to caputre the global trend and a local GP which is augmented with a flexible variance model to capture the change of local volatilities. For p input variables, such two GPs involve 2p+2 unknown parameters in total. As demonstrated in Ba and Joseph (2012), by assuming α_j=θ_j+κ for j=1,…,p, fitting the CGP model only requires estimating p+3 unknown parameters, which is comparable to fitting a stationary GP model (p unknown parameters).
This function fits the CGP model and returns an object of class “CGP
”. Function predict.CGP
can be further used for making new predictions and function summary.CGP
can be used to print a summary of the “CGP
” object.
An object of class “CGP
” is a list containing at least the following components:
lambda |
Estimated nugget value (λ) |
theta |
Vector of estimated correlation parameters (θ) in the global GP |
alpha |
Vector of estimated correlation parameters (α) in the local GP |
bandwidth |
Estimated bandwidth parameter (b) in the variance model |
rmscv |
Root mean squared (leave-one-out) cross validation error |
Yp_jackknife |
Vector of Jackknife (leave-one-out) predicted values |
mu |
Estimated mean value (μ) for global trend |
tau2 |
Estimated variance (τ^2) for global trend |
beststart |
Best starting value found for optimizing the log-likelihood |
objval |
Optimal objective value for the negative log-likelihood (up to a constant) |
var_names |
Vector of input variable names |
Sig_matrix |
Diagonal matrix containing standardized local variances at each of the design points |
sf |
Standardization constant for rescaling the local variance model |
res2 |
Vector of squared residuals from the estimated global trend |
invQ |
Matrix of (\mathbf{G}+λ\mathbf{Σ}^{1/2}\mathbf{L}\mathbf{Σ}^{1/2})^{-1} |
temp_matrix |
Matrix of (\mathbf{G}+λ\mathbf{Σ}^{1/2}\mathbf{L}\mathbf{Σ}^{1/2})^{-1} (\mathbf{y}- \hat{μ}\mathbf{1}) |
Shan Ba <shanbatr@gmail.com> and V. Roshan Joseph <roshan@isye.gatech.edu>
Ba, S. and V. Roshan Joseph (2012) “Composite Gaussian Process Models for Emulating Expensive Functions”. Annals of Applied Statistics, 6, 1838-1860.
predict.CGP
, print.CGP
, summary.CGP
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 | x1<-c(0,.02,.075,.08,.14,.15,.155,.156,.18,.22,.29,.32,.36,
.37,.42,.5,.57,.63,.72,.785,.8,.84,.925,1)
x2<-c(.29,.02,.12,.58,.38,.87,.01,.12,.22,.08,.34,.185,.64,
.02,.93,.15,.42,.71,1,0,.21,.5,.785,.21)
X<-cbind(x1,x2)
yobs<-sin(1/((x1*0.7+0.3)*(x2*0.7+0.3)))
## Not run:
#Fit the CGP model
#Increase the lower bound for nugget to 0.01 (Optional)
mod<-CGP(X,yobs,nugget_l=0.01)
summary(mod)
mod$objval
#-27.4537
mod$lambda
#0.6210284
mod$theta
#6.065497 8.093402
mod$alpha
#143.1770 145.2049
mod$bandwidth
#1
mod$rmscv
#0.5714969
## End(Not run)
|
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.