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 ( |

`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 ( |

`alpha_l` |
Optional. The lower bound for all correlation parameters in the local GP ( |

`kappa_u` |
Optional. The upper bound for |

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 |

`theta` |
Vector of estimated correlation parameters |

`alpha` |
Vector of estimated correlation parameters |

`bandwidth` |
Estimated bandwidth parameter |

`rmscv` |
Root mean squared (leave-one-out) cross validation error |

`Yp_jackknife` |
Vector of Jackknife (leave-one-out) predicted values |

`mu` |
Estimated mean value |

`tau2` |
Estimated variance |

`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 |

`temp_matrix` |
Matrix of |

Shan Ba <[email protected]> and V. Roshan Joseph <[email protected]>

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)
``` |

CGP documentation built on June 12, 2018, 5:19 p.m.

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