# The Fitting Function of `GPM` Package Fits a Gaussian process (GP) to a set of simulation data as described in `reference 1`. Both the inputs and outputs can be multi-dimensional. The outputs can be noisy in which case it is assumed that the noise is stationary (i.e., its variance is not a function of x). ```r Fit(X, Y, CorrType = 'G', Eps = 10^(seq(-1, -12)), AnaGr = NULL, Nopt = 5,TraceIt = 0, MaxIter = 100, Seed = 1, LowerBound = NULL, UpperBound = NULL, StopFlag = 1, Progress = 0, DoParallel = 0, Ncores = NULL) ``` ## Arguments - `X`: Matrix containing the training (aka design or input) data points. The rows and columns of `X` denote individual observation settings and input dimension, respectively. - `Y`: Matrix containing the output (aka response) data points. The rows and columns of `Y` denote individual observation responses and output dimension, respectively. - `CorrType`: The type of the correlation function of the GP model. Choices include `'G'` (default), `'PE'`, `'LBG'`, and `'LB'`. See the `references` for the details. For smooth (or analytic) functions, choose either `'G'` or `'LBG'`. Fitting is faster if `'G'` is chosen. - `Eps`: A vector containing the smallest eigen value(s) that the correlation matrix is allowed to have. The elements of Eps must be in [0, 1] and sorted in a descending order. - `AnaGr`: Flag indicating whether the gradient of the log-likelihood should be taken analytically (`!= 0`) or numerically (`== 0`). For now, only available when `CorrType == 'G'` or `CorrType == 'PE'`. If `AnaGr != 0`, the fitted model will generally be more accurate. - `Nopt`: The number of times the log-likelihood function is optimized when `Eps[1]` is used to constraint the smallest eigen value that the correlation matrix is allowed to have. Higher `Nopt` will increase fitting time as well as the chances of finding the global optimum. If `nrow(X)` is large (i.e., large training datasets), `Nopt` can be small.Analyzing the optimization results for `Eps[1]` and when `Progress != 0` will determine if `Nopt` has been large enough. - `TraceIt`: Non-negative integer. If positive, tracing information on the progress of the optimization is printed . There are six levels of tracing (see `optim`) and higher values will produce more tracing information. - `MaxIter`: Maximum number of iterations allowed for each optimization (see `optim`). - `Seed`: An integer for the random number generator. Use this to make the results reproducible. - `LowerBound, UpperBound`: To estimate the scale (aka roughness) parameters of the correlation function, the feasible range should be defined. `LowerBound` and `UpperBound` are vectors determining, resepectively, the lower and upper bounds. Their length depends on the parametric form of the correlation function (see `reference 1` for the details). - `StopFlag`: Flag indicating whether the optimization must be stopped if the negative log-likelihood increases with decreasing `Eps[i]`. - `Progress`: Flag indicating if the fitting process should be summarized. Set it to `!=0` to turn it on. - `DoParallel`: If `!= 0`, optimizations will be done in parallel. - `Ncores`: Number of cores to use if `DoParallel != 0`. The default is the maximum number of physical cores. ## Returns Model A list containing the following components: * `CovFunc` A list containing the type and estimated parameters of the correlation function. * `Data` A list storing the original (but scaled) data. * `Details` A list of some parameters (used in prediction) as well as some values reporting the total run-time (`cost`) and the added nugget (`Nug_opt`) for satisfying the constraint on the smallest eigen value of the correlation matrix. * `OptimHist` The optimization history. * `Setting` The default/provided settings for running the code. ## Examples ```r # 1D example: Fit a model (with default settings) and evaluate the performance # by computing the root mean squared error (RMSE) in prediction. library(lhs) X <- 5*maximinLHS(15, 1) Y <- 2*sin(2*X) + log(X+1) M <- Fit(X, Y) XF <- matrix(seq(0, 5, length.out = 100), 100, 1) YF <- Predict(XF, M) RMSE <- sqrt(mean((YF$YF - (2*sin(2*XF) + log(XF+1)))^2)) ## Not run: # 1D example: Fit a model, evaluate the performance, and plot the response # along with 95% prediction interval X <- 10*maximinLHS(10, 1) - 5 Y <- X*cos(X) M <- Fit(X, Y) XF <- matrix(seq(-5, 5, length.out = 500), 500, 1) YF <- Predict(XF, M) RMSE <- sqrt(mean((YF$YF - (XF*cos(XF)))^2)) Draw(M, 1, res = 20) # 2D example: Fit a model, evaluate the performance, and plot the response # surface along with 95% prediction interval X <- 2*maximinLHS(10, 2) - 1 Y <- X[, 1]^2 + X[, 2]^2 M <- Fit(X, Y, CorrType = "PE") XF <- 2*maximinLHS(100, 2) - 1 YF <- Predict(XF, M) RMSE <- sqrt(mean((YF$YF - (XF[, 1]^2 + XF[, 2]^2))^2)) library(lattice) Draw(M, c(1, 1), res = 15, PI95=1) # 2D example: Plot the previous model wrt X1 in the [-2, 2] # interval with X2=1 Draw(M, c(1, 0), LB = -2, UB = 2, res = 15, PI95=1) # 3D example: Compare the performance of Gaussian ("G") and lifted Browninan # with Gamma=1 ("LBG") X <- 2*maximinLHS(50, 3) - 1 Y <- cos(X[, 1]^2) + 2*sin(X[, 2]^2) + X[, 3]^2 M_G <- Fit(X, Y) M_LBG <- Fit(X, Y, CorrType = "LBG") XF <- 2*maximinLHS(500, 3) - 1 YF_G <- Predict(XF, M_G) YF_LBG <- Predict(XF, M_LBG) RMSE_G <- sqrt(mean((YF_G$YF - (cos(XF[, 1]^2) + 2*sin(XF[, 2]^2) + XF[, 3]^2))^2)) RMSE_LBG <- sqrt(mean((YF_LBG$YF - (cos(XF[, 1]^2) + 2*sin(XF[, 2]^2) + XF[, 3]^2))^2)) # 3D example: Draw the response in 2D using the M_G model when X3=0 Draw(M_G, c(1, 1, 0), PI95 = 0, Values = 0, X1Label = 'Input 1', X2Label = 'Input 2') # 3D example: 2D response X <- 2*maximinLHS(50, 3) - 1 Y <- cbind(cos(X[, 1]^2) + 2*sin(X[, 2]^2) + X[, 3]^2, rowSums(X)) M <- Fit(X, Y) Draw(M, c(0, 1, 1), Response_ID = 2, Values = 0.5) # 2D example with noise X <- 2*maximinLHS(100, 2) - 1 Y <- X[, 1]^2 + X[, 2]^2 + matrix(rnorm(nrow(X), 0, .5), nrow(X), 1) M <- Fit(X, Y) # Estimating the noise variance (should be close to 0.5^2) M$Details$Nug_opt*M$CovFunc$Parameters$Sigma2*M$Data$Yrange^2 ## End(Not run) ``` ## References 1. Bostanabad, R., Kearney, T., Tao, S., Apley, D. W. & Chen, W. (2018) Leveraging the nugget parameter for efficient Gaussian process modeling. Int J Numer Meth Eng, 114, 501-516. 2. Plumlee, M. & Apley, D. W. (2017) Lifted Brownian kriging models. Technometrics, 59, 165-177. ## See Also `optim` for the details on `L-BFGS-B` algorithm used in optimization. `Predict` to use the fitted GP model for prediction. `Draw` to plot the response via the fitted model.

Fit function