# SL2D: Squared L_{2} Discrepancy Approach for Estimating the... In DynamicGP: Modelling and Analysis of Dynamic Computer Experiments

## Description

This function fits an SVD-based GP model on the training dataset `design` and response matrix `resp`, and minimizes the squared L_{2} discrepancy between the target response and the predicted mean of the SVD-based GP model on the test set `candidate` to estimate the solution to the inverse problem. It is a naive approach for estimating the solution provided in Chapter 4 of Zhang (2018).

## Usage

 ```1 2 3``` ```SL2D(design,resp,yobs,candidate,frac=.95,nstarts=5, mtype=c("zmean","cmean","lmean"), gstart=0.0001) ```

## Arguments

 `design` An N by d matrix of N training/design inputs. `resp` An L by N response matrix of `design`, where L is the length of the time series outputs, N is the number of design points. `yobs` A vector of length L of the time-series valued field observations or the target response. `candidate` An M by d matrix of M candidate points on which the estimated solution to the inverse problem is extracted. `frac` The threshold in the cumulative percentage criterion to select the number of SVD bases. The default value is 0.95. `nstarts` The number of starting points used in the numerical maximization of the posterior density function. The larger `nstarts` will typically lead to more accurate prediction but longer computational time. The default value is 5. `mtype` The type of mean functions for the GP models. The choice "zmean" denotes zero-mean, "cmean" indicates constant-mean, "lmean" indicates linear-mean. The default choice is "zmean". `gstart` The starting number and upper bound for estimating the nugget parameter. If `gstart = sqrt(.Machine\$double.eps)`, the nugget parameter will be fixed at `sqrt(.Machine\$double.eps)`, since `sqrt(.Machine\$double.eps)` is the lower bound of the nugget term. The default value is 0.0001.

## Value

 `xhat` The estimated solution to the inverse problem obtained from the candidate set `candidate`.

## Author(s)

Ru Zhang heavenmarshal@gmail.com,

C. Devon Lin devon.lin@queensu.ca,

Pritam Ranjan pritamr@iimidr.ac.in

## References

Zhang, R. (2018) Modeling and Analysis of Dynamic Computer Experiments, PhD thesis, Queen's University, ON, Canada.

`ESL2D`, `saEI`, `svdGP`.
 ``` 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``` ``` library("lhs") forretal <- function(x,t,shift=1) { par1 <- x*6+4 par2 <- x*16+4 par3 <- x*6+1 t <- t+shift y <- (par1*t-2)^2*sin(par2*t-par3) } timepoints <- seq(0,1,len=200) design <- lhs::randomLHS(30,3) candidate <- lhs::randomLHS(500,3) candidate <- rbind(candidate,design) ## evaluate the response matrix on the design matrix resp <- apply(design,1,forretal,timepoints) x0 <- runif(3) y0 <- forretal(x0,timepoints) yobs <- y0+rnorm(200,0,sd(y0)/sqrt(50)) xhat <- SL2D(design,resp,yobs,candidate,nstarts=1) yhat <- forretal(xhat,timepoints) ## draw a figure to illustrate plot(y0,ylim=c(min(y0,yhat),max(y0,yhat))) lines(yhat,col="red") ```