# Function for new locations given a model object The function `spPredict` collects posterior predictive samples for a set of new locations given a `spLM`, `spMvLM`, `spGLM`, `spMvGLM`, `spMisalignLM`, `spMisalignGLM`, `bayesGeostatExact`, `bayesLMConjugate` `bayesLMRef` or `spSVC` object. ```r spPredict(sp.obj, pred.coords, pred.covars, joint=FALSE, start=1, end, thin=1, verbose=TRUE, n.report=100, n.omp.threads=1, ...) ``` ## Arguments - `sp.obj`: an object returned by `spLM`, `spMvLM`, `spGLM`, `spMvGLM`, `spMisalignLM`, `spMisalignGLM`, `bayesGeostatExact`, `bayesLMConjugate` or `bayesLMRef`. For `spSVC`, `sp.obj` is an object from `spRecover`. - `pred.coords`: for `spLM`, `spMvLM`, `spGLM`, `spMvGLM`, and `bayesGeostatExact` `pred.coords` is a $n* x 2$ matrix of $n*$ prediction location coordinates in $R^2$ (e.g., easting and northing). For `spMisalignLM` and `spMisalignGLM` `pred.coords` is a list of $q$ $n*_i x 2$ matrices of prediction location coordinates where $i=(1,2,\ldots,q)$. For `spSVC` `pred.coords` is an $n* x m$ matrix of $n*$ prediction location coordinates in $R^m$. - `pred.covars`: for `spLM`, `spMvLM`, `spGLM`, `spMvGLM`, `bayesGeostatExact`, `bayesLMConjugate`, `bayesLMRef`, and `spSVC` `pred.covars` is a $n* x p$ design matrix associated with the new locations (including the intercept if one is specified in `sp.obj`'s formula argument). If this is a multivariate prediction defined by $q$ models, i.e., for `spMvLM` or `spMvGLM`, the multivariate design matrix can be created by passing a list of the $q$ univariate design matrices to the `mkMvX` function. For `spMisalignLM` and `spMisalignGLM` `pred.covars` is a list of $q$ $n*_i x p_i$ design matrices where $i=(1,2,\ldots,q)$ - `joint`: specifies whether posterior samples should be drawn from the joint or point-wise predictive distribution. This argument is only implemented for `spSVC`. Prediction for all other models uses the point-wise posterior predictive distribution. - `start`: specifies the first sample included in the composition sampling. - `end`: specifies the last sample included in the composition. The default is to use all posterior samples in `sp.obj`. - `thin`: a sample thinning factor. The default of 1 considers all samples between `start` and `end`. For example, if `thin = 10` then 1 in 10 samples are considered between `start` and `end`. - `verbose`: if `TRUE`, model specification and progress of the sampler is printed to the screen. Otherwise, nothing is printed to the screen. - `n.report`: the interval to report sampling progress. - `n.omp.threads`: a positive integer indicating the number of threads to use for SMP parallel processing. The package must be compiled for OpenMP support. For most Intel-based machines, we recommend setting `n.omp.threads` up to the number of hyperthreaded cores. This argument is only implemented for `spSVC`. - `...`: currently no additional arguments. ## Returns - **p.y.predictive.samples**: a matrix that holds the response variable(s) posterior predictive samples. For multivariate models `spMvLM` or `spMvGLM` the rows of this matrix correspond to the predicted locations and the columns are the posterior predictive samples. If prediction is for $q$ response variables the `p.y.predictive.samples` matrix has $qn*$ rows, where $n*$ is the number of prediction locations. The predictions for locations are held in rows c("$1:q, (q+1):2q, \\ldots,\n$", "$ ((n*-1)q+1):qn*$") (i.e., the samples for the first location's $q$ response variables are in rows $1:q$, second location in rows $(q+1):2q$, etc.). For `spMisalignLM` and `spMisalignGLM` the posterior predictive samples are organized differently in `p.y.predictive.samples` with the first response variable $n_*1$ locations held in rows $1,\ldots,n*_1$ rows, then the next response variable samples held in the $(n*_1+1),\ldots,(n*_1+n*_2)$, etc. For `spSVC` given the $r$ space-varying coefficients, `p.y.predictive.samples` has $rn*$ rows and the columns are the posterior predictive samples. The predictions for coefficient are held in rows c("$1:r, (r+1):2r, \\ldots,\n$", "$ ((n-1)r+1):rn*$") (i.e., the samples for the first location's $r$ regression coefficients are in rows 1:r, second location in rows $(r+1):2r$, etc.). For `spGLM` and `spMisalignGLM` the `p.y.predictive.samples` matrix holds posterior predictive samples $1/1+(exp(-x(s)'B-w(s)))$ and $exp(x(s)'B+w(s))$ for `family` binomial and poisson, respectively. Here $s$ indexes the prediction location, $B$ is the vector of regression coefficients, and $w$ is the associated spatial random spatial effect. These values can be fed directly into `rbinom` or `rpois` to generate the realization from the respective distribution. - **p.w.predictive.samples**: a matrix organized the same as `p.y.predictive.samples`, that holds the spatial random effects posterior predictive samples. - **p.w.predictive.samples.list**: only returned for `spSVC`. This provides `p.w.predictive.samples` in a different (more convenient form). Elements in this list hold samples for each of the $r$ coefficients. List element names indicate either the coefficient index or name specified in `spSVC`'s `svc.cols` argument. The sample matrices are $n*$ rows and predictive samples along the columns. - **p.tilde.beta.predictive.samples.list**: only returned for `spSVC`. Like `p.w.predictive.samples.list` but with the addition of the corresponding $beta$ posterior samples (i.e., $beta+w(s)$). - **center.scale.pred.covars**: only returned for the `spSVC` when its `center.scale` argument is `TRUE`. This is the prediction design matrix centered and scaled with respect to column means and variances of the design matrix used to estimate model parameters, i.e., the one defined in `spSVC`'s formula argument. ## References Banerjee, S., Carlin, B.P., and Gelfand, A.E. (2004). Hierarchical modeling and analysis for spatial data. Chapman and Hall/CRC Press, Boca Raton, FL. Finley, A.O., S. Banerjee, and A.E. Gelfand. (2015) spBayes for large univariate and multivariate point-referenced spatio-temporal data models. **Journal of Statistical Software**, 63:1--28. [https://www.jstatsoft.org/article/view/v063i13](https://www.jstatsoft.org/article/view/v063i13). Finley, A.O. and S. Banerjee (2019) Bayesian spatially varying coefficient models in the spBayes R package. [https://arxiv.org/abs/1903.03028](https://arxiv.org/abs/1903.03028). ## Author(s) Andrew O. Finley finleya@msu.edu , Sudipto Banerjee sudipto@ucla.edu ## Examples ```r ## Not run: rmvn <- function(n, mu=0, V = matrix(1)){ p <- length(mu) if(any(is.na(match(dim(V),p)))) stop("Dimension problem!") D <- chol(V) t(matrix(rnorm(n*p), ncol=p)%*%D + rep(mu,rep(n,p))) } set.seed(1) n <- 200 coords <- cbind(runif(n,0,1), runif(n,0,1)) X <- as.matrix(cbind(1, rnorm(n))) B <- as.matrix(c(1,5)) p <- length(B) sigma.sq <- 10 tau.sq <- 0.01 phi <- 3/0.5 D <- as.matrix(dist(coords)) R <- exp(-phi*D) w <- rmvn(1, rep(0,n), sigma.sq*R) y <- rnorm(n, X%*%B + w, sqrt(tau.sq)) ##partition the data for out of sample prediction mod <- 1:100 y.mod <- y[mod] X.mod <- X[mod,] coords.mod <- coords[mod,] n.samples <- 1000 starting <- list("phi"=3/0.5, "sigma.sq"=50, "tau.sq"=1) tuning <- list("phi"=0.1, "sigma.sq"=0.1, "tau.sq"=0.1) priors <- list("beta.Flat", "phi.Unif"=c(3/1, 3/0.1), "sigma.sq.IG"=c(2, 5), "tau.sq.IG"=c(2, 0.01)) cov.model <- "exponential" m.1 <- spLM(y.mod~X.mod-1, coords=coords.mod, starting=starting, tuning=tuning, priors=priors, cov.model=cov.model, n.samples=n.samples) m.1.pred <- spPredict(m.1, pred.covars=X, pred.coords=coords, start=0.5*n.samples) y.hat <- apply(m.1.pred$p.y.predictive.samples, 1, mean) quant <- function(x){quantile(x, prob=c(0.025, 0.5, 0.975))} y.hat <- apply(m.1.pred$p.y.predictive.samples, 1, quant) plot(y, y.hat[2,], pch=19, cex=0.5, xlab="observed y", ylab="predicted y") arrows(y[-mod], y.hat[2,-mod], y[-mod], y.hat[1,-mod], angle=90, length=0.05) arrows(y[-mod], y.hat[2,-mod], y[-mod], y.hat[3,-mod], angle=90, length=0.05) ## End(Not run) ```

spPredict function