# btgp: Bayesian Nonparametric & Nonstationary Regression Models In tgp: Bayesian Treed Gaussian Process Models

## Description

The seven functions described below implement Bayesian regression models of varying complexity: linear model, linear CART, Gaussian process (GP), GP with jumps to the limiting linear model (LLM), treed GP, and treed GP LLM.

## Usage

 ``` 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 28 29 30 31 32 33 34 35``` ```blm(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", BTE = c(1000, 4000, 3), R = 1, m0r1 = TRUE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, trace = FALSE, verb = 1, ...) btlm(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", tree = c(0.5, 2), BTE = c(2000, 7000, 2), R = 1, m0r1 = TRUE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, trace = FALSE, verb = 1, ...) bcart(X, Z, XX = NULL, bprior = "bflat", tree = c(0.5, 2), BTE = c(2000, 7000, 2), R = 1, m0r1 = TRUE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv=FALSE, sens.p = NULL, trace = FALSE, verb = 1, ...) bgp(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", corr = "expsep", BTE = c(1000, 4000, 2), R = 1, m0r1 = TRUE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, nu = 1.5, trace = FALSE, verb = 1, ...) bgpllm(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", corr = "expsep", gamma=c(10,0.2,0.7), BTE = c(1000, 4000, 2), R = 1, m0r1 = TRUE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, nu = 1.5, trace = FALSE, verb = 1, ...) btgp(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", corr = "expsep", tree = c(0.5, 2), BTE = c(2000, 7000, 2), R = 1, m0r1 = TRUE, linburn = FALSE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, nu = 1.5, trace = FALSE, verb = 1, ...) btgpllm(X, Z, XX = NULL, meanfn = "linear", bprior = "bflat", corr = "expsep", tree = c(0.5, 2), gamma=c(10,0.2,0.7), BTE = c(2000, 7000, 2), R = 1, m0r1 = TRUE, linburn = FALSE, itemps = NULL, pred.n = TRUE, krige = TRUE, zcov = FALSE, Ds2x = FALSE, improv = FALSE, sens.p = NULL, nu = 1.5, trace = FALSE, verb = 1, ...) ```

## Arguments

Each of the above functions takes some subset of the following arguments...

 `X` `data.frame`, `matrix`, or vector of inputs `X` `Z` Vector of output responses `Z` of length equal to the leading dimension (rows) of `X`, i.e., `length(Z) == nrow(X)` `XX` Optional `data.frame`, `matrix`, or vector of predictive input locations with the same number of columns as `X`, i.e., `ncol(XX) == ncol(X)` `meanfn` A choice of mean function for the process. When `meanfn = "linear"` (default), then we have the process Z = cbind(rep(1,nrow(X), X)) %*% beta + W(X), where W(X) represents the Gaussian process part of the model (if present). Otherwise, when `meanfn = "constant"`, then Z = beta0 + W(X). `bprior` Linear (beta) prior, default is `"bflat"`; alternates include `"b0"` hierarchical Normal prior, `"bmle"` empirical Bayes Normal prior, `"b0not"` Bayesian treed LM-style prior from Chipman et al. (same as `"b0"` but without `tau2`), `"bmzt"` a independent Normal prior (mean zero) with inverse-gamma variance (`tau2`), and `"bmznot"` is the same as `"bmznot"` without `tau2`. The default `"bflat"` gives an “improper” prior which can perform badly when the signal-to-noise ratio is low. In these cases the “proper” hierarchical specification `"b0"` or independent `"bmzt"` or `"bmznot"` priors may perform better `tree` a 2-vector containing the tree process prior parameterization `c(alpha, beta)` specifying p(split leaf eta) = alpha*(1+depth(eta))^(-beta) automatically giving zero probability to trees with partitions containing less than `min(c(10,nrow(X)+1))` data points. You may also specify a longer vector, writing over more of the components of the `\$tree` output from `tgp.default.params` `gamma` Limiting linear model parameters `c(g, t1, t2)`, with growth parameter `g > 0` minimum parameter `t1 >= 0` and maximum parameter `t1 >= 0`, where `t1 + t2 <= 1` specifies p(b|d)= t1 + exp(-g*(t2-t1)/(d-0.5)) `corr` Gaussian process correlation model. Choose between the isotropic power exponential family (`"exp"`) or the separable power exponential family (`"expsep"`, default); the current version also supports the isotropic Matern (`"matern"`) and single-index Model (`"sim"`) as “beta” functionality. `BTE` 3-vector of Monte-carlo parameters (B)urn in, (T)otal, and (E)very. Predictive samples are saved every E MCMC rounds starting at round B, stopping at T. `R` Number of repeats or restarts of `BTE` MCMC rounds, default `R=1` is no restarts `m0r1` If `TRUE` (default) the responses `Z` will be scaled to have a mean of zero and a range of 1 `linburn` If `TRUE` initializes MCMC with `B` (additional) rounds of Bayesian Linear CART (`btlm`); default is `FALSE` `itemps` Importance tempering (IT) inverse temperature ladder, or powers to improve mixing. See `default.itemps`. The default is no IT `itemps = NULL` `pred.n` `TRUE` (default) value results in prediction at the inputs `X`; `FALSE` skips prediction at `X` resulting in a faster implementation `krige` `TRUE` (default) value results in collection of kriging means and variances at predictive (and/or data) locations; `FALSE` skips the gathering of kriging statistics giving a savings in storage `zcov` If `TRUE` then the predictive covariance matrix is calculated– can be computationally (and memory) intensive if `X` or `XX` is large. Otherwise only the variances (diagonal of covariance matrices) are calculated (default). See outputs `Zp.s2`, `ZZ.s2`, etc., below `Ds2x` `TRUE` results in ALC (Active Learning–Cohn) computation of expected reduction in uncertainty calculations at the `XX` locations, which can be used for adaptive sampling; `FALSE` (default) skips this computation, resulting in a faster implementation `improv` `TRUE` results in samples from the improvement at locations `XX` with respect to the observed data minimum. These samples are used to calculate the expected improvement over `XX`, as well as to rank all of the points in `XX` in the order that they should be sampled to minimize the expected multivariate improvement (refer to Schonlau et al, 1998). Alternatively, `improv` can be set to any positive integer 'g', in which case the ranking is performed with respect to the expectation for improvement raised to the power 'g'. Increasing 'g' leads to rankings that are more oriented towards a global optimization. The option `FALSE` (default) skips these computations, resulting in a faster implementation. Optionally, a two-vector can be supplied where `improv` is interpreted as the (maximum) number of points to rank by improvement. See the note below. If not specified, the entire `XX` matrix is ranked. `sens.p` Either `NULL` or a vector of parameters for sensitivity analysis, built by the function `sens`. Refer there for details `nu` “beta” functionality: fixed smoothness parameter for the Matern correlation function; `nu + 0.5` times differentiable predictive surfaces result `trace` `TRUE` results in a saving of samples from the posterior distribution for most of the parameters in the model. The default is `FALSE` for speed/storage reasons. See note below `verb` Level of verbosity of R-console print statements: from 0 (none); 1 (default) which shows the “progress meter”; 2 includes an echo of initialization parameters; up to 3 and 4 (max) with more info about successful tree operations `...` These ellipses arguments are interpreted as augmentations to the prior specification generated by `params <- tgp.default.params(ncol(X)+1)`. You may use these to specify a custom setting of any of default parameters in the output list `params` except those for which a specific argument is already provided (e.g., `params\$corr` or `params\$bprior`) or those which contradict the type of `b*` function being called (e.g., `params\$tree` or `params\$gamma`); these redundant or possibly conflicting specifications will be ignored. Refer to `tgp.default.params` for details on the prior specification

## Details

The functions and their arguments can be categorized by whether or not they use treed partitioning (T), GP models, and jumps to the LLM (or LM)

 blm LM Linear Model btlm T, LM Treed Linear Model bcart T Treed Constant Model bgp GP GP Regression bgpllm GP, LLM GP with jumps to the LLM btgp T, GP treed GP Regression btgpllm T, GP, LLM treed GP with jumps to the LLM

Each function implements a special case of the generic function `tgp` which is an interface to C/C++ code for treed Gaussian process modeling of varying parameterization. Documentation for `tgp` has been declared redundant, and has subsequently been removed. To see how the `b*` functions use `tgp` simply examine the function. In the latest version, with the addition of the ellipses “...” argument, there is nothing that can be done with the direct `tgp` function that cannot also be done with a `b*` function

Only functions in the T (tree) category take the `tree` argument; GP category functions take the `corr` argument; and LLM category functions take the `gamma` argument. Non-tree class functions omit the `parts` output, see below

`bcart` is the same as `btlm` except that only the intercept term in the LM is estimated; the others are zero, thereby implementing a Bayesian version of the original CART model

The `sens.p` argument contains a vector of parameters for sensitivity analysis. It should be `NULL` unless created by the `sens` function. Refer to `help(sens)` for details.

If `itemps =! NULL` then importance tempering (IT) is performed to get better mixing. After each restart (when `R > 1`) the observation counts are used to update the pseudo-prior. Stochastic approximation is performed in the first burn-in rounds (for `B-T` rounds, not `B`) when `c0` and `n0` are positive. Every subsequent burn-in after the first restart is for `B` rounds in order to settle-in after using the observation counts. See `default.itemps` for more details and an example

Please see `vignette("tgp")` for a detailed illustration

## Value

`bgp` returns an object of class `"tgp"`. The function `plot.tgp` can be used to help visualize results.

An object of class `"tgp"` is a list containing at least the following components... The `parts` output is unique to the T (tree) category functions. Tree viewing is supported by `tgp.trees`

 `X` Input argument: `data.frame` of inputs `X` `n` Number of rows in `X`, i.e., `nrow(X)` `d` Number of cols in `X`, i.e., `ncol(X)` `Z` Vector of output responses `Z` `XX` Input argument: `data.frame` of predictive locations `XX` `nn` Number of rows in `XX`, i.e., `nrow(XX)` `BTE` Input argument: Monte-carlo parameters `R` Input argument: restarts `linburn` Input argument: initialize MCMC with linear CART `params` `list` of model parameters generated by `tgp.default.params` and subsequently modified according to the calling `b*` function and its arguments `dparams` Double-representation of model input parameters used by the C-code `itemps` `data.frame` containing the importance tempering ladders and pseudo-prior: `\$k` has inverse inverse temperatures (from the input argument), `\$k` has an updated pseudo-prior based on observation counts and (possibly) stochastic approximation during burn-in and (input) stochastic approximation parameters c0 and n0. See `default.itemps` for more info `Zp.mean` Vector of mean predictive estimates at `X` locations `Zp.q1` Vector of 5% predictive quantiles at `X` locations `Zp.q2` Vector of 95% predictive quantiles at `X` locations `Zp.q` Vector of quantile norms `Zp.q2-Zp.q1` `Zp.s2` If input `zcov = TRUE`, then this is a predictive covariance matrix for the inputs at locations `X`; otherwise then this is a vector of predictive variances at the `X` locations (diagonal of the predictive covariance matrix). Only appears when input `pred.n = TRUE` `Zp.km` Vector of (expected) kriging means at `X` locations `Zp.vark` Vector of posterior variance for kriging surface (no additive noise) at `X` locations `Zp.ks2` Vector of (expected) predictive kriging variances at `X` locations `ZZ.mean` Vector of mean predictive estimates at `XX` locations `ZZ.q1` Vector of 5% predictive quantiles at `XX` locations `ZZ.q2` Vector of 95% predictive quantiles at `XX` locations `ZZ.q` Vector of quantile norms `ZZ.q2-ZZ.q1`, used by the ALM adaptive sampling algorithm `ZZ.s2` If input `zcov = TRUE`, then this is a predictive covariance matrix for predictive locations `XX`; otherwise then this is a vector of predictive variances at the `XX` locations (diagonal of the predictive covariance matrix). Only appears when input `XX != NULL` `ZpZZ.s2` If input `zcov = TRUE`, then this is a predictive `n * nn` covariance matrix between locations in `X` and `XX`; Only appears when `zcov = TRUE` and both `pred.n = TRUE` and `XX != NULL` `ZZ.km` Vector of (expected) kriging means at `XX` locations `ZZ.vark` Vector of posterior variance for kriging surface (no additive noise) at `XX` locations `ZZ.ks2` Vector of (expected) predictive kriging variances at `XX` locations `Ds2x` If argument `Ds2x=TRUE`, this vector contains ALC statistics for `XX` locations `improv` If argument `improv` is `TRUE` or a positive integer, this is a 'matrix' with first column set to the expected improvement statistics for `XX` locations, and the second column set to a ranking in the order that they should be sampled to minimize the expected multivariate improvement raised to a power determined by the argument `improv` `response` Name of response `Z` if supplied by `data.frame` in argument, or "z" if none provided `parts` Internal representation of the regions depicted by partitions of the maximum a' posteriori (MAP) tree `trees` `list` of trees (maptree representation) which were MAP as a function of each tree height sampled between MCMC rounds `B` and `T` `trace` If `trace==TRUE`, this `list` contains traces of most of the model parameters and posterior predictive distributions at input locations `XX`. Otherwise the entry is `FALSE`. See note below `ess` Importance tempering effective sample size (ESS). If `itemps==NULL` this corresponds to the total number of samples collected, i.e.. `R*(BTE-BTE)/BTE`. Otherwise the ESS will be lower due to a non-zero coefficient of variation of the calculated importance tempering weights `sens` See `sens` documentation for more details

## Note

Inputs `X, XX, Z` containing `NaN, NA`, or `Inf` are discarded with non-fatal warnings

Upon execution, MCMC reports are made every 1,000 rounds to indicate progress

Stationary (non-treed) processes on larger inputs (e.g., `X,Z`) of size greater than 500, *might* be slow in execution, especially on older machines. Once the C code starts executing, it can be interrupted in the usual way: either via Ctrl-C (Unix-alikes) or pressing the Stop button in the R-GUI. When this happens, interrupt messages will indicate which required cleanup measures completed before returning control to R.

Whereas most of the tgp models will work reasonably well with little or no change to the default prior specification, GP's with the `"mrexpsep"` correlation imply a very specific relationship between fine and coarse data, and a careful prior specification is usually required.

The ranks provided in the second column of the `improv` field of a `tgp` object are based on the expectation of a multivariate improvement that may or may not be raised to a positive integer power. They can thus differ significantly from a simple ranking of the first column of expected univariate improvement values.

Regarding `trace=TRUE`: Samples from the posterior will be collected for all parameters in the model. GP parameters are collected with reference to the locations in `XX`, resulting `nn=nrow{XX}` traces of `d,g,s2,tau2`, etc. Therefore, it is recommended that `nn` is chosen to be a small, representative, set of input locations. Besides GP parameters, traces are saved for the tree partitions, areas under the LLM, log posterior (as a function of tree height), and samples from the posterior predictive distributions. Note that since some traces are stored in files, multiple `tgp`/R sessions in the same working directory can clobber the trace files of other sessions

## Author(s)

Robert B. Gramacy, rbg@vt.edu, and Matt Taddy, mataddy@amazon.com

## References

Gramacy, R. B. (2020) Surrogates: Gaussian Process Modeling, Design and Optimization for the Applied Sciences. Boca Raton, Florida: Chapman Hall/CRC. (See Chapter 9.) https://bobby.gramacy.com/surrogates/

Gramacy, R. B. (2008). tgp: An R Package for Bayesian Nonstationary, Semiparametric Nonlinear Regression and Design by Treed Gaussian Process Models. Journal of Statistical Software, 19(9). https://www.jstatsoft.org/v19/i09

Robert B. Gramacy, Matthew Taddy (2010). Categorical Inputs, Sensitivity Analysis, Optimization and Importance Tempering with tgp Version 2, an R Package for Treed Gaussian Process Models. Journal of Statistical Software, 33(6), 1–48. https://www.jstatsoft.org/v33/i06/.

Gramacy, R. B., Lee, H. K. H. (2007). Bayesian treed Gaussian process models with an application to computer modeling. Journal of the American Statistical Association, 103(483), pp. 1119-1130. Also available as ArXiv article 0710.4536 https://arxiv.org/abs/0710.4536

Gramacy, R. B. and Lee, K.H. (2008). Gaussian Processes and Limiting Linear Models. Computational Statistics and Data Analysis, 53, pp. 123-136. Also available as ArXiv article 0804.4685 https://arxiv.org/abs/0804.4685

Gramacy, R. B., Lee, H. K. H. (2009). Adaptive design and analysis of supercomputer experiments. Technometrics, 51(2), pp. 130-145. Also avaliable on ArXiv article 0805.4359 https://arxiv.org/abs/0805.4359

Robert B. Gramacy, Heng Lian (2011). Gaussian process single-index models as emulators for computer experiments. Available as ArXiv article 1009.4241 https://arxiv.org/abs/1009.4241

Chipman, H., George, E., \& McCulloch, R. (1998). Bayesian CART model search (with discussion). Journal of the American Statistical Association, 93, 935–960.

Chipman, H., George, E., \& McCulloch, R. (2002). Bayesian treed models. Machine Learning, 48, 303–324.

M. Schonlau and Jones, D.R. and Welch, W.J. (1998). Global versus local search in constrained optimization of computer models. In "New Developments and applications in experimental design", IMS Lecture Notes - Monograph Series 34. 11–25.

## See Also

`plot.tgp`, `tgp.trees`, `predict.tgp`, `sens`, `default.itemps`

## Examples

 ``` 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 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76``` ```## ## Many of the examples below illustrate the above ## function(s) on random data. Thus it can be fun ## (and informative) to run them several times. ## # # simple linear response # # input and predictive data X <- seq(0,1,length=50) XX <- seq(0,1,length=99) Z <- 1 + 2*X + rnorm(length(X),sd=0.25) out <- blm(X=X, Z=Z, XX=XX) # try Linear Model plot(out) # plot the surface # # 1-d Example # # construct some 1-d nonstationary data X <- seq(0,20,length=100) XX <- seq(0,20,length=99) Z <- (sin(pi*X/5) + 0.2*cos(4*pi*X/5)) * (X <= 9.6) lin <- X>9.6; Z[lin] <- -1 + X[lin]/10 Z <- Z + rnorm(length(Z), sd=0.1) out <- btlm(X=X, Z=Z, XX=XX) # try Linear CART plot(out) # plot the surface tgp.trees(out) # plot the MAP trees out <- btgp(X=X, Z=Z, XX=XX) # use a treed GP plot(out) # plot the surface tgp.trees(out) # plot the MAP trees # # 2-d example # (using the isotropic correlation function) # # construct some 2-d nonstationary data exp2d.data <- exp2d.rand() X <- exp2d.data\$X; Z <- exp2d.data\$Z XX <- exp2d.data\$XX # try a GP out <- bgp(X=X, Z=Z, XX=XX, corr="exp") plot(out) # plot the surface # try a treed GP LLM out <- btgpllm(X=X, Z=Z, XX=XX, corr="exp") plot(out) # plot the surface tgp.trees(out) # plot the MAP trees # # Motorcycle Accident Data # # get the data require(MASS) # try a GP out <- bgp(X=mcycle[,1], Z=mcycle[,2]) plot(out) # plot the surface # try a treed GP LLM # best to use the "b0" beta linear prior to capture common # common linear process throughout all regions (using the # ellipses "...") out <- btgpllm(X=mcycle[,1], Z=mcycle[,2], bprior="b0") plot(out) # plot the surface tgp.trees(out) # plot the MAP trees ```

tgp documentation built on Jan. 13, 2021, 3:49 p.m.