# Bayesian Single-Index Regression with Gaussian Process Link and von Mises-Fisher Prior Fits a single–index model $Y_i \sim \mathcal{N}(f(X_i'\theta), \sigma^2), i = 1,\cdots,n$, where the index $\theta$ lies on the unit sphere with von Mises-Fisher prior, and the link $f(\cdot)$ is represented with Gaussian process. ```r gpFisher( formula, data, prior = NULL, init = NULL, monitors = NULL, niter = 10000, nburnin = 1000, thin = 1, nchain = 1, setSeed = FALSE ) gpFisher_setup( formula, data, prior = NULL, init = NULL, monitors = NULL, niter = 10000, nburnin = 1000, thin = 1, nchain = 1, setSeed = FALSE ) ``` ## Arguments - `formula`: an object of class formula . Interaction term is not allowed for single-index model. - `data`: an data frame. - `prior`: Optional named list of prior settings. Further descriptions are in Details section. - `init`: Optional named list of initial values. If the values are not assigned, they are randomly sampled from prior or designated value. Further descriptions are in Details section. - `monitors`: Optional character vector of additional monitor nodes. To check the names of the nodes, fit the `model_setup` function and then inspect the variable names stored in the model object using getVarMonitor . - `niter`: Integer. Total MCMC iterations (default `10000`). - `nburnin`: Integer. Burn-in iterations (default `1000`). - `thin`: Integer. Thinning for monitors (default `1`). - `nchain`: Integer. Number of MCMC chains (default `1`). - `setSeed`: Logical or numeric argument. Further details are provided in runMCMC `setSeed` argument. ## Returns A `list` typically containing: - **`coefficients`**: Mean estimates of index vector. Return list of `model_setup` does not include it. - **`ses`**: Mean standard error of index vector. Return list of `model_setup` does not include it. - **`residuals`**: Residuals with mean estimates of fitted values. Return list of `model_setup` does not include it. - **`fitted.values`**: Mean estimates of fitted values. Return list of `model_setup` does not include it. - **`linear.predictors`**: Mean estimates of single-index values. Return list of `model_setup` does not include it. - **`gof`**: Goodness-of-fit. Return list of `model_setup` function does not include it. - **`samples`**: Posterior draws of variables for computing fitted values of the model, including $\theta$, $\sigma^2$. Return list of `model_setup` does not include it. - **`input`**: List of data used in modeling, formula, input values for prior and initial values, and computation time without compiling. - **`defModel`**: Nimble model object. - **`defSampler`**: Nimble sampler object. - **`modelName`**: Name of the model. ## Details Model The single-index model uses Gaussian process with zero mean and and covariance kernel $\eta \cdot \text{exp}(-\frac{(t_i-t_j)^2}{l})$ as a link function, where $t_i, t_j, j = 1, \ldots, n$ is index value. Index vector should be length 1. Priors * von Mises–Fisher prior on the index $\theta$ with direction and concentration. * Covariance kernel: Amplitude, $\eta$, follows log normal distribution with mean $a_\eta$ and variance $b_\eta$. Length-scale parameter follows gamma distribution with shape parameter $\alpha_l$ and rate parameter $\beta_l$. * Inverse gamma prior on $\sigma^2$ with shape parameter $a_\sigma$ and rate parameter $b_\sigma$. Sampling All sampling parameters' samplers are assigned by nimble. Prior hyper-parameters These are the prior hyper-parameters set in the function. You can define new values for each parameter in prior_param . 1. Index vector: von Mises--Fisher prior for the projection vector $\theta$ (`index`). `index_direction` is a unit direction vector of the von Mises--Fisher distribution. By default, the estimated vector from projection pursuit regression is assigned. `index_dispersion` is the positive concentration parameter. By default, `150` is assigned. 2. Link function: * Length-scale:Gamma distribution is assigned for length-scale parameter, $l$. `link_lengthscale_shape` is shape parameter (default `1/8`) and `link_lengthscale_rate` is rate parameter of `lengthscale`. (default `1/8`) * Amplitude: Log-normal distribution is assigned for amplitude parameter, $\eta$. `link_amp_a` is mean (default `-1`), and `link_amp_b` is variance. (default `1`) 3. Error variance (`sigma2`): An inverse-gamma prior is assigned to $\sigma^2$ where `sigma2_shape` is shape parameter and `sigma2_rate` is rate parameter of inverse gamma distribution. (default `sigma2_shape = 1, sigma2_rate = 1`) Initial values These are the initial values set in the function. You can define new values for each initial value in init_param . 1. Index vector (`index`): Initial unit index vector $\theta$. By default, the vector is sampled from the von Mises--Fisher prior. 2. Link function: `link_lengthscale` is initial scalar length-scale parameter. (default: `0.1`) `link_amp` is initial scalar amplitude parameter. (default: `1`) 3. Error variance (`sigma2`): Initial scalar error variance. (default: `1`) ## Examples ```r set.seed(123) N <- 60; p <- 2 x1 <- runif(N, -3, 5) x2 <- runif(N, -3, 5) beta1 <- 0.45; beta2 <- sqrt(1-beta1^2) sigma <- sqrt(0.0036) xlin <- x1*beta1 + x2*beta2 eta <- 0.1*xlin + sin(0.5*xlin)^2 y <- rnorm(N, eta, sigma) x <- matrix(c(x1, x2), ncol = 2) simdata <- data.frame(x = x, y = y) colnames(simdata) <- c("X1", "X2", "y") # One tool version fit1 <- gpFisher(y ~ ., data = simdata, nchain = 1, niter = 1000, nburnin = 100) # Split version models <- gpFisher_setup(y ~ ., data = simdata, nchain = 1) Ccompile <- compileModelAndMCMC(models) nimSampler <- get_sampler(Ccompile) initList <- getInit(models) mcmc.out <- runMCMC(nimSampler, niter = 1000, nburnin = 100, thin = 1, nchains = 1, setSeed = TRUE, inits = initList, summary = TRUE, samplesAsCodaMCMC = TRUE) fit2 <- as_bsim(models, mcmc.out) summary(fit2) ``` ## References Antoniadis, A., Grégoire, G., & McKeague, I. W. (2004). Bayesian estimation in single-index models. **Statistica Sinica**, 1147-1164. Choi, T., Shi, J. Q., & Wang, B. (2011). A Gaussian process regression approach to a single-index model. **Journal of Nonparametric Statistics**, 23(1), 21-36. Hornik, K., & Grün, B. (2014). movMF: an R package for fitting mixtures of von Mises-Fisher distributions. **Journal of Statistical Software**, 58, 1-31.

gpFisher function