vbmp {vbmp} | R Documentation

## Variational Bayesian Multinomial Probit Regression with Gaussian Process Priors.

### Description

Used to fit a Multinomial Probit Regression model, specified by giving the matrix design `X`, the associated response variables `t.class`, kernel type and covariate scaling parameters. Covariance paramters can be inferred from the data.

### Usage

    vbmp(X, t.class, X.TEST, t.class.TEST, theta, control = list())

### Arguments

`X` | Feature matrix for parameter 'estimation'
`t.class` | Target values, integer number used for class labels.
`X.TEST` | Feature matrix to compute out-of-sample (test) prediction errors and likelihoods
`t.class.TEST` | Target values for test data
`theta` | The covariance function parameters (e.g. scaling coefficients for each dimension)
`control` | A list of control parameters. See Details

### Details

In this implementation a single covariance function is shared across all classes. Compute the predictive posteriors on the test set and the associated likelihood and test errors at each iteration.

The control argument is a list that can supply any of the following components:

InfoLevel
  0 to suppress tracing ( > 0 to print different levels of monitoring information)
sFILE.TRACE
  File name where to redirect output (default NULL)
bThetaEstimate
  if covariance parameter estimation switched on. Defaults to FALSE (switched off)
sKernelType
  Kernel function used in training and predicting. Currently implemented kernels are Gaussian (`"gauss"`), Cauchy (`"cauchy"`), Laplace (`"laplace"`), Polynomial (`"poly"`), Homogeneous polynomial (`"hpoly"`), 'Thin-plate' spline (`"tps"`), 'linear' spline (`"lsp"`) and Inner product(`"iprod"`). Defaults to `"gauss"`.
maxIts
  Maximum number of variational EM steps to take. Defaults to 50.
Thresh
  Convergence threshold on marginal likelihood lowerbound. Defaults to 1e-4.
nSampsTG
  Number of samples used in obtaining mean of truncated Gaussian. Defaults to 1000.
nSampsIS
  Number of samples used in the importance sampler. Defaults to 1000.
nSmallNo
  Small number used to prevent numerical problems (ill-conditioned covariance matrix). Defaults to 1e-10.
parGammaTau,parGammaSigma
  The location and scale parameters of the Gamma prior over covaraince params. Default to 1e-6.
bMonitor
  TRUE to collect monitor convergence diagnostics at each iteration. Defaults to FALSE.
bPlotFitting
  TRUE to plot test performance results at each iteration during model estimation (if TRUE it forces bMonitor to TRUE). Defaults to FALSE.

### Value

`vbmp` returns an object of class "VBMP.obj". An object of class "VBMP.obj" is a list containing at least the following components:

`Kc` | Number of classes
`Ptest` | Matrix of multinomial class predictive posterior probabilities for the test data
`X` | Feature matrix
`invPHI` | Inverse of the Kernel matrix
`Y` | Matrix of auxiliary variables
`M` | Matrix of GP random variables
`theta` | covariance kernel hyperparameters (estimates computed during model fitting, if inferred
`sKernelType` | Kernel function used in training and predicting
`Test.Err` | Out-of-Sample Percent Prediction error estimates computed during model fitting (0-1 error loss).
`PL` | Predictive Likelihood estimates computed during model fitting
`LOWER.BOUND` | Lower bound estimates computed during model fitting

### Author(s)

N Lama nicola.lama@unina2.it, MA Girolami girolami@dcs.gla.ac.uk

### References

Girolami M, Rogers S, Variational Bayesian Multinomial Probit Regression with Gaussian Process Priors, Neural Computation 18, 1790-1817 (2006).

http://www.dcs.gla.ac.uk/~girolami

### See Also

See Also as `lowerBound`, `predError`, `predLik`, `predClass`

### Examples

    ## -----------------------------------------------------------------------------
    ## EXAMPLE 1 - Theta estimate with synthetic data
    ## -----------------------------------------------------------------------------
    ## Samples of 2-D data points drawn from three nonlinearly separable
    ## classes which take the form of two annular rings and one zero-centered
    ## Gaussian are used in this little illustrative example.
    genSample <- function(n, noiseVar=0) {
        ## class 1 and 2 (x ~ U(0,1))
        u <- 4. * matrix(runif(2*n), nrow=n, ncol=2) - 2.;
        i <- which(((u[, 1]^2 + u[, 2]^2) > .1) & ((u[, 1]^2 + u[, 2]^2) < .5) );
        j <- which(((u[, 1]^2 + u[, 2]^2) > .6) & ((u[, 1]^2 + u[, 2]^2) < 1) );
        X <- u[c(i, j),];
        t.class <- c(rep(1, length(i)),rep(2, length(j)));
        ## class 3 (x ~ N(0,1))
        x <- 0.1 * matrix(rnorm(2*length(i)), ncol=2, nrow=length(i) );
        k <- which((x[, 1]^2 + x[, 2]^2) < 0.1);
        X <- rbind(X, x[k, ]);
        t.class <- c(t.class, rep(3, length(k)));
        ## add random coloumns
        if (noiseVar>0) X <- cbind(X, matrix(rnorm(noiseVar*nrow(X)), ncol=noiseVar, nrow=nrow(X)));
        structure( list( t.class=t.class, X=X), class="MultiNoisyData");
    }

    set.seed(100); ## Init random number generator

    ## Generate training and test samples as an independent
    ## test set to assess out-of-sample prediction error
    ## and predictive likelihoods.
    nNoisyInputs <- 0;       ## number of additional noisy input parameters
    Ntest <- Ntrain <- 500;  ## sample sizes
    dataXt.train <- genSample(Ntrain, nNoisyInputs);
    dataXt.test  <- genSample(Ntest,  nNoisyInputs);

    ## Not run:

     theta <- runif(ncol(dataXt.train$X));
     res <- vbmp( dataXt.train$X, dataXt.train$t.class,
            dataXt.test$X, dataXt.test$t.class, theta,
             control=list(bThetaEstimate = T,
             bPlotFitting=T, maxIts=50));
    ## End(Not run)

    ## set theta params (previously estimated)
    theta <- c(0.09488309, 0.16141604);
    ## Fit the vbmp
    res <- vbmp( dataXt.train$X, dataXt.train$t.class,
            dataXt.test$X, dataXt.test$t.class, theta,
            control=list(maxIts=5));
    ## print out-of-sample error estimate
    predError(res);

    ## Not run:
    ## ----------------------------------------------------------
    ## EXAMPLE 2 - BRCA12 genomic data
    ## ----------------------------------------------------------
    ## Leave-one-out (LOO) cross-validation prediction error of the probabilistic
    ## Gaussian process classifier used in Zsofia Kote-Jarai et al.
    ## Clin Cancer Res 2006;12(13);3896-3901

      if(any(installed.packages()[,1]=="Biobase")) {
        library("Biobase");
        data("BRCA12");
        brca.y <- BRCA12$Target.class;
        brca.x <- t(exprs(BRCA12));
      } else {
        print("Deprecated.....");
        load(url("http://www.dcs.gla.ac.uk/people/personal/girolami/pubs_2005/VBGP/BRCA12.RData"));
        brca.y <- as.numeric(BRCA12$y);
        brca.x <- as.matrix(BRCA12[,-1]);
      }

      sKernelType <- "iprod";  ## Covariance function type
      Thresh <- 1e-8;  ## Iteration threshold
      InfoLevel <- 1;
      theta <- rep(1.0, ncol(brca.x));
      ITER.THETA <- 24;
      n     <- nrow(brca.x) ;
      Kfold <- n; # number of folds , if equal to n then LOO
      samps <- sample(rep(1:Kfold, length=n), n, replace=FALSE);
      res   <- rep(NA, n);
      print(paste("LOO crossvalidation started...... (",n,"steps)"));
      for (x in 1:Kfold) {
          cat(paste(x,", ",sep="")); flush.console();
          resX <- vbmp( brca.x[samps!=x,], brca.y[samps!=x],
                        brca.x[samps==x,], brca.y[samps==x],
                        theta,  control=list(bThetaEstimate=F,
                        bPlotFitting=F, maxIts=ITER.THETA,
                        sKernelType=sKernelType, Thresh=Thresh));
          res[samps==x] <- predClass(resX);
      }
      print("(end)");
      print(paste("Crossvalidated error rate", round(sum(res!=brca.y)/n,2)));
    ## End(Not run)

[Package vbmp version 1.6.0 Index]