R: Tuning the learning rate for Gibbs posterior

tuneLearn {qgam} | R Documentation

## Tuning the learning rate for Gibbs posterior

### Description

The learning rate (sigma) of the Gibbs posterior is tuned either by calibrating the credible intervals for the fitted curve, or by minimizing the pinball loss on out-of-sample data. This is done by bootrapping or by k-fold cross-validation. Here the calibration loss function is evaluated on a grid of values provided by the user.

### Usage

    tuneLearn(
      form,
      data,
      lsig,
      qu,
      err = NULL,
      multicore = !is.null(cluster),
      cluster = NULL,
      ncores = detectCores() - 1,
      paropts = list(),
      control = list(),
      argGam = NULL
    )

### Arguments

`form` | A GAM formula, or a list of formulae. See ?mgcv::gam details.
`data` | A data frame or list containing the model response variable and covariates required by the formula. By default the variables are taken from environment(formula): typically the environment from which gam is called.
`lsig` | A vector of value of the log learning rate (log(sigma)) over which the calibration loss function is evaluated.
`qu` | The quantile of interest. Should be in (0, 1).
`err` | An upper bound on the error of the estimated quantile curve. Should be in (0, 1). Since qgam v1.3 it is selected automatically, using the methods of Fasiolo et al. (2017). The old default was `err=0.05`.
`multicore` | If TRUE the calibration will happen in parallel.
`cluster` | An object of class `c("SOCKcluster", "cluster")`. This allowes the user to pass her own cluster, which will be used if `multicore == TRUE`. The user has to remember to stop the cluster.
`ncores` | Number of cores used. Relevant if `multicore == TRUE`.
`paropts` | a list of additional options passed into the foreach function when parallel computation is enabled. This is important if (for example) your code relies on external data or packages: use the .export and .packages arguments to supply them so that all cluster nodes have the correct environment set up for computing.
`control` | A list of control parameters for `tuneLearn` with entries: `loss` = loss function use to tune log(sigma). If `loss=="cal"` is chosen, then log(sigma) is chosen so that credible intervals for the fitted curve are calibrated. See Fasiolo et al. (2017) for details. If `loss=="pin"` then log(sigma) approximately minimizes the pinball loss on the out-of-sample data. `sam` = sampling scheme use: `sam=="boot"` corresponds to bootstrapping and `sam=="kfold"` to k-fold cross-validation. The second option can be used only if `ctrl$loss=="pin"`. `K` = if `sam=="boot"` this is the number of boostrap datasets, while if `sam=="kfold"` this is the number of folds. By default `K=50`. `b` = offset parameter used by the mgcv::gauslss. By default `b=0`. `vtype` = type of variance estimator used to standardize the deviation from the main fit in the calibration. If set to `"m"` the variance estimate obtained by the full data fit is used, if set to `"b"` than the variance estimated produced by the bootstrap fits are used. By default `vtype="m"`. `epsB` = positive tolerance used to assess convergence when fitting the regression coefficients on bootstrap data. In particular, if `|dev-dev_old|/(|dev|+0.1)<epsB` then convergence is achieved. Default is `epsB=1e-5`. `verbose` = if TRUE some more details are given. By default `verbose=FALSE`. `link` = link function to be used. See `?elf` and `?elflss` for defaults. `progress` = argument passed to plyr::llply. By default `progress="text"` so that progress is reported. Set it to `"none"` to avoid it.
`argGam` | A list of parameters to be passed to `mgcv::gam`. This list can potentially include all the arguments listed in `?gam`, with the exception of `formula`, `family` and `data`.

### Value

A list with entries:

- `lsig` = the value of log(sigma) resulting in the lowest loss.

- `loss` = vector containing the value of the calibration loss function corresponding to each value of log(sigma).

- `edf` = a matrix where the first colums contain the log(sigma) sequence, and the remaining columns contain the corresponding effective degrees of freedom of each smooth.

- `convProb` = a logical vector indicating, for each value of log(sigma), whether the outer optimization which estimates the smoothing parameters has encountered convergence issues. `FALSE` means no problem.

### Author(s)

Matteo Fasiolo <matteo.fasiolo@gmail.com>.

### References

Fasiolo, M., Goude, Y., Nedellec, R. and Wood, S. N. (2017). Fast calibrated additive quantile regression. Available at https://arxiv.org/abs/1707.03307.

### Examples

    library(qgam); library(MASS)

    # Calibrate learning rate on a grid
    set.seed(41444)
    sigSeq <- seq(1.5, 5, length.out = 10)
    closs <- tuneLearn(form = accel~s(times,k=20,bs="ad"),
                       data = mcycle,
                       lsig = sigSeq,
                       qu = 0.5)

    plot(sigSeq, closs$loss, type = "b", ylab = "Calibration Loss", xlab = "log(sigma)")

    # Pick best log-sigma
    best <- sigSeq[ which.min(closs$loss) ]
    abline(v = best, lty = 2)

    # Fit using the best sigma
    fit <- qgam(accel~s(times,k=20,bs="ad"), data = mcycle, qu = 0.5, lsig = best)
    summary(fit)

    pred <- predict(fit, se=TRUE)
    plot(mcycle$times, mcycle$accel, xlab = "Times", ylab = "Acceleration",
         ylim = c(-150, 80))
    lines(mcycle$times, pred$fit, lwd = 1)
    lines(mcycle$times, pred$fit + 2*pred$se.fit, lwd = 1, col = 2)
    lines(mcycle$times, pred$fit - 2*pred$se.fit, lwd = 1, col = 2)

[Package qgam version 1.3.2 Index]