    GP-PRED:  Make predictions for test cases using Gaussian process model.

    GP-pred prints guesses at the target values for a set of test cases,
    as obtained using a Gaussian process, or set of Gaussian processes.
    If the true targets are known, performance of the guesses can also be
    evaluated.  Inputs can be printed as well.

    Usage:

        gp-pred options { log-file range } [ / test-inputs [ test-targets ] ]

    The final optional arguments give the source of inputs and targets for
    the cases to look at; they default to the test data specification in
    the first log file given.  The Gaussian processes to use in making
    guesses are taken from the records with the given ranges of indexes in
    the given log files.  The outputs of all these are combined to give a
    single guess for each case.  The Gaussian process models should all
    have the same specification and use the same data specifications.

    An index range can have one of the forms "[low][:[high]][%mod]" or
    "[low][:[high]]+num", or one of these forms preceded by "@".  When "@"
    is present, "low" and "high" are given in terms of cpu time, otherwise
    they are iteration numbers.  When just "low" is given, only that index
    is used.  If the colon is included, but "high" is not, the range
    extends to the highest index in the log file.  The "mod" form allows
    iterations to be selected whose numbers are multiples of "mod", with
    the default being "mod" of one.  The "num" form allows the total
    number of iterations used to be specified; they are distributed as
    evenly as possible within the specified range.  Note that it is
    possible that the number of Gaussian processes used in the end may not
    equal this number, if records with some indexes are missing.

    The 'options' argument consists of one or more of the following letters:

        i   Display the input values for each case
        t   Display the target values for each case
        r   Use the raw form of the target values, before transformation

        p   Display the log probability of the true targets (to base e)

        m   Display the guess based on the mode, and whether it is in error
        n   Display the guess based on the mean, and its squared error
        d   Display the guess based on the median, and its absolute error
        D   Display the guess based on the mean of the median for each iteration.
            Not really useful for gp-pred.

        q   Display the 10% and 90% quantiles of the predictive distributions
            for the targets.  Note that these distributions include the noise.
        Q   Display the 1% and 99% quantiles of the predictive distributions.

        l   Use stored latent values for predictions with a regression model
            (if available).  Normally, predictions for regression models are
            done as if no latent values were stored, since this is better,
            but suppressing this behaviour might be of interest for testing
            purposes.  (Stored latent values are essential for other models.)

        b   Suppress headings and averages - just bare numbers for each case.
            The numbers are printed in exponential format, to high precision.
        B   Bare numbers, but with blank lines whenever first input changes.

        a   Display only average log probabilities and errors, suppressing
            the results for individual cases (makes sense only in combination
            with one or more of 'p', 'm', 'n', and 'd', and not with 'i' or 't')

        E   Display the expected error for test cases, based on the predictive
            probabilities, as well as the actual error (if targets are known).
            Currently implemented only for "binary" data.

        1-9 If any of these characters are used, only the corresponding
            non-linear parts of the covariance is used when computing
            covariances involving test cases (other non-linear parts and the
            constant and linear parts are ignored).  This results in predictions
            for just these components of the function, allowing one to see the
            components of an additive model.  (But note that if the identity of
            a component can change during a run, this option will be meaningful
            only when the predictions are based on a single iteration.)

    Some of these options are illegal for some data models.  The illegal
    combinations are marked with an 'X' in the following table:

            binary  class  real-valued  no-model

        r     X       X
        p                                  X
        m                      X           X
        n
        d/D   X       X
        q/Q   X       X

    Furthermore, the 'a' option is incompatible with 'b', 'i', 't', or
    'q', and the 't', 'p', and 'a' options may be used only if the true
    targets are given.  The errors for individual cases are also displayed
    only if the true targets are known.  The 'n' option for class models
    displays the mean probabilities for each class, and computes a single
    figure for squared error that is the sum of the squares of the
    differences between these probabilities and the indicator variables
    that are one for the right class and zero for the wrong classes.

    Making predictions requires inverting the covariance matrix of the
    training points for each Gaussian process used, which can take a
    substantial amount of time if the number of training points is large.
    This could be avoided if the inverse covariance matrix were saved in
    the log file for each iteration, but this could take up a very large
    amount of disk space.  When the number of test cases is large, it will
    take much longer to compute the median or log probability predictions
    (options "d" or "p") for a regression model than to compute only the
    mean predictions (option "n").  For binary and class models, all
    sorts of predictions take about the same amount of time.

    The median and quantiles are calculated by Monte Carlo, using a sample
    consisting of 101 points from the predictive distribution for each
    Gaussian process.  A sample of 100 points for each Gaussian process is
    used to calculate predictive probabilities for binary and class
    models.  If the model has case-by-case noise variances, a single
    test-case variance is chosen randomly for each Gaussian process (this
    is a bit sub-optimal, as it would be better to pick a new variance for
    each of the 101 points used to compute the median, and to integrate
    the variance away to produce a t-distribution when computing the log
    predictive probability - but the programming was easier this way).

    These Monte Carlo estimates are found using a random number stream
    initialized by setting the seed to one at the start of the program.
    Accuracy can be increased by repeating the same log-file/range
    combination several times, effectively increasing the sample size use.

    The 'D' option is implemented, but is of no real use, since models for
    survival analysis aren't supported, and individual medians for other
    models are the same as means.  It's here only because of net-pred.

    Each average performance figure is accompanied by +- its standard
    error (as long as there is more than one test case).

    If only inputs and targets are to be displayed (no predictions), one
    may give just a single log file with no range.  Otherwise, at least
    one Gaussian process must be specified.

    An additional option of -use-inverse or -alt-mean may appear before
    the "options" argument above.  These control the way the matrix
    operations are done, and are primarily meant for testing purposes.
    However, users might have a use for -alt-mean, which forces a possibly
    more accurate computation of the mean prediction (which is, however,
    slower, if only the mean is being computed).

                Copyright (c) 1995-2003 by Radford M. Neal