    GP-EVAL:  Evaluate function drawn from a Gaussian process.

    GP-eval displays function values drawn at random from the Gaussian
    processes defined by values for hyperparameters stored at various
    iterations in a log file.  If no training data is specified, the
    functions are drawn from the prior defined by the hyperparameters; if
    there is training data, the functions are drawn from the conditional
    process given this training data, or from the conditional process
    given particular values for the function at the training points and/or
    values for the noise variance at each training point, if these are
    stored in the log file.  The points at which to evaluate the function
    are either taken from a file, or are generated on a grid.

    Note that although gp-eval is related to gp-pred, it differs in that
    it does NOT average over iterations, and it generates functions at
    random, rather than computing summaries such as the mean.

    Usage:

        gp-eval log-file range [ [-]N ] data-file [ "targets" ]

    or:

        gp-eval log-file range [ [-]N ] { / low high grid-size } [ "targets" ]

    N functions (default 1) are generated based on each iteration in the
    named log-file, within the specified range.  The range has the usual
    form, of "[low][:[high]][%mod]", with 'high' defaulting to 'low' if
    the colon is absent, and to the end of the file if the colon is
    present.  The random number seed used for the i'th function generated
    for iteration t is 100*t+i, if no "-" precedes the value N, or just i,
    if a "-" does precede N.  (The latter option may be useful to see how
    the function generated by one seed varies with varying hyperparameter
    values, though the effects are sensitive to the method used, based on
    a Cholesky decomposition.)

    For the first form, points at which to evaluate the functions are read
    from the given file specification (which can have any of the forms
    described in numin.doc), in the same way (and with the same
    transformations) as training data is read.  In the second form each
    group of the remaining arguments describes the grid along one input
    dimension, with 'low' and 'high' being the range of the grid along
    that dimension, and grid-size + 1 being the number of grid points
    spread across that range.  The number of such argument groups must be
    the same as the number of input dimensions, and currently cannot be
    more than two.

    The output consists of a section for each function drawn, with
    sections separated by blank lines.  Each section contains as many
    lines as are in the data file, or as there are grid points.  If the
    inputs are from a grid, each line contains the input values followed
    by the output values.  If the inputs come from a file, eachh line
    contains only the output values.  When a grid is used for more than
    one input variable, a blank line is output between groups of points
    where all but the last variable are the same.  There must be only one
    output value.

    If the final argument is "targets", then rather than writing the
    function values themselves, the program instead generates target
    values from these outputs, as defined by the data model.  This option
    is currently allowed only for real targets values.  The noise for
    these target values will not be autocorrelated, though if the data
    model so specifies, the generation of these values will be affected
    by autocorrelation in the noise for the training cases (if any).

    If the covariance does not include a jitter part, and there is no
    noise added (ie, "targets" is not specified for a regression model),
    then gp-eval will add a small amount (1e-6) to the diagonal of the
    covariance to try to prevent numerical problems.  Round off error may
    still make the covariance matrix appear to not be positive definite,
    however, in which case a message about not being able to find the
    Cholesky decomposition will appear.

    Note that the grid gives the actual values for the inputs.  Any
    transformations that may have been specified with "data-spec" are
    ignored.  (However, values from a data file are transformed as
    specified by data-spec.)  Similarly, it is the raw function value (or
    generated target value) that is displayed, without any transformation
    (regardless of where the input values come from).

    NOTE: The output produced with two inputs can be plotted as a surface
    using "gnuplot".  Save the output of gp-eval in a file (eg, "plt"),
    and then use the following gnuplot commands:

        set parametric
        set data style lines
        set hidden3d
        splot "plt"

    The viewpoint can be changed with the "set view" and "replot" commands.

                Copyright (c) 1995-2004 by Radford M. Neal