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.

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

    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

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

    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')

    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    X       X    
     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 is calculated by Monte Carlo, using a sample consisting of
eleven points from the predictive distribution for each Gaussian
process.  A sample of one hundred 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 eleven 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).

In all these cases, the random number seed used in making predictions
is 101*x+i, where x is the iteration number and i is the number of the
test case.

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 mean 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) 1996, 1998 by Radford M. Neal