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 (except
log-sum-exp 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
with other models, for which this option is ignored.)
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.
W-Z If one or more of W,X,Y,Z are present, only the corresponding
terms in a log-sum-exp model are used. This allows one to see
the parts of such a model. (But see the notes below.)
0-9 If one or more of these characters are present, only the
corresponding linear 0) or non-linear (1-9) parts of the covariance
are used when computing covariances involving test cases. This
results in predictions for just these components of the function,
allowing one to see the components of an additive model. (But see
the notes below.)
x Add ten to the number of a part specified with 0-9. The effect
is cumulative, so that, for instance, options of 1x2x3 specify
that only terms 1, 12, and 23 of the covariance will be used.
z Pretend that all the Gausian conditional distributions for targets
or latent values in test cases have zero variance. Useful with
0-9 and W-Z. (See notes below.)
Some of these options are illegal for some data models. The illegal
combinations are marked with an 'X' in the following table:
binary count class real-valued no-model
r X X
p X
m X X X
n
d/D X X X
q/Q X 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. The
'r' option for count data causes the 'n' option to output the mean
prediction for the log of the Poisson mean (and no squared errors are
computed, regardless of whether targets are available).
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, count, 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.
When using the 0-9 and W-Z options, note that if the identity of a
term or component can change during a run, these options will be
meaningful only when predictions are based on a single iteration.
Note also that predictions are done for each test case individually,
for in terms of the joint distribution of all test cases.
Consequently, the components of an additive model or terms in a
log-sum-exp model may appear "noisy", because any jitter or ambiguity
in the additive decomposition is resolved andomly for each case. This
can be avoided by using the 'z' option, but one should interpret the
results with care.
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) 1995-2004 by Radford M. Neal