    GP-QUANTITIES:  Quantities from log files relating to Gaussian processes.

    The quantities below relating to Gaussian process models can be
    obtained from log files (eg, for use in gp-plt).  As far as possible,
    these quantities have been defined analogously to those available for
    neural network modes.  However, since computations for test cases are
    computationally demanding for the Gaussian process models, quantities
    relating to test cases have not been included here.  Some other useful
    quantities that could take a long time to compute have also been omitted.

    Note that the generic quantities documented in quantities.doc and the
    quantities relating to Markov chain methods documented in mc-quantities.doc
    will also be available.

        xn   Array of n'th input values for training cases.
        in   Same as xn, except n must not be omitted

        on   Array of n'th output values for training cases.  This quantity
             can be found only if latent values are explicitly represented.

        yn   Array of n'th "guessed" values for training cases.  These will be
             the class probabilities for binary and class models.  This quantity
             can be found only if latent values are explicitly represented.

        z[n] Array of n'th target values for training cases
        tn   Same as zn, except n must not be omitted

        P    Log prior probability density of the current hyperparameter values.
             This is the density for the logarithmic form of the hyperparameters,
             as is used in the Markov chain methods.  NOTE: This is not entirely
             analogous to the P quantity for network models.

        l    Array of minus log probabilities for training cases, given the
             current latent values.  If not specified to be an array, the
             average minus log probability over training cases.  This quantity
             can be found only if latent values are explicitly represented.

        a[n] Array of absolute errors of n'th target for training cases,
             or sum of absolute errors if 'n' not specified.  If not
             specified to be an array, the average error over training
             cases.  This quantity can be found only if latent values are
             explicitly represented.

        b[n] Array of squared errors of n'th target for training cases, or
             sum of squared errors if 'n' not specified.  If not specified
             to be an array, the average error over training cases.  This
             quantity can be found only if latent values are explicitly
             represented.

        S[n] Scale hyperparameter for the n'th exponential part, except that
             with n of zero, it is the hyperparameter for the constant part.

        R[n] Top-level relevance hyperparameter for the n'th exponential part
             if scalar; associated lower-level hyperparameters if array.  With
             n of zero, gives the hyperparameters for the linear part.

        M[n] For n>0, Mn is an array in which the i'th element is equal to:

                  Mn@i = (Sn@i)^2 * (1 - exp(-(Rn@i)^P))

             Where P is the power specification for the n'th exponential part.
             M0 is an array in which the i'th element is (1/2)*(R0@i)^2, or it
             is an array of zeros if there is no linear part.

             With these definitions, Mn@i is half the prior expected squared
             difference in the value of the n'th component of the function,
             for any two input points that differ by one in the i'th input,
             with other inputs being equal.  It summarizes the "magnitude"
             information from the S and R quantities.

             When n is not specified, the value of M@i is the sum of Mn@i
             for all n from 0 (linear) to the number of exponential parts.
             This is a measure of the total relevance of input i.

             NOTE: This is not analogous to the M quantity for networks.

        G    Jitter (regularization) hyperparameter.  NOTE: This is not
             analogous to the G quantity for networks.

        n    For regression model: The common noise std. dev. if scalar,
             or array of output-specific noise levels.

        N    For regression model: The common noise variance if scalar, or
             array of output-specific noise levels.

        v[n] Array of case-by-case standard deviations for regression model.
             NOTE: This is not analogous to the v quantity for neural networks.

        V[n] Array of case-by-case variances for regression model. NOTE: This
             is not analogous to the v quantity for neural networks.

    When a value for 'n' is needed but not specified, it defaults to zero.
    This isn't allowed with 'i', and 't', however, since they have other
    meanings when 'n' is omitted.

    The 'o' values are the latent values defined by the Gaussian process,
    before interpretation by the model.  The 'y' values are the same as
    the 'o' values for regression models, but for binary and multi-class
    models, the 'y' values are the bit/class probabilities obtained from
    the 'o' values (note that these probabilities are based only on the
    present latent values, not on an integration over the "jitter").  For
    multi-class models, the 'z' quantity with a modifier ('n') refers to
    the representation of the class as a binary vector with the true class
    indicated by a one; with no modifier, it refers to the numeric value
    of the class.  (Since 'n' can't be omitted when using 't', this
    synonym can't be used for 'z' with no modifier.)  For multi-class
    models, the 'a' and 'b' quantities are based on the difference between
    the vector of probabilities for the classes and the vector of 0s and
    1s with a single 1 indicating the correct class.

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