GPCLA: Gaussian Process Classification Model Using the Laplace...
In duckmayr/gpmss: Gaussian Process Models for Social Science
Description
Details
Super class
Public fields
Methods
Description
Approximate inference for GP classification using the Laplace approximation;
this inference method can be used with a likelihood that accepts only values
-1 and 1, such as LikLogis.
Details
In Gaussian process (GP) classification, we consider the likelihood of
outcomes y \in \{-1, 1\} to be
∏_i σ ( y_i f(x_i) ),
where σ is a function mapping the reals to [0, 1].
In many dichotomous models in R, the dichotomous values are constrained to
be in {0, 1}; we follow the standard practice in GP models of instead using
{-1, 1} as this is more convenient and efficient for likelihood-related
computation. If y is provided with some other set of dichotomous
values, the outcomes are transformed to {-1, 1}, with a warning
(see the formula argument of the new() method for details).
We place a GP prior over the values of f(X),
which says that before observing y, we believe the values of
f(X) are distributed
f (X) ~ N ( m (X), K (X, X) ),
where m ( X ) is a mean function
and K ( X, X ) is a
covariance function.
In other words, the prior mean over function outputs is a function of the
predictors, as is the prior covariance over function outputs.
Unlike with GP regression, we cannot simply apply Bayes' rule
and derive an anlyitical posterior. However, we can use Laplace's method
to approximate the posterior with a normal distribution:
f | y, X ~ N(μ + K \nabla log p(y | \hat{f}), (K^{-1} + W)^{-1} ),
where K denotes the prior covariance function
(and we have suprressed in the notation that here it is evaluated at
X), μ is the prior mean of f, and
W = -\nabla \nabla log p(y | \hat{f})
.
For new test cases X*, f* is distributed
f* | y, X, X* ~ N(μ* + \nabla log p(y | \hat{f}),
K(X*) - K(X*, X) (K + W^{-1})^{-1} K(X, X*)),
where μ* is the prior mean function evaluated at
X*.
(See Rasmussen and Williams (2006), Section 3.4 for details).
Super class
gpmss::GPModel -> GPCLA
Public fields
yThe outcomes; should be a numeric vector.
The outcome variable should only have -1 and 1 values.
If the outcome has only two values that are not 1 and -1,
the highest value is reassigned the value 1 and the lowest
value is reassigned the value -1, with a warning.
This field is usually generated automatically during object
construction and does not generally need to be interacted
with directly by the user.
XThe predictors; should be a numeric matrix.
This field is usually generated automatically during object
construction and does not generally need to be interacted
with directly by the user.
termsThe terms object related to the
formula provided in model construction (this is useful for
later prediction).
force_all_levsA logical vector of length one recording the
user's preference for dropping unused factor levels
meanfunThe prior mean function; must be an object of class
MeanFunction
covfunThe prior covariance function; must be an object of
class CovarianceFunction
likfunThe likelihood function; must be an object of class
LikelihoodFunction
LA numeric matrix such that
L L^T = I + W^{1/2} K W^{1/2}
alphaA numeric vector equal to
\nabla log p(y | \hat{f})
sWA numeric matrix equal to W^{1/2}
post_meanA numeric vector giving the posterior mean at the
training data
post_covA numeric vector giving the posterior covariance at
the training data
prior_meanA numeric vector giving the prior mean at the
training data
marginal_effectsA list with an element for each predictor in
the model marginal effects have been requested for (via the
margins() method). Each element is itself a list.
If marginal effects on the latent function scale have been
requested, this is a list of length two,
with an element "mean", each entry i of which gives the mean of
the distribution of the marginal effect of the predictor on the
ith observation, and an element "covariance", giving the
covariance matrix for the distribution of marginal effects of
that predictor.
If marginal effects on the probability scale have been requested,
this is a list of length one with element "draws", giving the
draws of the requested marginal effects (see the Details
section of the margins() method).
average_marginal_effectsA dataframe with a row for each
predictor in the model that marginal effects have been requested
for (via the margins() method) and columns:
"Variable", giving the name of the predictor;
"Mean", giving the mean of the average marginal effect of that
predictor;
"LB", giving the lower bound on the requested confidence
interval (see the margins() method);
and "UB", giving the upper bound on the CI.
Methods
Public methods
Inherited methods
Method new()
Create a new GPModel object
Usage
GPCLA$new(
formula,
data,
likfun = LikLogis,
meanfun = MeanZero,
covfun = CovSEard,
optimize = FALSE,
force_all_levs = FALSE,
...
)
Arguments
formulaA formula object giving the variable name of the
outcomes on the left-hand side and the predictors on the
right-hand side, a la lm.
Note the outcome variable should only have -1 and 1 values.
If the outcome has only two values that are not 1 and -1,
the highest value is reassigned the value 1 and the lowest
value is reassigned the value -1, with a warning.
dataAn optional data frame where the variables in
formula are to be found. If not found there,
we search for the variables elsewhere, generally the
calling environment.
likfunAn object inheriting from class
LikelihoodFunction, or a generator for such a
class. This is the likelihood function for the GP model.
The default is LikLogis.
If a generator is provided rather than an object,
an object will be created with the default value for the
hyperparameters.
meanfunAn object inheriting from class
MeanFunction, or a generator for such a
class. This is the mean function for the GP prior (see Details).
The default is MeanZero.
If a generator is provided rather than an object,
an object will be created with the default value for the
hyperparameters.
covfunAn object inheriting from class
CovarianceFunction, or a generator for such a
class.
This is the covariance function for the GP prior (see Details).
The default is CovSEard.
If a generator is provided rather than an object,
an object will be created with the default value for the
hyperparameters.
optimizeA logical vector of length one; if TRUE,
the hyperparameters of the mean, covariance, and likelihood
functions are optimized automatically as part of the model
construction process (see Details).
The default is FALSE, meaning the hyperparameters given
at the time of creating those objects are used for inference.
force_all_levsA logical vector of length one; if TRUE,
unused factor levels in right-hand side variables are
not dropped. The default is FALSE.
...Other arguments specific to a particular model; unused.
Train the GP model, providing a characterization of the posterior
of the function of interest at the input values given in the
training data.
Method train()
Usage
GPCLA$train(...)
Arguments
...Additional arguments affecting the inference calculations
(usused).
Details
The implementation here follows very closely Algorithm 3.1 in
Rasmussen and Williams (2006). The main difference is that like
the MATLAB/Octave software GPML, we use successive over-relaxation
in the Newton steps, with an adaptive relaxation factor, though the
implementation differs somewhat from GPML.
Method predict()
Characterize the posterior predictive distribution of the function
of interest at new test points.
Usage
GPCLA$predict(newdata, ...)
Arguments
newdataA data frame containing the data for the new test
points
...Additional arguments affecting the predictions produced
Details
The implementation here follows very closely Algorithm 3.2 in
Rasmussen and Williams (2006).
Method nlml()
Caclulate the negative log marginal likelihood of the GP model.
Usage
GPCLA$nlml(...)
Arguments
...Additional arguments affecting the calculation
Method dnlml()
Caclulate the gradient of the negative log marginal likelihood of
the GP model with respect to the hyperparameters of the mean,
covariance, and likelihood functions.
Usage
GPCLA$dnlml(...)
Arguments
...Additional arguments affecting the calculation
Details
The implementation here follows very closely Algorithm 5.1 in
Rasmussen and Williams (2006).
Method margins()
Caclulate marginal effects of predictors.
Usage
GPCLA$margins(
variables = NULL,
base_categories = NULL,
differences = NULL,
indices = NULL,
ci = 0.95,
force = FALSE,
type = "link",
M = 1000,
...
)
Arguments
variablesA vector specifying the variables marginal effects
are desired for; can be an integer vector, giving the column
indices of X to get effects for, or a character vector;
if NULL (the default), effects are derived for all variables.
base_categoriesIf X contains contrasts,
the marginal effects will be the differences between the levels
and a base level. By default, the base level is the lowest
factor level of the contrast, but you can pass a named list to
change the base level for some or all of the variables assigned
to a contrast scheme.
differencesA named list of 2-length numeric vectors
may be provided giving the low (first element of each vector)
and high (second element) values to calculate the effect at for
continuous variables. Any elements giving values for binary or
categorical variables are ignored (this is meaningless for
binary variables as there are only two values to choose from,
and categorical variables should be controlled with the option
base_categories).
If NULL (the default), derivatives are used for marginal
effects of continuous predictors.
indicesA numeric vector of indices over which to average
marginal effects. If NULL (the default), all observations are
used.
ciA numeric vector of length one such that 0 < ci < 1
giving the width of the confidence interval for the average
marginal effects. The default is 0.95, corresponding to a
95% confidence interval.
forceA logical vector of length one; should marginal effects
for variables who have already had effects calculated be
re-calculated? The default is FALSE. (This is useful in case the
user has requested effects for continuous variables be
calculated as differences but would now prefer derivatives,
or vice versa).
typeA character vector of length one; should be one of
"link" or "response". If "link", marginal effects are calculated
on the latent function scale (i.e. change in f). If
"reponse", marginal effects are calculated on the probability
scale (i.e. change in probability of positive response).
The default is "link".
MThe number of marginal effect draws to take if
type == "response". The default is 1000.
...Other arguments affecting the calculations (unused)
Details
The derivative of a GP is also a GP, giving us easy access to the
distribution of marginal effects of predictors. The first time this
method is called, it calculates and stores the distribution of
pointwise partial derivatives of f for each specified predictor
in the model (if no predictors are specified, marginal effects are
calculated for all predictors).
A user can request marginal effects on the probability scale,
i.e. change in probability of positive outcome,
by specifying type = "response",
or marginal effects on the latent function scale,
i.e. change in f,
by specifying type = "link".
The latter are sometimes called "partial effects."
The partial effects can be calculated directly,
but the marginal effects must be simulated.
If a predictor is binary, instead the distribution of the difference
in f between the predictor taking the binary variable's
highest value and its lowest value is calculated.
(Or, if type = "response", the distribution in the
difference of a positive response is simulated).
If a predictor is categorical, a similar calculation is
made, but comparing each of the labels except one to a baseline
label. The user may specify to use a similar calculation for two
user specified values for continuous variables rather than using
the partial derivatives (in some cases this may be more easily
interpretable). Additional calls to margins() may calculate
marginal effects for predictors whose marginal effects had not yet
been requested, but marginal effects for variables that have already
been calculated are not re-calculated unless force = TRUE.
Every time the method is called, it stores a dataframe of average
marginal effects; it calculates the mean, lower bound, and upper
bound (the width of the confidence interval is specifiable) of the
marginal effect of each predictor over all observations, or over
specified indices of observations if provided.
Method clone()
The objects of this class are cloneable with this method.
Usage
GPCLA$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
duckmayr/gpmss documentation built on Nov. 8, 2021, 5:48 a.m.
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Description Details Super class Public fields Methods
Description
Approximate inference for GP classification using the Laplace approximation;
this inference method can be used with a likelihood that accepts only values
-1 and 1, such as LikLogis.
Details
In Gaussian process (GP) classification, we consider the likelihood of outcomes y \in \{-1, 1\} to be
∏_i σ ( y_i f(x_i) ),
where σ is a function mapping the reals to [0, 1].
In many dichotomous models in R, the dichotomous values are constrained to
be in {0, 1}; we follow the standard practice in GP models of instead using
{-1, 1} as this is more convenient and efficient for likelihood-related
computation. If y is provided with some other set of dichotomous
values, the outcomes are transformed to {-1, 1}, with a warning
(see the formula argument of the new() method for details).
We place a GP prior over the values of f(X), which says that before observing y, we believe the values of f(X) are distributed
f (X) ~ N ( m (X), K (X, X) ),
where m ( X ) is a mean function and K ( X, X ) is a covariance function. In other words, the prior mean over function outputs is a function of the predictors, as is the prior covariance over function outputs.
Unlike with GP regression, we cannot simply apply Bayes' rule and derive an anlyitical posterior. However, we can use Laplace's method to approximate the posterior with a normal distribution:
f | y, X ~ N(μ + K \nabla log p(y | \hat{f}), (K^{-1} + W)^{-1} ),
where K denotes the prior covariance function (and we have suprressed in the notation that here it is evaluated at X), μ is the prior mean of f, and W = -\nabla \nabla log p(y | \hat{f}) . For new test cases X*, f* is distributed
f* | y, X, X* ~ N(μ* + \nabla log p(y | \hat{f}), K(X*) - K(X*, X) (K + W^{-1})^{-1} K(X, X*)),
where μ* is the prior mean function evaluated at X*. (See Rasmussen and Williams (2006), Section 3.4 for details).
Super class
gpmss::GPModel -> GPCLA
Public fields
yThe outcomes; should be a numeric vector. The outcome variable should only have -1 and 1 values. If the outcome has only two values that are not 1 and -1, the highest value is reassigned the value 1 and the lowest value is reassigned the value -1, with a warning. This field is usually generated automatically during object construction and does not generally need to be interacted with directly by the user.
XThe predictors; should be a numeric matrix. This field is usually generated automatically during object construction and does not generally need to be interacted with directly by the user.
termsThe
termsobject related to the formula provided in model construction (this is useful for later prediction).force_all_levsA logical vector of length one recording the user's preference for dropping unused factor levels
meanfunThe prior mean function; must be an object of class
MeanFunctioncovfunThe prior covariance function; must be an object of class
CovarianceFunctionlikfunThe likelihood function; must be an object of class
LikelihoodFunctionLA numeric matrix such that L L^T = I + W^{1/2} K W^{1/2}
alphaA numeric vector equal to \nabla log p(y | \hat{f})
sWA numeric matrix equal to W^{1/2}
post_meanA numeric vector giving the posterior mean at the training data
post_covA numeric vector giving the posterior covariance at the training data
prior_meanA numeric vector giving the prior mean at the training data
marginal_effectsA list with an element for each predictor in the model marginal effects have been requested for (via the
margins()method). Each element is itself a list. If marginal effects on the latent function scale have been requested, this is a list of length two, with an element "mean", each entry i of which gives the mean of the distribution of the marginal effect of the predictor on the ith observation, and an element "covariance", giving the covariance matrix for the distribution of marginal effects of that predictor. If marginal effects on the probability scale have been requested, this is a list of length one with element "draws", giving the draws of the requested marginal effects (see the Details section of the margins() method).average_marginal_effectsA dataframe with a row for each predictor in the model that marginal effects have been requested for (via the
margins()method) and columns: "Variable", giving the name of the predictor; "Mean", giving the mean of the average marginal effect of that predictor; "LB", giving the lower bound on the requested confidence interval (see themargins()method); and "UB", giving the upper bound on the CI.
Methods
Public methods
Inherited methods
Method new()
Create a new GPModel object
Usage
GPCLA$new( formula, data, likfun = LikLogis, meanfun = MeanZero, covfun = CovSEard, optimize = FALSE, force_all_levs = FALSE, ... )
Arguments
formulaA formula object giving the variable name of the outcomes on the left-hand side and the predictors on the right-hand side, a la
lm. Note the outcome variable should only have -1 and 1 values. If the outcome has only two values that are not 1 and -1, the highest value is reassigned the value 1 and the lowest value is reassigned the value -1, with a warning.dataAn optional data frame where the variables in
formulaare to be found. If not found there, we search for the variables elsewhere, generally the calling environment.likfunAn object inheriting from class
LikelihoodFunction, or a generator for such a class. This is the likelihood function for the GP model. The default isLikLogis. If a generator is provided rather than an object, an object will be created with the default value for the hyperparameters.meanfunAn object inheriting from class
MeanFunction, or a generator for such a class. This is the mean function for the GP prior (see Details). The default isMeanZero. If a generator is provided rather than an object, an object will be created with the default value for the hyperparameters.covfunAn object inheriting from class
CovarianceFunction, or a generator for such a class. This is the covariance function for the GP prior (see Details). The default isCovSEard. If a generator is provided rather than an object, an object will be created with the default value for the hyperparameters.optimizeA logical vector of length one; if
TRUE, the hyperparameters of the mean, covariance, and likelihood functions are optimized automatically as part of the model construction process (see Details). The default isFALSE, meaning the hyperparameters given at the time of creating those objects are used for inference.force_all_levsA logical vector of length one; if
TRUE, unused factor levels in right-hand side variables are not dropped. The default isFALSE....Other arguments specific to a particular model; unused. Train the GP model, providing a characterization of the posterior of the function of interest at the input values given in the training data.
Method train()
Usage
GPCLA$train(...)
Arguments
...Additional arguments affecting the inference calculations (usused).
Details
The implementation here follows very closely Algorithm 3.1 in Rasmussen and Williams (2006). The main difference is that like the MATLAB/Octave software GPML, we use successive over-relaxation in the Newton steps, with an adaptive relaxation factor, though the implementation differs somewhat from GPML.
Method predict()
Characterize the posterior predictive distribution of the function of interest at new test points.
Usage
GPCLA$predict(newdata, ...)
Arguments
newdataA data frame containing the data for the new test points
...Additional arguments affecting the predictions produced
Details
The implementation here follows very closely Algorithm 3.2 in Rasmussen and Williams (2006).
Method nlml()
Caclulate the negative log marginal likelihood of the GP model.
Usage
GPCLA$nlml(...)
Arguments
...Additional arguments affecting the calculation
Method dnlml()
Caclulate the gradient of the negative log marginal likelihood of the GP model with respect to the hyperparameters of the mean, covariance, and likelihood functions.
Usage
GPCLA$dnlml(...)
Arguments
...Additional arguments affecting the calculation
Details
The implementation here follows very closely Algorithm 5.1 in Rasmussen and Williams (2006).
Method margins()
Caclulate marginal effects of predictors.
Usage
GPCLA$margins( variables = NULL, base_categories = NULL, differences = NULL, indices = NULL, ci = 0.95, force = FALSE, type = "link", M = 1000, ... )
Arguments
variablesA vector specifying the variables marginal effects are desired for; can be an integer vector, giving the column indices of X to get effects for, or a character vector; if NULL (the default), effects are derived for all variables.
base_categoriesIf X contains contrasts, the marginal effects will be the differences between the levels and a base level. By default, the base level is the lowest factor level of the contrast, but you can pass a named list to change the base level for some or all of the variables assigned to a contrast scheme.
differencesA named list of 2-length numeric vectors may be provided giving the low (first element of each vector) and high (second element) values to calculate the effect at for continuous variables. Any elements giving values for binary or categorical variables are ignored (this is meaningless for binary variables as there are only two values to choose from, and categorical variables should be controlled with the option base_categories). If NULL (the default), derivatives are used for marginal effects of continuous predictors.
indicesA numeric vector of indices over which to average marginal effects. If NULL (the default), all observations are used.
ciA numeric vector of length one such that 0 < ci < 1 giving the width of the confidence interval for the average marginal effects. The default is 0.95, corresponding to a 95% confidence interval.
forceA logical vector of length one; should marginal effects for variables who have already had effects calculated be re-calculated? The default is FALSE. (This is useful in case the user has requested effects for continuous variables be calculated as differences but would now prefer derivatives, or vice versa).
typeA character vector of length one; should be one of "link" or "response". If "link", marginal effects are calculated on the latent function scale (i.e. change in f). If "reponse", marginal effects are calculated on the probability scale (i.e. change in probability of positive response). The default is "link".
MThe number of marginal effect draws to take if
type == "response". The default is 1000....Other arguments affecting the calculations (unused)
Details
The derivative of a GP is also a GP, giving us easy access to the distribution of marginal effects of predictors. The first time this method is called, it calculates and stores the distribution of pointwise partial derivatives of f for each specified predictor in the model (if no predictors are specified, marginal effects are calculated for all predictors).
A user can request marginal effects on the probability scale,
i.e. change in probability of positive outcome,
by specifying type = "response",
or marginal effects on the latent function scale,
i.e. change in f,
by specifying type = "link".
The latter are sometimes called "partial effects."
The partial effects can be calculated directly,
but the marginal effects must be simulated.
If a predictor is binary, instead the distribution of the difference
in f between the predictor taking the binary variable's
highest value and its lowest value is calculated.
(Or, if type = "response", the distribution in the
difference of a positive response is simulated).
If a predictor is categorical, a similar calculation is
made, but comparing each of the labels except one to a baseline
label. The user may specify to use a similar calculation for two
user specified values for continuous variables rather than using
the partial derivatives (in some cases this may be more easily
interpretable). Additional calls to margins() may calculate
marginal effects for predictors whose marginal effects had not yet
been requested, but marginal effects for variables that have already
been calculated are not re-calculated unless force = TRUE.
Every time the method is called, it stores a dataframe of average marginal effects; it calculates the mean, lower bound, and upper bound (the width of the confidence interval is specifiable) of the marginal effect of each predictor over all observations, or over specified indices of observations if provided.
Method clone()
The objects of this class are cloneable with this method.
Usage
GPCLA$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
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