GPR: Gaussian Process Regression Model for Exact Inference
In duckmayr/gpmss: Gaussian Process Models for Social Science
Description
Details
Super class
Public fields
Methods
Examples
Description
Exact inference for GP regression; this inference method can only be used
with a Gaussian likelihood (see LikGauss)
Details
In Gaussian process (GP) regression, we consider that our outcomes y
are noisy observations of an unknown function of the predictors X:
y = f (X) + ε.
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.
Application of Bayes' rule in conjunction with Gaussian identities gives
us the posterior distribution of the values of f(X)
after observing y. This is a flexible Bayesian non-parametric
framework to reason about the relationship between predictors and outcomes
without making strong functional form assumptions.
For GP regression, the posterior distribution has an analytical solution:
f | y, X ~ N(μ + K K_y^{-1} (y - μ), K - K K_y^{-1} K),
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
K_y^{-1} = (K + σ_y^2 I)^{-1}.
For new test cases X*, f* is distributed
f* | y, X, X* ~ N(μ* + K(X*, X) K_y^{-1} (y - μ),
K(X*) - K(X*, X) K_y^{-1} K(X, X*)),
where μ* is the prior mean function evaluated at
X*.
Super class
gpmss::GPModel -> GPR
Public fields
yThe outcomes; should be a numeric vector.
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 = K + σ_y^2 I
alphaA numeric vector equal to
(K + σ_y^2 I)^{-1} (y - μ)
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 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.
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
GPR$new(
formula,
data,
likfun = LikGauss,
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
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.
This is restricted to be LikGauss.
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
GPR$train(...)
Arguments
...Additional arguments affecting the inference calculations
Method predict()
Characterize the posterior predictive distribution of the function
of interest at new test points.
Usage
GPR$predict(newdata, ...)
Arguments
newdataA data frame containing the data for the new test
points
...Additional arguments affecting the predictions produced
Method nlml()
Caclulate the negative log marginal likelihood of the GP model.
Usage
GPR$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
GPR$dnlml(...)
Arguments
...Additional arguments affecting the calculation
Method margins()
Caclulate marginal effects of predictors.
Usage
GPR$margins(
variables = NULL,
base_categories = NULL,
differences = NULL,
indices = NULL,
ci = 0.95,
force = FALSE,
...
)
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).
...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). 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. 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 and variance 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
GPR$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
Examples
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## Simulate data
set.seed(123)
n <- 100
x <- rnorm(n, sd = 2)
y <- sin(x) + rnorm(n, sd = sqrt(0.1))
## Train model
m <- GPR$new(y ~ x)
## Visualize posterior over f at training points
plot(x, y, pch = "+", col = "#33333387")
curve(sin(x), lty = 2, add = TRUE)
o <- order(x)
lines(x[o], m$post_mean[o])
s <- sqrt(diag(m$post_cov))[o]
h <- m$post_mean[o] + 1.96 * s
l <- m$post_mean[o] - 1.96 * s
polygon(x = c(x[o], rev(x[o])), y = c(h, rev(l)),
border = NA, col = "#87878733")
legend(x = -3, y = max(y) + 0.1, bty = "n",
pch = c(3, NA, NA, 15), lty = c(NA, 2, 1, NA),
col = c("#333333", "black", "black", "#87878733"),
legend = c("Observations", "True f(x)", "Post. mean", "95% CI"))
## Get negative log marginal likelihood of the model
m$nlml()
## Look at derivatives of negative log marginal likelihood wrt hypers
m$dnlml()
duckmayr/gpmss documentation built on Nov. 8, 2021, 5:48 a.m.
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Description Details Super class Public fields Methods Examples
Description
Exact inference for GP regression; this inference method can only be used
with a Gaussian likelihood (see LikGauss)
Details
In Gaussian process (GP) regression, we consider that our outcomes y are noisy observations of an unknown function of the predictors X:
y = f (X) + ε.
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. Application of Bayes' rule in conjunction with Gaussian identities gives us the posterior distribution of the values of f(X) after observing y. This is a flexible Bayesian non-parametric framework to reason about the relationship between predictors and outcomes without making strong functional form assumptions.
For GP regression, the posterior distribution has an analytical solution:
f | y, X ~ N(μ + K K_y^{-1} (y - μ), K - K K_y^{-1} K),
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 K_y^{-1} = (K + σ_y^2 I)^{-1}. For new test cases X*, f* is distributed
f* | y, X, X* ~ N(μ* + K(X*, X) K_y^{-1} (y - μ), K(X*) - K(X*, X) K_y^{-1} K(X, X*)),
where μ* is the prior mean function evaluated at X*.
Super class
gpmss::GPModel -> GPR
Public fields
yThe outcomes; should be a numeric vector. 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 = K + σ_y^2 I
alphaA numeric vector equal to (K + σ_y^2 I)^{-1} (y - μ)
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 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.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
GPR$new( formula, data, likfun = LikGauss, 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
lmdataAn 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. This is restricted to beLikGauss. 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
GPR$train(...)
Arguments
...Additional arguments affecting the inference calculations
Method predict()
Characterize the posterior predictive distribution of the function of interest at new test points.
Usage
GPR$predict(newdata, ...)
Arguments
newdataA data frame containing the data for the new test points
...Additional arguments affecting the predictions produced
Method nlml()
Caclulate the negative log marginal likelihood of the GP model.
Usage
GPR$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
GPR$dnlml(...)
Arguments
...Additional arguments affecting the calculation
Method margins()
Caclulate marginal effects of predictors.
Usage
GPR$margins( variables = NULL, base_categories = NULL, differences = NULL, indices = NULL, ci = 0.95, force = FALSE, ... )
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).
...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). 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. 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 and variance 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
GPR$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 | ## Simulate data
set.seed(123)
n <- 100
x <- rnorm(n, sd = 2)
y <- sin(x) + rnorm(n, sd = sqrt(0.1))
## Train model
m <- GPR$new(y ~ x)
## Visualize posterior over f at training points
plot(x, y, pch = "+", col = "#33333387")
curve(sin(x), lty = 2, add = TRUE)
o <- order(x)
lines(x[o], m$post_mean[o])
s <- sqrt(diag(m$post_cov))[o]
h <- m$post_mean[o] + 1.96 * s
l <- m$post_mean[o] - 1.96 * s
polygon(x = c(x[o], rev(x[o])), y = c(h, rev(l)),
border = NA, col = "#87878733")
legend(x = -3, y = max(y) + 0.1, bty = "n",
pch = c(3, NA, NA, 15), lty = c(NA, 2, 1, NA),
col = c("#333333", "black", "black", "#87878733"),
legend = c("Observations", "True f(x)", "Post. mean", "95% CI"))
## Get negative log marginal likelihood of the model
m$nlml()
## Look at derivatives of negative log marginal likelihood wrt hypers
m$dnlml()
|
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