CovSEard: Squared Exponential Covariance Function with Automatic...
In duckmayr/gpmss: Gaussian Process Models for Social Science
Description
Details
Warning
Super class
Public fields
Methods
Description
The anisometric squared exponential covariance function, also called the
squared exponential covariance function with automatic relevance
determination.
Details
The anisometric squared exponential covariance function has D + 1
hyperparameters (where D is the number of dimensions in X),
σ_f, the scale factor, and ell,
a vector of characteristic length scales.
The scale factor governs (on average) how far from the mean the function
values can be, while the length scales govern how quickly the
function can change given movement across a particular dimension of X;
or, in other words, as the function output covariance is given as a
function of distance in the covariate space,
the length scale governs what "closeness" means,
for each dimension.
This can also be interpreted as determining the relative importance of
dimensions, particularly for dichotomous dimensions of X
(hence the moniker "automatic relevance determination").
The covariance between f(x_i)
and f(x_j)
is given by
k ( x_i, x_j ) = σ_f^2 exp [ ( x_i - x_j )^T M ( x_i - x_j ) ],
where M is a matrix whose dth diagonal entry is
1 / ell_d^2.
Warning
Note that the hyperparameters should be stored on the log scale;
that is, you should supply the log of the scale factor and the log of the
length scale (in that order).
Super class
gpmss::CovarianceFunction -> CovSEard
Public fields
nameA character vector of length one giving the covariance
function's name; "anisotropic squared exponential"
hypersA numeric vector giving the covariance function's
hyperparameters; a vector of length D+1 giving the log of the
scale factor and the log of the length scale, in that order
Methods
Public methods
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-
-
-
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Inherited methods
Method cov()
Compute function covariance
Usage
CovSEard$cov(X, Z = X, hypers = NULL)
Arguments
XThe first set of input values (should be a numeric matrix)
ZThe second set of input values (should be a numeric matrix);
The default is Z = X.
hypersA numeric vector giving hyperparameters for the
covariance function. If NULL (the default), the hypers data
member is used.
Method parameter_derivative()
Compute partial derivatives of covariance function with respect to
its hyperparameters
Usage
CovSEard$parameter_derivative(X, Z = X, hypers = NULL, param = 1, K = NULL)
Arguments
XThe first set of input values (should be a numeric matrix)
ZThe second set of input values (should be a numeric matrix);
The default is Z = X.
hypersA numeric vector giving hyperparameters for the
covariance function. If NULL (the default), the hypers data
member is used.
paramAn integer vector of length one; which element of
hypers should the derivative be taken with respect to?
If 1 (the default), the derivative is taken with respect to the
(log of the) scale factor; if 2 or more, it is taken with
respect to the param - 1 element of the
(log of the) length scale.
KAn optional provision of the pre-computed kernel;
this is useful if parameter_derivative() will be called
repeatedly (for the different hypers) without the kernel
itself changing
Method input_derivative()
Compute partial derivatives of covariance function with respect to
its inputs
Usage
CovSEard$input_derivative(
X,
Z = X,
hypers = NULL,
dimension = 1,
order = 1,
K = NULL
)
Arguments
XThe first set of input values (should be a numeric matrix)
ZThe second set of input values (should be a numeric matrix);
The default is Z = X.
hypersA numeric vector giving hyperparameters for the
covariance function. If NULL (the default), the hypers data
member is used.
dimensionan integer vector of length one giving the dimension
of X with respect to which the derivative is being taken; the
default is 1
orderAn integer vector of length one indicating whether the
first partial derivative (order = 1) is desired, or the cross
partial (order = 2); the default is 1
KAn optional provision of the pre-computed kernel;
this is useful if parameter_derivative() will be called
repeatedly (for the different hypers) without the kernel
itself changing
Method new()
Create a new CovSEard object
Usage
CovSEard$new(hypers = c(0, 0))
Arguments
hypersA numeric vector giving hyperparameters for the
covariance function; a vector of length D+1 giving the log of
the scale factor and the log of the length scale, in that order.
If the provided hypers are of length two instead, the second
element will be recycled as necessary to match the number of
columns of X when used.
Method clone()
The objects of this class are cloneable with this method.
Usage
CovSEard$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
duckmayr/gpmss documentation built on Nov. 8, 2021, 5:48 a.m.
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Description Details Warning Super class Public fields Methods
Description
The anisometric squared exponential covariance function, also called the squared exponential covariance function with automatic relevance determination.
Details
The anisometric squared exponential covariance function has D + 1 hyperparameters (where D is the number of dimensions in X), σ_f, the scale factor, and ell, a vector of characteristic length scales. The scale factor governs (on average) how far from the mean the function values can be, while the length scales govern how quickly the function can change given movement across a particular dimension of X; or, in other words, as the function output covariance is given as a function of distance in the covariate space, the length scale governs what "closeness" means, for each dimension. This can also be interpreted as determining the relative importance of dimensions, particularly for dichotomous dimensions of X (hence the moniker "automatic relevance determination").
The covariance between f(x_i) and f(x_j) is given by
k ( x_i, x_j ) = σ_f^2 exp [ ( x_i - x_j )^T M ( x_i - x_j ) ],
where M is a matrix whose dth diagonal entry is 1 / ell_d^2.
Warning
Note that the hyperparameters should be stored on the log scale; that is, you should supply the log of the scale factor and the log of the length scale (in that order).
Super class
gpmss::CovarianceFunction -> CovSEard
Public fields
nameA character vector of length one giving the covariance function's name; "anisotropic squared exponential"
hypersA numeric vector giving the covariance function's hyperparameters; a vector of length D+1 giving the log of the scale factor and the log of the length scale, in that order
Methods
Public methods
Inherited methods
Method cov()
Compute function covariance
Usage
CovSEard$cov(X, Z = X, hypers = NULL)
Arguments
XThe first set of input values (should be a numeric matrix)
ZThe second set of input values (should be a numeric matrix); The default is Z = X.
hypersA numeric vector giving hyperparameters for the covariance function. If NULL (the default), the hypers data member is used.
Method parameter_derivative()
Compute partial derivatives of covariance function with respect to its hyperparameters
Usage
CovSEard$parameter_derivative(X, Z = X, hypers = NULL, param = 1, K = NULL)
Arguments
XThe first set of input values (should be a numeric matrix)
ZThe second set of input values (should be a numeric matrix); The default is Z = X.
hypersA numeric vector giving hyperparameters for the covariance function. If NULL (the default), the hypers data member is used.
paramAn integer vector of length one; which element of
hypersshould the derivative be taken with respect to? If 1 (the default), the derivative is taken with respect to the (log of the) scale factor; if 2 or more, it is taken with respect to the param - 1 element of the (log of the) length scale.KAn optional provision of the pre-computed kernel; this is useful if parameter_derivative() will be called repeatedly (for the different hypers) without the kernel itself changing
Method input_derivative()
Compute partial derivatives of covariance function with respect to its inputs
Usage
CovSEard$input_derivative( X, Z = X, hypers = NULL, dimension = 1, order = 1, K = NULL )
Arguments
XThe first set of input values (should be a numeric matrix)
ZThe second set of input values (should be a numeric matrix); The default is Z = X.
hypersA numeric vector giving hyperparameters for the covariance function. If NULL (the default), the hypers data member is used.
dimensionan integer vector of length one giving the dimension of X with respect to which the derivative is being taken; the default is 1
orderAn integer vector of length one indicating whether the first partial derivative (order = 1) is desired, or the cross partial (order = 2); the default is 1
KAn optional provision of the pre-computed kernel; this is useful if parameter_derivative() will be called repeatedly (for the different hypers) without the kernel itself changing
Method new()
Create a new CovSEard object
Usage
CovSEard$new(hypers = c(0, 0))
Arguments
hypersA numeric vector giving hyperparameters for the covariance function; a vector of length D+1 giving the log of the scale factor and the log of the length scale, in that order. If the provided hypers are of length two instead, the second element will be recycled as necessary to match the number of columns of X when used.
Method clone()
The objects of this class are cloneable with this method.
Usage
CovSEard$clone(deep = FALSE)
Arguments
deepWhether to make a deep clone.
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