GaussianProcess¶

class probnum.randprocs.GaussianProcess(mean, cov)¶

Bases: probnum.randprocs._random_process.RandomProcess[Union[numpy.floating, numpy.ndarray], Union[numpy.floating, numpy.ndarray]]

Gaussian processes.

A Gaussian process is a continuous stochastic process which if evaluated at a finite set of inputs returns a random variable with a normal distribution. Gaussian processes are fully characterized by their mean and covariance function.

Parameters

mean (Callable[[Union[floating, ndarray]], Union[floating, ndarray]]) – Mean function.
cov (Kernel) – Covariance function or kernel.

See also

RandomProcess: Random processes.
MarkovProcess: Random processes with the Markov property.

Examples

Define a Gaussian process with a zero mean function and RBF kernel.

>>> import numpy as np
>>> from probnum.kernels import ExpQuad
>>> from probnum.randprocs import GaussianProcess
>>> mu = lambda x : np.zeros_like(x)  # zero-mean function
>>> k = ExpQuad(input_dim=1)  # RBF kernel
>>> gp = GaussianProcess(mu, k)

Sample from the Gaussian process.

>>> x = np.linspace(-1, 1, 5)[:, None]
>>> rng = np.random.default_rng(seed=42)
>>> gp.sample(rng, x)
array([[-0.7539949 ],
       [-0.6658092 ],
       [-0.52972512],
       [ 0.0674298 ],
       [ 0.72066223]])
>>> gp.cov(x)
array([[1.        , 0.8824969 , 0.60653066, 0.32465247, 0.13533528],
       [0.8824969 , 1.        , 0.8824969 , 0.60653066, 0.32465247],
       [0.60653066, 0.8824969 , 1.        , 0.8824969 , 0.60653066],
       [0.32465247, 0.60653066, 0.8824969 , 1.        , 0.8824969 ],
       [0.13533528, 0.32465247, 0.60653066, 0.8824969 , 1.        ]])

Attributes Summary

`dtype`	Data type of (elements of) the random process evaluated at an input.
`input_dim`	Shape of inputs to the random process.
`output_dim`	Shape of the random process evaluated at an input.

Methods Summary

`__call__`(args)	Evaluate the random process at a set of input arguments.
`cov`(args0[, args1])	Covariance function or kernel.
`marginal`(args)	Batch of random variables defining the marginal distributions at the inputs.
`mean`(args)	Mean function.
`push_forward`(args, base_measure, sample)	Transform samples from a base measure into samples from the random process.
`sample`(rng[, args, size])	Sample paths from the random process.
`std`(args)	Standard deviation function.
`var`(args)	Variance function.

Attributes Documentation

dtype¶

Data type of (elements of) the random process evaluated at an input.

Return type: dtype

input_dim¶

Shape of inputs to the random process.

Return type: int

output_dim¶

Shape of the random process evaluated at an input.

Return type: int

Methods Documentation

__call__(args)[source]¶

Evaluate the random process at a set of input arguments.

Parameters: args (Union[floating, ndarray]) – shape=(input_dim,) or (n, input_dim) – Input(s) to evaluate random process at.
Returns: shape=(), (output_dim,) or (n, output_dim) – Random process evaluated at the inputs.
Return type: randvars.RandomVariable

cov(args0, args1=None)[source]¶

Covariance function or kernel.

Returns the covariance function \(\operatorname{Cov}(f(x_0), f(x_1)) = \mathbb{E}[(f(x_0) - \mathbb{E}[f(x_0)])(f(x_0) - \mathbb{E}[f( x_0)])^\top]\) of the process evaluated at \(x_0\) and \(x_1\). If only args0 is given the covariance among the components of the random process at the inputs defined by args0 is computed.

Parameters

args0 (Union[floating, ndarray]) – shape=(input_dim,) or (n0, input_dim) – First input to the covariance function.
args1 (Union[floating, ndarray, None]) – shape=(input_dim,) or (n1, input_dim) – Second input to the covariance function.

Returns

shape=(), (output_dim, output_dim), (n0, n1) or (n0, n1, output_dim, output_dim) – Covariance of the process at args0 and args1. If only args0 is given the kernel matrix \(K=k(x_0, x_0)\) is computed.

Return type

_OutputType

marginal(args)¶

Batch of random variables defining the marginal distributions at the inputs.

Parameters: args (~InputType) – shape=(input_dim,) or (n, input_dim) – Input(s) to evaluate random process at.
Return type: _RandomVariableList

mean(args)[source]¶

Mean function.

Returns the mean function evaluated at the given input(s).

Parameters: args (Union[floating, ndarray]) – shape=(input_dim,) or (n, input_dim) – Input(s) where the mean function is evaluated.
Returns: shape=(), (output_dim, ) or (n, output_dim) – Mean function of the process evaluated at inputs x.
Return type: _OutputType

push_forward(args, base_measure, sample)[source]¶

Transform samples from a base measure into samples from the random process.

This function can be used to control sampling from the random process by explicitly passing samples from a base measure evaluated at the input arguments.

Parameters

args (Union[floating, ndarray]) – Input arguments.
base_measure (Type[RandomVariable]) – Base measure. Given as a type of random variable.
sample (ndarray) – shape=(sample_size, output_dim) – Sample(s) from a base measure evaluated at the input arguments.

Return type

ndarray

sample(rng, args=None, size=())¶

Sample paths from the random process.

If no inputs are provided this function returns sample paths which are callables, otherwise random variables corresponding to the input locations are returned.

Parameters

rng (Generator) – Random number generator.
args (Optional[~InputType]) – shape=(input_dim,) or (n, input_dim) – Evaluation input(s) of the sample paths of the process. If None, sample paths, i.e. callables are returned.
size (Union[Integral, Iterable[Integral]]) – Size of the sample.

Return type

Union[Callable[[~InputType], ~OutputType], ~OutputType]

std(args)¶

Standard deviation function.

Parameters: args (~InputType) – shape=(input_dim,) or (n, input_dim) – Input(s) to the standard deviation function.
Returns: shape=(), (output_dim,) or (n, output_dim) – Standard deviation of the process at args.
Return type: _OutputType

var(args)¶

Variance function.

Returns the variance function which is the value of the covariance or kernel evaluated elementwise at args for each output dimension separately.

Parameters: args (~InputType) – shape=(input_dim,) or (n, input_dim) – Input(s) to the variance function.
Returns: shape=(), (output_dim,) or (n, output_dim) – Variance of the process at args.
Return type: _OutputType