GaussianProcess¶

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

Bases: RandomProcess[Union[_SupportsArray[dtype], _NestedSequence[_SupportsArray[dtype]], bool, int, float, complex, str, bytes, _NestedSequence[Union[bool, int, float, complex, str, bytes]]], 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 – Mean function.
cov – 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.functions import Zero
>>> from probnum.randprocs.kernels import ExpQuad
>>> from probnum.randprocs import GaussianProcess
>>> mu = Zero(input_shape=())  # zero-mean function
>>> k = ExpQuad(input_shape=())  # RBF kernel
>>> gp = GaussianProcess(mu, k)

Sample from the Gaussian process.

>>> x = np.linspace(-1, 1, 5)
>>> rng = np.random.default_rng(seed=42)
>>> gp.sample(rng, x)
array([-0.7539949 , -0.6658092 , -0.52972512,  0.0674298 ,  0.72066223])
>>> gp.cov.matrix(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

`cov`	Covariance function \(k(x_0, x_1)\) of the random process.
`dtype`	Data type of (elements of) the random process evaluated at an input.
`input_ndim`	Syntactic sugar for `len(input_shape)`.
`input_shape`	Shape of inputs to the random process.
`mean`	Mean function \(m(x) := \mathbb{E}[f(x)]\) of the random process.
`output_ndim`	Syntactic sugar for `len(output_shape)`.
`output_shape`	Shape of the random process evaluated at an input.

Methods Summary

`__call__`(args)	Evaluate the random process at a set of input arguments.
`marginal`(args)	Batch of random variables defining the marginal distributions at the inputs.
`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

cov¶: Covariance function \(k(x_0, x_1)\) of the random process.

\begin{equation} k(x_0, x_1) := \mathbb{E} \left[ (f(x_0) - \mathbb{E}[f(x_0)]) (f(x_1) - \mathbb{E}[f(x_1)])^\top \right] \end{equation}

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

input_ndim¶: Syntactic sugar for len(input_shape).

input_shape¶: Shape of inputs to the random process.

mean¶: Mean function \(m(x) := \mathbb{E}[f(x)]\) of the random process.

output_ndim¶: Syntactic sugar for len(output_shape).

output_shape¶: Shape of the random process evaluated at an input.

Methods Documentation

__call__(args)[source]¶

Evaluate the random process at a set of input arguments.

Parameters: args (ArrayLike) – shape= batch_shape + input_shape – (Batch of) input(s) at which to evaluate the random process. Currently, we require batch_shape to have at most one dimension.
Returns: shape= batch_shape + output_shape – Random process evaluated at the input(s).
Return type: randvars.RandomVariable

marginal(args)¶

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

Parameters: args (InputType) – shape= batch_shape + input_shape – (Batch of) input(s) at which to evaluate the random process. Currently, we require batch_shape to have at most one dimension.
Return type: _RandomVariableList

push_forward(args, base_measure, sample)¶

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 (InputType) – Input arguments.
base_measure (Type[RandomVariable]) – Base measure. Given as a type of random variable.
sample (ndarray) – shape= sample_shape + input_shape – (Batch of) input(s) at which to evaluate the random process. Currently, we require sample_shape to have at most one dimension.

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= size + input_shape – (Batch of) input(s) at which the sample paths will be evaluated. Currently, we require size to have at most one dimension. If None, sample paths, i.e. callables are returned.
size (ShapeLike) – Size of the sample.

Return type

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

std(args)¶

Standard deviation function.

Parameters: args (InputType) – shape= batch_shape + input_shape – (Batch of) input(s) at which to evaluate the standard deviation function.
Returns: shape= batch_shape + output_shape – 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= batch_shape + input_shape – (Batch of) input(s) at which to evaluate the variance function.
Returns: shape= batch_shape + output_shape – Variance of the process at args.
Return type: OutputType