# RandomProcess¶

class probnum.randprocs.RandomProcess(input_dim, output_dim, dtype)

Bases: Generic[typing.InputType, typing.OutputType], abc.ABC

Random processes represent uncertainty about a function.

Random processes generalize functions by encoding uncertainty over function values in their covariance function. They can be used to model (deterministic) functions which are not fully known or to define functions with stochastic output.

Parameters
• input_dim (Integral) – Input dimension of the random process.

• output_dim (Integral) – Output dimension of the random process.

• dtype (Union[dtype, str]) – Data type of the random process evaluated at an input. If object will be converted to numpy.dtype.

RandomVariable

Random variables.

GaussianProcess

Gaussian processes.

MarkovProcess

Random processes with the Markov property.

Notes

Random processes are assumed to have an (un-/countably) infinite domain. Random processes with a finite index set are represented by RandomVariable.

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([args, size, random_state]) 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

abstract __call__(args)[source]

Evaluate the random process at a set of input arguments.

Parameters

args (~InputType) – 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

abstract 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 (~InputType) – shape=(input_dim,) or (n0, input_dim) – First input to the covariance function.

• args1 (Optional[~InputType]) – 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)[source]

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

abstract mean(args)[source]

Mean function.

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

Parameters

args (~InputType) – 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

abstract 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 (~InputType) – 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(args=None, size=(), random_state=None)[source]

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
• 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.

• random_state (Union[None, int, RandomState, Generator]) – Random state of the random process. If None (or np.random), the global numpy.random state is used. If integer, it is used to seed the local RandomState instance.

Return type

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

std(args)[source]

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)[source]

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