# IntegratedWienerProcess¶

class probnum.randprocs.markov.integrator.IntegratedWienerProcess(initarg, num_derivatives=1, wiener_process_dimension=1, initrv=None, diffuse=False, forward_implementation='classic', backward_implementation='classic')

Integrated Wiener process.

Convenience access to $$\nu$$ times integrated ($$d$$ dimensional) Wiener processes.

Parameters
• initarg – Initial time point.

• num_derivatives – Number of modelled derivatives of the integrated process (‘’order’’, ‘’number of integrations’’). Optional. Default is $$\nu=1$$.

• wiener_process_dimension – Dimension of the underlying Wiener process. Optional. Default is $$d=1$$. The dimension of the integrated Wiener process itself is $$d(\nu + 1)$$.

• initrv – Law of the integrated Wiener process at the initial time point. Optional. Default is a $$d(\nu + 1)$$ dimensional standard-normal distribution.

• diffuse – Whether to instantiate a diffuse prior. A diffuse prior has large initial variances. Optional. Default is False. If True, and if an initial random variable is not passed, an initial random variable is created, where the initial covariance is of the form $$\kappa I_{d(\nu + 1)}$$ with $$\kappa=10^6$$. Diffuse priors are used when initial distributions are not known. They are common for filtering-based probabilistic ODE solvers.

• forward_implementation – Implementation of the forward-propagation in the underlying transitions. Optional. Default is classic. sqrt implementation is more computationally expensive, but also more stable.

• backward_implementation – Implementation of the backward-conditioning in the underlying transitions. Optional. Default is classic. sqrt implementation is more computationally expensive, but also more stable.

Raises

Warning – If initrv is not None and diffuse is True.

Examples

>>> iwp1 = IntegratedWienerProcess(initarg=0.)
>>> print(iwp1)
<IntegratedWienerProcess with input_dim=1, output_dim=2, dtype=float64>

>>> iwp2 = IntegratedWienerProcess(initarg=0., num_derivatives=2)
>>> print(iwp2)
<IntegratedWienerProcess with input_dim=1, output_dim=3, dtype=float64>

>>> iwp3 = IntegratedWienerProcess(initarg=0., wiener_process_dimension=10)
>>> print(iwp3)
<IntegratedWienerProcess with input_dim=1, output_dim=20, dtype=float64>

>>> iwp4 = IntegratedWienerProcess(initarg=0., num_derivatives=4, wiener_process_dimension=1)
>>> print(iwp4)
<IntegratedWienerProcess with input_dim=1, output_dim=5, dtype=float64>


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)

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)

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
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)

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)

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
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
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