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')

Bases: MarkovProcess

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_shape=(), output_shape=(2,), dtype=float64>

>>> iwp2 = IntegratedWienerProcess(initarg=0., num_derivatives=2)
>>> print(iwp2)
<IntegratedWienerProcess with input_shape=(), output_shape=(3,), dtype=float64>

>>> iwp3 = IntegratedWienerProcess(initarg=0., wiener_process_dimension=10)
>>> print(iwp3)
<IntegratedWienerProcess with input_shape=(), output_shape=(20,), dtype=float64>

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


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.

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

NotImplementedError – If no covariance function was assigned to 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.

Raises:

NotImplementedError – If no mean function was assigned to 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)

Evaluate the random process at a set of input arguments.

Parameters:

args (floating | ndarray) – 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:

randvars._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 (InputType | None) – 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.

Raises:

NotImplementedError – General path sampling is currently not supported.

Return type:

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 function 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