# IntegratedOrnsteinUhlenbeckProcess¶

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

Integrated Ornstein-Uhlenbeck process.

Convenience access to $$\nu$$ times integrated ($$d$$ dimensional) Ornstein-Uhlenbeck processes.

Parameters
• driftspeed – Drift-speed of the underlying OrnsteinUhlenbeck process.

• 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

>>> ioup1 = IntegratedOrnsteinUhlenbeckProcess(driftspeed=1., initarg=0.)
>>> print(ioup1)
<IntegratedOrnsteinUhlenbeckProcess with input_shape=(), output_shape=(2,), dtype=float64>

>>> ioup2 = IntegratedOrnsteinUhlenbeckProcess(driftspeed=1.,initarg=0., num_derivatives=2)
>>> print(ioup2)
<IntegratedOrnsteinUhlenbeckProcess with input_shape=(), output_shape=(3,), dtype=float64>

>>> ioup3 = IntegratedOrnsteinUhlenbeckProcess(driftspeed=1.,initarg=0., wiener_process_dimension=10)
>>> print(ioup3)
<IntegratedOrnsteinUhlenbeckProcess with input_shape=(), output_shape=(20,), dtype=float64>

>>> ioup4 = IntegratedOrnsteinUhlenbeckProcess(driftspeed=1.,initarg=0., num_derivatives=4, wiener_process_dimension=1)
>>> print(ioup4)
<IntegratedOrnsteinUhlenbeckProcess 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.

\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}
Return type

Kernel

dtype

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

Return type

dtype

input_ndim

Syntactic sugar for len(input_shape).

Return type

int

input_shape

Shape of inputs to the random process.

Return type
mean

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

Return type

Function

output_ndim

Syntactic sugar for len(output_shape).

Return type

int

output_shape

Shape of the random process evaluated at an input.

Return type

Methods Documentation

__call__(args)

Evaluate the random process at a set of input arguments.

Parameters

args (Union[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 (TypeVar(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
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[[TypeVar(InputType)], TypeVar(OutputType)], TypeVar(OutputType)]

std(args)

Standard deviation function.

Parameters

args (TypeVar(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 (TypeVar(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