nifty.re.gauss_markov module#

class GaussMarkovProcess(process: Callable, x0: float | jax.Array | nifty.re.model.LazyModel, dt: float | jax.Array, name='xi', N_steps: int = None, **kwargs)[source]#

Bases: Model

__init__(process: Callable, x0: float | Array | LazyModel, dt: float | Array, name='xi', N_steps: int = None, **kwargs)[source]#

Implements the Integrated Wiener Process.

The generalized IWP in continuous time takes the form:

\[\begin{split}\begin{align} \frac{d}{dt} x_t &= y_t + \sigma * \mathrm{asperity} * \xi^1_t , \\ \frac{d}{dt} y_t &= \sigma * \xi^2_t \end{align}\end{split}\]

where \(\xi^i_t\) are continuous time white noise processes. This is a standard IWP in x in the case that asperity is zero (None). If it is non-zero, a WP with relative strength set by asperity is added to the dynamics of the IWP.

For information on the parameters, please refer to WienerProcess. The asperity parameter may be defined analogously to sigma.

This is also the process used in the CorrelatedField to describe the deviations of the power spectra from power-laws on a double logarithmic scale. See also re.correlated_field.CorrelatedFieldMaker and references there for further information.

Parameters:

x0 (tuple, array, LazyModel) – Initial position of the IWP and its derivative. In case you set a tuple, make sure the structure is: tuple(array(mean, mean), array(std, std))
sigma (tuple, array, LazyModel) – standard deviation of the integrated Wiener process.
dt (float, array) – time steps
name (str)
asperity (tuple, float, array, LazyModel)
N_steps (int (optional)) – Option to set the number of steps in case dt is a scalar.

Notes

\(\sigma\) and asperity may also be sequences. See notes on WienerProcess for further information.

Implements the Ornstein Uhlenbeck process (OUP).

The stochastic differential equation of the OUP takes the form:

\[\frac{d}{dt} x_t + \gamma x_t = \sigma \xi_t\]

where \(\xi_t\) is continuous time white noise.

For information on the parameters, please refer to WienerProcess. The \(\gamma\) parameter may be defined analogously to \(\sigma\). Unlike the WP and IWP, the OUP is a proper process, i.E. allows for a proper steady state solution. Therefore, x0 may not be provided at all, in which case it defaults to a random variable drawn from the steady state distribution of the OUP.

Notes

sigma and gamma may also be sequences. See notes on WienerProcess for further information.

Implements the Wiener process (WP).

The WP in continuous time takes the form:

\[\frac{d}{dt} x_t = \sigma \xi_t ,\]

where \(\xi_t\) is continuous time white noise.

Parameters:

x0 (tuple, float, or LazyModel) – Initial position of the WP. Can be passed as a fixed value, or a generative Model. Passing a tuple is a shortcut to set a normal prior with mean and std equal to the first and second entry of the tuple respectively on x0.
sigma (tuple, float, Array, LazyModel) – Standard deviation of the WP. Analogously to x0 may also be passed on as a model. May also be passed as a sequence of length equal to dt in which case a different sigma is used for each time interval.
dt (float or Array of float) – Stepsizes of the process. In case it is a single float, N_steps must be provided to indicate the number of steps taken.
name (str) – Name of the key corresponding to the parameters of the WP. Default wp.
N_steps (int (optional)) – Option to set the number of steps in case dt is a scalar.

Notes

In case \(\sigma\) is time dependent, i.E. passed on as a sequence of length equal to \(\xi\), it is assumed to be constant within each timebin, i.E. \(\sigma_t = \sigma_i\) for \(t_i <= t < t_{i+1}\).

discrete_gauss_markov_process(xi: Array, x0: Array, drift: Array, diffamp: Array)[source]#

Generator for a Gauss-Markov process (GMP).

Given the discrete transition probabilities via the drift and diffamp matrices, this function produces a series res of the form:

\[\mathrm{res}_{i+1} = \mathrm{drift}_i @ \mathrm{res}_i + \mathrm{diffamp}_i @ \xi_i\]

which corresponds to a random realization of a GMP in case xi is standard normal distributed.

Parameters:

xi (Array) – Input random variables for the GMP.
x0 (Array) – Initial state of the process. The first entry of the result res_0 is always equal to x0.
drift (Array) – Matrix or sequence of matrices that describe the mean of the transition probabilities of the GMP.
diffamp (Array) – Matrix or sequence of matrices that correspond to the amplitude of the transition probabilities. For the GMP they imply that their matrix product diffamp_i @ diffamp_i.T is equal to the covariance of the transition probability iff xi is standard normal.

Returns:

Sequence of vectors of the Gauss-Markov series. The sequence is returned
as an array where the first axis corresponds to the steps of the
sequence and the second axis are the entries of the vector.

Notes

In case the sequence xi has length N (i.E. xi.shape[0] = N) the result sequence is of length N+1 as x0 is returned as the first entry of the final sequence, followed by N transitions.

integrated_wiener_process(xi: Array, x0: Array, sigma: Array, dt: Array, asperity: float | Array = None)[source]#: Implements the (generalized) Integrated Wiener process (IWP).

ornstein_uhlenbeck_process(xi: Array, x0: float, sigma: float | Array, gamma: float | Array, dt: float | Array)[source]#: Implements the Ornstein Uhlenbeck process (OUP).

scalar_gauss_markov_process(xi, x0, drift, diffamp)[source]#: Simple wrapper of discrete_gm_general for 1D scalar processes.

wiener_process(xi: Array, x0: float | Array, sigma: float | Array, dt: float | Array)[source]#: Implements the Wiener process (WP).