markovflow.sde.sde_utils

Utility functions for SDE

Module Contents

euler_maruyama(sde: markovflow.sde.SDE, x0: tf.Tensor, time_grid: tf.Tensor) → tf.Tensor

Numerical Simulation of SDEs of type: dx(t) = f(x,t)dt + L(x,t)dB(t) using the Euler-Maruyama Method.

..math:: x(t+1) = x(t) + f(x,t)dt + L(x,t)*sqrt(dt*q)*N(0,I)

Parameters

• sde – Object of SDE class

• x0 – state at start time, t0, with shape `[num_batch, state_dim]`

• time_grid – A homogeneous time grid for simulation, `[num_transitions, ]`

Returns

Simulated SDE values, `[num_batch, num_transitions+1, state_dim]`

Note: evaluation time grid is [t0, tn], x0 value is appended for t0 time. Thus, simulated values are (num_transitions+1).

handle_tensor_shape(tensor: tf.Tensor, desired_dimensions=2)

Handle shape of the tensor according to the desired dimensions.

• if the shape is 1 more and at dimension 0 there is nothing then drop it.

• if the shape is 1 less then add a dimension at the start.

• else raise an Exception

linearize_sde(sde: markovflow.sde.SDE, transition_times: gpflow.base.TensorType, linearization_path: gpflow.probability_distributions.Gaussian, initial_state: gpflow.probability_distributions.Gaussian) → markovflow.state_space_model.StateSpaceModel

Linearizes the SDE (with fixed diffusion) on the basis of the Gaussian over states.

Note: this currently only works for sde with a state dimension of 1.

..math:: q(cdot) sim N(q_{mean}, q_{covar})

..math:: A_{i}^{*} = (E_{q(.)}[d f(x)/ dx]) * dt + I ..math:: b_{i}^{*} = (E_{q(.)}[f(x)] - A_{i}^{*} E_{q(.)}[x]) * dt

Parameters

• sde – SDE to be linearized.

• transition_times – Transition_times, [num_transitions, ]

• linearization_path – Gaussian of the states over transition times.

• initial_state – Gaussian over the initial state.

Returns

the state-space model of the linearized SDE.

squared_drift_difference_along_Gaussian_path(sde_p: markovflow.sde.SDE, linear_drift: markovflow.sde.drift.LinearDrift, q: gpflow.probability_distributions.Gaussian, dt: float, quadrature_pnts: int = 20) → tf.Tensor

Expected Square Drift difference between two SDEs

• a first one denoted by p, that can be any arbitrary SDE.

• a second which is linear, denoted by p_L, with a drift defined as f_L(x(t)) = A_L(t) x(t) + b_L(t)

Where the expectation is over a third distribution over path summarized by its mean (m) and covariance (S) for all times given by a Gaussian q.

Formally, the function calculates:

0.5 * E_{q}[||f_L(x(t)) - f_p(x(t))||^{2}_{Σ^{-1}}].

This function corresponds to the expected log density ratio: E_q log [p_L || p].

When the linear drift is of q, then the function returns the KL[q || p].

NOTE:

1. The function assumes that both the SDEs have same diffusion.

Gaussian quadrature method is used to approximate the expectation and integral over time is approximated as Riemann sum.

Parameters

• sde_p – SDE p.

• linear_drift – Linear drift representing the drift of the second SDE.

• q – Gaussian states of the path along which the drift difference is calculated.

• dt – Time-step value, float.

• quadrature_pnts – Number of quadrature points used.

Note: Batching isn’t supported.