markovflow.sde.drift

Module containing classes for different drifts.

Module Contents

class LinearDrift(A: gpflow.base.TensorType = None, b: gpflow.base.TensorType = None)

A linear drift of the form, f(x_t, t) = A_t * x_t + b_t.

Initialize the linear drift with the parameters A and b defining it as f(x_t, t) = A_t * x_t + b_t.

Parameters

• A – A parameter of the linear drift with shape batch_shape + [num_transitions, state_dim, state_dim]`

• b – b parameter of the linear drift with shape batch_shape + [num_transitions, state_dim]

set_from_ssm(ssm: markovflow.state_space_model.StateSpaceModel, dt: float)

Approximately converts a StateSpaceModel into a linear drift of the form: f(x(t), t) = A(t) * x(t) + b(t),

Given the linear drift f(x(t), t) = A(t) * x(t) + b(t), of a continuous time SDE. The conditional distribution of two states x(t_{k+1}), x(t_{k}) is given by

x(t_{k+1}) | x(t_{k}) = C_k x(t_{k}) + b_k + n_k,

with C_k = exp( int_{t_{k} < t < t_{k+1}} A(t) dt) and b_k = [C_k - I] A(t)^{-1} b(t).

Taking the limit of small-time intervals ( dt = t_{k+1} - t_{k} -> 0). The first order Taylor expansion of the exponential for small-time intervals ( dt = t_{k+1} - t_{k} -> 0), gives the approximation:

C_k ~= I + A(t_{k}) dt and b_k ~= b(t_{k}) dt

Therefore,

A = (SSM.state_transitions - I)/dt b = SSM.state_offsets / dt

to_ssm(q: gpflow.base.TensorType, transition_times: gpflow.base.TensorType, initial_mean: gpflow.base.TensorType, initial_chol_covariance: gpflow.base.TensorType)

Approximately converts a linear drift SDE of the form: f(x(t), t) = A(t) * x(t) + b(t) to a StateSpaceModel.

For details, look into the docstring of set_from_ssm function.

SSM.state_transitions = (A + I) * dt SSM.state_offsets = b * dt