markovflow.mean_function

Module containing mean functions.

Module Contents

class MeanFunction(name=None)

Bases: tf.Module, abc.ABC

Abstract class for mean functions.

Represents the action $u(t)$ added to the latent states:

$$dx(t)/dt = F x(t) + u(t) + L w(t)$$

…resulting in the the mean function:

$$dμ(t)/dt = F μ(t) + u(t)$$

We can then solve the pure SDE:

$$dg(t)/dt = F g(t)+ L w(t)$$

…where:

$$g(t) = x(t) - μ(t)$$

This class provides a very general interface for the function $μ(t)$.

Note

Implementations of this class should typically avoid performing computation in their __init__ method. Performing computation in the constructor conflicts with running in TensorFlow’s eager mode.

__call__(time_points: tf.Tensor) → tf.Tensor

Return the mean function evaluated at the given time points.

Parameters

time_points – A tensor with shape [... num_data].

Returns

The mean function evaluated at the time points, with shape [... num_data, obs_dim].

class ZeroMeanFunction(obs_dim: int)

Bases: MeanFunction

Represents a mean function that is zero everywhere.

Parameters

obs_dim – The dimension of the zeros to output.

__call__(time_points: tf.Tensor) → tf.Tensor

Return the mean function evaluated at the given time points.

Parameters

time_points – A tensor with shape [... num_data].

Returns

The mean function evaluated at the time points, with shape [... num_data, obs_dim].

class LinearMeanFunction(coefficient: float, obs_dim: int = 1)

Bases: MeanFunction

Represents a mean function that is linear. That is, where $m(t) = a * t$.

Parameters

• coefficient – The linear coefficient.

• obs_dim – The output dimension of the mean function.

__call__(time_points: tf.Tensor) → tf.Tensor

Return the mean function evaluated at the given time points.

Parameters

time_points – A tensor with shape [... num_data].

Returns

The mean function evaluated at the time points with shape [... num_data, obs_dim].

class ImpulseMeanFunction(action_times: tf.Tensor, state_perturbations: tf.Tensor, kernel: markovflow.kernels.SDEKernel)

Bases: MeanFunction

Represents a mean function that is an impulse action. This is given by:

$$u(t) = Σₖ uₖ δ(t - tₖ)$$

…in:

$$dx(t)/dt = F x(t) + u(t) + L w(t)$$

…and then:

$$\begin{split}&μ_(t) = 0 & t ≤ t₀\\ &μ₀(t) = exp(F (t - t₀)) u₀ &t₀ < t ≤ t₁\\ &μₖ(t) = exp(F (t - tₖ))(μₖ₋₁(tₖ) + uₖ) &tₖ < t ≤ tₖ₊₁\end{split}$$

Or:

$$\begin{split}&μ₋₁(t) = 0 & t ≤ t₀\\ &μₖ(t) = exp(F (t - tₖ))aₖ &tₖ < t ≤ tₖ₊₁\end{split}$$

If we let:

$$\begin{split}&a₋₁ = 0\\ &a₀ = u₀\\ &aₖ = Aₖaₖ₋₁ + uₖ\end{split}$$

…where:

$$Aₖ = exp(F (tₖ - tₖ₋₁))$$

…then we can write this as a LowerTriangularBlockTriDiagonal equation:

[ I             ] a₀   u₀
[-A₁, I         ] a₁   u₁
[    -A₂, I     ] a₂ = u₂
[         ᨞  ᨞  ] ⋮     ⋮
[         -Aₙ, I] aₙ   uₙ

We can then determine the $aₖ$ using a matrix solve.

Note

The effect of the action is not seen until fractionally after it is applied. That is, if an impulse is applied at time $t$, $μ(t)$ will not see the effect but $μ(t + ε)$ will.

Parameters

• action_times – The times at which actions occur, with shape [... num_actions].

• state_perturbations – The magnitude of the impulse, with shape [... num_actions, state_dim].

• kernel – The kernel that is used to generate this mean function.

__call__(time_points: tf.Tensor) → tf.Tensor

Return the mean function evaluated at the given time points.

For each time point, we find the index of the function associated with it; that is, the closest previous impulse.

This index is then used to find the parameters of the function:

$$μₖ(t) = exp(F (t - tₖ))aₖ$$

…where $tₖ < t ≤ tₖ₊₁$.

Parameters

time_points – A tensor with shape [... num_data].

Returns

The mean function evaluated at the time points, with shape [... num_data, state_dim].

class StepMeanFunction(action_times: tf.Tensor, state_perturbations: tf.Tensor, kernel: markovflow.kernels.SDEKernel)

Bases: MeanFunction

Represents a mean function that is a step action. This is given by:

$$\begin{split}&u(t) = 0 &t ≤ t₀\\ &u(t) = uₖ &tₖ < t ≤ tₖ₊₁\end{split}$$

…in:

$$dx(t)/dt = F x(t) + u(t) + L w(t)$$

Then:

$$\begin{split}&μ_(t) = 0 & t ≤ t₀\\ &μ₀(t) = -F⁻¹u₀ + exp(F (t - t₀))F⁻¹u₀ &t₀ < t ≤ t₁\\ &μₖ(t) = -F⁻¹uₖ + exp(F (t - tₖ))(F⁻¹uₖ + μₖ₋₁(tₖ)) &tₖ < t ≤ tₖ₊₁\end{split}$$

Or:

$$\begin{split}&μ₋₁(t) = 0 & t ≤ t₀\\ &μₖ(t) = aₖ + exp(F (t - tₖ))bₖ &tₖ < t ≤ tₖ₊₁\end{split}$$

If we let:

$$\begin{split}&a₋₁ = b₋₁ = 0\\ &aₖ = -F⁻¹uₖ\\ &bₖ = -aₖ + μₖ₋₁(tₖ) = Aₖbₖ₋₁ + aₖ₋₁ - aₖ\end{split}$$

…where:

$$Aₖ = exp(F (tₖ - tₖ₋₁))$$

…we can write this as a LowerTriangularBlockTriDiagonal equation:

[ I             ] b₀   [a₋₁ - a₀]
[-A₁, I         ] b₁   [a₀ - a₁]
[    -A₂, I     ] b₂ = [a₁ - a₂]
[         ᨞  ᨞  ] ⋮         ⋮
[         -Aₙ, I] bₙ   [aₙ₋₁ - aₙ]

We can then determine the $bₖ$ using a matrix solve.

Parameters

• action_times – The times at which actions occur, with shape [... num_actions].

• state_perturbations – The magnitude of the impulse, with shape [... num_actions, obs_dim].

• kernel – The kernel that is used to generate this mean function.

__call__(time_points: tf.Tensor) → tf.Tensor

Return the mean function evaluated at the given time points.

For each time point, we find the index of the function associated with it; that is, the closest previous step (call it $k$).

This index is then used to find the parameters of the function:

$$μₖ(t) = aₖ + exp(F (t - tₖ))bₖ$$

…where $tₖ < t ≤ tₖ₊₁$.

Parameters

time_points – A tensor with shape [... num_data].

Returns

The mean function evaluated at the time points, with shape [... num_data, obs_dim].