# Linear Gaussian filtering and smoothing (discrete)¶

Provided is an example of discrete, linear state-space models on which one can perform Bayesian filtering and smoothing in order to obtain a posterior distribution over a latent state trajectory based on noisy observations. In order to understand the theory behind these methods in detail we refer to [1] and [2].

References: > [1] Särkkä, Simo, and Solin, Arno. Applied Stochastic Differential Equations. Cambridge University Press, 2019.
> > [2] Särkkä, Simo. Bayesian Filtering and Smoothing. Cambridge University Press, 2013.
[1]:

import numpy as np

import probnum as pn
from probnum import filtsmooth, randvars, randprocs
from probnum.problems import TimeSeriesRegressionProblem

[2]:

rng = np.random.default_rng(seed=123)

[3]:

# Make inline plots vector graphics instead of raster graphics
%matplotlib inline
from IPython.display import set_matplotlib_formats

set_matplotlib_formats("pdf", "svg")

# Plotting
import matplotlib.pyplot as plt
import matplotlib.gridspec as gridspec

plt.style.use("../../probnum.mplstyle")

/tmp/ipykernel_125474/236124620.py:5: DeprecationWarning: set_matplotlib_formats is deprecated since IPython 7.23, directly use matplotlib_inline.backend_inline.set_matplotlib_formats()
set_matplotlib_formats("pdf", "svg")


## Linear Discrete State-Space Model: Car Tracking¶

We showcase the arguably most simple case in which we consider the following state-space model. Consider matrices $$A \in \mathbb{R}^{d \times d}$$ and $$H \in \mathbb{R}^{m \times d}$$ where $$d$$ is the state dimension and $$m$$ is the dimension of the measurements. Then we define the dynamics and the measurement model as follows:

For $$k = 1, \dots, K$$ and $$x_0 \sim \mathcal{N}(\mu_0, \Sigma_0)$$:

\begin{split}\begin{align} \boldsymbol{x}_k &\sim \mathcal{N}(\boldsymbol{A} \, \boldsymbol{x}_{k-1}, \boldsymbol{Q}) \\ \boldsymbol{y}_k &\sim \mathcal{N}(\boldsymbol{H} \, \boldsymbol{x}_k, \boldsymbol{R}) \end{align}\end{split}
This defines a dynamics model that assumes a state $$\boldsymbol{x}_k$$ in a discrete sequence of states arising from a linear projection of the previous state $$x_{k-1}$$ corrupted with additive Gaussian noise under a process noise covariance matrix $$Q$$.
Similarly, the measurements $$\boldsymbol{y}_k$$ are assumed to be linear projections of the latent state under additive Gaussian noise according to a measurement noise covariance $$R$$. In the following example we consider projections and covariances that are constant over the state and measurement trajectories (linear time invariant, or LTI). Note that this can be generalized to a linear time-varying state-space model, as well. Then $$A$$ is a function $$A: \mathbb{T} \rightarrow \mathbb{R}^{d \times d}$$ and $$H$$ is a function $$H: \mathbb{T} \rightarrow \mathbb{R}^{m \times d}$$ where $$\mathbb{T}$$ is the “time dimension”.

In other words, here, every relationship is linear and every distribution is a Gaussian distribution. Under these simplifying assumptions it is possible to obtain a filtering posterior distribution over the state trajectory $$(\boldsymbol{x}_k)_{k=1}^{K}$$ by using a Kalman Filter. The example is taken from Example 3.6 in [2].

### Define State-Space Model¶

#### I. Discrete Dynamics Model: Linear, Time-Invariant, Gaussian Transitions¶

[4]:

state_dim = 4
observation_dim = 2

[5]:

delta_t = 0.2

# Define linear transition operator
dynamics_transition_matrix = np.eye(state_dim) + delta_t * np.diag(np.ones(2), 2)

# Define process noise (covariance) matrix
noise_matrix = (
np.diag(np.array([delta_t ** 3 / 3, delta_t ** 3 / 3, delta_t, delta_t]))
+ np.diag(np.array([delta_t ** 2 / 2, delta_t ** 2 / 2]), 2)
+ np.diag(np.array([delta_t ** 2 / 2, delta_t ** 2 / 2]), -2)
)


To create a discrete, LTI Gaussian dynamics model, probnum provides the LTIGaussian class.

[6]:

# Create discrete, Linear Time-Invariant Gaussian dynamics model
noise = randvars.Normal(mean=np.zeros(state_dim), cov=noise_matrix)
dynamics_model = randprocs.markov.discrete.LTIGaussian(
transition_matrix=dynamics_transition_matrix,
noise=noise,
)


#### II. Discrete Measurement Model: Linear, Time-Invariant, Gaussian Measurements¶

[7]:

measurement_marginal_variance = 0.5
measurement_matrix = np.eye(observation_dim, state_dim)
measurement_noise_matrix = measurement_marginal_variance * np.eye(observation_dim)

[8]:

noise = randvars.Normal(mean=np.zeros(observation_dim), cov=measurement_noise_matrix)
measurement_model = randprocs.markov.discrete.LTIGaussian(
transition_matrix=measurement_matrix,
noise=noise,
)


#### III. Initial State Random Variable¶

[9]:

mu_0 = np.zeros(state_dim)
sigma_0 = 0.5 * measurement_marginal_variance * np.eye(state_dim)
initial_state_rv = randvars.Normal(mean=mu_0, cov=sigma_0)

[10]:

prior_process = randprocs.markov.MarkovSequence(
transition=dynamics_model, initrv=initial_state_rv, initarg=0.0
)


### Generate Data for the State-Space Model¶

Next, sample both latent states and noisy observations from the specified state-space model.

[11]:

time_grid = np.arange(0.0, 10.0, step=delta_t)

[12]:

latent_states, observations = randprocs.markov.utils.generate_artificial_measurements(
rng=rng,
prior_process=prior_process,
measmod=measurement_model,
times=time_grid,
)

[13]:

regression_problem = TimeSeriesRegressionProblem(
observations=observations,
locations=time_grid,
measurement_models=[measurement_model] * len(time_grid),
)


### Kalman Filtering¶

#### I. Kalman Filter¶

[14]:

kalman_filter = filtsmooth.gaussian.Kalman(prior_process)


#### II. Perform Kalman Filtering + Rauch-Tung-Striebel Smoothing¶

[15]:

state_posterior, _ = kalman_filter.filtsmooth(regression_problem)


The method filtsmooth returns a KalmanPosterior object which provides convenience functions for e.g. sampling and interpolation. We can also extract the just computed posterior smoothing state variables. This yields a list of Gaussian random variables from which we can extract the statistics in order to visualize them.

[16]:

grid = state_posterior.locations
posterior_state_rvs = (
state_posterior.states
)  # List of <num_time_points> Normal Random Variables
posterior_state_means = posterior_state_rvs.mean  # Shape: (num_time_points, state_dim)
posterior_state_covs = (
posterior_state_rvs.cov
)  # Shape: (num_time_points, state_dim, state_dim)


### Visualize Results¶

[17]:

state_fig = plt.figure()
state_fig_gs = gridspec.GridSpec(ncols=2, nrows=2, figure=state_fig)

# Plot means
mu_x_1, mu_x_2, mu_x_3, mu_x_4 = [posterior_state_means[:, i] for i in range(state_dim)]

ax_00.plot(grid, mu_x_1, label="posterior mean")
ax_01.plot(grid, mu_x_2)
ax_10.plot(grid, mu_x_3)
ax_11.plot(grid, mu_x_4)

# Plot marginal standard deviations
std_x_1, std_x_2, std_x_3, std_x_4 = [
np.sqrt(posterior_state_covs[:, i, i]) for i in range(state_dim)
]

ax_00.fill_between(
grid,
mu_x_1 - 1.96 * std_x_1,
mu_x_1 + 1.96 * std_x_1,
alpha=0.2,
label="1.96 marginal stddev",
)
ax_01.fill_between(grid, mu_x_2 - 1.96 * std_x_2, mu_x_2 + 1.96 * std_x_2, alpha=0.2)
ax_10.fill_between(grid, mu_x_3 - 1.96 * std_x_3, mu_x_3 + 1.96 * std_x_3, alpha=0.2)
ax_11.fill_between(grid, mu_x_4 - 1.96 * std_x_4, mu_x_4 + 1.96 * std_x_4, alpha=0.2)

# Plot groundtruth
obs_x_1, obs_x_2 = [observations[:, i] for i in range(observation_dim)]

ax_00.scatter(time_grid, obs_x_1, marker=".", label="measurements")
ax_01.scatter(time_grid, obs_x_2, marker=".")

ax_00.set_xlabel("t")
ax_01.set_xlabel("t")
ax_10.set_xlabel("t")
ax_11.set_xlabel("t")

ax_00.set_title(r"$x_1$")
ax_01.set_title(r"$x_2$")
ax_10.set_title(r"$x_3$")
ax_11.set_title(r"$x_4$")
handles, labels = ax_00.get_legend_handles_labels()
state_fig.legend(handles, labels, loc="center left", bbox_to_anchor=(1, 0.5))

state_fig.tight_layout()