Linear Gaussian filtering and smoothing¶
Provided are two examples of 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].
We provide examples for two different types of state-space model: 1. Linear, Discrete State-Space Model: Car Tracking 2. Linear, Continuous-Discrete State-Space Model: The Ornstein-Uhlenbeck Process
[1]:
import numpy as np
import probnum as pn
from probnum import filtsmooth, randvars, statespace, randprocs
from probnum.problems import TimeSeriesRegressionProblem
[2]:
np.random.seed(12345)
[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")
1. Linear Discrete State-Space Model: Car Tracking¶
We begin showcasing 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 \(x_0 \sim \mathcal{N}(\mu_0, \Sigma_0)\):
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
process_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 DiscreteLTIGaussian
class that takes - state_trans_mat
: the linear transition matrix (above: \(A\)) - shift_vec
: a force vector for affine transformations of the state (here: zero) - proc_noise_cov_mat
: the covariance matrix for the Gaussian process noise
[6]:
# Create discrete, Linear Time-Invariant Gaussian dynamics model
dynamics_model = statespace.DiscreteLTIGaussian(
state_trans_mat=dynamics_transition_matrix,
shift_vec=np.zeros(state_dim),
proc_noise_cov_mat=process_noise_matrix,
)
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]:
measurement_model = statespace.DiscreteLTIGaussian(
state_trans_mat=measurement_matrix,
shift_vec=np.zeros(observation_dim),
proc_noise_cov_mat=measurement_noise_matrix,
)
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.MarkovProcess(
transition=dynamics_model, initrv=initial_state_rv, initarg=0.0
)
Generate Data for the State-Space Model¶
statespace.generate_samples()
is used to 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 = statespace.generate_samples(
dynmod=dynamics_model,
measmod=measurement_model,
initrv=initial_state_rv,
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.Kalman(prior_process)
II. Perform Kalman Filtering + Rauch-Tung-Striebel Smoothing¶
[15]:
state_posterior, _ = kalman_filter.filtsmooth(regression_problem)
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 by querying the .state_rvs
property.[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)
ax_00 = state_fig.add_subplot(state_fig_gs[0, 0])
ax_01 = state_fig.add_subplot(state_fig_gs[0, 1])
ax_10 = state_fig.add_subplot(state_fig_gs[1, 0])
ax_11 = state_fig.add_subplot(state_fig_gs[1, 1])
# 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=".")
# Add labels etc.
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()
2. Linear Continuous-Discrete State-Space Model: Ornstein-Uhlenbeck Process¶
Now, consider we have a look at continuous dynamics. We assume that there is a continuous process that defines the dynamics of our latent space from which we collect discrete linear-Gaussian measurements (as above). Only the dynamics model becomes continuous. In particular, we formulate the dynamics as a stochastic process in terms of a linear time-invariant stochastic differential equation (LTISDE). We refer to [1] for more details. Consider matrices \(\boldsymbol{F} \in \mathbb{R}^{d \times d}\), \(\boldsymbol{L} \in \mathbb{R}^{s \times d}\) and \(H \in \mathbb{R}^{m \times d}\) where \(d\) is the state dimension and \(m\) is the dimension of the measurements. We define the following continuous-discrete state-space model:
Let \(x(t_0) \sim \mathcal{N}(\mu_0, \Sigma_0)\).
where \(\boldsymbol{\omega} \in \mathbb{R}^s\) denotes a vector of driving forces (often Brownian Motion).
Note that this can be generalized to a linear time-varying state-space model, as well. Then \(\boldsymbol{F}\) is a function \(\mathbb{T} \rightarrow \mathbb{R}^{d \times d}\), \(\boldsymbol{L}\) is a function \(\mathbb{T} \rightarrow \mathbb{R}^{s \times d}\), and \(H\) is a function \(\mathbb{T} \rightarrow \mathbb{R}^{m \times d}\) where \(\mathbb{T}\) is the “time dimension”. In the following example, however, we consider a LTI SDE, namely, the Ornstein-Uhlenbeck Process from which we observe discrete linear Gaussian measurements.
Define State-Space Model¶
I. Continuous Dynamics Model: Linear, Time-Invariant Stochastic Differential Equation (LTISDE)¶
[18]:
state_dim = 1
observation_dim = 1
[19]:
delta_t = 0.2
# Define Linear, time-invariant Stochastic Differential Equation that models
# the (scalar) Ornstein-Uhlenbeck Process
drift_constant = 0.21
dispersion_constant = np.sqrt(0.5)
drift = -drift_constant * np.eye(state_dim)
force = np.zeros(state_dim)
dispersion = dispersion_constant * np.eye(state_dim)
The continuous counterpart to the discrete LTI Gaussian model from above is provided via the LTISDE
class. It is initialized by the state space components - driftmat
: the drift matrix \(\boldsymbol{F}\) - forcevec
: a force vector that is added to the state (note that this is not \(\boldsymbol{\omega}\).) Here: zero. - dispmat
: the dispersion matrix \(\boldsymbol{L}\)
[20]:
# Create dynamics model
dynamics_model = statespace.LTISDE(
driftmat=drift,
forcevec=force,
dispmat=dispersion,
)
II. Discrete Measurement Model: Linear, Time-Invariant Gaussian Measurements¶
[21]:
measurement_marginal_variance = 0.1
measurement_matrix = np.eye(observation_dim, state_dim)
measurement_noise_matrix = measurement_marginal_variance * np.eye(observation_dim)
As above, the measurement model is discrete, LTI Gaussian. Only the dymanics are continuous (i.e. continuous-discrete).
[22]:
measurement_model = statespace.DiscreteLTIGaussian(
state_trans_mat=measurement_matrix,
shift_vec=np.zeros(observation_dim),
proc_noise_cov_mat=measurement_noise_matrix,
)
III. Initial State Random Variable¶
[23]:
mu_0 = 10.0 * np.ones(state_dim)
sigma_0 = np.eye(state_dim)
initial_state_rv = randvars.Normal(mean=mu_0, cov=sigma_0)
[24]:
prior_process = randprocs.MarkovProcess(
transition=dynamics_model, initrv=initial_state_rv, initarg=0.0
)
Generate Data for the State-Space Model¶
statespace.generate_samples()
is used to sample both latent states and noisy observations from the specified state space model.
[25]:
time_grid = np.arange(0.0, 10.0, step=delta_t)
[26]:
latent_states, observations = statespace.generate_samples(
dynmod=dynamics_model,
measmod=measurement_model,
initrv=initial_state_rv,
times=time_grid,
)
[27]:
regression_problem = TimeSeriesRegressionProblem(
observations=observations,
locations=time_grid,
measurement_models=[measurement_model] * len(time_grid),
)
Kalman Filtering¶
In fact, since we still consider a linear model, we can apply Kalman Filtering in this case again. According to Section 10 in [1], the moments of the filtering posterior in the continuous-discrete case are solutions to linear differential equations, which probnum
solves for us when invoking the <Kalman_object>.filtsmooth(...)
method.
I. Kalman Filter¶
[28]:
kalman_filter = filtsmooth.Kalman(prior_process)
II. Perform Kalman Filtering + Rauch-Tung-Striebel Smoothing¶
[29]:
state_posterior, _ = kalman_filter.filtsmooth(regression_problem)
filtsmooth
returns a KalmanPosterior
object which provides convenience functions for e.g. sampling and prediction. We can also extract the just computed posterior smoothing state variables by querying the .state_rvs
property.[30]:
grid = np.linspace(0, 11, 500)
posterior_state_rvs = state_posterior(
grid
) # List of <num_time_points> Normal Random Variables
posterior_state_means = posterior_state_rvs.mean.squeeze() # Shape: (num_time_points, )
posterior_state_covs = posterior_state_rvs.cov # Shape: (num_time_points, )
samples = state_posterior.sample(size=3, t=grid)
Visualize Results¶
[31]:
state_fig = plt.figure()
ax = state_fig.add_subplot()
# Plot means
ax.plot(grid, posterior_state_means, label="posterior mean")
# Plot samples
for smp in samples:
ax.plot(
grid,
smp[:, 0],
color="gray",
alpha=0.75,
linewidth=1,
linestyle="dashed",
label="sample",
)
# Plot marginal standard deviations
std_x = np.sqrt(np.abs(posterior_state_covs)).squeeze()
ax.fill_between(
grid,
posterior_state_means - 1.96 * std_x,
posterior_state_means + 1.96 * std_x,
alpha=0.2,
label="1.96 marginal stddev",
)
ax.scatter(time_grid, observations, marker=".", label="measurements")
# Add labels etc.
ax.set_xlabel("t")
ax.set_title(r"$x$")
# These two lines just remove duplicate labels (caused by the samples) from the legend
handles, labels = ax.get_legend_handles_labels()
by_label = dict(zip(labels, handles))
ax.legend(
by_label.values(), by_label.keys(), loc="center left", bbox_to_anchor=(1, 0.5)
)
state_fig.tight_layout()