# Linear Gaussian SSMs — State Space Models: A Modern Approach

# Linear Gaussian SSMs

    ### Import standard libraries

    import abc
    from dataclasses import dataclass
    import functools
    from functools import partial
    import itertools
    import matplotlib.pyplot as plt
    import numpy as np
    from typing import Any, Callable, NamedTuple, Optional, Union, Tuple

    import jax
    import jax.numpy as jnp
    from jax import lax, vmap, jit, grad
    #from jax.scipy.special import logit
    #from jax.nn import softmax
    import jax.random as jr

    import distrax
    import optax

    import jsl
    import ssm_jax

    import inspect
    import inspect as py_inspect
    import rich
    from rich import inspect as r_inspect
    from rich import print as r_print

    def print_source(fname):
        r_print(py_inspect.getsource(fname))

# Linear Gaussian SSMs

Consider the state space model in (1) where we assume the observations are conditionally iid given the hidden states and inputs (i.e. there are no auto-regressive dependencies between the observables). We can rewrite this model as a stochastic \(\keyword{nonlinear dynamical system}\) or \(\keyword{NLDS}\) by defining the distribution of the next hidden state \(\hidden_t \in \real^{\nhidden}\) as a deterministic function of the past state \(\hidden_{t-1}\), the input \(\inputs_t \in \real^{\ninputs}\), and some random \(\keyword{process noise}\) \(\transNoise_t \in \real^{\nhidden}\)

(4)\[\begin{align} \hidden_t &= \dynamicsFn(\hidden_{t-1}, \inputs_t, \transNoise_t) \end{align}\]

where \(\transNoise_t\) is drawn from the distribution such that the induced distribution on \(\hidden_t\) matches \(p(\hidden_t|\hidden_{t-1}, \inputs_t)\). Similarly we can rewrite the observation distribution as a deterministic function of the hidden state plus \(\keyword{observation noise}\) \(\obsNoise_t \in \real^{\nobs}\):

(5)\[\begin{align} \obs_t &= \measurementFn(\hidden_{t}, \inputs_t, \obsNoise_t) \end{align}\]

If we assume additive Gaussian noise, the model becomes

(6)\[\begin{align} \hidden_t &= \dynamicsFn(\hidden_{t-1}, \inputs_t) + \transNoise_t \\ \obs_t &= \measurementFn(\hidden_{t}, \inputs_t) + \obsNoise_t \end{align}\]

where \(\transNoise_t \sim \gauss(\vzero,\transCov_t)\) and \(\obsNoise_t \sim \gauss(\vzero,\obsCov_t)\). We will call these \(\keyword{Gaussian SSMs}\).

If we additionally assume the transition function \(\dynamicsFn\) and the observation function \(\measurementFn\) are both linear, then we can rewrite the model as follows:

(7)\[\begin{align} p(\hidden_t|\hidden_{t-1},\inputs_t) &= \gauss(\hidden_t|\ldsDyn \hidden_{t-1} + \ldsDynIn \inputs_t, \transCov) \\ p(\obs_t|\hidden_t,\inputs_t) &= \gauss(\obs_t|\ldsObs \hidden_{t} + \ldsObsIn \inputs_t, \obsCov) \end{align}\]

This is called a \(\keyword{linear-Gaussian state space model}\) or \(\keyword{LG-SSM}\); it is also called a \(\keyword{linear dynamical system}\) or \(\keyword{LDS}\). We usually assume the parameters are independent of time, in which case the model is said to be time-invariant or homogeneous.

## Example: tracking a 2d point

Consider an object moving in \(\real^2\). Let the state be the position and velocity of the object, \(\hidden_t =\begin{pmatrix} u_t & \dot{u}_t & v_t & \dot{v}_t \end{pmatrix}\). (We use \(u\) and \(v\) for the two coordinates, to avoid confusion with the state and observation variables.) If we use Euler discretization, the dynamics become

(8)\[\begin{align} \underbrace{\begin{pmatrix} u_t\\ \dot{u}_t \\ v_t \\ \dot{v}_t \end{pmatrix}}_{\hidden_t} = \underbrace{ \begin{pmatrix} 1 & 0 & \Delta & 0 \\ 0 & 1 & 0 & \Delta\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} }_{\ldsDyn} \underbrace{\begin{pmatrix} u_{t-1} \\ \dot{u}_{t-1} \\ v_{t-1} \\ \dot{v}_{t-1} \end{pmatrix}}_{\hidden_{t-1}} + \transNoise_t \end{align}\]

where \(\transNoise_t \sim \gauss(\vzero,\transCov)\) is the process noise. We assume that the process noise is a white noise process added to the velocity components of the state, but not to the location, so \(\transCov = \diag(0, q, 0, q)\). This is known as a random accelerations model. (See [Sar13] p60 for a more accurate way to convert the continuous time process to discrete time.)

Now suppose that at each discrete time point we observe the location, corrupted by Gaussian noise. Thus the observation model becomes

(9)\[\begin{align} \underbrace{\begin{pmatrix} \obs_{1,t} \\ \obs_{2,t} \end{pmatrix}}_{\obs_t} &= \underbrace{ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{pmatrix} }_{\ldsObs} \underbrace{\begin{pmatrix} u_t\\ \dot{u}_t \\ v_t \\ \dot{v}_t \end{pmatrix}}_{\hidden_t} + \obsNoise_t \end{align}\]

where \(\obsNoise_t \sim \gauss(\vzero,\obsCov)\) is the observation noise. We see that the observation matrix \(\ldsObs\) simply ``extracts’’ the relevant parts of the state vector.

Suppose we sample a trajectory and corresponding set of noisy observations from this model, \((\hidden_{1:T}, \obs_{1:T}) \sim p(\hidden,\obs|\params)\). (We use diagonal observation noise, \(\obsCov = \diag(\sigma_1^2, \sigma_2^2)\).) The results are shown below.

    key = jax.random.PRNGKey(314)
    timesteps = 15
    delta = 1.0
    A = jnp.array([
        [1, 0, delta, 0],
        [0, 1, 0, delta],
        [0, 0, 1, 0],
        [0, 0, 0, 1]
    ])

    C = jnp.array([
        [1, 0, 0, 0],
        [0, 1, 0, 0]
    ])

    state_size, _ = A.shape
    observation_size, _ = C.shape

    Q = jnp.eye(state_size) * 0.001
    R = jnp.eye(observation_size) * 1.0
    # Prior parameter distribution
    mu0 = jnp.array([8, 10, 1, 0]).astype(float)
    Sigma0 = jnp.eye(state_size) * 1.0

    from jsl.lds.kalman_filter import LDS, smooth, filter

    lds = LDS(A, C, Q, R, mu0, Sigma0)
    print(lds)

    WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)

    LDS(A=DeviceArray([[1., 0., 1., 0.],
                 [0., 1., 0., 1.],
                 [0., 0., 1., 0.],
                 [0., 0., 0., 1.]], dtype=float32), C=DeviceArray([[1, 0, 0, 0],
                 [0, 1, 0, 0]], dtype=int32), Q=DeviceArray([[0.001, 0.   , 0.   , 0.   ],
                 [0.   , 0.001, 0.   , 0.   ],
                 [0.   , 0.   , 0.001, 0.   ],
                 [0.   , 0.   , 0.   , 0.001]], dtype=float32), R=DeviceArray([[1., 0.],
                 [0., 1.]], dtype=float32), mu=DeviceArray([ 8., 10.,  1.,  0.], dtype=float32), Sigma=DeviceArray([[1., 0., 0., 0.],
                 [0., 1., 0., 0.],
                 [0., 0., 1., 0.],
                 [0., 0., 0., 1.]], dtype=float32), state_offset=None, obs_offset=None, nstates=4, nobs=2)

    from jsl.demos.plot_utils import plot_ellipse

    def plot_tracking_values(observed, filtered, cov_hist, signal_label, ax):
        timesteps, _ = observed.shape
        ax.plot(observed[:, 0], observed[:, 1], marker="o", linewidth=0,
                markerfacecolor="none", markeredgewidth=2, markersize=8, label="observed", c="tab:green")
        ax.plot(*filtered[:, :2].T, label=signal_label, c="tab:red", marker="x", linewidth=2)
        for t in range(0, timesteps, 1):
            covn = cov_hist[t][:2, :2]
            plot_ellipse(covn, filtered[t, :2], ax, n_std=2.0, plot_center=False)
        ax.axis("equal")
        ax.legend()

    z_hist, x_hist = lds.sample(key, timesteps)

    fig_truth, axs = plt.subplots()
    axs.plot(x_hist[:, 0], x_hist[:, 1],
            marker="o", linewidth=0, markerfacecolor="none",
            markeredgewidth=2, markersize=8,
            label="observed", c="tab:green")

    axs.plot(z_hist[:, 0], z_hist[:, 1],
            linewidth=2, label="truth",
            marker="s", markersize=8)
    axs.legend()
    axs.axis("equal")

    (7.24486608505249, 23.857812213897706, 8.042076778411865, 11.636079120635987)

The main task is to infer the hidden states given the noisy observations, i.e., \(p(\hidden_t|\obs_{1:t},\params)\) or \(p(\hidden_t|\obs_{1:T}, \params)\) in the offline case. We discuss the topic of inference in States estimation (inference). We will usually represent this belief state by a Gaussian distribution, \(p(\hidden_t|\obs_{1:s},\params) = \gauss(\hidden_t| \mean_{t|s}, \covMat_{t|s})\), where usually \(s=t\) or \(s=T\). Sometimes we use information form, \(p(\hidden_t|\obs_{1:s},\params) = \gaussInfo(\hidden_t|\precMean_{t|s}, \precMat_{t|s})\).

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