Autoregressive (AR) HMM Demo

This notebook demonstrates how to construct and fit a linear autoregressive HMM. Let \(y_t\) denote the observation at time \(t\). Let \(z_t\) denote the corresponding discrete latent state.

The autoregressive hidden Markov model has the following likelihood,

\[\begin{align*} y_t \mid y_{t-1}, z_t &\sim \mathcal{N}\left(A_{z_t} y_{t-1} + b_{z_t}, Q_{z_t} \right). \end{align*}\]

(Higher-order autoregressive processes are also supported.)

This notebook will also show how inputs are passed into SSMs in Dynamax.

Setup

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%%capture
try:
    import dynamax
except ModuleNotFoundError:
    print('installing dynamax')
    %pip install -q dynamax[notebooks]
    import dynamax

import jax.numpy as jnp
import jax.random as jr
import matplotlib.pyplot as plt
import seaborn as sns

from dynamax.hidden_markov_model import LinearAutoregressiveHMM
from dynamax.utils.plotting import gradient_cmap
from dynamax.utils.utils import random_rotation

Helper functions for plotting

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sns.set_style("white")
sns.set_context("talk")

color_names = [
    "windows blue",
    "red",
    "amber",
    "faded green",
    "dusty purple",
    "orange",
    "brown",
    "pink"
]

colors = sns.xkcd_palette(color_names)
cmap = gradient_cmap(colors)

Manually construct an ARHMM

# Make a transition matrix
num_states = 5
transition_probs = (jnp.arange(num_states)**10).astype(float)
transition_probs /= transition_probs.sum()
transition_matrix = jnp.zeros((num_states, num_states))
for k, p in enumerate(transition_probs[::-1]):
    transition_matrix += jnp.roll(p * jnp.eye(num_states), k, axis=1)
    
plt.imshow(transition_matrix, vmin=0, vmax=1, cmap="Greys")
plt.xlabel("next state")
plt.ylabel("current state")
plt.title("transition matrix")
plt.colorbar()

<matplotlib.colorbar.Colorbar at 0x7fb1ac786b20>

# Make observation distributions
emission_dim = 2
num_lags = 1

keys = jr.split(jr.PRNGKey(0), num_states)
angles = jnp.linspace(0, 2 * jnp.pi, num_states, endpoint=False)
theta = jnp.pi / 25 # rotational frequency
weights = jnp.array([0.8 * random_rotation(key, emission_dim, theta=theta) for key in keys])
biases = jnp.column_stack([jnp.cos(angles), jnp.sin(angles), jnp.zeros((num_states, emission_dim - 2))])
covariances = jnp.tile(0.001 * jnp.eye(emission_dim), (num_states, 1, 1))

# Compute the stationary points
stationary_points = jnp.linalg.solve(jnp.eye(emission_dim) - weights, biases)

/opt/hostedtoolcache/Python/3.9.18/x64/lib/python3.9/site-packages/jax/_src/numpy/linalg.py:708: FutureWarning: jnp.linalg.solve: batched 1D solves with b.ndim > 1 are deprecated, and in the future will be treated as a batched 2D solve. Use solve(a, b[..., None])[..., 0] to avoid this warning.
  warnings.warn("jnp.linalg.solve: batched 1D solves with b.ndim > 1 are deprecated, "

Plot dynamics functions

if emission_dim == 2:
    lim = 5
    x = jnp.linspace(-lim, lim, 10)
    y = jnp.linspace(-lim, lim, 10)
    X, Y = jnp.meshgrid(x, y)
    xy = jnp.column_stack((X.ravel(), Y.ravel()))

    fig, axs = plt.subplots(1, num_states, figsize=(3 * num_states, 6))
    for k in range(num_states):
        A, b = weights[k], biases[k]
        dxydt_m = xy.dot(A.T) + b - xy
        axs[k].quiver(xy[:, 0], xy[:, 1],
                    dxydt_m[:, 0], dxydt_m[:, 1],
                    color=colors[k % len(colors)])

        
        axs[k].set_xlabel('$x_1$')
        axs[k].set_xticks([])
        if k == 0:
            axs[k].set_ylabel("$x_2$")
        axs[k].set_yticks([])
        axs[k].set_aspect("equal")
                        

    plt.tight_layout()

Sample emissions from the ARHMM

# Make an Autoregressive (AR) HMM
true_arhmm = LinearAutoregressiveHMM(num_states, emission_dim, num_lags=num_lags)
true_params, _ = true_arhmm.initialize(initial_probs=jnp.ones(num_states) / num_states,
                                       transition_matrix=transition_matrix,
                                       emission_weights=weights,
                                       emission_biases=biases,
                                       emission_covariances=covariances)

time_bins = 10000
true_states, emissions = true_arhmm.sample(true_params, jr.PRNGKey(0), time_bins)

# Compute the lagged emissions (aka inputs)
inputs = true_arhmm.compute_inputs(emissions)

# Plot the sampled data
fig = plt.figure(figsize=(8, 8))
for k in range(num_states):
    plt.plot(*emissions[true_states==k].T, 'o', color=colors[k],
         alpha=0.75, markersize=3)
    
plt.plot(*emissions[:1000].T, '-k', lw=0.5, alpha=0.2)
plt.xlabel("$x_1$")
plt.ylabel("$x_2$")

Text(0, 0.5, '$x_2$')

Below, we visualize each component of of the observation variable as a time series. The colors correspond to the latent state. The dotted lines represent the stationary point of the the corresponding AR state while the solid lines are the actual observations sampled from the HMM.

# Plot the emissions and the smoothed emissions
plot_slice = (0, 200)
lim = 1.05 * abs(emissions).max()
plt.figure(figsize=(8, 6))
plt.imshow(true_states[None, :],
           aspect="auto",
           cmap=cmap,
           vmin=0,
           vmax=len(colors)-1,
           extent=(0, time_bins, -lim, (emission_dim)*lim))


Ey = jnp.array(stationary_points)[true_states]
for d in range(emission_dim):
    plt.plot(emissions[:,d] + lim * d, '-k')
    plt.plot(Ey[:,d] + lim * d, ':k')

plt.xlim(plot_slice)
plt.xlabel("time")
plt.yticks(lim * jnp.arange(emission_dim), ["$y_{{{}}}$".format(d+1) for d in range(emission_dim)])

plt.tight_layout()

Fit an ARHMM

# Now fit an HMM to the emissions
key1, key2 = jr.split(jr.PRNGKey(0), 2)
test_num_states = num_states

# Initialize with K-Means
arhmm = LinearAutoregressiveHMM(num_states, emission_dim, num_lags=num_lags)
params, props = arhmm.initialize(key=jr.PRNGKey(1), method="kmeans", emissions=emissions)

# Fit with EM
fitted_params, lps = arhmm.fit_em(params, props, emissions, inputs=inputs)

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Plot the log likelihoods against the true likelihood, for comparison

true_lp = true_arhmm.marginal_log_prob(true_params, emissions, inputs=inputs)
plt.plot(lps, label="EM")
plt.plot(true_lp * jnp.ones(len(lps)), ':k', label="True")
plt.xlabel("EM Iteration")
plt.ylabel("Log Probability")
plt.legend(loc="lower right")
plt.show()

Find the most likely states

posterior = arhmm.smoother(fitted_params, emissions, inputs=inputs)
most_likely_states = arhmm.most_likely_states(fitted_params, emissions, inputs=inputs)

if emission_dim == 2:
    lim = abs(emissions).max()
    x = jnp.linspace(-lim, lim, 10)
    y = jnp.linspace(-lim, lim, 10)
    X, Y = jnp.meshgrid(x, y)
    xy = jnp.column_stack((X.ravel(), Y.ravel()))

    fig, axs = plt.subplots(2, max(num_states, test_num_states), figsize=(3 * num_states, 6))
    for i, model in enumerate([true_arhmm, arhmm]):
        for j in range(model.num_states):
            A = fitted_params.emissions.weights[j]
            b = fitted_params.emissions.biases[j]
            dxydt_m = xy.dot(A.T) + b - xy
            axs[i,j].quiver(xy[:, 0], xy[:, 1],
                        dxydt_m[:, 0], dxydt_m[:, 1],
                        color=colors[j % len(colors)])


            axs[i,j].set_xlabel('$x_1$')
            axs[i,j].set_xticks([])
            if j == 0:
                axs[i,j].set_ylabel("$x_2$")
            axs[i,j].set_yticks([])
            axs[i,j].set_aspect("equal")
                        

    plt.tight_layout()

Plot the true and inferred discrete states

plot_slice = (0, 1000)
plt.figure(figsize=(8, 4))
plt.subplot(211)
plt.imshow(true_states[None,num_lags:], aspect="auto", interpolation="none", cmap=cmap, vmin=0, vmax=len(colors)-1)
plt.xlim(plot_slice)
plt.ylabel("$z_{\\mathrm{true}}$")
plt.yticks([])

plt.subplot(212)
plt.imshow(posterior.smoothed_probs.T, aspect="auto", interpolation="none", cmap="Greys", vmin=0, vmax=1)
plt.xlim(plot_slice)
plt.ylabel("$z_{\\mathrm{inferred}}$")
plt.yticks([])
plt.xlabel("time")

plt.tight_layout()

Sample new data from the fitted model

A good (and difficult!) test of a generative model is its ability to simulate data that looks like the real data. Let’s simulate new data from an ARHMM with the fitted parameter and see what it looks like.

sampled_states, sampled_emissions = arhmm.sample(fitted_params, jr.PRNGKey(0), time_bins)

fig = plt.figure(figsize=(8, 8))
for k in range(test_num_states):
    plt.plot(*sampled_emissions[sampled_states==k].T, 'o', color=colors[k % len(colors)],
         alpha=0.75, markersize=3)
    
plt.plot(*sampled_emissions.T, '-k', lw=0.5, alpha=0.2)
plt.xlabel("$x_1$")
plt.ylabel("$x_2$")
plt.gca().set_aspect("equal")

Conclusion

This notebook showed how to sample and fit an autoregressive HMM. These models can produce complex multivariate time series by switching between different autoregressive regimes. In this model, the each discrete state has linear autoregressive dynamics, but one could imagine extending this model to nonlinear dynamics (perhaps in a future version of Dynamax!). For now, this notebook should provide a good launchpad for fitting ARHMMs to real data.