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Poisson-Zero Hidden Markov Model¶

Summary¶

The following exposition uses pymc3-hmm to both simulate and estimate a hidden Markov model (HMM) with emissions consisting of a Poisson random variable and the point at zero.

In [3]:

import numpy as np

import theano.tensor as tt

import pymc3 as pm

from pymc3_hmm.utils import compute_steady_state
from pymc3_hmm.distributions import SwitchingProcess, HMMStateSeq
from pymc3_hmm.step_methods import FFBSStep, TransMatConjugateStep

Introduction¶

Our observation model can be described as follows:

\begin{align} \label{eq:pois-zero-model} Y_t &= S_t \epsilon_t,\quad \epsilon_t \sim \operatorname{Pois}\left( \mu \right) \\ S_t &\sim \operatorname{Bern}\left( \pi_t \right) \;, \end{align}

where $y_t \sim Y_t$ are the observed values sampled from the observation distribution, $Y_t$, spanning $t \in \left\{0, \dots, T \right\}$.

The "hidden" state sequence, $\{S_t\}$, is driven by the following Markov relationship:

\begin{equation*} \operatorname{P}\left(S_t \mid S_{t-1}\right) \triangleq \Gamma_{t,t-1} \in \mathbb{R}^{2 \times 2}_{[0, 1]} \end{equation*}

The marginal state probability, $\pi_t$, is then given by

\begin{equation*} \begin{aligned} \operatorname{P}\left( S_t \right) &= \int_{S_{t-1}} \operatorname{P}\left(S_t \mid S_{t-1}\right) \operatorname{dP}\left(S_{t-1}\right) \\ &= \begin{pmatrix} \Gamma^{(0, 0)}_{t,t-1} & \Gamma^{(0, 1)}_{t,t-1} \\ \Gamma^{(1, 0)}_{t,t-1} & \Gamma^{(1, 1)}_{t,t-1} \end{pmatrix}^\top \begin{pmatrix} \pi_{t-1} \\ 1 - \pi_{t-1} \end{pmatrix} \\ &= \begin{pmatrix} \pi_{t} \\ 1 - \pi_{t} \end{pmatrix} \;. \end{aligned} \end{equation*}

Our model is effectively a zero-inflated Poisson with a simple first-order Markov process as the mixing distribution.

In the remainder of this exposition, we will assume Dirichlet priors for each row of $\Gamma_{t, t-1}$, the conjugate Gamma prior for $\mu$, and the steady-state value of $\Gamma_{t, t-1}$ for $\pi_0$.

Simulation¶

In [4]:

def create_poisson_zero_hmm(mu, observed, p_0_a=np.r_[5, 1], p_1_a=np.r_[3, 5]):

    p_0_rv = pm.Dirichlet("p_0", p_0_a)
    p_1_rv = pm.Dirichlet("p_1", p_1_a)

    P_tt = tt.stack([p_0_rv, p_1_rv])
    P_rv = pm.Deterministic("P_tt", tt.shape_padleft(P_tt))

    pi_0_tt = compute_steady_state(P_rv)

    S_rv = HMMStateSeq("S_t", P_rv, pi_0_tt, shape=np.shape(observed)[-1])

    Y_rv = SwitchingProcess(
        "Y_t", [pm.Constant.dist(0), pm.Poisson.dist(mu)], S_rv, observed=observed
    )

    return Y_rv

In [5]:

np.random.seed(2032)

mu_true = 5000

with pm.Model() as sim_model:
    _ = create_poisson_zero_hmm(mu_true, np.zeros(10000))

sim_point = pm.sample_prior_predictive(samples=1, model=sim_model)
sim_point["Y_t"] = sim_point["Y_t"].squeeze()

y_t = sim_point["Y_t"]

Estimation¶

We will use the "true" data-generating observation model to estimate the parameters $\mu$ and $\Gamma_{t, t-1}$ (the latter as rows denoted by p_0 and p_1). For demonstration purposes, we choose hyper-parameters for the $\mu$ prior that are "far" from the true $\mu$ value.

The sampling steps for $S_t$ are performed using forward-filtering backward-sampling (FFBS). The steps for $\Gamma_{t, t-1}$ are conjugate samples conditional on the sampled state sequence $\{S_t\}$ provided by FFBS. Both of these step methods are available in pymc3-hmm.

While $\mu$ could also be sampled exactly using a conjugate step—conditional on $\{S_t\}$—we instead sample this parameter using PyMC3's built-in NUTS.

In [6]:

with pm.Model() as test_model:
    E_mu, Var_mu = 1000.0, 10000.0
    mu_rv = pm.Gamma("mu", E_mu ** 2 / Var_mu, E_mu / Var_mu)

    _ = create_poisson_zero_hmm(mu_rv, y_t, p_0_a=np.r_[1, 1], p_1_a=np.r_[1, 1])

with test_model:
    transmat_step = TransMatConjugateStep([test_model.p_0, test_model.p_1], test_model.S_t)
    states_step = FFBSStep([test_model.S_t])
    mu_step = pm.NUTS([test_model.mu])

    posterior_trace = pm.sample(
        step=[transmat_step, states_step, mu_step],
        return_inferencedata=True,
        chains=1,
        progressbar=True,
    )

Sequential sampling (1 chains in 1 job)
CompoundStep
>TransMatConjugateStep: [p_0, p_1]
>FFBSStep: [S_t]
>NUTS: [mu]

100.00% [2000/2000 10:45<00:00 Sampling chain 0, 0 divergences]

Sampling 1 chain for 1_000 tune and 1_000 draw iterations (1_000 + 1_000 draws total) took 645 seconds.
Only one chain was sampled, this makes it impossible to run some convergence checks

Posterior Samples¶

In [7]:

import matplotlib.pyplot as plt


ax = pm.traceplot(
    posterior_trace.posterior,
    var_names=[
        "mu",
        "p_0",
        "p_1",
    ],
    lines=[
        ("mu", {}, [mu_true]),
        ("p_0", {}, [sim_point["p_0"]]),
        ("p_1", {}, [sim_point["p_1"]]),
    ],
    compact=True,
    figsize=(15, 15),
)

plt.tight_layout()

/home/bwillard/apps/anaconda3/envs/pymc3-hmm/lib/python3.7/site-packages/ipykernel_launcher.py:20: UserWarning: This figure was using constrained_layout==True, but that is incompatible with subplots_adjust and or tight_layout: setting constrained_layout==False.

Posterior Predictive Samples¶

In [8]:

with test_model:
    posterior_pred_trace = pm.sample_posterior_predictive(
        posterior_trace.posterior, var_names=["Y_t"]
    )

100.00% [1000/1000 01:22<00:00]

In [13]:

import pandas as pd

import arviz as az

from matplotlib.ticker import MaxNLocator

from pymc3_hmm.utils import plot_split_timeseries


plot_data = pd.DataFrame(
    {"y_t": y_t}, index=pd.date_range(start="1/1/2020", periods=len(y_t), freq="H")
)

with test_model:
    az_post_trace = az.from_pymc3(
        posterior_predictive=posterior_pred_trace,
        dims={"Y_t": ["dt"]},
        coords={"dt": plot_data.index},
    )

hdi_data = az.hdi(
    az_post_trace, hdi_prob=0.97, group="posterior_predictive", var_names=["Y_t"]
).to_dataframe()
hdi_data = hdi_data.unstack(level="hdi")

plot_data["E[Y_t]"] = az_post_trace.posterior_predictive["Y_t"].mean(axis=1).squeeze()
plot_data["E[S_t]"] = posterior_trace.posterior.S_t.mean(axis=1).squeeze()

plot_data = plot_data.merge(hdi_data, right_index=True, left_index=True)


def plot_fn(ax, data, **kwargs):
    ax.plot(
        data["y_t"],
        label="y_t",
        alpha=0.7,
        color="red",
        linewidth=0.8,
        drawstyle="steps",
    )

    ax.plot(
        data["E[Y_t]"],
        label="E[Y_t]",
        alpha=0.3,
        color="blue",
        linewidth=0.8,
        drawstyle="steps",
    )

    ax.fill_between(
        data.index,
        data[("Y_t", "lower")],
        data[("Y_t", "higher")],
        label=r"95% Y_t CI",
        color="b",
        step="pre",
        alpha=0.3,
    )


axes_split_data = plot_split_timeseries(
    plot_data,
    twin_column_name="E[S_t]",
    twin_plot_kwargs={
        "color": "green",
        "drawstyle": "steps",
        "linestyle": "--",
        "alpha": 0.4,
    },
    figsize=(15, 20),
    plot_fn=plot_fn,
)

for (_, twin_ax), _ in axes_split_data:
    _ = twin_ax.set_ylabel("E[S_t]", color=twin_ax.get_lines()[0].get_color())
    _ = twin_ax.yaxis.set_major_locator(MaxNLocator(integer=True))


plt.tight_layout()