Example: AR2 process

In this example we show how to use jax.lax.scan to avoid writing a (slow) Python for-loop. In this toy example, with --num-data=1000, the improvement is of almost 10x.

To demonstrate, we will be implementing an AR2 process. The idea is that we have some times series

\[y_0, y_1, ..., y_T\]

and we seek parameters \(c\), \(\alpha_1\), and \(\alpha_2\) such that for each \(t\) between \(2\) and \(T\), we have

\[y_t = c + \alpha_1 y_{t-1} + \alpha_2 y_{t-2} + \epsilon_t\]

where \(\epsilon_t\) is an error term.

import argparse
import os
import time

import matplotlib
import matplotlib.pyplot as plt
import numpy as np

import jax
from jax import random
import jax.numpy as jnp

import numpyro
import numpyro.distributions as dist


def ar2(y, unroll_loop=False):
    alpha_1 = numpyro.sample("alpha_1", dist.Normal(0, 1))
    alpha_2 = numpyro.sample("alpha_2", dist.Normal(0, 1))
    const = numpyro.sample("const", dist.Normal(0, 1))
    sigma = numpyro.sample("sigma", dist.Normal(0, 1))

    def transition_fn(carry, y):
        y_1, y_2 = carry
        pred = const + alpha_1 * y_1 + alpha_2 * y_2
        return (y, y_1), pred

    if unroll_loop:
        preds = []
        for i in range(2, len(y)):
            preds.append(const + alpha_1 * y[i - 1] + alpha_2 * y[i - 2])
        preds = jnp.asarray(preds)
        _, preds = jax.lax.scan(transition_fn, (y[1], y[0]), y[2:])

    mu = numpyro.deterministic("mu", preds)
    numpyro.sample("obs", dist.Normal(mu, sigma), obs=y[2:])

def run_inference(model, args, rng_key, y):
    start = time.time()
    sampler = numpyro.infer.NUTS(model)
    mcmc = numpyro.infer.MCMC(
        progress_bar=False if "NUMPYRO_SPHINXBUILD" in os.environ else True,
    mcmc.run(rng_key, y=y, unroll_loop=args.unroll_loop)
    print("\nMCMC elapsed time:", time.time() - start)
    return mcmc.get_samples()

def main(args):
    # generate artifical dataset
    num_data = args.num_data
    t = np.arange(0, num_data)
    y = np.sin(t) + np.random.randn(num_data) * 0.1

    # do inference
    rng_key, _ = random.split(random.PRNGKey(0))
    samples = run_inference(ar2, args, rng_key, y)

    # do prediction
    mean_prediction = samples["mu"].mean(axis=0)

    # make plots
    fig, ax = plt.subplots(figsize=(8, 6), constrained_layout=True)

    # plot training data
    ax.plot(t, y, color="blue", label="True values")
    # plot mean prediction
    # note that we can't make predictions for the first two points,
    # because they don't have lagged values to use for prediction.
    ax.plot(t[2:], mean_prediction, color="orange", label="Mean predictions")
    ax.set(xlabel="time", ylabel="y", title="AR2 process")


if __name__ == "__main__":
    assert numpyro.__version__.startswith("0.8.0")
    parser = argparse.ArgumentParser(description="AR2 example")
    parser.add_argument("--num-data", nargs="?", default=142, type=int)
    parser.add_argument("-n", "--num-samples", nargs="?", default=1000, type=int)
    parser.add_argument("--num-warmup", nargs="?", default=1000, type=int)
    parser.add_argument("--num-chains", nargs="?", default=1, type=int)
    parser.add_argument("--device", default="cpu", type=str, help='use "cpu" or "gpu".')
        help="whether to unroll for-loop (note: slower)",
    args = parser.parse_args()



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