How to Naïvely Generate Autoregressive Time Series Data in Python

In this post I will show you how to simulate an autoregressive (AR) process using Matplotlib and NumPy.

First, let’s define a function which pseudorandomly generates the data sequence. We’ll set an intercept parameter to set a consistent offset of the series from zero, and a sequence of autoregressive coefficients. The order of the coefficients entails the lag order that they are each applied to, so if you want to skip a lag order you can just set it to zero. Some AR series are noisier than others, and you can set that level of noise using the noise parameter. In this model we assume that the noise is a stationary normal distribution:

\[\epsilon_t \sim \mathcal{N}(0,\sigma)\]

for all \(t\).

The following implementation is a generalization of the implementation provided in Generate Fake Autoregressive Data in which I have allowed for any number of time lags. I have also allowed a user-provided random number generator to be passed.

"""Naively simulate autoregressive (AR) time series data."""

import numpy as np
from typing import List

def simulate_ar(
    intercept: float,
    coefs: List[float],
    noise: float = 0.3,
    warmup: int = 10,
    steps: int = 200,
    rng=None
    ) -> np.ndarray:
    """Simulates an autoregressive (AR) time series.

    Args:
        intercept (float): The intercept term of the AR process.
        coefs (List[float]): List of coefficients for the lagged terms.
        noise (float, optional): The standard deviation of the Gaussian noise. Default is 0.3.
        warmup (int, optional): Number of initial steps to discard to allow the process to stabilize. Default is 10.
        steps (int, optional): Number of steps to simulate after the warmup period. Default is 200.

    Returns:
        np.ndarray: Simulated AR time series data of length `steps`.
    """
    rng = np.random.default_rng() if rng is None else rng
    max_lag = len(coefs)
    draws = np.zeros(warmup + steps)
    draws[:max_lag] = intercept
    for step in range(max_lag, warmup + steps):
        draws[step] = intercept + rng.normal(0, noise)
        for lag, coef in enumerate(coefs, start=1):
            draws[step] += coef * draws[step - lag]
    return draws[warmup:]

Let us set a seed for reproducibility.

"""Set a seed for reproducible simulation."""

import numpy as np

RANDOM_SEED = 2018
rng = np.random.default_rng(RANDOM_SEED)

Now let’s choose some parameters.

"""Choose the true parameters of the autoregressive process."""

# True parameters of the AR process
intercept = 10
coefs = [-0.3, 0.1, -0.3]  # Example coefficients for AR process with 3 lags

Now let us simulate the AR process.

"""Simulate the autoregressive process using the seeded generator and parameters."""

import os
import sys

sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))

from simulate_ar import simulate_ar
from seed import rng
from parameters import intercept, coefs

ar_data = simulate_ar(intercept, coefs, warmup=2018, steps=200, rng=rng)

Finally, let us plot the simulated data using Matplotlib.

"""Plot the simulated autoregressive data and save the figure."""

import os
import sys

from matplotlib import pyplot as plt

sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))

from simulate import ar_data

fig, ax = plt.subplots(figsize=(8, 3))
ax.set_title("Generated Autoregressive Timeseries", fontsize=15)
ax.plot(ar_data)
ax.set_xlabel('Time')
ax.set_ylabel('Signal')

os.makedirs("figures", exist_ok=True)
fig.savefig("figures/autoregressive_timeseries.png", dpi=300, bbox_inches="tight")
plt.show()

This implementation is not particularly efficient in terms of computing resources. Although performance is influenced by many things, a large factor is the presence of an ordinary for loop. Upon every iteration of the loop the Python interpreter will check that all types are still valid, which is a waste of computing resources when you can assume that they are. Perhaps I will write a more efficient function, but this example should be suitable for examples and tinkering for now.

Here is the output data for the example:

"""Print the simulated autoregressive data for the example."""

import os
import sys

sys.path.insert(0, os.path.dirname(os.path.abspath(__file__)))

from simulate import ar_data

print(ar_data)

[6.941708   6.3155875  6.79896769 6.62115125 6.64690478 6.79654732
 6.32163015 6.72852581 6.35004628 6.89721308 6.92979629 6.97466588
 6.47919687 6.24134393 6.91045418 6.43927381 6.55083547 6.53331397
 6.44255904 6.58915744 6.4923775  7.07488836 6.32039603 7.21035304
 6.27709664 7.0134047  6.68872494 6.97816857 6.75025929 7.00950853
 6.39465926 6.68118717 6.85268958 6.46779684 6.76181945 7.29674199
 6.64142503 6.58699206 6.72707155 5.87192126 6.49363922 6.28287734
 7.18342675 6.70507765 6.94382677 6.39717353 6.49306093 7.02621907
 6.8610425  6.87523356 6.7267247  6.46888024 6.8280871  6.7789368
 6.77176061 6.58257732 6.33784174 6.808397   6.62658717 7.53212642
 6.59001865 7.09541224 6.4240351  6.40443905 6.08217251 7.13828813
 7.09221513 7.08928927 6.21026942 6.76872715 6.28272682 6.77029136
 6.46984318 7.0331383  6.85807573 6.64286023 6.3238929  7.03278515
 6.54674528 6.97528036 7.11517396 6.22263056 6.74882539 6.66139963
 7.0783666  6.23980551 6.67212967 6.46792713 6.66990209 6.77251836
 6.94614928 6.83357381 6.89816709 6.15734245 6.86025094 6.93488203
 6.46304183 6.91241082 6.47560317 6.89390062 7.12772216 6.41811335
 7.40498293 6.42844055 7.2251552  6.87492573 6.15158164 6.82079686
 6.790811   6.90070107 6.37238571 6.42500936 6.56438666 6.35378324
 7.26239073 6.54888149 6.80957892 6.20860224 7.48779131 6.84955546
 6.94570356 6.72697226 6.27613628 6.89544594 6.75888508 7.07931692
 6.66666166 6.3180758  6.19398727 7.09355878 6.49304741 7.55478488
 6.35099042 6.92931065 6.00868397 6.86902339 6.99545602 6.95783505
 6.64467603 6.47337945 6.78201172 6.63753378 7.48158873 6.51522663
 7.05250727 6.39296225 7.3519113  6.66425586 6.48041592 6.17408598
 6.45883478 6.22486773 6.54137246 6.34066406 6.96759892 6.72473722
 6.98529234 6.58060658 6.59542768 6.54721287 6.88949907 6.43898143
 7.00231541 6.1544984  6.68328599 6.12233025 6.16702292 6.64338878
 7.31158216 6.90580583 6.62631565 6.80756359 6.87177271 6.57651991
 6.29329644 6.78533065 6.8948178  6.61792592 7.0182514  6.22628874
 6.86051622 6.49121734 7.16799008 6.20211378 7.30671013 6.12323977
 7.34758837 6.16358606 6.88596075 6.71944393 6.36211338 7.12466492
 6.27294854 7.01723785 6.84008439 6.80771285 5.93900768 6.51765076
 6.92400731 6.54072599]