# ARIMAX models¶

## Introduction¶

Autoregressive integrated moving average (ARIMAX) models extend ARIMA models through the inclusion of exogenous variables $$X$$. We write an $$ARIMAX(p,d,q)$$ model for some time series data $$y_{t}$$ and exogenous data $$X_{t}$$, where $$p$$ is the number of autoregressive lags, $$d$$ is the degree of differencing and $$q$$ is the number of moving average lags as:

$\Delta^{D}y_{t} = \sum^{p}_{i=1}\phi_{i}\Delta^{D}y_{t-i} + \sum^{q}_{j=1}\theta_{j}\epsilon_{t-j} + \sum^{M}_{m=1}\beta_{m}X{m,t} + \epsilon_{t}$
$\epsilon_{t} \sim N\left(0,\sigma^{2}\right)$

## Example¶

We will combine ARIMA dynamics with intervention analysis for monthly UK driver death data. There are two interventions we are interested in: the 1974 oil crisis and the introduction of the seatbelt law in 1983. We will model the effects of these events as structural breaks.

import numpy as np
import pandas as pd
import pyflux as pf
from datetime import datetime
import matplotlib.pyplot as plt
%matplotlib inline

data.index = data['time'];
data.loc[(data['time']>=1983.05), 'seat_belt'] = 1;
data.loc[(data['time']<1983.05), 'seat_belt'] = 0;
data.loc[(data['time']>=1974.00), 'oil_crisis'] = 1;
data.loc[(data['time']<1974.00), 'oil_crisis'] = 0;
plt.figure(figsize=(15,5));
plt.plot(data.index,data['drivers']);
plt.ylabel('Driver Deaths');
plt.title('Deaths of Car Drivers in Great Britain 1969-84');
plt.plot();

The structural breaks can be included via patsy notation. Below we estimate a point mass estimate $$z^{MLE}$$ of the latent variables:

model = pf.ARIMAX(data=data, formula='drivers~1+seat_belt+oil_crisis',
ar=1, ma=1, family=pf.Normal())
x = model.fit("MLE")
x.summary()

ARIMAX(1,0,1)
======================================== ================================================
Dependent Variable: drivers              Method: MLE
Start Date: 1969.08333333                Log Likelihood: -1278.7616
End Date: 1984.91666667                  AIC: 2569.5232
Number of observations: 191              BIC: 2589.0368
=========================================================================================
Latent Variable          Estimate   Std Error  z        P>|z|    95% C.I.
======================== ========== ========== ======== ======== ========================
AR(1)                    0.5002     0.0933     5.3607   0.0      (0.3173 | 0.6831)
MA(1)                    0.1713     0.0991     1.7275   0.0841   (-0.023 | 0.3656)
Beta 1                   946.585    176.9439   5.3496   0.0      (599.7749 | 1293.3952)
Beta seat_belt           -57.8924   57.8211    -1.0012  0.3167   (-171.2217 | 55.437)
Beta oil_crisis          -151.673   44.119     -3.4378  0.0006   (-238.1462 | -65.1998)
Sigma                    195.6042
=========================================================================================

We can plot the in-sample fit using plot_fit():

model.plot_fit(figsize=(15,10))

To forecast forward we need exogenous variables for future dates. Since the interventions carry forward, we can just use a slice of the existing dataframe and use func:plot_predict:

model.plot_predict(h=10, oos_data=data.iloc[-12:], past_values=100, figsize=(15,5))

## Class Description¶

class ARIMAX(data, formula, ar, ma, integ, target, family)

Autoregressive Integrated Moving Average Exogenous Variable Models (ARIMAX).

Parameter Type Description
data pd.DataFrame or np.ndarray Contains the univariate time series
formula string Patsy notation specifying the regression
ar int The number of autoregressive lags
ma int The number of moving average lags
integ int How many times to difference the data (default: 0)
target string or int Which column of DataFrame/array to use.
family pf.Family instance The distribution for the time series, e.g pf.Normal()

Attributes

latent_variables

A pf.LatentVariables() object containing information on the model latent variables, prior settings. any fitted values, starting values, and other latent variable information. When a model is fitted, this is where the latent variables are updated/stored. Please see the documentation on Latent Variables for information on attributes within this object, as well as methods for accessing the latent variable information.

Methods

Adjusts the priors for the model latent variables. The latent variables and their indices can be viewed by printing the latent_variables attribute attached to the model instance.

Parameter Type Description
index int Index of the latent variable to change
prior pf.Family instance Prior distribution, e.g. pf.Normal()

Returns: void - changes the model latent_variables attribute

fit(method, **kwargs)

Estimates latent variables for the model. User chooses an inference option and the method returns a results object, as well as updating the model’s latent_variables attribute.

Parameter Type Description
method str Inference option: e.g. ‘M-H’ or ‘MLE’

See Bayesian Inference and Classical Inference sections of the documentation for the full list of inference options. Optional parameters can be entered that are relevant to the particular mode of inference chosen.

Returns: pf.Results instance with information for the estimated latent variables

plot_fit(**kwargs)

Plots the fit of the model against the data. Optional arguments include figsize, the dimensions of the figure to plot.

Returns : void - shows a matplotlib plot

plot_ppc(T, nsims)

Plots a histogram for a posterior predictive check with a discrepancy measure of the user’s choosing. This method only works if you have fitted using Bayesian inference.

Parameter Type Description
T function Discrepancy, e.g. np.mean or np.max
nsims int How many simulations for the PPC

Returns: void - shows a matplotlib plot

plot_predict(h, oos_data, past_values, intervals, **kwargs)

Plots predictions of the model, along with intervals.

Parameter Type Description
h int How many steps to forecast ahead
oos_data pd.DataFrame Exogenous variables in a frame for h steps
past_values int How many past datapoints to plot
intervals boolean Whether to plot intervals or not

To be clear, the oos_data argument should be a DataFrame in the same format as the initial dataframe used to initialize the model instance. The reason is that to predict future values, you need to specify assumptions about exogenous variables for the future. For example, if you predict h steps ahead, the method will take the h first rows from oos_data and take the values for the exogenous variables that you asked for in the patsy formula.

Optional arguments include figsize - the dimensions of the figure to plot. Please note that if you use Maximum Likelihood or Variational Inference, the intervals shown will not reflect latent variable uncertainty. Only Metropolis-Hastings will give you fully Bayesian prediction intervals. Bayesian intervals with variational inference are not shown because of the limitation of mean-field inference in not accounting for posterior correlations.

Returns : void - shows a matplotlib plot

plot_predict_is(h, fit_once, fit_method, **kwargs)

Plots in-sample rolling predictions for the model. This means that the user pretends a last subsection of data is out-of-sample, and forecasts after each period and assesses how well they did. The user can choose whether to fit parameters once at the beginning or every time step.

Parameter Type Description
h int How many previous timesteps to use
fit_once boolean Whether to fit once, or every timestep
fit_method str Which inference option, e.g. ‘MLE’

Optional arguments include figsize - the dimensions of the figure to plot. h is an int of how many previous steps to simulate performance on.

Returns : void - shows a matplotlib plot

plot_sample(nsims, plot_data=True)

Plots samples from the posterior predictive density of the model. This method only works if you fitted the model using Bayesian inference.

Parameter Type Description
nsims int How many samples to draw
plot_data boolean Whether to plot the real data as well

Returns : void - shows a matplotlib plot

plot_z(indices, figsize)

Returns a plot of the latent variables and their associated uncertainty.

Parameter Type Description
indices int or list Which latent variable indices to plot
figsize tuple Size of the matplotlib figure

Returns : void - shows a matplotlib plot

ppc(T, nsims)

Returns a p-value for a posterior predictive check. This method only works if you have fitted using Bayesian inference.

Parameter Type Description
T function Discrepancy, e.g. np.mean or np.max
nsims int How many simulations for the PPC

Returns: int - the p-value for the discrepancy test

predict(h, oos_data, intervals=False)

Returns a DataFrame of model predictions.

Parameter Type Description
h int How many steps to forecast ahead
oos_data pd.DataFrame Exogenous variables in a frame for h steps
intervals boolean Whether to return prediction intervals

To be clear, the oos_data argument should be a DataFrame in the same format as the initial dataframe used to initialize the model instance. The reason is that to predict future values, you need to specify assumptions about exogenous variables for the future. For example, if you predict h steps ahead, the method will take the 5 first rows from oos_data and take the values for the exogenous variables that you specified as exogenous variables in the patsy formula.

Please note that if you use Maximum Likelihood or Variational Inference, the intervals shown will not reflect latent variable uncertainty. Only Metropolis-Hastings will give you fully Bayesian prediction intervals. Bayesian intervals with variational inference are not shown because of the limitation of mean-field inference in not accounting for posterior correlations.

Returns : pd.DataFrame - the model predictions

predict_is(h, fit_once, fit_method)

Returns DataFrame of in-sample rolling predictions for the model.

Parameter Type Description
h int How many previous timesteps to use
fit_once boolean Whether to fit once, or every timestep
fit_method str Which inference option, e.g. ‘MLE’

Returns : pd.DataFrame - the model predictions

sample(nsims)

Returns np.ndarray of draws of the data from the posterior predictive density. This method only works if you have fitted the model using Bayesian inference.

Parameter Type Description
nsims int How many posterior draws to take

Returns : np.ndarray - samples from the posterior predictive density.

## References¶

Box, G; Jenkins, G. (1970). Time Series Analysis: Forecasting and Control. San Francisco: Holden-Day.

Hamilton, J.D. (1994). Time Series Analysis. Taylor & Francis US.