ARIMA models

Introduction

Autoregressive integrated moving average (ARIMA) models were popularised by Box and Jenkins (1970). An ARIMA model describes a univariate time series as a combination of autoregressive (AR) and moving average (MA) lags which capture the autocorrelation within the time series. The order of integration denotes how many times the series has been differenced to obtain a stationary series.

We write an \(ARIMA(p,d,q)\) model for some time series data \(y_{t}\), where \(p\) is the number of autoregressive lags, \(d\) is the degree of differencing and \(q\) is the number of moving average lags as:

\[\Delta^{D}y_{t} = \sum^{p}_{i=1}\phi_{i}\Delta^{D}y_{t-i} + \sum^{q}_{j=1}\theta_{j}\epsilon_{t-j} + \epsilon_{t}\]

\[\epsilon_{t} \sim N\left(0,\sigma^{2}\right)\]

ARIMA models are associated with a Box-Jenkins approach to time series. According to this approach, you should difference the series until it is stationary, and then use information criteria and autocorrelation plots to choose the appropriate lag order for an \(ARIMA\) process. You then apply inference to obtain latent variable estimates, and check the model to see whether the model has captured the autocorrelation in the time series. For example, you can plot the autocorrelation of the model residuals. Once you are happy, you can use the model for retrospection and forecasting.

Example

We’ll run an ARIMA Model for yearly sunspot data. First we load the data:

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

data = pd.read_csv('https://vincentarelbundock.github.io/Rdatasets/csv/datasets/sunspot.year.csv')
data.index = data['time'].values

plt.figure(figsize=(15,5))
plt.plot(data.index,data['sunspot.year'])
plt.ylabel('Sunspots')
plt.title('Yearly Sunspot Data');

We can build an ARIMA model as follows, specifying the order of model we want, as well as a pandas DataFrame or numpy array carrying the data. Here we specify an arbitrary \(ARIMA(4,0,4)\) model:

model = pf.ARIMA(data=data, ar=4, ma=4, target='sunspot.year', family=pf.Normal())

Next we estimate the latent variables. For this example we will use a maximum likelihood point mass estimate \(z^{MLE}\):

x = model.fit("MLE")
x.summary()

ARIMA(4,0,4)
======================================== ============================================
Dependent Variable: sunspot.year         Method: MLE
Start Date: 1704                         Log Likelihood: -1189.488
End Date: 1988                           AIC: 2398.9759
Number of observations: 285              BIC: 2435.5008
=====================================================================================
Latent Variable      Estimate   Std Error  z        P>|z|    95% C.I.
==================== ========== ========== ======== ======== ========================
Constant             8.0092     3.2275     2.4816   0.0131   (1.6834 | 14.3351)
AR(1)                1.6255     0.0367     44.2529  0.0      (1.5535 | 1.6975)
AR(2)                -0.4345    0.2455     -1.7701  0.0767   (-0.9157 | 0.0466)
AR(3)                -0.8819    0.2295     -3.8432  0.0001   (-1.3317 | -0.4322)
AR(4)                0.5261     0.0429     12.2515  0.0      (0.4419 | 0.6103)
MA(1)                -0.5061    0.0383     -13.2153 0.0      (-0.5812 | -0.4311)
MA(2)                -0.481     0.1361     -3.533   0.0004   (-0.7478 | -0.2142)
MA(3)                0.2511     0.1093     2.2979   0.0216   (0.0369 | 0.4653)
MA(4)                0.2846     0.0602     4.7242   0.0      (0.1665 | 0.4027)
Sigma                15.7944
=====================================================================================

We can plot the latent variables \(z^{MLE}\): using the plot_z(): method:

model.plot_z(figsize=(15,5))

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

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

We can get an idea of the performance of our model by using rolling in-sample prediction through the plot_predict_is(): method:

model.plot_predict_is(h=50, figsize=(15,5))

If we want to plot predictions, we can use the plot_predict(): method:

model.plot_predict(h=20,past_values=20,figsize=(15,5))

If we want the predictions in a DataFrame form, then we can just use the predict(): method.

Class Description

class ARIMA(data, ar, ma, integ, target, family)

Autoregressive Integrated Moving Average Models (ARIMA).

Parameter	Type	Description
data	pd.DataFrame or np.ndarray	Contains the univariate time series
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

adjust_prior(index, prior)

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, past_values, intervals, **kwargs)

Plots predictions of the model, along with intervals.

Parameter	Type	Description
h	int	How many steps to forecast ahead
past_values	int	How many past datapoints to plot
intervals	boolean	Whether to plot intervals or not

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, intervals=False)

Returns a DataFrame of model predictions.

Parameter	Type	Description
h	int	How many steps to forecast ahead
intervals	boolean	Whether to return prediction intervals

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.