Introduction¶

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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}]

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.

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

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

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:

Class Description¶

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

Autoregressive Integrated Moving Average Exogenous Variable Models (ARIMAX).

ParameterTypeDescription
datapd.DataFrame or np.ndarrayContains the univariate time series
formulastringPatsy notation specifying the regression
arintThe number of autoregressive lags
maintThe number of moving average lags
integintHow many times to difference the data(default: 0)
targetstring or intWhich column of DataFrame/array to use.
familypf.Family instanceThe 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 variableinformation. 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 thisobject, as well as methods for accessing the latent variable information.

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Methods

adjust_prior(index, prior)

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

ParameterTypeDescription
indexintIndex of the latent variable to change
priorpf.Family instancePrior 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 themethod returns a results object, as well as updating the model’s latent_variablesattribute.

ParameterTypeDescription
methodstrInference option: e.g. ‘M-H’ or ‘MLE’
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See Bayesian Inference and Classical Inference sections of the documentation for thefull list of inference options. Optional parameters can be entered that are relevantto 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 theuser’s choosing. This method only works if you have fitted using Bayesian inference.

ParameterTypeDescription
TfunctionDiscrepancy, e.g. np.mean or np.max
nsimsintHow 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.

ParameterTypeDescription
hintHow many steps to forecast ahead
oos_datapd.DataFrameExogenous variables in a frame for h steps
past_valuesintHow many past datapoints to plot
intervalsbooleanWhether to plot intervals or not

To be clear, the oos_data argument should be a DataFrame in the same format as the initialdataframe 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 youpredict h steps ahead, the method will take the h first rows from oos_data and take thevalues for the exogenous variables that you asked for in the patsy formula.

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Optional arguments include figsize - the dimensions of the figure to plot. Please notethat if you use Maximum Likelihood or Variational Inference, the intervals shown will notreflect latent variable uncertainty. Only Metropolis-Hastings will give you fully Bayesianprediction intervals. Bayesian intervals with variational inference are not shown becauseof 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 alast subsection of data is out-of-sample, and forecasts after each period and assesseshow well they did. The user can choose whether to fit parameters once at the beginningor every time step.

ParameterTypeDescription
hintHow many previous timesteps to use
fit_oncebooleanWhether to fit once, or every timestep
fit_methodstrWhich 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 worksif you fitted the model using Bayesian inference.

ParameterTypeDescription
nsimsintHow many samples to draw
plot_databooleanWhether 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.

ParameterTypeDescription
indicesint or listWhich latent variable indices to plot
figsizetupleSize 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 havefitted using Bayesian inference.

ParameterTypeDescription
TfunctionDiscrepancy, e.g. np.mean or np.max
nsimsintHow 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.

ParameterTypeDescription
hintHow many steps to forecast ahead
oos_datapd.DataFrameExogenous variables in a frame for h steps
intervalsbooleanWhether to return prediction intervals

To be clear, the oos_data argument should be a DataFrame in the same format as the initialdataframe 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 youpredict h steps ahead, the method will take the 5 first rows from oos_data and take thevalues 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 shownwill not reflect latent variable uncertainty. Only Metropolis-Hastings will give you fullyBayesian prediction intervals. Bayesian intervals with variational inference are not shownbecause 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.

ParameterTypeDescription
hintHow many previous timesteps to use
fit_oncebooleanWhether to fit once, or every timestep
fit_methodstrWhich inference option, e.g. ‘MLE’

Returns : pd.DataFrame - the model predictions

sample(nsims)
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Returns np.ndarray of draws of the data from the posterior predictive density. Thismethod only works if you have fitted the model using Bayesian inference.

ParameterTypeDescription
nsimsintHow 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.

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