## 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:

## 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.

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).**

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 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.

**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.

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 themethod 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 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.

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 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.

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.

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 worksif 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 havefitted 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 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.

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. Thismethod 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.

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