# DAR models¶

## Introduction¶

Gaussian state space models - often called structural time series or unobserved component models - provide a way to decompose a time series into several distinct components. These components can be extracted in closed form using the Kalman filter if the errors are jointly Gaussian, and parameters can be estimated via the prediction error decomposition and Maximum Likelihood.

We can write a dynamic autoregression model in this framework as:

$y_{t} = \sum^{p}_{i=1}\phi_{i,t}y_{t-i} + \epsilon_{t}$
$\phi_{i,t}= \phi_{i,t-1} + \eta_{i,t}$
$\epsilon_{t} \sim N\left(0,\sigma^{2}\right)$
$\eta_{i,t} \sim N\left(0,\sigma_{\eta_{i}}^{2}\right)$

In other words the dynamic autoregression coefficients follow a random walk.

## Example¶

We’ll run an Dynamic Autoregressive (DAR) Model for yearly sunspot 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.index = data['time'].values

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


Here we specify an arbitrary DAR(9) model (note: which is probably overspecified).

model = pf.DAR(data=data, ar=9, integ=0, target='sunspot.year')


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()

DAR(9, integrated=0)
====================================== =================================================
Dependent Variable: sunspot.year       Method: MLE
Start Date: 1709                       Log Likelihood: -1179.097
End Date: 1988                         AIC: 2380.194
Number of observations: 280            BIC: 2420.1766
========================================================================================
Latent Variable         Estimate   Std Error  z        P>|z|    95% C.I.
======================= ========== ========== ======== ======== ========================
Sigma^2 irregular       0.301
Constant                60.0568    23.83      2.5202   0.0117   (13.3499 | 106.7637)
Sigma^2 AR(1)           0.005
Sigma^2 AR(2)           0.0
Sigma^2 AR(3)           0.0005
Sigma^2 AR(4)           0.0001
Sigma^2 AR(5)           0.0002
Sigma^2 AR(6)           0.0011
Sigma^2 AR(7)           0.0002
Sigma^2 AR(8)           0.0003
Sigma^2 AR(9)           0.032
=========================================================================================


Note we have no standard errors in the results table because it shows the transformed parameters. If we want standard errors, we can call x.summary(transformed=False). Next we will plot the in-sample fit and the dynamic coefficients using plot_fit():

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


The sharp changes at the beginning reflect the diffuse initialization; together with high initial uncertainty, this leads to stronger updates towards the beginning of the series. We can predict forward using plot_predict:

We can predict forwards through the plot_predict(): method:

model.plot_predict(h=50, past_values=40, figsize=(15,5))


The prediction intervals here are unrealistic and reflect the Gaussian distributional assumption we’ve chosen – we can’t have negative sunspots! – but if we are just want the predictions themselves, we can use the predict(): method.

## Class Description¶

class DAR(data, ar, integ, target, family)

Dynamic Autoregression Models (DAR).

Parameter Type Description
data pd.DataFrame or np.ndarray Contains the univariate time series
ar int The number of autoregressive 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_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_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

predict(h)

Returns a DataFrame of model predictions.

Parameter Type Description
h int How many steps to forecast ahead

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

simulation_smoother(beta)

Returns np.ndarray of draws of the data from the Durbin and Koopman (2002) simulation smoother.

Parameter Type Description
beta np.array np.array of latent variables

Recommended just to use model.latent_variables.get_z_values() for the beta input, if you have already fit a model.

Returns : np.ndarray - samples from simulation smoother

## References¶

Durbin, J. and Koopman, S. J. (2002). A simple and efficient simulation smoother for state space time series analysis. Biometrika, 89(3):603–615.

Harvey, A. C. (1989). Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press, Cambridge.