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:
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 = 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');
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¶
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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
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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
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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_variablesattribute 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_variablesattribute
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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_variablesattribute.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
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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
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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
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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
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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
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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
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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
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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
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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.