Skip to content
Permalink
Branch: master
Find file Copy path
Find file Copy path
Fetching contributors…
Cannot retrieve contributors at this time
274 lines (190 sloc) 13.3 KB

Non-Gaussian Local Linear Trend Models

Introduction

With Non-Gaussian state space models, we have the same basic setup as Gaussian state space models, but now a potentially non-Gaussian measurement density. That is we are interested in problems of the form:

p\left(y_{t}\mid{z}_{t}\right)
\theta_{t} = f\left(\alpha_{t}\right)
\alpha_{t} =  \alpha_{t-1} + \eta_{t}
\eta_{t} \sim N\left(0,\Sigma\right)

Usually MCMC based schemes are the right way to tackle this problem. Currently PyFlux uses BBVI for speed, but the mean-field approximation means there can be some bias in the states (although the results are generally okay for prediction). In the future, PyFlux will use a more structured approximation.

The Non-Gaussian local linear trend model has the same form as a Gaussian local linear trend model, but with a non-Gaussian measurement density.

Example

For fun, and since it’s topical, we’ll apply a Poisson local level model to count data on the number of goals the football team Leicester have scored since they rejoined the Premier League. Each index represents a match they have played. This is a short dataset, but it shows the principle behind the model.

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

leicester = pd.read_csv('http://www.pyflux.com/notebooks/leicester_goals_scored.csv')
leicester.columns= ["Time","Goals","Season2"]
plt.figure(figsize=(15,5))
plt.plot(leicester["Goals"])
plt.ylabel('Goals Scored')
plt.title('Leicester Goals Since Joining EPL');
plt.show()

http://www.pyflux.com/notebooks/NonGaussianStateSpace/output_11_0.png

We can fit a Poisson local linear trend model as follows:

model = pf.NLLT(data=leicester, target='Goals', family=pf.Poisson())

We can also use the higher-level wrapper which allows us to specify the family. If you pick a Normal distribution, then the Kalman filter will be used:

model = pf.LocalTrend(data=leicester, target='Goals', family=pf.Poisson())

Next we estimate the latent variables through a BBVI estimate z^{BBVI}:

x = model.fit(iterations=5000)
x.summary()

10% done : ELBO is -27837.965202
20% done : ELBO is -10667.1947315
30% done : ELBO is -5150.42573307
40% done : ELBO is -2567.54029949
50% done : ELBO is -1291.29282788
60% done : ELBO is -578.99494029
70% done : ELBO is -251.124996408
80% done : ELBO is -100.355592594
90% done : ELBO is -49.3752685727
100% done : ELBO is -13.9899801048

Final model ELBO is 46.2333499244
Poisson Local Linear Trend Model
======================================================= ================================================
Dependent Variable: Goals                               Method: BBVI
Start Date: 0                                           Unnormalized Log Posterior: 235.942
End Date: 74                                            AIC: -467.884097447
Number of observations: 75                              BIC: -463.24912122
========================================================================================================
Latent Variable                          Median             Mean               95% Credibility Interval
======================================== ================== ================== =========================
Sigma^2 level                            0.1738             0.1739             (0.1539 | 0.197)
Sigma^2 trend                            0.0                0.0                (0.0 | 0.0)
========================================================================================================

We can plot the evolution parameter with :py:func:`plot_z`:

model.plot_z([0])
model.plot_z([1])

http://www.pyflux.com/notebooks/NonGaussianStateSpace/output_25_1.png

http://www.pyflux.com/notebooks/NonGaussianStateSpace/output_25_2.png

Next we will plot the in-sample fit using :py:func:`plot_fit`:

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

http://www.pyflux.com/notebooks/NonGaussianStateSpace/output_26_0.png

Class Description

.. py:class:: NLLT(data, ar, integ, target, family)

   **Non-Gaussian Local Linear Trend Models (NLLT).**

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

   .. py:attribute:: 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**

   .. py:method:: 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

   .. py:method:: 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

   .. py:method:: 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

   .. py:method:: 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

   .. py:method:: 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

   .. py:method:: 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

   .. py:method:: predict(h)

      Returns a DataFrame of model predictions.

      ==================   ========================    ======================================
      Parameter            Type                        Description
      ==================   ========================    ======================================
      h                    int                         How many steps to forecast ahead
      ==================   ========================    ======================================

      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

   .. py:method:: 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

References

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

Ranganath, R., Gerrish, S., and Blei, D. M. (2014). Black box variational inference. In Artificial Intelligence and Statistics.

You can’t perform that action at this time.