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# Non-Gaussian Local Level 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 level model has the same form as a Gaussian local level 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.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()

We can fit a Poisson local level model as follows:

model = pf.NLLEV(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.LocalLevel(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 -107.599165657
20% done : ELBO is -127.571498111
30% done : ELBO is -136.25857363
40% done : ELBO is -137.626516299
50% done : ELBO is -137.539662707
60% done : ELBO is -137.321490055
70% done : ELBO is -137.518451697
80% done : ELBO is -137.311382466
90% done : ELBO is -136.3580387
100% done : ELBO is -137.346927749

Final model ELBO is -135.76799195

Poisson Local Level Model
======================================== =================================================
Dependent Variable: Goals                Method: BBVI
Start Date: 0                            Unnormalized Log Posterior: -56.8409
End Date: 74                             AIC: 115.681720125
Number of observations: 75               BIC: 117.999208239
==========================================================================================
Latent Variable           Median             Mean               95% Credibility Interval
========================= ================== ================== ==========================
Sigma^2 level             0.0406             0.0406             (0.0353 | 0.0467)
==========================================================================================

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

model.plot_z()

Next we will plot the in-sample fit using :py:func: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 get an idea of the performance of our model by prediction through the :py:func:plot_predict: method:

model.plot_predict(h=5,figsize=(15,5))

If we just want the predictions themselves, we can use the :py:func:predict: method.

## Class Description

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

**Non-Gaussian Local Level Models (NLLEV).**

==================   ===============================    ======================================
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**

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.

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