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Non-Gaussian Dynamic Regression 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 dynamic regression model has the same form as a dynamic linear regression model, but with a non-Gaussian measurement density.

Example

See the notebook at https://github.com/RJT1990/talks/blob/master/PyDataTimeSeriesTalk.ipynb and the example for non-Gaussian estimation of a beta coefficient for finance. The API is from an old version here, but shows a use of this model type.

Class Description

.. py:class:: NDynReg(formula, data, family)

**Non-Gaussian Dynamic Regression models**

==================   ===============================    ======================================
Parameter            Type                                Description
==================   ===============================    ======================================
formula              string                             Patsy notation specifying the regression
data                 pd.DataFrame                       Contains the univariate time series
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_ppc(T, nsims)

Plots a histogram for a posterior predictive check with a discrepancy measure of the
user'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

.. py:method:: 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 initial
dataframe 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 you
predict *h* steps ahead, the method will take the h first rows from *oos_data* and take the
values for the exogenous variables that you asked for in the patsy formula.

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, oos_data)

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
==================   ========================    ======================================

To be clear, the *oos_data* argument should be a DataFrame in the same format as the initial
dataframe 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 you
predict *h* steps ahead, the method will take the 5 first rows from *oos_data* and take the
values 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 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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