Gaussian Local Linear Trend 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.

One classic univariate structural time series model is the local linear trend model. We can write this as a combination of a time-varying level, time-varying trend and an irregular term:

\[y_{t} = \mu_{t} + \epsilon_{t}\]

\[\mu_{t} = \beta_{t-1} + \mu_{t-1} + \eta_{t}\]

\[\beta_{t} = \beta_{t-1} + \zeta_{t}\]

\[\epsilon_{t} \sim N\left(0,\sigma_{\epsilon}^{2}\right)\]

\[\eta_{t} \sim N\left(0,\sigma_{\eta}^{2}\right)\]

\[\zeta_{t} \sim N\left(0,\sigma_{\zeta}^{2}\right)\]

Example

We will use data on logged US Real GDP.

growthdata = pd.read_csv('http://www.pyflux.com/notebooks/GDPC1.csv')
USgrowth = pd.DataFrame(np.log(growthdata['VALUE']))
USgrowth.index = pd.to_datetime(growthdata['DATE'])
USgrowth.columns = ['Logged US Real GDP']
plt.figure(figsize=(15,5))
plt.plot(USgrowth.index, USgrowth)
plt.ylabel('Real GDP')
plt.title('US Real GDP');

Here define a Local Linear Trend model as follows:

model = pf.LLT(data=USgrowth)

We can also use the higher-level wrapper which allows us to specify the family, although if we pick a non-Gaussian family then the model will be estimated in a different way (not through the Kalman filter):

model = pf.LocalTrend(data=USgrowth, family=pf.Normal())

Next we estimate the latent variables. For this example we will use a maximum likelihood point mass estimate \(z^{MLE}\):

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

LLT
======================================== =================================================
Dependent Variable: Logged US Real GDP   Method: MLE
Start Date: 1947-01-01 00:00:00          Log Likelihood: 877.3334
End Date: 2015-10-01 00:00:00            AIC: -1748.6667
Number of observations: 276              BIC: -1737.8055
==========================================================================================
Latent Variable           Estimate   Std Error  z        P>|z|    95% C.I.
========================= ========== ========== ======== ======== ========================
Sigma^2 irregular         3.306e-07
Sigma^2 level             5.41832e-0
Sigma^2 trend             1.55224e-0
==========================================================================================

We can plot the in-sample fit using plot_fit():

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

The trend picks out the underlying local growth rate in the series. Together with the local level this constitutes the smoothed series. We can use the simulation smoother to simulate draws from the states, using py:func:simulation_smoother:. We draw from the local trend state:

plt.figure(figsize=(15,5))
for i in range(10):
    plt.plot(model.index,model.simulation_smoother(
            model.latent_variables.get_z_values())[1][0:model.index.shape[0]])
plt.show()

If we want to plot rolling in-sample predictions, we can use the plot_predict_is(): method:

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

We can view out-of-sample predictions using plot_predict():

model.plot_predict(h=20, past_values=17*4, figsize=(15,6))

If we want the predictions in a DataFrame form, then we can just use the predict(): method.

Class Description

class LLT(data, integ, target)

Local Linear Trend Models.

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.

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_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

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_sample(nsims, plot_data=True)

Plots samples from the posterior predictive density of the model. This method only works if you fitted the model using Bayesian inference.

Parameter	Type	Description
nsims	int	How many samples to draw
plot_data	boolean	Whether to plot the real data as well

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

ppc(T, nsims)

Returns a p-value for a posterior predictive check. 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: int - the p-value for the discrepancy test

predict(h, intervals=False)

Returns a DataFrame of model predictions.

Parameter	Type	Description
h	int	How many steps to forecast ahead
intervals	boolean	Whether to return prediction intervals

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

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

sample(nsims)

Returns np.ndarray of draws of the data from the posterior predictive density. This method only works if you have fitted the model using Bayesian inference.

Parameter	Type	Description
nsims	int	How many posterior draws to take

Returns : np.ndarray - samples from the posterior predictive density.

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.