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)
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 :py:func:`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 :py:func:`plot_predict_is`: method:
model.plot_predict_is(h=15,figsize=(15,5))
We can view out-of-sample predictions using :py:func:`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 :py:func:`predict`: method.
.. py: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**
.. 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_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, 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_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
.. 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:: 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
.. py:method:: 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
.. 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
.. py:method:: 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.
.. py:method:: 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
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.