# Gaussian Local Level 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 level model. We can write this as a combination of a time-varying level and an irregular term:

$y_{t} = \mu_{t} + \epsilon_{t}$
$\mu_{t} = \mu_{t-1} + \eta_{t}$
$\epsilon_{t} \sim N\left(0,\sigma_{\epsilon}^{2}\right)$
$\eta_{t} \sim N\left(0,\sigma_{\eta}^{2}\right)$

## Example¶

We will use data on the number of goals scored by soccer teams Nottingham Forest and Derby in their head-to-head matches from the beginning of their competitive history. We are interested to know whether these games have become more or less high scoring over time.

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

nile.index = pd.to_datetime(nile['time'].values,format='%Y')
plt.figure(figsize=(15,5))
plt.plot(nile.index,nile['Nile'])
plt.ylabel('Discharge Volume')
plt.title('Nile River Discharge');
plt.show()


Here define a Local Level model as follows:

model = pf.LLEV(data=nile, target='Nile')


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.LocalLevel(data=nile, target='Nile', 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()

LLEV
======================================== =================================================
Dependent Variable: Nile                 Method: MLE
Start Date: 1871-01-01 00:00:00          Log Likelihood: -641.5238
End Date: 1970-01-01 00:00:00            AIC: 1287.0476
Number of observations: 100              BIC: 1292.258
==========================================================================================
Latent Variable           Estimate   Std Error  z        P>|z|    95% C.I.
========================= ========== ========== ======== ======== ========================
Sigma^2 irregular         15098.5722
Sigma^2 level             1469.11317
==========================================================================================


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

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


The model adapts to the lower level at the beginning of the 20th century.

We can use the Durbin and Koopman (2002) simulation smoother to simulate draws from the local level state, using simulation_smoother():

plt.figure(figsize=(15,5))
for i in range(10):
plt.plot(model.index, model.simulation_smoother(
model.latent_variables.get_z_values())[0][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=20,figsize=(15,5))


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

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


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

## Class Description¶

class LLEV(data, integ, target)

Local Level 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.