# pybuc
`pybuc` (Python Bayesian Unobserved Components) is a version of R's Bayesian structural time
series package, `bsts`, written by Steven L. Scott. The source paper can be found
[here](https://people.ischool.berkeley.edu/~hal/Papers/2013/pred-present-with-bsts.pdf) or in the *papers*
directory of this repository. While there are plans to expand the feature set of `pybuc`, currently there is no roadmap
for the release of new features. The syntax for using `pybuc` closely follows `statsmodels`' `UnobservedComponents`
module.
The current version of `pybuc` includes the following options for modeling and
forecasting a structural time series:
- Stochastic or non-stochastic level
- Damped level
- Stochastic or non-stochastic trend
- Damped trend <sup/>*</sup>
- Multiple stochastic or non-stochastic periodic-lag seasonality
- Multiple damped periodic-lag seasonality
- Multiple stochastic or non-stochastic "dummy" seasonality
- Multiple stochastic or non-stochastic trigonometric seasonality
- Regression with static coefficients<sup/>**</sup>
<sup/>*</sup> `pybuc` dampens trend differently than `bsts`. The former assumes an AR(1) process **without**
drift for the trend state equation. The latter assumes an AR(1) **with** drift. In practice this means that the trend,
on average, will be zero with `pybuc`, whereas `bsts` allows for the mean trend to be non-zero. The reason for
choosing an autoregressive process without drift is to be conservative with long horizon forecasts.
<sup/>**</sup> `pybuc` estimates regression coefficients differently than `bsts`. The former uses a standard Gaussian
prior. The latter uses a Bernoulli-Gaussian mixture commonly known as the spike-and-slab prior. The main
benefit of using a spike-and-slab prior is its promotion of coefficient-sparse solutions, i.e., variable selection, when
the number of predictors in the regression component exceeds the number of observed data points.
Fast computation is achieved using [Numba](https://numba.pydata.org/), a high performance just-in-time (JIT) compiler
for Python.
# Installation
```
pip install pybuc
```
See `pyproject.toml` and `poetry.lock` for dependency details. This module depends on NumPy, Numba, Pandas, and
Matplotlib. Python 3.9 and above is supported.
# Motivation
The Seasonal Autoregressive Integrated Moving Average (SARIMA) model is perhaps the most widely used class of
statistical time series models. By design, these models can only operate on covariance-stationary time series.
Consequently, if a time series exhibits non-stationarity (e.g., trend and/or seasonality), then the data first have to
be stationarized. Transforming a non-stationary series to a stationary one usually requires taking local and/or seasonal
time-differences of the data, but sometimes a linear trend to detrend a trend-stationary series is sufficient.
Whether to stationarize the data and to what extent differencing is needed are things that need to be determined
beforehand.
Once a stationary series is in hand, a SARIMA specification must be identified. Identifying the "right" SARIMA
specification can be achieved algorithmically (e.g., see the Python package `pmdarima`) or through examination of a
series' patterns. The latter typically involves statistical tests and visual inspection of a series' autocorrelation
(ACF) and partial autocorrelation (PACF) functions. Ultimately, the necessary condition for stationarity requires
statistical analysis before a model can be formulated. It also implies that the underlying trend and seasonality, if
they exist, are eliminated in the process of generating a stationary series. Consequently, the underlying time
components that characterize a series are not of empirical interest.
Another less commonly used class of model is structural time series (STS), also known as unobserved components (UC).
Whereas SARIMA models abstract away from an explicit model for trend and seasonality, STS/UC models do not. Thus, it is
possible to visualize the underlying components that characterize a time series using STS/UC. Moreover, it is relatively
straightforward to test for phenomena like level shifts, also known as structural breaks, by statistical examination of
a time series' estimated level component.
STS/UC models also have the flexibility to accommodate multiple stochastic seasonalities. SARIMA models, in contrast,
can accommodate multiple seasonalities, but only one seasonality/periodicity can be treated as stochastic. For example,
daily data may have day-of-week and week-of-year seasonality. Under a SARIMA model, only one of these seasonalities can
be modeled as stochastic. The other seasonality will have to be modeled as deterministic, which amounts to creating and
using a set of predictors that capture said seasonality. STS/UC models, on the other hand, can accommodate both
seasonalities as stochastic by treating each as distinct, unobserved state variables.
With the above in mind, what follows is a comparison between `statsmodels`' `SARIMAX'` module, `statsmodels`'
`UnobservedComponents` module, and `pybuc`. The distinction between `statsmodels.UnobservedComponents` and `pybuc` is
the former is a maximum likelihood estimator (MLE) while the latter is a Bayesian estimator. The following code
demonstrates the application of these methods on a data set that exhibits trend and multiplicative seasonality.
The STS/UC specification for `statsmodels.UnobservedComponents` and `pybuc` includes stochastic level, stochastic trend
(trend), and stochastic trigonometric seasonality with periodicity 12 and 6 harmonics.
# Usage
## Example: univariate time series with level, trend, and multiplicative seasonality
A canonical data set that exhibits trend and seasonality is the airline passenger data used in
Box, G.E.P.; Jenkins, G.M.; and Reinsel, G.C. Time Series Analysis, Forecasting and Control. Series G, 1976. See plot
below.
This data set gave rise to what is known as the "airline model", which is a SARIMA model with first-order local and
seasonal differencing and first-order local and seasonal moving average representations.
More compactly, SARIMA(0, 1, 1)(0, 1, 1) without drift.
To demonstrate the performance of the "airline model" on the airline passenger data, the data will be split into a
training and test set. The former will include all observations up until the last twelve months of data, and the latter
will include the last twelve months of data. See code below for model assessment.
### Import libraries and prepare data
```
from pybuc import buc
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.tsa.statespace.sarimax import SARIMAX
from statsmodels.tsa.statespace.structural import UnobservedComponents
# Convenience function for computing root mean squared error
def rmse(actual, prediction):
act, pred = actual.flatten(), prediction.flatten()
return np.sqrt(np.mean((act - pred) ** 2))
# Import airline passenger data
url = "https://raw.githubusercontent.com/devindg/pybuc/master/examples/data/airline-passengers.csv"
air = pd.read_csv(url, header=0, index_col=0)
air = air.astype(float)
air.index = pd.to_datetime(air.index)
hold_out_size = 12
# Create train and test sets
y_train = air.iloc[:-hold_out_size]
y_test = air.iloc[-hold_out_size:]
```
### SARIMA
```
''' Fit the airline data using SARIMA(0,1,1)(0,1,1) '''
sarima = SARIMAX(y_train, order=(0, 1, 1),
seasonal_order=(0, 1, 1, 12),
trend=[0])
sarima_res = sarima.fit(disp=False)
print(sarima_res.summary())
# Plot in-sample fit against actuals
plt.plot(y_train)
plt.plot(sarima_res.fittedvalues)
plt.title('SARIMA: In-sample')
plt.xticks(rotation=45, ha="right")
plt.show()
# Get and plot forecast
sarima_forecast = sarima_res.get_forecast(hold_out_size).summary_frame(alpha=0.05)
plt.plot(y_test)
plt.plot(sarima_forecast['mean'])
plt.fill_between(sarima_forecast.index,
sarima_forecast['mean_ci_lower'],
sarima_forecast['mean_ci_upper'], alpha=0.2)
plt.title('SARIMA: Forecast')
plt.legend(['Actual', 'Mean', '95% Prediction Interval'])
plt.show()
# Print RMSE
print(f"SARIMA RMSE: {rmse(y_test.to_numpy(), sarima_forecast['mean'].to_numpy())}")
```
```
SARIMA RMSE: 21.09028021383853
```
The SARIMA(0, 1, 1)(0, 1, 1) forecast plot.
### MLE Unobserved Components
```
''' Fit the airline data using MLE unobserved components '''
mle_uc = UnobservedComponents(y_train, exog=None, irregular=True,
level=True, stochastic_level=True,
trend=True, stochastic_trend=True,
freq_seasonal=[{'period': 12, 'harmonics': 6}],
stochastic_freq_seasonal=[True])
# Fit the model via maximum likelihood
mle_uc_res = mle_uc.fit(disp=False)
print(mle_uc_res.summary())
# Plot in-sample fit against actuals
plt.plot(y_train)
plt.plot(mle_uc_res.fittedvalues)
plt.title('MLE UC: In-sample')
plt.show()
# Plot time series components
mle_uc_res.plot_components(legend_loc='lower right', figsize=(15, 9), which='smoothed')
plt.show()
# Get and plot forecast
mle_uc_forecast = mle_uc_res.get_forecast(hold_out_size).summary_frame(alpha=0.05)
plt.plot(y_test)
plt.plot(mle_uc_forecast['mean'])
plt.fill_between(mle_uc_forecast.index,
mle_uc_forecast['mean_ci_lower'],
mle_uc_forecast['mean_ci_upper'], alpha=0.2)
plt.title('MLE UC: Forecast')
plt.legend(['Actual', 'Mean', '95% Prediction Interval'])
plt.show()
# Print RMSE
print(f"MLE UC RMSE: {rmse(y_test.to_numpy(), mle_uc_forecast['mean'].to_numpy())}")
```
```
MLE UC RMSE: 17.961873327622694
```
The MLE Unobserved Components forecast and component plots.
As noted above, a distinguishing feature of STS/UC models is their explicit modeling of trend and seasonality. This is
illustrated with the components plot.
Finally, the Bayesian analog of the MLE STS/UC model is demonstrated. Default parameter values are used for the priors
corresponding to the variance parameters in the model. See below for default priors on variance parameters.
**Note that because computation is built on Numba, a JIT compiler, the first run of the code could take a while.
Subsequent runs (assuming the Python kernel isn't restarted) should execute considerably faster.**
### Bayesian Unobserved Components
```
''' Fit the airline data using Bayesian unobserved components '''
bayes_uc = buc.BayesianUnobservedComponents(response=y_train,
level=True, stochastic_level=True,
trend=True, stochastic_trend=True,
trig_seasonal=((12, 0),), stochastic_trig_seasonal=(True,),
seed=123)
post = bayes_uc.sample(5000)
mcmc_burn = 100
# Print summary of estimated parameters
for key, value in bayes_uc.summary(burn=mcmc_burn).items():
print(key, ' : ', value)
# Plot in-sample fit against actuals
bayes_uc.plot_post_pred_dist(burn=mcmc_burn)
plt.title('Bayesian UC: In-sample')
plt.show()
# Plot time series components
bayes_uc.plot_components(burn=mcmc_burn, smoothed=True)
plt.show()
# Plot trace of posterior
bayes_uc.plot_trace(burn=mcmc_burn)
plt.show()
# Get and plot forecast
forecast, _ = bayes_uc.forecast(hold_out_size, mcmc_burn)
forecast_mean = np.mean(forecast, axis=0)
forecast_l95 = np.quantile(forecast, 0.025, axis=0).flatten()
forecast_u95 = np.quantile(forecast, 0.975, axis=0).flatten()
plt.plot(y_test)
plt.plot(bayes_uc.future_time_index, forecast_mean)
plt.fill_between(bayes_uc.future_time_index, forecast_l95, forecast_u95, alpha=0.2)
plt.title('Bayesian UC: Forecast')
plt.legend(['Actual', 'Mean', '95% Prediction Interval'])
plt.show()
# Print RMSE
print(f"BAYES-UC RMSE: {rmse(y_test.to_numpy(), forecast_mean)}")
```
```
BAYES-UC RMSE: 16.620400857034113
```
The Bayesian Unobserved Components forecast plot, components plot, and RMSE are shown below.
#### Component plots
##### Smoothed
##### Filtered
#### Trace plots
# Model
A structural time series model with level, trend, seasonal, and regression components takes the form:
$$
y_t = \mu_t + \boldsymbol{\gamma}^\prime _t \mathbb{1}_p + \mathbf x_t^\prime \boldsymbol{\beta} + \epsilon_t
$$
where $\mu_t$ specifies an unobserved dynamic level component, $\boldsymbol{\gamma}_ t$ is a $p \times 1$ vector of
unobserved dynamic seasonal components that represent unique periodicities, $\mathbf x_t^\prime \boldsymbol{\beta}$ a
partially unobserved regression component (the regressors $\mathbf x_t$ are observed, but the coefficients
$\boldsymbol{\beta}$ are not), and $\epsilon_t \sim N(0, \sigma_{\epsilon}^2)$ an unobserved irregular component. The
equation describing the outcome $y_t$ is commonly referred to as the observation equation, and the transition equations
governing the evolution of the unobserved states are known as the state equations.
## Level and trend
The unobserved level evolves according to the following general transition equations:
$$
\begin{align}
\mu_{t+1} &= \kappa \mu_t + \delta_t + \eta_{\mu, t} \\
\delta_{t+1} &= \phi \delta_t + \eta_{\delta, t}
\end{align}
$$
where $\eta_{\mu, t} \sim N(0, \sigma_{\eta_\mu}^2)$ and $\eta_{\delta, t} \sim N(0, \sigma_{\eta_\delta}^2)$ for all
$t$. The state equation for $\delta_t$ represents the local trend at time $t$.
The parameters $\kappa$ and $\phi$ represent autoregressive coefficients. In general, $\kappa$ and $\phi$ are expected
to be in the interval $(-1, 1)$, which implies a stationary process. In practice, however, it is possible for either
$\kappa$ or $\phi$ to be outside the unit circle, which implies an explosive process. While it is mathematically
possible for an explosive process to be stationary, the implication of such a result implies that the future predicts
the past, which is not a realistic assumption. If an autoregressive level or trend is specified, no hard constraints
(by default) are placed on the bounds of the autoregressive parameters. Instead, the default prior for these parameters
is vague (see section on priors below).
Note that if $\sigma_{\eta_\mu}^2 = \sigma_{\eta_\delta}^2 = 0$ and $\phi = 1$ and $\kappa = 1$, then the level
component in the observation equation, $\mu_t$, collapses to a deterministic intercept and linear time trend.
## Seasonality
### Periodic-lag form
For a given periodicity $S$, a seasonal component in $\boldsymbol{\gamma}_t$, $\gamma^S_t$, can be modeled in three
ways. One way is based on periodic lags. Formally, the seasonal effect on $y$ is modeled as
$$
\gamma^S_t = \rho(S) \gamma^S_{t-S} + \eta_{\gamma^S, t},
$$
where $S$ is the number of periods in a seasonal cycle, $\rho(S)$ is an autoregressive parameter expected to lie in the
unit circle (-1, 1), and $\eta_{\gamma^S, t} \sim N(0, \sigma_{\eta_\gamma^S}^2)$ for all $t$. If damping is not
specified for a given periodic lag, $\rho(S) = 1$ and seasonality is treated as a random walk process.
This specification for seasonality is arguably the most robust representation (relative to dummy and trigonometric)
because its structural assumption on periodicity is the least complex.
### Dummy form
Another way is known as the "dummy" variable approach. Formally, the seasonal effect on the outcome $y$ is modeled as
$$
\sum_{j=0}^{S-1} \gamma^S_{t-j} = \eta_{\gamma^S, t} \iff \gamma^S_t = -\sum_{j=1}^{S-1} \gamma^S_{t-j} + \eta_{\gamma^S, t},
$$
where $j$ indexes the number of periods in a seasonal cycle, and $\eta_{\gamma^S, t} \sim N(0, \sigma_{\eta_\gamma^S}^2)$
for all $t$. Intuitively, if a time series exhibits periodicity, then the sum of the periodic effects over a cycle
should, on average, be zero.
### Trigonometric form
The final way to model seasonality is through a trigonometric representation, which exploits the periodicity of sine and
cosine functions. Specifically, seasonality is modeled as
$$
\gamma^S_t = \sum_{j=1}^h \gamma^S_{j, t}
$$
where $j$ indexes the number of harmonics to represent seasonality of periodicity $S$ and
$1 \leq h \leq \lfloor S/2 \rfloor$ is the highest desired number of harmonics. The state transition equations for each
harmonic, $\gamma^S_{j, t}$, are represented by a real and imaginary part, specifically
$$
\begin{align}
\gamma^S_ {j, t+1} &= \cos(\lambda_j) \gamma^S_{j, t} + \sin(\lambda_j) \gamma^{S*}_ {j, t} + \eta_{\gamma^S_ j, t} \\
\gamma^{S*}_ {j, t+1} &= -\sin(\lambda_j) \gamma^S_ {j, t} + \cos(\lambda_j) \gamma^{S*}_ {j, t} + \eta_{\gamma^{S*}_ j , t}
\end{align}
$$
where frequency $\lambda_j = 2j\pi / S$. It is assumed that $\eta_{\gamma^S_j, t}$ and $\eta_{\gamma^{S*}_ j , t}$ are
distributed $N(0, \sigma^2_{\eta^S_\gamma})$ for all $j, t$. Note that when $S$ is even, $\gamma^{S*}_ {S/2, t+1}$ is not
needed since
$$
\begin{align}
\gamma^S_{S/2, t+1} &= \cos(\pi) \gamma^S_{S/2, t} + \sin(\pi) \gamma^{S*}_ {S/2, t} + \eta_{\gamma^S_{S/2}, t} \\
&= (-1) \gamma^S_{S/2, t} + (0) \gamma^{S*}_ {S/2, t} + \eta_{\gamma^S_{S/2}, t} \\
&= -\gamma^S_{S/2, t} + \eta_{\gamma^S_{S/2}, t}
\end{align}
$$
Accordingly, if $S$ is even and $h = S/2$, then there will be $S - 1$ state equations. More generally, the number of
state equations for a trigonometric specification is $2h$, except when $S$ is even and $h = S/2$.
## Regression
There are two ways to configure the model matrices to account for a regression component with static coefficients.
The canonical way (Method 1) is to append $\mathbf x_t^\prime$ to $\mathbf Z_t$ and $\boldsymbol{\beta}_t$ to the
state vector, $\boldsymbol{\alpha}_t$ (see state space representation below), with the constraints
$\boldsymbol{\beta}_0 = \boldsymbol{\beta}$ and $\boldsymbol{\beta}_t = \boldsymbol{\beta} _{t-1}$ for all $t$.
Another, less common way (Method 2) is to append $\mathbf x_t^\prime \boldsymbol{\beta}$ to $\mathbf Z_t$ and 1 to the
state vector.
While both methods can be accommodated by the Kalman filter, Method 1 is a direct extension of the Kalman filter as it
maintains the observability of $\mathbf Z_t$ and treats the regression coefficients as unobserved states. Method 2 does
not fit naturally into the conventional framework of the Kalman filter, but it offers the significant advantage of only
increasing the size of the state vector by one. In contrast, Method 1 increases the size of the state vector by the size
of $\boldsymbol{\beta}$. This is significant because computational complexity is quadratic in the size of the state
vector but linear in the size of the observation vector.
The unobservability of $\mathbf Z_t$ under Method 2 can be handled with maximum likelihood or Bayesian estimation by
working with the adjusted series
$$
y_t^* \equiv y_t - \tau_t = \mathbf x_ t^\prime \boldsymbol{\beta} + \epsilon_t
$$
where $\tau_t$ represents the time series component of the structural time series model. For example, assuming a level
and seasonal component are specified, this means an initial estimate of the time series component
$\tau_t = \mu_t + \boldsymbol{\gamma}^\prime_ t \mathbb{1}_p$ and $\boldsymbol{\beta}$ has to be acquired first. Then
$\boldsymbol{\beta}$ can be estimated conditional on $\mathbf y^ * \equiv \left(y_1^ *, y_2^ *, \cdots, y_n^ *\right)^\prime$.
`pybuc` uses Method 2 for estimating static coefficients.
## Default priors
### Irregular and state variances
If no priors are given for variances corresponding to stochastic states (i.e., level, trend, and seasonality),
the following defaults are used:
$$
\begin{align}
\sigma^2_{\mathrm{level}} &\sim \mathrm{IG}(0.01, (0.01 * \mathrm{Std.Dev}(y))^2) \\
\sigma^2_{\mathrm{seasonal}} &\sim \mathrm{IG}(0.01, (10 * 0.01 * \mathrm{Std.Dev}(y))^2) \\
\sigma^2_{\mathrm{trend}} &\sim \mathrm{IG}(0.01, (0.1 * 0.01 * \mathrm{Std.Dev}(y))^2) \\
\end{align}
$$
The level prior matches the default level prior in R's `bsts` package. However, the default seasonal and trend priors
are different. While the default trend prior in `bsts` is the same as the level and seasonal priors, `pybuc` makes a
more conservative assumption about the variance associated with trend. This is reflected by a standard deviation
that is one-tenth (one-hundredth) the magnitude of the level (seasonal) standard deviation. In other words,
this prior assumes that variation in trend is small relative to variation in level and seasonality. The objective is to
mitigate the impact that noise in the data could have on producing an overly aggressive trend.
In a similar vein, the default seasonal prior is different from `bsts`'s in that it allows for more flexibility in scale.
Specifically, the seasonal standard deviation is ten (one-hundred) times larger than the standard deviation for level
(trend). Intuitively, changes from one cycle to another, as opposed to one period to the next, are likely to be larger.
The default prior for irregular variance is:
$$
\sigma^2_{\mathrm{irregular}} \sim \mathrm{IG}(0.01, (0.01 * \mathrm{Std.Dev}(y))^2)
$$
### Damped/autoregressive state coefficients
Damping can be applied to level, trend, and periodic-lag seasonality state components. By default, if no prior is given
for an autoregressive (i.e., AR(1)) coefficient, the prior takes the form
$$
\phi \sim N(1, 1)
$$
where $\phi$ represents some autoregressive coefficient. Thus, the prior encodes the belief that the process (level,
trend, seasonality) is a random walk. A unit variance is assumed to accommodate different types of dynamics, including
stationary or explosive processes that may oscillate, grow, or decay.
### Regression coefficients
The default prior for regression coefficients is
$$
\boldsymbol{\beta} \sim N\left(\mathbf 0, \frac{\kappa}{n} \left(\frac{1}{2} \mathbf X^\prime \mathbf X +
\frac{1}{2} \mathrm{diag}(\mathbf X^\prime \mathbf X) \right)\right)
$$
where $\mathbf X$ is the design matrix, $n$ is the number of response observations, and $\kappa = 0.000001$ is the number
of default prior observations given to the mean prior of $\mathbf 0$. This prior is a slight modification of Zellner's
g-prior (to guard against potential singularity of the design matrix). The number of prior observations, $\kappa$, can be
changed by passing a value to the argument `zellner_prior_obs` in the `sample()` method. If Zellner's g-prior is not
desired, then a custom precision matrix can be passed to the argument `reg_coeff_prec_prior`. Similarly, if a zero-mean
prior is not wanted, a custom mean prior can be passed to `reg_coeff_mean_prior`.
## State space representation (example)
The unobserved components model can be rewritten in state space form. For example, suppose level, trend, seasonal,
regression, and irregular components are specified, and the seasonal component takes a trigonometric form with
periodicity $S=4$ and $h=2$ harmonics. Let $\mathbf Z_t \in \mathbb{R}^{1 \times m}$,
$\mathbf T \in \mathbb{R}^{m \times m}$, $\mathbf R \in \mathbb{R}^{m \times q}$, and
$\boldsymbol{\alpha}_ t \in \mathbb{R}^{m \times 1}$ denote the observation matrix, state transition matrix,
state error transformation matrix, and unobserved state vector, respectively, where $m$ is the number of state equations
and $q$ is the number of state parameters to be estimated (i.e., the number of stochastic state equations,
which is defined by the number of positive state variance parameters).
There are $m = 1 + 1 + (h * 2 - 1) + 1 = 6$ state equations and $q = 1 + 1 + (h * 2 - 1) = 5$ stochastic state equations
, where the term $(h * 2 - 1)$ follows from $S=4$ being even and $h = S/2$. Outside of this case, there would generally
be $h * 2$ state equations for trigonometric seasonality. Note also that there are 5 stochastic state equations because
the state value for the regression component is not stochastic; it is 1 for all $t$ by construction. The observation,
state transition, and state error transformation matrices may be
written as
$$
\begin{align}
\mathbf Z_t &= \left(\begin{array}{cc}
1 & 0 & 1 & 0 & 1 & \mathbf x_t^{\prime} \boldsymbol{\beta}
\end{array}\right) \\
\mathbf T &= \left(\begin{array}{cc}
1 & 1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & \cos(2\pi / 4) & \sin(2\pi / 4) & 0 & 0 \\
0 & 0 & -\sin(2\pi / 4) & \cos(2\pi / 4) & 0 & 0 \\
0 & 0 & 0 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{array}\right) \\
\mathbf R &= \left(\begin{array}{cc}
1 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0
\end{array}\right)
\end{align}
$$
Given the definitions of $\mathbf Z_t$, $\mathbf T$, and $\mathbf R$, the state space representation of the unobserved
components model above can compactly be expressed as
$$
\begin{align}
y_t &= \mathbf Z_t \boldsymbol{\alpha}_ t + \epsilon_t \\
\boldsymbol{\alpha}_ {t+1} &= \mathbf T \boldsymbol{\alpha}_ t + \mathbf R \boldsymbol{\eta}_ t, \hspace{5pt}
t=1,2,...,n
\end{align}
$$
where
$$
\begin{align}
\boldsymbol{\alpha}_ t &= \left(\begin{array}{cc}
\mu_t & \delta_t & \gamma^4_{1, t} & \gamma^{4*}_ {1, t} & \gamma^4_{2, t} & 1
\end{array}\right)^\prime \\
\boldsymbol{\eta}_ t &= \left(\begin{array}{cc}
\eta_{\mu, t} & \eta_{\delta, t} & \eta_{\gamma^4_ 1, t} & \eta_{\gamma^{4*}_ 1, t} &
\eta_{\gamma^4_ 2, t}
\end{array}\right)^\prime
\end{align}
$$
and
$$
\mathrm{Cov}(\boldsymbol{\eta}_ t) = \mathrm{Cov}(\boldsymbol{\eta}_ {t-1}) = \boldsymbol{\Sigma}_ \eta =
\mathrm{diag}(\sigma^2_{\eta_\mu}, \sigma^2_{\eta_\delta}, \sigma^2_{\eta_{\gamma^4_ 1}}, \sigma^2_{\eta_{\gamma^{4*}_ 1}},
\sigma^2_{\eta_{\gamma^4_ 2}}) \in \mathbb{R}^{5 \times 5} \hspace{5pt} \textrm{for all } t=1,2,...,n
$$
# Estimation
`pybuc` mirrors R's `bsts` with respect to estimation method. The observation vector, state vector, and regression
coefficients are assumed to be conditionally normal random variables, and the error variances are assumed to be
conditionally independent inverse-Gamma random variables. These model assumptions imply conditional conjugacy of the
model's parameters. Consequently, a Gibbs sampler is used to sample from each parameter's posterior distribution.
To achieve fast sampling, `pybuc` follows `bsts`'s adoption of the Durbin and Koopman (2002) simulation smoother. For
any parameter $\theta$, let $\theta(s)$ denote the $s$-th sample of parameter $\theta$. Each sample $s$ is drawn by
repeating the following three steps:
1. Draw $\boldsymbol{\alpha}(s)$ from
$p(\boldsymbol{\alpha} | \mathbf y, \boldsymbol{\sigma}^2_\eta(s-1), \boldsymbol{\beta}(s-1), \sigma^2_\epsilon(s-1))$
using the Durbin and Koopman simulation state smoother, where
$\boldsymbol{\alpha}(s) = (\boldsymbol{\alpha}_ 1(s), \boldsymbol{\alpha}_ 2(s), \cdots, \boldsymbol{\alpha}_ n(s))^\prime$
and $\boldsymbol{\sigma}^2_\eta(s-1) = \mathrm{diag}(\boldsymbol{\Sigma}_\eta(s-1))$. Note that `pybuc` implements a
correction (based on a potential misunderstanding) for drawing $\boldsymbol{\alpha}(s)$ per "A note on implementing
the Durbin and Koopman simulation smoother" (Marek Jarocinski, 2015).
2. Draw $\boldsymbol{\sigma}^2(s) = (\sigma^2_ \epsilon(s), \boldsymbol{\sigma}^2_ \eta(s))^\prime$ from
$p(\boldsymbol{\sigma}^2 | \mathbf y, \boldsymbol{\alpha}(s), \boldsymbol{\beta}(s-1))$ using Durbin and Koopman's
simulation disturbance smoother.
3. Draw $\boldsymbol{\beta}(s)$ from
$p(\boldsymbol{\beta} | \mathbf y^ *, \boldsymbol{\alpha}(s), \sigma^2_\epsilon(s))$, where $\mathbf y^ *$ is defined
above.
By assumption, the elements in $\boldsymbol{\sigma}^2(s)$ are conditionally independent inverse-Gamma distributed random
variables. Thus, Step 2 amounts to sampling each element in $\boldsymbol{\sigma}^2(s)$ independently from their
posterior inverse-Gamma distributions.
Raw data
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"keywords": "structural, time series, unobserved components, bayesian, bsts, forecast, state space",
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"description": "# pybuc\n`pybuc` (Python Bayesian Unobserved Components) is a version of R's Bayesian structural time \nseries package, `bsts`, written by Steven L. Scott. The source paper can be found \n[here](https://people.ischool.berkeley.edu/~hal/Papers/2013/pred-present-with-bsts.pdf) or in the *papers* \ndirectory of this repository. While there are plans to expand the feature set of `pybuc`, currently there is no roadmap \nfor the release of new features. The syntax for using `pybuc` closely follows `statsmodels`' `UnobservedComponents` \nmodule.\n\nThe current version of `pybuc` includes the following options for modeling and \nforecasting a structural time series: \n\n- Stochastic or non-stochastic level\n- Damped level\n- Stochastic or non-stochastic trend\n- Damped trend <sup/>*</sup>\n- Multiple stochastic or non-stochastic periodic-lag seasonality\n- Multiple damped periodic-lag seasonality\n- Multiple stochastic or non-stochastic \"dummy\" seasonality\n- Multiple stochastic or non-stochastic trigonometric seasonality\n- Regression with static coefficients<sup/>**</sup>\n\n<sup/>*</sup> `pybuc` dampens trend differently than `bsts`. The former assumes an AR(1) process **without** \ndrift for the trend state equation. The latter assumes an AR(1) **with** drift. In practice this means that the trend, \non average, will be zero with `pybuc`, whereas `bsts` allows for the mean trend to be non-zero. The reason for \nchoosing an autoregressive process without drift is to be conservative with long horizon forecasts.\n\n<sup/>**</sup> `pybuc` estimates regression coefficients differently than `bsts`. The former uses a standard Gaussian \nprior. The latter uses a Bernoulli-Gaussian mixture commonly known as the spike-and-slab prior. The main \nbenefit of using a spike-and-slab prior is its promotion of coefficient-sparse solutions, i.e., variable selection, when \nthe number of predictors in the regression component exceeds the number of observed data points.\n\nFast computation is achieved using [Numba](https://numba.pydata.org/), a high performance just-in-time (JIT) compiler \nfor Python.\n\n# Installation\n```\npip install pybuc\n```\nSee `pyproject.toml` and `poetry.lock` for dependency details. This module depends on NumPy, Numba, Pandas, and \nMatplotlib. Python 3.9 and above is supported.\n\n# Motivation\n\nThe Seasonal Autoregressive Integrated Moving Average (SARIMA) model is perhaps the most widely used class of \nstatistical time series models. By design, these models can only operate on covariance-stationary time series. \nConsequently, if a time series exhibits non-stationarity (e.g., trend and/or seasonality), then the data first have to \nbe stationarized. Transforming a non-stationary series to a stationary one usually requires taking local and/or seasonal \ntime-differences of the data, but sometimes a linear trend to detrend a trend-stationary series is sufficient. \nWhether to stationarize the data and to what extent differencing is needed are things that need to be determined \nbeforehand.\n\nOnce a stationary series is in hand, a SARIMA specification must be identified. Identifying the \"right\" SARIMA \nspecification can be achieved algorithmically (e.g., see the Python package `pmdarima`) or through examination of a \nseries' patterns. The latter typically involves statistical tests and visual inspection of a series' autocorrelation \n(ACF) and partial autocorrelation (PACF) functions. Ultimately, the necessary condition for stationarity requires \nstatistical analysis before a model can be formulated. It also implies that the underlying trend and seasonality, if \nthey exist, are eliminated in the process of generating a stationary series. Consequently, the underlying time \ncomponents that characterize a series are not of empirical interest.\n\nAnother less commonly used class of model is structural time series (STS), also known as unobserved components (UC). \nWhereas SARIMA models abstract away from an explicit model for trend and seasonality, STS/UC models do not. Thus, it is \npossible to visualize the underlying components that characterize a time series using STS/UC. Moreover, it is relatively \nstraightforward to test for phenomena like level shifts, also known as structural breaks, by statistical examination of \na time series' estimated level component.\n\nSTS/UC models also have the flexibility to accommodate multiple stochastic seasonalities. SARIMA models, in contrast, \ncan accommodate multiple seasonalities, but only one seasonality/periodicity can be treated as stochastic. For example, \ndaily data may have day-of-week and week-of-year seasonality. Under a SARIMA model, only one of these seasonalities can \nbe modeled as stochastic. The other seasonality will have to be modeled as deterministic, which amounts to creating and \nusing a set of predictors that capture said seasonality. STS/UC models, on the other hand, can accommodate both \nseasonalities as stochastic by treating each as distinct, unobserved state variables.\n\nWith the above in mind, what follows is a comparison between `statsmodels`' `SARIMAX'` module, `statsmodels`' \n`UnobservedComponents` module, and `pybuc`. The distinction between `statsmodels.UnobservedComponents` and `pybuc` is \nthe former is a maximum likelihood estimator (MLE) while the latter is a Bayesian estimator. The following code \ndemonstrates the application of these methods on a data set that exhibits trend and multiplicative seasonality.\nThe STS/UC specification for `statsmodels.UnobservedComponents` and `pybuc` includes stochastic level, stochastic trend \n(trend), and stochastic trigonometric seasonality with periodicity 12 and 6 harmonics.\n\n# Usage\n\n## Example: univariate time series with level, trend, and multiplicative seasonality\n\nA canonical data set that exhibits trend and seasonality is the airline passenger data used in\nBox, G.E.P.; Jenkins, G.M.; and Reinsel, G.C. Time Series Analysis, Forecasting and Control. Series G, 1976. See plot \nbelow.\n\n\n\nThis data set gave rise to what is known as the \"airline model\", which is a SARIMA model with first-order local and \nseasonal differencing and first-order local and seasonal moving average representations. \nMore compactly, SARIMA(0, 1, 1)(0, 1, 1) without drift.\n\nTo demonstrate the performance of the \"airline model\" on the airline passenger data, the data will be split into a \ntraining and test set. The former will include all observations up until the last twelve months of data, and the latter \nwill include the last twelve months of data. See code below for model assessment.\n\n### Import libraries and prepare data\n\n```\nfrom pybuc import buc\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\nfrom statsmodels.tsa.statespace.sarimax import SARIMAX\nfrom statsmodels.tsa.statespace.structural import UnobservedComponents\n\n\n# Convenience function for computing root mean squared error\ndef rmse(actual, prediction):\n act, pred = actual.flatten(), prediction.flatten()\n return np.sqrt(np.mean((act - pred) ** 2))\n\n\n# Import airline passenger data\nurl = \"https://raw.githubusercontent.com/devindg/pybuc/master/examples/data/airline-passengers.csv\"\nair = pd.read_csv(url, header=0, index_col=0)\nair = air.astype(float)\nair.index = pd.to_datetime(air.index)\nhold_out_size = 12\n\n# Create train and test sets\ny_train = air.iloc[:-hold_out_size]\ny_test = air.iloc[-hold_out_size:]\n```\n\n### SARIMA\n\n```\n''' Fit the airline data using SARIMA(0,1,1)(0,1,1) '''\nsarima = SARIMAX(y_train, order=(0, 1, 1),\n seasonal_order=(0, 1, 1, 12),\n trend=[0])\nsarima_res = sarima.fit(disp=False)\nprint(sarima_res.summary())\n\n# Plot in-sample fit against actuals\nplt.plot(y_train)\nplt.plot(sarima_res.fittedvalues)\nplt.title('SARIMA: In-sample')\nplt.xticks(rotation=45, ha=\"right\")\nplt.show()\n\n# Get and plot forecast\nsarima_forecast = sarima_res.get_forecast(hold_out_size).summary_frame(alpha=0.05)\nplt.plot(y_test)\nplt.plot(sarima_forecast['mean'])\nplt.fill_between(sarima_forecast.index,\n sarima_forecast['mean_ci_lower'],\n sarima_forecast['mean_ci_upper'], alpha=0.2)\nplt.title('SARIMA: Forecast')\nplt.legend(['Actual', 'Mean', '95% Prediction Interval'])\nplt.show()\n\n# Print RMSE\nprint(f\"SARIMA RMSE: {rmse(y_test.to_numpy(), sarima_forecast['mean'].to_numpy())}\")\n```\n\n```\nSARIMA RMSE: 21.09028021383853\n```\n\nThe SARIMA(0, 1, 1)(0, 1, 1) forecast plot.\n\n\n\n### MLE Unobserved Components\n\n```\n''' Fit the airline data using MLE unobserved components '''\nmle_uc = UnobservedComponents(y_train, exog=None, irregular=True,\n level=True, stochastic_level=True,\n trend=True, stochastic_trend=True,\n freq_seasonal=[{'period': 12, 'harmonics': 6}],\n stochastic_freq_seasonal=[True])\n\n# Fit the model via maximum likelihood\nmle_uc_res = mle_uc.fit(disp=False)\nprint(mle_uc_res.summary())\n\n# Plot in-sample fit against actuals\nplt.plot(y_train)\nplt.plot(mle_uc_res.fittedvalues)\nplt.title('MLE UC: In-sample')\nplt.show()\n\n# Plot time series components\nmle_uc_res.plot_components(legend_loc='lower right', figsize=(15, 9), which='smoothed')\nplt.show()\n\n# Get and plot forecast\nmle_uc_forecast = mle_uc_res.get_forecast(hold_out_size).summary_frame(alpha=0.05)\nplt.plot(y_test)\nplt.plot(mle_uc_forecast['mean'])\nplt.fill_between(mle_uc_forecast.index,\n mle_uc_forecast['mean_ci_lower'],\n mle_uc_forecast['mean_ci_upper'], alpha=0.2)\nplt.title('MLE UC: Forecast')\nplt.legend(['Actual', 'Mean', '95% Prediction Interval'])\nplt.show()\n\n# Print RMSE\nprint(f\"MLE UC RMSE: {rmse(y_test.to_numpy(), mle_uc_forecast['mean'].to_numpy())}\")\n```\n\n```\nMLE UC RMSE: 17.961873327622694\n```\n\nThe MLE Unobserved Components forecast and component plots.\n\n\n\n\n\nAs noted above, a distinguishing feature of STS/UC models is their explicit modeling of trend and seasonality. This is \nillustrated with the components plot.\n\nFinally, the Bayesian analog of the MLE STS/UC model is demonstrated. Default parameter values are used for the priors \ncorresponding to the variance parameters in the model. See below for default priors on variance parameters.\n\n**Note that because computation is built on Numba, a JIT compiler, the first run of the code could take a while. \nSubsequent runs (assuming the Python kernel isn't restarted) should execute considerably faster.**\n\n### Bayesian Unobserved Components\n```\n''' Fit the airline data using Bayesian unobserved components '''\nbayes_uc = buc.BayesianUnobservedComponents(response=y_train,\n level=True, stochastic_level=True,\n trend=True, stochastic_trend=True,\n trig_seasonal=((12, 0),), stochastic_trig_seasonal=(True,),\n seed=123)\npost = bayes_uc.sample(5000)\nmcmc_burn = 100\n\n# Print summary of estimated parameters\nfor key, value in bayes_uc.summary(burn=mcmc_burn).items():\n print(key, ' : ', value)\n\n# Plot in-sample fit against actuals\nbayes_uc.plot_post_pred_dist(burn=mcmc_burn)\nplt.title('Bayesian UC: In-sample')\nplt.show()\n\n# Plot time series components\nbayes_uc.plot_components(burn=mcmc_burn, smoothed=True)\nplt.show()\n\n# Plot trace of posterior\nbayes_uc.plot_trace(burn=mcmc_burn)\nplt.show()\n\n# Get and plot forecast\nforecast, _ = bayes_uc.forecast(hold_out_size, mcmc_burn)\nforecast_mean = np.mean(forecast, axis=0)\nforecast_l95 = np.quantile(forecast, 0.025, axis=0).flatten()\nforecast_u95 = np.quantile(forecast, 0.975, axis=0).flatten()\n\nplt.plot(y_test)\nplt.plot(bayes_uc.future_time_index, forecast_mean)\nplt.fill_between(bayes_uc.future_time_index, forecast_l95, forecast_u95, alpha=0.2)\nplt.title('Bayesian UC: Forecast')\nplt.legend(['Actual', 'Mean', '95% Prediction Interval'])\nplt.show()\n\n# Print RMSE\nprint(f\"BAYES-UC RMSE: {rmse(y_test.to_numpy(), forecast_mean)}\")\n```\n\n```\nBAYES-UC RMSE: 16.620400857034113\n```\n\nThe Bayesian Unobserved Components forecast plot, components plot, and RMSE are shown below.\n\n\n\n#### Component plots\n\n##### Smoothed\n\n\n\n##### Filtered\n\n\n\n#### Trace plots\n\n\n\n# Model\n\nA structural time series model with level, trend, seasonal, and regression components takes the form: \n\n$$\ny_t = \\mu_t + \\boldsymbol{\\gamma}^\\prime _t \\mathbb{1}_p + \\mathbf x_t^\\prime \\boldsymbol{\\beta} + \\epsilon_t\n$$ \n\nwhere $\\mu_t$ specifies an unobserved dynamic level component, $\\boldsymbol{\\gamma}_ t$ is a $p \\times 1$ vector of \nunobserved dynamic seasonal components that represent unique periodicities, $\\mathbf x_t^\\prime \\boldsymbol{\\beta}$ a \npartially unobserved regression component (the regressors $\\mathbf x_t$ are observed, but the coefficients \n$\\boldsymbol{\\beta}$ are not), and $\\epsilon_t \\sim N(0, \\sigma_{\\epsilon}^2)$ an unobserved irregular component. The \nequation describing the outcome $y_t$ is commonly referred to as the observation equation, and the transition equations \ngoverning the evolution of the unobserved states are known as the state equations.\n\n## Level and trend\n\nThe unobserved level evolves according to the following general transition equations:\n\n$$\n\\begin{align}\n \\mu_{t+1} &= \\kappa \\mu_t + \\delta_t + \\eta_{\\mu, t} \\\\ \n \\delta_{t+1} &= \\phi \\delta_t + \\eta_{\\delta, t} \n\\end{align}\n$$ \n\nwhere $\\eta_{\\mu, t} \\sim N(0, \\sigma_{\\eta_\\mu}^2)$ and $\\eta_{\\delta, t} \\sim N(0, \\sigma_{\\eta_\\delta}^2)$ for all \n$t$. The state equation for $\\delta_t$ represents the local trend at time $t$. \n\nThe parameters $\\kappa$ and $\\phi$ represent autoregressive coefficients. In general, $\\kappa$ and $\\phi$ are expected \nto be in the interval $(-1, 1)$, which implies a stationary process. In practice, however, it is possible for either \n$\\kappa$ or $\\phi$ to be outside the unit circle, which implies an explosive process. While it is mathematically \npossible for an explosive process to be stationary, the implication of such a result implies that the future predicts \nthe past, which is not a realistic assumption. If an autoregressive level or trend is specified, no hard constraints \n(by default) are placed on the bounds of the autoregressive parameters. Instead, the default prior for these parameters \nis vague (see section on priors below).\n\nNote that if $\\sigma_{\\eta_\\mu}^2 = \\sigma_{\\eta_\\delta}^2 = 0$ and $\\phi = 1$ and $\\kappa = 1$, then the level \ncomponent in the observation equation, $\\mu_t$, collapses to a deterministic intercept and linear time trend.\n\n## Seasonality\n\n### Periodic-lag form\nFor a given periodicity $S$, a seasonal component in $\\boldsymbol{\\gamma}_t$, $\\gamma^S_t$, can be modeled in three \nways. One way is based on periodic lags. Formally, the seasonal effect on $y$ is modeled as\n\n$$\n\\gamma^S_t = \\rho(S) \\gamma^S_{t-S} + \\eta_{\\gamma^S, t},\n$$\n\nwhere $S$ is the number of periods in a seasonal cycle, $\\rho(S)$ is an autoregressive parameter expected to lie in the \nunit circle (-1, 1), and $\\eta_{\\gamma^S, t} \\sim N(0, \\sigma_{\\eta_\\gamma^S}^2)$ for all $t$. If damping is not \nspecified for a given periodic lag, $\\rho(S) = 1$ and seasonality is treated as a random walk process.\n\nThis specification for seasonality is arguably the most robust representation (relative to dummy and trigonometric) \nbecause its structural assumption on periodicity is the least complex.\n\n### Dummy form\nAnother way is known as the \"dummy\" variable approach. Formally, the seasonal effect on the outcome $y$ is modeled as \n\n$$\n\\sum_{j=0}^{S-1} \\gamma^S_{t-j} = \\eta_{\\gamma^S, t} \\iff \\gamma^S_t = -\\sum_{j=1}^{S-1} \\gamma^S_{t-j} + \\eta_{\\gamma^S, t},\n$$ \n\nwhere $j$ indexes the number of periods in a seasonal cycle, and $\\eta_{\\gamma^S, t} \\sim N(0, \\sigma_{\\eta_\\gamma^S}^2)$ \nfor all $t$. Intuitively, if a time series exhibits periodicity, then the sum of the periodic effects over a cycle \nshould, on average, be zero.\n\n### Trigonometric form\nThe final way to model seasonality is through a trigonometric representation, which exploits the periodicity of sine and \ncosine functions. Specifically, seasonality is modeled as\n\n$$\n\\gamma^S_t = \\sum_{j=1}^h \\gamma^S_{j, t}\n$$\n\nwhere $j$ indexes the number of harmonics to represent seasonality of periodicity $S$ and \n$1 \\leq h \\leq \\lfloor S/2 \\rfloor$ is the highest desired number of harmonics. The state transition equations for each \nharmonic, $\\gamma^S_{j, t}$, are represented by a real and imaginary part, specifically\n\n$$\n\\begin{align}\n \\gamma^S_ {j, t+1} &= \\cos(\\lambda_j) \\gamma^S_{j, t} + \\sin(\\lambda_j) \\gamma^{S*}_ {j, t} + \\eta_{\\gamma^S_ j, t} \\\\\n \\gamma^{S*}_ {j, t+1} &= -\\sin(\\lambda_j) \\gamma^S_ {j, t} + \\cos(\\lambda_j) \\gamma^{S*}_ {j, t} + \\eta_{\\gamma^{S*}_ j , t}\n\\end{align}\n$$\n\nwhere frequency $\\lambda_j = 2j\\pi / S$. It is assumed that $\\eta_{\\gamma^S_j, t}$ and $\\eta_{\\gamma^{S*}_ j , t}$ are \ndistributed $N(0, \\sigma^2_{\\eta^S_\\gamma})$ for all $j, t$. Note that when $S$ is even, $\\gamma^{S*}_ {S/2, t+1}$ is not \nneeded since \n\n$$\n\\begin{align}\n \\gamma^S_{S/2, t+1} &= \\cos(\\pi) \\gamma^S_{S/2, t} + \\sin(\\pi) \\gamma^{S*}_ {S/2, t} + \\eta_{\\gamma^S_{S/2}, t} \\\\\n &= (-1) \\gamma^S_{S/2, t} + (0) \\gamma^{S*}_ {S/2, t} + \\eta_{\\gamma^S_{S/2}, t} \\\\\n &= -\\gamma^S_{S/2, t} + \\eta_{\\gamma^S_{S/2}, t}\n\\end{align}\n$$\n \nAccordingly, if $S$ is even and $h = S/2$, then there will be $S - 1$ state equations. More generally, the number of \nstate equations for a trigonometric specification is $2h$, except when $S$ is even and $h = S/2$.\n\n## Regression\nThere are two ways to configure the model matrices to account for a regression component with static coefficients. \nThe canonical way (Method 1) is to append $\\mathbf x_t^\\prime$ to $\\mathbf Z_t$ and $\\boldsymbol{\\beta}_t$ to the \nstate vector, $\\boldsymbol{\\alpha}_t$ (see state space representation below), with the constraints \n$\\boldsymbol{\\beta}_0 = \\boldsymbol{\\beta}$ and $\\boldsymbol{\\beta}_t = \\boldsymbol{\\beta} _{t-1}$ for all $t$. \nAnother, less common way (Method 2) is to append $\\mathbf x_t^\\prime \\boldsymbol{\\beta}$ to $\\mathbf Z_t$ and 1 to the \nstate vector. \n\nWhile both methods can be accommodated by the Kalman filter, Method 1 is a direct extension of the Kalman filter as it \nmaintains the observability of $\\mathbf Z_t$ and treats the regression coefficients as unobserved states. Method 2 does \nnot fit naturally into the conventional framework of the Kalman filter, but it offers the significant advantage of only \nincreasing the size of the state vector by one. In contrast, Method 1 increases the size of the state vector by the size \nof $\\boldsymbol{\\beta}$. This is significant because computational complexity is quadratic in the size of the state \nvector but linear in the size of the observation vector.\n\nThe unobservability of $\\mathbf Z_t$ under Method 2 can be handled with maximum likelihood or Bayesian estimation by \nworking with the adjusted series \n\n$$\ny_t^* \\equiv y_t - \\tau_t = \\mathbf x_ t^\\prime \\boldsymbol{\\beta} + \\epsilon_t\n$$\n\nwhere $\\tau_t$ represents the time series component of the structural time series model. For example, assuming a level \nand seasonal component are specified, this means an initial estimate of the time series component \n$\\tau_t = \\mu_t + \\boldsymbol{\\gamma}^\\prime_ t \\mathbb{1}_p$ and $\\boldsymbol{\\beta}$ has to be acquired first. Then \n$\\boldsymbol{\\beta}$ can be estimated conditional on $\\mathbf y^ * \\equiv \\left(y_1^ *, y_2^ *, \\cdots, y_n^ *\\right)^\\prime$.\n\n`pybuc` uses Method 2 for estimating static coefficients.\n\n## Default priors\n\n### Irregular and state variances\n\nIf no priors are given for variances corresponding to stochastic states (i.e., level, trend, and seasonality), \nthe following defaults are used:\n\n$$\n\\begin{align}\n \\sigma^2_{\\mathrm{level}} &\\sim \\mathrm{IG}(0.01, (0.01 * \\mathrm{Std.Dev}(y))^2) \\\\\n \\sigma^2_{\\mathrm{seasonal}} &\\sim \\mathrm{IG}(0.01, (10 * 0.01 * \\mathrm{Std.Dev}(y))^2) \\\\\n \\sigma^2_{\\mathrm{trend}} &\\sim \\mathrm{IG}(0.01, (0.1 * 0.01 * \\mathrm{Std.Dev}(y))^2) \\\\\n\\end{align}\n$$\n\nThe level prior matches the default level prior in R's `bsts` package. However, the default seasonal and trend priors\nare different. While the default trend prior in `bsts` is the same as the level and seasonal priors, `pybuc` makes a \nmore conservative assumption about the variance associated with trend. This is reflected by a standard deviation \nthat is one-tenth (one-hundredth) the magnitude of the level (seasonal) standard deviation. In other words, \nthis prior assumes that variation in trend is small relative to variation in level and seasonality. The objective is to \nmitigate the impact that noise in the data could have on producing an overly aggressive trend.\n\nIn a similar vein, the default seasonal prior is different from `bsts`'s in that it allows for more flexibility in scale. \nSpecifically, the seasonal standard deviation is ten (one-hundred) times larger than the standard deviation for level \n(trend). Intuitively, changes from one cycle to another, as opposed to one period to the next, are likely to be larger.\n\nThe default prior for irregular variance is:\n\n$$\n\\sigma^2_{\\mathrm{irregular}} \\sim \\mathrm{IG}(0.01, (0.01 * \\mathrm{Std.Dev}(y))^2)\n$$\n\n### Damped/autoregressive state coefficients\n\nDamping can be applied to level, trend, and periodic-lag seasonality state components. By default, if no prior is given \nfor an autoregressive (i.e., AR(1)) coefficient, the prior takes the form \n\n$$\n\\phi \\sim N(1, 1)\n$$\n\nwhere $\\phi$ represents some autoregressive coefficient. Thus, the prior encodes the belief that the process (level, \ntrend, seasonality) is a random walk. A unit variance is assumed to accommodate different types of dynamics, including \nstationary or explosive processes that may oscillate, grow, or decay.\n\n\n### Regression coefficients\n\nThe default prior for regression coefficients is\n\n$$\n\\boldsymbol{\\beta} \\sim N\\left(\\mathbf 0, \\frac{\\kappa}{n} \\left(\\frac{1}{2} \\mathbf X^\\prime \\mathbf X + \n\\frac{1}{2} \\mathrm{diag}(\\mathbf X^\\prime \\mathbf X) \\right)\\right)\n$$\n\nwhere $\\mathbf X$ is the design matrix, $n$ is the number of response observations, and $\\kappa = 0.000001$ is the number \nof default prior observations given to the mean prior of $\\mathbf 0$. This prior is a slight modification of Zellner's \ng-prior (to guard against potential singularity of the design matrix). The number of prior observations, $\\kappa$, can be \nchanged by passing a value to the argument `zellner_prior_obs` in the `sample()` method. If Zellner's g-prior is not \ndesired, then a custom precision matrix can be passed to the argument `reg_coeff_prec_prior`. Similarly, if a zero-mean \nprior is not wanted, a custom mean prior can be passed to `reg_coeff_mean_prior`.\n\n## State space representation (example)\nThe unobserved components model can be rewritten in state space form. For example, suppose level, trend, seasonal, \nregression, and irregular components are specified, and the seasonal component takes a trigonometric form with \nperiodicity $S=4$ and $h=2$ harmonics. Let $\\mathbf Z_t \\in \\mathbb{R}^{1 \\times m}$, \n$\\mathbf T \\in \\mathbb{R}^{m \\times m}$, $\\mathbf R \\in \\mathbb{R}^{m \\times q}$, and \n$\\boldsymbol{\\alpha}_ t \\in \\mathbb{R}^{m \\times 1}$ denote the observation matrix, state transition matrix, \nstate error transformation matrix, and unobserved state vector, respectively, where $m$ is the number of state equations \nand $q$ is the number of state parameters to be estimated (i.e., the number of stochastic state equations, \nwhich is defined by the number of positive state variance parameters). \n\nThere are $m = 1 + 1 + (h * 2 - 1) + 1 = 6$ state equations and $q = 1 + 1 + (h * 2 - 1) = 5$ stochastic state equations\n, where the term $(h * 2 - 1)$ follows from $S=4$ being even and $h = S/2$. Outside of this case, there would generally \nbe $h * 2$ state equations for trigonometric seasonality. Note also that there are 5 stochastic state equations because \nthe state value for the regression component is not stochastic; it is 1 for all $t$ by construction. The observation, \nstate transition, and state error transformation matrices may be \nwritten as\n\n$$\n\\begin{align}\n \\mathbf Z_t &= \\left(\\begin{array}{cc} \n 1 & 0 & 1 & 0 & 1 & \\mathbf x_t^{\\prime} \\boldsymbol{\\beta}\n \\end{array}\\right) \\\\\n \\mathbf T &= \\left(\\begin{array}{cc} \n 1 & 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & \\cos(2\\pi / 4) & \\sin(2\\pi / 4) & 0 & 0 \\\\\n 0 & 0 & -\\sin(2\\pi / 4) & \\cos(2\\pi / 4) & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & -1 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 1\n \\end{array}\\right) \\\\\n \\mathbf R &= \\left(\\begin{array}{cc} \n 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 1 & 0 \\\\\n 0 & 0 & 0 & 0 & 1 \\\\\n 0 & 0 & 0 & 0 & 0\n \\end{array}\\right)\n\\end{align}\n$$\n\nGiven the definitions of $\\mathbf Z_t$, $\\mathbf T$, and $\\mathbf R$, the state space representation of the unobserved \ncomponents model above can compactly be expressed as\n\n$$\n\\begin{align}\n y_t &= \\mathbf Z_t \\boldsymbol{\\alpha}_ t + \\epsilon_t \\\\\n \\boldsymbol{\\alpha}_ {t+1} &= \\mathbf T \\boldsymbol{\\alpha}_ t + \\mathbf R \\boldsymbol{\\eta}_ t, \\hspace{5pt} \n t=1,2,...,n\n\\end{align}\n$$\n\nwhere\n\n$$\n\\begin{align}\n \\boldsymbol{\\alpha}_ t &= \\left(\\begin{array}{cc} \n \\mu_t & \\delta_t & \\gamma^4_{1, t} & \\gamma^{4*}_ {1, t} & \\gamma^4_{2, t} & 1\n \\end{array}\\right)^\\prime \\\\\n \\boldsymbol{\\eta}_ t &= \\left(\\begin{array}{cc} \n \\eta_{\\mu, t} & \\eta_{\\delta, t} & \\eta_{\\gamma^4_ 1, t} & \\eta_{\\gamma^{4*}_ 1, t} & \n \\eta_{\\gamma^4_ 2, t}\n \\end{array}\\right)^\\prime\n\\end{align}\n$$\n\nand \n\n$$\n\\mathrm{Cov}(\\boldsymbol{\\eta}_ t) = \\mathrm{Cov}(\\boldsymbol{\\eta}_ {t-1}) = \\boldsymbol{\\Sigma}_ \\eta = \n\\mathrm{diag}(\\sigma^2_{\\eta_\\mu}, \\sigma^2_{\\eta_\\delta}, \\sigma^2_{\\eta_{\\gamma^4_ 1}}, \\sigma^2_{\\eta_{\\gamma^{4*}_ 1}}, \n\\sigma^2_{\\eta_{\\gamma^4_ 2}}) \\in \\mathbb{R}^{5 \\times 5} \\hspace{5pt} \\textrm{for all } t=1,2,...,n\n$$\n\n# Estimation\n`pybuc` mirrors R's `bsts` with respect to estimation method. The observation vector, state vector, and regression \ncoefficients are assumed to be conditionally normal random variables, and the error variances are assumed to be \nconditionally independent inverse-Gamma random variables. These model assumptions imply conditional conjugacy of the \nmodel's parameters. Consequently, a Gibbs sampler is used to sample from each parameter's posterior distribution.\n\nTo achieve fast sampling, `pybuc` follows `bsts`'s adoption of the Durbin and Koopman (2002) simulation smoother. For \nany parameter $\\theta$, let $\\theta(s)$ denote the $s$-th sample of parameter $\\theta$. Each sample $s$ is drawn by \nrepeating the following three steps:\n\n1. Draw $\\boldsymbol{\\alpha}(s)$ from \n $p(\\boldsymbol{\\alpha} | \\mathbf y, \\boldsymbol{\\sigma}^2_\\eta(s-1), \\boldsymbol{\\beta}(s-1), \\sigma^2_\\epsilon(s-1))$ \n using the Durbin and Koopman simulation state smoother, where \n $\\boldsymbol{\\alpha}(s) = (\\boldsymbol{\\alpha}_ 1(s), \\boldsymbol{\\alpha}_ 2(s), \\cdots, \\boldsymbol{\\alpha}_ n(s))^\\prime$ \n and $\\boldsymbol{\\sigma}^2_\\eta(s-1) = \\mathrm{diag}(\\boldsymbol{\\Sigma}_\\eta(s-1))$. Note that `pybuc` implements a \n correction (based on a potential misunderstanding) for drawing $\\boldsymbol{\\alpha}(s)$ per \"A note on implementing \n the Durbin and Koopman simulation smoother\" (Marek Jarocinski, 2015).\n2. Draw $\\boldsymbol{\\sigma}^2(s) = (\\sigma^2_ \\epsilon(s), \\boldsymbol{\\sigma}^2_ \\eta(s))^\\prime$ from \n $p(\\boldsymbol{\\sigma}^2 | \\mathbf y, \\boldsymbol{\\alpha}(s), \\boldsymbol{\\beta}(s-1))$ using Durbin and Koopman's \n simulation disturbance smoother.\n3. Draw $\\boldsymbol{\\beta}(s)$ from \n $p(\\boldsymbol{\\beta} | \\mathbf y^ *, \\boldsymbol{\\alpha}(s), \\sigma^2_\\epsilon(s))$, where $\\mathbf y^ *$ is defined \n above.\n\nBy assumption, the elements in $\\boldsymbol{\\sigma}^2(s)$ are conditionally independent inverse-Gamma distributed random \nvariables. Thus, Step 2 amounts to sampling each element in $\\boldsymbol{\\sigma}^2(s)$ independently from their \nposterior inverse-Gamma distributions.\n",
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