Overview of Bayesian Structural Time Series Models and Examples of Application and Implementation

Bayesian Structural Time Series Models

Bayesian Structural Time Series Model (BSTS) is a type of statistical model that models phenomena that change over time and is used for forecasting and causal inference purposes.

BSTS can model the effects of trends, seasonality, events, and exogenous variables on time series data. The parameters of the model are obtained by an estimation method based on Bayesian statistics.

BSTS is used in a wide range of applications such as economic forecasting, web traffic forecasting, and sales force forecasting.

On the algorithms used in Bayesian structural time series models

Various algorithms exist for Bayesian structural time series models. Typical algorithms are described below.

• Kalman filter and Kalman smoothing: The Kalman filter, described in “Overview of State Space Models and Examples of Implementations for Analyzing Time Series Data Using R and Python” is an algorithm for filtering and predicting time series data in a system with linear dynamics. The Kalman filter is an algorithm for filtering and predicting time series data in systems with linear dynamics, and is a method for simultaneously filtering and smoothing data using all previous observations.
• Dynamic Linear Models (DLMs): DLMs are models for Bayesian inference on time series data with linear dynamics. DLMs are based on Kalman filters and Kalman smoothing, also called state space models.
• Gaussian Processes: Gaussian processes, which are also discussed in “Nonparametric Bayes and Gaussian Processes” provide a way to perform Bayesian inference on nonlinear time series data. Gaussian processes can be used to model data uncertainty and estimate predictive distributions.
• Hilbert-Wang Transform: The Hilbert-Wang transform is a method used to analyze nonlinear time series data. In this method, the signal is subjected to an analytical signal and covariance analysis to extract temporal characteristics.
• Nonparametric Bayesian Models: Nonparametric Bayesian models, also described in “Nonparametric Bayesian and Gaussian Processes” are methods for constructing models from data without a prior distribution. Dirichlet processes and Dirichlet process mixture models are examples of this method.

These algorithms are selected according to the characteristics of the time series data and the problem, and Bayesian structural time series models are an important method for reliable forecasting and analysis while appropriately handling data uncertainty and model complexity.

The procedure for implementing a Bayesian structural time series model

The procedure for implementing a Bayesian structural time series model may vary depending on the complexity of the model and the algorithm used, but the general steps are as follows. For the sake of simplicity, we will assume that we are implementing a Dynamic Linear Model (DLM).

1. Data Preparation: The first step is to collect the time series data to be analyzed and perform the necessary preprocessing. This includes smoothing the data, handling missing values, normalization, etc.
2. Model design: Select a Bayesian structured time series model appropriate to the problem to be analyzed. In the case of a dynamic linear model (DLM), the equations of state (linear dynamics) and the observation equations (observation model) need to be designed.
3. Setting up a prior distribution: In Bayesian statistics, it is important to set up a prior distribution. Examine the data and the model, select a prior distribution for the parameters, and set the hyperparameters. This allows prior knowledge of the data to be incorporated into the model.
4. Selecting an inference algorithm: Bayesian models require an inference algorithm to estimate the posterior distribution. These include the Markov Chain Monte Carlo (MCMC) method described in “Overview and Implementation of Markov Chain Monte Carlo“, Variational inference as described in “Overview of Variational Bayesian Learning and Various Implementations” and particle filters described in “Implementing Particle Filters on Time Series Data“, Select the appropriate algorithm.
5. Model fitting: Using the inference algorithm selected, estimate the posterior distribution for the data. This provides an estimate of the parameters and state of the model.
6. Model evaluation: Evaluate the inferred model to assess the model’s goodness of fit and predictive performance. Model diagnostics and sampling convergence checks are performed to confirm the goodness of fit of the model and the reliability of the parameters.
7. Perform forecasting: Forecast future time-series data using the estimated model. This allows the prediction performance of the model to be compared with actual data.
8. Interpret and report results: Interpret the results obtained, output them in an appropriate form, and provide semantic interpretations of the model parameters and forecast results for use in business decision making.

Libraries and platforms used for Bayesian structural time series models

Below we describe the libraries and platforms available for implementing Bayesian Structural Time Series models.

• Stan: Stan is a probabilistic programming language that provides a powerful tool for describing and estimating a variety of Bayesian statistical models, including Bayesian Structured Time Series models; Stan uses Hamiltonian Monte Carlo (HMC) sampling to enable advanced Bayesian inference.
• PyMC3: PyMC3 is a Python-based library used to build and estimate Bayesian models; PyMC3 supports HMC, nonparametric Bayesian models, and more, and can define models using relatively intuitive notation.
• TensorFlow Probability: TensorFlow Probability is a library developed by Google that supports probabilistic modeling and Bayesian inference. statistical modeling can also be combined.
• Edward: Edward is another library developed by Google that supports probabilistic graphical models and Bayesian statistics; Edward, like PyMC3 and Stan, is designed to make model building and estimation easy.
• JAGS: JAGS (Just Another Gibbs Sampler) is a program for estimating Bayesian models using Markov Chain Monte Carlo (MCMC) methods, and can be used by calling JAGS from R or Python.
• Anglican: Anglican is a statistical programming language for easy probabilistic modeling and inference, and is a probabilistic programming language based on Clojure. probabilistic programming capabilities, making it easy to describe probabilistic models and implement inference algorithms.

Bayesian Structural Time Series Model Application Examples

Bayesian structural time series models have been widely applied in various domains. Some of the applications are described below.

• Finance: Bayesian time series models are used in areas such as stock and exchange rate forecasting, risk management, and portfolio optimization. They are useful for modeling dynamic market fluctuations and predicting future prices.
• Weather Forecasting: In time series analysis of meteorological data, Bayesian time series models are used to forecast weather and analyze weather patterns. They can help model and make precise forecasts of variations in precipitation, temperature, wind speed, etc.
• Medical data analysis: Bayesian Structural Time Series models are applied to monitor patient health, predict disease progression, and evaluate drug effects. They are used to capture the variability of biological data over time.
• Marketing analysis: Bayesian structured time series models are used in the analysis of sales data and consumer behavior data to forecast demand and evaluate campaign effectiveness. This allows for the capture and forecasting of sales trends and seasonality.
• Media analysis: Bayesian time-series models are used in media analysis to forecast viewership ratings and measure the effectiveness of advertising campaigns. This includes, for example, building a model that takes into account viewer attributes and past viewing history in order to predict viewership ratings for TV programs.
• Traffic forecasting: In the analysis of traffic volume data and movement patterns, Bayesian structured time series models are used to forecast road congestion and plan public transportation operations. This enables the prediction of traffic peaks and the occurrence of traffic congestion.
• Energy Forecasting: Models of fluctuations in electricity consumption and renewable energy generation are used to forecast electricity supply and demand and to evaluate energy policies. This helps forecast the balance between supply and demand.
• IoT Analysis: Bayesian Structural Time Series models are used in IoT analysis to analyze sensor data and detect anomalies. For example, sensor data such as temperature and humidity can be used to efficiently control an air conditioning system.

Bayesian structural time series models are considered to be particularly effective for problems where causality is important over time, and are being applied.

Implementation of Bayesian structural time series models in various languages

<python>

• pymc3

A common Python implementation of Bayesian structural time series models is to use the pymc3 library, an open source library for performing Bayesian statistical modeling in Python, using either theano or tensorflow as a backend. It uses either theano or tensorflow as a backend. The following is an example of a Bayesian structural time series model implemented in pymc3.

import numpy as np
import pandas as pd
import pymc3 as pm

# Loading Data
data = pd.read_csv('data.csv')

# Model Definition
with pm.Model() as model:
    # Setting the prior distribution
    mu_alpha = pm.Normal('mu_alpha', mu=0, sd=10)
    sigma_alpha = pm.Exponential('sigma_alpha', lam=1)
    mu_beta = pm.Normal('mu_beta', mu=0, sd=10)
    sigma_beta = pm.Exponential('sigma_beta', lam=1)

    # Parameter Definitions
    alpha = pm.GaussianRandomWalk('alpha', mu=mu_alpha, sd=sigma_alpha, shape=len(data))
    beta = pm.GaussianRandomWalk('beta', mu=mu_beta, sd=sigma_beta, shape=len(data))

    # Model Definition
    mu = alpha + beta * data['x']
    sigma = pm.Exponential('sigma', lam=1)
    y = pm.Normal('y', mu=mu, sd=sigma, observed=data['y'])

    # sampling
    trace = pm.sample(1000, tune=1000, chains=2)

In this example, pymc3 is used to define a Bayesian structural time series model. The data are read from the data.csv file, and alpha and beta are modeled with a random walk process described in “Overview of Random Walks, Algorithms, and Examples of Implementations“, respectively, and y is assumed to follow a normal distribution. The sample method can be used to generate a trace object from which the posterior distribution can be analyzed.

• pydmn

Dynamic Linear Models (dlm) is one means of implementing Bayesian structural time series models using PyDLM, a library in Python. Below is a basic example of implementing a dynamic linear model (DLM) using dlm. First, install PyDLM.

pip install pydlm

Next, the dlm is used to model the time series data and make predictions. In this example, the standard deviation of the observed noise is changed to adjust the goodness of fit of the model.

from pydlm import dlm, trend, dynamic

# Sample Data Preparation
data = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]

# DLM Settings
myDLM = dlm(data)
myDLM = myDLM + trend(degree=1, discount=0.98, name='lineTrend')
myDLM = myDLM + dynamic(features=[[i] for i in data], discount=0.9, name='dynamic')

# Model fitting
myDLM.fit()

# Predicting the future
future = myDLM.predictN(N=5)

print("Prediction Results:", future)

In this example, the observed data is stored in data and a dlm model is constructed. The model is constructed by including trend and dynamic elements (in this case, the observed data itself), fitting the model with the fit method, and using the predictN method to predict the future.

<R>

Several packages exist for implementing Bayesian structural time series models in R. Some of the most representative ones are described below.

• bsts

The bsts package will be a package to implement the Bayesian Structural Time Series (BSTS) model. the BSTS model is a framework for modeling time series data by combining elements such as trend, seasonality, and regression, and MCMC sampling to estimate the posterior distribution of parameters. Below is an example of modeling the AirPassengers dataset using the bsts package.

library(bsts)
data(AirPassengers)
model <- bsts(y = AirPassengers, state.specification = "ar(2)", niter = 500)
summary(model)

• bvarsv

The bvarsv package implements the Bayesian Vector Autoregression (BVAR) model, a framework for simultaneously modeling multiple time series variables, used for forecasting and causal estimation of time series data. The bvarsv package uses MCMC sampling to estimate the posterior distribution of parameters. The following is an example of modeling a VAR model using the bvarsv package.

library(bvarsv)
data("Canada")
model <- bvar(y = Canada, p = 2, niter = 500)
summary(model)

• Dynamic Linear Models (dlm)

An example of implementing a Bayesian structural time series model using the dlm package in R is shown. This package is specialized for modeling dynamic linear models (DLMs) and supports state space models and Kalman filters. First, install the dlm package in R.

install.packages("dlm")

Next, the following code is an example of implementing a dynamic linear model using the dlm package. Here, a simple local linear trend model is used to model the observed data.

library(dlm)

# Sample Data Preparation
data <- c(10, 20, 30, 40, 50, 60, 70, 80, 90, 100)

# Setting up a local linear trend model
model <- dlmModPoly(order = 1)
build <- dlmBuild(data, model)

# Filtering and Smoothing
filtered <- dlmFilter(build)
smoothed <- dlmSmooth(build)

# Predicting the future
future <- dlmForecast(build, nAhead = 5)

print("Prediction Results:")
print(future)

In this example, the observed data is stored in data and a local linear trend model is built using the dlm package. dlmFilter function is used for filtering, dlmSmooth function for smoothing, and dlmForecast function for forecasting the future. dlm package is very flexible and versatile, allowing for complex models and data.

• bayesm

The bayesm package is a powerful tool for implementing Bayesian statistical models using R. Below is a basic example of implementing a Bayesian structural time series model using the bayesm package. Here, a dynamic linear model (DLM) is used. First, install the bayesm package.

install.packages("bayesm")

The following code then becomes an example of implementing a dynamic linear model using the bayesm package. In this example, a simple local linear trend model is used to model the observed data.

library(bayesm)

# Sample Data Preparation
data <- c(10, 20, 30, 40, 50, 60, 70, 80, 90, 100)

# Setting up a dynamic linear model
y <- data
T <- length(y)
nfac <- 1  # Number of Factors
X <- matrix(1, T, nfac)
Z <- matrix(1, T, nfac)
G <- diag(nfac)
W <- diag(nfac)
m0 <- rep(0, nfac)
C0 <- 1e6 * diag(nfac)
a0 <- rep(0, nfac)
R0 <- diag(nfac)
delta = 2
kappa = 1

# Model fitting
dlmout <- dlm(y = y, X = X, Z = Z, G = G, W = W,
              m0 = m0, C0 = C0, a0 = a0, R0 = R0,
              delta = delta, kappa = kappa)

# Prediction Execution
forecast <- dlmForecast(dlmout, nAhead = 5)

print("Prediction Results:")
print(forecast)

In this example, the observed data are stored in data and a dynamic linear model is built using the bayesm package. dlm functions are used to fit the model and dlmForecast functions are used to predict the future.

Since the bayesm package is very flexible and multifunctional, the construction of the model and the adjustment of the parameters must be tailored to the specific problem.

• marss

The marss (Multivariate Autoregressive State-Space Models) package is a package for implementing Bayesian statistical models using R. Below is an example of implementing a Bayesian structural time series model using the marss package. Here is an example of handling multivariate time series data. First, install the marss package.

install.packages("marss")

The following code is then an example of implementing a Bayesian structural time series model using the marss package. In this example, a bivariate multivariate autoregressive model (VAR model) is used to model the observed data.

library(marss)

# Sample Data Preparation
set.seed(123)
T <- 100
y <- matrix(rnorm(2 * T), ncol = 2)

# Setting up a VAR model
A <- array(0, dim = c(2, 2, 1))
A[, , 1] <- diag(2)
Q <- array(0, dim = c(2, 2, 1))
Q[, , 1] <- diag(2)
Z <- array(0, dim = c(2, 2, 1))
Z[, , 1] <- diag(2)
H <- array(0, dim = c(2, 2, 1))
H[, , 1] <- diag(2)
m0 <- array(0, dim = c(2, 1, 1))
C0 <- diag(2)

# Model Setup
model <- MARSS(y, model = "diagonal", A = A, Q = Q, Z = Z, H = H,
               m0 = m0, C0 = C0)

# fitting
fit <- fitMARSS(model)

# Prediction Execution
forecast <- predictMARSS(fit, nAhead = 5)

print("Prediction Results:")
print(forecast$statesPred)

In this example, bivariate multivariate time series data are stored in y and a VAR model is built using the marss package: the model is set up with the MARSS function, the model is fitted with the fitMARSS function, and the predictMARSS function predicts the future.

References and Bibliography

The details of time series data analysis are described in “Time Series Data Analysis” and Bayesian inference is discussed in “Probabilistic Generative Models” “Bayesian Inference and Machine Learning with Graphical Models” “Nonparametric Bayesian and Gaussian Processes” “Markov Chain Monte Carlo (MCMC) Method and Bayesian Inference“. See also.

Reference book is “State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications”

“Time Series Analysis for the State-Space Model with R/Stan”

“State-Space Models: Applications in Economics and Finance”

“Testing for Random Walk Coefficients in Regression and State Space Models”

“The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, & Emerged Triumphant from Two Centuries of C”

“Think Bayes: Bayesian Statistics in Python“

“Bayesian Modeling and Computation in Python“

“Bayesian Analysis with Python: Introduction to statistical modeling and probabilistic programming using PyMC3 and ArviZ, 2nd Edition”