Bayesian Forecast: Credible Interval With Predicted Regressors

Mar 1, 2025 by ADMIN 63 views

Introduction

In the realm of time series forecasting, Bayesian methods have gained significant attention in recent years due to their ability to incorporate prior knowledge and uncertainty into the forecasting process. One of the key aspects of Bayesian forecasting is the estimation of credible intervals, which provide a range of possible values for the forecasted variable. In this article, we will explore how to perform a Bayesian linear regression forecast with a credible interval, where the forecasted variable (orders) depends on multiple regressors, including time and another variable (accounts).

Problem Formulation

Let's consider a scenario where we want to forecast the number of orders at time $t$ , denoted as $\text{orders}_{t}$ . We assume that the orders depend on two regressors: time $t$ and the number of accounts at time $t$ , denoted as $\text{accounts}_{t}$ . The relationship between the orders and the regressors can be modeled using a linear regression equation:

\text{orders}_{t} = \beta_{0} + \beta_{1}t + \beta_{2}\text{accounts}_{t} + \epsilon_{t}

where $\beta_{0}$ , $\beta_{1}$ , and $\beta_{2}$ are the regression coefficients, and $\epsilon_{t}$ is the error term.

Bayesian Linear Regression

In a Bayesian framework, we treat the regression coefficients as random variables with prior distributions. We assume that the prior distributions for the regression coefficients are normal with mean 0 and variance $\tau^{2}$ :

\beta_{0}, \beta_{1}, \beta_{2} \sim \mathcal{N}(0, \tau^{2})

The likelihood function for the linear regression model is given by:

p(\text{orders}_{t} | \beta_{0}, \beta_{1}, \beta_{2}, \sigma^{2}) = \mathcal{N}(\text{orders}_{t} | \beta_{0} + \beta_{1}t + \beta_{2}\text{accounts}_{t}, \sigma^{2})

where $\sigma^{2}$ is the variance of the error term.

Posterior Distribution

Using Bayes' theorem, we can update the prior distributions for the regression coefficients with the likelihood function to obtain the posterior distribution:

p(\beta_{0}, \beta_{1}, \beta_{2} | \text{orders}_{t}, \sigma^{2}) \propto p(\text{orders}_{t} | \beta_{0}, \beta_{1}, \beta_{2}, \sigma^{2}) \times p(\beta_{0}, \beta_{1}, \beta_{2})

The posterior distribution is also normal with mean $\mu$ and variance $\Sigma$ :

\beta_{0}, \beta_{1}, \beta_{2} | \text{orders}_{t}, \sigma^{2} \sim \mathcal{N}(\mu, \Sigma)

Predicted Regressors

To perform a forecast, we need to predict the values of the regressors at future time points. We can use the posterior distribution of the regression coefficients to predict the values of the regressors:

\text{accounts}_{t+1} | \text{orders}_{t}, \sigma^{2} \sim \mathcal{N}(\mu_{\text{accounts}}, \Sigma_{\text{accounts}})

where $\mu_{\text{accounts}}$ and $\Sigma_{\text{accounts}}$ are the mean and variance of the posterior distribution of the accounts regressor.

Credible Interval

A credible interval is a range of possible values for the forecasted variable, which is obtained by sampling from the posterior distribution of the regression coefficients. We can use the posterior distribution to compute the credible interval for the forecasted orders:

\text{orders}_{t+1} | \text{orders}_{t}, \sigma^{2} \sim \mathcal{N}(\mu_{\text{orders}}, \Sigma_{\text{orders}})

where $\mu_{\text{orders}}$ and $\Sigma_{\text{orders}}$ are the mean and variance of the posterior distribution of the orders.

Implementation

To implement the Bayesian linear regression forecast with a credible interval, we can use a programming language such as Python with libraries such as PyMC3 or scikit-learn. We can define the model using the following code:

import pymc3 as pm
import numpy as np

# Define the model
with pm.Model() as model:
    # Define the prior distributions for the regression coefficients
    beta0 = pm.Normal('beta0', mu=0, sigma=10)
    beta1 = pm.Normal('beta1', mu=0, sigma=10)
    beta2 = pm.Normal('beta2', mu=0, sigma=10)
    
    # Define the likelihood function
    orders = pm.Normal('orders', mu=beta0 + beta1 * t + beta2 * accounts, sigma=1, observed=True)
    
    # Sample from the posterior distribution
    trace = pm.sample(1000)

# Compute the credible interval
credible_interval = pm.hpd(trace, varnames=['orders'])

Conclusion

In this article, we have discussed how to perform a Bayesian linear regression forecast with a credible interval, where the forecasted variable depends on multiple regressors, including time and another variable. We have shown how to define the model using a programming language such as Python with libraries such as PyMC3 or scikit-learn. We have also demonstrated how to compute the credible interval using the posterior distribution of the regression coefficients. The Bayesian linear regression forecast with a credible interval provides a powerful tool for forecasting and decision-making in a wide range of applications.

References

Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2004). Bayesian data analysis. Chapman and Hall/CRC.
Kruschke, J. K. (2011). Doing Bayesian data analysis: A tutorial with R and BUGS. Academic Press.
McElreath, R. (2016). Statistical rethinking: A Bayesian course with R and Stan. Chapman and Hall/CRC.
Bayesian Forecast: Credible Interval with Predicted Regressors - Q&A ====================================================================

Introduction

In our previous article, we discussed how to perform a Bayesian linear regression forecast with a credible interval, where the forecasted variable depends on multiple regressors, including time and another variable. In this article, we will answer some frequently asked questions (FAQs) related to Bayesian forecasting and credible intervals.

Q: What is the difference between a Bayesian and a frequentist approach to forecasting?

A: The main difference between a Bayesian and a frequentist approach to forecasting is the way they handle uncertainty. In a frequentist approach, uncertainty is typically handled using confidence intervals, which are based on the sampling distribution of the estimator. In a Bayesian approach, uncertainty is handled using posterior distributions, which are based on the combination of prior knowledge and data.

Q: What is a credible interval, and how is it different from a confidence interval?

A: A credible interval is a range of possible values for the forecasted variable, which is obtained by sampling from the posterior distribution of the regression coefficients. It is different from a confidence interval, which is based on the sampling distribution of the estimator. A credible interval is more informative than a confidence interval, as it takes into account the prior knowledge and the data.

Q: How do I choose the prior distribution for the regression coefficients?

A: The choice of prior distribution for the regression coefficients depends on the specific problem and the available data. In general, a non-informative prior distribution is a good starting point, as it allows the data to dominate the posterior distribution. However, if there is prior knowledge about the regression coefficients, a more informative prior distribution can be used.

Q: How do I handle missing data in Bayesian forecasting?

A: Missing data can be handled in Bayesian forecasting using a variety of techniques, including multiple imputation and data augmentation. Multiple imputation involves creating multiple versions of the data set with the missing values imputed, and then analyzing each version separately. Data augmentation involves adding additional variables to the model to account for the missing data.

Q: Can I use Bayesian forecasting for non-linear models?

A: Yes, Bayesian forecasting can be used for non-linear models. In fact, Bayesian methods are particularly well-suited for non-linear models, as they can handle complex relationships between the variables. However, the choice of prior distribution and the specification of the model may need to be adjusted to accommodate the non-linear relationships.

Q: How do I evaluate the performance of a Bayesian forecasting model?

A: The performance of a Bayesian forecasting model can be evaluated using a variety of metrics, including mean absolute error (MAE), mean squared error (MSE), and root mean squared percentage error (RMSPE). These metrics can be used to compare the performance of different models and to evaluate the robustness of the results.

Q: Can I use Bayesian forecasting for real-time forecasting?

A: Yes, Bayesian forecasting can be used for real-time forecasting. In fact, Bayesian methods are particularly well-suited for real-time forecasting, as they can handle changing relationships between the variables and provide up-to-date estimates of the forecasted variable.

Q: How do I implement Bayesian forecasting in practice?

A: Bayesian forecasting can be implemented in practice using a variety of software packages, including R, Python, and Julia. The specific implementation will depend on the problem and the available data, but in general, it involves specifying the model, choosing the prior distribution, and sampling from the posterior distribution.

Conclusion

In this article, we have answered some frequently asked questions related to Bayesian forecasting and credible intervals. We hope that this article has provided a useful overview of the key concepts and techniques involved in Bayesian forecasting, and has provided a starting point for further exploration and implementation.

References

Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2004). Bayesian data analysis. Chapman and Hall/CRC.
Kruschke, J. K. (2011). Doing Bayesian data analysis: A tutorial with R and BUGS. Academic Press.
McElreath, R. (2016). Statistical rethinking: A Bayesian course with R and Stan. Chapman and Hall/CRC.