In statistics, Bayesian multivariate linear regression is a
Bayesian approach to multivariate linear regression, i.e. linear regression where the predicted outcome is a vector of correlated random variables rather than a single scalar random variable. A more general treatment of this approach can be found in the article MMSE estimator.
Consider a regression problem where the dependent variable to be predicted is not a single real-valued scalar but an m-length vector of correlated real numbers. As in the standard regression setup, there are n observations, where each observation i consists of k−1 explanatory variables, grouped into a vector of length k (where a dummy variable with a value of 1 has been added to allow for an intercept coefficient). This can be viewed as a set of m related regression problems for each observation i:
where the set of errors are all correlated. Equivalently, it can be viewed as a single regression problem where the outcome is a row vector and the regression coefficient vectors are stacked next to each other, as follows:
The coefficient matrix B is a matrix where the coefficient vectors for each regression problem are stacked horizontally:
The noise vector for each observation i is jointly normal, so that the outcomes for a given observation are correlated:
We can write the entire regression problem in matrix form as:
where Y and E are matrices. The design matrix X is an matrix with the observations stacked vertically, as in the standard linear regression setup:
The classical, frequentists linear least squares solution is to simply estimate the matrix of regression coefficients using the Moore-Penrose pseudoinverse:
To obtain the Bayesian solution, we need to specify the conditional likelihood and then find the appropriate conjugate prior. As with the univariate case of linear Bayesian regression, we will find that we can specify a natural conditional conjugate prior (which is scale dependent).
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
This course aims to introduce the basic principles of machine learning in the context of the digital humanities. We will cover both supervised and unsupervised learning techniques, and study and imple
Machine learning and data analysis are becoming increasingly central in sciences including physics. In this course, fundamental principles and methods of machine learning will be introduced and practi
Ce cours présentera les bases de l'analyse des données et de l'apprentissage à partir des données, l'estimation des erreurs et la stochasticité en physique. Les concepts seront introduits théoriquemen
In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.
Correlated errors of experimental data are a common but often neglected problem in physical sciences. Various tools are provided here for thorough propagation of uncertainties in cases of correlated errors. Discussed are techniques especially applicable to ...
ELSEVIER2023
In the rapidly evolving landscape of machine learning research, neural networks stand out with their ever-expanding number of parameters and reliance on increasingly large datasets. The financial cost and computational resources required for the training p ...
With the significant increase in photovoltaic (PV) electricity generation, more attention has been given to PV power forecasting. Indeed, accurate forecasting allows power grid operators to better schedule and dispatch their assets, such as energy storage ...
Pergamon-Elsevier Science Ltd2024
Covers supervised learning with a focus on linear regression, including topics like digit classification, spam detection, and wind speed prediction.
Introduces pKa values and explains how to derive them from the Z plot using linear regression.
Covers the basics of linear regression, focusing on minimizing errors and predicting outputs.