Lecture

Gaussian Mixture Regression: Modeling and Prediction

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

In course

MICRO-455: Machine learning I

Real-world engineering applications must cope with a large dataset of dynamic variables, which cannot be well approximated by classical or deterministic models. This course gives an overview of methods from Machine Learning for the analysis of non-linear, highly noisy and multi dimensional data

Description

This lecture introduces Gaussian Mixture Regression (GMR), a method to estimate joint and conditional densities from multimodal datasets. It covers the principles of GMR with single and multiple Gaussian components, explaining how to estimate marginals, conditionals, and regressions. The lecture also discusses the computation of variances and the interpretation of noise in state space. The key takeaway is that GMR allows for predicting multi-dimensional outputs and capturing correlations across dimensions, although it requires computing the full distribution and careful initialization of the Expectation-Maximization algorithm.

Instructor

Aude Billard

Expertise RoboticsMachine LearningArtificial IntelligenceHuman-Robot InteractionMechatronicsDynamical Systems ControlComputational Neuroscience Aude Billard is full professor and head of the LASA laboratory at the School of Engineering at the Swiss Institute of Technology Lausanne (EPFL), and the Director of the Swiss Innovation Booster on Robotics. Dr. Billard acts as the President of the IEEE Robotics and Automation Society (RAS), after serving in several roles in the administrative and executive committe

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Data analysis: Regression analysis

Related concepts (32)

Machine learning

Machine learning (ML) is an umbrella term for solving problems for which development of algorithms by human programmers would be cost-prohibitive, and instead the problems are solved by helping machines 'discover' their 'own' algorithms, without needing to be explicitly told what to do by any human-developed algorithms. Recently, generative artificial neural networks have been able to surpass results of many previous approaches.

Maximum likelihood estimation

In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable. The point in the parameter space that maximizes the likelihood function is called the maximum likelihood estimate. The logic of maximum likelihood is both intuitive and flexible, and as such the method has become a dominant means of statistical inference.

Linear regression

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.

Correlation

In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.

Pearson correlation coefficient

In statistics, the Pearson correlation coefficient (PCC) is a correlation coefficient that measures linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviations; thus, it is essentially a normalized measurement of the covariance, such that the result always has a value between −1 and 1. As with covariance itself, the measure can only reflect a linear correlation of variables, and ignores many other types of relationships or correlations.

Related lectures (64)

Multivariate Statistics: Normal Distribution

Covers the multivariate normal distribution, properties, and sampling methods.

Supervised Learning: Linear Regression

Covers supervised learning with a focus on linear regression, including topics like digit classification, spam detection, and wind speed prediction.

Probabilistic Linear Regression

Explores probabilistic linear regression, covering joint and conditional probability, ridge regression, and overfitting mitigation.

Nonparametric Statistics: Bayesian Approach

Explores non-parametric statistics, Bayesian methods, and linear regression with a focus on kernel density estimation and posterior distribution.

Linear Regression: Statistical Inference and Regularization