Lecture

Fitting data with one Gauss function

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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 covers the uni-dimensional and multi-dimensional Gaussian functions, including their pdf representations, mean, and variance. It also explains how to model data using a Gaussian function, decompose the covariance matrix, and derive optimal parameters. The likelihood function and maximum likelihood optimization are discussed, emphasizing the importance of finding the optimal mean and covariance of the data.

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

Related concepts (30)

Variance

In probability theory and statistics, variance is the squared deviation from the mean of a random variable. The variance is also often defined as the square of the standard deviation. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by , , , , or .

Covariance

In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (that is, the variables tend to show opposite behavior), the covariance is negative.

Sample mean and covariance

The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables. The sample mean is the average value (or mean value) of a sample of numbers taken from a larger population of numbers, where "population" indicates not number of people but the entirety of relevant data, whether collected or not. A sample of 40 companies' sales from the Fortune 500 might be used for convenience instead of looking at the population, all 500 companies' sales.

Likelihood principle

In statistics, the likelihood principle is the proposition that, given a statistical model, all the evidence in a sample relevant to model parameters is contained in the likelihood function. A likelihood function arises from a probability density function considered as a function of its distributional parameterization argument.

Algorithms for calculating variance

Algorithms for calculating variance play a major role in computational statistics. A key difficulty in the design of good algorithms for this problem is that formulas for the variance may involve sums of squares, which can lead to numerical instability as well as to arithmetic overflow when dealing with large values.

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