**Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?**

Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur GraphSearch.

Concept# Partition of sums of squares

Résumé

The partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors. Mathematically, the sum of squared deviations is an unscaled, or unadjusted measure of dispersion (also called variability). When scaled for the number of degrees of freedom, it estimates the variance, or spread of the observations about their mean value. Partitioning of the sum of squared deviations into various components allows the overall variability in a dataset to be ascribed to different types or sources of variability, with the relative importance of each being quantified by the size of each component of the overall sum of squares.
The distance from any point in a collection of data, to the mean of the data, is the deviation. This can be written as , where is the ith data point, and is the estimate of the mean. If all such deviations are squared, then summed, as in \sum_{i=1}^n\left(y_i-\overline{y},\right)^2, this gives the "sum of squares" for these data.
When more data are added to the collection the sum of squares will increase, except in unlikely cases such as the new data being equal to the mean. So usually, the sum of squares will grow with the size of the data collection. That is a manifestation of the fact that it is unscaled.
In many cases, the number of degrees of freedom is simply the number of data points in the collection, minus one. We write this as n − 1, where n is the number of data points.
Scaling (also known as normalizing) means adjusting the sum of squares so that it does not grow as the size of the data collection grows. This is important when we want to compare samples of different sizes, such as a sample of 100 people compared to a sample of 20 people. If the sum of squares were not normalized, its value would always be larger for the sample of 100 people than for the sample of 20 people. To scale the sum of squares, we divide it by the degrees of freedom, i.e.

Source officielle

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Cours associés (12)

Concepts associés (7)

Lack-of-fit sum of squares

In statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the pure-error sum of squares. The pure-error sum of squares is the sum of squared deviations of each value of the dependent variable from the average value over all observations sharing its independent variable value(s).

Total sum of squares

In statistical data analysis the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. For a set of observations, , it is defined as the sum over all squared differences between the observations and their overall mean .: For wide classes of linear models, the total sum of squares equals the explained sum of squares plus the residual sum of squares. For proof of this in the multivariate OLS case, see partitioning in the general OLS model.

Partition of sums of squares

The partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors. Mathematically, the sum of squared deviations is an unscaled, or unadjusted measure of dispersion (also called variability). When scaled for the number of degrees of freedom, it estimates the variance, or spread of the observations about their mean value.

PHYS-467: Machine learning for physicists

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

MATH-413: Statistics for data science

Statistics lies at the foundation of data science, providing a unifying theoretical and methodological backbone for the diverse tasks enountered in this emerging field. This course rigorously develops

MATH-408: Regression methods

General graduate course on regression methods

Séances de cours associées (108)

Régression linéairePHYS-467: Machine learning for physicists

Couvre la régression linéaire pour estimer la vitesse du train en utilisant les moindres carrés et la régularisation.

Estimation des moindres carrés pondérés : Algorithme IRLSMATH-413: Statistics for data science

Explore l'algorithme IRLS pour l'estimation pondérée des moindres carrés dans GLM.

Régression linéaire : bases et estimationPHYS-467: Machine learning for physicists

Couvre les bases de la régression linéaire et la façon de résoudre les problèmes d'estimation en utilisant les moindres carrés et la notation matricielle.