R-squared

In statistics, the coefficient of determination, denoted R² or r² and pronounced "R squared", is a number that indicates the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

It is a statistic used in the context of statistical models whose main purpose is either the prediction of future outcomes or the testing of hypotheses, on the basis of other related information. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model.

There are several definitions of R² that are only sometimes equivalent. One class of such cases includes that of simple linear regression where r² is used instead of R². When an intercept is included, then r² is simply the square of the sample correlation coefficient (i.e., r) between the observed outcomes and the observed predictor values. If additional regressors are included, R² is the square of the coefficient of multiple correlation. In both such cases, the coefficient of determination ranges from 0 to 1.

Important cases where the computational definition of R² can yield negative values, depending on the definition used, arise where the predictions that are being compared to the corresponding outcomes have not been derived from a model-fitting procedure using those data, and where linear regression is conducted without including an intercept. Additionally, negative values of R² may occur when fitting non-linear functions to data. In cases where negative values arise, the mean of the data provides a better fit to the outcomes than do the fitted function values, according to this particular criterion.

A data set has n values marked y₁,...,y_n (collectively known as y_i or as a vector y = [y₁,...,y_n]^T), each associated with a predicted (or modeled) value f₁,...,f_n (known as f_i, or sometimes ŷ_i, as a vector f).

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