Explained variation

In statistics, explained variation measures the proportion to which a mathematical model accounts for the variation (dispersion) of a given data set. Often, variation is quantified as variance; then, the more specific term explained variance can be used. The complementary part of the total variation is called unexplained or residual variation; likewise, when discussing variance as such, this is referred to as unexplained or residual variance.

In statistics, explained variation measures the proportion to which a mathematical model accounts for the variation (dispersion) of a given data set. Often, variation is quantified as variance; then, the more specific term explained variance can be used. The complementary part of the total variation is called unexplained or residual variation; likewise, when discussing variance as such, this is referred to as unexplained or residual variance. F(\theta) = \int \textrm{d}r\,g(r)\,\ln f(r;\theta) where g(r) is the probability density of a random variable R\, , and f(r;\theta)\, with \theta\in\Theta_i ( i=0,1\, ) are two families of parametric models. Model family 0 is the simpler one, with a restricted parameter space \Theta_0\subset\Theta_1 . Parameters are determined by maximum likelihood estimation, \theta_i = \operatorname{arg max}_{\theta\in\Theta_i} F(\theta). The information gain of model 1 over model 0 is written as \Gamma(\theta_1:\theta_0) = 2 [ F(\theta_1)-F(\theta_0) ]\, where a factor of 2 is included for convenience. Γ is always nonnegative; it measures the extent to which the best model of family 1 is better than the best model of family 0 in explaining g(r). Assume a two-dimensional random variable R=(X,Y) where X shall be considered as an explanatory variable, and Y as a dependent variable. Models of family 1 "explain" Y in terms of X, f(y\mid x;\theta) , whereas in family 0, X and Y are assumed to be independent. We define the randomness of Y by D(Y)=\exp[-2F(\theta_0)] , and the randomness of Y, given X, by D(Y\mid X)=\exp[-2F(\theta_1)] . Then, \rho_C^2 = 1-D(Y\mid X)/D(Y) can be interpreted as proportion of the data dispersion which is "explained" by X. Let X be a random vector, and Y a random variable that is modeled by a normal distribution with centre \mu=\Psi^\textrm{T}X . In this case, the above-derived proportion of explained variation \rho_C^2 equals the squared correlation coefficient R^2 . Note the strong model assumptions: the centre of the Y distribution must be a linear function of X, and for any given x, the Y distribution must be normal. In other situations, it is generally not justified to interpret R^2 as proportion of explained variance.

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