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The general regression model with n observations and k explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is = + where y is an n × 1 vector of dependent variable observations, each column of the n × k matrix X is a vector of observations on one of the k explanators, is a k × 1 vector of true coefficients, and e is an n× 1 vector of the ...
To have a lack-of-fit sum of squares that differs from the residual sum of squares, one must observe more than one y-value for each of one or more of the x-values. One then partitions the "sum of squares due to error", i.e., the sum of squares of residuals, into two components:
Since this is a biased estimate of the variance of the unobserved errors, the bias is removed by dividing the sum of the squared residuals by df = n − p − 1, instead of n, where df is the number of degrees of freedom (n minus the number of parameters (excluding the intercept) p being estimated - 1). This forms an unbiased estimate of the ...
It is calculated as the sum of squares of the prediction residuals for those observations. [1] [2] [3] Specifically, the PRESS statistic is an exhaustive form of cross-validation, as it tests all the possible ways that the original data can be divided into a training and a validation set.
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.
The sum of the residuals (unlike the sum of the errors) is necessarily 0. If one knows the values of any n − 1 of the residuals, one can thus find the last one. That means they are constrained to lie in a space of dimension n − 1. One says that there are n − 1 degrees of freedom for errors.
It is common to make the additional stipulation that the ordinary least squares (OLS) method should be used: the accuracy of each predicted value is measured by its squared residual (vertical distance between the point of the data set and the fitted line), and the goal is to make the sum of these squared deviations as small as possible.
Using matrix notation, the sum of squared residuals is given by S ( β ) = ( y − X β ) T ( y − X β ) . {\displaystyle S(\beta )=(y-X\beta )^{T}(y-X\beta ).} Since this is a quadratic expression, the vector which gives the global minimum may be found via matrix calculus by differentiating with respect to the vector β {\displaystyle \beta ...