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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:
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., calculate the sum of squares per degree of freedom, or variance. Standard deviation, in turn, is the square root of the variance.
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.
Greedy number partitioning – loops over the numbers, and puts each number in the set whose current sum is smallest. If the numbers are not sorted, then the runtime is O( n ) and the approximation ratio is at most 3/2 ("approximation ratio" means the larger sum in the algorithm output, divided by the larger sum in an optimal partition).
Prefix sums are trivial to compute in sequential models of computation, by using the formula y i = y i − 1 + x i to compute each output value in sequence order. However, despite their ease of computation, prefix sums are a useful primitive in certain algorithms such as counting sort, [1] [2] and they form the basis of the scan higher-order function in functional programming languages.
The explained sum of squares (ESS) is the sum of the squares of the deviations of the predicted values from the mean value of a response variable, in a standard regression model — for example, y i = a + b 1 x 1i + b 2 x 2i + ... + ε i, where y i is the i th observation of the response variable, x ji is the i th observation of the j th ...
The sum of squared deviations needed to calculate sample variance (before deciding whether to divide by n or n − 1) is most easily calculated as = From the two derived expectations above the expected value of this sum is
The sum of the a 2-column and the b 2-column must be bigger than the sum within entries of the a 2-column, since all the entries within the b 2-column are positive (except when the population mean is the same as the sample mean, in which case all of the numbers in the last column will be 0). Therefore: