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Best rational approximants for π (green circle), e (blue diamond), ϕ (pink oblong), (√3)/2 (grey hexagon), 1/√2 (red octagon) and 1/√3 (orange triangle) calculated from their continued fraction expansions, plotted as slopes y/x with errors from their true values (black dashes)
Any non-linear differentiable function, (,), of two variables, and , can be expanded as + +. If we take the variance on both sides and use the formula [11] for the variance of a linear combination of variables (+) = + + (,), then we obtain | | + | | +, where is the standard deviation of the function , is the standard deviation of , is the standard deviation of and = is the ...
For many common constellations including BPSK, QPSK, and 8PSK, these two methods for finding the reference give the same result, but for higher-order QAM constellations including 16QAM, Star 32QAM, 32APSK, and 64QAM the RMS average and the maximum produce different reference values.
The angle between two term frequency vectors cannot be greater than 90°. If the attribute vectors are normalized by subtracting the vector means (e.g., ¯), the measure is called the centered cosine similarity and is equivalent to the Pearson correlation coefficient. For an example of centering,
In information theory, the Hamming distance between two strings or vectors of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of substitutions required to change one string into the other, or equivalently, the minimum number of errors that could have transformed one string into the other.
The relative entropy was introduced by Solomon Kullback and Richard Leibler in Kullback & Leibler (1951) as "the mean information for discrimination between and per observation from ", [6] where one is comparing two probability measures ,, and , are the hypotheses that one is selecting from measure , (respectively).
A seminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin. [1] A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings.
In bioinformatics, the root mean square deviation of atomic positions is the measure of the average distance between the atoms of superimposed proteins. In structure based drug design, the RMSD is a measure of the difference between a crystal conformation of the ligand conformation and a docking prediction.