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In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure.For a real-valued continuous function f, defined on an interval [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x ...
Total variation distance is half the absolute area between the two curves: Half the shaded area above. In probability theory, the total variation distance is a statistical distance between probability distributions, and is sometimes called the statistical distance, statistical difference or variational distance.
Connection with total variation distance. The Hellinger distance (,) and the total variation distance (or statistical distance) (,) are related as follows: [8 ...
The total variation distance of two distributions and over a finite domain , (often referred to as statistical difference [2] or statistical distance [3] in ...
Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [l] is defined as the linear part of the change in the functional, and the second variation [m] is defined as the quadratic part. [22]
In cases where (,) are such that the conditional expected value is linear; that is, in cases where = +, it follows from the bilinearity of covariance that = (,) and = (,) and the explained component of the variance divided by the total variance is just the square of the correlation between and ; that is, in such ...
This follows from the fact that the variance and mean are independent of the ordering of x. Scale invariance: c v (x) = c v (αx) where α is a real number. [22] Population independence – If {x,x} is the list x appended to itself, then c v ({x,x}) = c v (x). This follows from the fact that the variance and mean both obey this principle.
Notably, except for total variation distance, all others are special cases of -divergence, or linear sums of -divergences. For each f-divergence D f {\displaystyle D_{f}} , its generating function is not uniquely defined, but only up to c ⋅ ( t − 1 ) {\displaystyle c\cdot (t-1)} , where c {\displaystyle c} is any real constant.