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Covariance function. In probability theory and statistics, the covariance function describes how much two random variables change together (their covariance) with varying spatial or temporal separation. For a random field or stochastic process Z (x) on a domain D, a covariance function C (x, y) gives the covariance of the values of the random ...
The covariance is the sum of the volumes of the cuboids in the 1st and 3rd quadrants (red) minus those in the 2nd and 4th (blue). Suppose that and have the following joint probability mass function, [6] in which the six central cells give the discrete joint probabilities of the six hypothetical realizations :
where E is the expected value operator. Notably, correlation is dimensionless while covariance is in units obtained by multiplying the units of the two variables. If Y always takes on the same values as X, we have the covariance of a variable with itself (i.e. ), which is called the variance and is more commonly denoted as the square of the ...
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations. The form of the definition involves a "product moment", that is, the mean (the first moment about the origin) of the product of the mean-adjusted random variables; hence the modifier product-moment in the name.
The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in Rp×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. [1]
Indicator function. A three-dimensional plot of an indicator function, shown over a square two-dimensional domain (set X): the "raised" portion overlays those two-dimensional points which are members of the "indicated" subset (A). In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps ...
In probability theory, the law of total covariance, [1] covariance decomposition formula, or conditional covariance formula states that if X, Y, and Z are random variables on the same probability space, and the covariance of X and Y is finite, then. {\displaystyle \operatorname {cov} (X,Y)=\operatorname {E} (\operatorname {cov} (X,Y\mid Z ...
Correlation and covariance. In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the process in question.