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The sign of the covariance, therefore, shows the tendency in the linear relationship between the variables. If greater values of one variable mainly correspond with greater values of the other variable, and the same holds for lesser values (that is, the variables tend to show similar behavior), the covariance is positive. [2]
With any number of random variables in excess of 1, the variables can be stacked into a random vector whose i th element is the i th random variable. Then the variances and covariances can be placed in a covariance matrix, in which the (i, j) element is the covariance between the i th random variable and the j th one.
In statistics, the Pearson correlation coefficient (PCC) [a] is a correlation coefficient that measures linear correlation between two sets of data. It is the ratio between the covariance of two variables and the product of their standard deviations; thus, it is essentially a normalized measurement of the covariance, such that the result always ...
The same C(x, y) is called the autocovariance function in two instances: in time series (to denote exactly the same concept except that x and y refer to locations in time rather than in space), and in multivariate random fields (to refer to the covariance of a variable with itself, as opposed to the cross covariance between two different ...
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose j th element (j = 1, ..., K) is one of the random variables. [1] The sample covariance matrix has in the denominator rather than due to a variant of Bessel's correction: In short, the sample ...
In statistics and in probability theory, distance correlation or distance covariance is a measure of dependence between two paired random vectors of arbitrary, not necessarily equal, dimension. The population distance correlation coefficient is zero if and only if the random vectors are independent .
Note: The conditional expected values E( X | Z) and E( Y | Z) are random variables whose values depend on the value of Z. Note that the conditional expected value of X given the event Z = z is a function of z. If we write E( X | Z = z) = g(z) then the random variable E( X | Z) is g(Z). Similar comments apply to the conditional covariance.
Analysis of covariance (ANCOVA) is a general linear model that blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of one or more categorical independent variables (IV) and across one or more continuous variables.