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A set of two or more random variables , …, is called uncorrelated if each pair of them is uncorrelated. This is equivalent to the requirement that the non-diagonal elements of the autocovariance matrix K X X {\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {X} }} of the random vector X = [ X 1 …
When the errors on x are uncorrelated, the general expression simplifies to =, where = is the variance of k-th element of the x vector. Note that even though the errors on x may be uncorrelated, the errors on f are in general correlated; in other words, even if Σ x {\displaystyle {\boldsymbol {\Sigma }}^{x}} is a diagonal matrix, Σ f ...
The observations on the dependent variable are stacked into a column vector y; the observations on each independent variable are also stacked into column vectors, and these latter column vectors are combined into a design matrix X (not denoting a random vector in this context) of observations on the independent variables. Then the following ...
In general, random variables may be uncorrelated but statistically dependent. But if a random vector has a multivariate normal distribution then any two or more of its components that are uncorrelated are independent. This implies that any two or more of its components that are pairwise independent are independent.
Throughout this article, boldfaced unsubscripted and are used to refer to random vectors, and Roman subscripted and are used to refer to scalar random variables.. If the entries in the column vector = (,, …,) are random variables, each with finite variance and expected value, then the covariance matrix is the matrix whose (,) entry is the covariance [1]: 177 ...
In a linear regression, the true parameters are =, = which are reliably estimated in the case of uncorrelated and (black case) but are unreliably estimated when and are correlated (red case). Perfect multicollinearity refers to a situation where the predictors are linearly dependent (one can be written as an exact linear function of the others ...
For example, to calculate the autocorrelation of the real signal sequence = (,,) (i.e. =, =, =, and = for all other values of i) by hand, we first recognize that the definition just given is the same as the "usual" multiplication, but with right shifts, where each vertical addition gives the autocorrelation for particular lag values: +
Pairwise independent random variables with finite variance are uncorrelated. A pair of random variables X and Y are independent if and only if the random vector (X, Y) with joint cumulative distribution function (CDF) , (,) satisfies , (,) = (),