Covariance

covariantcovariationcovarycovariance matrixcovariation biascovariesco-varycovariabilitycovariantlycovariation principle
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.wikipedia
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Covariance and correlation

correlationscovariancenormalized version of the covariance
The normalized version of the covariance, the correlation coefficient, however, shows by its magnitude the strength of the linear relation.
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Variance

sample variancepopulation variancevariability
where is the expected value of X, also known as the mean of X. The covariance is also sometimes denoted \sigma_{XY} or \sigma(X,Y), in analogy to variance.
The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by \sigma^2, s^2, or.

Uncorrelatedness (probability theory)

uncorrelated
Random variables whose covariance is zero are called uncorrelated.
In probability theory and statistics, two real-valued random variables, X, Y, are said to be uncorrelated if their covariance, is zero.

Covariance matrix

variance-covariance matrixcovariance matricescovariance
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a vector whose jth element is one of the random variables.
In probability theory and statistics, a covariance matrix, also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix, is a matrix whose element in the i, j position is the covariance between the i-th and j-th elements of a random vector.

Expected value

expectationexpectedmean
where is the expected value of X, also known as the mean of X. The covariance is also sometimes denoted \sigma_{XY} or \sigma(X,Y), in analogy to variance.
The amount by which the multiplicativity fails is called the covariance:

Pearson correlation coefficient

correlation coefficientcorrelationPearson correlation
The normalized version of the covariance, the correlation coefficient, however, shows by its magnitude the strength of the linear relation.
Pearson's correlation coefficient is the covariance of the two variables divided by the product of their standard deviations.

Independence (probability theory)

independentstatistically independentindependence
If X and Y are independent, then their covariance is zero.
and the covariance is zero, since we have

Cross-covariance matrix

cross-covariance matrices
For real random vectors and, the m \times n cross-covariance matrix is equal to
In probability theory and statistics, a cross-covariance matrix is a matrix whose element in the i, j position is the covariance between the i-th element of a random vector and j-th element of another random vector.

Normally distributed and uncorrelated does not imply independent

here for an examplein general, not sufficientindividually normally distributed
However, if two variables are jointly normally distributed (but not if they are merely individually normally distributed), uncorrelatedness does imply independence.
To see that X and Y are uncorrelated, one may consider the covariance : by definition, it is

Cauchy–Schwarz inequality

Cauchy Schwarz inequalityCauchy's inequalityCauchy-Schwarz inequality
holds via the Cauchy–Schwarz inequality.
where denotes variance, and denotes covariance.

Multivariate random variable

random vectorvectormultivariate
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a vector whose jth element is one of the random variables. For real random vectors and, the m \times n cross-covariance matrix is equal to
The covariance matrix (also called second central moment or variance-covariance matrix) of an n \times 1 random vector is an n \times n matrix whose (i,j) th element is the covariance between the i th and the j th random variables.

Financial economics

financial economistfinancial economistsfinance
Covariances play a key role in financial economics, especially in portfolio theory and in the capital asset pricing model.
Then, given this CML, the required return on risky securities will be independent of the investor's utility function, and solely determined by their covariance ("beta") with aggregate, i.e. market, risk.

Kalman filter

unscented Kalman filterKalmanInformation filter
This is an example of its widespread application to Kalman filtering and more general state estimation for time-varying systems.
The weights are calculated from the covariance, a measure of the estimated uncertainty of the prediction of the system's state.

Correlation and dependence

correlationcorrelatedcorrelate
The sign of the covariance therefore shows the tendency in the linear relationship between the variables.
It is obtained by dividing the covariance of the two variables by the product of their standard deviations.

Covariance function

covariancespatial covariance function
Covariance function
In probability theory and statistics, covariance is a measure of how much two variables change together, and the covariance function, or kernel, describes the spatial or temporal covariance of a random variable process or field.

Autocovariance

autocovariance functionautocovariance matrix
Autocovariance
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.

Eddy covariance

boundary layer eddieseddy covariance techniqueflux tower
The eddy covariance technique is a key atmospherics measurement technique where the covariance between instantaneous deviation in vertical wind speed from the mean value and instantaneous deviation in gas concentration is the basis for calculating the vertical turbulent fluxes.
In mathematical terms, "eddy flux" is computed as a covariance between instantaneous deviation in vertical wind speed (w') from the mean value (w-overbar) and instantaneous deviation in gas concentration, mixing ratio (s'), from its mean value (s-overbar), multiplied by mean air density (ρa).

Modern portfolio theory

portfolio theoryportfolio analysismean-variance analysis
Covariances play a key role in financial economics, especially in portfolio theory and in the capital asset pricing model.
The risk, return, and correlation measures used by MPT are based on expected values, which means that they are mathematical statements about the future (the expected value of returns is explicit in the above equations, and implicit in the definitions of variance and covariance).

Algorithms for calculating variance

computational algorithmsNumerically stable algorithmsnumerically stable alternatives
Numerically stable algorithms should be preferred in this case.
One can also find there similar formulas for covariance.

Propagation of uncertainty

error propagationtheory of errorspropagation of error
Propagation of uncertainty
If the uncertainties are correlated then covariance must be taken into account.

Distance correlation

distance standard deviationdistance covariance
Distance covariance, or Brownian covariance.
Distance covariance can be expressed in terms of the classical Pearson’s covariance,

Law of total covariance

conditional correlation
Law of total covariance
In probability theory, the law of total covariance, 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

Probability theory

theory of probabilityprobabilityprobability theorist
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.

Statistics

statisticalstatistical analysisstatistician
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.

Random variable

random variablesrandom variationrandom
In probability theory and statistics, covariance is a measure of the joint variability of two random variables.