# Covariance

**covariantcovariationcovarycovariance matrixcovariation biascovariesco-varycovariabilitycovariantlycovariation principle**

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

234 Related Articles

### 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.

|covariance||

### 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.