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[[File:covariance_trends.svg|covariance trends|thumb]] [[File:Covariance_geometric_visualisation.svg|Covariance geometric visualisation|thumb|left]] '''Covariance''' is a measure used in [[statistics]] to quantify the relationship between two [[random variables]]. It indicates the direction of the linear relationship between variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values, the covariance is positive. In contrast, a negative covariance indicates that the greater values of one variable mainly correspond with the lesser values of the other variable.

==Definition==
The covariance between two random variables \(X\) and \(Y\) can be defined as the expected value of the product of their deviations from their respective means. Mathematically, it is represented as:

\[ \text{Cov}(X, Y) = E\left[ (X - \mu_X)(Y - \mu_Y) \right] \]

where \(E\) denotes the expected value, \(\mu_X\) is the mean of \(X\), and \(\mu_Y\) is the mean of \(Y\).

==Properties==
Covariance has several important properties:
* It is symmetric: \(\text{Cov}(X, Y) = \text{Cov}(Y, X)\).
* The covariance of a variable with itself is its variance: \(\text{Cov}(X, X) = \text{Var}(X)\).
* Covariance can take any value between negative infinity and positive infinity, where a value of zero indicates no linear relationship between the variables.

==Interpretation==
The sign of the covariance provides insight into the direction of the relationship between the variables:
* A positive covariance indicates a positive linear relationship.
* A negative covariance indicates a negative linear relationship.
* A covariance of zero indicates that no linear relationship exists, although it does not imply independence unless the variables are jointly normally distributed.

==Limitations==
One limitation of covariance is that it does not standardize the degree of relationship between the variables. This makes it difficult to compare the strength of the relationships across different pairs of variables. To address this limitation, the concept of [[correlation]] is used, which normalizes the covariance by the standard deviations of the variables, providing a dimensionless measure of linear relationship.

==Applications==
Covariance is widely used in various fields such as [[finance]] for portfolio optimization, in [[econometrics]], and in [[machine learning]] for feature selection and dimensionality reduction techniques like Principal Component Analysis (PCA).

==See Also==
* [[Variance]]
* [[Correlation coefficient]]
* [[Expected value]]
* [[Principal Component Analysis (PCA)]]

[[Category:Statistics]]
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Covariance - Revision history

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