Covariance

Definition

In Statistical Theory of probability and , one names covariance of two random variable with actual values X and Y the value:

$\ sigma_ {xy} = \ operatorname {cov} (X, there) = \ sum_ {i=1} ^n \ sum_ {j=1} ^m (x_i y_j p (x_i) p (y_j/x=x_i))- \ bar {X} \ bar {there} = \ sum_ {i=1} ^n \ sum_ {j=1} ^m (x_i y_j p (y_j) p (x_i/y=y_j))- \ bar {X} \ bar {there}$

with

\ sigma_x^2 = \ sum_ {i=1} ^n x_i^2 p (x_i) - \ bar {X} ^2

and

\ sigma_y^2 = \ sum_ {j=1} ^m y_j^2 p (y_j) - \ bar {there} ^2

Intuitively, covariance is a measurement of the simultaneous variation of two random variables. I.e. covariance becomes more positive for each couple of values which differ from their average in the same direction, and more negative for each couple of values which differ from their average in the opposite direction.

The measuring unit of covariance cov ( X , Y ) is the product of the unit of the random variables X and Y . On the other hand, the Corrélation, which depends on covariance, is a linear measurement of dependence without unit.

The definition above is equivalent to the following formula, more often used for calculations:

$\ operatorname {cov} (X, Y) = E (X Y) - E (X) E (Y)$

where E indicates the mathematical Espérance. If X and Y is independent variables , then their covariance is null. Indeed, one has then:

$E (X \ cdot Y) =E (X) \ cdot E (Y) =E (X) E (Y)$ ,

The reciprocal one, however, is not true. It is indeed possible that X and Y is not independent, and that their covariance is null. Random variables whose covariance is null known as are not correlated.

If X and Y is random variables with actual values, and C a constant (“constant”, in this context, meaning not-random), then the following formulas are direct consequences of the definition of covariance:

$\ operatorname {cov} (X, X) = \ operatorname {VAr} (X)$

\ operatorname {cov} (X, Y) = \ operatorname {cov} (Y, X)

\ operatorname {cov} (cX, Y) = C \, \ operatorname {cov} (X, Y)

\ operatorname {cov} \ left (\ sum_i {X_i}, \ sum_j {Y_j} \ right) = \ sum_i {\ sum_j {\ operatorname {cov} \ left (X_i, Y_j \ right)}}

Example

In a forum Internet, somebody affirms that the activities are more intense the days of full moon. One can not have the calendar of full moons, but if this assertion is exact and if one names NR (T) the number of contributions to the day T, covariance between NR (T) and NR (t+28) cumulated on all the values of T will be higher in theory than covariances between NR (T) and NR (t+x) for the values of X different from 28.

Example of results

The curve of covariance indeed shows a light peak for value 28, but of other peaks of increasing importance for 21,14 and 7. It is thus more plausible to put forth the assumption of a weekly point of activity (weekend, for example) for this forum, as an explanatory variable (this case was observed in 1995 on the forum CompuServe of the newspaper Le Monde).

Technical details

Covariance $v_ {AB}$ of two random variable $A$ and $B$ jointly observed $N$ time is defined by:

$v_ {AB} = \ frac {\ sum a_i \ cdot b_i} {NR} - \ frac {\ sum a_i} {NR} \ cdot \ frac {\ sum b_i} {NR}$

The matrix of covariance $V$ of a random variable vectorial $X$ observed $N$ time is defined by:

$V = \ frac {\ sum x_i \ cdot x_i^T} {NR} - \ frac {\ sum x_i} {NR} \ cdot \ left (\ frac {\ sum x_i} {NR} \ right) ^T$

It will be noticed that if $A$ is the random variable corresponding to the $a$ -ième coordinate of $X$ and if $B$ is the random variable corresponding to the $b$ -ième coordinate of $X$ then there is $v_ {AB} = V_ {ab}$ .

The Inverse of the matrix of covariance is sometimes indicated by the term of “matrix of precision”.

Bilinéarité

Are $X$ and $Y$ two random variables definite on same probabilized space. The formula var (X+Y) = VAr (X) + VAr (Y) + 2cov (X, Y) is the analog of $(x+y) ^2=x^2+y^2+ 2xy$ . In fact, the majority of the properties of covariance are similar to those of the product of two realities or the scalar product of two vectors.

Use

The knowledge of covariances is generally essential in the functions of Estimation, filtering and Lissage. They allow, inter alia in Photographie, to manage to correct in a spectacular way the Flou S of development as well as the blurs of moved, which is extremely important for the astronomical stereotypes. One also uses them in Automatique. In Sociolinguistique, covariance indicates the correspondence between the membership of some Social class and some to speak inherent in this social condition.

See too

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