covariance

American Heritage Dictionary:

co·var·i·ance

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(kō-vâr'ē-əns) pronunciation

n.
A statistical measure of the variance of two random variables that are observed or measured in the same mean time period. This measure is equal to the product of the deviations of corresponding values of the two variables from their respective means.

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The covariance of two random variables is the difference between the expected value of their product and the product of their separate expected values. For random variables X and Y,Cov(X, Y)=E(XY)-E(X)×E(Y). If X and Y are independent then Cov(X, Y)=0. However, if Cov(X, Y)=0 then X and Y may not be independent. A useful result isVar(aX+bY)=a²Var(X)+2abCov(X, Y)+b²Var(Y),where Var denotes variance, and a and b are constants. The term 'covariance' was used by Sir Ronald Fisher in 1930. See also correlation.

Barron's Business Dictionary:

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Statistical term for the correlation between two variables multiplied by the standard deviation for each of the variables. A positive covariance indicates that the two variables tend to move up and down together; a negative covariance indicates that when one moves higher, the other tends to go lower.

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Oxford Dictionary of Geography:

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An index of the degree of association between two groups of variables. It is calculated in samples by transforming the scores in both groups to deviations from their respective means, multiplying pairs of scores from each group together, and calculating the mean. If the variables are x and y then the covariance of x and y is S(x_i - x)(y_i - y).

In the World Bank Poverty Profile, the degree to which values co-vary with a poverty indicator determines how much the total variation has been ‘explained’, and which variables contribute to that explanation.

Investopedia Financial Dictionary:

Covariance

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A measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns move inversely.

One method of calculating covariance is by looking at return surprises (deviations from expected return) in each scenario. Another method is to multiply the correlation between the two variables by the standard deviation of each variable.

Investopedia Says:
Possessing financial assets that provide returns and have a high covariance with each other will not provide very much diversification.

For example, if stock A's return is high whenever stock B's return is high and the same can be said for low returns, then these stocks are said to have a positive covariance. If an investor wants a portfolio whose assets have diversified earnings, he or she should pick financial assets that have low covariance to each other.

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See why investors today still follow this old set of principles that reduce risk and increase returns through diversification. Modern Portfolio Theory: Why It's Still Hip
Volatility is not the only way to measure risk. Learn about the "new science of risk management". An Introduction To Value at Risk (VAR)
Volatility is not the only way to measure risk. Learn about the "new science of risk management". How To Convert Value At Risk To Different Time Periods
What to know about stationary and non-stationary processes before you try to model or forecast. Introduction To Stationary And Non-Stationary Processes

Oxford Dictionary of Biochemistry:

covariance

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a measure of the association between two variables. For n pairs of values of two random variables, x and y, this is given by:
Cov. (x,y) = (x − x̄)(y − ȳ)/(n − 1)
where x̄ and ȳ are the means of the populations of x and y, respectively. Compare correlation coefficient, regression coefficient, variance.

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Saunders Veterinary Dictionary:

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The expected value of the product of the deviations of corresponding values of two random variables from their respective means.

c. method — used for the calculation of relationship and inbreeding in large populations.

Wikipedia on Answers.com:

Covariance

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This article is about the measure of linear relation between random variables. For other uses, see Covariance (disambiguation).

In probability theory and statistics, covariance is a measure of how much two random variables change together. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the smaller values, i.e. the variables tend to show similar behavior, the covariance is a positive number. In the opposite case, when the greater values of one variable mainly correspond to the smaller values of the other, i.e. the variables tend to show opposite behavior, the covariance is negative. The sign of the covariance therefore shows the tendency in the linear relationship between the variables. The magnitude of the covariance is not that easy to interpret. The normalized version of the covariance, the correlation coefficient, however shows by its magnitude the strength of the linear relation.

A distinction must be made between (1) the covariance of two random variables, which is a population parameter that can be seen as a property of the joint probability distribution, and (2) the sample covariance, which serves as an estimated value of the parameter.

Contents

1 Definition
2 Properties
3 Calculating the sample covariance
4 Comments
5 See also
6 References
7 External links

Definition

The covariance between two jointly distributed real-valued random variables X and Y with finite second moments is

$\operatorname{Cov}(X,Y) = \operatorname{E}{\big[(X - \operatorname{E}[X])(Y - \operatorname{E}[Y])\big]}$

where E[X] is the expected value of X. By using the linearity property of expectations, this can be simplified to

$\operatorname{Cov}(X,Y) = \operatorname{E}\big[X Y\big] - \operatorname{E}[X]\operatorname{E}[Y]$

For random vectors X and Y (of dimension m and n respectively) the m×n covariance matrix is equal to

$\begin{align} \operatorname{Cov}(X,Y) & = \operatorname{E}\left[(X - \operatorname{E}[X])(Y - \operatorname{E}[Y])^T\right]\\ & = \operatorname{E}\left[X Y^T\right] - \operatorname{E}[X]\operatorname{E}[Y]^T \end{align}$

where M^T is the transpose of a matrix (or vector) M.

The (i,j)-th element of this matrix is equal to the covariance Cov(X_i, Y_j) between the i-th scalar component of X and the j-th scalar component of Y. In particular, Cov(Y, X) is the transpose of Cov(X, Y).

For a vector $X=(X_1,\dots . X_n)$ of n jointly distributed random variables with finite second moments, its covariance matrix is defined as:

$\operatorname{Cov}(X) = \operatorname{Cov}(X,X) .$

Random variables whose covariance is zero are called uncorrelated.

The units of measurement of the covariance Cov(X, Y) are those of X times those of Y. By contrast, correlation, which depends on the covariance, is a dimensionless measure of linear dependence. (In fact, correlation can simply be understood as a normalized version of covariance.)

Properties

Variance is a special case of the covariance when the two variables are identical:

$\operatorname{Cov}(X, X) =\operatorname{Var}(X).$

If X, Y, W, and V are real-valued random variables and a, b, c, d are constant ("constant" in this context means non-random), then the following facts are a consequence of the definition of covariance:

$\begin{align} \operatorname{Cov}(X, a) &= 0 \\ \operatorname{Cov}(X, X) &= \operatorname{Var}(X) \\ \operatorname{Cov}(X, Y) &= \operatorname{Cov}(Y, X) \\ \operatorname{Cov}(aX, bY) &= ab\, \operatorname{Cov}(X, Y) \\ \operatorname{Cov}(X+a, Y+b) &= \operatorname{Cov}(X, Y) \\ \operatorname{Cov}(aX+bY, cW+dV) &= ac\,\operatorname{Cov}(X,W)+ad\,\operatorname{Cov}(X,V)+bc\,\operatorname{Cov}(Y,W)+bd\,\operatorname{Cov}(Y,V) \end{align}$

For sequences X₁, ..., X_n and Y₁, ..., Y_m of random variables, we have

$\operatorname{Cov}\left(\sum_{i=1}^n {X_i}, \sum_{j=1}^m{Y_j}\right) = \sum_{i=1}^n{\sum_{j=1}^m{\operatorname{Cov}\left(X_i, Y_j\right)}}.\,$

For a sequence X₁, ..., X_n of random variables, and constants a₁, ..., a_n, we have

$\operatorname{Var}\left(\sum_{i=1}^n a_iX_i \right) = \sum_{i=1}^n a_i^2\operatorname{Var}(X_i) + 2\sum_{i,j\,:\,i<j} a_ia_j\operatorname{Cov}(X_i,X_j).$

A more general identity for covariance matrices

Let $v$ be a random vector, and let $\operatorname{Cov}(v)$ denote its covariance matrix, and let $A$ be a matrix that can act on $v$ . Then $\operatorname{Cov}(Av) = A \operatorname{Cov}(v) A^T$

Uncorrelatedness and independence

If X and Y are independent, then their covariance is zero. This follows because under independence,

$\operatorname{E}\left[X \cdot Y\right] = E[X] \cdot E[Y].$

The converse, however, is not generally true. For example, let X be uniformly distributed in [-1, 1] and let Y = X². Clearly, X and Y are dependent, but

$\begin{align} \operatorname{Cov}(X, Y) &= \operatorname{Cov}(X, X^2) \\ &= \operatorname{E}\!\left[X \cdot X^2\right] - \operatorname{E}[X] \cdot \operatorname{E}\!\left[X^2\right] \\ &= \operatorname{E}\!\left[X^3\right] - \operatorname{E}[X]\operatorname{E}\!\left[X^2\right] \\ &= 0 - 0 \cdot \operatorname{E}\!\left[X^2\right] \\ &= 0. \end{align}$

Relationship to inner products

Many of the properties of covariance can be extracted elegantly by observing that it satisfies similar properties to those of an inner product:

bilinear: for constants a and b and random variables X, Y, and U, Cov(aX + bY, U) = a Cov(X, U) + b Cov(Y, U)
symmetric: Cov(X, Y) = Cov(Y, X)
positive semi-definite: Var(X) = Cov(X, X) ≥ 0, and Cov(X, X) = 0 implies that X is a constant random variable (K).

In fact these properties imply that the covariance defines an inner product over the quotient vector space obtained by taking the subspace of random variables with finite second moment and identifying any two that differ by a constant. (This identification turns the positive semi-definiteness above into positive definiteness.) That quotient vector space is isomorphic to the subspace of random variables with finite second moment and mean zero; on that subspace, the covariance is exactly the L² inner product of real-valued functions on the sample space.

As a result for random variables with finite variance the following inequality holds via the Cauchy–Schwarz inequality:

$|\operatorname{Cov}(X,Y)| \le \sqrt{\operatorname{Var}(X) \operatorname{Var}(Y)}$

Proof: If Var(Y) = 0, then it holds trivially. Otherwise, let random variable

$Z = X - \frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(Y)} Y$

Then we have:

$\begin{align} 0 \le \operatorname{Var}(Z) & = \operatorname{Cov}\left(X - \frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(Y)} Y,X - \frac{\operatorname{Cov}(X,Y)}{\operatorname{Var}(Y)} Y \right) \\[12pt] & = \operatorname{Var}(X) - \frac{ (\operatorname{Cov}(X,Y))^2 }{\operatorname{Var}(Y)} \end{align}$

QED.

Calculating the sample covariance

Main article: Sample mean and sample covariance

The sample covariance of N observations of K variables is the K-by-K matrix $\textstyle \mathbf{Q}=\left[ q_{jk}\right]$ with the entries given by

$q_{jk}=\frac{1}{N-1}\sum_{i=1}^{N}\left( x_{ij}-\bar{x}_j \right) \left( x_{ik}-\bar{x}_k \right)$

The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector $\textstyle \mathbf{X}$ , a row vector whose j^th element (j = 1, ..., K) is one of the random variables. The reason the sample covariance matrix has $\textstyle N-1$ in the denominator rather than $\textstyle N$ is essentially that the population mean $E(X)$ is not known and is replaced by the sample mean $\mathbf{\bar{x}}$ . If the population mean $E(X)$ is known, the analogous unbiased estimate is given by

$q_{jk}=\frac{1}{N}\sum_{i=1}^N \left( x_{ij}-E(X_j)\right) \left( x_{ik}-E(X_k)\right)$

Comments

The covariance is sometimes called a measure of "linear dependence" between the two random variables. That does not mean the same thing as in the context of linear algebra (see linear dependence). When the covariance is normalized, one obtains the correlation matrix. From it, one can obtain the Pearson coefficient, which gives us the goodness of the fit for the best possible linear function describing the relation between the variables. In this sense covariance is a linear gauge of dependence.

References

External links

Look up covariance in Wiktionary, the free dictionary.

Statistics

Descriptive statistics

Continuous data

Location	Mean (Arithmetic, Geometric, Harmonic) Median Mode

Dispersion	Range Standard deviation Coefficient of variation Percentile Interquartile range

Shape	Variance Skewness Kurtosis Moments L-moments

Count data

Index of dispersion

Summary tables

Dependence

Statistical graphics

Data collection

Designing studies	Effect size Standard error Statistical power Sample size determination

Survey methodology	Sampling Stratified sampling Opinion poll Questionnaire

Controlled experiment	Design of experiments Randomized experiment Random assignment Replication Blocking Factorial experiment Optimal design

Uncontrolled studies	Natural experiment Quasi-experiment Observational study

Statistical inference

Statistical theory	Sampling distribution Sufficient statistic Meta-analysis

Bayesian inference	Bayesian probability Prior Posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator

Frequentist inference	Confidence interval Hypothesis testing Likelihood-ratio

Specific tests	Z-test (normal) Student's t-test F-test Chi-squared test Wald test Mann–Whitney U Shapiro–Wilk Signed-rank Kolmogorov–Smirnov test

General estimation	Bias Robustness Efficiency Maximum likelihood Method of moments Minimum distance Density estimation

Correlation and regression analysis

Correlation	Pearson product-moment correlation Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust

Generalized linear model	Exponential families Logistic (Bernoulli) Binomial Poisson

Partition of variance	Analysis of variance (ANOVA) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical, multivariate, time-series, or survival analysis

Categorical data

Multivariate statistics

Time series analysis

General	Decomposition Trend Stationarity Seasonal adjustment

Time domain	ACF PACF XCF ARMA model ARIMA model Vector autoregression

Frequency domain	Spectral density estimation

Survival analysis

Applications

Biostatistics	Bioinformatics Biometrics Clinical trials & studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process & Quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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covariance

co·var·i·ance

covariance

covariance

covariance

Covariance

covariance

covariance

Covariance

Definition

Properties

A more general identity for covariance matrices

Uncorrelatedness and independence

Relationship to inner products

Calculating the sample covariance

Comments

See also

References

External links

covariance

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