Covariance Definition and 171 Threads

In probability theory and statistics, covariance is a measure of the joint variability of two random 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 (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (that is, 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 easy to interpret because it is not normalized and hence depends on the magnitudes of the variables. 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 in addition to serving as a descriptor of the sample, also serves as an estimated value of the population parameter.

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  1. Coffee_

    Contravariant & Covariant Vectors: Electric Field Case

    From what I understood, a vector can be either covariant or contravariant. Which one this is will depend on how the coordinates of this vector transform under a coordinate transformation. Let's take a look at the electric field then: ##\vec{E}=-\nabla{V}##, so here it looks as if the electric...
  2. fezster

    Connection between Lorentz covariance and special relativity

    s2 = t2 - x2 - y2 - z2 This equation is covariant (Lorentz covariance). The interval "s" is invariant (Lorentz invariance). Can you derive everything in special relativity from these facts? Or am I mistaken about that?
  3. T

    Covariance betw scalar amplitude & spectral index in Planck?

    I am reading some of "Planck 2013 results. XXII. Constraints on inflation." The paper is full of values for various inflationary parameters under various models, with their confidence intervals. For instance, in Table 5 on page 13, the authors report that — for a model including both running of...
  4. SpiderET

    Lorenz covariance vs proofs of relativity theory

    I have been studying history of relativity theory and now it seems to me, that it is wrong to automatically assume that proofs of Lorentz covariance are proofs of Special relativity theory. It seems to me, that there is broader group of theories, that are compatible with Lorentz covariance but...
  5. ElijahRockers

    Calculating Covariance with a Random Vector

    Homework Statement Let ##X## be a random variable such that ##\mu_X = 0## and ##K_{XX} = I##. Find ##Cov(a^T X, b^T X)## for ##a = (1, 1, 0, 0)## and ##b = (0, 1, 1, 0)##. The Attempt at a Solution I guess I am assuming that ##X## is a 4 element random vector. I can't know values of the random...
  6. D

    Parallel Transport & Covariant Derivative: Overview

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  7. I

    MHB Find the expectation and covariance of a stochastic process

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  8. W

    What Are the Constraints of a Valid Covariance Matrix?

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  9. jk22

    Variation on Quantum cross covariance and CHSH

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  10. T

    MHB Covariance and Correlation matrix

    I would love to learn more about those two matrices. What do they tell us,how to calculate them? Maybe in R Studio? I was searching for some good explanations on google,but i didnt find them. And another question,i apologize if is not in right forum... How do i know how much dispersion can i...
  11. T

    Proving the Property of Covariance Function: (r(n)-r(m))^2≤2r(0)(r(0-r(n-m)))

    Hi all. My task is to prove the property of covariance function: ##(r(n)-r(m))^2≤2r(0)(r(0-r(n-m)))## My solution: ##1) (r(n)-r(m))^2=r(n)^2-2r(n)r(m)+r(m)^2## ##2) 2r(0)(r(0)-r(n-m)))=2r(0)^2-2r(0)r(n-m)## From covariance function properties I know that ##2r(0)^2≥r(n)^2+r(m)^2## So now I...
  12. A

    MHB Proof that covariance matrix is positive semidefinite

    Hello, i am having a hard time understanding the proof that a covariance matrix is "positive semidefinite" ... i found a numbe of different proofs on the web, but they are all far too complicated / and/ or not enogh detailed for me. Such as in the last anser of the link : probability -...
  13. J

    Deriving Covariance between S and N: E(SN) - ìXìN

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  14. jk22

    Sum of the probabilities equals 3 in bipartite covariance ?

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  15. C

    Random vector mean and covariance

    Homework Statement Random vector Y = [Y_1 Y_2 Y_3 …. Y_m]' where ' = transpose mean = u and and ∑ = covariance Z = N_1 * Y_1 + N_2 * Y_2 + …. + N_m*Y_m all N are numbers Find the covariance of Z E[ (Y- E[Y] )(Y - E[Y] ) ] = E[YY'] -E[Y]E[Y]'= [N_1 N_2 .. N_m] [∑ - u^2 ….∑ -u^2] ' This...
  16. P

    Covariance matrix does not always exist?

    Hey guys. I am going through the PRM (risk manager) material and there is a sample question that is bugging me. The PRM forum is relatively dead, and they don't usually go that deep into the theory anyway. So wanted to ask you guys. Shouldn't a random vector always have a covariance matrix? Why...
  17. T

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  18. K

    Covariance involving Expectation

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  19. K

    What can we say about Covariance?

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  20. E

    Covariance between functions of 3 random variables

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  21. G

    Calculating covariance from variances

    Homework Statement Suppose that X1, X2, and X3 are independent random variables with variances 3, 4, and 8, respectively. Let Y1 = 2X1 + 3X2, Y2 = X3 – X2, and Y3 = X1 + X2 + X3. (a) Using the general relationship Cov(W+X, Y+Z) = Cov(W,Y) + Cov(W, Z) + Cov(X, Y) + Cov(X, Z), find...
  22. D

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  23. D

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  24. E

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  25. dexterdev

    MATLAB How to find covariance matrix of 3 or more vectors in matlab?

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  26. dexterdev

    Physical significance of eigen vectors of Covariance matrix

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  27. S

    General covariance vs. locality

    hi i read an article by S.carlip about quantum gravity, arXiv:gr-qc/0108040v1 , in this article carlip stated: why we need quantum gravity what's problems of quantum gravity and two ways of quantization of GR. I couldn't realize some clues in section of "the problems of quantum gravity"...
  28. C

    Covariance Matrix of a Vector Random Variable w/ Components Related

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  29. marcus

    Quant. stat. mech. based on general covariance (BHR's paper)

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  30. P

    Solving Covariance: X and Y Binomial Variance

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  31. P

    Does Covariance Remain Unchanged Under Variable Transformations?

    Let X1 and Y1 be two random variables. We have Cov(X1,Y1) = 0. Does this extend to any transformation X2 = g(X1) and Y2 = g(Y1), such that Cov(X2,Y2)? Here, g is a continuous function. For example, if we set X2 = X1^2 and Y2 = Y1^2. Do we then from Cov(X1,Y1) = 0 that Cov(X1^2,Y1^2) = 0?
  32. jegues

    Covariance of two dependent variables

    Homework Statement See figure attached Homework Equations The Attempt at a Solution I am not concerned with part (a), I have deduced that indeed X and Y are dependent. I'm not sure if I have done part (b) correctly, and I am quite certain I have done part (c) incorrectly, but...
  33. B

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  34. G

    How Do Eigenvectors of Block Covariance Matrices Interrelate?

    Hello everybody, I’d like to present this math problem that I’ve trying to solve… This matter is important because the covariance matrix is widely use and this leads to new interpretations of the cross covariance matrices. Considering the following covariance block matrix ...
  35. O

    MHB Significance of the Eigenvalues of a covariance matrix

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  36. jk22

    Uncovering Uncertainty: The Experimental Covariance Curve for Entangled Photons

    Some experimental covariance curve for entangled photons gives abs(Cov(0)) less than 1. For example : Violation of Bell inequalities by photons more than 10km apart by Gisin's group in Geneva. Does this mean that experimentally we can't predict with certainty in this case ? In order to...
  37. T

    Are Spinors Truly Invariant Under Coordinate Transformations Beyond Lorentz?

    Is the Dirac Equation generally covariant and if not, what is the accepted version that is? For general coordinate changes beyond just Lorentz, how do spinous transform?
  38. V

    What is the significance of manifest Lorentz covariance for field theories?

    My question is: what does "manifest" Lorentz covariance means for a field theory, as opposed to simply Lorentz invariance. Thanks for the replies!
  39. W

    MATLAB Estimating the variance of eigenvalues of sample covariance matrices in Matlab

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  40. M

    Covariance of Maxwell Eqn. - conceptual question

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  41. zonde

    Understanding General Covariance & Relativity Principle

    I am trying to understand what exactly general covariance states. As I understand general covariance appeared as generalization of relativity principle so I will try to state relativity principle in a manner that I consider more convenient for my purpose. So let's say we have inertial...
  42. T

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    Suppose we have a mxn matrix, where each row is an observation and each column is a variable. The (i,j)-element of its covariance matrix is \mathrm{E}\begin{bmatrix}(\vec{X_i} - \vec{\mu_i})^t*(\vec{X_j} - \vec{\mu_j})\end{bmatrix}, where \vec{X_i} is the column vector corresponding to a...
  43. S

    Covariance - Bernoulli Distribution

    1. Consider the random variables X,Y where X~B(1,p) and f(y|x=0) = 1/2 0<y<2 f(y|x=1) = 1 0<y<1 Find cov(x,y) Homework Equations Cov(x,y) = E(XY) - E(X)E(Y) = E[(x-E(x))(y-E(y))] E(XY)=E[XE(Y|X)] The Attempt at a Solution E(X) = p (known since it's Bernoulli, can also...
  44. iVenky

    How does covariance and correlation coefficient actually work?

    We all know that if the covariance is positive then it means that if one increases then the other one also increases. If the covariance is negative it is the other way round. I know to calculate the covariance and deduce the relation between them. But I don't get an intuitive feeling regarding...
  45. ShayanJ

    Lorentz covariance and Noether's theorem

    Not sure its in the right place or not.If its not,sorry. The relativity postulate of special relativity says that all physical equations should remain invariant under lorentz transformations And that includes Lagrangian too. So it seems we have a symmetry(which is continuous),So by Noether's...
  46. O

    Mahalanobis Distance using Eigen-Values of the Covariance Matrix

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  47. V

    MATLAB Generating covariance matrices as defined in MATLAB

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  48. O

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    Hello Everyone! What $b$ minimizes $E[(X-b)^2]$ where $b$ is some constant, isn't it $b=E[X]$? Is it right to go about the proof as follows: $E[(X-b)^2] = E[(X^2+b^2-2bX)] = E[X^2] + E[b^2]-2bE[X]$, but $E[b] = b$, we differentiate with respect to $b$ and set to zero, we obtain that $b=E[X]$...
  49. F

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    The following question appeared on a practice exam: For f(x,y) = 24xy if 0<x+y<1 , 0<x,y 0 elsewhere find Cov(X,Y) I used Cov(X,Y) = E(XY) - E(X)E(Y) to calculate covariance, with E(XY) = \int^{1}_{0}\int^{1-y}_{0}24x^{2}y^{2}dxdy but for some reason I didn't get the...
  50. M

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    I know what gauge invariance is, but I'm not sure what gauge covariance is. Is it that a given field has a gauge covariant derivative? And under which circumstances do we get a field that is gauge invariant but not gauge covariant? And I would appreciate an example (other than the one...
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