Covariance Definition and 173 Threads

Hello! I am reading GR and got confused about the principle of General Covariance. The principle of General Covariance says that the laws of physics take the same form in all reference frames. Since the laws are same in all reference frames, any experiment performed should give identical...

Homework Statement [/B] So, I need to show Lorentz covariance of a Proca field E-L equation, conceptually I have no problems with this, I just have to make one final step that I cannot really justify. Homework Equations "Proca" (quotation marks because of the minus next to the mass part, I...

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...

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?

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...

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...

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...

I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators \partial_{a} are dependent on the coordinate system one chooses and thus not naturally associated with the...

The problem is:Let $W(t)$, $t ≥ 0$, be a standard Wiener process. Define a new stochastic process $Z(t)$ as $Z(t)=e^{W(t)-(1/2)\cdot t}$, $t≥ 0$. Show that $\mathbb{E}[Z(t)] = 1$ and use this result to compute the covariance function of $Z(t)$. I wonder how to compute and start with the...

I'm trying to understand what makes a valid covariance matrix valid. Wikipedia tells me all covariance matrices are positive semidefinite (and, in fact, they're positive definite unless one signal is an exact linear combination of others). I don't have a very good idea of what this means in...

I tried another approach to the problem of covariance like in Bell's theorem :from the definition ##Cov(A,B)=\langle\Psi|A\otimes B|\Psi\rangle-\langle\Psi|A\otimes 1|\Psi\rangle\langle\Psi|1\otimes B|\Psi\rangle## (##A=diag(1,-1)=B##) we can see that this 'average' is in fact a quadratic form...

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...

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...

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 -...

consider the following model for aggregate claim amounts S: S=X1+X2+...+XN where the Xi are independent, identically distributed random variables representing individual claim amounts and N is a random variable,independent of the Xi and representing the number of claims.let X has ìx and...

If we consider a bipartite system as in EPRB experiment we get the probabilities : p(++)=p(--)=1/4*(1-cos(theta)) p(+-)=p(-+)=1/4*(1+cos(theta)) p(+A)=p(+B)=p(-A)=p(-B)=1/2 Thus the sum of all the probabilities equals 3... How does that come ? Is it because in fact there are only...

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...

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...

Consider a co-variance matrix A such that each element ai,j = E(Xi Xj) - E(Xi) E(Xj) where Xi,Xj are random variables. Consider the case that each variable X has a different sample size. Let's say that Xi contains the elements xi,1, …, xi,N, and Xj contains the elements xj,1, ..., xj,n where...

Homework Statement Suppose ##X,Y## are random variables and ##\varepsilon = Y - E(Y|X)##. Show that ##Cov(\varepsilon , E(Y|X)) = 0##. Homework Equations ##E(\varepsilon) = E(\varepsilon | X) = 0## ##E(Y^2) < \infty## The Attempt at a Solution ##Cov(\varepsilon , E(Y|X)) =...

I'm working on a problem that wants me to show that $$Cov(X,Y) = 0$$ and I am up to the point where I simplified it down to $$Cov(X,Y) = E(XY)$$. In other words, $$E(X)E(Y) = 0$$ to make the above true. My question is, what can we conclude if we have that the covariance of two random variables...

Find cov(Y,Z) where Y = 2X_1 - 3X_2 + 4X_3 and Z = X_1 + 2X_2 - X_3 Information given E(X_1) =4 E(X_2) = 9 E(X_3) = 5 E(Y) = -7 E(Z) = 26 I tried expanding cov(Y,Z) = E(YZ) - E(Y)E(Z) but can't figure out how to calculate E(YZ)

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...

http://en.wikipedia.org/wiki/Covariance I read from the above link that- The sample covariance of N observations of K variables is the K-by-K matrix. I am wondering- Should "K-by-K" be "N-by-K" or not?

if X= (3, 5, 7) & Y = (2, 4, 1) What is the 3x3 covariance matrix for X & Y?

Let xij be the ith independently drawn observation (i=1,...,N) on the jth random variable (j=1,...,K). These observations can be arranged into N column vectors, each with K entries, with the K ×1 column vector giving the ith observations of all variables being denoted xi (i=1,...,N). I have...

Hi all, I know how to find covariance of 2 vectors and variance too. If covariance matrix is to be found of 3 vectors x,y and z, then then the cov matrix is given by cov_matrix(x,y,z) =[var(x) cov(x,y) cov(x,z); cov(x,y) var(y) cov(y,z); cov(x,z) cov(y,z) var(z) ]; Is this...

Hi all, I have a doubt regarding the physical significance of eigen vectors of the covariance matrix. I came to know that eigen vectors of covariance matrix are the principal components for dimensionality reduction etc, but how to prove it?

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"...

Homework Statement Given X=ZU+Y where (i) U,X,Y, and Z are random variables (ii) U~N(0,1) (iii) U is independent of Z and Y (iv) f(z) = \frac{3}{4} z2 if 1 \leq z \leq 2 , f(z)=0 otherwise (v) fY|Z=z(y) = ze-zy (i.e. Y depends conditionally on...

Bianchi Haggard Rovelli just posted a landmark paper showing that QSM rises automatically from the GR requirement of general covariance. Yesterday in another thread Atyy identified their paper as especially interesting. I agree. http://arxiv.org/abs/1306.5206 The boundary is mixed Eugenio...

Hi, Homework Statement A fair die is rolled n times. X denotes the number of times '1' is obtained. Y denotes the number of times '6' is obtained. I am first asked to state how X and Y are distributed (marginally) and to find their variance.Homework Equations The Attempt at a Solution Aren't X...

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?

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...

So I'm trying to take the expectation of the covariance estimate. I'm stuck at this point. I know I have to separate the instances where i=j for the terms of the form E[XiYj], but I'm not quite sure how to in this instance. The answer at the end should be biased, and I'm trying to...

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 ...

Hello everyone! I'm curious to know what is the significance of the Eigenvalues of a covariance matrix. I'm not interested to find an answer in terms of PCA (as you of you may be familiar with the term). I'm thinking of a Gaussian vector, whose variance represent some notion of power or...

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...

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?

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

I am trying to investigate the statistical variance of the eigenvalues of sample covariance matrices using Matlab. To clarify, each sample covariance matrix, \hat{\mathbb{R}}_{nn}, is constructed from a finite number, N, of vector snapshots, each sized (L_{vec} \times 1) (afflicted with random...

Hey all, This has been bugging me for quite a while now. My question is essentially about how one shows that Maxwell equations are invariant under Lorentz transforms. Writing them in index notation, it is usually appealed to that all terms involved are Lorentz tensors (or contractions thereof)...

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...

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...

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...

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...

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...

Given the formula of Mahalanobis Distance: D^2_M = (\mathbf{x} - \mathbf{\mu})^T \mathbf{S}^{-1} (\mathbf{x} - \mathbf{\mu}) If I simplify the above expression using Eigen-value decomposition (EVD) of the Covariance Matrix: S = \mathbf{P} \Lambda \mathbf{P}^T Then, D^2_M =...

Hi, I'm fairly new to MATLAB and I was wondering if you guys could help me out. If I have an N*N matrix, C where the (k,l)-entry is defined as: http://a3.sphotos.ak.fbcdn.net/hphotos-ak-ash3/556394_10151031836051952_2120388553_n.jpg Where x_i is from an N-vector where x_i is normally...

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 $b=E[X]$...