stochastic processes – Decomposition of Gaussian spaces with respect to covariance function

Let $K(t,s):T^2 to mathbb{R}$ be a kernel symmetric and type positive (for every $n$ $sum^n_{i,j}u_iu_jK(t_i,t_j) geq 0$ and $(u_1,dots,u_n) in mathbb{R}^n$) where $T$ is any set. Thus, it is possible get a suitable probability space ($Omega, mathcal{F},P$) and a Gaussian process $(X_t)_{t in D}:Omega to mathbb{R}$ with values in $mathcal{H}$ (a closed subspace of $L^2(Omega, mathcal{F},P$) of Gaussian process with mean 0) and $K$ is the covariance function of $X_t$. I know that if I can write $X_t = Z^{1}_t + Z^2_t$ where $Z^{1}_t$ and $ Z^2_t$ are independent centered Gaussian processes then
$$K(t,s) = K_1(t,s) + K_2(t,s),$$
where $K_1$ and $K_2$ are the covariance function of $Z^{1}_t$ and $Z^2_t$ respectively. I’m asking for a converse of it: If I have $K(t,s) = K_1(t,s) + K_2(t,s)$, then I can get $Z^{1}_t$ and $ Z^2_t$ independent centered Gaussian processes. Thanks for any help.

Is this the return type covariance issue PHP faced when type declarations launched, violation of Liskov principle, or flaw in my pattern?

I’ve been using a somewhat odd yet effective pattern for a current use case. The one issue is that I’m getting an undefined method notice on a method that is unique to the subclass. The method of course works fine but the notice leads me to believe that either my architecture is shotty or something else is going on.

I don’t believe this violates Liskov Principle as the subclass satisfies all contracts with identical signatures, put simply, it could replace any other subclass and work reliably.

Is this an instance of the variance issue PHP has stated when releasing type declarations, a violation of Liskov principle, or do I need to clean up my architecture?

Quick code example and brief explanation.

interface Quiz
    public function generate() : void;

class LessonQuiz implements Quiz
    public function generate() : void
        //stuff here

    public function lessonId() : int 
        //stuff here

class QuizClient //Determines/returns instantiated quiz subclass & other permission stuff.
    public static function create(string $case) : Quiz
            case('lesson'): return new LessonQuiz;
            //more logic     

/** In another class that builds a lesson template.
* ... 
$quiz = QuizClient::create('lesson');
$lessonId = $quiz->lessonId(); //undefined method.

I have a client that determines and returns a subclass object of an interface. The return type is set to the interface as I need to have some instance of that interface returned.

All subclasses satisfy the interface contracts with identical signatures. However, the subclass specific method is undefined. I don’t understand why this is incorrect. Class ‘LessonQuiz’ is an instance of ‘QuizFactory’ but also ‘LessonQuiz’.

This does work but the fact that it reports method undefined makes me suspicious about my architecture.

Would really appreciate any help.

Expected value and covariance matrix

E (V) E (V) ^ T $

$ Z = ∑_v ^ {- 1/2} V∑_v ^ {+ 1/2} $

What is $ E (ZZ ^ T)? $

I did
$ E (∑ ^ {- 1/2} V ∑ ^ {+ 1/2} (∑ ^ {- 1/2} V∑ ^ {+ 1/2}) ^ T) $

$ E (∑ ^ {- 1/2} V∑ ^ {+ 1/2} ∑ ^ {+ 1/2} V ^ T∑ ^ {- 1/2}) $
But I can't go on. How can i proceed?

it happens $ E (∑ ^ {- 1/2} V∑ ^ {+ 1} V ^ T∑ ^ {- 1/2}) = ∑ ^ {- 1/2} E (V∑ ^ {+ 1} V ^ T) ∑ ^ {- 1/2} $ is it?

How can I do that? Can you guide me

Inequality for the total law of covariance

Suppose that $ 0 <g (Z) <1 $ and $ 0 <h (Z) <1 $ pretty sure.

Is it possible to form inequalities (larger / smaller) from?

mathbb {E} big ( mathbb {E} (XY | Z) , g (Z) , h (Z) big) – mathbb {E} big ( mathbb {E} (X | ) Z) g (Z) big) , mathbb {E} big ( mathbb {E} (Y | Z) h (Z) big) \
leq ( geq) \
mathbb {E} big ( mathbb {E} (XY | Z) big) – mathbb {E} big ( mathbb {E} (X | Z) big) mathbb {E} , big ( mathbb {E} (Y | Z) big) \
= mathrm {cov} (X, Y)?

in terms of $ mathrm {cov} (X, Y) $ On the right side?

Integration – covariance of a rectified (relu) Gaussian

Given a normal random vector $$ X sim N ( mu, Sigma) $$ for spd $ Sigma $I am interested in the covariance matrix $ K = mathrm {cov} (Y) $ of the variables $$ Y = mathrm {relu} (X) $$ where the relu is carried out element by element $ Y_i = mathrm {Max} (0, x_i) $, so $ Y $ is distributed according to the rectified Gaussian distribution.

Assuming I know everything about $ Sigma $how can I calculate $ K $?

The mean and variance of each $ Y_i $ has been covered in other issues on this page, but the non-diagonal elements of $ K $ The calculation seems pretty difficult, and I haven't found anything about it on SO or anywhere else on the Internet.

I'm actually after that Eigenvectors of $ K $, so if someone can relate the eigenvectors between $ Sigma $ and $ K $ without directly calculating $ K $, that would be even more interesting.

Thanks a lot!

Reference Requirement – Bound to eigenvalues ​​of sample covariance matrices related to $ d, n $, where $ n = $ sample size, $ d = $ dimension of the data

To let $ Z = (z_1, dots z_n) $ be a $ d times n $ Matrix where the $ z_i $They are random vectors with mean values $ mu in mathbb {R} ^ d $ and $ d times d $ (Population) covariance matrix $ Sigma $but the entries $ z_ {ij} is $ are not necessarily iid. Consider the (non-scaled) sample covariance matrix $ C: = ZZ & # 39; in mathbb {R} ^ {d times d} $. I was wondering if we have any results in the top and bottom of $ lambda_1 (C), lambda_d {(C)} $ as a function of $ n $ and $ d $. I do not start from something like the Egyptian regime in random matrix theory, so for example from no assumption like $ n, d to infty, d / n to c $. I'd rather have a non-asymtotic result that shows dependency on $ d, n $. Thank you very much and references would be very helpful!

Statistics – Prove that the characteristic function of the normal matrix is ​​exp (-2 tr (YΣYΣ)), where Σ means covariance matrix and tr is the trace

Given the matrix B = (b (k, l)), where

b (k, l) has the distribution N (0.1), k <l independently

b (k, k) has the distribution N (0.2) and

b (k, l) = b (l, k), k <l

Z = Σ ^ (1/2) BΣ ^ (1/2) and Z is a normal matrix

We have to prove that φz (Y) = exp ((- 2 tr (YΣYΣ))

Covariance matrix in Google Sheets for stocks

I try to use a template to see what it says about my stock portfolio. A covariance matrix with the data from excess returns is used. The template instructions state that an add-on should be used to calculate the matrix, but the add-on does not work. I tried a formula but it didn't work. =MMULT(TRANSPOSE('Excess Returns'!B3:F252),'Excess Returns'!B3:F252)/251

I have 5 shares and 249 data. In a video I looked at how to create a matrix that says the 5×5 matrix should be highlighted, and holding down the Shift key applies the formula to the grid, but it doesn't work.

Doesn't anyone know what I'm doing wrong or how I'm doing better?