What is Gaussian process: Definition and 13 Discussions

In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.
The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions.
Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quantities include the average value of the process over a range of times and the error in estimating the average using sample values at a small set of times. While exact models often scale poorly as the amount of data increases, multiple approximation methods have been developed which often retain good accuracy while drastically reducing computation time.

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  1. B

    How are hyperparameters determined in Bayesian optimization?

    Hello, I am better studying the theory that is the basis of Bayesian optimization with a Gaussian Process and the acquisition function EI. I would like to expose what I think I understand and ask you to correct me if I'm wrong. The aim is to find the best ##\theta## parameters for a parametric...
  2. L

    MATLAB Gaussian process and climate model in Matlab

    I'll admit I am very new to Gaussian processes, but from what I know a Gaussian process is completely determined by a mean vector E(Y(θ)) and a covariance function Cov[Y(θ1), Y(θ2)]. E(Y(θ)) is given, and we have the correlation, which is just the covariance divided by Var(θ1)*Var(θ2).The...
  3. M

    A Probability of Surface Realization Above Black Line in Gaussian Process Puzzle

    I am looking at a surface where the height is described by a zero-mean gaussian process with cov(h_1,h_2)=\sigma^2\exp[-0.5\frac{(x_1-x_2)}{a^2}]. Given that h(x=0) = 0, what is the probability that a surface realization will go above the black line going from x=0 to infinity in the figure...
  4. G

    A Expanding CCRs, and their underlying meaning

    Hi, I remember seeing a few months ago, at a lecture about statistical signal processing, something which looked similar to commutation relations, only with a gaussian, instead of a delta function. Basically, it looked like this: $$\left[\phi(x),\phi(y)\right] = ie^{-\alpha(x-y)^2}$$ This...
  5. H

    About stochastic process....Help please

    Given a Gaussian process X(t), identify which of the following , if any, are gaussian processes. (a)X(2t) solution said that X(2t) is not gaussian process, since and similarly Given Poisson process X(t) (a) X(2t) soultion said that X(2t) is not poisson process, since same reason above...
  6. H

    If X(t) is gaussian process, How about X(2t)?

    written as title, 1. If X(t) is gaussian process, then Can I say that X(2t) is gaussian process? of course, 2*X(t) is gaussian process 2. If X(t) is poisson process, then X(2t) is also poisson process?
  7. R

    Comparing curves using gaussian process regression

    Hi guys, I have run multiple simulations on networks that are all slightly perturbed from each other. They produce slightly different curve outputs onto an x-y graph which I need to now analyse (it has been about 5 years since I did statistical analysis hence why I am here). A couple of the...
  8. M

    Simulating Gaussian Process in R w/ Mean & Variance

    Hi everybody Any ideas how to simulate a Gaussian stationary process in R language using predefined variance and mean? I have a uni-variate normal distribution for my real life process Thank you
  9. A

    Weakly stationary Gaussian process?

    X = (Y(t))^2 where Y(t) is zero mean Gaussian process and correlation function R_YY = exp(-λ|τ|) i want to check if X is weakly stationary.So i guess for the first part, i checked if mean is constant σ^2=R_YY = exp(-λ|0|) = 1 E(X^2) = μ^2+ σ^2 = 1 since μ is zero and σ = 1 I wanted to check if...
  10. G

    Gaussian process with linear correlation

    Hi I have a stationary gaussian process { X_t } and I have the correlation function p(t) = Corr(X_s, X_(s+t)) = 1 - t/m, where X_t is in {1,2,..,m}, and I want to simulate this process. The main idea is to write X_t as sum from 1 to N of V_i, X_t+1 as sum from p+1 to N+p V_i and so on, where...
  11. P

    Covariance of Gaussian Process

    This is probably a stupid question, but here goes: based on the covariance function of some (centered, stationary) Gaussian process - how can one determine non-degeneracy (here I mean for any choice of a finite number of sampling times, the resulting RV is AC). Ideas?
  12. S

    Gaussian process & Brownian motion

    Homework Statement Let \{ B(t) \}_{t \geq 0} be a standard Brownian motion and U \sim U[0,1] and {Y(t)}_t\geq0 be defined by Y(t) = B(t) + I_{t=U}. Verify that Y(t) is a Gaussian process and state its mean and covariance functions. Is Y(t) a standard Brownian motion?Homework Equations The...
  13. C

    Proving Linear Filtering of Gaussian Process Still Gaussian

    How to prove the output of Linear Filtering a Gaussian Process is still Gaussian? It has been stated in many books I read, but none of them actually prove it. One even stated that "The technical mechinery to prove this property is beyond the scope of this book..." By definition, a Gaussian...