Gaussian processes refer to stochastic processes whose realization consists of normally distributed random variables, with the additional property that any finite collection of these random variables have a multivariate normal distribution. The machinery of Gaussian processes can be employed in ...

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What are the various basic kernels available?

I am currently following the book Gaussian Processes for Machine Learning by C.E. Rasmussen and C.K.I. Williams and I have come across various kernels in their Chapter 4 I have also gone through the ...
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Product of two gaussian processes

Given, $\ {y}_{i} = N({\mu}_{i}, {\Sigma }_{i}) $ If we go by the link http://www.tina-vision.net/docs/memos/2003-003.pdf then we can understand that the product of many multivariate gaussians can ...
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Polynomial transform of a Gaussian Random Variable

I have been trying to find how the PDF of a polynomial transform of a Gaussian Random Variable could be found. Eg: If \begin{equation} \ X\sim N(\mu,\sigma^2) \end{equation} Then what would be the ...
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If $E[X(t)X(s)]=t \land s $.Show that this process has independent increments

Let $X(t), t\ge0$ be a real-valued Gaussian process with mean zero and covariance function $E[X(t)X(s)]=t \land s $.Show that this process has independent increments.
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I want Find finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions

Write the finite dimensional densities for an $\mathbb R^d-$ valued Gaussian process $X$with specified mean and covariance functions.
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51 views

What is the probability of observing some function given a gaussian process? [on hold]

I would like to compare a parametric function to a Gaussian Process. This may sound weird, but read on: Data description. I am looking at projections of a 3D object. However I expect a certain amount ...
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Detecting mean difference between two observed stochastic processes

Suppose to collect real time two set of samples coming from two sources so defined: S1: gaussian(m1, s1) S2: gaussian(m1+m2, s1+s2) with probability p S2: gaussian(m1, s1) with probability ...
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23 views

Gaussian Process Prediction Uncertainty

I am using Gaussian Process Regression to interpolate my input points. I would like to measure the total uncertainty of my prediction thus I sum up the GPR prediction variances at all the testing ...
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7 views

Gaussian Process Regression with positive prediction weights

I want to do Gaussian Process Regression for a density function $f(x)$ with a gaussian kernel function $k(x, x')$. Given the training data $\mathbf{x} = (x_1, x_2, \dots, x_N)$ and $\mathbf{f} = ...
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Why does the density function has product of variance and covariance for higher model order time series

In my previous question Density function for AR model, the density function of AR model has the covariance-variance matrix given as $\sigma^2 *V_p$. In multivariate Gaussian distribution, the pdf ...
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38 views

Gradients of marginal likelihood of Gaussian Process with squared exponential covariance, for learning hyper-parameters

The derivation of gradient of the marginal likelihood is given in this pdf, equation 5.9. But the gradient for the most commonly used covariance function, squared exponential covariance, is not ...
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Gaussian based clustering/partitioning, does it make sense with not much data?

I have a dataset about hourly aggregated mobile phone usage (#calls, #sms, #internetConnections) in one mobile cell. For example I have this data about activity at 8:00am: ...
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Is it posible to find a derivative of the mean function of Gaussian process regression?

The mean function $\hat{\mu}(x_*)$ of GPR is $k(x_*, X)(k(X, X) + \sigma^2_w I)^{-1}Y$ where $k(\cdot, \cdot)$ is a kernel matrix or vector of appropriate size and is parametrized by some ...
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27 views

How to measure goodness of fit in a simple quadratic Gaussian GLM?

I hope this question will be specific enough, I went through many of the other questions about GLM but now I am even more confused because my sample size is small and it seems that R square (or pseudo ...
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49 views

Gaussian mixture vs. Gaussian process

As far as I know, both Gaussian mixtures as well as Gaussian processes can be used for regression. My question is: what is better and why? The answers might contain theoretic insights, practical ...
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meanZero error when trying to use GPML gp function to perform regression to predict y

I've trying to learn how to use Gaussian Process for Machine learning (GPML)to do some prediction stuff. My goal is to get the returned output ...
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42 views

overfit gaussian regression

I have recently read a lot of posts about how maximizing the marginal likelihood in Gaussian Regression can cause overfitting. What is the best way out then? If we let the hyperparameters take values ...
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14 views

Gaussian Process Regression Variance

While doing regression using Gaussian Processes, isn't the variance of the posterior supposed to be low where training data has already been observed?
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27 views

Stationary function

I am reading Karl Rasmussen's book on Gaussian processes and in the introductory chapter he has the following statement: ...
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61 views

Estimating correlation hyperparameters of a Gaussian Process

I have an actual function that I need to simulate using a GP model. I've not done this before so I'm unclear of the steps. I have used the true function at different values of the inputs ($\vec X1, ...
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83 views

Gaussian Process Regression/ Classification

How do we estimate parameters of the model while performing Gaussian Process Regression or Classification? While performing regression, we estimate parameters such that the model is the best fit to ...
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19 views

Digit recognition using GPML package

Is there any code in GPML package (http://www.gaussianprocess.org/) using which I can execute 10-class digit recognition problem? Because I notices that this is for binary classification. For n-class ...
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46 views

R Bayesian prediction of a Gaussian process

I have a Gaussian model with mean zero, variance is arbitrary constant, and correlation function $e^{-\theta(x-x')^2}$ where $\theta$ is again an arbitrary constant. I've plotted some realizations of ...
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periodic covariance function

"Gaussian processes can be completely defined by their second-order statistics.Thus, if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the ...
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Determine a distribution of a gaussian stochastic with different time

I would like to determine the autocorrelation function of a Gaussian stochastic. Let see my problem So my solution is The distribution of $y=x(t_1)-x(t_2)$ is also a Gaussian stochastic with ...
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31 views

Gaussian Processes: 2-output [latitude longitude] as regression output

I'm trying to model a problem where inputs are 100-d vectors and outputs are 2-d vector [latitude, longitude]. I need to perform prediction on new unseen 100-d vectors and find out the latitude and ...
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39 views

Stochastic linearization by irregular waves of ship roll motion equation

I'm interested in finding some way for doing "stochastic linearisation by irregular waves of ship roll motion equation". I found some publications about it but its hard for me to understand, how to ...
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How many models I need?

I am doing an estimation using a bunch of sparse data. Suppose that I have a 100x100 grid and 20 data is available on this 2D grid. One solution is to use a determiastic method and estimate the other ...
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80 views

Jaynes' Derivation of Herschel-Maxwell for Normal Distribution

I am reading the following paper: http://www-biba.inrialpes.fr/Jaynes/cc07s.pdf and cannot seem to figure out how Jaynes is deriving (P2) and below (specifically the log arithmetic ...
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8 views

Can a kernel function for GP-regression use measurement information?

when building a kernel function for a Gaussian-Process-Regression I am asking myself whether the kernel function is allowed to contain information from the measurements. To ask a little more general, ...
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51 views

How to apply Gaussian kernel to smooth density of points on 2D (theoretically)

I have a set of discrete points on a 2D surface and need to build a heat map or a distribution of the density of the points. However, I also need to smooth out the density/distribution by applying ...
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66 views

Why we check the residuals of ARIMA model for white Gaussian?

I have problem about the assumptions and model verification of ARIMA models. I know that Gaussian distributed assumption is not necessary for fitting ARIMA models but I wonder why a lot of people ...
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33 views

issue in using GP_fit() of GPfit package in R

I am trying to work with Gaussian process through GPfit package of R but every time I use GP_fit() of this package, either it ...
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44 views

Gaussian Process Regression with additional Basis Functions

I'm working through Rasmussen's Gaussian Processes book, and I have a question about the possibility of optimizing additional basis function hyperparameters (in section 2.7 ...
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31 views

Low pass filter to maintain edge information

I am looking for a kernel as low pass filter that satisfy as:I must find a kernel that statisfies as follows: In the my reference paper, the author suggest gaussian kernel that is The gaussian ...
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73 views

How to find $E[x|y]$ when distributions of y and x are separately known,(p.s. they are both Gaussian)?

In detail, I have these relations (in order of causality): $u_1 = ax_0$ $x_1 = u_1 + x_0$ $y = x_1 + w$ where $w = N(0,1), x_0 = N(0,\sigma^2)$. This was my approach: I know the distribution of ...
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268 views

Which is kernel similar gaussian kernel?

I must find a kernel that statisfies as follows: In the my reference paper, the author suggest gaussian kernel that is The purpose of that kernel is that it will take a weight for each points ...
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1answer
90 views

How to understand the log marginal likelihood of a Gaussian Process?

I'm trying to understand Gaussian Processes. Could anyone tell me: Why we need to use the log marginal likelihood? Why using log, the marginal likelihood can be decomposed to 3 terms (including a ...
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27 views

Combining independent Gaussian probabilities

I am using three Gaussian distributions with which I generate random numbers to represent many candidate xyz points. I use some selection criteria (details not particularly relevant) to decide on ...
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20 views

Is my interpretation of a Hierarchical Gaussian Process model correct?

I'm learning about GPs in hopes that it'll be the tool I need to deal with the common scenario I encounter where I have a continuous predictor variable whose effect might not be linear. I think I ...
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How can one understand the projection of a general Gaussian onto the sphere?

Let $N(0,\Sigma)$ be a Gaussian distribution with mean $0$ and covariance $\Sigma$, a $p\times p$ matrix. Is there an understanding the distribution $\mathcal{P}_{\mathbb{S}^{p-1}} (N(0,\Sigma))$? Is ...
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83 views

How to implement a 2-D Gaussian Processes Regression through GPML (MATLAB)?

I just touched Gaussian processes two weeks ago. I am not very familiar with the selection of a model and its hyperparameters. Here is the demo code that I run for a 2-D Gaussian processes regression. ...
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Confusion related to derivation of a formula in gaussian process

I was reading this paper related to online Gaussian processes. I didn't get how equation set 2 was derived. Any suggestions?
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25 views

Confusion related to posterior gaussian process

I was reading this [paper][1] related to sparse online gaussian processes. However, I didn't get how the denominator in the equation 1 was derived? It was supposed ...
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246 views

Why is this likelihood function equal to the noise PDF?

My professor has this slide up here: Here, $y$ is an observed signal. $H$ is a deterministic transformation, which is assumed known. $f$ is the original signal (which we dont know), and $w$ is ...
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22 views

Deviation from Gaussian smoothed curve and relation of its derivatives to the results [closed]

I was plotting a deviation of my data points from a Gaussian smoothed curve depending on different properties of this Gaussian curve that smooths the series. So I had a series of: ...
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1answer
44 views

Differential Entropy of Gaussian Process

I have $N$ datapoints that have $d$ features in a GP and their covariance matrix $K$ and I want to calculate the differential entropy of that GP. Is this formula right? $E(I)= \frac{1}{2} ...
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74 views

Random forest ML algorithm suitable for use on cluster based HPC?

I have developed a script using pythons scipy package to analyse a rather large model that I wish to solve, the model contains over 12gb of data, including over 500 parameters. Now running small ...
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Likelihood vs. noise kernel hyperparameter in GPML Toolbox

I'm using GPML toolbox by C.E.Rasmussen to solve the basic GP regression problem (presented in the book) with noisy observations. That is to say, estimate the underlying function $f$ of a static noisy ...
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51 views

Gaussian sampling in high dimension

I have a covariance function $f(x)$, where $x = (x_1, x_2, x_3)$ is a point in three-dimensional space. I need to generate a Gaussian field with given covariance function on a 3D grid of points, that ...