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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Sub-space / latent-space covariance

I am not entirely sure what I should be googling for this in my present context. Basically I am working with a set of latent variable models such as the Gaussian Processes latent variable model ...
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Covariance function to draw an inverse function

In a Gaussian Process (GP), we know that choice of the covariance function determines the shape of function that can be drawn from the GP. eg. Constant : $\sigma _{o}^{2}$ Draws constant function ...
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How to prove that Radial Basis Function can be derived by mapping function?

How to prove the radial basis function $k(u,v) = \int_{\mathbb{R}^d} \phi_t(u)\phi_t(v)dt $ can be integrated out by mapping function? $$\phi_{t}(u) = \frac{1}{(2\pi\Sigma)^{d/2}} ...
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20 views

How to fit a Gaussian Mixture Model to data with correlated errors?

I'm restating this question in the hope of getting more interest. The usual function for scoring a Gaussian mixture model assumes independent measurements. But what if we have correlated measurements? ...
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27 views

How to estimate the noise variance of covSEiso covariance function for GPML code?

For the covariance function covSEiso: $$k(x^i,x^j)=\sigma_f^2\exp(-(x^i-x^j)^T{\rm diag}(l)^{-2}(x^i-x^j))+\sigma_n^2\delta_{ij}$$ in ...
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12 views

A binomial bandit v. a Gaussian process optimizer

I am reading about multi-armed bandits for web optimization, and I have come across a couple of options, two of which are the binomial bandit and the Gaussian process (an implementation here). Am I ...
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1answer
37 views

Determining whether time series is Gaussian from autocorrelations

Can one say anything about the "Gaussiness" of a time series merely by looking at its autocorrelations? I find this hard to reconcile. Let us say I find that there are significant autocorrelations in ...
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1answer
17 views

How to optimize Gaussian-process parameters for multiple tasks with GPML?

I have a lot of test curves and I want to optimize the length and scale parameters simultaneously for all curves. Is this possible?
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15 views

Does the curse of dimensionality apply to discriminative models?

I want to understand the curse of dimensionality better and where does/doesn't apply. I know about exponential volume growth and also the distances becoming not distinguishable from each other in ...
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62 views

How to reduce dimension of the sampling procedure?

I am stuck with this problem for a long time, hopefully I can get help here! Basically, I want to sample from a posterior distribution that looks like, \begin{align*} X &\sim ...
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1answer
15 views

How to implement Gaussian process using GPML toolbox with known output noise?

I want to implement a simple regression model using Gaussian process. I chose GPML toolbox of Rasmussen for simplicity. My question is how we can let the toolbox know that we already have a different ...
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27 views

Gaussian or Dirichlet Process from aggregated samples

For N individuals, I have the number of measurements taken per individual, the individual's mean measurement value, and the standard deviation of the individual's measurements, but I do not have ...
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41 views

Gaussian Process: Using partitions of a Cholesky decomposition solution for conditional deduction

If I define a GP over observed values, $y$, of a sensor reading over time, $t$, as (for simplicity assuming discrete time series e.g lets say readings after every 5 mins) : $y=f(t)+\epsilon$ where ...
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14 views

Find the bounding rectangle of a covariance matrix based on mahalanobis distance

I'm trying to develop an algorithm that makes use of the Mahalanobis distance from an arbitrary test point to assign a score to each observation in a dataset. I want to only consider observations ...
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13 views

Is derivative of a Gaussian Signal also Gaussian? How to find variance of signal that is obtained from differentiation of a Gaussian signal?

Could someone please let me know or give appropriate references for the question I have posed above. My main interest lies in applying Kalman filter for state estimation. The noise on sensor ...
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1answer
40 views

Posterior covariance from GPML toolbox

I am currently using the GPML toolbox to perform regression. Generally, after learning the hyperparameters we can extract the posterior mean and variance by using the function in the toolbox as ...
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1answer
35 views

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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2answers
57 views

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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55 views

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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19 views

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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23 views

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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1answer
105 views

What is the probability of observing some function given a gaussian process?

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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16 views

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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1answer
48 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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2answers
32 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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94 views

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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68 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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19 views

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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1answer
15 views

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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46 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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1answer
97 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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30 views

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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2answers
49 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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1answer
20 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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1answer
33 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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1answer
80 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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1answer
94 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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21 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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1answer
50 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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21 views

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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20 views

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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39 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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54 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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30 views

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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1answer
106 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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65 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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1answer
71 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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1answer
68 views

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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1answer
51 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 ...