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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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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6 views

Problem with calculating the avarage power of a vector? [migrated]

I am calculating the average power of a vector. I would like to compare it the final expression with the simulation. However, they are not equal. Please help me to point out which steps are wrong. ...
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19 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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10 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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5 views

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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23 views

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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17 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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80 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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16 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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32 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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1answer
36 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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2answers
39 views

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 ...
2
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2answers
43 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 ...
2
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1answer
27 views

Gaussian Process and Expectation Propagation time complexity?

What's the time complexity of training a Gaussian process and its Expectation Propagation approximation? (Before studying them, I'd like to understand if they are even feasible for my application)
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22 views

How to create a new covariance function using GPML toolbox in Matlab?

I want to create my own covariance function based on squared exponential or Matern that treats each dimension differently i.e. having a hyperparameter for each dimension, not just ell. How do I need ...
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1answer
31 views

Gaussian Process for Closed Curves

A Gaussian process gives a probability distribution over functions that pass through the data points. Is there a way to parameterize the Gaussian process to give a probability distribution over closed ...
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24 views

Glm gaussian Vs Glm Binomial Vs s log-linked GLM Gaussian

I am trying to do a study on death of malaria in certain in order to estimate the best way to predict how dangerous is this disease. I don't have a strong background in statistics, I am auto-learner ...
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25 views

What Gaussian process covariance function captures an affine mean?

In the documentation for GPML, the author trained a GP with an affine mean function and isotropic squared exponential covariance function. Then there is an exercise to the reader: Try training a ...
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1answer
56 views

Linear Kernel in Baysian Linear Regression

I came up with http://mlg.eng.cam.ac.uk/duvenaud/cookbook/index.html and it is actually very useful. At some point it says If you use just a linear kernel in a GP, you're simply doing Bayesian ...
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41 views

Derive squared exponential covariance function

In Gaussian Processes, SVMs, kernels are used (as to my understanding) as similarity measure. However, they have the constraint that any kernel has to be represented as a dot product. i.e. ...
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87 views

Gaussian Process , selecting the hyperparameters

I am using Gaussian Process regression toolbox from the site http://www.gaussianprocess.org/gpml/code/matlab/doc/ I was able to use implement the code in matlab easily, following the guide lines. ...
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29 views

Difference between summing and multiplying covariance matrices?

Say we have an RBF covariance matrix A and some periodic covariance matrix B for a given dataset. Covariance matrix A says that you believe that points that are close together are somewhat similar, ...
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22 views

Variance estimation from two gaussian distributions

Assume a stochastic process with observation $r$, and two hypotheses : $X \sim (0, \sigma^2)$ and $Y \sim (0, \sigma^2 + \tau^2)$. When we observe/receive $r$ we don't know which hypothesis $X$ or ...
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12 views

Neighborhood in Gaussian graphical models

A gaussian random vector $X$ can be represented using a graph where two nodes $a$ and $b$ are connected $\Leftrightarrow X_a$ is dependent on $X_b$ given all the remaining random variables. I have two ...
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1answer
175 views

Gaussian Process Kernel and Ridge Regression

Can a Dual Ridge Regression produce the same prediction results as a Gaussian Process with a polynomial kernel $K(x,x')=(x^Tx'+1)^2$ in less time complexity (GP is $O(n^3)$ ) using Cholesky ...
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1answer
75 views

Gaussian Regression With Multiple Inputs?

Is it possible to use a Gaussian Process to relate multiple independent input variables (X1, X2, X3) to an output variable (Y)? More specifically, I would like to produce a regression graph like the ...
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232 views

Multi-target Tracking: calculate the association gate from Kalman filter

I'm trying to implement a multi target tracking with Kalman filter. Each object has an instance of Kalman Filter. The true position of the objects $(x,Y)$ are the corrected state out of the KF after ...
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315 views

Is it possible to convert a Rayleigh distribution into a Gaussian distribution?

...and how might we do this? If possible, I am curious if outliers in the Rayleigh distributed data would also remain outliers in the new Gaussian distributed data. Thanks.
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177 views

Kalman filter equation derivation

I'm studying the Kalman Filter for tracking and smoothing. Even if I have understood the Bayesian filter concept, and I can efficiently use some of Kalman Filter implementation I'm stucked on ...
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1answer
35 views

Why X-process is called a process?

I have recently learnt about kernels in machine learning. And I have been introduced to many different processes e.g. Gaussian process, Wiener process. Now my question is why a set of functions has ...
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493 views

Is expectation the same as mean?

I am doing ML at my university, and the professor mentioned the term Expectation (E), while he was trying to explain us some things on Gaussian processes. But from the way he explained it, I ...
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44 views

Shouldn't a function of data from a PDF repeated over and over on new data eventually yield a Gaussian PDF?

I got into an interesting discussion with a co-worker today and we are not sure what the answer is: We have $N=1000$ samples from a Rayleigh PDF. We take those $N$ samples, and compute their ...
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1answer
90 views

Is Gaussian process regression a Bayesian method?

Actually I thought Gaussian Process is a kind of Bayesian method, since I read many tutorials in which GP is presented in Bayesian context, for example, in this tutorial, just pay attention to page ...
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44 views

derivation of predictive distribution of Gaussian Process

There is a duplicate, and the reason why I still ask this question is that, the answer to that duplicate doesn't answer the question well. The Gaussian Process prior is $$u\sim GP(0,k(x,x'))$$, I ...
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22 views

Large scale 1-Dimensional Gaussian Process Classification

I have a dataset with a small number of input points (e.g. +- 300), but millions of boolean outcomes. ...
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16 views

generating covariance matrices from multiple priors

In many optimisation problems, one typically uses many forms of regularisations over the parameters that is being estimated. For example, a typical cost function (to maximise) may look like this: $$ ...
2
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1answer
134 views

Trying to understand Gaussian Process

I'm reading the GPML book and in Chapter 2 (page 15), it tells how to do regression using Gaussian Process(GP), but I'm having a hard time figuring how it works. In Bayesian inference for parametric ...
2
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1answer
79 views

What's the point of estimating the posterior distribution $Y | X $ using a Gaussian Process?

So classic regression methods (ridge regression, LASSO) only predict the posterior mean $E[ Y | X ]$, while Gaussian Processes give you the full posterior distribution $Y | X$. It would be very ...
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1answer
75 views

Relationship between average and characteristic function of a Gaussian process

I'm having trouble understanding an equality given in a book ("Speckle Phenomena in Optics" by Joseph Goodman p.145) for a zero mean, stationary Gaussian process: $\overline{\exp(i ...
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27 views

Is a linear mixed model a special case of Gaussian process regression?

Are there any introductory references or resources that discuss this connection if there is one? I have not been able to find any ones suitable for beginners.
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28 views

Is a Stationary VAR Process with Zero Mean Gaussian Innovations a Gaussian Stationary Process?

Consider the stationary VAR process $${\bf X}_t = \sum_{\tau = 1}^{L} A_\tau {\bf X}_{t-\tau} +{\bf \epsilon}_t$$ If the innovations $\epsilon_t \sim MVN({\bf 0},\Sigma)$ then is ${\bf X}_t$ a ...
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178 views

Why are kernel methods with RBFs effective for handwritten digits (letters) classification?

The question emerged while reading Ch. 3 of Rasmussen & Williams . In the end of this chapter, the authors gave results for the problem of handwritten digits classification (16x16 greyscale ...
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71 views

GPML gives too large length scale when optimising hyperparameters

I recently started trying to apply Gaussian process regression to a problem, using the MATLAB GPML toolbox. The problem has five (or more) input variables, but for now I'm just looking at one of them. ...
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1answer
50 views

Gaussian process estimation

The stochastic process $(X_t)_{t\in T}$ is called Gaussian if for all $t_1,\dots,t_k\in T$, for all $k$, the joint distribution of $X_{t_1},\dots,X_{t_k}$ is multivariate normal. The process is ...
3
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1answer
487 views

Scikit-learn's Gaussian Processes: How to include multiple hyperparameters in kernel/cov function?

I'm using the scikit-learn's implementation of Gaussian processes. A simple thing to do is to combine multiple kernels as a linear combination to describe your time series properly. So I'd like to ...
0
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1answer
74 views

Gaussian Distribution Prediction

What are some common methods of making distribution predictions? I have a set of features $x_1,x_2,x_3$ which map to Gaussian distributions ($\mu,\sigma^2$). That is, the feature vector of a single ...
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2answers
278 views

Output of a system with a white Gaussian process as an input

I want to present a question which seems to predict a theorem I know. Hence I guess I'm missing something and would be happy to understand what I'm missing. Here's is the system: The signals $ ...
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136 views

I am still confused with Gaussian kernel in SVM

From the slides http://www.csie.ntu.edu.tw/~cjlin/talks/kuleuven_svm.pdf, $$\min \frac{1}{2}w^Tw $$ subject to $$y_i(w^T\phi(x_i)+b)\ge 1,i=1,\cdots,n$$ I think most people are very familiar with ...
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95 views

Fully Bayesian hyper-parameter selection in GPML

Is it possible to perform an approximated fully Bayesian (1) selection of hyper-parameters (e.g. covariance scale) with the GPML code, instead of maximizing the marginal likelihood (2) ? I think using ...
6
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1answer
138 views

Constructing an interval estimate for a multivariate output

I am building a multivariate Gaussian Process model that predicts output $X_i$ and $Y_i$ jointly from input location $z_i$. At the end of it all I want to be able to plot my best estimate of $X_i$ ...