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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Simulating a Brownian Excursion using a Brownian Bridge?

I would like to simulate a Brownian excursion process (a Brownian motion that is conditioned always be positive when $0 \lt t \lt 1$ to $0$ at $t=1$). Since a Brownian excursion process is a Brownian ...
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7 views

Gaussian Mixture Model: bandwidth parameter versus variogram fitting?

I'm estimating a stationary, spatially random variable over a 2-dimensional domain. I have ground-truth measurements in several locations, over time. I need some way of spatially-interpolating ...
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44 views

The squared-norm of the projection of a Gaussian vector onto an independent $d$-dimensional subspace is a $\chi^2_{2d}$

How we can prove that: The squared-norm of the projection of a $N$-dimensional complex vector with i.i.d. unit-variance and zero mean Gaussian components onto an independent $d$-dimensional subspace ...
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21 views

gaussian process - missing data

One approach to deal with missing data is be to define a joint gaussian distribution / Gaussian process, and then define the (conditional) distribution of the unknown values on the known values. (e.g. ...
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28 views

interpolation of noisy data

Kevin Murphy discusses an approach to interpolate 1D data in his book "Machine Learning - a Probabilistic Approach". I have been staring at the page below for a while -- but am struggling to see the ...
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225 views

Gaussian Processes for Stock Market

I'm reading this paper and I would like to implement it. It gives me a matrix in this way: $$ \begin{array}{lcr} \mbox{Year} & \mbox{day 1} & \mbox{day 2} & \dots & \mbox{day 250} ...
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20 views

Gaussian Process for periodic data?

Assuming that I have data with repeated measures (or, in other words, multiple time series of different realizations of the same process), can I train a Gaussian Process on this data? In fact, is ...
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1answer
26 views

Regression on multiple independent (noisy) inputs [closed]

I have a data set that is highly variable and made up of 17 different inputs and 1 output. I'm currently using a Gaussian Process Regression toolbox. However I'm curious to see if they're other types ...
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19 views

What is the integrand in a Gaussian process covariance function?

I have a very basic question about Gaussian process covariance functions. When I specify a covariance function like $k(x,y)$, what exactly is assumed to be co-varying? Is it correlated noise at the ...
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10 views

Fitting of non-linear GP-LVM

I'm trying to perform dimension reduction using the Gaussian process latent variable model. I read the paper from N. Lawrence http://jmlr.org/papers/volume6/lawrence05a/lawrence05a.pdf and there are ...
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1answer
19 views

GP regression - Matern kernel gradient issue

I'm trying to use a Matern 5/2 kernel for GP regression, so my kernel function is $ K(x,x')\triangleq\theta_0(1+\sqrt{5r(x,x')}+5/3r)\exp(-\sqrt{5r}), $ where $r(x,x')\triangleq\sum_{d=1}^D ...
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How to select a subset of variables from a large dimensional dataset for Gaussian Process Regression?

I am currently working on GPR(gaussian process regression) on a large dimensional input dataset(around 300). I am pretty sure that some of these variables have weak correlation with target output. If ...
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34 views

optimal sequential sampling in gaussian process models

Let's say we have a one dimensional dataset of 24 points along with their responses. I am reserving three boundary points for testing (i=1,23,24) and i am fitting a Gaussian process model based on a ...
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1answer
44 views

Gaussian Process Regression for piecewise linear response functions

I am performing Gaussian Process Regression (without noise) for response functions which are piecewise linear. My question: Does there exist a covariance function, such that sample paths from a ...
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1answer
34 views

Meaning of syntax $N(\mathbf{y} \mid \mathbf{0}, \mathbf{K})$ (multivariate normal distribution)

So I'm reading notes on Gaussian Processses, and came across syntax $p(\mathbf{y} \mid \text{stuff}) = N(\mathbf{y} \mid \mathbf{0}, \mathbf{K})$ for multivariate normal distribution, and I'm not ...
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1answer
34 views

Gaussian process regression implementaion

I am using the GPML matlab code found here. I have been using a squared exponential cov function with ARD. I am finding that if I use the minimise function to train the process I get uniform large ...
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1answer
32 views

Gaussian Process Regression

After implementing a Gaussian process regression model I am getting a negative log marginal likelihood figure that is very high (~100). What does this mean?
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1answer
79 views

Gaussian process regression: leave-one-out prediction

According to Dubrule's Cross validation of kriging in a unique neighborhood, it is possible to compute leave-one-out the gaussian process prediction $\hat{Y}_{-i}(x_i)$ at a point $x_i$ from the ...
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69 views

Hyperparameter tuning in Gaussian Process Regression

I am trying to tune the hyperparameters of the gaussian process regression algorithm I've implemented. I simply want to maximize the log marginal likelihood given by the formula ...
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25 views

What is the shape of the decision surface of a Gaussian Process classifier?

I've got a binary classification problem, which I am trying to solve using a generative classifier. If I use Gaussian Discriminant Analysis, and fit two Gaussian distributions to my two classes, the ...
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1answer
40 views

Incremental Gaussian Process Regression

I want to implement an incremental gaussian process regression using a sliding window over the data points which arrives one by one through a stream. Let $d$ denote the dimensionality of the input ...
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25 views

Bayesian multivariate extrapolation

I'm not sure "Bayesian multivariate extrapolation " is the right formulation of what I want to do but here is my problem: I have a set of observations in a state $k$ (having a multivariate Gaussian ...
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1answer
168 views

Unscented Kalman Filter with Gaussian Process regression for time series prediction

I've trained a gaussian process which will take X (x1:5) and predict Y (x6). I'm trying to do 1step ahead prediction with Unscented Kalman filter with this GP as my state transition funtion. The ...
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1answer
16 views

GPML producing wrong output using correct target labels

I am using the GPML code found here. The key function in the aforementioned library is the gp function described below: Two modes are possible: training or ...
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55 views

An example for Gaussian Process: Singular covariance matrix?

I follow Christopher Bishop's book "Pattern Recognition and Machine Learning" and I am studying the section on Gaussian Processes. As an introduction, a simple model is given with the following ...
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1answer
269 views

How does the mean function work for a Gaussian Process?

I was reading the notes on Gaussian Processes by Choung B. Do (stanford course CS229) however was unsure of how the mean function worked and what a random variable was on the Gaussian Process So ...
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33 views

What is the realization value of f(x) in Gaussian process regression?

This is related to the Gaussian process regression. We propose the model as $y_i = f(\mathbf{x}_i) + \epsilon_i$ with $i = 1 \ldots n$ and $\epsilon_i \sim \mathcal{N}(0, \sigma^2)$. Here ...
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52 views

How to compare the relative importance of features in GP regression?

Kernel function with different length scales, such as the squared exponential function, is said to be able to quantify the relative importance among the input (predictor) features. The idea is to ...
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1answer
31 views

Reference for definition of multiple-output Gaussian process

Does anyone know any good reference that has a clear and precise definition of multiple-output Gaussian process? Something like the definition of the Gaussian process in the third page of this set of ...
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1answer
27 views

Marginal Likelihood of a Gaussian Process Model, Duplicate entries in kernel matrix

I am trying to fit a Gaussian process model using the toolbox and I got stuck in the following problem. Assuming that I have some duplicated data points in my training data, then those will map to ...
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1answer
58 views

Estimating the variance of the noise in Gaussian Process prediction

I've been trying to use leave-one-out cross-validation to estimate the $\sigma_n$, the variance of the signal noise when doing prediction according to $E[f_*] = k_*^T(K+\sigma_n^2I)^{-1}y$ (GPML ...
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1answer
33 views

scale of variance in Gaussian process

When performing Gaussian process regression, the variance at a prediction point is given by $var[f_*] = k(x_*,x_*) - k_*^T(k+\sigma_n^2I)^{-1}k_*$ (Equation 2.26 from GPML) The variance is not ...
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99 views

What mathematical background do I need for the Gaussian Process book by Rasmussen and Williams?

I started reading the book today, but right off the bat, he mentions infinite Hilbert spaces in the notation, so I feel that it might be out of my league. I am familiar with linear algebra, ...
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307 views

Calculate PLS Xscores for predicting new data

I wish to extract Partial Least Squares (PLS) components to apply non-linear regression (Gaussian Process Regression (GPR)) on the scores of the predictors (Xscores). The reason is my data is very ...
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1answer
88 views

Splines vs Gaussian Process Regression

I'm know that Gaussian Process Regression (GPR) is an alternative to using splines for fitting flexible nonlinear models. I would like to know in which situations would one be more suitable than the ...
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1answer
97 views

Kernel methods in machine learning?

I am beginning to tackle geostatistics problems where I tried to apply kriging(gaussian processes) to interpolate demographical water drop. According to my understanding, kernel methods are something ...
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1answer
54 views

How could the predictive mean in a GP become negative when both the prior and the training target values are non-negative?

I am training a Gaussian process regression where the training target values are between 0 and 1 and the prior mean is the fixed zero function. The predictive mean sometimes becomes negative e.g. ...
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78 views

Handling missing data in Gaussian Process Regression

I am trying to handle missing data in a model using Gaussian Processes. I have two spatial dimensions sharing the same length hyper-parameter, one dimension for time. Additionally I have split one ...
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32 views

Kriging, Gaussian Processes with categorical data

Theoretically it is possible to use Kriging also for categorical features by using a kernel function which supports factors. Does anybody know some references on this topic or whether they are ...
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44 views

How to reduce the number of features for Gaussian Process regression?

Ridge regression reduces complexity of the model by scaling down the coefficient. Lasso reduces the complexity of the model by selecting the features used. For Gaussian Process, is there similar way ...
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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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1answer
42 views

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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42 views

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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67 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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1answer
106 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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27 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
51 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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54 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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23 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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70 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 ...