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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Negative quadratic data-fit term of marginal likelihood in Gaussian process regression

I am trying to implement an auto-tuner for hyperparameters of a Gaussian process regression. A way of doing this is optimizing the marginal likelihood function. The marginal likelihood contains the ...
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25 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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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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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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21 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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18 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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69 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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58 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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22 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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37 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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22 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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143 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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11 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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44 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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261 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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29 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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48 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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21 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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48 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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30 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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94 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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270 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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61 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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75 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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52 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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62 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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31 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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43 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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11 views

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
31 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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41 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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59 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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92 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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25 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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49 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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48 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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22 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 ...
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2answers
49 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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34 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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87 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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22 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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1answer
45 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
54 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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59 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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93 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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74 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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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.