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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Gaussian Processes - cubic correlation model [closed]

Is there anyone that could explain the cubic correlation model in the sklearn gaussian processes?
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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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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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26 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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246 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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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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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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26 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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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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34 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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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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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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197 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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43 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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59 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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45 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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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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28 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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Why does inversion of a covariance matrix yield partial correlations between random variables?

I heard that partial correlations between random variables can be found by inverting the covariance matrix and taking appropriate cells from such resulting precision matrix (this fact is mentioned in ...
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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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27 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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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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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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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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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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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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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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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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67 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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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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32 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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74 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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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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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
53 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
53 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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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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73 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.
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111 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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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
85 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
46 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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110 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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118 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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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 ...