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Questions tagged [gaussian-process]

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 regression and classification problems.

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Possible error in evaluating kernel gradient in scikit-learn's GPR

Perhaps I am missing something very obvious, but in the standard kernels associated with scikit-learn's Gaussian process regression framework, the radial basis function (RBF), $$f = e^{-x^2/2l^2},$$ ...
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Gaussian process likelihood function

I'm trying to understand the likelihood function in Gaussian Process. The book by Rasmussen et al. defined Gaussian Process lml as $$log~p(y|X) = -\frac{1}{2}y^T\alpha-\sum log L_{ii} - \frac{N}{2}...
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Learning a Gaussian Process from function observations (not GP regression)

Suppose we have a set of observations, where each observation represents a function. For example, our set is $\{f_1, f_2, ..., f_n\}$ where each $f_i = \{(x_1, y_1), (x_2, y_2), ..., (x_{p_i}, y_{p_i})...
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How to prevent heteroskedastic models from overfitting?

I'm trying to fit neuroscience data using a Gaussian Process, but noticed that it behaves poisson-like (var = mean). Since classic GP models assume iid noise, I figured I could get a better fit by ...
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34 views

How to calculate the Product between Gaussian and exponential distribution in Matlab?

I require to calculate the Product of Gaussian distributed and exponential distribution. My work clamp force = (force of the wheel * friction coefficient) where the force of the wheel is Gaussian ...
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98 views

The Spectral Density for the Matérn Covariance Function [closed]

In Paper1, we're working with a linear functional approximation to a Gaussian Process, shown below. In equation (8) of this paper, we have $$V=\mathrm{Diag}(S^{-1}(\sqrt{\lambda_j}))$$ (there's ...
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41 views

Is it possible to apply a monotonicity constraint on a Gaussian process regression fit?

Below is a code using scikit-learn where I simply apply Gaussian process regression (GPR) on a set of observed data to produce an expected fit. I know physically that this curve should be ...
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Condition on the covariance matrix of a gaussian process needed to have the Markov property

Let suppose to have a realization $\mathbf{X}=(\mathbf{X}_1,\dots, \mathbf{X}_n)$, where $\mathbf{X}_i \in \mathcal{R}^d$, from a $d-$variate Gaussian process. Let also suppose that $E(\mathbf{X}_i)= ...
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Is it valid to compare the likelihood of different models in Gaussian process regression?

When applying different kernel's through scikit-learn's Gaussian process regression, I observe certain instances with positive log-likelihood outputs which indicate a likelihood that is greater than 1....
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Is there a Gaussian Process Kernel that limits functions to sigmoids?

I am modeling a large number of Dose-response curves. I have strong reason to believe that the generating function will be sigmoidal against the concentration of the assay (Michaelis-Menten kinetics). ...
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Is this a valid Gaussian Process kernel?

$\mathcal{K}\Big( \; (x,y), (x',y') \; \Big) = \sigma_f^2 \exp{ \frac{(x-x')^2}{2l^2 \cdot (y+y')^2} } $, where $l > 0$ The variance associated with each training point (given by a vector) is a ...
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44 views

Gaussian Process: why is my data only explained by noise?

I'm trying to explain a very simple 10-pt dataset by a Gaussian Process (part of a larger bayesian optimization framework) and I don't understand why it is only being explained by noise. Here is the ...
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What multivariable model should I use?

I am trying to predict the slope of the learning curve for 3d data (you can consider it to be locations). In 1d, people have used linear regression for this type of task. The idea is, you get (...
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Generating fractional Brownian motion in R [closed]

I was trying to generate fractional Brownian motion in R using fbm of the package somebm. However, in this package, I can not ...
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How do I know how to combine kernels in e.g. Gaussian Process Regressions

Looking for optimal parameters usually is a pain for most Machine Learning tasks. In most cases, however, we can perform a grid search to find out about which parameters do the job better or worse. ...
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Bayesian estimation of traffic flow - Help with methodology

I need help setting up a model for estimation of traffic flow. I shall do the analysis with a Bayesian approach. Data: I have sensor data from ten sensors. The sensors are installed at three main ...
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44 views

Covariance Matrix of HIERARCHICAL MULTITASK GAUSSIAN PROCESS

I'm currently trying to develop a Gaussian Process to predict different levels of different individuals over time. So it is a Time Regression Problem in which we have multiple tasks, but also ...
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Marginal Distribution from Bivariate Distribution Matrix

I am doing some practice problems to prepare for my statistics exam, and I just want to know if my reasoning is correct on one problem, and if not, I want to know how I should reason through this. The ...
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Gaussian Process regression: does there exist a conjugate prior over hyperparameters?

When adopting a fully Bayesian hierarchical setting in Gaussian Process regression is there a choice of kernel (covariance) function such that there exist a conjugate prior? If so which?
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67 views

Deriving non-Gaussian uncertainties using Gaussian process regression

I came across this work on "Gaussian Process Regression with Heteroscedastic or Non-Gaussian Residuals" which is intriguing, but I believe I am failing to properly interpret its results. Namely, it ...
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45 views

Conditional Entropy in the context of Gaussian Processes

I've got a question regarding the conditional entropy of a discrete random variable. According to this paper the conditional entropy of a Gaussian random variable conditioned on a set of variables can ...
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Equation (3.23) GP for ML book

This is the computation of the variance when we do Laplace Approximation for inference in binary classification. I do not understand why the variance is decomposed into these two terms.
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Are there kernel functions available for categorical variables where matches between different variables would also raise the similarity?

For my master thesis I have to apply bayesian optimization on the development of modular endolysins. This endolysin consists of 3 building blocks that are linked together (variables). Each of these ...
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Convergence of the Matérn covariance function to the squared exponential

The Matérn covariance function converges to the squared exponential covariance function. Many sources, amongst them the GPML book and Wikipedia, state this result. None of them provide details. I ...
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141 views

Bayesian inference for non-Gaussian errors

Following from a previously unanswered question, regression tasks involving measurements with normally distributed noise apply Gaussian processes. But are there any recommended approaches for ...
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64 views

Use the RBF kernel to construct a positive definite covariance matrix

A commonly used kernel in Gaussian processes is the RBF kernel: $$ \kappa(x,x') = \exp\left(-\frac{|| x-x'||^2}{2\sigma^2}\right) $$ In the context of a Gaussian process, a kernel is used to ...
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Gaussian Processes, Prior function of constant, Linear and Polynomial Kernel

In my university course slide, regarding the exponential and Gaussian kernel there are these two relevant pictures. They show the prior function of each kernel: You can see that the resulting prior ...
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Gaussian Processes, basic question about how the prior is computed

I'm approaching the topic of GP, and I have a question regarding how functions are sampled. On my textbook is stated that to represent a distribution over a function (the prior): we only need to ...
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How to choose my next sample point in a 2D boundary problem?

I've got a real-world problem that I'm trying to solve with as few computations as possible. In this 2D problem, everything is parameterized on a unit square, $f(x,y); x, y \in [0, 1]$. Anecdotal ...
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Additive Gaussian Processes with Penalized Likelihood

I have a problem with many - say $D$ - input variables, $\mathbf x=(x_1,\dotsc,x_D)^\top$. I have have dataset $\mathcal D$ of $n$ input/outputs, with $n<D$. Only $\delta<<D$ should suffice ...
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Why mean in Gaussian Process is not so important? [duplicate]

Source of my doubt is the section 2.7 of GPML book by Rasmussen, an screenshot of the book is attached below. Much of my confusion is clarified by this discussion. If mean of GP is not estimated and ...
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Accounting for errors in independent variable through Gaussian process regression

In Gaussian process regression (GPR), one applies a kernel (i.e. covariance function) to describe the similarity between observed and predicted data in the domain. The diagonal of the covariance ...
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How to get a predicted value from a gaussian regression model [closed]

I have a set of the times where the trash level is greater than 60 % during 6 days and i want to predict the time in the 7th day. I choosed to work with gaussian regression and after training the ...
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Error propagation in Gaussian processes

I am trying to predict a time-series measurements, $x_{t+1},x_{t+2},...$ using a Gaussian process model, where the kernel is a Gaussian kernel with small noise-level $\sigma_w^2$, i.e. $k_{ij}=e^{-\...
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Natural Gradients in Stochastic Variational Inference (SVI) for Gaussian Process Regression

Currently, I've hard times in understanding the natural gradients update in SVI method for Gaussian Process. I'm learning the SVI method for Gaussian Process through Gaussian Process for Big Data ...
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Gaussian process for machine learning consistent property explanation

I am currently reading Gaussian process for machine learning book from Christopher Williams, and I encounter a note on function-space view where consistency property is explained, what I am having ...
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237 views

Hyperparameter Optimization Using Gaussian Processes

I have a dataset that is divided into training and validation dataset. I am using Gaussian Processes to perform hyperparameter optimization. So I am using the accuracy recorded on the validation ...
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50 views

Reducibility between Gaussian Mixture Models and Gaussian Processes

I am studying gaussian processes and I have already discrete amount of knowledge in gaussian mixture models. I am here to undersrtand if with a gaussian process you can fit a gaussian mixture model. ...
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41 views

why does ignoring spatial autocorrelation lead to spurious significance

In spatial statistics one often hears the statements like the following: unaccounted for spatial autocorrelation may lead to spurious significance / understimated uncertainty / too narrow ...
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67 views

Generate Gaussian process with squared exponential covariance function

In a (stationary) Gaussian Process, values which are closeby are more similar than values far away from each other. The correlation function tends to zero as distance increases. Often, one models the ...
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Guidance on time-series change point detection or identification of contributions

Let me preface this by saying that I am not a data scientist. Please excuse any imprecision in my use of subject specific terms or notations. Please feel free to edit my question, to improve any ...
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Gaussian random fields: matrix and convolution sampling

I should be able to generate a stationary GRF from white noise in two different ways: multiplying the white noise vector by the square root of a covariance matrix with appropriate kernel; taking the ...
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1answer
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Estimate a probability distribution of target values using features

In my particular problem, I have $$t \in \{1,...N\}$$ time periods, and feature vectors $$x_t \in R^m $$ which I hypothesize predict something about the probability distribution that the targets $...
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41 views

Covariance kernel of a Gaussian process

I just started studying the theory of Gaussian processes. I'm mainly interested in studying functional data and I haven't found the answer to my doubt. Let's say I have some curves that I consider as ...
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2answers
162 views

Cross-validation and building a final model when using hyperparameter optimization

I am trying to build a Gaussian process (GP) regression for a problem in which each experiment is computationally expensive, using cross-validation. Here is how I do it: Build the GP regressor on the ...
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1answer
48 views

Imposing constraints on a Gaussian process

I am trying to model a univariate function $f(x)$ (whose functional form is unknown) by a Gaussian process. The function is defined for $x>0$ and function evaluation for growing values of $x$ ...
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1answer
37 views

Fitting a NN model on one-to-many function

Given $f(x) = y$ as a surjective (many-to-one) function, we know that $f^{-1} (y) = x$ is a one-to-many mapping for function $f^{-1}$. In my application, $x$ is a spatial data represented by a 2D ...
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Infill likelihood for a continuously observed continuous-time process

Consider a continuous-time stochastic process $y(t)$ having the following linear (Gaussian) state-space representation for $t \geq 0$ $$ \left\{ \begin{array}{c c l} \text{d}{\boldsymbol{\...
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What is the posterior kernel lengthscale of a Gaussian process?

If I have access to multiple samples from a Gaussian process with known covariance kernel but unknown parameters (i.e. unknown lengthscale), it is straightforward to estimate the lengthscale using ...
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Choice of Gaussian process in non-parametric regression

I have been trying to understand non-parametric regression using Gaussian processes (GP), which are used to represent prior distributions over the space of functions. The linear model considered is $$ ...