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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How to compare two bayesian linear models?

Let say, we have two bayesian linear models learned from some data (not necessarily same but from the same data distribution), how can we compare or what is the notion of similarity between the two ...
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Can Laplace Approximation be used when Gaussian Process is only a part of bigger model?

I felt in love with Gaussian Process models in Species Distribution Modelling, especially with the Laplace Approximation (LA), which is able to compute these models very fast. This way, I am able to ...
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How can I calibrate my sigmas for Gaussian process regression?

I have a large-ish set of unevenly-spaced time-series data from instruments around the world, for which I'm using Gaussian process regression to do interpolation and short-term future prediction. My ...
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Are there reasonable ways to do bidirectional dimensionality reduction?

Are there any sensible techniques for dimensionality reduction, for example from 20 to 5 dimensions, and then being able (albeit with loss of information) go back from 5 to 20? Algorithms like t-SNE, ...
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Gradient of the variance of a Gaussian process

I'm trying to compute the spatial derivatives of the expected improvement acquisition function in Gaussian-process optimisation, and doing so requires the spatial derivatives of the predictive ...
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Are there examples of covariance functions used in Gaussian processes with negative non-diagonal elements?

I've been searching through numerous kernels used in Gaussian processes, and one common feature is that the covariance matrices always have only positive elements. Yet the only requirement on the ...
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Probability distribution over geographic coordinates

I would like to model some quantity y that varies over different locations on the Earth using a Gaussian process. My concern with just using ...
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How to calculate the mean of standard deviation when data are drawn from a Gaussian population?

This is a pure algebraic question. Drawing a sample of size $n$ from a Gaussian population $N(0, \sigma^2)$, the posterior probability of $\sigma$ is proportional to $\frac{1}{\sigma^{n+1}} e^{-(s/\...
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What is the formula for the conditional variance when taking the derivative of a Gaussian process?

The formulae for the conditional mean and variance of a Gaussian process is given by equations (2.23) and (2.24): Also, the formula for the covariance of the derivative of a Gaussian process is given ...
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scikt-learn how to incorporate explicit basis functions in gaussian process regression?

I'm kinda new on Gaussian Process regression and I'm having some trouble understanding on how to build a model when implementing a explicit mean model function with scikit-learn. From what I ...
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Finding the bottleneck in a set of trajectories

I have some 2D data which looks like the following: This can be seen as a plot of trajectories in 2D space, where the trajectories all pass through a point called the "bottleneck". What I want to do,...
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Measuring the uncertainty of points along a trajectory

I have a number of 2D trajectories, and I want to be able to estimate the variance of each step along the mean trajectory. I thought that this might be suited to Gaussian process modelling, but then I ...
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Estimating Gaussians with (Gaussians+cross entropy) or (output+MSE)?

model used :- neural nets x is an N-dimensional vector. y is a real number. p(x,y) is the joint probability distribution over x and y. I know p(y|x) is a Gaussian probability distribution for ...
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Multi-output Hardamard Gaussian processes for Heterotopic data

I am using gpytorch to implement Hadamard gaussian processes for Heterotopic data sets, see https://gpytorch.readthedocs.io/en/latest/examples/03_Multitask_GP_Regression/...
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Computing realizations of a gaussian process in parallel

I have to evaluate many realizations of the same Gaussian process in parallel, using R. Currently, I am calling mvrnorm ...
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32 views

how to recognize the Gaussian distribtuion in a formula?

For someone is the question quite easy but I would like to know how to detect in a formula the Gaussian distribution? For instance in the equation in attached picture, what are the keys to say there ...
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Which variable to predict when using Gaussian Processes: $y_*$ or $f_*$?

We have the following model: $y_t=f(x_t)+\epsilon_t$, where $f$ follows a gaussian process, and $\epsilon$ a normal distribution. Which quantity should I predict, $y_*$ or $f(x_*)$? The difference, ...
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In Gaussian processes, why does the conditional Gaussian “agree” with data?

I'm learning about GPs, and one thing I don't quite understand is how the posterior works. Consider this figure: Rasmussen and Williams say: Graphically in Figure 2.2 you may think of generating ...
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Variance of an integral of a product of functions with Gaussian distributions

Let \begin{align} I(t) = K \int_{-T/2}^{T/2} e^{i 2\pi f_X \left(t + X(t) \right)} \cos\,(2\pi f_Y (t + Y(t)) - \phi))\,dt, \end{align} where $X, Y$ are independent random variables which are the ...
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Stationarization of 2-dimensional Time-Series

I'm trying to perform a Gaussian Process Regression on time-varying data of the form (t, x, y, z), where t is the time when ...
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Gaussian process regression model for comparing two groups

I have a data set consisting of functional observations, where $Y_{mi}$ is the response of the $m^{th}$ functional observation from the $i^{th}$ group, $m=1,...,M$ and $i=1,2,$. The observations are ...
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Gaussian process kernels for predict trend and seasonality in Python

I referred to below 2 posts: [[1]: Forecasting a seasonal time series in R [2] predict seasonality and trend combined, better approach? Do you have similar examples in Python? I'm looking at tuning ...
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Predictive Distribution in Gaussian Process Derivation

In Gaussian Process for Machine Learning (Rasmussen and Williams), on p.11, we are given the following predictive distribution: $$p\left(f_{*} | \mathbf{x}_{*}, X, \mathbf{y}\right)=\int p\left(f_{*} ...
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Empirical Comparison: which ideal data characteristics are best captured by each type of machine learning model?

I have reached the point as a data scientist where the empirical differences between the different types of regression models (leaving out classification only for simplicity) have started to matter ...
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Beta Coefficient for Gaussian Process with Non-Zero Mean

I'm reading through Rasmussen & Williams (2006)'s book on Gaussian Processes, specifically on section 2.7 on incorporating explicit basis function. I am confused on how the analytical solution $\...
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Gaussian process - Why adding data points cannot increase the predictive bias?

I've seen this question here: How to increase variance in Gaussian Process regression? And trying to complete the proof. I'm looking at this book: Rasmussen & Williams 2006: Gaussian Processes ...
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26 views

Concentration inequality for max component of a multivariate Gaussian in the general case

I am looking to bound the variance of the maximum component of a vector distributed multivariate Gaussian in the general case where the Gaussian distribution has arbitrary mean and full covariance ...
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Error when computing the gradient of a derivative with Gaussian process regression in scikit

Gaussian process regression in scikit-learn provides a conditional mean and variance, $(y_*,\sigma_*^2)$, based upon the observed data, $(y,\sigma^2)$. But given $(y_*,\sigma_*^2)$, how can you ...
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Confirming the formula for the Spectral Density of a Matern Covariance Function

The Matérn Class of functions is given by $$k_M(r)=\frac{2^{1-\nu}}{\Gamma(\nu)}\left(\frac{r\sqrt{2\nu}}{l}\right)^{\nu} K_{\nu}\left(\frac{r\sqrt{2\nu}}{l}\right)$$ where $r=\Vert x-x'\Vert$. The ...
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Proper way to use Gaussian Process results

This is my first time using ML. In the past I only used polynomial fitting to describe my test results, and I used those polynomial mainly as deliverables to other teams for use in their work. Now I ...
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Eigenvalue decomposition of a covariance matrix using a fast Cholesky decomposition

Let $\mathbf{C}$ be a $n \times n$ covariance matrix and assume that the LDL' Cholesky decomposition can be obtained efficiently. Can we take advantage of this to obtain a fast eigenvalue ...
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Gaussian process and its limitations

I once saw the following statement on Gaussian process, ...
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What is the relation between a surrogate function and an acquisition function?

A surrogate function is a simpler function than the objective function to evaluate. An acquisition function is used to propose sampling points. In the context of Bayesian optimisation and Gaussian ...
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Bayesian Linear Regression to Gaussian Process

I'm trying to understand how a Gaussian Process with a squared exponential covariance function can be obtained from Bayesian Linear Regression with a Gaussian prior $N(0,\sigma_p^2 I)$ on the ...
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What is a sparse Gaussian process?

In the paper Junction Tree Variational Autoencoder for Molecular Graph Generation, section 3.2, the authors state that they train a sparse Gaussian process to predict a chemical property, $y(m)$, of a ...
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Dirac delta function for multivariate input (in the context of Gaussian processes)

Let's say we have a set of $N$ observations $D = \{\bf X, t\}$ where ${\bf X} = [{\bf x}_1, ..., {\bf x}_N]^T$ are the locations and ${\bf t} = [t_1, ..., t_N]^T$ are the targets. When applying a ...
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Gaussian process where the output is constrained to be 0 or greater

I am trying my hand at simple GP regression and in my case, the output variable can be greater than or equal to zero. How can one constrain GPs, so that the predictions always stay in the specified ...
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Intuition behind the length-scale of the Rational Quadratic Kernel

What is the meaning of the length-scale in a rational quadratic? \begin{equation} k_{\textrm{RQ}}(t, t') = \sigma^2 \left( 1 + \frac{(t - t')^2}{2 \alpha \ell^2} \right)^{-\alpha} \label{eq:...
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Gaussian Processes: advice on proper optimization settings for simple model?

I am trying my hand at Gaussian Processes with GPflow (basically using this basic example as my guide), and am experiencing difficulties fitting some basic periodic data which I generated. My code: <...
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posterior distribution of f for Gaussian process model given existed observation data and input

In the chapter 2 of [Gaussian Process], equations (2.22-2.24) gives the predictive equations for Gaussian process regression, shown as follows. My question is how to derive f|X,y. It seems that the ...
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bayesian analysis for learning hyper parameters of gaussian process model

For the Gaussian process, what are the approaches to learn the hyper-parameters? Are there any ways to apply Bayesian analysis over these hyper parameters based on new observations?
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Bound of the bias of a Gaussian Process by its standard deviation in Gaussian Process Regression

In Gaussian Process Regression (GPR), intuitively, the bias of the conditioned Gaussian Process (posterior) at a location $x^*$ gets smaller if the variance at $x^*$ is getting smaller, for example in ...
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Gradient of marginal likelihood of Gaussian Process w.r.t likelihood parameters with Laplace approximation

The derivation of gradient of the marginal likelihood w.r.t covariance function hyperparameters $\theta$ is given in http://www.gaussianprocess.org/gpml/chapters/RW5.pdf, page 125. However, the ...
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Comparison of two normal distribution

I have two normally distributed samples. I want to know how close or similar it is. I tried few methods to find the similarity, like z-score and bhattacharyya distance. Bhattacharyya distance didn't ...
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How does one know when burn-in doesn't need to be discarded from an MCMC simulation?

I'm reading a paper about Bayesian model calibration (https://cfwebprod.sandia.gov/cfdocs/CompResearch/docs/McFCalib0307.pdf). The authors fitted a Bayesian Gaussian process model and sampled from the ...
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Do GGMs model partial correlation or conditional independence or both?

Post that says partial correlation != conditional dependence/independence: https://www.quora.com/Does-a-partial-correlation-of-0-imply-conditional-independence Post above points to following paper: ...
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Should the predicted variance in Gaussian process regression include experimental error?

Suppose I have an experiment where I measure the temperature of water in a cup, $y$, as a function of time, $x$. My measurement is normally distributed with an experimental uncertainty given by $\...
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Calculating travel distance from GPS updates

I am learning about Kalman filters but struggle to apply them for the following problem: tracking the distance traveled from GPS data. The GPS provides position updates every second and an estimate of ...
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Multivariate Theory: How does the new mean only depend on the conditioned variable?

I'm doing some review of Gaussian Processes and Multivariate Normal Theory. I found a really helpful website here, but I have run into a snag. What does the author mean in the sentence below this ...
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Gaussian process vs. Bayesian linear regression / computational cost in weight space

Gaussian process (GP) regression with the linear covariance function $$k(x_i, x_j) = \sigma_0^2 + \sigma_1^2 x_i x_j + \delta(i=j)$$ can be seen as a Bayesian linear regression (BLR) model $$ y_i = ...