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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What does it mean that a variogram keeps increasing with distance?

I am modeling my 3D dataset with a Gaussian Process with square-exponential covariance. To test whether this is a good model, I subtract the mean from the observed data and then calculate the ...
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Gaussian process with boundaries on unobserved variables

I'm trying to use Gaussian process regression of which I only have basic knowledge and the problem I have to deal with has a bit of a twist relative to the natural set-up, so I was wondering if there ...
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In Matlab 2016 GP Tool Box ,significance of regression loss ?`

I used GP tool Box (Gaussian Process Regression) to predict and trained my model. While making so I found one term called 'regression loss'. I just used this one for my calculation and my regression ...
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Decomposing a locally stationary covariance matrix

Say I have a non-stationary Gaussian Process with a square exponential covariance whose shape varies throughout space. The covariance entries are: $$ K_{ij} = N(|x_i-x_j|,\sigma_i^2+\sigma_j^2) $$ ...
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Relationship between Gaussian process and Regression by supervised learning model like SVR

Recently, I often see studies about Gaussian Process, and started to learn it. I'm familiar with regression models using machine learning such as Support Vector Regression. SVR learns a training ...
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a question on sequential estimation

I am reading Chris Bishop's Pattern Recognition and Machine Learning. In Section 2.3.5 he introduces some ideas on the contribution of the $n$th observation in a data set to the maximum likelihood ...
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Gaussian processes with finite sampling area

I apologize in advance if this question is poorly-posed: I'm an astronomer, not a statistician. My question is specifically aimed to help me figure out whether Gaussian processes is an appropriate ...
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26 views

Bayesian Optimization for a Stochastic Target that changes over time

Let's say there is a single slot machine that: costs zero to play can only be played once per day has a payout that is conditionally normal and is a function of the date and time. I want to use ...
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questions about predicting time series using gaussian process regression

I am using gaussian process regression to predict a time series. The time series is number of daily active users(DAU) of an APP, and takes daily numbers of installing users and uninstalling users as ...
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Main advantages of Gaussian process models

The Gaussian process has been widely used, especially in emulation. It is known that the computational demand is high ($0(n^3)$). What makes them popular? What are their main and hidden ...
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Can we calculate the predictive process of the derivative of a stochastic process from the data?

I asked this question, Calculating the expression for the derivative of a Gaussian process, some time ago, and now I am interested in an extension to the question. So originally I wanted to know the ...
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26 views

Learning matrix with gaussian process

Assume we have a matrix $A$ and that its rows are normally distributed (we assume a gaussian prior for the rows of A). Now, we want to learn the matrix A. The problem I find is in determining the mean ...
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Sample size needed for Gaussian process classification

I recently read the paper by Loeppky et al. on Choosing the sample size of a computer experiment: a practical guide and was curious to know if there were rules of thumb about the sample size ...
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What is the value of kernel width parameter (Gamma and σ ) in RBF for KFCM?

I am using the RBF Kernel which is of form: G(x)=exp(−x2/2σ2) and K(x,y)=exp(−γ||x−y||2) I got to know that in clustering ...
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31 views

Is there a good equivalent of multivariate normal distribution for strictly positive data?

More precisely: the distribution of data for each variate is similar to gamma/exponential distribution; and there are strong inter-variate correlations which I would like to take into account. A ...
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Is removing duplicate data necessary for Gaussian Process Regression (GPR)?

I will consider Non-Noisy Observations i.e. $y=f(x)$ Lets say we have the following data set of 5 training examples with one of the examples duplicated $(1,2,3,4,4)$ maps to $(2,4,6,8,8)$. Since for ...
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How to compute posterior density in infinite mixture models

I am learning infinite mixture models and I tried to derive the posterior density. I am confused by the paper the infinite mixture model by Rasmussen. He writes: Eq.1 $$ ...
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31 views

force tgp to use a zero mean GP prior

I'm using the tgp package in R for fully Bayesian Gaussian Process Regression, and it's great! I'm currently performing regression for experimental data coming from ...
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44 views

Most suitable optimizer for the Gaussain process likelihood function

The Gaussian process (GP) log Likelihood function can be expressed as Where K is a positive definite covariance matrix. The hyperparameters can be obtained through maximizing the likelihood ...
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53 views

Gaussian process likelihood with binned data

I have some binned data (no access to underlying info) and prior knowledge that the value in each bin smoothly varies in space. So I am modeling using a Gaussian Process prior, which according to ...
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20 views

How to calculate the value for a multivariate Gaussian

How to evaluate the value for a multivariate Gaussian. For instance, to evaluate the 20 dimensional Gaussian function value with respect to a 20 dimensional input vector x, I need calculate a 20 by 20 ...
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42 views

Marginal likelihood of a Gaussian Process

I have been trying to figure out how to get the marginal likelihood of a GP model. I am working on a regression problem, where my target is $y$ and my inputs are denoted by $x$. The model is ...
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144 views

Sampling from Gaussian Process Posterior

Anyone know of a Python package that both fits a Gaussian Process to data, and also lets you sample paths from the posterior? I'm interested in sampling the colorful lines on right (b) of the ...
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Gaussian Process Regression in R

I'm not sure this is the right Stack Exchange Community for the question: in case I'm wrong, please let me know. I would like to try using GP for regression: here is an example of data I need to fit ...
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39 views

Multiplication of two normals Gaussian Processes

I am working with Bayesian statistics for gaussian processes and I want to derive the posterior distribution. In general, I am clear about how to derive a posterior using Bayes rule. However in this ...
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48 views

Gaussian Process FITC approximation

I'm reading up on Gaussian Processes and trying to understand sparse Gaussian processes; in particular the FITC approximation (http://papers.nips.cc/paper/3351-the-generalized-fitc-approximation.pdf) ...
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Implementation of FITC approximation for Gaussian Processes [closed]

I'm attempting to use Gaussian processes for classification. When using a large number of observations, sparse approaches are used to deal with the scalability issue of O(N^3). Sparse approaches ...
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Corresponding RKHS of Common Kernels

A kernel, $k(x_1, x_2)$, has the interesting property that it may be represented as the dot product in a reproducing kernel hilbert space (RKHS), $\phi(x_0)\phi(x_1)$. I know that for the gaussian ...
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practical implementation detail of Bayesian Optimization

I'm giving Bayesian Optimization a go, following Snoek, Larochelle, and Adams [http://arxiv.org/pdf/1206.2944.pdf], using GPML [http://www.gaussianprocess.org/gpml/code/matlab/doc/]. I've implemented ...
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Diagonal linear discriminant analysis

On p110 of Murphy's machine learning book, he derives the discriminant function for the diagonal LDA model by simplifying the full linear discriminant analysis equation (4.33): It seems that he's ...
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What does surrogate mean in this context?

I'm trying to learn about Gaussian Processes and ran into an interesting example on the scikit-learn documentation but am having trouble interpreting the line below. Say we want to surrogate the ...
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Gaussian Process: Parameter of Kernel function

I am quite new to kernel method, I am trying to estimate $y'$ corresponding to $x'$, given [x, y] data. I am using Gaussian Method for analysis with Kernel function: $k(x_1,x_2 ) =p_1\exp\{-p_2(x_1 ...
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52 views

Gaussian process resources

I'm willing to learn about Gaussian process (I've a special interest in Gaussian Process for classification). Do you know some resources from which I can learn (aside of "Gaussian Processes for ...
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66 views

Calculating the probability a predicted point is 0

I have a deterministic function $f(x)$ and have evaluated some points $x_1,...,x_n$. So essentially I have pairs of data $(x_1,f(x_1)),...,(x_n,f(x_n))$. I am modeling the function $f(x)$ using a ...
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Customization of a standard Bell Curve

Hopefully this isn't a duplicate, I've tried to search for similar things, but to no luck. I'm curious on how you would computationally compute a random distribution of numbers that follows a bell ...
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Effect of similiar data on variance in a Gaussian Process

Given data ($\mathcal{D}$) of the form $\langle x, y \rangle$ where $y \in \mathbb{R}$, am I right to assume that greater variance in $y$ values for similar $x$ values, where similarity is defined by ...
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44 views

Calculating the marginal likelihood with multiple observations in a multivariate Normal-Normal model

Given $f, y_1, \ldots, y_n \in \mathcal{R}^d$ and $V$ fixed: $f \sim N(f; 0, V)$ $y_i | f \sim N(y_i; f, \sigma^2I_d)$ for $i = 1, \ldots, n$ [so they're iid] Find the marginal likelihood: $p(y_1, ...
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Stochastic Differential Equation Interpretation of Squared Exponential Kernel

As far as I understand, many Gaussian Processes can be either described by their corresponding mean and kernel functions or by a stochastic differential equation (SDE). For my purposes it is ...
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43 views

Estimate mean and variance from multiple realizations of Gaussian process

I have a certain number of realizations of the same Gaussian process. I want to get the mean and the variance of this process. How can I do that? To better explain the question lets suppose I have my ...
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Ill-conditioned covariance matrix in GP regression for Bayesian optimization

Background and problem I am using Gaussian Processes (GP) for regression and subsequent Bayesian optimization (BO). For regression I use the gpml package for MATLAB with several custom-made ...
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228 views

Calculating the expression for the derivative of a Gaussian process

I know based on the answers to this question Derivative of a Gaussian Process that the derivative of a Gaussian process is another Gaussian process, but I was wondering if someone could tell (or show) ...
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Gaussian-Process (scikit-learn) Prediction Confidence Interval Oddities - Stats Question

I'm doing some particle physics analysis and was hoping someone out there could give me some insight on a Gaussian-Process fit I'm trying to use to extrapolate some data. I have data with ...
3
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50 views

Effect of measurement variance on predictive variance in Gaussian Processes

When performing Gaussian process regression, the variance at a prediction point is given by $\operatorname{var}[f_*] = k(x_*,x_*) - k_*^T(K+\sigma_n^2I)^{-1}k_*$ (Equation 2.26 from GPML) Basic ...
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Gaussian Kernel to Continuous Spaces

How can we adapt the Gaussian Kernel to discontinuous spaces,such as that of strings over a fi nite alphabet, for which we already have a kernel (say K(s, t))??
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Efficient, stable inverse of a patterned covariance matrix for gridded data

I have computed a stationary covariance matrix defined for data on a grid. The data y are regularly spaced in 3D, lexicographically ordered in the covariance matrix, and I'm using a using a square ...
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36 views

Baysian Models Test Set performance metric

I am wondering which kind of test set performance metric is used to access the predictive performance of Bayesian models. All I can find in the literature are measures that are suitable for model ...
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29 views

Stationary covariance function three times continuously differentiable with finit support?

Suppose we have an Euclidiean space $\mathbb{R}^p$ and Gaussian process with covariance function of the form $k(x_1, x_2) = k(\|x_1 - x_2\|), x_1, x_2 \in \mathbb{R}^p$. I am looking for a covariance ...
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How to show that any Gaussian time-series is linear one?

In this paper I saw the following statement: If the time series is Gaussian (i.e., normally distributed) then the best linear forecast is in fact the best of all possible forecasts: No ...
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Jointed distribution of normal random variables

Given two models: $Y_i = \alpha + \beta x_i + Z_i$, $i = 1, 2, 3, \dots , n$ $Y_i = A + B x_i + Z_i$, $i = 1, 2, 3, \dots , n$ where, $\alpha$, $\beta$, $x_i$ are fixed and $A$ , $B$, $Z_i$'s are ...
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$L_1$ distance between two Gaussian processes

In Brown's famous paper (1996), the $L_1$ norm between two Gaussian processes defined on time domain $[0,1]$ $$dY_t = f(t)dt + \sigma(t)dB_t\quad\text{and}\quad dZ_t = g(t)dt + \sigma(t)dB_t$$ is ...