International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056 Volume: 11 Issue: 06 | Jun 2024
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p-ISSN: 2395-0072
A Warped Gaussian Process Tutorial Using a Simple Example Emma Foley1,2 1Graduate Research Assistant, Bredesen Center, University of Tennessee, TN, United States 2Grid Communication and Security, Oak Ridge National Laboratory, TN, United States
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Abstract –This paper describes an implementation of
where µ is the mean and Σ is the covariance matrix. Then there is a corresponding collection of function values that also follows a Gaussian distribution:
warped Gaussian process using a simple example. The results show the advantages of this method over a traditional Gaussian process. The flexibility of the warped Gaussian process maintains interpretability and mathematical foundation while accounting for nonGaussian and/or noisy data. These features provide a basis for using this method in power grid applications, where this method is underrepresented in the literature. As exemplified by measurements taken in the Autonomous Intelligence Measurement Sensors and Systems (AIMS) project, data collected from sensors to support grid operations is nonGaussian and noisy in nature. Warped Gaussian process is a flexible method suitable for analyzing multiple different types of power grid data.
f(x) = [f(x1) f(x2) … f(xn) ]T By definition, a GP is a collection of random variables which have a joint Gaussian distribution [4]. Thus, f(x) is a Gaussian process and can be further defined as follows: f(x) ∼ GP(m(x),k(x,x′)) m(x) = E[f(x)] k(x,x′) = E[(f(x)−m(x))(f(x′)−m(x′))] +δijσ2 ϵ ∼ N(0, σ2) where m(x) is the mean function, k(x,x′) is the covariance function (also known as a kernel), δ is the Kronecker delta, and ϵ is the independent and identically distributed Gaussian noise.
Key Words: Gaussian, warping function, tutorial, sensing, smart grid
1.INTRODUCTION
2.2 Implementing the Warping Function
A Warped Gaussian Process (WGP) is a variation of the traditional Gaussian Process (GP). This method introduces a warping function that can account for non-Gaussian data, non-Gaussian noise, and capture more of the uncertainty in the data than the traditional Gaussian Process [1]. This tutorial will walk through the mathematical background and present a straightforward example using Python [2].
In order to warp the observation space to latent space, let z be a vector of latent target values modelled by a GP. The nonlinear monotonic function g maps all the entries from the actual target space to the latent space: zn = g(f(x); Ψ)
2. WARPED GAUSSIAN PROCESS
where Ψ is a parameter. The function g can be any function but must be monotonic to maintain the validity of the probability distribution over f(x). Once the target values have been warped into latent space, GP progresses as normal using z. The conditional distribution determines the predictive equations accounting for noise:
2.1 Define the Gaussian Process In a traditional GP, we define a prior distribution over multiple functions so that we can infer a posterior distribution that better represents the actual underlying distribution. Once the posterior is determined, it can be used to calculate mean, variance, etc. GPs are particularly useful from a function space perspective [3], which is how we will proceed in this tutorial. First, define X to be a vector of random variables that follow a Gaussian distribution, then [
f∗|Z,f + ϵ,Z∗ ∼ N(f∗,cov(f∗)) ̅ ≜E[Z,f + ϵ,Z∗] = K(Z∗,Z)[K(Z,Z)+σ2I]−1[f + ϵ] cov(f∗) = K(Z∗,Z∗)−K(Z∗,Z)[K(Z,Z)+σ2I]−1K(Z,Z∗) where the star subscript denotes data from the test subset. The marginal likelihood is defined below:
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