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Pavel Erofeev edited GPR.tex
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\section{Gaussian Processes Regression}
\label{sec:GaussianProcessesRegression}
In multidimensional regression problem we assume that $f: \mathcal{X} \rightarrow \mathbb{R}, \mathcal{X}\subset\mathbb{R}^m$ is an unknow dependency function. We are given a noisy \textit{learning set} $D = \left\{\left(\mathbf{x}_i, y_i\right)\right\}$, where $y_i = f(\mathbf{x}_i) + \varepsilon_i, \mathbf{x}_i\in\mathcal{X},
\varepsilon_i\sim\mathcal{N}(0,\sigma^2), i = \overline{1, N}$. \varepsilon_i\sim\mathcal{N}(0,\sigma^2)$ for $i=1,\dots,N$ sampled independently and identically distributed (i.i.d.). The problem is to construct function $\hat{f}$ from specific class $(\vecX) = \hat{f}(\vecX | D_{learn})$ для исходной зависимости $\vecY = f(\vecX)$ по обучающей выборке $D_{learn}$.
Если для всех $\vecX \in \XX$ (не только для $\vecX \in D_{learn}$) имеет место примерное равенство
\begin{equation}\label{eq:good_approx}