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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
so-called noisy \textit{learning set}
whic is probably noisy $D = \left(X, Y\right) = \left\{\left(\mathbf{x}_i, y_i =
f(\mathbf{x}_i)\right), f(\mathbf{x}_i)\right) + \varepsilon_i, \mathbf{x}_i\in\mathcal{X}, \varepsilon_i\sim\mathcal{N}(0,\sigma^2), i = \overline{1,
N}, \mathbf{x}\in\mathcal{X}\right\}$. N}\right\}$. 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}