The ?Actual? Nadaraya Watson Model
We can do kernel regression starting with kernel density estimation.
where latex($f(\bar{x},t)$) is the component density function, and there is one such component centered on each data point.
The regression function, latex($y(\bar{x})$), which is the conditional average of the target variable conditioned on the input variable:
![\Large
\begin{eqnarray}
y(\bar{x}) = \mathbb{E}
\left[t|\mathbf{x}\right] = \int_{\infty}^{\infty}tp(t|\mathbf{x})dt \\
= \frac{\Sigma_{n}\int{tf(\mathbf{x}-\mathbf{x}_{n}, t-t_{n})dt}}{\Sigma_{m}\int{tf(\mathbf{x}-\mathbf{x}_{m}, t-t_{m})dt}}
\end{eqnarray}](/wiki/Classes/BMTRY790/KernelMethods/015_Nadaraya_Watson_Model?action=AttachFile&do=get&target=latex_0d221fb825f4ccb3727721f25465f78f156577b7_p1.png "\Large
\begin{eqnarray}
y(\bar{x}) = \mathbb{E}
\left[t|\mathbf{x}\right] = \int_{\infty}^{\infty}tp(t|\mathbf{x})dt \\
= \frac{\Sigma_{n}\int{tf(\mathbf{x}-\mathbf{x}_{n}, t-t_{n})dt}}{\Sigma_{m}\int{tf(\mathbf{x}-\mathbf{x}_{m}, t-t_{m})dt}}
\end{eqnarray}")
We then assume that the component density functions have zero mean so that:
