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[regression] all equations are numbered

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Jan Benda 2019-12-11 18:46:23 +01:00
parent 14160e2f8d
commit a7e64a5c6d
2 changed files with 12 additions and 4 deletions
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5
regression/lecture/regression-chapter.tex
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@ -16,6 +16,11 @@
\include{regression} \include{regression}
\subsection{Notes}
\begin{itemize}
\item Fig 8.2 right: this should be a chi-squared distribution with one degree of freedom!
\end{itemize}
\subsection{Start with one-dimensional problem!} \subsection{Start with one-dimensional problem!}
\begin{itemize} \begin{itemize}
\item Just the root mean square as a function of the slope \item Just the root mean square as a function of the slope

11
regression/lecture/regression.tex
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@ -283,10 +283,11 @@ the partial derivatives using the difference quotient
(Box~\ref{differentialquotientbox}) for small steps $\Delta m$ and (Box~\ref{differentialquotientbox}) for small steps $\Delta m$ and
$\Delta b$. For example, the partial derivative with respect to $m$ $\Delta b$. For example, the partial derivative with respect to $m$
can be computed as can be computed as
\[\frac{\partial f_{cost}(m,b)}{\partial m} = \lim\limits_{\Delta m \to \begin{equation}
\frac{\partial f_{cost}(m,b)}{\partial m} = \lim\limits_{\Delta m \to
0} \frac{f_{cost}(m + \Delta m, b) - f_{cost}(m,b)}{\Delta m} 0} \frac{f_{cost}(m + \Delta m, b) - f_{cost}(m,b)}{\Delta m}
\approx \frac{f_{cost}(m + \Delta m, b) - f_{cost}(m,b)}{\Delta m} \; \approx \frac{f_{cost}(m + \Delta m, b) - f_{cost}(m,b)}{\Delta m} \; .
. \] \end{equation}
The length of the gradient indicates the steepness of the slope The length of the gradient indicates the steepness of the slope
(\figref{gradientquiverfig}). Since want to go down the hill, we (\figref{gradientquiverfig}). Since want to go down the hill, we
choose the opposite direction. choose the opposite direction.
@ -341,7 +342,9 @@ descent works as follows:
sufficiently close to zero (e.g. \varcode{norm(gradient) < 0.1}). sufficiently close to zero (e.g. \varcode{norm(gradient) < 0.1}).
\item \label{gradientstep} If the length of the gradient exceeds the \item \label{gradientstep} If the length of the gradient exceeds the
threshold we take a small step into the opposite direction: threshold we take a small step into the opposite direction:
\[p_{i+1} = p_i - \epsilon \cdot \nabla f_{cost}(m_i, b_i)\] \begin{equation}
p_{i+1} = p_i - \epsilon \cdot \nabla f_{cost}(m_i, b_i)
\end{equation}
where $\epsilon = 0.01$ is a factor linking the gradient to where $\epsilon = 0.01$ is a factor linking the gradient to
appropriate steps in the parameter space. appropriate steps in the parameter space.
\item Repeat steps \ref{computegradient} -- \ref{gradientstep}. \item Repeat steps \ref{computegradient} -- \ref{gradientstep}.