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Last time I talked about the beautiful linear Algebra interpretation of
Least Squares which adds a wonderful geometric interpretation to the
analytical footwork we had to do before. This time I will borrow from a
video I saw on Khan Academy a long time ago. I connects some of the
elemental principles of statistics with regression.
The basic idea is similar to the analytical approach. But this time, we will
only try to fit a line to $$M$$ $$x$$/$$y$$ pairs. So, we
have a model of the form
\begin{equation*}y(x) = m x + c.\end{equation*}
The squared errors between model and data is then
\begin{align*}S := \sum_i \epsilon_i^2 &= \sum_i (m x_i + c - y_i)^2 \\
&= \sum_i m^2 x_i^2 + 2 c m x_i + c^2 - 2 m x_i y_i - 2 c y_i + y_i^2\end{align*}
Of course, we search for the global minima of this solution with respect to
the parameters $$m$$ and $$c$$. So let's find the derivations:
\begin{align*}0 \stackrel{!}{=} \frac{\partial S}{\partial m} &= \sum_i 2 m x_i^2 + 2 c x_i - 2 x_i y_i \\
0 \stackrel{!}{=} \frac{\partial S}{\partial c} &= \sum_i 2 m x_i + 2 c - 2 y_i\end{align*}
We can conveniently drop the 2 that appears in front of all terms.
We will now rewrite these equations by using the definition of the sample mean:
\begin{align*}0 \stackrel{!}{=} m M \overline{x^2} + M c \overline{x} - M \overline{x y} \\
0 \stackrel{!}{=} m M \overline{x} + M c - M \overline{y}\end{align*}
Let's loose the $$M$$ s and solve for $$m$$ and $$c$$.
\begin{align*}m &= \frac{\overline{x}\,\overline{y}- \overline{x y}}{\overline{x}^2 - \overline{x^2}} \\
c &= \overline{y} - m \overline{x}\end{align*}
If you look closely at the term for the slope $$m$$ you see that this is
actually just the covariance of $$x$$ and $$y$$ divided by the variance of $$x$$.
I find this very intriguing.
I just release UltiSnips 2.0 together with a short screencast to run you
through the new features.
2.0 is a killer-feature release for me: Having normal mode editing and $VISUAL
support adds a lot of value to UltiSnips for me.
UltiSnips now has all the features I initially expected from a snippet plugin
when I started out writing it. It was not an easy road getting here but I am
proud what we have achieved. I consider it now mostly feature complete - I
will not object to good ideas for new features that enhance the user experience
but we will try as best as we can to avoid feature creep. UltiSnips
wants to be only one thing - the ultimate solution for snippets in Vim.
