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solution for regression exercises

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Jan Benda
2018-01-08 17:52:48 +01:00
parent 9265f75d12
commit 9c0559ab03
10 changed files with 243 additions and 36 deletions
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8
statistics/lecture/statistics.tex
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@@ -348,14 +348,14 @@ probability density functions like the one of the normal distribution
\subsection{Kernel densities}
A problem of using histograms for estimating probability densities is
that the have hard bin edges. Depending on where the bin edges are placed
that they have hard bin edges. Depending on where the bin edges are placed
a data value falls in one or the other bin.
\begin{figure}[t]
\includegraphics[width=1\textwidth]{kerneldensity}
\titlecaption{\label{kerneldensityfig} Kernel densities.}{Left: The
histogram-based estimation of the probability density is dependent
also on the position of the bins. In the bottom plot the bins have
on the position of the bins. In the bottom plot the bins have
bin shifted by half a bin width (here $\Delta x=0.4$) and as a
result details of the probability density look different. Look,
for example at the height of the largest bin. Right: In contrast,
@@ -366,7 +366,7 @@ a data value falls in one or the other bin.
To avoid this problem one can use so called \enterm {kernel densities}
for estimating probability densities from data. Here every data point
is replaced by a kernel (a function with integral one, like for
example the Gaussian function) that is moved exactly to the position
example the Gaussian) that is moved exactly to the position
indicated by the data value. Then all the kernels of all the data
values are summed up, the sum is divided by the number of data values,
and we get an estimate of the probability density.
@@ -417,7 +417,7 @@ and percentiles can be determined from the inverse cumulative function.
100 data values drawn from a normal distribution (red) in
comparison to the true cumulative distribution function computed
by numerically integrating the normal distribution function
(blue). From the cumulative distribution function one can read of
(blue). From the cumulative distribution function one can read off
the probabilities of getting values smaller than a given value
(here: $P(x \ge -1) \approx 0.15$). From the inverse cumulative
distribution the position of percentiles can be computed (here: