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fixed some index entries

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Jan Benda 2019-12-06 19:29:20 +01:00
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9
likelihood/lecture/likelihood.tex
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@ -6,8 +6,9 @@
A common problem in statistics is to estimate from a probability
distribution one or more parameters $\theta$ that best describe the
data $x_1, x_2, \ldots x_n$. \enterm{Maximum likelihood estimators}
(\enterm[mle|see{Maximum likelihood estimators}]{mle},
data $x_1, x_2, \ldots x_n$. \enterm[maximum likelihood
estimator]{Maximum likelihood estimators} (\enterm[mle|see{maximum
likelihood estimator}]{mle},
\determ{Maximum-Likelihood-Sch\"atzer}) choose the parameters such
that they maximize the likelihood of the data $x_1, x_2, \ldots x_n$
to originate from the distribution.
@ -228,7 +229,7 @@ maximized respectively.
\begin{figure}[t]
\includegraphics[width=1\textwidth]{mlepropline}
\titlecaption{\label{mleproplinefig} Maximum likelihood estimation
of the slope of line through the origin.}{The data (blue and
of the slope of a line through the origin.}{The data (blue and
left histogram) originate from a straight line $y=mx$ trough the origin
(red). The maximum-likelihood estimation of the slope $m$ of the
regression line (orange), \eqnref{mleslope}, is close to the true
@ -257,6 +258,7 @@ respect to $\theta$ and equate it to zero:
This is an analytical expression for the estimation of the slope
$\theta$ of the regression line (\figref{mleproplinefig}).
\subsection{Linear and non-linear fits}
A gradient descent, as we have done in the previous chapter, is not
necessary for fitting the slope of a straight line, because the slope
can be directly computed via \eqnref{mleslope}. More generally, this
@ -275,6 +277,7 @@ exponential decay
Such cases require numerical solutions for the optimization of the
cost function, e.g. the gradient descent \matlabfun{lsqcurvefit()}.
\subsection{Relation between slope and correlation coefficient}
Let us have a closer look on \eqnref{mleslope}. If the standard
deviation of the data $\sigma_i$ is the same for each data point,
i.e. $\sigma_i = \sigma_j \; \forall \; i, j$, the standard deviation drops

2
plotting/lecture/plotting.tex
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@ -93,7 +93,7 @@ number of datasets.
\subsection{Simple plotting}
Creating a simple line-plot is rather easy. Assuming there exists a
variable \varcode{y} in the \codeterm{Workspace} that contains the
variable \varcode{y} in the \codeterm{workspace} that contains the
measurement data it is enough to call \code[plot()]{plot(y)}. At the
first call of this function a new \codeterm{figure} will be opened and
the data will be plotted with as a line plot. If you repeatedly call

6
pointprocesses/lecture/pointprocesses.tex
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@ -3,7 +3,7 @@
\chapter{Spiketrain analysis}
\exercisechapter{Spiketrain analysis}
\enterm[Action potentials]{action potentials} (\enterm{spikes}) are
\enterm[action potential]{Action potentials} (\enterm{spikes}) are
the carriers of information in the nervous system. Thereby it is the
time at which the spikes are generated that is of importance for
information transmission. The waveform of the action potential is
@ -110,9 +110,9 @@ describing the statistics of stochastic real-valued variables:
\frac{1}{n}\sum\limits_{i=1}^n T_i$.
\item Standard deviation of the interspike intervals: $\sigma_{ISI} = \sqrt{\langle (T - \langle T
\rangle)^2 \rangle}$\vspace{1ex}
\item \enterm{Coefficient of variation}: $CV_{ISI} =
\item \enterm[coefficient of variation]{Coefficient of variation}: $CV_{ISI} =
\frac{\sigma_{ISI}}{\mu_{ISI}}$.
\item \enterm{Diffusion coefficient}: $D_{ISI} =
\item \enterm[diffusion coefficient]{Diffusion coefficient}: $D_{ISI} =
\frac{\sigma_{ISI}^2}{2\mu_{ISI}^3}$.
\end{itemize}

2
programmingstyle/lecture/programmingstyle.tex
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@ -356,7 +356,7 @@ should take care when defining nested functions.
\section{Specifics when using scripts}
A similar problem as with nested function arises when using scripts
(instead of functions). All variables that are defined within a script
become available in the global \codeterm{Workspace}. There is the risk
become available in the global \codeterm{workspace}. There is the risk
of name conflicts, that is, a called sub-script redefines or uses the
same variable name and may \emph{silently} change its content. The
user will not be notified about this change and the calling script may

4
regression/lecture/regression.tex
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@ -76,8 +76,8 @@ large deviations.
\begin{exercise}{meanSquareError.m}{}\label{mseexercise}%
Implement a function \code{meanSquareError()}, that calculates the
\emph{mean square distance} bewteen a vector of observations ($y$)
and respective predictions ($y^{est}$). \pagebreak[4]
\emph{mean square distance} between a vector of observations ($y$)
and respective predictions ($y^{est}$).
\end{exercise}

34
statistics/lecture/statistics.tex
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@ -147,11 +147,11 @@ data are smaller than the 3$^{\rm rd}$ quartile.
% from a normal distribution.}
% \end{figure}
\enterm{Box-whisker plots} are commonly used to visualize and compare
the distribution of unimodal data. A box is drawn around the median
that extends from the 1$^{\rm st}$ to the 3$^{\rm rd}$ quartile. The
whiskers mark the minimum and maximum value of the data set
(\figref{displayunivariatedatafig} (3)).
\enterm[box-whisker plots]{Box-whisker plots} are commonly used to
visualize and compare the distribution of unimodal data. A box is
drawn around the median that extends from the 1$^{\rm st}$ to the
3$^{\rm rd}$ quartile. The whiskers mark the minimum and maximum value
of the data set (\figref{displayunivariatedatafig} (3)).
\begin{exercise}{univariatedata.m}{}
Generate 40 normally distributed random numbers with a mean of 2 and
@ -175,7 +175,7 @@ The distribution of values in a data set is estimated by histograms
\subsection{Histograms}
\enterm[Histogram]{Histograms} count the frequency $n_i$ of
\enterm[histogram]{Histograms} count the frequency $n_i$ of
$N=\sum_{i=1}^M n_i$ measurements in each of $M$ bins $i$
(\figref{diehistogramsfig} left). The bins tile the data range
usually into intervals of the same size. The width of the bins is
@ -193,8 +193,9 @@ categories $i$ is the \enterm{histogram}, or the \enterm{frequency
with the expected theoretical distribution of $P=1/6$.}
\end{figure}
Histograms are often used to estimate the \enterm{probability
distribution} of the data values.
Histograms are often used to estimate the
\enterm[probability!distribution]{probability distribution} of the
data values.
\subsection{Probabilities}
In the frequentist interpretation of probability, the probability of
@ -254,12 +255,13 @@ In the limit to very small ranges $\Delta x$ the probability of
getting a measurement between $x_0$ and $x_0+\Delta x$ scales down to
zero with $\Delta x$:
\[ P(x_0<x<x_0+\Delta x) \approx p(x_0) \cdot \Delta x \; . \]
In here the quantity $p(x_00)$ is a so called \enterm{probability
density} that is larger than zero and that describes the
distribution of the data values. The probability density is not a
unitless probability with values between 0 and 1, but a number that
takes on any positive real number and has as a unit the inverse of the
unit of the data values --- hence the name ``density''.
In here the quantity $p(x_00)$ is a so called
\enterm[probability!density]{probability density} that is larger than
zero and that describes the distribution of the data values. The
probability density is not a unitless probability with values between
0 and 1, but a number that takes on any positive real number and has
as a unit the inverse of the unit of the data values --- hence the
name ``density''.
\begin{figure}[t]
\includegraphics[width=1\textwidth]{pdfprobabilities}
@ -280,14 +282,14 @@ the probability density over the whole real axis must be one:
\end{equation}
The function $p(x)$, that assigns to every $x$ a probability density,
is called \enterm{probability density function},
is called \enterm[probability!density function]{probability density function},
\enterm[pdf|see{probability density function}]{pdf}, or just
\enterm[density|see{probability density function}]{density}
(\determ{Wahrscheinlichkeitsdichtefunktion}). The well known
\enterm{normal distribution} (\determ{Normalverteilung}) is an example of a
probability density function
\[ p_g(x) = \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} \]
--- the \enterm{Guassian distribution}
--- the \enterm{Gaussian distribution}
(\determ{Gau{\ss}sche-Glockenkurve}) with mean $\mu$ and standard
deviation $\sigma$.
The factor in front of the exponential function ensures the normalization to