155 lines
4.2 KiB
TeX
155 lines
4.2 KiB
TeX
\documentclass[12pt]{book}
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\input{../../header}
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\lstset{inputpath=../code}
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\graphicspath{{figures/}}
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\typein[\pagenumber]{Number of first page}
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\typein[\chapternumber]{Chapter number}
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\setcounter{page}{\pagenumber}
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\setcounter{chapter}{\chapternumber}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\include{statistics}
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\end{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Statistics}
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What is "a statistic"? % dt. Sch\"atzfunktion
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\begin{definition}[statistic]
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A statistic (singular) is a single measure of some attribute of a
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sample (e.g., its arithmetic mean value). It is calculated by
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applying a function (statistical algorithm) to the values of the
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items of the sample, which are known together as a set of data.
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\source{http://en.wikipedia.org/wiki/Statistic}
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\end{definition}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Data types}
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\subsection{Nominal scale}
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\begin{itemize}
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\item Binary
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\begin{itemize}
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\item ``yes/no'',
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\item ``true/false'',
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\item ``success/failure'', etc.
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\end{itemize}
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\item Categorial
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\begin{itemize}
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\item cell type (``rod/cone/horizontal cell/bipolar cell/ganglion cell''),
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\item blood type (``A/B/AB/0''),
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\item parts of speech (``noun/veerb/preposition/article/...''),
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\item taxonomic groups (``Coleoptera/Lepidoptera/Diptera/Hymenoptera''), etc.
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\end{itemize}
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\item Each observation/measurement/sample is put into one category
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\item There is no reasonable order among the categories.\\
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example: [rods, cones] vs. [cones, rods]
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\item Statistics: mode, i.e. the most common item
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\end{itemize}
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\subsection{Ordinal scale}
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\begin{itemize}
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\item Like nominal scale, but with an order
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\item Examples: ranks, ratings
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\begin{itemize}
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\item ``bad/ok/good'',
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\item ``cold/warm/hot'',
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\item ``young/old'', etc.
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\end{itemize}
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\item {\bf But:} there is no reasonable measure of {\em distance}
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between the classes
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\item Statistics: mode, median
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\end{itemize}
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\subsection{Interval scale}
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\begin{itemize}
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\item Quantitative/metric values
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\item Reasonable measure of distance between values, but no absolute zero
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\item Examples:
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\begin{itemize}
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\item Temperature in $^\circ$C ($20^\circ$C is not twice as hot as $10^\circ$C)
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\item Direction measured in degrees from magnetic or true north
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\end{itemize}
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\item Statistics:
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\begin{itemize}
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\item Central tendency: mode, median, arithmetic mean
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\item Dispersion: range, standard deviation
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\end{itemize}
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\end{itemize}
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\subsection{Absolute/ratio scale}
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\begin{itemize}
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\item Like interval scale, but with absolute origin/zero
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\item Examples:
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\begin{itemize}
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\item Temperature in $^\circ$K
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\item Length, mass, duration, electric charge, ...
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\item Plane angle, etc.
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\item Count (e.g. number of spikes in response to a stimulus)
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\end{itemize}
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\item Statistics:
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\begin{itemize}
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\item Central tendency: mode, median, arithmetic, geometric, harmonic mean
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\item Dispersion: range, standard deviation
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\item Coefficient of variation (ratio standard deviation/mean)
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\item All other statistical measures
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\end{itemize}
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\end{itemize}
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\subsection{Data types}
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\begin{itemize}
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\item Data type selects
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\begin{itemize}
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\item statistics
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\item type of plots (bar graph versus x-y plot)
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\item correct tests
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\end{itemize}
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\item Scales exhibit increasing information content from nominal
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to absolute.\\
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Conversion ,,downwards'' is always possible
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\item For example: size measured in meter (ratio scale) $\rightarrow$
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categories ``small/medium/large'' (ordinal scale)
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\end{itemize}
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\subsection{Examples from neuroscience}
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\begin{itemize}
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\item {\bf absolute:}
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\begin{itemize}
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\item size of neuron/brain
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\item length of axon
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\item ion concentration
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\item membrane potential
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\item firing rate
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\end{itemize}
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\item {\bf interval:}
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\begin{itemize}
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\item edge orientation
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\end{itemize}
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\item {\bf ordinal:}
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\begin{itemize}
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\item stages of a disease
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\item ratings
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\end{itemize}
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\item {\bf nominal:}
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\begin{itemize}
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\item cell type
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\item odor
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\item states of an ion channel
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\end{itemize}
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\end{itemize}
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