First matlab codes for statistics

statistics/code/quartiles.m

@@ -0,0 +1,25 @@
+% generate data:
+x = randn( 1, 100000 );
+
+% histogram:
+[h,b] = hist( x, 100 );
+% normalize:
+bs = b(2)-b(1);
+h = h/sum(h)/bs;
+
+% plot:
+bar( b, h );
+xlabel( 'x' );
+
+% median, quartile:
+xs = sort( x )
+q = [ xs(length(xs)/4), xs(length(xs)/2), xs(3*length(xs)/4) ];
+%q = quantile( x, [0.25, 0.5, 0.75 ] );
+
+% plot:
+bar( b(b<q(1)), h(b<q(1)), 'FaceColor', [0.5 0 0.5] );
+hold on;
+bar( b((b>=q(1)) & (b<q(2))), h((b>=q(1)) & (b<q(2))), 'FaceColor', [0.9 0 0] );
+bar( b((b>=q(2)) & (b<q(3))), h((b>=q(2)) & (b<q(3))), 'FaceColor', [0 0 0.9] );
+bar( b(b>=q(3)), h(b>=q(3)), 'FaceColor', [0.5 0 0.5] );
+hold off;

statistics/code/randomwalk.m

@@ -0,0 +1,6 @@
+function x = randomwalk(n,p)
+    r = rand(n,1);
+    r(r<p) = -1.0;
+    r(r>=p) = +1.0;
+    x = cumsum(r);
+end

statistics/code/randomwalkstatistics.m

@@ -0,0 +1,25 @@
+p = 0.5;
+nsteps = 100;
+nwalks = 1000;
+
+y = zeros( nwalks, nsteps/10 );
+for k = 1:length( y )
+    x = randomwalk( nsteps, p );
+    for j = 1:nsteps/10
+        y(k,j) = x((j-1)*10+1);
+    end
+    %plot( x )
+    %pause( 1 )
+    if rem(k,100) == 0
+        %[h1,b1] = hist( y(1:k,1), [-50:2:50] );
+        %[h2,b2] = hist( y(1:k,2), [-50:2:50] );
+        %bar( b1, h1, 1.0, 'b' );
+        %hold on;
+        %bar( b2, h2, 'FaceColor', 'r' );
+        %hold off;
+        sdev = var( y(1:k,:), 1 );
+        plot( sdev )
+        pause( 1.0 );
+    end
+end
+

statistics/lecture/descriptivestatistics.tex

@@ -89,108 +89,190 @@
   \tableofcontents
 \end{frame}
 
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Descriptive statistics}
 
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-\subsection{types of data}
+\subsection{Statistics of ratio data}
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
+%-------------------------------------------------------------
 \begin{frame}
-  \frametitle{data scales}
+  \frametitle{Statistics of ratio data}
-  \framesubtitle{What data types are distinguished in statistics?}
-  \Large
-  {\bf Why are data types important?}
-  \pause
   \begin{itemize}
-  \item selection of statistics 
+  \item Location, central tendency
-  \item selection of plots
+    \begin{itemize}
-  \item selection of correct tests
+    \item arithmetic mean
+    \item median
+    \item mode
+    \end{itemize}
+
+  \item Spread, dispersion
+    \begin{itemize}
+    \item variance
+    \item standard deviation
+    \item interquartile range
+    \item coefficient of variation
+    \item minimum, maximum
+    \end{itemize}
+
+  \item Shape
+    \begin{itemize}
+    \item skewnees
+    \item kurtosis
+    \end{itemize}
+
+  \item Dependence
+    \begin{itemize}
+    \item Pearson correlation coefficient
+    \item Spearman's rank correlation coefficient
+    \end{itemize}
+
   \end{itemize}
 \end{frame}
-%-------------------------------------------------------------
 
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\subsection{Data types}
+
+%-------------------------------------------------------------
 \begin{frame}
-  \frametitle{data scales}
+  \frametitle{Data types: nominal scale}
-  \framesubtitle{nominal/categorial scale}
   \begin{itemize}
-  \item properties like cell type, experimental group (i.e. treatment
+  \item Binary
-    1, treatment 2, control)
+    \begin{itemize}
-  \item each observation/sample is put into one category
+    \item ``yes/no'',
-  \item there is no reasonable order among the categories
+    \item ``true/false'',
-  \item  example: [rods, cones] vs. [cones, rods]
+    \item ``success/failure'', etc.
+    \end{itemize}
+  \item Categorial
+    \begin{itemize}
+    \item cell type (``rod/cone/horizontal cell/bipolar cell/ganglion cell''),
+    \item blood type (``A/B/AB/0''),
+    \item parts of speech (``noun/veerb/preposition/article/...''),
+    \item taxonomic groups (``Coleoptera/Lepidoptera/Diptera/Hymenoptera''), etc.
+    \end{itemize}
+  \item Each observation/measurement/sample is put into one category
+  \item There is no reasonable order among the categories.\\
+    example: [rods, cones] vs. [cones, rods]
+  \pause
+  \item Statistics: mode, i.e. the most common item
   \end{itemize}
 \end{frame}
-%-------------------------------------------------------------
 
+%-------------------------------------------------------------
 \begin{frame}
-  \frametitle{data scales}
+  \frametitle{Data types: ordinal scale}
-  \framesubtitle{ordinal scale}
   \begin{itemize}
-  \item like nominal scale, but there is an order
+  \item Like nominal scale, but with an order
-  \item {\bf but:} there is no reasonable measure of {\em distance}
+  \item Examples: ranks, ratings
+    \begin{itemize}
+    \item ``bad/ok/good'',
+    \item ``cold/warm/hot'',
+    \item ``young/old'', etc.
+    \end{itemize}
+  \item {\bf But:} there is no reasonable measure of {\em distance}
     between the classes
-  \item examples: ranks, ratings
+  \pause
+  \item Statistics: mode, median
   \end{itemize}
 \end{frame}
-%-------------------------------------------------------------
 
+%-------------------------------------------------------------
 \begin{frame}
-  \frametitle{data scales}
+  \frametitle{Data types: interval scale}
-  \framesubtitle{interval scale}
   \begin{itemize}
-  \item quantitative/metric values
+  \item Quantitative/metric values
-  \item reasonable measure of distance between values but no absolute zero
+  \item Reasonable measure of distance between values, but no absolute zero
-  \item examples: temperature in $^\circ$C
+  \item Examples: 
+    \begin{itemize}
+    \item Temperature in $^\circ$C ($20^\circ$C is not twice as hot as $10^\circ$C)
+    \item Direction measured in degrees from magnetic or true north
+    \end{itemize}
+  \pause
+  \item Statistics:
+    \begin{itemize}
+    \item Central tendency: mode, median, arithmetic mean
+    \item Dispersion: range, standard deviation
+    \end{itemize}
   \end{itemize}
 \end{frame}
-%-------------------------------------------------------------
 
+%-------------------------------------------------------------
 \begin{frame}
-  \frametitle{data scales}
+  \frametitle{Data types: absolute/ratio scale}
-  \framesubtitle{absolut/ratio scale}
   \begin{itemize}
-  \item like interval scale but with absolute zero
+  \item Like interval scale, but with absolute origin/zero
-  \item example: temperature in $^\circ$K
+  \item Examples: 
-  \end{itemize}
+    \begin{itemize}
+    \item Temperature in $^\circ$K
+    \item Length, mass, duration, electric charge, ...
+    \item Plane angle, etc.
+    \item Count (e.g. number of spikes in response to a stimulus)
+    \end{itemize}
   \pause
-  %\begin{emphasize}{relationsships between scales}
+  \item Statistics:
     \begin{itemize}
-    \item scales exhibit increasing information content from nominal
+    \item Central tendency: mode, median, arithmetic, geometric, harmonic mean
-      to absolute
+    \item Dispersion: range, standard deviation
-    \item conversion  ,,downwards'' always possible
+    \item Coefficient of variation (ratio standard deviation/mean)
+    \item All other statistical measures
     \end{itemize}
-  %\end{emphasize}
+  \end{itemize}
 \end{frame}
 
 %-------------------------------------------------------------
 \begin{frame}
-  \frametitle{examples from neuroscience and psychology}
+  \frametitle{Data types}
   \begin{itemize}
-  \item {\bf nominal:}\pause 
+  \item Data type selects
     \begin{itemize}
-    \item treatment group
+    \item statistics 
-    \item stimulus class
+    \item type of plots (bar graph versus x-y plot)
-    \item cell type
+    \item correct tests
+    \end{itemize}
+  \item Scales exhibit increasing information content from nominal
+    to absolute.\\
+    Conversion  ,,downwards'' is always possible
+  \item For example: size measured in meter (ratio scale) $\rightarrow$
+    categories ``small/medium/large'' (ordinal scale)
+  \end{itemize}
+\end{frame}
+
+%-------------------------------------------------------------
+\begin{frame}
+  \frametitle{Examples from neuroscience}
+  \begin{itemize}
+
+  \item {\bf absolute:}\pause
+    \begin{itemize}
+    \item size of neuron/brain
+    \item length of axon
+    \item ion concentration
+    \item membrane potential
+    \item firing rate
+    \end{itemize}
+
+  \item {\bf interval:}\pause
+    \begin{itemize}
+    \item edge orientation
     \end{itemize}
 
   \item {\bf ordinal:} \pause 
     \begin{itemize}
+    \item stages of a disease
     \item ratings
-    \item clinical stages of a disease
-    \item states of an ion channel
     \end{itemize}
-  \item {\bf Absolut-/Ratioskala:}\pause
+
+  \item {\bf nominal:}\pause 
     \begin{itemize}
-    \item firing rate
+    \item cell type
-    \item membrane potential
+    \item odor
-    \item ion concentration
+    \item states of an ion channel
     \end{itemize}
+
   \end{itemize}
 \end{frame}
 
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\end{document}
-\subsection{Real-valued data}