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changeset 43:3f643e9fead0

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Added Andrew's bits, added to fig 2, fixed some spellings, added some section crossrefs.
author samer
date Thu, 15 Mar 2012 15:08:46 +0000
parents 1161caf0bdda
children 244b74fb707d
files draft.pdf draft.tex
diffstat 2 files changed, 86 insertions(+), 25 deletions(-) [+]
line wrap: on
line diff
Binary file draft.pdf has changed
--- a/draft.tex	Thu Mar 15 12:14:59 2012 +0000
+++ b/draft.tex	Thu Mar 15 15:08:46 2012 +0000
@@ -416,6 +416,7 @@
 	be in competition with other sensory streams.
 
 	\subsection{Information measures for stationary random processes}
+	\label{s:process-info}
 
 
  	\begin{fig}{predinfo-bg}
@@ -433,7 +434,34 @@
 		\newcommand\longblob{\ovoid{\axis}}
 		\newcommand\shortblob{\ovoid{1.75em}}
 		\begin{tabular}{c@{\colsep}c}
-			\subfig{(a) excess entropy}{%
+			\subfig{(a) multi-information and entropy rates}{%
+				\begin{tikzpicture}%[baseline=-1em]
+					\newcommand\rc{1.75em}
+					\newcommand\throw{2.5em}
+					\coordinate (p1) at (180:1.5em);
+					\coordinate (p2) at (0:0.3em);
+					\newcommand\bound{(-7em,-2.6em) rectangle (7em,3.0em)}
+					\newcommand\present{(p2) circle (\rc)}
+					\newcommand\thepast{(p1) ++(-\throw,0) \ovoid{\throw}}
+					\newcommand\fillclipped[2]{%
+						\begin{scope}[even odd rule]
+							\foreach \thing in {#2} {\clip \thing;}
+							\fill[black!#1] \bound;
+						\end{scope}%
+					}%
+					\fillclipped{30}{\present,\bound \thepast}
+					\fillclipped{15}{\present,\bound \thepast}
+					\fillclipped{45}{\present,\thepast}
+					\draw \thepast;
+					\draw \present;
+					\node at (barycentric cs:p2=1,p1=-0.3) {$h_\mu$};
+					\node at (barycentric cs:p2=1,p1=1) [shape=rectangle,fill=black!45,inner sep=1pt]{$\rho_\mu$};
+					\path (p2) +(90:3em) node {$X_0$};
+					\path (p1) +(-3em,0em) node  {\shortstack{infinite\\past}};
+					\path (p1) +(-4em,\rad) node [anchor=south] {$\ldots,X_{-1}$};
+				\end{tikzpicture}}%
+			\\[1.25em]
+			\subfig{(b) excess entropy}{%
 				\newcommand\blob{\longblob}
 				\begin{tikzpicture}
 					\coordinate (p1) at (-\offs,0em);
@@ -451,7 +479,7 @@
 				\end{tikzpicture}%
 			}%
 			\\[1.25em]
-			\subfig{(b) predictive information rate $b_\mu$}{%
+			\subfig{(c) predictive information rate $b_\mu$}{%
 				\begin{tikzpicture}%[baseline=-1em]
 					\newcommand\rc{2.1em}
 					\newcommand\throw{2.5em}
@@ -468,6 +496,7 @@
 							\fill[black!#1] \bound;
 						\end{scope}%
 					}%
+					\fillclipped{80}{\future,\thepast}
 					\fillclipped{30}{\present,\future,\bound \thepast}
 					\fillclipped{15}{\present,\bound \future,\bound \thepast}
 					\draw \future;
@@ -494,7 +523,8 @@
 		In (b), the circle represents the `present'. Its total area is
 		$H(X_0)=\rho_\mu+r_\mu+b_\mu$, where $\rho_\mu$ is the multi-information
 		rate, $r_\mu$ is the residual entropy rate, and $b_\mu$ is the predictive
-		information rate. The entropy rate is $h_\mu = r_\mu+b_\mu$.
+		information rate. The entropy rate is $h_\mu = r_\mu+b_\mu$. The small dark
+		region  below $X_0$ in (c) is $\sigma_\mu = E-\rho_\mu$.
 		}
 	\end{fig}
 
@@ -534,6 +564,9 @@
 	\begin{equation}
 		E = I(\past{X}_t; X_t,\fut{X}_t).
 	\end{equation}
+	Both the excess entropy and the multi-information rate can be thought
+	of as measures of \emph{redundancy}, quantifying the extent to which 
+	the same information is to be found in all parts of the sequence.
 	
 
 	The \emph{predictive information rate} (or PIR) \cite{AbdallahPlumbley2009}  
@@ -549,7 +582,7 @@
 	as 
 	\begin{equation}
 %		\IXZ_t 
-		I(X_t;\fut{X}_t|\past{X}_t) = h_\mu - r_\mu,
+b_\mu = H(X_t|\past{X}_t) - H(X_t|\past{X}_t,\fut{X}_t) = h_\mu - r_\mu,
 %		\label{<++>}
 	\end{equation}
 %	If $X$ is stationary, then 
@@ -565,21 +598,10 @@
 	$\sigma_\mu$, the difference between the multi-information rate and the excess
 	entropy, as an interesting quantity that measures the predictive benefit of
 	model-building (that is, maintaining an internal state summarising past 
-	observations in order to make better predictions). They also identify
-	$w_\mu = \rho_\mu + b_{\mu}$, which they call the \emph{local exogenous
-	information} rate.
-
-  \begin{fig}{wundt}
-    \raisebox{-4em}{\colfig[0.43]{wundt}}
- %  {\ \shortstack{{\Large$\longrightarrow$}\\ {\scriptsize\emph{exposure}}}\ }
-    {\ {\large$\longrightarrow$}\ }
-    \raisebox{-4em}{\colfig[0.43]{wundt2}}
-    \caption{
-      The Wundt curve relating randomness/complexity with
-      perceived value. Repeated exposure sometimes results
-      in a move to the left along the curve \cite{Berlyne71}.
-    }
-  \end{fig}
+	observations in order to make better predictions). 
+%	They also identify
+%	$w_\mu = \rho_\mu + b_{\mu}$, which they call the \emph{local exogenous
+%	information} rate.
 
 
 	\subsection{First and higher order Markov chains}
@@ -622,6 +644,10 @@
 	the multi-information rate $\rho_\mu$ and the excess entropy $E$. These are identical
 	for first order Markov chains, but for order $N$ chains, $E$ can be up to $N$ times larger 
 	than $\rho_\mu$.
+
+	[Something about what kinds of Markov chain maximise $h_\mu$ (uncorrelated `white'
+	sequences, no temporal structure), $\rho_\mu$ and $E$ (periodic) and $b_\mu$. We return
+	this in \secrf{composition}.]
 		
 
 \section{Information Dynamics in Analysis}
@@ -728,8 +754,43 @@
 
 \subsection{Beat Tracking}
 
+A probabilistic method for drum tracking was presented by Robertson 
+\cite{Robertson11c}. The algorithm is used to synchronise a music 
+sequencer to a live drummer. The expected beat time of the sequencer is 
+represented by a click track, and the algorithm takes as input event 
+times for discrete kick and snare drum events relative to this click 
+track. These are obtained using dedicated microphones for each drum and 
+using a percussive onset detector (Puckette 1998). The drum tracker 
+continually updates distributions for tempo and phase on receiving a new 
+event time. We can thus quantify the information contributed of an event 
+by measuring the difference between the system's prior distribution and 
+the posterior distribution using the Kullback-Leiber divergence.
+
+Here, we have calculated the KL divergence and entropy for kick and 
+snare events in sixteen files. The analysis of information rates can be 
+considered \emph{subjective}, in that it measures how the drum tracker's 
+probability distributions change, and these are contingent upon the 
+model used as well as external properties in the signal. We expect, 
+however, that following periods of increased uncertainty, such as fills 
+or expressive timing, the information contained in an individual event 
+increases. We also examine whether the information is dependent upon 
+metrical position.
+
 
 \section{Information dynamics as compositional aid}
+\label{s:composition}
+
+  \begin{fig}{wundt}
+    \raisebox{-4em}{\colfig[0.43]{wundt}}
+ %  {\ \shortstack{{\Large$\longrightarrow$}\\ {\scriptsize\emph{exposure}}}\ }
+    {\ {\large$\longrightarrow$}\ }
+    \raisebox{-4em}{\colfig[0.43]{wundt2}}
+    \caption{
+      The Wundt curve relating randomness/complexity with
+      perceived value. Repeated exposure sometimes results
+      in a move to the left along the curve \cite{Berlyne71}.
+    }
+  \end{fig}
 
 In addition to applying information dynamics to analysis, it is also possible
 to apply it to the generation of content, such as to the composition of musical
@@ -773,14 +834,14 @@
 predictability of its output.
 
 The triangle is `populated' with possible parameter values for melody generators.
-These are plotted in a 3d statistical space of redundancy, entropy rate and
-predictive information rate.
- In our case we generated thousands of transition matrixes, representing first-order
+These are plotted in a 3D information space of $\rho_\mu$ (redundancy), $h_\mu$ (entropy rate) and
+$b_\mu$ (predictive information rate), as defined in \secrf{process-info}.
+ In our case we generated thousands of transition matrices, representing first-order
  Markov chains, by a random sampling method.  In figure \ref{InfoDynEngine} we
- see a representation of how these matrixes are distributed in the 3d statistical
+ see a representation of how these matrices are distributed in the 3d statistical
  space; each one of these points corresponds to a transition matrix.
 
-The distribution of transition matrixes plotted in this space forms an arch shape
+The distribution of transition matrices plotted in this space forms an arch shape
 that is fairly thin.  It thus becomes a reasonable approximation to pretend that
 it is just a sheet in two dimensions; and so we stretch out this curved arc into
 a flat triangle.  It is this triangular sheet that is our `Melody Triangle' and
@@ -879,5 +940,5 @@
 \section{Conclusion}
 
 \bibliographystyle{unsrt}
-{\bibliography{all,c4dm,nime}}
+{\bibliography{all,c4dm,nime,andrew}}
 \end{document}