# HG changeset patch # User samer # Date 1331824126 0 # Node ID 3f643e9fead005684a027889deed483d4032481b # Parent 1161caf0bddad984c8091000b2f8f61b0454149d Added Andrew's bits, added to fig 2, fixed some spellings, added some section crossrefs. diff -r 1161caf0bdda -r 3f643e9fead0 draft.pdf Binary file draft.pdf has changed diff -r 1161caf0bdda -r 3f643e9fead0 draft.tex --- 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}