comparison draft.tex @ 64:a18a4b0517e8

Finished sec 3B.

       informative of notes at different periodicities (\ie hypothetical
       bar lengths) and phases (\ie positions within a bar).
       }
     \end{fig}
 
-    \subsection{Content analysis/Sound Categorisation}
-	 Using analogous definitions of differential entropy, the methods outlined
-	 in the previous section are equally applicable to continuous random variables.
+    \subsection{Real-valued signals and audio analysis}
+	 Using analogous definitions based on the differential entropy
+	 \cite{CoverThomas}, the methods outlined
+	 in \secrf{surprise-info-seq} and \secrf{process-info}
+	 are equally applicable to random variables taking values in a continuous domain.
 	 In the case of music, where expressive properties such as dynamics, tempo,
 	 timing and timbre are readily quantified on a continuous scale, the information
-	 dynamic framework thus may also be considered.
-
-	 In \cite{Dubnov2006}, Dubnov considers the class of stationary Gaussian
+	 dynamic framework may thus be applied.
+
+	 Dubnov \cite{Dubnov2006} considers the class of stationary Gaussian
 	 processes. For such processes, the entropy rate may be obtained analytically
-	 from the power spectral density of the signal, allowing the multi-information
-	 rate to be subsequently obtained. 
+	 from the power spectral density of the signal. Dubnov found that the
+	 multi-information rate (which he refers to as `information rate') can be
+	 expressed as a function of the spectral flatness measure. For a given variance,
+	 Gaussian processes with maximal multi-information rate are those with maximally
+	 non-flat spectra. These are essentially consist of a single
+	 sinusoidal component and hence are completely predictable and periodic once
+	 the parameters of the sinusoid have been inferred.
 %	 Local stationarity is assumed, which may be achieved by windowing or 
 %	 change point detection \cite{Dubnov2008}. 
 	 %TODO
-	 mention non-gaussian processes extension Similarly, the predictive information
-	 rate may be computed using a Gaussian linear formulation CITE. In this view,
-	 the PIR is a function of the correlation  between random innovations supplied
-	 to the stochastic process.  %Dubnov, MacAdams, Reynolds (2006) %Bailes and
-	 Dean (2009)
-
-	% !!! FIXME
-		[ Continuous domain information ]
-		[Audio based music expectation modelling]
-		[ Gaussian processes]
+
+	 We are currently working towards methods for the computation of predictive information
+	 rate in some restricted classes of Gaussian processes including finite-order
+	 autoregressive models and processes with power-law spectra (fractional Brownian
+	 motions).
+
+%	 mention non-gaussian processes extension Similarly, the predictive information
+%	 rate may be computed using a Gaussian linear formulation CITE. In this view,
+%	 the PIR is a function of the correlation  between random innovations supplied
+%	 to the stochastic process.  %Dubnov, MacAdams, Reynolds (2006) %Bailes and Dean (2009)
+
 
 
 \subsection{Beat Tracking}
 
 A probabilistic method for drum tracking was presented by Robertson 

 The triangle is populated with first order Markov chain transition
 matrices as illustrated in \figrf{mtriscat}.
 The distribution of transition matrices plotted in this space forms an arch shape
 that is fairly thin. Thus, it is a reasonable simplification to project out the 
 third dimension (the PIR) and present an interface that is just two dimensional. 
-The right-angled triangle is rotated and stretched to form an equilateral triangle with
-the $h_\mu=0, \rho_\mu=0$ vertex at the top, the `redundancy' axis down the right-hand
-side, and the `entropy rate' axis down the left, as shown in \figrf{TheTriangle}.
+The right-angled triangle is rotated, reflected and stretched to form an equilateral triangle with
+the $h_\mu=0, \rho_\mu=0$ vertex at the top, the `redundancy' axis down the left-hand
+side, and the `entropy rate' axis down the right, as shown in \figrf{TheTriangle}.
 This is our `Melody Triangle' and
 forms the interface by which the system is controlled. 
 %Using this interface thus involves a mapping to information space; 
 The user selects a position within the triangle, the point is mapped into the 
 information space, and a corresponding transition matrix is returned. The third dimension,

 	
 \begin{fig}{mtri-results}
 	\def\scat#1{\colfig[0.42]{mtri/#1}}
 	\def\subj#1{\scat{scat_dwells_subj_#1} & \scat{scat_marks_subj_#1}}
 	\begin{tabular}{cc}
-		\subj{a} \\
+%		\subj{a} \\
 		\subj{b} \\
-		\subj{c} \\
-		\subj{d}
+		\subj{c} 
+%		\subj{d}
 	\end{tabular}
 	\caption{Dwell times and mark positions from user trials with the
-	on-screen Melody Triangle interface. The left-hand column shows
+	on-screen Melody Triangle interface, for two subjects. The left-hand column shows
 	the positions in a 2D information space (entropy rate vs multi-information rate
-	in bits) where spent their time; the area of each circle is proportional
+	in bits) where each spent their time; the area of each circle is proportional
 	to the time spent there. The right-hand column shows point which subjects
-	`liked'.}
+	`liked'; the area of the circles here is proportional to the duration spent at
+	that point before the point was marked.}
 \end{fig}
 
 Information measures on a stream of symbols can form a feedback mechanism; a
 rudimentary `critic' of sorts.  For instance symbol by symbol measure of predictive
 information rate, entropy rate and redundancy could tell us if a stream of symbols