Syncing to home.

main.tex

@@ -105,6 +105,7 @@
 \newcommand{\pc}{p(c,\,T)} % Probability density (general interval)
 \newcommand{\pclp}{p(c,\,\tlp)} % Probability density (lowpass interval)
 \newcommand{\pci}{p(c_i,\,\tlp)} % Kernel-specific probability density (lowpass interval)
+\newcommand{\tstat}{T_{\text{total}}} % Time interval where c(t) is stationary
 \newcommand{\muf}{\mu_{f_i}} % Average feature value
 
 \section{Introduction}
@@ -827,11 +828,10 @@ between the resulting $\env(t)$, $\db(t)$, and $\adapt(t)$. It is necessary to
 use $\filt(t)$ as input for this analysis instead of $\env(t)$, because
 $\env(t)$ results from a nonlinear transformation and hence cannot be
 synthesized as an additive mixture of song component $\soc(t)$ and noise
-component $\noc(t)$. % <-- Sentence may be methods section material.
+component $\noc(t)$. However, it is much easier to conceive a mathematical
-However, it is much easier to conceive a mathematical description of the
+description of the effects of logarithmic compression and adaptation if
-effects of logarithmic compression and adaptation if $\env(t)$ itself is
+$\env(t)$ itself is assumed to be composed of $\soc(t)$ and $\noc(t)$. In the
-assumed to be composed of $\soc(t)$ and $\noc(t)$. In the noiseless
+noiseless case~(Fig.\,\ref{fig:log-hp}a), $\env(t)$ takes the form of
-case~(Fig.\,\ref{fig:log-hp}a), $\env(t)$ takes the form of
 \begin{equation}
     \env(t)\,=\,\sca\,\cdot\,\soc(t), \qquad \env(t)\,>\,0\enspace\forall\enspace t\,\in\,\mathbb{R}
     \label{eq:toy_env_pure}
@@ -1000,35 +1000,44 @@ corresponding $\tlp$:
     f(t)\,\approx\,\int_{\Theta}^{+\infty} \pclp\,dc\,=\,P(c\,>\,\Theta,\,\tlp)
     \label{eq:feat_prop}
 \end{equation}
+% Little bit of patch-work here...
+% 1) Interpretation of the feature value:
 In a sense, $f(t)$ can be interpreted as some sort of duty cycle of $c(t)$ with
 respect to $\Theta$. For example, a feature value of $f(t)=0.4$ means that
 $c(t)$ exceeds $\Theta$ for approximately 40\,\% of the time within $\tlp$. In
 the most extreme cases, $\Theta$ lays either above the maximum of $c(t)$ or
 below the minimum of $c(t)$, which results in a minimum or maximum possible
 feature value of $f(t)=0$~(Fig.\,\ref{fig:thresh-lp_single}d, left column) or
-$f(t)=1$, respectively. Furthermore, if $c(t)$ is stationary --- so that its
+$f(t)=1$, respectively. 
-statistics do not change substantially over time --- and if $\tlp$ is much
-longer than the relevant time scales of $c(t)$, then $\pclp$ is largely
-independent of $t$. In this case, $f(t)$ is approximately constant across
-$t$~(Fig.\,\ref{fig:stages_feat}c).
 
-Importantly, $f(t)$ neither retains information about the timing of individual
+% 2) Constant feature values across t:
-threshold crossings nor the precise values of $c(t)$ apart from their relation
+If the time $T_1$ where $c(t)>\Theta$ within $\tlp$ is approximately constant
-to $\Theta$. Different $\sca$ can hence result in similar feature values by
+across $t$ for some time interval $\tstat>\tlp$, then $f(t)$ is approximately
-producing similar $T_1$ segments. The most reliable way of exploiting this
+constant across $t\in\tstat$ as well~(Fig.\,\ref{fig:stages_feat}c). This is
-invariant property of $f(t)$ is to set $\Theta$ to a value near 0, because
+fulfilled if $c(t)$ is stationary in the sense that its distribution $\pclp$
-these values are least affected by different scales of $c(t)$. For sufficiently
+does not change substantially within $\tstat$, which requires that $\tlp$ is
-large $\sca$, $f(t)$ then approaches the same constant $\mu_f$ in both the
+much longer than the relevant time scales of $c(t)$. However, stationarity of
-noiseless and the noisy case~(Fig.\,\ref{fig:thresh-lp_single}e, saturation
+$c(t)$ is not a necessary condition for $f(t)$ to be constant because $f(t)$
-regime).
+depends only on the total $T_1$ --- irrespective of the timing of individual
+threshold crossings --- and different $\pclp$ can, in principle, still result
+in similar $T_1$.
 
-The saturation level of $f(t)$ is independent of the precise value of $\Theta$,
+% 3) Constant feature values across alpha:
-but the saturation point decreases with
+Similarly, $f(t)$ retains no information about the precise values of $c(t)$
-$\Theta$~(Fig.\,\ref{fig:thresh-lp_single}e). Therefore, a threshold value of
+apart from their relation to $\Theta$. Different scales $\sca$ can hence result
-$\Theta=0$ would be the optimal choice for achieving intensity invariance at
+in similar values of $f(t)$ as long as $T_1$ remains similar across $\sca$. The
-the lowest possible $\sca$. In stark contrast, the closer $\Theta$ is to 0, the
+most reliable way of exploiting this invariant property of $f(t)$ is to set
-higher $\mu_f$ in response to the pure noise component $\noc(t)$ and the lower
+$\Theta$ to a value near 0, because these values are least affected by
-the resulting SNR of $f(t)$ between noise regime and saturation
+different scales of $c(t)$. For sufficiently large $\sca$, $f(t)$ then
+approaches the same constant $\mu_f$ in both the noiseless and the noisy
+case~(Fig.\,\ref{fig:thresh-lp_single}e, saturation regime). The saturation
+level of $f(t)$ is independent of the precise value of $\Theta$, but the
+saturation point decreases with $\Theta$~(Fig.\,\ref{fig:thresh-lp_single}e).
+Therefore, a threshold value of $\Theta=0$ would be the optimal choice for
+achieving intensity invariance at the lowest possible $\sca$. In stark
+contrast, the closer $\Theta$ is to 0, the higher $\mu_f$ in response to the
+pure noise component $\noc(t)$ and the lower the resulting SNR of $f(t)$
+between noise regime and saturation
 regime~(Fig.\,\ref{fig:thresh-lp_single}b-d, left column, and
 Fig.\,\ref{fig:thresh-lp_single}e). This trade-off between intensity invariance
 and SNR has already been observed during the previous analysis on logarithmic
@@ -1581,89 +1590,30 @@ constitutes the basis for song recognition. The songs of different species are
 represented by specific combinations of feature values, which should be as
 constant as possible for the duration of a song to fasciliate recognition. The
 fundamental requirement for a constant feature $f_i(t)$ is that the time where
-kernel response $c_i(t)$ exceeds the threshold value $\thr$ within averaging
+kernel response $c_i(t)$ exceeds the threshold value $\thr$ is approximately
-interval $\tlp$ is the same for all time points $t$. This is fulfilled if
-$c_i(t)$ is stationary within a certain time window and $\tlp$ is much longer
-than the relevant time scales of $c_i(t)$, so that the distribution $\pci$ of
-$c_i(t)$ and hence the value of $f_i(t)$ remain stable across $t$.
-
-% Practical cases that allow for approximately constant features:
-There are two very different practical cases in which $c_i(t)$ could fulfill
-the stationarity requirement. First, $c_i(t)$ is 
-
-can be assumed to be
-stationary: Either $c_i(t)$ is entirely unstructured on most time scales, or
-$c_i(t)$ is periodic.
 
 
-First, $c_i(t)$ is entirely unstructured on most time scales, which
+Each
+feature $f_i(t)$ approximately quantifies the proportion of time where kernel
+response $c_i(t)$ exceeds the threshold value $\thr$ within the averaging
+interval $\tlp$. The value of $f_i(t)$ at time point $t$ is hence determined by
+the distribution $\pci$ of $c_i(t)$ around $t$.
 
+Accordingly, if $c_i(t)$ is
+stationary within some time interval $T>\tlp$ --- so that $\pci$ does not
+change substantially with $t$ --- then the value of $f_i(t)$ is approximately
+constant across $t$.
 
-
+If the time $T_1$ where $c(t)>\Theta$ within $\tlp$ is approximately constant
-The structure of noise-evoked $c_i(t)$ is largely random with an
+across $t$ for some time interval $\tstat>\tlp$, then $f(t)$ is approximately
-approximately normal $\pci$ with constant mean and variance across $t$.
+constant across $t\in\tstat$ as well~(Fig.\,\ref{fig:stages_feat}c). This is
-
+fulfilled if $c(t)$ is stationary in the sense that its distribution $\pclp$
-Either the structure of $c_i(t)$ is largely random, or 
+does not change substantially within $\tstat$, which requires that $\tlp$ is
-
+much longer than the relevant time scales of $c(t)$. However, stationarity of
-Noise-evoked $c_i(t)$ are 
+$c(t)$ is not a necessary condition for $f(t)$ to be constant because $f(t)$
-
+depends only on the total $T_1$ --- irrespective of the timing of individual
-Noise-evoked $c_i(t)$ are largely unstructured and follow a roughly
+threshold crossings --- and different $\pclp$ can, in principle, still result
-normal $\pci$ with constant mean and variance across $t$. In contrast,
+in similar $T_1$.
-song-evoked $c_i(t)$ are highly 
-
-Noise-evoked $c_i(t)$ fulfill the stationary requirement because their $\pci$
-is approximately a normal distribution with constant mean and variance across
-$t$.
-
-Each feature $f_i(t)$ approximately
-quantifies the proportion of time where the respective kernel response $c_i(t)$
-exceeds the threshold value $\thr$ within the averaging interval $\tlp$. The
-songs of different species are represented by specific combinations of feature
-values, which should preferably be as constant as possible during a song to
-fasciliate recognition. The fundamental requirement for constant $f_i(t)$ is
-that the time where $c_i(t)>\thr$ within $\tlp$ is the same for all $t$, which
-is fulfilled if the distribution $\pci$ of 
-
-
-The
-value of $f_i(t)$ is hence determined by $\thr$ with respect to the
-distribution $\pci$ of $c_i(t)$.
-
-
-$c_i(t)>\thr$.
-
-The feature set is the final song representation along the model pathway and
-constitutes the basis for song recognition. Each feature $f_i(t)$ results from
-the thresholding of the respective kernel response $c_i(t)$ by $\nl$ and the
-subsequent temporal averaging of binary response $b_i(t)$ by a lowpass filter
-with cutoff frequency $\fc$, which specifies the averaging interval $\tlp$.
-Feature $f_i(t)$ approximately quantifies the proportion of time during which
-$c_i(t)$ exceeds the threshold value $\thr$ within $\tlp$. The value of
-$f_i(t)$ at time point $t$ is hence determined by $\thr$ with respect to the
-distribution $\pci$ of $c_i(t)$ around $t$ and restricted to the interval
-$[0,1]$.
-
-% Theoretical constraints for constant features:
-The songs of different species are represented by specific combinations of
-values across the feature set, which should preferably be constant for the
-duration of a song to fasciliate recognition. The fundamental requirement for
-constant $f_i(t)$ is that the time where $c_i(t)>\thr$ within $\tlp$ is the
-same for all $t$.
-
-This is fulfilled if $c_i(t)$ is stationary across $t$ and
-$\tlp$ is much longer than the relevant time scales of $c_i(t)$, so that $\pci$
-is independent of $t$.
-
-
-, which is fulfilled if $\pci$ is stable across $t$.
-
-The most
-straightforward way to achieve a stable $\pci$ is that $c_i(t)$ is stationary
-and $\tlp$ is sufficiently long to average over the stationary distribution of
-
-The most
-straightforward way to achieve a stable $\pci$ is that $c_i(t)$ is periodic and
-$\tlp$ is sufficiently long to average over multiple cycles of $c_i(t)$.
 
 Most song-evoked $c_i(t)$ are indeed highly repetitive, albeit not perfectly
 periodic, which is largely an inherited property of the song itself.