Finished the simulation part of the method sections.

main.tex

@@ -103,6 +103,9 @@
 \newcommand{\xvar}{\sigma_{x}^{2}} % Variance of synthetic mixture
 \newcommand{\svar}{\sigma_{\text{s}}^{2}} % Song component variance
 \newcommand{\nvar}{\sigma_{\eta}^{2}} % Noise component variance
+\newcommand{\xsig}{\sigma_x} % Standard deviation of synthetic mixture
+\newcommand{\ssig}{\sigma_{\text{s}}} % Song component standard deviation
+\newcommand{\nsig}{\sigma_{\eta}} % Noise component standard deviation
 \newcommand{\pc}{p(c,\,T)} % Probability density (general interval)
 \newcommand{\pclp}{p(c,\,\tlp)} % Probability density (lowpass interval)
 \newcommand{\muf}{\mu_{f_i}} % Average feature value
@@ -558,7 +561,170 @@ can be read out by a simple linear classifier.
 \end{figure}
 \FloatBarrier
 
-\subsubsection{Simulation-based analysis of the model pathway}
+\subsection{Simulation-based analysis of the model pathway}
+
+\subsubsection{Data sourcing}
+
+All simulations were based on a dataset that was assembled from five different
+sources, each of which is an established reference for the identification of
+European grasshopper species. The dataset was limited to six species from the
+species-rich \textit{Gomphocerinae} sub-family that are known to be common
+throughout Central and Southern Europe. All recordings were converted to
+standard~\textit{.wav}~format with a sampling rate of~44.1\,kHz and an
+amplitude scale in arbitrary units. Individual songs were then cut from each
+recording. The dataset includes a total of 31 recordings across species, which
+amounts to a total of 153 isolated songs. However, the number of available
+species-specific songs varies greatly across species, with a maximum of 48
+songs for \textit{C. biguttulus} and a minimum of 6 songs for \textit{C.
+mollis}~(Tab.\,\ref{tab:species_list}).
+
+\begin{itemize}
+    \item "Heuschrecken beobachten, bestimmen" by~Heiko~Bellmann\\
+    1$^{\text{st}}$\,edition, 1993, Naturbuch, Augsburg
+    \item "Gesänge der heimischen Heuschrecken. Akustisch-optische
+    Bestimmungshilfe."\\
+    by~Karl-Heinz~Garberding, Deutscher Jugendbund für Naturbeobachtung\\
+    1$^{\text{st}}$\,edition, 2001, DJN, Göttingen
+    \item "Heuschrecken -- Die Stimmen von 61 heimischen Arten"
+    by~Heiko~Bellmann\\
+    1$^{\text{st}}$\,edition, 2004, AMPLE, Germering
+    \item "Fauna d'Italia XLVIII -- Orthoptera" by~Bruno~Massa, Paolo~Fontana,
+    Filippo~M.~Buzzetti, Roy~M.J.C.~Kleukers, Baudewijn~Odé\\
+    1$^{\text{st}}$\,edition, 2012, edagricola, Milano
+    \item "Singing Orthoptera of Slovenia" by~Stanislav~Gomboc, Blaz~Segula\\
+    1$^{\text{st}}$\,edition, 2014, EGEA, Ljubljana
+\end{itemize}
+
+\begin{table}[!ht]
+    \centering
+    \captionsetup{width=.75\textwidth}
+    \caption{Overview of the six grasshopper species from the
+    \textit{Gomphocerinae} sub-family, the number of sources per species, the
+    number of available recordings across sources, and the number of isolated
+    songs across recordings.}
+    \begin{tabular}{|lccc|}
+        \hline
+        \textbf{Species} & \textbf{Sources} & \textbf{Recordings} & \textbf{Songs}\\
+        \hline
+        \textit{Chorthippus biguttulus} & 5 & 6 & 48\\ 
+        \textit{Chorthippus mollis} & 3 & 3 & 6\\ 
+        \textit{Chrysochraon dispar} & 4 & 5 & 45\\ 
+        \textit{Gomphocerippus rufus} & 4 & 8 &  16\\ 
+        \textit{Omocestus rufipes} & 4 & 5 & 14\\ 
+        \textit{Pseudochorthippus parallelus} & 4 & 4 & 24\\ 
+        \hline
+    \end{tabular}
+    \label{tab:species_list}
+\end{table}
+
+\subsubsection{Generating synthetic input signals}
+
+Different processing steps along the model pathway were tested for intensity
+invariance by generating synthetic input signals $x(t)$ of varying intensity,
+transforming them through the respective processing steps, and comparing the
+resulting signal representations. Inputs were generated for two distinct cases.
+In the idealized, noiseless case, $x(t)$ consists of a song component $\soc(t)$
+with $\ssig=1$ and a multiplicative scale $\sca$:
+\begin{equation}
+    x(t)\,=\,\sca\,\cdot\,\soc(t), \qquad \sca\,\geq\,0
+    \label{eq:noiseless}
+\end{equation}
+In the noiseless case, $x(t)$ is hence only a scaled version of $\soc(t)$ with
+$\xsig=\sca$. In the more realistic, noisy case, $x(t)$ consists of the same
+song component $\soc(t)$ scaled by $\sca$ and an additive noise component
+$\noc(t)$ with $\nsig=1$:
+\begin{equation}
+    x(t)\,=\,\sca\,\cdot\,\soc(t)\,+\,\noc(t), \qquad \sca\,\geq\,0
+    \label{eq:noisy}
+\end{equation}
+Accordingly, the SNR of input $x(t)$ in the noisy case equals the squared
+$\sca$ value:
+% Make sure that SNR = signal-to-noise ratio is introduced somewhere!
+\begin{equation}
+    \text{SNR}_x(\sca)\,=\,\frac{(\sca\,\cdot\,\ssig)^2}{\nsig^2}\,=\,\sca^2, \qquad \ssig\,=\,\nsig\,=\,1
+    \label{eq:input_snr}
+\end{equation}
+For most analyses, it would be sufficient if input $x(t)$ corresponds to the
+signal representation immediately before the first of the tested
+transformations. For instance, when testing the effects of logarithmic
+compression~(Eq.\,\ref{eq:log}), $x(t)$ would correspond to the signal envelope
+$\env(t)$. However, in this particular case, $\env(t)$ results from a nonlinear
+transformation~(Eq.\,\ref{eq:env}), which cannot be synthesized as an additive
+mixture of $\soc(t)$ and $\noc(t)$. For this reason, any input $x(t)$ across
+all analyses corresponds not to the representation immediately before the
+tested transformations but its predecessor representation instead. Therefore,
+when testing logarithmic compression, $x(t)$ corresponds to the tympanal signal
+$\filt(t)$ instead of $\env(t)$.
+
+The raw $\soc(t)$ was drawn from the dataset of isolated species-specific song
+recordings, whereas the raw $\noc(t)$ consists of a segment of normally
+distributed white noise. Both $\soc(t)$ and $\noc(t)$ were normalized to unit
+standard deviation. These can be used without further processing for all
+analyses where input $x(t)$ corresponds to $\raw(t)$. For analyses where $x(t)$
+corresponds to a later representation, $\soc(t)$ and $\noc(t)$ were first
+processed along the model pathway up to the required representation, again
+normalized to unit standard deviation, and then used to generate $x(t)$
+according to either Eq.\,\ref{eq:noiseless} in the noiseless case or
+Eq.\,\ref{eq:noisy} in the noisy case.
+
+\subsubsection{Quantifying signal intensity across representations}
+
+All intensity measures were calculated over a manually labeled segment within
+each song. Segments always excluded the first and last few syllables to allow
+slowly changing representations such as $f_i(t)$ to stabilize. The duration of
+each segment and the number of contained syllables depends on the duration of
+the species-specific song. Care was taken to ensure that the segment contained
+a sufficient number of syllables to obtain a reliable estimate of the intensity
+measures.
+
+The standard deviation $\sigma$ was used as a measure of intensity for all
+representations resulting from the transformation of input $x(t)$ up to and
+including the kernel responses $c_i(t)$, for which individual $\sigma_{c_i}$
+were used as kernel-specific intensity measures. The binary responses $b_i(t)$
+were deemed to similar to the features $f_i(t)$ to warrant their own intensity
+measure and were hence omitted from all related analyses. For $f_i(t)$,
+$\sigma$ is not an appropriate intensity measure because each $f_i(t)$ is
+ideally constant with $\sigma=0$ for the duration of a song. Therefore, the
+average value $\muf$ of each $f_i(t)$ was used as a kernel-specific intensity
+measure instead.
+
+It is arguably not ideal to quantify the intensity of $c_i(t)$ and $f_i(t)$
+separately for each kernel. Overall, these representations are not separate
+signals bundled together but rather a set that acts as a unit with a single
+intensity measure. However, there is no straightforward way to quantify the
+intensity of $c_i(t)$ or $f_i(t)$ as a whole that would not entail a certain
+ambiguity, e.\,g by averaging across kernels. In this sense, we opted for the
+kernel-specific approach because it allows to asses differences in the
+dependency on $\sca$ between individual members of either $c_i(t)$ and
+$f_i(t)$.
+
+The absolute intensity measures allow to compare the intensity of a
+representation across different $\sca$ values. Additionally, ratios were
+calculated between the intensity measures for $\sca>0$ and the respective
+pure-noise reference measure for $\sca=0$ to better compare the intensities of
+different representations. This is only possible in the noisy case, where input
+$x(t)=\noc(t)$ for $\sca=0$, whereas $x(t)=0$ for $\sca=0$ in the noiseless
+case. At the level of input $x(t)$, the ratio of intensity measures depends on
+the square root of $\sca$:
+\begin{equation}
+    \frac{\xsig}{\nsig}\,=\,\sqrt{\frac{\xsig^2}{\nsig^2}}\,=\,\sqrt{\frac{(\sca\,\cdot\,\ssig)^2\,+\,\nsig^2}{\nsig^2}}\,=\,\sqrt{\sca^2\,+\,1}, \qquad \ssig\,=\,\nsig\,=\,1
+    \label{eq:input_ratio}
+\end{equation}
+This holds only if $\soc(t)\perp\noc(t)$, so that $\xsig^2=\ssig^2+\nsig^2$,
+which is a reasonable assumption for the raw $\soc(t)$ and $\noc(t)$. However,
+the dependency of the ratio on $\sca$ is not necessarily the same for
+representations that are transformed from $x(t)$ by nonlinear operations, since
+these change the relationship of $\soc(t)$ and $\noc(t)$ in an unpredictable
+fashion. Furthermore, the ratio is not a proper SNR of the representation
+because it does not relate $\soc(t)$ to $\noc(t)$ within the representation but
+rather the entire representation to $\noc(t)$ alone. However, it still provides
+a useful measure of the relative intensity of a representation with and without
+$\soc(t)$, which is the closest we can get to the SNR of the representation. As
+such, the ratio of intensity measures is referred to as SNR in the following.
+% Is this legal? "SNR" is much shorter than "ratio of intensity measure to the pure-noise reference measure".
+% Haven't used it much yet, sticked to "ratio" in most cases.
+
+\subsection{Field data-based analysis of the model pathway}
 
 \section{Results}