diff --git a/figures/fig_invariance_thresh_lp_single.pdf b/figures/fig_invariance_thresh_lp_single.pdf index 75af34a..372a9b9 100644 Binary files a/figures/fig_invariance_thresh_lp_single.pdf and b/figures/fig_invariance_thresh_lp_single.pdf differ diff --git a/main.aux b/main.aux index f1d68b7..66d8a5a 100644 --- a/main.aux +++ b/main.aux @@ -252,14 +252,14 @@ \newlabel{eq:toy_highpass_pure}{{13}{13}{}{}{}} \newlabel{eq:toy_env_noise}{{14}{13}{}{}{}} \newlabel{eq:toy_log_noise}{{15}{13}{}{}{}} -\newlabel{eq:toy_highpass_noise}{{16}{13}{}{}{}} +\newlabel{eq:toy_highpass_noise}{{16}{14}{}{}{}} \@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces \textbf {Intensity invariance through logarithmic compression and adaptation is restricted by the noise floor and decreases SNR.} Input $x_{\text {filt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is successively transformed into envelope $x_{\text {env}}(t)$, logarithmically compressed envelope $x_{\text {log}}(t)$, and intensity-adapted envelope $x_{\text {adapt}}(t)$. \textbf {Top}:~Example representations of $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, and $x_{\text {adapt}}(t)$ for different $\alpha $. \textbf {a}:~Noiseless case. \textbf {b}:~Noisy case. \textbf {Bottom}:~Intensity metrics over a range of $\alpha $. \textbf {c}:~Noiseless case: Standard deviations $\sigma _x$ of $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, and $x_{\text {adapt}}(t)$. \textbf {d}:~Noisy case: Ratios of $\sigma _x$ of $x_{\text {env}}(t)$, $x_{\text {log}}(t)$, and $x_{\text {adapt}}(t)$ to the respective reference standard deviation $\sigma _{\eta }$ for input $x_{\text {filt}}(t)=\eta (t)$. Shaded areas indicate $5\,\%$ (dark grey) and $95\,\%$ (light grey) curve span for $x_{\text {adapt}}(t)$. \textbf {e}:~Ratios of $\sigma _x$ to $\sigma _{\eta }$ of $x_{\text {adapt}}(t)$ as in \textbf {d} for different species (averaged over songs and recordings, see appendix Fig\,\ref {fig:app_log-hp_curves}). Dots indicate $95\,\%$ curve span per species. }}{15}{}\protected@file@percent } \newlabel{fig:log-hp}{{5}{15}{}{}{}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Thresholding nonlinearity \& temporal averaging}{16}{}\protected@file@percent } \newlabel{eq:pdf_split}{{17}{16}{}{}{}} \newlabel{eq:feat_avg}{{18}{16}{}{}{}} \newlabel{eq:feat_prop}{{19}{16}{}{}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces \textbf {Intensity invariance through thresholding and temporal averaging is mediated by the interaction of threshold value and noise floor.} Input $x_{\text {adapt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is transformed into single kernel response $c(t)$, binary response $b(t)$, and feature $f(t)$. Different color shades indicate different threshold values $\Theta $ (multiples of reference standard deviation $\sigma _{\eta }$ of $c(t)$ for input $x_{\text {adapt}}(t)=\eta (t)$, with darker colors for higher $\Theta $). \textbf {Left}:~Noisy case: Example representations of $x_{\text {adapt}}(t)$ as well as $c(t)$, $b(t)$, and $f(t)$ for different $\alpha $. \textbf {a}:~$x_{\text {adapt}}(t)$ with kernel $k(t)$ in black. \textbf {b\,-\,d}: $c(t)$, $b(t)$, and $f(t)$ based on the same $x_{\text {adapt}}(t)$ from \textbf {a} but with different $\Theta $. \textbf {Right}:~Average value $\mu _f$ of $f(t)$ for each $\Theta $ from \textbf {b\,-\,d}, once for the noisy case (solid lines) and once for the noiseless case (dotted lines). Dots indicate $95\,\%$ curve span (noisy case). \textbf {e}:~$\mu _f$ over a range of $\alpha $. \textbf {f}:~$\mu _f$ over the standard deviation of noisy input $x_{\text {adapt}}$ corresponding to the values of $\alpha $ shown in \textbf {e}. }}{18}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces \textbf {Intensity invariance through thresholding and temporal averaging is mediated by the interaction of threshold value and noise floor.} Input $x_{\text {adapt}}(t)$ consists of song component $s(t)$ scaled by $\alpha $ with optional noise component $\eta (t)$ and is transformed into single kernel response $c(t)$, binary response $b(t)$, and feature $f(t)$. Different color shades indicate different threshold values $\Theta $ (multiples of reference standard deviation $\sigma _{\eta }$ of $c(t)$ for input $x_{\text {adapt}}(t)=\eta (t)$, with darker colors for higher $\Theta $). \textbf {Left}:~Noisy case: Example representations of $x_{\text {adapt}}(t)$ as well as $c(t)$, $b(t)$, and $f(t)$ for different $\alpha $. \textbf {a}:~$x_{\text {adapt}}(t)$ with kernel $k(t)$ in black. \textbf {b\,-\,d}: $c(t)$, $b(t)$, and $f(t)$ based on the same $x_{\text {adapt}}(t)$ from \textbf {a} but with different $\Theta $. \textbf {Right}:~Average value $\mu _f$ of $f(t)$ for each $\Theta $ from \textbf {b\,-\,d}. Dots indicate $95\,\%$ curve span (noisy case). \textbf {e}:~$\mu _f$ over a range of $\alpha $, once for the noisy case (solid lines) and once for the noiseless case (dotted lines). \textbf {f}:~Noisy case: $\mu _f$ over the standard deviation of input $x_{\text {adapt}}$ corresponding to the values of $\alpha $ shown in \textbf {e}. Shaded area indicates standard deviations that would be capped in the output $x_{\text {adapt}}(t)$ of the previous transformation pair (see Fig.\,\ref {fig:log-hp}cd). }}{18}{}\protected@file@percent } \newlabel{fig:thresh-lp_single}{{6}{18}{}{}{}} \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces \textbf {Feature representation of different species-specific songs saturates at different points in feature space.} Same input and processing as in Fig.\,\ref {fig:thresh-lp_single} but with three different kernels $k_i$, each with a single kernel-specific threshold value $\Theta _i=0.5\cdot \sigma _{\eta _i}$. \textbf {a}:~Examples of species-specific grasshopper songs. \textbf {Middle}:~Average value $\mu _{f_i}$ of each feature $f_i(t)$ over $\alpha $ per species (averaged over songs and recordings, see appendix Figs.\,\ref {fig:app_thresh-lp_pure} and \ref {fig:app_thresh-lp_noise}). Different color shades indicate different kernels $k_i$. Dots indicate $95\,\%$ curve span per $k_i$. \textbf {b}:~Noiseless case. \textbf {c}:~Noisy case. \textbf {Bottom}:~2D feature spaces spanned by each pair of $f_i(t)$. Each trajectory corresponds to a species-specific combination of $\mu _{f_i}$ that develops with $\alpha $ (colorbars). Horizontal dashes in the colorbar indicate $5\,\%$ (dark grey) and $95\,\%$ (light grey) curve span of the norm across all three $\mu _{f_i}$ per species. \textbf {d}:~Noiseless case. \textbf {e}:~Noisy case. Shaded areas indicate the average minimum $\mu _{f_i}$ across all species-specific trajectories. }}{19}{}\protected@file@percent } \newlabel{fig:thresh-lp_species}{{7}{19}{}{}{}} diff --git a/main.blg b/main.blg index b7f3f35..52ce8c2 100644 --- a/main.blg +++ b/main.blg @@ -1,71 +1,71 @@ [0] Config.pm:307> INFO - This is Biber 2.19 [0] Config.pm:310> INFO - Logfile is 'main.blg' -[37] biber:340> INFO - === Di Mai 5, 2026, 14:08:58 -[45] Biber.pm:419> INFO - Reading 'main.bcf' -[76] Biber.pm:979> INFO - Found 55 citekeys in bib section 0 -[82] Biber.pm:4419> INFO - Processing section 0 -[87] Biber.pm:4610> INFO - Looking for bibtex file 'cite.bib' for section 0 -[88] bibtex.pm:1713> INFO - LaTeX decoding ... -[119] bibtex.pm:1519> INFO - Found BibTeX data source 'cite.bib' -[302] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable' -[302] UCollate.pm:68> INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized' -[302] Biber.pm:4239> INFO - 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[28 <./figures/fig_kernel_sd_perc_thresh_lp_appendix.pdf> <./figures/fig_kernel_sd_perc_full_appendix.pdf>] File: figures/fig_kernel_sd_perc_field_appendix.pdf Graphic file (type pdf) -Package pdftex.def Info: figures/fig_kernel_sd_perc_field_appendix.pdf used on input line 1233. +Package pdftex.def Info: figures/fig_kernel_sd_perc_field_appendix.pdf used on input line 1250. (pdftex.def) Requested size: 483.69687pt x 241.84782pt. File: figures/fig_invariance_cross_species_thresh_appendix.pdf Graphic file (type pdf) -Package pdftex.def Info: figures/fig_invariance_cross_species_thresh_appendix.pdf used on input line 1243. +Package pdftex.def Info: figures/fig_invariance_cross_species_thresh_appendix.pdf used on input line 1260. (pdftex.def) Requested size: 483.69687pt x 483.69566pt. 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PDF statistics: 2556 PDF objects out of 2984 (max. 8388607) 1129 compressed objects within 12 object streams diff --git a/main.pdf b/main.pdf index 7989fe4..27478e9 100644 Binary files a/main.pdf and b/main.pdf differ diff --git a/main.synctex.gz b/main.synctex.gz index 7e9196b..7a2de5f 100644 Binary files a/main.synctex.gz and b/main.synctex.gz differ diff --git a/main.tex b/main.tex index 0769e6e..e2f0f9c 100644 --- a/main.tex +++ b/main.tex @@ -610,16 +610,16 @@ This effect is more pronounced for lower $\fc$ of the lowpass filter and is presumably caused by the attenuation of high-frequency components in the signal, which are more prominent in the noise component $\noc(t)$ than in the song component $\soc(t)$. The effect also appears relatively consistent across -different species, although small variations based on different song structures -and distributions exist~(Fig.\,\ref{fig:rect-lp}e). In summary, the standard -deviation of $\env(t)$ has never been observed to transition into a saturation -regime for larger $\sca$ but rather continues to increase proportionally to -$\sca$ for all tested $\fc$, in both the noiseless and the noisy case and -across different species. Consequently, the combination of rectification and -lowpass filtering does not contribute to intensity invariance. However, this -transformation pair does improve the SNR of $\env(t)$ relative to $\filt(t)$ -and thus provides subsequent processing stages with a more robust input -representation and higher input SNR. +different species, although small variations exist~(Fig.\,\ref{fig:rect-lp}e) +that are presumably based on different song structures and frequency spectra. +In summary, the standard deviation of $\env(t)$ has never been observed to +transition into a saturation regime for larger $\sca$ but rather continues to +increase proportionally to $\sca$ for all tested $\fc$, in both the noiseless +and the noisy case and across different species. Consequently, the combination +of rectification and lowpass filtering does not contribute to intensity +invariance. However, this transformation pair does improve the SNR of $\env(t)$ +relative to $\filt(t)$ and thus provides subsequent processing stages with a +more robust input representation and higher input SNR. \begin{figure}[!ht] \centering @@ -880,7 +880,21 @@ the SNR of $\adapt(t)$ are much less understood and likely relate to properties of the signal, whereas the SNR of $f(t)$ depends on the choice of $\Theta$ and can be more directly manipulated by the system. -Finally, +Finally, the effects of thresholding and temporal averaging must be seen in the +context of the previous transformation pair of logarithmic compression and +adaptation. + +Finally, the question remains whether the intensity-invariant output $\adapt(t)$ +of the previous transformation pair allows feature + +Finally, the output $\adapt(t)$ of the previous transformation +pair~(Fig.\,\ref{fig:log-hp}cd) can be related to the input $\adapt(t)$ of the +current transformation pair by plotting the values of $f(t)$ over the standard +deviation of input $\adapt(t)$ instead of +$\sca$~(Fig.\,\ref{fig:thresh-lp_single}f). This is relevant because, unlike +$\sca$, the standard deviation of $\adapt(t)$ is capped to a maximum value of +around 10\,dB by the previous transformation pair~(Fig.\,\ref{fig:log-hp}cd) + \begin{figure}[!ht] \centering @@ -904,14 +918,17 @@ Finally, same $\adapt(t)$ from \textbf{a} but with different $\Theta$. \textbf{Right}:~Average value $\mu_f$ of $f(t)$ for each - $\Theta$ from \textbf{b\,-\,d}, once for the noisy case - (solid lines) and once for the noiseless case (dotted - lines). Dots indicate $95\,\%$ curve span (noisy case). - \textbf{e}:~$\mu_f$ over a range of $\sca$. - \textbf{f}:~$\mu_f$ over the standard deviation of noisy - input $\adapt$ corresponding to the values of $\sca$ shown - in \textbf{e}. - % Why plot noiseless case over SD of noisy input? Omit? + $\Theta$ from \textbf{b\,-\,d}. Dots indicate $95\,\%$ + curve span (noisy case). + \textbf{e}:~$\mu_f$ over a range of $\sca$, once for the + noisy case (solid lines) and once for the noiseless case + (dotted lines). + \textbf{f}:~Noisy case: $\mu_f$ over the standard + deviation of input $\adapt$ corresponding to the values of + $\sca$ shown in \textbf{e}. Shaded area indicates standard + deviations that would be capped in the output $\adapt(t)$ + of the previous transformation pair (see + Fig.\,\ref{fig:log-hp}cd). } \label{fig:thresh-lp_single} \end{figure} diff --git a/python/fig_invariance_thresh-lp_single.py b/python/fig_invariance_thresh-lp_single.py index a81bdf2..733f83a 100644 --- a/python/fig_invariance_thresh-lp_single.py +++ b/python/fig_invariance_thresh-lp_single.py @@ -266,6 +266,12 @@ leg_kwargs = dict( handlelength=1.5, columnspacing=1, ) +cap_kwargs = dict( + color='k', + alpha=0.5, + lw=0, + zorder=5, +) plateau_settings = dict( low=0.05, high=0.95, @@ -293,6 +299,12 @@ kern_specs = np.array([ [3, 0.002], ])[np.array([1])] +# PREPARATION: + +# Get saturation level of invariant envelope from log-hp analysis: +inv_path = search_files(example_file, dir='../data/inv/log_hp/')[0] +sigma_cap = load_data(inv_path, files='measure_inv')[0]['measure_inv'][-1] + # EXECUTION: print(f'Processing {data_path}') @@ -444,31 +456,45 @@ for i in range(thresh_rel.size): ind = get_saturation(noise_data['measure_feat'][:, i], **plateau_settings)[1] saturation_inds.append(ind) -# Plot analysis results: -for ax, x in zip([alpha_ax, sigma_ax], [scales, noise_data['measure_inv']]): - # Plot pure-song analysis results: - handles = ax.plot(x, pure_data['measure_feat'], lw=lw['big'], ls='dotted') - [h.set_color(c) for h, c in zip(handles, shaded['feat'])] +# Plot pure-song analysis results over alpha: +handles = alpha_ax.plot(scales, pure_data['measure_feat'], lw=lw['big'], ls='dotted') +[h.set_color(c) for h, c in zip(handles, shaded['feat'])] - # Plot noise-song analysis results: - handles = ax.plot(x, noise_data['measure_feat'], lw=lw['big']) - [h.set_color(c) for h, c in zip(handles, shaded['feat'])] +# Plot noise-song analysis results over alpha: +handles = alpha_ax.plot(scales, noise_data['measure_feat'], lw=lw['big']) +[h.set_color(c) for h, c in zip(handles, shaded['feat'])] - # Indicate threshold-specific saturation: - for i, ind in enumerate(saturation_inds): - color = shaded['feat'][i] - ax.plot(x[ind], 0, c='w', alpha=1, zorder=5.5, **plateau_dot_kwargs, - transform=ax.get_xaxis_transform()) - ax.plot(x[ind], 0, mfc=color, mec='k', alpha=0.75, zorder=6, - **plateau_dot_kwargs, transform=ax.get_xaxis_transform()) - ax.vlines(x[ind], ax.get_ylim()[0], noise_data['measure_feat'][ind, i], - color=color, **plateau_line_kwargs) +# Indicate threshold-specific saturation: +for i, ind in enumerate(saturation_inds): + color = shaded['feat'][i] + alpha_ax.plot(scales[ind], 0, c='w', alpha=1, zorder=5.5, **plateau_dot_kwargs, + transform=alpha_ax.get_xaxis_transform()) + alpha_ax.plot(scales[ind], 0, mfc=color, mec='k', alpha=0.75, zorder=6, + **plateau_dot_kwargs, transform=alpha_ax.get_xaxis_transform()) + alpha_ax.vlines(scales[ind], alpha_ax.get_ylim()[0], noise_data['measure_feat'][ind, i], + color=color, **plateau_line_kwargs) - # Add proxy legend: - if ax == alpha_ax: - h1 = ax.plot([], [], c='k', lw=lw['big'], label='$\\alpha\\cdot s(t) + \\eta(t)$')[0] - h2 = ax.plot([], [], c='k', lw=lw['big'], ls='dotted', label='$\\alpha\\cdot s(t)$')[0] - ax.legend(handles=[h1, h2], **leg_kwargs) +# Add proxy legend: +h1 = alpha_ax.plot([], [], c='k', lw=lw['big'], label='$\\alpha\\cdot s(t) + \\eta(t)$')[0] +h2 = alpha_ax.plot([], [], c='k', lw=lw['big'], ls='dotted', label='$\\alpha\\cdot s(t)$')[0] +alpha_ax.legend(handles=[h1, h2], **leg_kwargs) + +# Plot noise-song analysis results over sigma: +handles = sigma_ax.plot(noise_data['measure_inv'], noise_data['measure_feat'], lw=lw['big']) +[h.set_color(c) for h, c in zip(handles, shaded['feat'])] + +# Indicate threshold-specific saturation: +for i, ind in enumerate(saturation_inds): + color = shaded['feat'][i] + sigma_ax.plot(scales[ind], 0, c='w', alpha=1, zorder=5.5, **plateau_dot_kwargs, + transform=sigma_ax.get_xaxis_transform()) + sigma_ax.plot(scales[ind], 0, mfc=color, mec='k', alpha=0.75, zorder=6, + **plateau_dot_kwargs, transform=sigma_ax.get_xaxis_transform()) + sigma_ax.vlines(scales[ind], sigma_ax.get_ylim()[0], noise_data['measure_feat'][ind, i], + color=color, **plateau_line_kwargs) + +# Indicate sigma range capped by log-hp mechanism: +sigma_ax.axvspan(sigma_cap, sigma_ax.get_xlim()[1], **cap_kwargs) if save_path is not None: fig.savefig(save_path)