For the present investigation, time–voltage recordings of the EOD containing chirps generated spontaneously or evoked pharmacologically were analyzed. These data had been collected as part of a previous study examining the effect of urethane anesthesia on EOD frequency and chirping behavior in A. leptorhynchus (Eske et al. 2023).

Eight fish (total lengths: median, 116 mm; range 107–143 mm; body weights: median, 2.9 g; range 2.5–4.8 g) were used. Their EOD baseline frequencies varied between 683 Hz and 868 Hz (normalized to frequency values expected at 26 \(^\)C, using a Q\(_\) of 1.56). The morphological data and EOD frequencies indicate that the fish were approximately 1 year old and included both males and females (Ilieş et al. 2014; Zupanc et al. 2014).

Details of the experiments and the recording technique are given in Eske et al. (2023). Briefly, each fish was kept in an isolation tank in which a cylindrical plastic tube provided shelter. Differential recording of the fish’s EOD was done through a pair of stainless-steel electrodes mounted on the inside of the tube. During recording, the two open ends of the tube were closed with a coarse plastic mesh netting to ensure that the fish did not leave the tube.

The EOD of each fish was recorded for 30 min before, and 180 min immediately after, general anesthesia. State of anesthesia was induced by transferring the fish into a glass beaker containing 2.5% urethane dissolved in water from the fish’s isolation tank. During the pre-anesthesia session, spontaneous chirps occurred but at very low rates of approximately 1 chirp/30 min. Anesthesia induced a tremendous increase in chirping behavior, resulting, on average, in 1500 chirps during the 30 min immediately following anesthesia.

For the present analysis, the 30-min-pre-anesthesia recordings, and the 180-min-post-anesthesia recordings, of the 8 fish were combined, yielding a total of 1680 min of EOD recording. Employing the supervised learning algorithm, a total of 30,734 chirps were detected in these combined recordings.

The sampled voltage data \(\left( t_i, v_i\right) \), \(i=1, \ldots , M_\textrm\), were exported from Spike 2 and processed in MATLAB version R2021b. These data were filtered in 3-s windows with 2-s overlap using a bandpass filter with frequency band \([0.5, 1.5]\times f_0\), where the fundamental frequency \(f_0\) in each 3-s window was determined based on the power spectrum of the signal using fast Fourier transform and the “findpeaks” function of MATLAB.

Based on the zero-crossings of the filtered signal, we then computed the time, frequency, and amplitude values \(\left( T_k, f_k, A_k\right) \) associated with each \(k=1, \ldots , M,\) oscillation interval (for details, see Appendix A). An example of computed time-series data of frequency and amplitude is shown in Fig. 1.

EOD frequency f (a) and amplitude A (b) with respect to time T in a recording involving urethane anesthesia (for details of computation see Sect. “Calculation of EOD frequency and amplitude”). After baseline recording, the tube with the fish was transferred from the home tank to a glass beaker containing 2.5% urethane solution dissolved in aquarium water. As soon as the fish stopped undulating its anal fin and moving its opercula, it was returned to the home tank (arbitrarily defined as time point \(T=0\)). The gray bar indicates the time during which the fish was exposed to the anesthetic. Changes in the orientation and position of the fish relative to the recording electrodes result in noisy amplitude signals (pre-anesthesia, and \(T>\sim 2000\) s as shown in b). The reduction of noise immediately after anesthesia is related to the ceased movement of the fish. Note onset of type 2 chirping at higher rates immediately after anesthesia (a/A1, b/B1) that persists to approximately \(T=4600\) s after exposure to the anesthetic (a/A2, b/B2). The recorded signal contains both type 2 (a/A1’, b/B1’) and type 1 (a/A2’, b/B2’) chirps. The latter is characterized by large rise and negative undershoot in frequency (a/A2’), as well as a large drop in amplitude (b/B2’). By contrast, the former is characterized by a smaller rise without undershoot in frequency (a/A1’) and a smaller reduction in amplitude (b/B1’)

Data collection

Tuples of equal-time-length time-series data segments

$$\begin \textbf_(r-1)+j} \!= & \!\left( \!\left\, f_k^, A_k^\right\} : T_k^\!\in \!\left[ T_\textrm + (j-1)\Delta T, T_\textrm + j\Delta T\right] ,\right. \nonumber \\ & \left. k=1,\ldots , M-1 \right) , \quad j=1, \ldots , n_\textrm, \end$$

(1)

were collected from each recording \(r=1, \ldots , n_\textrm\), where \(n_\textrm\) is the total number of EOD recordings, and superscript \(\square ^\) indicates association with recording r. The time length of segments was determined as \(\Delta T = \left( T_\textrm-T_\textrm\right) \!/n_\textrm\). The values of parameters \(T_\textrm, T_\textrm, n_\textrm, n_\textrm\), used for the generation of time-series data segments are provided in Table 1.

Matlab tool built for collecting chirp samples from time-series frequency data (black dots). The user can select data points associated with a chirp by moving the cursor (intersection of black lines in a and c) to the two end points of the time interval delimiting the chirp instance. After selecting the time interval (red lines in b), the user must confirm the current selection before proceeding to collect further data points (see dialog box in b). Following the confirmation of the selection, data points associated with the selected time interval are stored and removed from the displayed data set (c). Once all displayed chirp instances have been collected, the user can move to the next (or previous), overlapping, time segment to collect the remaining chirp data points from the time-series frequency data segment

Using the MATLAB tool shown in Fig. 2, a person previously trained to identify chirps collected all chirp instances from each segment \(\textbf_i\) for all indices \(i\in \textbf_\textrm\), where the elements of subset \(\textbf_}\subset \left\n_\textrm\right\} \), with \(n_\textrm=\left| \textbf_}\right| \) (see Table 1), were randomly chosen, without replacement.

Although for each data point only time and frequency values were displayed during data collection (see Fig. 2), the associated amplitude values were also stored in the GT set of chirps

$$\begin \textbf = \left\, f_, A_\right\} \right) _^\right\} _^, \end$$

(2)

where \(\left\, f_, A_\right\} \) is the j-th data point of the i-th GT chirp sample, \(l_i\) denotes the number of data points in the i-th sample, and n is the total number of samples.

Data processing

The person who collected chirp samples was instructed to include, in each sample, data points prior to and after chirping, associated with the non-modulated, instantaneous “base” frequency of the fish. Hence, we assumed that each sample includes both pre and post-chirp data points and estimated the “base” frequency and amplitude of each sample i as

$$\begin f_, i}&= \textrm\left( \left\\right\} _^}, \left\\right\} _^}\right) , \end$$

(3)

$$\begin A_, i}&= \textrm\left( \left\\right\} _^}, \left\\right\} _^}\right) , \end$$

(4)

where \(n_\textrm < \underset(l_i/2)\) is an arbitrarily chosen positive integer which we set to \(n_\textrm=10\). We normalized each sample \(i=1, \ldots , n\) with respect to the maximum frequency rise according to

$$\begin \varphi _ = \frac - f_, i}} }\!\left( f_\right) - f_, i}}, \quad j=1, \ldots , l_i , \end$$

(5)

and with respect to the base amplitude as

$$\begin a_ = \frac - A_, i}}, i}}, \quad j=1, \ldots , l_i . \end$$

(6)

Then, we centered the time values of each sample according to

$$\begin & \tilde_:= T_ - T_, i}}, \quad j=1, \ldots , l_i, \end$$

(7)

$$\begin & j_, i} = \underset}\left( \left| H_-\frac\right| \right) , \end$$

(8)

$$\begin & H_ = \frac^h\!\left( \varphi _\right) }^h\!\left( \varphi _\right) }, \quad k=1, \ldots , l_i, \end$$

(9)

where rectifier

$$\begin h\!\left( \varphi _\right) = \frac\!\left( 1+e^\vert -\bar_)}\right) }_}, \end$$

(10)

with

$$\begin \bar_=4\max \!\left( \textrm\left( \left\\right\} _^}\right) \!,\, \textrm\left( \left\\right\} _^}\right) \right) , \end$$

(11)

was applied for the elimination of noise and to highlight “meaningful” parts of the frequency sample. Here \(\textrm\!\left( \cdot \right) \) denotes the standard deviation, \(\bar_\) is the cutoff value of normalized frequency associated with sample i and \(\delta =50\) is an arbitrarily chosen smoothing parameter.

Using the empirical cumulative distribution \(H_\) of rectified frequency values \(h\!\left( \varphi _\right) \), we trimmed each sample, such that only the data points j within interval \(\tilde_\in \left[ -3\Delta \tilde_i, 3\Delta \tilde_i\right] \) were kept, with

$$\begin & \Delta \tilde_i = \tilde__i} - \tilde__i}, \end$$

(12)

$$\begin & j^_i =\!\underset }}\!\left( \left| H_-0.9\right| \right) ,\nonumber \\ & j^_i =\!\underset }}\!\left( \left| H_-0.1\right| \right) . \end$$

(13)

Note that here \(\Delta \tilde_i\) is the difference between the 90% and 10% percentile estimates of the empirical cumulative distribution \(H_\). The above described data processing method is illustrated in Fig. 3.

Processing of “ground truth” samples (see Sect. ““Ground Truth” data set”). Data points \(\left\, f_\right) \right\} _^\) of the i-th sample are plotted in a as black dots. The frequency values \(\left\\right\} _^\) are normalized according to Eq. 5 and passed through the rectifier function (red curve) displayed in b. The green dashed lines in b and c display the cutoff value \(\bar_i\) of the rectifier function. The centered and normalized data points \(\left\_, \varphi _\right) \right\} _^\) of the i-th sample (see Eqs. 5–11) are displayed in c as black dots together with the rectified normalized frequencies (red curve) and their empirical cumulative distribution (blue curve). The 10% and 90% percentile estimates (blue, dashed lines in c) of this cumulative distribution determine the time width of the sample: \(\Delta \tilde_i = \tilde_-\tilde_\). The sample is trimmed based on this time width (d) such that data points outside interval \(\tilde_\in \left[ -3\Delta \tilde_i, 3\Delta \tilde_i\right] \) (delimited by black, dashed lines and marked by gray dots) are eliminated

Grouping and resampling

Because our supervised learning method requires uniform size among GT samples, we grouped and resampled all GT samples according to the number of data points that formed the individual GT samples.

After trimming, the size of each GT sample was roughly commensurate with the length of the associated chirp. To distinguish between chirps whose duration have different time scales, we divided GT samples into three groups and resampled the members of each r group such that associated samples contained \(10^r+1\) number of points:

$$\begin \textbf_r= & \left\, i}+j}, f_, i}+j}, A_, i}+j}\right) \right\} _^: \right. \nonumber \\ & \quad \left. \left| \left\_: \tilde_\!\in \!\!\left[ -3\Delta \tilde_i, 3\Delta \tilde_i\right] \!, 1\le j\le l_i\right\} \right| \! \in \!\left( 10^\!+\!1, 10^\!+\!1\right] \!,\right. \nonumber \\ & \quad \left. 1\le i\le n\right\} , \quad r=1, 2, 3. \end$$

(14)

Here we utilized the fact that all data points inside any GT sample can be located within the associated recording’s time-frequency-amplitude data. For example, if we know that \(T_\) and \(T_q\) are from the same recording and that \(T_ = T_q\), then we can find any other point j associated with sample i: \(\left( T_, f_, A_\right) = \left( T_, f_, A_\right) \).

Note that chirps typically have a duration shorter than 0.5 s, and the highest EOD frequency in A. leptorhynchus is approximately 1000 Hz, therefore GT sample groups \(\textbf_r\), \(r=1, 2, 3,\) are able to capture the full length of all chirps.

Principal component analysis

After resampling, we recomputed, according to Eqs. 3–6, the normalized frequencies and amplitudes \(\left( \varphi _, i}+j}, a_, i}+j}\right) , j=-10^r/2, \ldots , 10^r/2\), of each chirp sample i in each GT group \(\textbf_r\). For ease of notation, in the following, we drop the shift \(j_, i}\) in the second subscript index.

For each r, we collected from \(\textbf_r\) the normalized frequency and amplitude values

$$\begin \textbf_r^&= \left[ \varphi _, \ldots , \varphi _,\right] ^\textrm, \end$$

(15)

$$\begin \textbf_r^&= \left[ a_, \ldots , a_\right] ^\textrm, \end$$

(16)

of each sample i associated with the training set (for details about the training set, see Sect. “Cross-validation”) into a matrix \(\textbf_r\in }^\) such that

$$\begin \textbf_r^\textrm = \left[ \begin \textbf_r^ & \cdots & \textbf_r^\\ \textbf_r^ & \cdots & \textbf_r^ \end \right] , \end$$

(17)

where \(m_r\) is the total number of samples in \(\textbf_r\) associated with the training set. For the further ease of notation, in the following, we drop index r, as well.

We determined the principal components (PCs) \(\textbf_1, \ldots , \textbf_,\) of \(\textbf\) by performing the spectral decomposition of \(\textbf^\textrm\textbf\). Then we projected the training data set onto the space of the first N PCs, i.e., we computed

$$\begin \textbf = \textbf\textbf_, \end$$

(18)

where \(\textbf_N=\left[ \textbf_1, \ldots , \textbf_N\right] \).

Gaussian mixture model fitting

We modeled the projected data \(\textbf^\textrm=\left[ \textbf^, \ldots , \textbf^\right] \) using the Gaussian mixture model (GMM)

$$\begin \textbf^ \sim }\left( \varvec_c, \varvec_c\right) , \quad c\sim M_C\left( p_1, \ldots , p_C\right) , \end$$

(19)

where \(}\left( \varvec_c, \varvec_c\right) \) is the multivariate normal distribution of the c-th mixture component with mean \(\varvec_c\in }^\) and covariance \(\varvec_c\in }^\), while \(M_C\left( p_1, \ldots , p_C\right) \) is a multinomial distribution with C number of categories and mixing proportions \(p_1, \ldots , p_C\). We estimated the unknown parameters \(\varvec=\left\_1, \ldots , \varvec_C, \varvec_1, \ldots , \varvec_C\right\} \) of this GMM based on data \(\textbf\) using the “fitgmdist” function of MATLAB.

Elimination of outliers

After fitting the GMM, we assigned each data sample i to the cluster with maximum posterior probability, i.e., we computed the cluster of sample i according to

$$\begin c_i = \underset }}\!\left( P\left( c\vert i\right) \right) , \end$$

(20)

for each \(i=1, \ldots , m\), where \(P\!\left( c\vert i\right) \) is the probability that sample i belongs to cluster c, given the observation \(\textbf^\). Then, we computed the coefficient of determination (CoD) of the frequency component of each sample with respect to its assigned cluster mean as

$$\begin R^_ = 1 - \frac^-\bar}_\right\| ^2}^-\bar}^\right\| ^2}. \end$$

(21)

Here \(\left\| \cdot \right\| \) denotes the L2 norm and

$$\begin & \left[ \bar}_c, \bar}_c\right] ^\textrm = \textbf_N\hat}_c , \end$$

(22)

$$\begin & \bar}^=\frac\left( \textbf^\textrm\textbf^\right) \textbf , \end$$

(23)

with \(\textbf\) being a vector of 1-s.

We eliminated each cluster c for which the 5% percentile of associated CoD values \(\left\_: c_i=c, 1\le i\le m\right\} \) was below threshold \(\delta _=0.3\). Additionally, we eliminated each cluster c whose size \(\left| \left\ \right| \) was below threshold \(\delta _c=30\).

Training data set projected to the space spanned by the first two principal components (PC1 and PC2). Circles with different colors correspond to clusters identified by the algorithm. Gray crosses correspond to samples in an eliminated cluster. The percentage-wise size of kept (circles) and eliminated (crosses) clusters is indicated at the top left corner, relative to the size of the training set

Figure 4 illustrates the projected training data \(\textbf\) from \(\textbf_2\), with parameters \(N=2\) and \(C=5\); note the eliminated cluster.

Training yields PCs \(\textbf_N\) and GMM

$$\begin \textbf^ \sim }\left( \hat}_c, \hat}_c\right) , \quad c\sim M_\!\left( \tilde_1, \ldots , \tilde_\right) , \end$$

(24)

where \(C^*\le C\) is the number of kept clusters, with \(\tilde_c=\hat_c/\sum _^\hat_q\), and \(\hat_c,\hat}_c, \hat}_c\), being the estimated parameters of kept clusters \(c=1, \ldots , C^*\).

Illustration of the chirp detection methods described in Sect. “Detection”. Different rows correspond to different time instants (\(t_1<t_2<t_3<t_4\)) of the “sliding” time window indicated by vertical green lines in a. At each time instant, the Mahalanobis-distance-based detection algorithm (b) normalizes the data set inside the time window (green crosses in a) and projects it to the space spanned by the principal components of the training set (green cross in b). If the squared Mahalanobis distance value \(d^2\) associated with this projected point (indicated at the top of each row in b) is below the limit of the cluster with highest posterior probability (corresponding to the color-coded ellipse in b), then the Mahalanobis-distance-based algorithm may detect a chirp (2nd and 3rd row). At each time instant, the coefficient-of-determination-based algorithm (c) normalizes the data set inside the time window and computes its coefficient of determination with respect to each cluster mean. The highest coefficient-of-determination value \(R^2\) among all cluster means is indicated at the top of each row in c, and the related cluster mean is plotted (color-coded line in c). If this value is above a threshold, then the coefficient-of-determination-based algorithm may detect a chirp

To detect chirps in recordings, we analyzed data points \(\left\, f_, A_ \right) \right\}_^\), \(i=1, \ldots , M-10^r\) in a moving time window containing \(10^r+1\) samples (see Fig. 5a). At each instance i, we computed normalized frequency and amplitude values

$$\begin \textbf^&= \left[ \varphi _, \ldots , \varphi _,\right] ^\textrm, \end$$

(25)

$$\begin \textbf^&= \left[ a_, \ldots , a_\right] ^\textrm, \end$$

(26)

according to formulas Eqs. 3–6 with \(\left( T_, f_, A_\right) = \left( T_, f_, A_\right) \) and \(l_i = 10^r+1\).

Mahalanobis-distance-based detection

At each instance i, our Mahalanobis-distance-based (MDB) detection method first projects the normalized frequency and amplitude data onto the PCs according to

$$\begin \textbf^ = \textbf_N^\textrm \left[ \begin \textbf^\\ \textbf^ \end\right] , \end$$

(27)

then it determines the kept cluster which is most likely to generate \(\textbf^\):

$$\begin c_i = \underset }}\!\left( P\!\left( j\vert i\right) \right) . \end$$

(28)

Afterward, our method computes the Mahalanobis distance

$$\begin d_i = \sqrt^-\hat}_\right) ^\textrm \hat}_^\left( \textbf^-\hat}_\right) }. \end$$

(29)

For any point generated by kept cluster \(c_i\), realizations \(d_i^2\) follow a chi-squared distribution with N degrees of freedom: \(D_i^2\sim \chi ^2_N\).

The MDB method collects all i instances, where the squared Mahalanobis distance is below threshold \(\varepsilon _\) and the maximum frequency rise is above threshold \(\varepsilon _\), into the tuple

$$\begin \textbf_}= & \left( i: d_i^2<\varepsilon _, \underset\!\left( f_\right) -f_, i}>\varepsilon _,\right. \nonumber \\ & \quad \left. \left. i=1, \ldots , M - 10^r - 1 \right. \right) . \end$$

(30)

Each contiguous segment in \(\textbf_}\) corresponds to an identified chirp. In each contiguous segment, we associate the identified chirp with the instance i that has lowest distance \(d_i\). Threshold \(\varepsilon _\) is determined based on a chosen level of significance \(\alpha \) such that \(P\left( D_i^2<\varepsilon _\right) =1-\alpha \). The MDB method is illustrated in Fig. 5b.

Coefficient-of-determination-based detection

At each instance i, our coefficient-of-determination-based (CDB) detection method computes the CoD of the frequency component with respect to each kept cluster mean according to

$$\begin R^_ = 1 - \frac^-\bar}_\right\| ^2}^-\bar}^\right\| ^2}, \quad c=1, \ldots , C^*, \end$$

(31)

using formulae Eqs. 22–23, and assigns instance i to the cluster with highest CoD value:

$$\begin c_i = \underset }}\!\left( R^2_\right) . \end$$

(32)

Afterward, the CDB method collects all instances into the tuple \(\textbf_}\) where the CoD value and the maximum frequency rise are both above thresholds \(\varepsilon _\) and \(\varepsilon _\), respectively:

$$\begin \textbf_}= & \left( i: R^2_>\varepsilon _, \underset\!\left( f_\right) -f_, i}>\varepsilon _,\right. \nonumber \\ & \left. \left. i=1, \ldots , M - 10^r - 1 \right. \right) . \end$$

(33)

Similarly to the MDB method, identified chirps are associated with contiguous segments in \(\textbf_}\). In each contiguous segment, the identified chirp is assigned to the instance i that has the highest \(R^2_\) value. The CDB method is illustrated in Fig. 5c.

In order to assess the performance of the two algorithms detailed above, we chose, as a reference, the time-frequency-shape-based (TFSB) chirp detection algorithm described in (Eske et al. 2023). This algorithm is based on the chirp waveform function

$$\begin \varphi \!\left( \tilde;\tilde\right) = \dfrac \tilde}} \tilde}}, \end$$

(34)

which is assumed to characterize, during chirps, the normalized frequency \(\varphi \) with respect to time \(\tilde\). This function is parameterized by a single parameter \(\tilde\) that controls chirp duration (see Fig. 6).

Time–frequency shape function used for chirp detection in Eske et al. (2023). The time course of normalized frequency (see Eq. 5) during chirping is modeled by a single-parameter function \(\varphi \!\left( \tilde; \tilde \right) \). Different colors correspond to different shape parameter values \(\tilde \)

The TFSB algorithm has 7 hyper-parameters, out of which we fixed 5 (see Table 2), and the remaining 2 we determined via cross-validation (see Sect. “Cross-validation”).

To determine the optimal hyper-parameter values \(\textbf_\textrm\) of detection algorithms, we used k-fold cross-validation. In particular, we randomized indices \(i\in \textbf_\textrm\) associated with time-series data segments \(\textbf_i\) and split them onto k number of equal-size folds: \(\textbf_, q}\subset \textbf_\textrm\), \(q=1, \ldots , k\). For each iteration step \(q=1, \ldots , k,\) of cross validation, a single fold \(\textbf_, q}\) was used as a test set for determining the performance of the algorithm, while the rest of the folds were used as a training set. Note that only the two supervised algorithms were trained (for details, see Sect. “Training”), while the TFSB algorithm did not involve any training (Eske et al. 2023). The performance of each algorithm was determined by computing the false positive and false negative rates for each iteration step \(q=1, \ldots , k\), as

$$\begin \textrm_q&= \!\frac_\!, q}}\!\sum \limits _^, s}}\!\mathbbm \!\left( \left| \left\_j^\!\notin \!\left[ T_^, T_^\right] \!, 1 \le i \le m_,s}\right\} \right| \!\!=\!m_, s}\right) }_\!, q}}\!\!m_, s}}, \end$$

(35)

$$\begin \textrm_q&= 1 - \frac_\!, q}}\!\!\!\sum \limits _^, s}}\!\mathbbm \!\left( \left| \left\_j^\!\in \!\left[ T_^, T_^\right] \!, 1 \le j \le m_,s}\right\} \right| \!>\!0\right) }_\!, q}}\!\!m_, s}}, \end$$

(36)

where \(\mathbbm \!\left( \cdot \right) \) is the indicator function, \(\hat^_j\) denotes the j-th time instance of chirps detected by the algorithm in time-series data segment \(\textbf_s\), while \(T_^\) and \(T_^\) correspond to the first and last data point of the i-th chirp sample in \(\textbf_r\) collected from data segment \(\textbf_s\). Parameters \(m_, s}\) and \(m_, s}\) denote the total number of chirps detected by the algorithm in \(\textbf_s\), and collected manually from \(\textbf_s\), respectively. The overall performance of the algorithm was determined by averaging over all folds:

$$\begin \overline}(\textbf) = \frac\sum _^\textrm_q(\textbf), \quad \overline}(\textbf) = \frac\sum _^\textrm_q(\textbf). \end$$

(37)

Note that false positive and false negative rates depend on hyper-parameters \(\textbf\). We tuned the hyper-parameters such that for a given maximum tolerated average false positive rate \(r_\textrm\), the average false negative rate is minimized, i.e.,

$$\begin \textbf_\textrm\!\left( r_\textrm\right) = \underset\in \textbf\left( r_\textrm\right) }}\!\left( \overline}(\textbf)\right) , \quad \textbf(r_\textrm)=\left\\in \mathbf : \overline}(\textbf)\le r_\textrm\right\} , \end$$

(38)

where \(\mathbf \) is the search domain of hyper-parameters. At the maximum tolerated average false positive rate \(r_\textrm\), the lowest achievable average false negative rate is

$$\begin r_\textrm\!\left( r_\textrm\right) = \overline}\!\left( \textbf_\textrm\!\left( r_\textrm\right) \right) . \end$$

(39)

The implemented search domains of hyper-parameters are summarized in Table 3.