Categorical Loudness Scaling (CLS)

• Categorical Loudness Scaling is a psychophysical method that assigns auditory stimuli to ordered categories to depict loudness-growth functions.
• The ACALOS procedure applies an 11-point scale to derive key parameters such as threshold-adjacent, medium, and uncomfortable loudness levels from narrowband signals.
• CLS also serves as a structured feature space for cochlear implant modeling, audiogram classification, and calibration offset estimation in mobile hearing assessments.

Searching arXiv for the cited papers and closely related CLS work. Categorical loudness scaling (CLS) is a psychophysical framework for representing how perceived loudness changes with stimulus level by assigning stimuli to ordered loudness categories rather than relying only on detection thresholds. In the material considered here, CLS appears in two closely related forms. In rehabilitative audiology, adaptive categorical loudness scaling (ACALOS; ISO 16832) uses an 11-point categorical scale to measure loudness-growth functions from narrowband-noise presentations, yielding parameters such as threshold-adjacent levels, medium loudness levels, uncomfortable loudness levels, slopes, and dynamic range (Xu et al., 20 Aug 2025). In cochlear-implant (CI) modeling, categorical loudness is treated as a computationally derived perceptual label obtained from a modeled loudness-growth function, with threshold of hearing (THL) and most comfortable loudness (MCL) serving as category anchors (Alvarez et al., 29 Jan 2025). Recent work therefore positions CLS both as a direct psychophysical procedure and as a structured feature space for machine learning, mobile hearing assessment, and physiologically grounded simulation (Xu et al., 4 Dec 2025).

1. Conceptual basis and perceptual anchors

CLS operationalizes loudness as a category-valued outcome of a loudness-growth function. In the CI modeling framework, categorical loudness is explicitly described as a single-valued perceptual label derived from a stimulus-dependent loudness-growth function, and the resulting loudness index is used to define categorical regions such as THL and MCL (Alvarez et al., 29 Jan 2025). This makes THL and MCL more than descriptive landmarks: they are anchors for relating a modeled or measured growth curve to perceptual categories.

Within ACALOS, the loudness-growth function is the main output and describes how perceived loudness changes with sound level. The procedure asks listeners to rate loudness on an 11-point categorical scale comprising 7 labeled categories—“not heard,” very soft, soft, medium, loud, very loud, and “too loud”—plus 4 unlabeled intermediate categories (Xu et al., 20 Aug 2025). The numerical landmarks used in the cited work connect specific categorical units to derived parameters, including 2.5 categorical units for hearing threshold level, 25 categorical units for medium loudness level, and 50 categorical units for uncomfortable loudness level (Xu et al., 20 Aug 2025).

The cited studies also distinguish CLS from pure threshold audiometry. Traditional pure-tone audiometry requires calibrated equipment, controlled background noise, and supervised test conditions, whereas supra-threshold measures such as CLS are less dependent on exact absolute calibration because much of the information comes from relative loudness growth rather than threshold detection alone (Xu et al., 4 Dec 2025). At the same time, CLS is not presented as equivalent to threshold audiometry: the machine-learning study reports that ACALOS can approximate audiogram class only to a limited and clinically coarse extent, and the CI study states that its computational framework does not replace human CLS (Xu et al., 4 Dec 2025).

2. ACALOS procedure and parameterization

The ACALOS implementations described in the cited work use narrowband noises at 1500 Hz and 4000 Hz and derive a compact set of parameters from the loudness-growth function (Xu et al., 20 Aug 2025). In the audiogram-classification study, 12 features were extracted: for each of the two test frequencies, the level at 2.5 categorical units, the level at 25 categorical units, the level at 50 categorical units, the slope of the lower segment, the slope of the upper segment, and the breakpoint between lower and upper segments (Xu et al., 4 Dec 2025). In that formulation, the feature vector is

$$x = [L2.5_{1500}, L25_{1500}, L50_{1500}, Mlow_{1500}, Mhigh_{1500}, Lcut_{1500}, L2.5_{4000}, \ldots, Lcut_{4000}].$$

A central methodological distinction is between level-dependent and level-independent parameters. The calibration-offset study identifies Lcut, HTL (L2.5), MLL (L25), and UCL (L50) as level-dependent or calibration-variant, because they shift with absolute presentation level. By contrast, mlow, mhigh, and DR are treated as level-independent or calibration-invariant, with dynamic range defined as

$DR = UCL - HTL.$

This distinction is the basis for later machine-learning and calibration-estimation applications (Xu et al., 20 Aug 2025).

Parameter	Definition in the cited work	Calibration behavior
HTL (L2.5)	level at 2.5 categorical units	level-dependent
MLL (L25)	level at 25 categorical units	level-dependent
UCL (L50)	uncomfortable loudness level at 50 categorical units	level-dependent
Lcut	transition level between low- and high-level slopes	level-dependent
mlow, mhigh	low-level and high-level slopes	level-independent
DR	$UCL - HTL$	level-independent

The calibration argument is geometric. A calibration offset is an additive shift applied to all presented levels. Absolute level quantities therefore move with the offset, whereas slopes and level differences are comparatively preserved because they reflect the shape of the loudness-growth function rather than its absolute position on the level axis (Xu et al., 20 Aug 2025). This property is why CLS can be used not only to quantify loudness perception, but also to infer hidden variables such as device calibration error.

3. Computational categorical loudness in cochlear implants

In CI research, categorical loudness has been cast as a model-based perceptual surrogate derived from simulated peripheral neural activity rather than directly from the electrical stimulus. The cited model begins with a 3D peripheral auditory model of the implanted cochlea in which electrical pulses are transformed into a voltage field through a finite-element electrode–nerve interface model and then into simulated auditory nerve fiber (ANF) spikes through a phenomenological spiking model (Alvarez et al., 29 Jan 2025). The ANF spikes are grouped by cochlear place into 40 groups, each formed by 750 adjacent ANFs, which are interpreted as a physiological analog of auditory filters.

Peripheral neural excitation is defined as

$E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$

where $\text{SA}_j(t)$ is the spike-activity signal for ANF $j$. Loudness contribution is then modeled as

$\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$

with $k=0.525$, and with the nonlinear term

$$\epsilon(t)=e^{\frac{E_T(t)-E_0}{m}}, \qquad \text{where } E_T(t)=\sum_x E_x(t),$$

using $E_0=12$ and $DR = UCL - HTL.$0 (Alvarez et al., 29 Jan 2025). The cited interpretation is that loudness grows near-linearly at low stimulation but increases exponentially as activity is recruited across more cochlear places, thereby representing spectral loudness summation.

Temporal integration is applied to loudness contribution rather than to raw neural activity:

$DR = UCL - HTL.$1

where $DR = UCL - HTL.$2 is an asymmetric temporal window with parameters $DR = UCL - HTL.$3 ms, $DR = UCL - HTL.$4 ms, $DR = UCL - HTL.$5 ms, and $DR = UCL - HTL.$6. Instantaneous loudness is then

$DR = UCL - HTL.$7

For scalar comparison across stimuli, the loudness index $DR = UCL - HTL.$8 is defined as the 99th percentile of $DR = UCL - HTL.$9 over stimulus duration (Alvarez et al., 29 Jan 2025). The model notes that $UCL - HTL$0 can be interpreted as twice as loud as $UCL - HTL$1, and the logarithm of $UCL - HTL$2 is used to emphasize the roughly linear growth with stimulation current at lower levels and the faster rise once the exponential term dominates.

THL and MCL are defined from the loudness-growth function generated by the model for each electrode and several stimulation rates, specifically 250, 500, and 1000 pps. The knee point between linear and exponential growth is identified on the 500 pps curve, and the scaling factor $UCL - HTL$3 is adjusted so that the $UCL - HTL$4 point lies above that knee. The paper also discusses an alternative geometric criterion used in prior work: THL as the level that excites an equivalent basilar-membrane width of about 1 mm and MCL as about 4 mm. The model’s THL is reported as roughly consistent with the 1 mm criterion for mid and apical electrodes, whereas its MCL corresponds to a substantially broader excitation spread than 4 mm, suggesting that the 4 mm criterion may underestimate the excitation needed for comfortable loudness in CI users (Alvarez et al., 29 Jan 2025).

Validation was carried out against psychophysical experiments in real CI users. For stimulation-rate effects, simulated THL and MCL were examined across 156 to 5000 pps and compared with data from Zhou et al.; the model captured the general increase in dynamic range with increasing rate and matched MCL trends particularly well, while THL at high rates showed some discrepancy. For electrode separation, the model reproduced McKay’s two-electrode interleaved-stimulation experiment and found effects below 1 dB, consistent with the human data. For amplitude modulation, it reproduced McKay’s loudness-balancing experiment and captured the qualitative result that modulated stimuli are influenced mainly by peak pulses, especially at high stimulation levels, deeper modulation, and lower carrier rates (Alvarez et al., 29 Jan 2025).

4. CLS as a feature space for audiogram-type classification

A distinct use of CLS is to infer coarse hearing-profile classes from loudness-growth features. The cited study used a large auditory reference database containing ACALOS data and targeted standard Bisgaard audiogram types (Xu et al., 4 Dec 2025). After merging audiogram data and ACALOS loudness-scaling data by participant and ear side, and after exclusions, the final modeling dataset contained $UCL - HTL$5 ear-specific records and six retained Bisgaard classes: N2, N3, N4, S1, S2, and S3. The original 10 Bisgaard types were reduced by excluding normal-hearing listeners with PTA $UCL - HTL$6 dB HL and removing rare classes N1, N5, N6, and N7, each with fewer than 5 percent of cases (Xu et al., 4 Dec 2025).

Unsupervised structure was examined with principal component analysis on the standardized 12 loudness-scaling features. With $UCL - HTL$7 as the standardized data matrix, PCA is written as

$UCL - HTL$8

and the covariance matrix is

$UCL - HTL$9

The first two principal components explained more than 50 percent of the variance, specifically 43.2 percent for PC1 and 16.2 percent for PC2, or roughly 59.4 percent together (Xu et al., 4 Dec 2025). The loading plot showed most features loading positively on PC1, while $E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$0 and $E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$1 were among the few negative contributors. $E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$2 and $E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$3 clustered near each other, $E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$4, $E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$5, and $E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$6 formed another cluster, and $E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$7 lay closest to the origin. The score plot showed substantial overlap among classes rather than clean separation.

Supervised classification then treated the task as a six-class problem and evaluated seven classifiers: Decision Tree, Gradient Boosting, K-Nearest Neighbors, Logistic Regression, Neural Network, Random Forest, and Support Vector Machine. Features were standardized to mean 0 and SD 1, evaluation used 10-fold cross-validation, and multiclass handling used One-vs-Rest, with each class treated as positive versus all others negative and the final label chosen by highest predicted probability (Xu et al., 4 Dec 2025). Performance was assessed with balanced accuracy,

$E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$8

and classwise

$E_x(t)=\frac{1}{N}\sum_{j\in x}\text{SA}_j(t),$9

together with confusion matrices, ROC curves and AUC, precision-recall curves and average precision, and micro-average One-vs-Rest AUC and AP for multiclass evaluation.

On the test set, all models performed only moderately, with balanced accuracy and weighted F1 roughly in the 0.32–0.53 range. Logistic regression performed best, with balanced accuracy $\text{SA}_j(t)$0 and weighted F1 $\text{SA}_j(t)$1, while KNN performed worst, with $\text{SA}_j(t)$2 and $\text{SA}_j(t)$3 (Xu et al., 4 Dec 2025). The training–test gap was substantial for several flexible models—for example, Random Forest had training BA 0.98 and test BA 0.45, Decision Tree had training BA 0.90 and test BA 0.39, and SVM had training BA 0.95 and test BA 0.39—indicating that the weak separability of the underlying class structure, rather than model class alone, limited performance.

Classwise behavior reflected overlap between neighboring Bisgaard profiles. N3 was the easiest class to classify overall; S1 was relatively well classified by logistic regression and SVM; N4 was one of the hardest classes; and S3 showed difficulty and large variability. The confusion matrix for logistic regression highlighted ambiguity especially around N2 versus S1 and N4 versus S3 (Xu et al., 4 Dec 2025). For the best model, the micro-average AUC was 0.84 and the micro-average AP was 0.48, while per-class ROC AUC ranged roughly from 0.77 for N2 and N4 to 0.90 for S3.

Explainable modeling further localized what CLS features carried the most information. SHAP and permutation feature importance, applied mainly to the logistic-regression model, identified $\text{SA}_j(t)$4, $\text{SA}_j(t)$5, and $\text{SA}_j(t)$6 as the most important features, while $\text{SA}_j(t)$7, $\text{SA}_j(t)$8, and $\text{SA}_j(t)$9 were among the least important (Xu et al., 4 Dec 2025). The central finding was that threshold-proximal loudness measures, especially at 4 kHz, were most informative, whereas high-level slope measures were least informative.

5. Calibration-offset estimation in mobile hearing tests

CLS has also been used to estimate smartphone or device calibration offsets when the playback chain is not clinically calibrated. The cited study used data from the Oldenburg Hearing Health Repository (OHHR), comprising $j$0 participants with mean age 70.0 years (SD 8.7), including 556 male and 291 female listeners with diverse hearing profiles; all had PTA4 $j$1 dB HL and were categorized according to Bisgaard et al. (2010) (Xu et al., 20 Aug 2025). Because all data were originally collected under fully calibrated laboratory conditions, the repository served as a reference set for simulating uncalibrated mobile tests.

Calibration offsets were simulated from Gaussian distributions with means of 5 dB or 10 dB and SDs of 5 dB or 10 dB, giving four scenarios: $j$2, $j$3, $j$4, and $j$5. These offsets were added only to the calibration-dependent CLS parameters—Lcut, HTL, MLL, and UCL—while $j$6, $j$7, and $j$8 were left unchanged (Xu et al., 20 Aug 2025). The predictive feature set therefore consisted of six level-independent variables across the two frequencies: $j$9, $\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$0, $\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$1, $\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$2, $\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$3, and $\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$4.

The primary model was Bayesian regression implemented with the brms package in R. The regression was trained on calibrated OHHR data to predict $\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$5 from the invariant CLS features, conceptually as

$\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$6

separately for $\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$7 (Xu et al., 20 Aug 2025). For uncalibrated data, the estimated calibration offset was taken as the discrepancy between predicted and observed $\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$8; when both frequencies were used, the predicted $\lambda_x(t)=k\cdot E_x(t)\cdot(1+\epsilon(t)),$9 values were averaged across 1500 Hz and 4000 Hz. A simpler nearest-neighbor method computed the RMSE between a participant’s six invariant CLS features and those of all participants in the calibrated database, selected the nearest neighbor, and estimated offset from the difference in $k=0.525$0 (Xu et al., 20 Aug 2025).

Evaluation used the Pearson correlation coefficient, p-values, median absolute error in dB, and a calibration uncertainty reduction factor defined as

$k=0.525$1

For predicting empirical $k=0.525$2 from invariant features, the regression correlations were weak: $k=0.525$3 at 1500 Hz and $k=0.525$4 at 4000 Hz in one reported analysis, and $k=0.525$5 at 1500 Hz and $k=0.525$6 at 4000 Hz in the study’s self-consistency summary (Xu et al., 20 Aug 2025). By contrast, prediction of threshold-like $k=0.525$7 was much stronger, with $k=0.525$8 at 1500 Hz and $k=0.525$9 at 4000 Hz. The paper interprets this as evidence that the CLS function contains a strong relationship between shape descriptors and threshold-related parameters, even when medium loudness level is only weakly predictable.

For offset estimation itself, the Bayesian regression model achieved $$\epsilon(t)=e^{\frac{E_T(t)-E_0}{m}}, \qquad \text{where } E_T(t)=\sum_x E_x(t),$$0 when the true offset SD was 5 dB and $$\epsilon(t)=e^{\frac{E_T(t)-E_0}{m}}, \qquad \text{where } E_T(t)=\sum_x E_x(t),$$1 when the true offset SD was 10 dB, with the same pattern across mean offsets of 5 dB and 10 dB (Xu et al., 20 Aug 2025). The nearest-neighbor model also worked, but with lower correlations around $$\epsilon(t)=e^{\frac{E_T(t)-E_0}{m}}, \qquad \text{where } E_T(t)=\sum_x E_x(t),$$2–0.69. In the $$\epsilon(t)=e^{\frac{E_T(t)-E_0}{m}}, \qquad \text{where } E_T(t)=\sum_x E_x(t),$$3 scenario, the Bayesian model had MAE about 5.06 dB overall, compared with about 6.59 dB overall for nearest neighbor. Calibration uncertainty reduction factors were 0.55–0.79 for Bayesian regression and 0.41–0.68 for nearest neighbor. The study further reports that using both 1500 Hz and 4000 Hz improved estimation relative to single-frequency models and reduced calibration-offset estimation error by about 2.5 dB (Xu et al., 20 Aug 2025).

6. Interpretive limits and broader significance

Across the cited work, CLS is consistently portrayed as informative but not exhaustive. In audiogram classification, the principal limitation is overlap in loudness patterns across audiogram types. The PCA score plot showed no clean class boundaries, and the study explicitly argues that loudness scaling is only indirectly related to pure-tone thresholds; even the strongest predictors, especially $$\epsilon(t)=e^{\frac{E_T(t)-E_0}{m}}, \qquad \text{where } E_T(t)=\sum_x E_x(t),$$4, are imperfect threshold surrogates (Xu et al., 4 Dec 2025). This is why the study characterizes its results as a form of performance ceiling: even a perfect machine-learning model cannot recover all audiogram information from supra-threshold loudness data alone.

The practical conclusion in that domain is correspondingly narrow. ACALOS data can approximate audiogram class and may support initial hearing-aid fitting, rough stratification, or remote and self-administered workflows, but they are not accurate enough to replace traditional audiometry for full diagnostic use (Xu et al., 4 Dec 2025). In mobile calibration, the conclusion is more favorable because the target is different: CLS is used not to replace threshold testing, but to estimate the missing calibration factor from relative loudness-growth structure. The cited paper therefore presents CLS as more robust than threshold-based approaches in uncontrolled environments, because slopes and dynamic range are largely independent of absolute level (Xu et al., 20 Aug 2025).

In CI modeling, the limitation is again one of role rather than validity. The computational framework is explicitly described as a complement to psychophysical categorical loudness scaling, not a substitute for human data (Alvarez et al., 29 Jan 2025). Its value lies in generating a model-based categorical loudness scale from an electrode–nerve interface plus ANF simulation, thereby allowing in silico estimates of perceptual loudness categories and comparisons among stimulation strategies. This suggests a broader methodological position for CLS: it can function as a psychophysical protocol, a feature representation for statistical learning, and a target construct for physiologically grounded simulation, provided that its inferential limits are kept explicit.