Recently, sound quality assessment using subjective measures has focused on identifying sound quality metrics that can predict subjective responses. The goal is to design a numerical prediction model that can replace listening tests. In practice, it then becomes possible to predict the sound quality perception of a panel of representative users for a new sound or a new sound design.

The principle is to link the detailed explanation of the properties of the sound (subjective evaluation) with the psychoacoustic indicators of the stimuli used in listening tests (objective evaluation). Two types of approaches are used for the objective prediction of sound quality.

The first approach involves correlation and regression analyses using a pool of preselected metrics that can provide meaningful models for engineers. Most of the research reported in the literature on the creation of objective models of sound quality is based on the theory of multiple linear regression (Otto et al., 2001; Kwon et al., 2018; Jiang and Zeng, 2014).

The second approach leverages recent advancements in machine learning and deep learning. For instance, Huang et al. demonstrated that convolutional neural networks (CNN) can be used for the sound quality prediction of interior noise (Huang et al., 2020). Other neural networks have been applied to sound quality prediction, including back propagation neural networks (BPNN) (Huang et al., 2021), radial basis function (RBF) neural networks (Xiong et al., 2015), or even genetic algorithms (Chen et al., 2022). However, despite the promise of these approaches, their practical application in product design to improve sound quality perception remains challenging (Lee, 2008; Paulraj et al., 2013; Wang et al., 2014) because the integration of machine learning and neural network models into the product design process often requires extensive and high-quality training data, which may not always be available or easy to obtain.

This paper presents an attempt to overcome these polarized limitations by seeking a realistic and pragmatic in-between solution that can be used by NVH engineers without requiring extensive and high-quality training data.

The linear regression method is based on a least-square (LS) approach that minimizes the prediction error. However, as with any classical LS solution to a problem with many potential predictors, all the predictor coefficients will be part of the solution, which can lead to overfitting. To address this issue and to produce a more parsimonious predictive model, more advanced methods adapted to the problem of sound quality have been developed (Gauthier et al., 2017). For instance, techniques such as regularization are used to simplify the model by performing a pseudo-inversion of the matrix of potential predictors, thereby selecting only the most relevant predictors.

Parsimonious models are preferred due to their simplicity, interpretability, and reduced risk of overfitting, allowing for a better understanding of data and more efficient, generalizable, and computationally manageable analyses. In this research work, the authors investigated parsimonious selection and extraction of sound quality/significance predictors using Lasso/Elastic-net (Tibshirani, 1996; Zou and Hastie, 2005) and Group-Lasso (Tibshirani and Taylor, 2011). The Lasso corresponds to a convex optimal problem with regularization of the 1-norm of the solution. The Elastic-net is a convex minimization problem with a weighted regularization of the 2-norm and 1-norm of the solution. Finally, the Group-Lasso is a structured parsimony approach that promotes inter-group parsimony via the 1-norm of the 2-norm of the (predefined) predictor groups (Friedman et al., 2010).

Despite its effectiveness in many applications, Lasso regressions, introduced by Tibshirani (Tibshirani, 1996), are known to have some limitations. Mainly, if there is a group of highly correlated predictors, the Lasso tends to select only any one of these predictors in the group. The Elastic-net approach proposed by Zou and Hastie (Zou and Hastie, 2005) is a variant of the Lasso regression. It improves the Lasso predictions and enhances the ability to do grouped selection. The Lasso and Elastic Net approaches used in this research work incorporate cross-validation steps, ensuring the selection of optimal models and reducing the risk of overfitting (see Section 3.2). This cross-validation process enhances the reliability of the predictive models for sound quality assessment.

Two families of metrics were used as predictors: 1) physical and psychoacoustic metrics, hereafter referred to as engineering metrics, and 2) the metrics from the MIR (Music Information Retrieval) library used for the extraction of audio and musical characteristics from digital audio files. Among the engineering metrics, the global loudness in sone, the specific loudness on the Bark scale in the 24 frequency bands between 20 Hz and 1,550 Hz, the fluctuation strength, the sound pressure level in dB of the third-octave band spectrum in the 29 frequency bands between 20 Hz and 16,000 Hz, roughness and sharpness were chosen. The MIR metrics used in this research project come from the library MIRtoolbox 1.6.1 for Matlab (Lartillot, 2014). These metrics are grouped into categories: tonality, timbre, rhythm, dynamics, and pitch.

Three statistical variants of each of the metrics were calculated and used as predictors: the mean value over the time duration of the sound sample (Mean), the standard deviation (Std), and the slope over time (Slope). The slope is defined as the linear trend along frames, which is the derivative of the line that best fits the curve. Specifically, the slope S is computed using a normalized representation of the curve C —centered, with unit variance, and scaled to a temporal series T of values between 0 and 1. This is solved as a least-squares solution to the equation S ∗ T = C . The choice of using such metrics is motivated by the need to include significant variables, such as the slope, for acceleration sounds (time-varying signals).

For metric calculation and and their variants, the signal was decomposed into frames, with a frame length of 50 ms and 50% overlap for the MIR metrics, and a 30 ms frame length for the engineering metrics. This frame-based analysis enables the extraction of metrics that are representative of the signal’s composition in both the time and spectral domains, ensuring an accurate capture of its characteristics.

Specific loudness and global loudness values were calculated according to the ISO532B model for stationary sounds (Zwicker et al., 1991) (for constant speed and idle) and according to the model of (Zwicker and Fastl, 1999) for non-stationary sounds (acceleration). Also, the value of the sharpness of a sound, and the value of variable sharpness in time, were calculated according to the procedure proposed by Fastl (derived from Zwicker) with the correction of (Aures, 1985). A total number of 182 metrics (engineering, MIR, including variants for non-stationary signals) were available in the bank of potential model candidates. The temporal variations of the metrics for stationary sounds (idle, constant speed) were then removed, resulting in a total of 127 metrics for these two stationary conditions.

The linear model used to create sound quality models is defined in matrix form and indices: v i = ∑ m = 1 M F i m b m + b + e i , (1) with v i the ith component of the vector of scores obtained by listening test statistics ( I sounds), F i m the ith component of the matrix of metrics or scalar measures of M potential predictors for the I sounds, b m the coefficients to be solved as a vector (linear regression coefficients), b the intercept (the score when all the coefficients b m are null) and e i the prediction errors for each ith sound. Equation 1 can be solved using linear regression. In the case of wide data with much fewer observations than predictive metrics ( I < < M ) , classical linear regression based on a least-square approach will typically achieve over-fitting and all the coefficients b m will be nonzero. As mentioned in the introduction, this strongly limits the interpretation of the resulting model for any engineering use or design guidelines. The aim of the Lasso and Elastic-net is to circumvent this issue by rewriting the problem as a composite cost function (Tibshirani, 1996): J λ , α = 1 2 I ∑ i = 1 I v i − b − ∑ m = 1 M b m F i m 2 + λ ∑ m = 1 M 1 − α 2 b m 2 + α | b m | , (2)

The first right-hand side term corresponds to the quadratic sum of the predictor errors and the second right-hand side term is a regularization term with regularization amount λ > 0 . The regularization combines 2-norm regularization b m 2 and 1-norm regularization | b m | based on the Elastic-net parameter 0 ≤ α ≤ 1 . For the Lasso, α = 1 , only the 1-norm regularization is included. This induces solution sparsity, i.e., few coefficients b m will be non-zero and will be therefore selected in the model (Tibshirani, 1996). The sparsity is controlled by the regularization amount λ . The Elastic-net (Zou and Hastie, 2005) involves 0 < α < 1 and will typically smooth out the selection towards the extreme case of 2-norm-only regularization with α = 0 (this corresponds to Tikhonov regularization). The minimum of the composite cost function cannot be found by analytical means since it is nonlinear. Therefore, this is solved iteratively using coordinate descent algorithms (Wright, 2015). In Equation 2, α is a user-selected parameter. Note that in the following, to create the simplest and parsimonious models, only the Lasso results are shown ( α = 1 ) . The penalization amount λ is set rigorously using the following procedure: The experimenter defines a limit on the maximum number of metrics that should be included in the model. Here, the aim is to make sure that the model is readable and is meaningful for engineers that can latter use it for product modification. Next, the problem is solved for a wide range of λ starting with the largest λ so that all coefficients b m are zeros, and decreasing in successive iterations. The λ iteration stops if: 1) the maximum number of non-zero coefficients b m is reached or 2) the minimum prediction error is obtained for k -fold cross-validation of the prediction. Cross-validation is used to avoid overfitting of the model. In this paper, the maximum number of predictive metrics is 7 and the cross-validation is 5-fold.

Cross-validation is a statistical technique used in machine learning and model evaluation. It involves dividing a dataset into subsets, typically a training set to build a predictive model and a validation set to assess its performance. This process is repeated multiple times, each time with a different subset as the validation set and the rest as the training set. The results are then averaged, providing a robust estimate of the model’s performance, helping to mitigate overfitting, and ensuring the model’s generalizability to unseen data. Cross-validation is crucial for selecting the best model and optimizing its hyperparameters while avoiding data leakage and providing a more accurate assessment of predictive performance.

Note that in this study, all seven sounds were retained for the training of the models. While this approach may limit our ability to test the predictive power of the models on new sounds, it is important to highlight that this was not a primary focus or requirement of the study. As a reminder, the main objective of this research is to develop simple objective models that can provide an objective understanding of the measured sounds of side-by-side vehicles. By including all available sounds in the training dataset, we aimed to capture the full range of sound characteristics and ensure a comprehensive analysis within the scope of this study.

As can be seen in Table 1, the Lasso models were able to predict the Powerful attribute with a coefficient of determination of 91% using only three predictors, the Aggressive attribute with a coefficient of determination of 86% using only one predictor, the Metallic attribute with a coefficient of determination of 84% using only two predictors, the Soft attribute with a coefficient of determination of 97% using only two predictors, the Vibrating attribute with a coefficient of determination of 85% using only three predictors, the Noisy attribute with a coefficient of determination of 95% using only two predictors, and the overall Desire-to-Buy with a coefficient of determination of 88% using only two predictors. The models derived from Lasso were also able to predict the average scores of PC1 with a coefficient of determination of 81% with only two predictors, and the average scores of PC2 with a coefficient of determination of 88% with only two predictors. All predictors were selected by the Lasso algorithm, which identifies the most relevant metrics (i.e., those that minimize prediction error) from the 127 available, with a maximum of three metrics per model.

The high adjusted coefficients of determination (adjusted R 2 ) and low p-values indicate that each of these models can be considered statistically reliable. Based on the model’s estimated coefficients, the following conclusions can be drawn regarding the constant speed condition:

• Powerful attribute: the specific loudness of Bark 4 positively impacts this model, while the specific loudness of Bark 14 and the level (dB) per 1/3 octave band of the 4,000 Hz band (3,548 Hz–4,467 Hz) negatively impact it. This means that 1) an increase in the energy in this low-frequency range (Bark 4) makes the sound feel more powerful and robust, 2) excessive energy in this higher frequency range (Bark 14) can make the sound feel less powerful, and 3) higher levels of energy in the 4,000 Hz band can reduce the overall sense of power.

As can be seen in Table 2, the Lasso models were able to predict the Powerful attribute with a coefficient of determination of 91% using only one predictor, the Aggressive attribute with a coefficient of determination of 91% using only two predictors, the Metallic attribute with a coefficient of determination of 86% using only two predictors, the Soft attribute with a coefficient of determination of 95% using only two predictors, the Vibrating attribute with a coefficient of determination of 83% using only one predictor, the Noisy attribute with a coefficient of determination of 76% using only one predictor, and the overall Desire-to-Buy with a coefficient of determination of 75% using only two predictors. The models derived from Lasso were also able to predict the average scores of PC1 with a coefficient of determination of 83% with only one predictor, and the average scores of PC2 with a coefficient of determination of 93% with only two predictors. Here again, all predictors were selected by the Lasso algorithm, which identifies the most relevant metrics (i.e., those that minimize prediction error) from the 127 available, with a maximum of three metrics per model.

The high adjusted coefficients of determination (adjusted R 2 ) and low p-values indicate that each of these models can be considered statistically reliable. Based on the model’s estimated coefficients, the following conclusions can be drawn regarding the acceleration condition:

• Powerful attribute: the slope (over time) of the specific loudness of Bark 11 positively impacts this model. This suggests that a steady increase in the energy in the Bark 11 band correlates with a more powerful perception of the sound during acceleration. This indicates that an increase in energy over time within this mid-frequency range (Bark 11) contributes to a more powerful and dynamic perception of the sound during acceleration.

In this study, we were also interested in communicating simply the meaning of the objective models using a simple holistic visual representation.

To this end, Figures 4, 5 present a summary of the sound signature and sound quality models for the constant speed and the acceleration conditions, respectively. These figures provide a visual illustration of the different models for predicting the sound signature and sound quality of SSVs with the metrics selected in each model. These figures also show the effectiveness of the Lasso in selecting only a few metrics to build these parsimonious models from a large metric bank.

From the models developed in this paper, it is now possible to retrieve the sound signature and sound quality of current SSVs and predict those of sounds measured on new SSVs or with virtually modified sounds. However, comparing vehicles based on all these models (all metrics in all three conditions) is likely to be confusing. Therefore, to simplify this, i.e., to make it easier to grasp the contribution of each metric and to compare between sounds, we created a tool named “attribute wheel of SSVs.” This tool allows one to easily visualize the different metrics, their contributions, the corresponding perceptual attributes, and the operating conditions for a given vehicle. It is typically read from the center to the circumference. This wheel of attributes is equivalent to the wine aroma wheel (Noble et al., 1987). As an example, Figure 6 shows the attribute wheel for the second vehicle (named V2 hereafter). The attribute wheel has three levels (the three circular rings in Figure 6):

1. The operational conditions are represented by three different categories: Constant speed (Cst speed), idle, and acceleration. Note that models for the idle condition are not reported in this paper.

For instance, for the acceleration condition of V2, shown in Figure 6, the Aggressive attribute has the largest score compared to the other attributes. The Aggressive objective model involves the two metrics N′ B12 Slope and N′ B11 Slope. These two metrics are the band-specific loudness slopes of Bark bands 11 and 12 (1,270 Hz–1720 Hz), respectively. This result suggests that the aggressiveness of the acceleration sound is predicted by the time variation of the spectral content in these two Bark bands. The graph also suggests that the effect of the N′ B12 Slope predictor is much larger than the N′ B11 Slope predictor in the model. The graph also shows that the same two metrics are involved in the Desire-to-buy objective model for this condition. Therefore, any positive variation in these two metrics will result in positive variations of the Aggressive attribute and Desire-to-buy. The inclusion of Desire-to-buy as one of the perceptual attributes in Figure 6 may seem odd. Specifically, in the idle condition, the features selected for the Desire-to-buy model do not match those of any of the perceptual attribute models. However, it should be noted that the idle condition may not be as critical in influencing the overall “desire to purchase” factor as the acceleration condition. Interestingly, in the acceleration condition, the two parameters used in the Desire-to-buy model are identical to those of the Aggressive perceptual attribute. This observation suggests a potential correlation between the perception of aggressiveness and the Desire-to-buy during acceleration, which could be a valuable element for further study.