## 4 Discussion  
In this study, we aimed to demonstrate the potential of the detrended cross-correlation function for capturing neural entrainment in continuous speech. The EEG signals were predicted from the speech envelope (univariate forward model) and speech spectrogram (multivariate forward model). The detrended cross-correlation function performed significantly better within a meaningful lag range than the baseline model computed from the baseline period. Furthermore, the proposed model has computational advantages while achieving the same level of prediction performance as the widely used ridge regression model. Ridge regression implemented in the mTRF toolbox [18] served as a reference for our study. It is widely applied in sensory neuroscience to model brain responses to continuous stimuli [31, 32]. Our results showed that the detrended cross-correlation function resulted in TRFs or speech STRFs comparable to those of ridge regression. It also exhibited a similar performance to individual models in terms of the Pearson correlation coefficient between the predicted EEG and original EEG. Moreover, the detrended cross-correlation function can more efficiently interpret neurophysiological data. The detrended cross-correlation function has a significant advantage in terms of computational time for both the univariate and multivariate forward models (Figure 4A). Moreover, ridge regression requires cross-validation to determine the optimal value of the regularization parameter, which can be computationally expensive if a wide range of parameter values are considered. Specifically, the TRFs estimated by ridge regression are sensitive to the regularization parameter, which can lead to unstable results, depending on the range of values chosen for regularization. In contrast, the detrended cross-correlation function is a parameter-free model that does not require a parameter optimization procedure. Previous studies applied cross-correlation analysis to investigate how the brain perceives attended and unattended speech [33, 34]. Although cross-correlation analysis is more straightforward than the ridge regression method, it is better suited for inputs modulated by a stochastic process. Because of this limitation, ridge regression has been widely applied in sensory neuroscience for continuous stimuli [31, 33, 35, 36]. In this study, we proposed a detrended cross-correlation function to investigate neural entrainment to speech envelopes. This method overcomes the limitations of conventional cross-correlation analysis in analyzing nonstationary or slow-modulating signals while maintaining its straightforwardness. Additionally, the estimations obtained from the detrended cross-correlation function were comparable to those of ridge regression, demonstrating the utility of the detrended cross-correlation function in auditory scene analysis research. The short trial duration may lead to overfitting, particularly in the large response to speech onset. Thus, even though the results of this study revealed a significant degree \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Computational Time (s)**} & \multicolumn{3}{c}{**Validation Pearson Correlation**} \\ \cline{2-7}  & \begin{tabular}{c} **Detrended Cross-** \\ **Correlation** \\ \end{tabular} & \begin{tabular}{c} **Ridge** \\ **Regression** \\ \end{tabular} & \(p\)**-Value** & \begin{tabular}{c} **Detrended Cross-** \\ **Correlation** \\ \end{tabular} & \begin{tabular}{c} **Ridge** \\ **Regression** \\ \end{tabular} & \(p\)**-Value** \\ \hline \
\begin{tabular}{c} Univariate \\ forward model \\ \end{tabular} & 1.09 \(\pm\) 0.27 & 1.29 \(\pm\) 0.33 & * & 0.024 \(\pm\) 0.010 & 0.027 \(\pm\) 0.009 & n.s \\ \hline
\begin{tabular}{c} Multivariate \\ forward model \\ \end{tabular} & 3.73 \(\pm\) 1.56 & 21.69 \(\pm\) 2.12 & ** & 0.038 \(\pm\) 0.012 & 0.038 \(\pm\) 0.011 & n.s \\ \hline \hline \end{tabular}
\end{table}