# MFCC Selection by LASSO for Honey Bee Classification
## 3 Results
### Features Selected by LASSO with AIC and BIC  
After feature extraction, the coefficients representing the audio recordings were chosen from the set **MFCC + Delta + DeltaDelta** of Mel-frequency cepstral coefficients and their first and second derivatives. The values of the Akaike Information Criterion and Bayesian Information Criterion obtained for different values of \(\alpha\) regularization parameter are shown in Figure 6. The number of features selected by LASSO for different values of the Akaike Information Criterion and Bayesian Information Criterion are presented in Table 1, and the exact indices of the coefficients selected for AIC and BIC can be found in Figure 7.  
To better explain how the proposed procedure works, let us analyze a particular case for regularization parameter \(\alpha=0.03\). The chosen feature set in that specific case consists of 29 coefficients obtained from the Formula (6). The feature set **MFCC + Delta + DeltaDelta** from which the coefficients were chosen includes: 40 Mel-frequency cepstral coefficients (**MFCC**), 40 first derivatives (**Delta**) and 40 second derivatives (**DeltaDelta**). The 29 selected coefficients, shown in Figure 7, have the following indices: from **MFCC**--1, 2, 4, 5, 7, 8, 9, 12, 13, 14, 18, 21, 22, 27, 28, 31, 32, 33, 35, 36, 39; from **Delta**--42, 44, 45, 47, 49, 53, 61; and finally from **DeltaDelta**--only 84.  
The information criteria associated with that \(\alpha=0.03\) value, calculated for the training set, are equal: 824.2 for the Akaike Information Criterion and 826.4 for the Bayesian Information Criterion. The values of the criteria were very close--compare also Figure 6--causing the numbers of selected-by-LASSO features to be equal for the chosen cases of this presentation. And in the exemplary case for \(\alpha=0.03\), both information criteria AIC and BIC led to the same set of selected-by-LASSO coefficients representing the audio recordings.  
The results chosen to be shown in Table 1 include the values of regularization parameter \(\alpha\) from the expression (6) and the related values of AIC and BIC; in that way, the jump in the number of features selected by LASSO is approximately 10 coefficients. The LASSO procedure sometimes adds more than one coefficient to the chosen representation in one step--see the LASSO path in Figure 4--and this is a reason why we could not present the exact increase in the feature number as being equal to 10 as planned.  
Figure 6: Information criteria: Akaike (AIC) and Bayesian (BIC), calculated for the feature sets selected by LASSO.  
Figure 7: Coefficients selected by LASSO from the set **MFCC + Delta + DeltaDelta** for different values \(\alpha\) of the regularization parameter.  
As we stated in Section 2.3, the information criteria are statistical measures for the comparison of different models, i.e., different coefficient sets, which are a trade-off between the complexity of a model and its goodness-of-fit to the data, with higher predictive power for the lower information criterion value.  
The values of AIC and BIC in Figure 6 drop swiftly together with the decrease in the regularization parameter \(\alpha\) value until the inflection point on the graph for the \(\alpha\approx 0.01\). For the lower alpha values, from \(\alpha=0.065\) to 0.0001, there is almost no significant change in value, which indicates that the particular models (for 69, 80, 93, 101, 111 and 120 selected coefficients) carry the same measure of information and are the best candidates to be chosen for the training of the neural network.