# Harris Hawks Sparse Auto-Encoder Networks for Automatic Speech Recognition System
## 3 Methodology
### Harris Hawks Sparse Auto-Encoder Networks (HHSAE)-ASR Framework
#### 3.2.3 Speech Recognition  
The convolution network trains the extracted features to recognize the speech signal in different noisy and loud environments. The learning process is done in the language and acoustic models because the introduced ASR framework should react perfectly in different speech environments. Therefore, only the system ensures a higher recognition rate.  
Consider that the extracted features are had at T-length, and the features are defined as \(X=\big{\{}\ X_{T}\in\mathbb{R}^{D}\big{|}t=1,\ldots,T\big{|}\big{\}}\). The features are extracted for the length of the spoken word and defined as \(W=\big{\{}\ W_{n}\in v|n=1,\ldots,N|\big{\}}\). The features \(X\) are derived from \(t\) frame and \(W\) word position \(n\) and \(v\) vocabulary in D-dimension. The derived features are further examined to get the acoustic features that are obtained from the most likely appearing words:  
\[\hat{W}=argmax_{w}P(W|X) \tag{15}\]  
The acoustic feature \(P(W|X)\) is computed from the word sequence \(W\) from \(X\) using Bayes' rules, defined in Equation (16). During the computation, \(P(X)\) is omitted when the word is constant, belonging to the word \(W\).  
\[\hat{W}=argmax_{w}\frac{P(X|W).P(W)}{P(X)} \tag{16}\]  
Figure 3: Feature extraction using level decomposition, wavelet computation, entropy calculation, and MFCC coefficient computation.  
\[\hat{W}=argmax_{w}P(X|W).P(W) \tag{17}\]  
Then, the sequence of features \(P(X|W)\) is computed from the acoustic model and the priori knowledge of the word \(P(W)\) is computed from the language model. The sequence of features, words, and the respective analysis is performed using Equation (18).  
\[\left.\begin{array}{c}\underset{w\in v^{*}}{\arg max_{w\in v^{*}}P(W|X)}\\ =\underset{w\in v^{*}}{\arg max_{w\in v^{*}}\sum\limits_{S}P(X|S,\ W),\ P(S|W)\ P(W)}\\ \approx\underset{w\in v^{*}}{\arg max_{w\in v^{*}}\sum\limits_{S}P(X|S),\ P(S|W)\ P(W)}\end{array}\right\} \tag{18}\]  
\(P(X|S)\) is derived from the acoustic model, which helps make the Markov assumption concerning the probabilistic chain rules (Equation (19)).  
\[\left.\begin{array}{c}P(X|S)=\prod\limits_{t=1}^{T}P(x_{t}|x_{1},x_{2},x_{3 },\ldots\ldots x_{t-1},S)\\ \approx\prod\limits_{t=1}^{T}P(x_{t}|S_{t})\propto\prod\limits_{t=1}^{T}\frac {P(S_{t}|X_{t})}{P(S_{t})}\end{array}\right\} \tag{19}\]  
The convolution network changed the \(P(x_{t}|S_{t})\) frame-wise likelihood function into the frame-wise posterior distribution \(\frac{P(S_{t}|X_{t})}{P(S_{t})}\). The frame-wise analysis helps to resolve the decision-making issues and the system's performance is improved by considering the lexicon model \(P(S|W)\). This lexicon model process is factorized according to the [PERSON] assumption and probabilistic model.  
\[\left.\begin{array}{c}P(S|W)=\prod\limits_{t=1}^{T}P(s_{t}|s_{1},s_{2},s_{3 }\ldots\ldots s_{t-1},W)\\ \approx\prod\limits_{t=1}^{T}P(s_{t}|s_{t-1},W)\end{array}\right\} \tag{20}\]  
The extracted phoneme features and respective Markov probability value helps to identify the lexicon information from the speech. Finally, the language model \(P(W)\) is computed using the [PERSON] assumption and probabilistic chain rule for a word in speech.  
\[\left.\begin{array}{c}P(W)=\prod\limits_{n=1}^{N}P(w_{n}|w_{1},w_{2},w_{3} \ldots\ldots w_{n-1})\\ \approx\prod\limits_{t=1}^{T}P(w_{n}|w_{n-m-1,\ldots,\ldots\ldots w_{m-1})} \end{array}\right\} \tag{21}\]  
The Appendix A is explained the sparse encoder and model Fine-Tuning using Haris Hawk optimization.