# An Enhanced Genetic Algorithm for Parameter Estimation of Sinusoidal Signals
## 5 Parameter Estimation of Sinusoidal Signals
### The Circadian Rhythms  
The Circadian Rhythms was an experiment that recorded the temperature of a long-tail pocket mouse every two minutes over three months. As some problems arose during the experiment, a proportion of outliers remained in the dataset. Though these outliers make the estimation quite difficult, they are meaningful and necessary for verifying the performance of an algorithm, because noise is ubiquitous in real-life signals. In this work, a 20 min temperature sample averaged from the dataset was extracted, and a four-component sinusoid model was used for the estimation, expressed as Equation (9):  
\[y(t)=K+A\sin(\omega t+\varphi), \tag{9}\]  
where \(K\) is the offset, \(A\) denotes the amplitude of sinusoid signal, \(\omega\) and \(\varphi\) are the frequency and phase, respectively.  
The parameter initialization for the estimation was set as \(K\in[y_{m}-50,y_{m}+50]\), \(A\in[-100,100]\), \(\omega\in[0,\pi]\), and \(\varphi\in[0,\pi]\). Parameter \(y_{m}\) is the median of the real data. The parameter configurations for the algorithms were the same as those set in Section 5.1, except the ones for the EGA, whose new  
Figure 7: Plots of voice data and fitting curves.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Parameter**} & \multicolumn{1}{c}{**SGA**} & \multicolumn{1}{c}{**PSO**} & \multicolumn{1}{c}{**CS**} & \multicolumn{1}{c}{**CMGA**} & \multicolumn{1}{c}{**EGA**} \\ \hline Offset & \(K\) & \(-0.8437\) & \(0.1416\) & \(-0.9918\) & \(-0.9774\) & \(-0.9423\) \\ \cline{2-7}  & \(A_{1}\) & \(22.1220\) & \(22.7440\) & \(20.9413\) & \(22.1339\) & \(22.5130\) \\ Amplitude & \(A_{2}\) & \(25.2844\) & \(26.5268\) & \(24.8365\) & \(24.9434\) & \(25.0975\) \\  & \(A_{3}\) & \(65.8691\) & \(66.8565\) & \(65.3737\) & \(64.6844\) & \(66.8589\) \\ \cline{2-7}  & \(\omega_{1}\) & \(0.2299\) & \(0.2320\) & \(0.2315\) & \(0.2293\) & \(0.2300\) \\ Frequency & \(\omega_{2}\) & \(0.3404\) & \(0.3395\) & \(0.3418\) & \(0.3407\) & \(0.3407\) \\  & \(\omega_{3}\) & \(0.1134\) & \(0.1126\) & \(0.1120\) & \(0.1133\) & \(0.1135\) \\ \cline{2-7}  & \(\varphi_{1}\) & \(-0.0569\) & \(-0.1467\) & \(-0.1465\) & \(-0.0216\) & \(-0.0609\) \\ Phase & \(\varphi_{2}\) & \(-0.2428\) & \(-0.2082\) & \(-0.3285\) & \(-0.2656\) & \(-0.2611\) \\  & \(\varphi_{3}\) & \(0.1771\) & \(0.1797\) & \(0.2008\) & \(0.1409\) & \(0.1135\) \\ \hline \multicolumn{2}{c}{\(R^{2}\)} & \multicolumn{1}{c}{0.9909} & \multicolumn{1}{c}{0.9912} & \multicolumn{1}{c}{0.9916} & \multicolumn{1}{c}{0.9922} & \multicolumn{1}{c}{**0.9931**} \\ \hline \hline \end{tabular}
\end{table}
Table 5: Estimation results of SGA, PSO, CS, CMGA, and EGA for voice dataset.  
settings were: \([P_{c1},P_{c2}]=[0.7,0.7]\), and \([P_{m1},P_{m2}]=[0.15,0.35]\). The corresponding evaluated results are illustrated in Figure 8 and Table 6.  
Figure 8 displays the original data and the best fitting curve. From this graph, it is evident to see that the dataset contains a large number of outliers, but the fitting curve successfully ignores them and follows nicely the periodic trend. In addition, the less evident outliers close to the fitting curve did not distort the data fitting, either. To examine the results in Table 6, we found that the sum of squared residuals \((SSR)\) of the EGA is less than that of the others, indicating that the performance of the EGA surpasses its competitors. Considering the same dataset, it was reported that the fitted frequency by the elemental set method [48] was 0.0873, which once again is close to our frequency estimation.