## 2 Optimization Model for the Adaptive Sine Multi-Taper Algorithm
### Focus of Optimization for the Adaptive Sine Multi-Taper Algorithm  
In response to the aforementioned issues, this paper proposes an optimal algorithm evaluation criterion based on the mean squared error (MSE) metric, aimed at enhancing the performance of the adaptive sine multi-window power spectral density estimation algorithm in the analysis of low-frequency and sub-low-frequency signals. The optimal algorithm evaluation criterion essentially addresses the relationship between bias and variance, which is intrinsically linked to the selection of the number of tapers [35]. Therefore, the adaptive sine multi-taper algorithm focuses on two key areas: (1) optimizing taper number selection using the MSE criterion to balance bias and variance, and (2) improving computational efficiency. Bias analysis reveals the following [36; 37; 38]:\[Bias\big{|}\hat{S}_{x}(f)\big{|}\approx\frac{\left(K(f)/2T\right)^{2}}{6}S^{\prime \prime}(f)=\frac{K(f)^{2}}{24T^{2}}S^{\prime\prime}(f) \tag{8}\]  
The cost function \(J(\alpha)\) is defined as follows:  
\[J(\alpha)=\varepsilon var\big{[}\hat{S}_{x}(f)\big{]}+(1-\varepsilon)bias^{2} \tag{9}\]  
The relative impact of adjusting bias and variance in the mean squared error (MSE) is represented by \(\varepsilon\), a weighting parameter used to balance the influence of bias and variance in the cost function. The mean squared error (\(MSE\)) is defined as:  
\[MSE\approx\ \ (1-\varepsilon)\Bigg{\{}\frac{Bias\big{|}\hat{S}_{x}(f)\big{|}}{ E\big{|}\hat{S}_{x}(f)\big{|}}\Bigg{\}}^{2}+\varepsilon\Bigg{\{}\frac{Var\big{|} \hat{S}_{x}(f)\big{|}}{E\big{|}\hat{S}_{x}(f)\big{|}^{2}}\Bigg{\}}\approx\ \ \ \frac{(1-\varepsilon)\,|\,K(f)\,|\,^{4}}{576T^{4}}\left[\frac{S^{\prime\prime}( f)}{S(f)}\right]^{2/5}+\varepsilon\frac{1}{K(f)} \tag{10}\]  
Minimizing \(MSE\) results in the ratio \(\frac{\varepsilon}{1-\varepsilon}=\beta\) which yields the optimal taper count:  
\[K(f)=2.70192\beta(\frac{T^{2}S(f)}{1\,S^{\prime\prime}(f)\,|})^{2/5} \tag{11}\]  
where \(\beta\) = 0.7 is determined via the bisection method. For the spectral analysis at the low frequency of 1 mHz, this paper reduces bias and enhances resolution to some extent by adjusting and decreasing the taper degree. A segmented improvement approach is primarily adopted: on one hand, a fixed small taper degree is applied to the low-frequency segment; on the other hand, to mitigate the potential instability in power spectral density (PSD) estimation caused by abrupt changes in the taper degree, a gradient transition is introduced. The optimization of the power spectral density (PSD) estimation is accomplished through a systematic sequence of steps, as detailed below:  
1. **Initial Setup:** The adaptive sine multi-taper algorithm starts by initializing the tapering parameter, setting all frequencies to a degree of 20. 2. **Spectral Estimate Processing:** * For frequencies below 1 mHz (_f_ < 1 mHz): The value of \(K(f)\) is set to 1. * For frequencies between 1 mHz and 0.01 Hz (1 mHz \(<\)_f_ < 0.01 Hz): A robust filtering technique is applied to smooth \(K(f)\). * For frequencies above 0.01 Hz (_f_ > 0.01 Hz): K(_f_) is determined using parabolic weighting as specified in Equation (11). * For frequencies ranging from 0.01 Hz to 1 Hz (0.01 Hz \(<\)_f_ < 1 Hz): A mild filtering technique is applied to further enhance the smoothness of \(K(f)\). 3. **Outlier Removal and Iteration:** * Outliers are identified and removed through linear interpolation. The \(K(f)\) values are smoothed over three iterations. The PSD estimation process is iterated 15 times to yield optimal estimates for both the PSD and the amplitude spectral density (ASD). Execute a total of 15 iterations for the PSD estimation process to derive the optimal estimates for both the PSD and the Amplitude Spectral Density (ASD). For frequencies below 1 mHz, the taper degree is set to 1 to reduce bias and enhance resolution, preventing low-frequency fluctuations from being misidentified as noise.