## 2 Optimization Model for the Adaptive Sine Multi-Taper Algorithm  
### Adaptive Sine Multi-Taper Power Spectral Density Estimation  
The adaptive sine multi-taper algorithm begins with the initialization of the tapering parameter, where all frequencies are initially set to a tapering degree of 20. The zero-mean real-valued stationary random process is represented as \(x(t)=[x_{1}(t)\cdots x_{N}(t)]^{T}\), over the integration interval \([-f_{N^{\prime}}f_{N}]\). The refined adaptive sine multi-taper spectral estimate is formulated as:  
\[\hat{S}_{x}(f)=\sum\
olimits_{i=0}^{K(f)-1}\alpha_{i}\hat{S}_{i}(f)=\frac{1}{ \mathrm{K}(f)\Delta t}\sum\
olimits_{i=0}^{\mathrm{K}(f)-1}\alpha_{i}\,|\,v(f )\,|^{2} \tag{1}\]  
where \(\hat{S}_{x}(f)\) denotes the power spectral density (PSD) estimate at frequency \(f\), \(\alpha_{i}\) is the taper weighting factor for the \(i\)-th window, \(\hat{S}_{i}(f)\) is the periodogram derived from the selected data window, and \(\mathrm{K}(f)\) is the optimal taper count, calculated as follows:  
\[\alpha_{i}=\left(K^{2}-i^{2}\right)/[K(4K-1)(K+1)/6] \tag{2}\]  
The overall taper \(v(f)\) is defined as follows:  
\[v(f)=\Delta t\sum\
olimits_{i=0}^{L}x(r\Delta t)h_{i}(r\Delta t)e^{-\beta\pi f \tau\Delta t} \tag{3}\]where \(h_{i}(t)\) is the sine taper function, \(r\) is the sample sequence number ranging from 1 to \(L\), \(L\) is the total number of observations, and \(\Delta t\) is the sampling interval. The sine window function is given by the following:  
\[h_{i}(t)=\sqrt{\frac{2}{T}}\text{sin}\bigg{(}\frac{i\pi t}{T}\bigg{)},0\leq t\leq T \tag{4}\]  
To ensure energy normalization in the frequency domain, the overall taper function satisfies [31; 32]:  
\[\int_{-f_{N}}^{f_{N}}v(f)df=1 \tag{5}\]  
The WOSA algorithm divides the data into shorter sequences by distributing \(N\) samples into \(K\) segments, with each segment containing about \(N/K\) samples. When spectral estimates \(S_{i}\) and \(S_{k}\) are minimally correlated, the variance decreases. For low correlations, the variance of the equal-weighted taper is approximated as follows [7; 33]:  
\[Var\big{[}\hat{S}_{x}(f)\big{]}=\frac{1}{K^{2}}\sum_{i=0}^{K-1}Var\big{[}\hat {S}_{i}(f)\big{]}\approx\frac{1}{K}\hat{S}_{x}^{2}(f) \tag{6}\]  
The LPSD algorithm addresses the discontinuity issue in WOSA but does not account for the multi-taper problem [34]. In 2021, [PERSON] et al. enhanced the [PERSON] multi-taper method by proposing \(K=2 NW-O(log_{2}(NW))\) tapers, reducing spectral leakage. However, the Thomson algorithm remains constrained by its single time-bandwidth product and high computational cost due to numerical eigenvalue decomposition.  
[PERSON] and [PERSON] introduced an adaptive multi-taper algorithm based on sine tapers, which eliminates the need for eigenvalue equations. The optimal taper count \(K_{opt}(f)\) is given by following:  
\[K_{opt}(f)=(\frac{12T^{2}S(f)}{|S^{\prime}(f)\,|})^{2/5}=2.701(\frac{T^{2}S(f) }{|S^{\prime}(f)\,|})^{2/5} \tag{7}\]  
where \(S(f)\) is the true power spectral density estimate, and \(S^{\prime\prime}(f)\) is the second-order derivative.  
While this method reduces variance, it increases bias. To minimize mean squared error (MSE), an iterative procedure optimizes taper selection. However, challenges remain in achieving sufficient resolution for low-frequency PSD estimation in gravitational wave detection.  
For noise analysis in the 0.1 mHz to 1 Hz band, a sampling rate of 20 Hz was selected to balance data acquisition efficiency and hardware performance. Despite substantial data collection, resolution in the 0.1 mHz to 1 mHz range remains inadequate. The LPSD algorithm's complexity \(O(N\cdot J)\) and the LISA-LPSD algorithm's complexity \(O(Nlog_{2}(N/K)+J)+O(M)\) highlight the need for computationally efficient solutions.