## 2 Theoretical Descriptions
### The Proposed Optimized Convex 1-D Second-Order Total Variation Denoising Algorithm  
In the basic theory of traditional total variation denoising algorithm, enhanced sparsity penalty function is usually defined as \(\phi(x)=\left\|\mathbf{x}\right\|_{1}\). However, it has been proven that the non-convex penalties have more obvious advantages in signal reconstruction over convex penalties [16]. Therefore, it is necessary to introduce an appropriate enhanced sparsity penalty function \(\phi(x)\) to improve the sparsity of \(\mathbf{TV}_{2}(\mathbf{x})\) and ensure the convexity of the objective function, which is conducive to solving the problem. Typical non-convex penalty functions should be continuous, symmetrical and twice differentiable. The most common non-convex penalty functions are the logarithmic penalty and arctangent penalty [17], which are shown in Equations (9) and (10):  
\[\phi(x;a)=\frac{1}{a}\text{log}(1+a\left|x\right|) \tag{9}\]  
\[\phi(x;a)=\frac{2}{a\sqrt{3}}\left(\tan^{-1}\left(\frac{1+2a\left|x\right|}{ \sqrt{3}}\right)-\frac{\pi}{6}\right) \tag{10}\]  
Considering the convexity condition, the parameter \(a\) should be selected in \(0<a<\frac{1}{4\lambda}\). After the identification of new sparsity upgrade penalty function \(\phi(x)\), this paper uses majorization-minimization (MM) approach to achieve fast convergence and Equation (11) is the basic iterative solver equation. \[x^{(k+1)}=y-D_{2}^{\intercal}\left(\frac{1}{\lambda}\left[W\left(D_{2}x^{(k) }\right)\right]^{-1}+D_{2}D_{2}^{\intercal}\right)^{-1}\!\!\!D_{2}y \tag{11}\]  
where \(W\left(D_{2}x\right)=\frac{\phi^{\prime}(D_{2}x)}{D_{2}x}\), \(x^{(0)}=y\), \(k=1,..,N\). Generally, the number of iterations \(N\) is selected as 20, which are usually sufficient. Equation (11) indicates that the regularization parameter \(\lambda\) plays a very important role in iterative solution processes, which is directly related to the actual noise elimination effect. Therefore, the parameter \(\lambda\) optimization method is introduced based on permutation entropy and cross correlation in this paper. Permutation entropy (PE) was proposed to detect random time series and dynamics mutations. PE has a series of advantages: simple concept, fast computing speed, strong anti-interference ability and good robustness, especially in nonlinear data analysis. Given the time series \(x(i)\) (\(i=1,2,...,N\)) its phase space is reconstructed to obtain Hankel matrix. \[\mathbf{Y}= \begin{pmatrix}x(1)&x(1+\tau)&\cdots&x(1+(m-1)\tau)\\ x(2)&x(2+\tau)&\cdots&x(2+(m-1)\tau)\\ x(j)&x(j+\tau)&\cdots&x(j+(m-1)\tau)\\ \vdots&\vdots&&\vdots\\ x(K)&x(K+\tau)&\cdots&x(K+(m-1)\tau)\end{pmatrix}  \tag{12}\]  
where \(j=1,2,...,K\), \(m\) and \(\tau\) are denoted as embedding dimension and delay time respectively, \(K=n-(m-1)\tau\). Each row in the matrix can be seen as a reconstruction component and there are \(K\) reconstruction components. The reconstructed matrix \(\mathbf{Y}\) in each row \(\{x(j),x(j+\tau),...,x(j+(m-1)\tau)\}\) are rearranged in ascending order, and \(j_{1},j_{2},...,j_{m}\) indicate the index of each element in the column of reconstruction components, namely:  
\[x(i+(j_{1}-1)\tau)\leq x(i+(j_{2}-1)\tau)\leq\cdots\leq x(i+(j_{m}-1)\tau) \tag{13}\]  
If there is an equal value in the reconstruction component such as \(x(i+(j_{p}-1)\tau)=x(i+(j_{q}-1)\tau)\), the elements should be aligned according to the values of \(j_{p}\) and \(j_{q}\). In other words, when \(j_{p}<j_{q}\), we can obtain the following Equation:  
\[x(i+(j_{p}-1)\tau)\leq x(i+(j_{q}-1)\tau) \tag{14}\]  
A set of symbols sequence can be obtained for each row in the matrix \(\mathbf{Y}\). \[S(l)=\{j_{1},j_{2},\ldots,j_{m}\} \tag{15}\]  
where \(l=1,2,\)\(\ldots,k\), and \(k\leq m!\). There is a total of \(m!\) different symbol sequences \(\{j_{1},j_{2},\ldots,j_{m}\}\) in \(m\) dimensional phase space, and symbol sequence \(S(l)\) is just one of them. The probability of each symbol sequence was calculated and named as \(P_{1}\), \(P_{2}\),..., \(P_{k}\).