写在前面 介绍循环奇异谱分析(Circulant Singular Spectrum Analysis)。
循环奇异谱分析 对于无限零均值平稳时间序列{ x t } \{x_{t}\} { x t } γ m = E ( x t x t − m ) \gamma_{m}=E(x_{t}x_{t-m}) γ m = E ( x t x t − m ) 2 π 2\pi 2 π f f f
Γ L ( f ) = ( γ 0 γ 1 γ 2 . . . γ L − 1 γ 1 γ 0 γ 1 . . . γ L − 2 ⋮ ⋮ ⋮ ⋮ ⋮ γ L − 1 γ L − 2 γ L − 3 . . . γ 0 ) \mathbf{\Gamma }_{L}(f)=\left(
\begin{array}{ccccc}
\gamma _{0} & \gamma _{1} & \gamma _{2} & ... & \gamma _{L-1} \\
\gamma _{1} & \gamma _{0} & \gamma _{1} & ... & \gamma _{L-2} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
\gamma _{L-1} & \gamma _{L-2} & \gamma _{L-3} & ... & \gamma _{0}%
\end{array}%
\right)
Γ L ( f ) = ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ γ 0 γ 1 ⋮ γ L − 1 γ 1 γ 0 ⋮ γ L − 2 γ 2 γ 1 ⋮ γ L − 3 . . . . . . ⋮ . . . γ L − 1 γ L − 2 ⋮ γ 0 ⎠ ⎟ ⎟ ⎟ ⎟ ⎞
矩阵Γ L ( f ) \mathbf{\Gamma }_{L}(f) Γ L ( f ) f f f γ m = ∫ 0 1 f ( w ) exp ( i 2 π m w ) d w \gamma _{m}=\int_{0}^{1}f(w)\exp (i2\pi mw)dw γ m = ∫ 0 1 f ( w ) exp ( i 2 π m w ) d w w ∈ [ 0 , 1 ] w \in \left[ 0,\ 1 \right] w ∈ [ 0 , 1 ]
C L ( f ) = ( c 0 c 1 c 2 . . . c L − 1 c L − 1 c 0 c 1 . . . c L − 2 ⋮ ⋮ ⋮ ⋮ ⋮ c 1 c 2 c 3 . . . c 0 ) \mathbf{C}_{L}(f)=\left(
\begin{array}{ccccc}
c_{0} & c_{1} & c_{2} & ... & c_{L-1} \\
c_{L-1} & c_{0} & c_{1} & ... & c_{L-2} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
c_{1} & c_{2} & c_{3} & ... & c_{0}%
\end{array}%
\right)
C L ( f ) = ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ c 0 c L − 1 ⋮ c 1 c 1 c 0 ⋮ c 2 c 2 c 1 ⋮ c 3 . . . . . . ⋮ . . . c L − 1 c L − 2 ⋮ c 0 ⎠ ⎟ ⎟ ⎟ ⎟ ⎞
循环矩阵的特征值和特征向量存在闭解(closed form)
C L ( f ) \mathbf{C}_{L}(f) C L ( f ) k k k λ L , k = ∑ m = 0 L − 1 c m exp ( i 2 π m k − 1 L ) \lambda _{L,k}=\sum_{m=0}^{L-1}c_{m}\exp \left( i2\pi m\frac{k-1}{L}\right)
λ L , k = m = 0 ∑ L − 1 c m exp ( i 2 π m L k − 1 )
C L ( f ) \mathbf{C}_{L}(f) C L ( f ) k k k u k = L − 1 / 2 ( u k , 1 , . . . , u k , L ) ′ \mathbf{u}_{k}=L^{-1/2}(u_{k,1,}...,u_{k,L})^{\prime }
u k = L − 1 / 2 ( u k , 1 , . . . , u k , L ) ′
其中
u k , j = exp ( − i 2 π ( j − 1 ) k − 1 L ) u_{k,j}=\exp \left( -i2\pi (j-1)\frac{k-1}{L}\right)
u k , j = exp ( − i 2 π ( j − 1 ) L k − 1 )
特别的，如果循环矩阵C L ( f ) \mathbf{C}_{L}(f) C L ( f ) c m c_{m} c m
c m = 1 L ∑ j = 0 L − 1 f ( j L ) exp ( i 2 π m j L ) c_{m}=\frac{1}{L}\sum_{j=0}^{L-1}f\left( \frac{j}{L}\right) \exp \left(i2\pi m\frac{j}{L}\right)
c m = L 1 j = 0 ∑ L − 1 f ( L j ) exp ( i 2 π m L j )
那么特征值与w k = k − 1 L w_k=\frac{k-1}{L} w k = L k − 1
λ L , k = f ( k − 1 L ) \lambda _{L,k}=f\left( \frac{k-1}{L}\right)
λ L , k = f ( L k − 1 )
此外，当L → ∞ L\rightarrow \infty L → ∞ Γ L ( f ) \mathbf{\Gamma }_{L}(f) Γ L ( f ) C L ( f ) \mathbf{C}_{L}(f) C L ( f ) Γ L ( f ) ∼ C L ( f ) \mathbf{\Gamma }_{L}(f)\sim \mathbf{C}_{L}(f) Γ L ( f ) ∼ C L ( f )
lim L → ∞ ∥ Γ L ( f ) − C L ( f ) ∥ F L = 0 \underset{L\rightarrow \infty }{\lim }\frac{\left\Vert \mathbf{\Gamma }_{L}(f)-\mathbf{C}_{L}(f)\right\Vert _{F}}{\sqrt{L}}=0
L → ∞ lim L ∥ Γ L ( f ) − C L ( f ) ∥ F = 0
由于C L ( f ) \mathbf{C}_{L}(f) C L ( f ) C L ( f ~ ) {\mathbf{C}}_{L}(\widetilde{f}) C L ( f )
c ~ m = L − m L γ m + m L γ L − m \widetilde{c}_{m}=\frac{L-m}{L}\gamma _{m}+\frac{m}{L}\gamma _{L-m}
c m = L L − m γ m + L m γ L − m
对于生成的f ~ \widetilde{f} f
Γ L ( f ) ∼ C L ( f ~ ) ∼ C L ( f ) \mathbf{\Gamma }_{L}(f)\sim{\mathbf{C}}_{L}(\widetilde{f})\sim\mathbf{C}_{L}(f)
Γ L ( f ) ∼ C L ( f ) ∼ C L ( f )
因此，通过求解循环矩阵C L ( f ~ ) {\mathbf{C}}_{L}(\widetilde{f}) C L ( f ) w k = k − 1 L w_k=\frac{k-1}{L} w k = L k − 1
另一方面，用样本统计量代替总体统计量
γ ^ m = 1 T − m ∑ t = 1 T − m x t x t + m \widehat{\gamma }_{m}=\frac{1}{T-m}\sum_{t=1}^{T-m}x_{t}x_{t+m}
γ m = T − m 1 t = 1 ∑ T − m x t x t + m
对应的样本自协方差矩阵S C \mathbf{S}_{C} S C
c ^ m = L − m L γ ^ m + m L γ ^ L − m \widehat{c}_{m}=\frac{L-m}{L}\widehat{\gamma }_{m}+\frac{m}{L}\widehat{\gamma }_{L-m}
c m = L L − m γ m + L m γ L − m
以上流程完成奇异谱分析的嵌入和分解步骤。因为谱密度的对称性，即λ ^ k = λ ^ L + 2 − k \widehat{\lambda }_{k} = \widehat{\lambda }_{L+2-k} λ k = λ L + 2 − k u k = u ‾ L + 2 − k \mathbf{u}_{k}=\overline{\mathbf{u}}_{L+2-k} u k = u L + 2 − k B k = { k , L + 2 − k } B_{k}=\{k,L+2-k\} B k = { k , L + 2 − k } B 1 = { 1 } B_{1}=\{1\} B 1 = { 1 } L L L B L 2 + 1 = { L 2 + 1 } B_{\frac{L}{2}+1}=\left\{ \frac{L}{2}+1\right\} B 2 L + 1 = { 2 L + 1 } k k k λ ^ k \widehat{\lambda }_{k} λ k λ ^ L + 2 − k \widehat{\lambda }_{L+2-k} λ L + 2 − k X B k \mathbf{X}_{B_{k}} X B k X k \mathbf{X}_{k} X k X L + 2 − k \mathbf{X}_{L+2-k} X L + 2 − k w k = k − 1 L w_k=\frac{k-1}{L} w k = L k − 1
X B k = X k + X L + 2 − k = u k u ‾ k ′ X + u L + 2 − k u ‾ L + 2 − k ′ X = ( u k u ‾ k ′ + u ‾ k u k ′ ) X = 2 [ ℜ ( u k ) ℜ ( u k ′ ) + ℑ ( u k ) ℑ ( u k ′ ) ] X \begin{aligned}
\mathbf{X}_{B_{k}} &=\mathbf{X}_{k}+\mathbf{X}_{L+2-k} \\
&=\mathbf{u}_{k}\overline{\mathbf{u}}_{k}^{\prime }\mathbf{X+u}_{L+2-k}\overline{\mathbf{u}}_{L+2-k}^{\prime }\mathbf{X} \\
&=(\mathbf{u}_{k}\overline{\mathbf{u}}_{k}^{\prime }+\overline{\mathbf{u}}_{k}\mathbf{u}_{k}^{\prime })\mathbf{X} \\
&=2[\Re(\mathbf{u}_{k})\Re(\mathbf{u}_{k}^{\prime })+\Im(\mathbf{u}_{k})\Im(\mathbf{u}_{k}^{\prime })]\mathbf{X}
\end{aligned}
X B k = X k + X L + 2 − k = u k u k ′ X + u L + 2 − k u L + 2 − k ′ X = ( u k u k ′ + u k u k ′ ) X = 2 [ ℜ ( u k ) ℜ ( u k ′ ) + ℑ ( u k ) ℑ ( u k ′ ) ] X
因此，X B k \mathbf{X}_{B_{k}} X B k
多维推广 不久，原班人马给出了循环奇异谱分析的多维推广，是将循环的机制变成块循环的方式。
给定时间序列x t = ( x t ( 1 ) , ⋯ , x t ( M ) ) ′ \mathbf{x}_t=\left(x_t^{\left(1\right)},\cdots,x_t^{\left(M\right)}\right)' x t = ( x t ( 1 ) , ⋯ , x t ( M ) ) ′
X = ( x 1 x 2 ⋯ x N x 2 x 3 ⋯ x N + 1 ⋮ ⋮ ⋮ ⋮ x L x L + 1 ⋯ x T ) = ( x 1 ( 1 ) x 2 ( 1 ) ⋯ x N ( 1 ) ⋮ ⋮ ⋮ ⋮ x 1 ( M ) x 2 ( M ) ⋯ x N ( M ) x 2 ( 1 ) x 3 ( 1 ) ⋯ x N + 1 ( 1 ) ⋮ ⋮ ⋮ ⋮ x 2 ( M ) x 3 ( M ) ⋯ x N + 1 ( M ) ⋮ ⋮ ⋮ ⋮ x L ( 1 ) x L + 1 ( 1 ) ⋯ x T ( 1 ) ⋮ ⋮ ⋮ ⋮ x L ( M ) x L + 1 ( M ) ⋯ x T ( M ) . ) \begin{aligned}
\mathbf{X}&=\left(\begin{matrix}\mathbf{x}_1&\mathbf{x}_2&\cdots&\mathbf{x}_N\\\mathbf{x}_2&\mathbf{x}_3&\cdots&\mathbf{x}_{N+1}\\\vdots&\vdots&\vdots&\vdots\\\mathbf{x}_L&\mathbf{x}_{L+1}&\cdots&\mathbf{x}_T\\\end{matrix}\right)\\
&=\left(\begin{matrix}x_1^{\left(1\right)}&x_2^{\left(1\right)}&\cdots&x_N^{\left(1\right)}\\\vdots&\vdots&\vdots&\vdots\\x_1^{\left(M\right)}&x_2^{\left(M\right)}&\cdots&x_N^{\left(M\right)}\\x_2^{\left(1\right)}&x_3^{\left(1\right)}&\cdots&x_{N+1}^{\left(1\right)}\\\vdots&\vdots&\vdots&\vdots\\x_2^{\left(M\right)}&x_3^{\left(M\right)}&\cdots&x_{N+1}^{\left(M\right)}\\\vdots&\vdots&\vdots&\vdots\\x_L^{\left(1\right)}&x_{L+1}^{\left(1\right)}&\cdots&x_T^{\left(1\right)}\\\vdots&\vdots&\vdots&\vdots\\x_L^{\left(M\right)}&x_{L+1}^{\left(M\right)}&\cdots&x_T^{\left(M\right)}.\\\end{matrix}\right)
\end{aligned}
X = ⎝ ⎜ ⎜ ⎜ ⎜ ⎛ x 1 x 2 ⋮ x L x 2 x 3 ⋮ x L + 1 ⋯ ⋯ ⋮ ⋯ x N x N + 1 ⋮ x T ⎠ ⎟ ⎟ ⎟ ⎟ ⎞ = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ x 1 ( 1 ) ⋮ x 1 ( M ) x 2 ( 1 ) ⋮ x 2 ( M ) ⋮ x L ( 1 ) ⋮ x L ( M ) x 2 ( 1 ) ⋮ x 2 ( M ) x 3 ( 1 ) ⋮ x 3 ( M ) ⋮ x L + 1 ( 1 ) ⋮ x L + 1 ( M ) ⋯ ⋮ ⋯ ⋯ ⋮ ⋯ ⋮ ⋯ ⋮ ⋯ x N ( 1 ) ⋮ x N ( M ) x N + 1 ( 1 ) ⋮ x N + 1 ( M ) ⋮ x T ( 1 ) ⋮ x T ( M ) . ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞
二阶统计量记为Γ k = E [ x t + k x t ′ ] \mathbf{\Gamma}_k= E[\mathbf{x}_{t+k}\mathbf{x}_t^{\prime}] Γ k = E [ x t + k x t ′ ] L × L L\times L L × L M × M M\times M M × M T L = [ Γ i j = Γ i − j ; i , j = 1 , ⋯ , L ] \mathbf{T}_L=\left[\mathbf{\Gamma}_{ij}=\mathbf{\Gamma}_{i-j};i,j=1,\cdots,L\right] T L = [ Γ i j = Γ i − j ; i , j = 1 , ⋯ , L ] L M × L M LM\times LM L M × L M
Γ k = ∫ 0 1 F ( ω ) exp ( − i 2 π k ω ) d ω , ∀ k ∈ Z \mathbf{\Gamma}_k=\int_{0}^{1}\mathbf{F}\left(\omega\right)\exp{\left(-i2\pi k\omega\right)}d\omega,\;\forall k\in\mathbb{Z}
Γ k = ∫ 0 1 F ( ω ) exp ( − i 2 π k ω ) d ω , ∀ k ∈ Z
其中ω ∈ [ 0 , 1 ] \omega\in\left[0,1\right] ω ∈ [ 0 , 1 ] F ( ω ) \mathbf{F}\left(\omega\right) F ( ω )
F ( ω ) = ∑ k = − ∞ ∞ Γ k exp ( i 2 π k ω ) , ω ∈ [ 0 , 1 ] \mathbf{F}\left(\omega\right)=\sum_{k=-\infty}^{\infty}{\mathbf{\Gamma}_k\exp{\left(i2\pi k\omega\right)}},\; \omega\in\left[0,1\right]
F ( ω ) = k = − ∞ ∑ ∞ Γ k exp ( i 2 π k ω ) , ω ∈ [ 0 , 1 ]
Toeplitz矩阵T L \mathbf{T}_L T L 块循环矩阵 来近似求解，其好处在于块循环矩阵可块对角化，乃至对角化，这利于求解特征值。但Toeplitz矩阵无法达到这点。
C L = ( Ω 0 Ω 1 Ω 2 ⋯ Ω L − 1 Ω L − 1 Ω 0 Ω 1 ⋱ ⋮ Ω L − 2 Ω L − 1 Ω 0 ⋱ Ω 2 ⋮ ⋱ ⋱ ⋱ Ω 1 Ω 1 ⋯ Ω L − 2 Ω L − 1 Ω 0 ) \mathbf{C}_L=\left(\begin{matrix}\mathbf{\Omega}_0&\mathbf{\Omega}_1&\mathbf{\Omega}_2&\cdots&\mathbf{\Omega}_{L-1}\\\mathbf{\Omega}_{L-1}&\mathbf{\Omega}_0&\mathbf{\Omega}_1&\ddots&\vdots\\\mathbf{\Omega}_{L-2}&\mathbf{\Omega}_{L-1}&\mathbf{\Omega}_0&\ddots&\mathbf{\Omega}_2\\\vdots&\ddots&\ddots&\ddots&\mathbf{\Omega}_1\\\mathbf{\Omega}_1&\cdots&\mathbf{\Omega}_{L-2}&\mathbf{\Omega}_{L-1}&\mathbf{\Omega}_0\\\end{matrix}\right)
C L = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ Ω 0 Ω L − 1 Ω L − 2 ⋮ Ω 1 Ω 1 Ω 0 Ω L − 1 ⋱ ⋯ Ω 2 Ω 1 Ω 0 ⋱ Ω L − 2 ⋯ ⋱ ⋱ ⋱ Ω L − 1 Ω L − 1 ⋮ Ω 2 Ω 1 Ω 0 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞
对于k = 0 , ⋯ , L − 1 k=0,\cdots,L-1 k = 0 , ⋯ , L − 1
Ω k = 1 L ∑ j = 0 L − 1 F ( j L ) exp ( i 2 π j k L ) \mathbf{\Omega}_k=\frac{1}{L}\sum_{j=0}^{L-1}{\mathbf{F}\left(\frac{j}{L}\right)\exp{\left(\frac{i2\pi jk}{L}\right)}}
Ω k = L 1 j = 0 ∑ L − 1 F ( L j ) exp ( L i 2 π j k )
类似一维，多维推广的Toeplitz矩阵与循环矩阵也具有渐进等价性质：当L → ∞ L\rightarrow \infty L → ∞ T L ( F ) ∼ C L ( F ) \mathbf{T}_{L}(\mathbf{F}) \sim \mathbf{C}_{L}(\mathbf{F}) T L ( F ) ∼ C L ( F ) F \mathbf{F} F { Γ k } k ∈ Z \left\{\mathbf{\Gamma}_k\right\}_{k\in\mathbb{Z}} { Γ k } k ∈ Z Γ k \mathbf{\Gamma}_k Γ k
Ω ~ k = k L Γ L − k + L − k L Γ − k {\widetilde{\mathbf{\Omega}}}_k=\frac{k}{L}\mathbf{\Gamma}_{L-k}+\frac{L-k}{L}\mathbf{\Gamma}_{-k}
Ω k = L k Γ L − k + L L − k Γ − k
对应的矩阵函数F ~ {\widetilde{\mathbf{F}}} F 2 π 2\pi 2 π
F ~ ( ω ) = 1 L ∑ m = 1 L ∑ l = 1 L Γ l − m exp ( i 2 π ( l − m ) ω ) {\widetilde{\mathbf{F}}}\left(\omega\right)=\frac{1}{L}\sum_{m=1}^{L}\sum_{l=1}^{L}{\mathbf{\Gamma}_{l-m}\exp{\left(i2\pi\left(l-m\right)\omega\right)}}
F ( ω ) = L 1 m = 1 ∑ L l = 1 ∑ L Γ l − m exp ( i 2 π ( l − m ) ω )
同样，理论保证了渐进性：T L ( F ) ∼ C L ( F ~ ) \mathbf{T}_{L}(\mathbf{F})\sim \mathbf{C}_{L}(\widetilde{\mathbf{F}}) T L ( F ) ∼ C L ( F )
C L ( F ) = ( U L ⊗ I M ) diag ( F 1 , ⋯ , F L ) ( U L ⊗ I M ) ∗ \mathbf{C}_L\left(\mathbf{F}\right)=\left(\mathbf{U}_L\otimes\mathbf{I}_M\right)\operatorname{diag}{\left(\mathbf{F}_1,\cdots,\mathbf{F}_L\right)}\left(\mathbf{U}_L\otimes\mathbf{I}_M\right)^*
C L ( F ) = ( U L ⊗ I M ) d i a g ( F 1 , ⋯ , F L ) ( U L ⊗ I M ) ∗
其中U L \mathbf{U}_L U L
U L = L 1 2 [ exp ( − i 2 π ( j − 1 ) ( k − 1 ) L ) ] \mathbf{U}_L=L^\frac{1}{2}\left[\exp{\left(\frac{-i2\pi\left(j-1\right)\left(k-1\right)}{L}\right)}\right]
U L = L 2 1 [ exp ( L − i 2 π ( j − 1 ) ( k − 1 ) ) ]
每个块矩阵F k = F ( k − 1 L ) \mathbf{F}_k=\mathbf{F}\left(\frac{k-1}{L}\right) F k = F ( L k − 1 ) x t \mathbf{x}_t x t ω k = k − 1 L , k = 1 , ⋯ , L \omega_k=\frac{k-1}{L}, k=1,\cdots,L ω k = L k − 1 , k = 1 , ⋯ , L F k = E k D k E k ∗ \mathbf{F}_k=\mathbf{E}_k\mathbf{D}_k\mathbf{E}_k^* F k = E k D k E k ∗ E k = [ e k , 1 ∣ ⋯ ∣ e k , M ] \mathbf{E}_k=\left[\mathbf{e}_{k,1}|\cdots|\mathbf{e}_{k,M}\right] E k = [ e k , 1 ∣ ⋯ ∣ e k , M ] D k \mathbf{D}_k D k
C L ( F ) = V D V ∗ \mathbf{C}_L\left(\mathbf{F}\right)=\mathbf{VD}\mathbf{V}^*
C L ( F ) = V D V ∗
V = ( U L ⊗ I M ) diag ( E 1 , ⋯ , E L ) ∈ C L M × L M \mathbf{V}=\left(\mathbf{U}_L\otimes\mathbf{I}_M\right)\operatorname{diag}{\left(\mathbf{E}_1,\cdots,\mathbf{E}_L\right)}\in\mathbb{C}^{LM\times L M}
V = ( U L ⊗ I M ) d i a g ( E 1 , ⋯ , E L ) ∈ C L M × L M
D = diag ( D 1 , ⋯ , D L ) \mathbf{D}=\operatorname{diag}{\left(\mathbf{D}_1,\cdots,\mathbf{D}_L\right)}
D = d i a g ( D 1 , ⋯ , D L )
由于谱的对称性，F k = F L + 2 − k T ; k = 2 , ⋯ , ⌊ L + 1 2 ⌋ \mathbf{F}_k=\mathbf{F}_{L+2-k}^T;\; k=2,\cdots,\left\lfloor\frac{L+1}{2}\right\rfloor F k = F L + 2 − k T ; k = 2 , ⋯ , ⌊ 2 L + 1 ⌋ D k = D L + 2 − k \mathbf{D}_k=\mathbf{D}_{L+2-k} D k = D L + 2 − k E k = E ‾ L + 2 − k \mathbf{E}_k=\overline{\mathbf{E}}_{L+2-k} E k = E L + 2 − k
前面给了一些必要的基础知识，下面开始介绍多元循环奇异谱分析 。嵌入阶段构造一个大的轨道矩阵，前面已经叙述过，下面着重介绍分解与分组步骤。
利用有限的二阶统计量构造块循环矩阵来近似Toeplitz矩阵：
S C = ( Ω ^ 0 Ω ^ 1 ⋯ Ω ^ L − 1 Ω ^ L − 1 Ω ^ 0 ⋱ ⋮ ⋮ ⋱ ⋱ Ω ^ 1 Ω ^ 1 ⋯ Ω ^ L − 1 Ω ^ 0 ) = ( U L ⊗ I M ) E ^ D ^ E ^ ∗ ( U L ⊗ I M ) ∗ = V ^ D ^ V ^ ∗ \begin{aligned}
\mathbf{S}_\mathbf{C}&=\left(\begin{matrix}{\hat{\mathbf{\Omega}}}_0&{\hat{\mathbf{\Omega}}}_1&\cdots&{\hat{\mathbf{\Omega}}}_{L-1}\\{\hat{\mathbf{\Omega}}}_{L-1}&{\hat{\mathbf{\Omega}}}_0&\ddots&\vdots\\\vdots&\ddots&\ddots&{\hat{\mathbf{\Omega}}}_1\\{\hat{\mathbf{\Omega}}}_1&\cdots&{\hat{\mathbf{\Omega}}}_{L-1}&{\hat{\mathbf{\Omega}}}_0\\\end{matrix}\right)\\
&=\left(\mathbf{U}_L\otimes\mathbf{I}_M\right)\hat{\mathbf{E}}\hat{\mathbf{D}}{\hat{\mathbf{E}}}^*\left(\mathbf{U}_L\otimes\mathbf{I}_M\right)^*=\hat{\mathbf{V}}\hat{\mathbf{D}}{\hat{\mathbf{V}}}^*
\end{aligned}
S C = ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎛ Ω ^ 0 Ω ^ L − 1 ⋮ Ω ^ 1 Ω ^ 1 Ω ^ 0 ⋱ ⋯ ⋯ ⋱ ⋱ Ω ^ L − 1 Ω ^ L − 1 ⋮ Ω ^ 1 Ω ^ 0 ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎞ = ( U L ⊗ I M ) E ^ D ^ E ^ ∗ ( U L ⊗ I M ) ∗ = V ^ D ^ V ^ ∗
由特征三元组得到对应于频率ω k = k − 1 L , k = 1 , ⋯ , L \omega_k=\frac{k-1}{L},\;k=1, \cdots,L ω k = L k − 1 , k = 1 , ⋯ , L k k k
X k , m = v ~ k , m w k , m ′ = v ~ k , m v ~ k , m ′ X ∈ R L M × N \mathbf{X}_{k,m}={\widetilde{\mathbf{v}}}_{k,m}\mathbf{w}_{k,m}'={\widetilde{\mathbf{v}}}_{k,m}{\widetilde{\mathbf{v}}}_{k,m}'\mathbf{X}\in\mathbb{R}^{LM\times N}
X k , m = v k , m w k , m ′ = v k , m v k , m ′ X ∈ R L M × N
特征向量v ~ k , m {\widetilde{\mathbf{v}}}_{k,m} v k , m i i i v ~ k , m ( i ) \widetilde{\mathbf{v}}_{k,m}^{\left(i\right)} v k , m ( i )
v ~ k , m ( i ) = ( I L ⊗ 1 M , i ′ ) v ~ k , m \widetilde{\mathbf{v}}_{k,m}^{\left(i\right)}=\left(\mathbf{I}_L\otimes\mathbf{1}_{M,i}'\right){\widetilde{\mathbf{v}}}_{k,m}
v k , m ( i ) = ( I L ⊗ 1 M , i ′ ) v k , m
第k k k i i i ω k \omega_k ω k
X k , m ( i ) = v ~ k , m ( i ) w k , m ′ = v ~ k , m ( i ) v ~ k , m ′ X ∈ R L × N \mathbf{X}_{k,m}^{\left(i\right)}={\widetilde{\mathbf{v}}}_{k,m}^{\left(i\right)}\mathbf{w}_{k,m}'={\widetilde{\mathbf{v}}}_{k,m}^{\left(i\right)}{\widetilde{\mathbf{v}}}_{k,m}'\mathbf{X}\in\mathbb{R}^{L\times N}
X k , m ( i ) = v k , m ( i ) w k , m ′ = v k , m ( i ) v k , m ′ X ∈ R L × N
因此，轨道矩阵可按照频率、特征值、序列进行分解
X = ∑ k , m X k , m = ∑ k , m P ( X k , m ( 1 ) ⋮ X k , m ( M ) ) \mathbf{X}=\sum_{k,m}\mathbf{X}_{k,m}=\sum_{k,m}\mathbf{P}\left(\begin{matrix}\mathbf{X}_{k,m}^{\left(1\right)}\\\vdots\\\mathbf{X}_{k,m}^{\left(M\right)}\\\end{matrix}\right)
X = k , m ∑ X k , m = k , m ∑ P ⎝ ⎜ ⎜ ⎜ ⎛ X k , m ( 1 ) ⋮ X k , m ( M ) ⎠ ⎟ ⎟ ⎟ ⎞
其中P \mathbf{P} P
分组步骤仍然由于成对出现特征值与特征向量的缘故，子成分需要叠加两个对应的基本矩阵，大体步骤与上面循环奇异谱分析一致，就不细说了。
X B k , m = X k , m + X L + 2 − k , m . \mathbf{X}_{B_{k,m}}=\mathbf{X}_{k,m}+\mathbf{X}_{L+2-k,m}.
X B k , m = X k , m + X L + 2 − k , m .
重构步骤涉及去hankel化，仅需注意对分组的矩阵进行反对角平均。
参考文献