Firstly, a POD is simply a decomposition of a spatiotemporal field into two disaparate components: the temporal features and the spatial features.
The hope is that they will approx
u ( s , t ) = ∑ k = 1 ∞ α k ( t ) ϕ k ( s )
\boldsymbol{u}(\mathbf{s},t) = \sum_{k=1}^{\infty}\alpha_k(t)\boldsymbol{\phi}_k(\mathbf{s}) u ( s , t ) = k = 1 ∑ ∞ α k ( t ) ϕ k ( s ) where ϕ k ( s ) \boldsymbol{\phi}_k(\mathbf{s}) ϕ k ( s ) a k ( t ) a_k(t) a k ( t ) D z D_z D z
u ( s , t ) ≈ ∑ k = 1 D z α k ( t ) ϕ k ( s ) \boldsymbol{u}(\mathbf{s},t) \approx \sum_{k=1}^{D_z}\alpha_k(t)\boldsymbol{\phi}_k(\mathbf{s}) u ( s , t ) ≈ k = 1 ∑ D z α k ( t ) ϕ k ( s ) In linear algebra form we can decompose the signal:
Singular Value Decomposition : U = L S R ⊤ \begin{aligned}
\text{Singular Value Decomposition}: && &&
\mathbf{U} &= \mathbf{LSR}^\top \\
\end{aligned} Singular Value Decomposition : U = LSR ⊤ We notice that the time coefficients vector is given by:
Time Coefficients : A s = L Spatial Covariance Matrix : C s = 1 m − 1 U U ⊤ \begin{aligned}
\text{Time Coefficients}: && &&
\mathbf{A_s} &= \mathbf{L} \\
\text{Spatial Covariance Matrix}: && &&
\mathbf{C_s} &= \frac{1}{m-1}\mathbf{UU}^\top \\
\end{aligned} Time Coefficients : Spatial Covariance Matrix : A s C s = L = m − 1 1 UU ⊤ $$
$$
U = A ϕ − 1 = A ϕ ⊤ \mathbf{U} = \mathbf{A}\boldsymbol{\phi}^{-1}=\mathbf{A}\boldsymbol{\phi}^\top U = A ϕ − 1 = A ϕ ⊤ where A s = S T Λ S T ⊤ \mathbf{A_s}=\mathbf{S}_T\boldsymbol{\Lambda}\mathbf{S}_T^\top A s = S T Λ S T ⊤
Snapshot ¶ # calculate correlation matrix
C_s = cov(Y, rowvar=True)
# solve eigenvalue problem
A_s, lam_s = eig(C_s)
# sort eigenvalues and eigenvectors
ilam_s = argsort(lam_s)[::-1]
lam_s = lam_s[ilam_s]
A_s = A_s[:, ilam_s]
# calculate spatial coefficientsSVD ¶ # normalize
Y_norm = Y / sqrt(Nt - 1)
# calculate SVD
U, eigs, VT = svd(Y_norm)
# get spatial nodes
PHI = VT
# Calculate the time coefficients
A = Y @ PHI
# calculate the eigenvalues
Eigs = diag(eigs) ** 2