Principal component analysis of complex-valued time series

Question

My problem is the following: I have many, say around 800 complex-valued time-series, but I have a hunch that with a proper unitary transformation they can actually be transformed into another 800 time series where any significant change happens in only two of them, and the rest of the series should be close to constant zero. (The series are the coefficients of basis vectors from a quantum mechanics simulation, and based on the physics of the problem I have this feeling.) Can you give me hints on how to find this unitary transformation? I was thinking about principal component analysis and singular value decomposition, but I am not sure that they are satisfactory in case of complex numbers.

Answer 1

Suppose $m$ is the number of time series you have (800 in your example), and $n$ is the size of each time series. You can construct a complex data matrix $\mathbf{A}\in\mathbb{C}^{m\times n}$ containing your time series. Then you can apply SVD to diagonalize and obtain: $$ \mathbf{A} = \mathbf{U}\mathbf{\Sigma}\mathbf{V}^H $$ where $\mathbf{U}\in\mathbb{C}^{m\times m}$ and $\mathbf{V}\in\mathbb{C}^{n\times n}$ are unitary matrices and $\mathbf{\Sigma}\in\mathbb{C}^{m\times n}$ is a matrix containing the singular values in its diagonal (there is also the reduced-form representation of SVD where $\mathbf{\Sigma}$ is square!).

Then you can represent your data using the eigenvectors with largest singular values.