Extend algorithm to multivariable data sets

[elise1993]

The current implementation only allows for single-variable time series. Research and implement how multivariable data may be included.

[elise1993]

I believe this requires a complete refactor of the whole code, and thus won't be added at this time.

While DMD is naturally well-suited for multivariable data sets, I do not know how Hankel matrices, and thus the HAVOK model itself, generalizes to multivariable data sets. Perhaps we can construct a separate HAVOK model for each variable and couple them using the forcing functions.

[elise1993]

I cannot find any information on how to generalize HAVOK to multivariate datasets. A few ideas on how to do this:

1. Since DMD treats each column of data as its own positional argument (variable), the order of the Hankel matrix may not be as important. However, re-ordering the columns of the Hankel matrix will have an impact on how the coordinate system is represented, and may not be ideal. For testing, when multivariate data (e.g., $x$ and $y$) is available, we could stack the Hankel matrix as follows:

H = \begin{bmatrix}
x(t_1) & y(t_1) & x(t_2) & y(t_2) & \cdots & x(t_p) & y(t_p) \\
x(t_2) & y(t_2) & x(t_3) & y(t_3) & \cdots & x(t_{p+1}) & y(t_{p+1}) \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
x(t_q) & y(t_q) & x(t_{q+1}) & y(t_{q+1}) & \cdots & x(t_{p+q}) & y(t_{p+q}) 
\end{bmatrix}

2. We can also develop a HAVOK model for each variable ($x$ and $y$) separately, and then link them through the nonlinear term. For example, if we have the following models for $x$ and $y$:

$$ \dot{\textbf{v}} = \textbf{A}_x \textbf{v} + \textbf{B}_x v_r(x,y)$$

$$ \dot{\textbf{w}} = \textbf{A}_y \textbf{w} + \textbf{B}_y w_r(x,y)$$

We can train the ML models for $v_r$ and $w_r$ as functions of both $x$ and $y$.