Matrix Regression for linear ODE system

Background

I have the following homogeneous ODE system as an Initial Value Problem: $$y'=A\cdot y\quad\wedge\quad y(0)=y_0$$ where $$y\in\mathbb{R}^{N\times 1}$$ is the unknown vector and $$A\in\mathbb{R}^{N\times N}$$ is a known, constant coefficient nonsingular, diagonalizable matrix with only $$N$$ independent entries.

For instance, if $$N=3$$ it could be: $$A=\begin{pmatrix} -x_1 & 0 & 0\\ x_1 & -x_2 & 0\\ 0 & x_2 & -x_3 \end{pmatrix}$$ with $$x_1,x_2,x_3$$ known. It is possible to express a closed form solution of this system as: $$y(t)=\sum_{i=1}^NK_i e^{\lambda_it}\cdot u_i$$ where $$\lambda_i$$ is the i-th eigenvalue of $$A$$ and $$u_i$$ is the i-th eigenvector of $$A$$. The $$K_i$$ are constants of integration such that the Initial Condition is followed.

Question

1. Suppose to have the above ODE system, with the matrix $$A$$ with a known structure but the entries' values $$x_1,\ldots,x_N$$ are unknown.

2. Suppose also to have $$t_k$$ and $$y(t_k)=y_k$$ values for $$k$$ "experiments" with $$k\gg N$$, including the initial condition state, $$k=0\to t_{k=0}=0$$.

Is there a way to calculate such entries values by means of a regression? If so, what kind of regression? Also we assume that $$y_k \sim \mathcal{N}(0,\sigma^2)$$.

Above all, I think the answer to the first question being "yes" since the number of unknown parameters $$N$$ is way less than the number of experimental data $$k$$.

Edit 1 - The Brute Force Approach

Since the eigenbasis of $$A$$ is invariant under uniform scaling, meaning that for any constant $$K\neq 0$$ if $$u_i$$ is an eigenvector of $$A$$ (with eigenvalue $$\lambda_i$$) then $$w_i=Ku_i$$ is also an eigenvector (with eigenvalue $$\lambda_i$$), since: $$Au_i=\lambda_iu_i\quad\to\quad AKu_i=\lambda_iKu_i\quad\to\quad Aw_i=\lambda_iw_i$$ the general solution can be rewritten as a model: $$\hat{y}(t,\lambda,W)=\sum_{i=1}^N e^{\lambda_it}\cdot w_i$$ or, by component: $$y_j(t) = \sum_{i=1}^N e^{\lambda_it}\cdot w_{ji}$$ This yields $$N$$ equations with a total of $$N+N^2$$ parameters ($$N$$ eigenvalues and $$N^2$$ eigenvector components of the matrix $$W=\{w\}_{ji}$$) and thus $$k\gg N(N+1)$$ is required at least.

The regression is expressed as a minimization problem: $$\min_{\lambda, W}\sum_{j=1}^{k} \left\lvert y_{j} - \hat{y} (t_j,\lambda,W)\right\rvert_2$$ and once $$\lambda$$ and $$W$$ are known, then: $$A=W^{-1}\mathrm{diag}(\lambda)W$$ This process is however tedious, boring, and does not exploit the structure and properties of A. What are the improvements?

• The asker could do that if the $y_k$'s were taken at a high enough sampling rate (close enough together) relative to the speed of the dynamic so that said finite differences are reasonably accurate. The resulting fit of the dynamic would be a simple linear regression then, with a known closed-form solution. However, this is not done in practice, as it greatly amplifies the effect of any noise present in the data. Fitting in the solution space is preferred. Fitting a linear system is a fairly well studied thing, but for some reason the asker wants a solution that leverages that specific $A$. – jnez71 Sep 25 '18 at 23:58
• @AhmadBazzi exactly, I have both the values $y_k$ and the times $t_k$, such that $y_k=y(t_k)$ – alandella Sep 26 '18 at 10:54
• @jnez71 "The resulting fit of the dynamic would be a simple linear regression then, with a known closed-form solution" Can you point out the sources? I am sorry, but through research I wasn't able to find any – alandella Sep 26 '18 at 10:57
• @AhmadBazzi the $y'_k$ can be known using a finite difference scheme $\Delta y/\Delta t$ – alandella Sep 26 '18 at 12:29
• If you want to exploit the structure of $A$, we need to know what $A$ is like, right? Abstractly, say $A$ is parameterized by $\vec{x}$, then we can write $\lambda_i(\vec{x})$ and $u_i(\vec{x})$. Depending on structure of $A(\vec{x})$, these might be solvable in closed form, and the minimization can become $\min_{\vec{x}} \sum |y - \hat{y}(\lambda(\vec{x}), W(\vec{x}))|^2$ etc. – antkam Sep 26 '18 at 16:05

From the following system at the generic time $$t$$, the model and experimental values: $$\hat{\dot{y}}_t=\hat{A}y_t\quad\wedge\quad \dot{y}_t=\frac{y_{t+1}-y_t}{\Delta t}$$ Then, the error vector is quadratic with $$e_t=\dot{y}_t-\hat{\dot{y}}_t$$, omitting $$t$$ for clarity: $$ε_t\left(A\right)=e^Te=\left(\dot{y}-Ay\right)^T(\dot{y}-Ay)$$ The modified error $$ε'$$ is obtained without all terms that do not include $$A$$, so $$\arg\min ε = \arg \min ε'$$ $$ε'\left(A\right)=-{\dot{y}}^TAy-y^TA^T\dot{y}+y^TA^TAy$$ We have$$\left(Ay\right)^T=y^TA^T$$, so $$Ay=\left(y^TA^T\right)^T$$, and moreover $${\dot{y}}^TAy=\left(Ay\right)^T\dot{y}$$, so: $$\varepsilon^\prime\left(A\right)=-{\dot{y}}^TAy-\left(Ay\right)^T\dot{y}+\left(Ay\right)^TAy=-2\left(Ay\right)^T\dot{y}+\left(Ay\right)^TAy$$ So we have a way to obtain the objective function to be minimized: $$\varepsilon^\prime\left(A\right)=\sum_{i=1}^{N}{\left(Ay\right)_i\left(\left(Ay\right)_i-2{\dot{y}}_i\right)}=\sum_{i=1}^{N}\left(Ay\right)_i^2-2\sum_{i=1}^{N}{\left(Ay\right)_i{\dot{y}}_i}$$ We let $$\left(Ay\right)_i$$ be the new variable, or better have the vector $$\hat{B}=\hat{\dot{y}}=\hat{A}y$$: $$\varepsilon^\prime\left(B\right)={\hat{B}}^T\hat{B}-2{\hat{B}}^T{\dot{y}}_i$$ The minimum condition is respected: $$\frac{\partial\varepsilon^\prime\left(B\right)}{\partial{\hat{B}}_i}=2{\hat{B}}_i-2{\dot{y}}_i=0\ \ \ \rightarrow\ \ \ \ \frac{\partial^2\varepsilon^\prime\left(B\right)}{\partial{\hat{B}}_i^2}=2>0\ \ \ \rightarrow\ \ \ \text{min}\ \ \forall\ i$$ So, we have $$\left(Ay\right)_i={\dot{y}}_i$$, when the objective function is minimized, it corresponds to the actual objective linear system for all components. We have reduced this to a linear regression, where $$A_i^T$$ is the transposed $$i$$-th row of A: $${\dot{y}}_i=\sum_{j=1}^{N}{A_{ij}y_j}=\left(\begin{matrix}y_1&\cdots&y_N\\\end{matrix}\right)\begin{pmatrix}A_{i1} \\\vdots\\A_{iN}\end{pmatrix}=Y_tA_i^T$$ At each time instant, we have a matrix expression for each $$K$$ time sample point: $$\begin{pmatrix}\dot{y}_{i,t_1} \\\vdots\\\dot{y}_{i,t_K}\end{pmatrix}=\begin{pmatrix}% y_{1,t_1} & \ldots & y_{N,t_1}\\ \vdots & \ddots & \vdots\\ y_{1,t_K} & \ldots & y_{N,t_K}\\ \end{pmatrix}\begin{pmatrix}A_{i1} \\\vdots\\A_{iN}\end{pmatrix}$$ Thus $${\dot{Y}}_i\in\mathbb{R}^{K\times1}$$ and $$Y\in\mathbb{R}^{K\times N}$$. We can finally obtain: $$\dot{Y}_i=YA_i^T\ \ \ \rightarrow\ \ \ Y^T\dot{Y}_i=Y^TYA_i^T\ \ \ \rightarrow\ \ \ A_i^T=\left(Y^TY\right)^{-1}Y^T\dot{Y}_i$$ We can repeat this procedure for all $$i$$, and obtain the rate coefficients per row directly for each time step. We have constructed the matrix A from the data, and the effective rate coefficients are taken directly by concatenating the rows: $$A=\mathop{\mathrm{cat}}\left(A_i^T\right)$$ This method depends on the choice of $$K$$ and can lead to numerical instabilities if $$K\gg N$$ or $$K\simeq N\gg 0$$.