What is Data 2 Linear Dynamics (Data2LD)?

Data 2 Linear Dynamics (Data2LD) estimates the solution and the parameters of linear dynamical systems from incomplete and noisy observations of the underlying processes.

It uses a linear combination of spline basis functions to approximate the implicitly defined solution of the dynamical system. While also estimating the systems' parameters by requiring that this approximating solution adheres to the data.

Papers:

Carey, M., Gath, E., Hayes, K. (2016) 'A generalized smoother for linear ordinary differential equations'. Journal of Computational and Graphical Statistics.

Carey, M., Gath, E., Hayes, K. (2014) 'Frontiers in financial dynamics'. Research in International Business and Finance

Using Data2LD to model the acceleration of brain tissue

Let's consider a study on traumatic brain injury (TBI), which contributes to just under a third (30.5%) of all injury-related deaths in the US and is caused by a blow to the head. Figure (1) shows the 133 accelerometer readings taken over 55.2 milliseconds. The dashed line represents the impulse function which denotes the blow to the head.

Figure (1)

The laws of motion tell us that the acceleration f(t) can be modelled by a second-order linear ordinary differential equation (ODE) with input a unit pulse u(t) representing the blow to the head and shown in the dashed lines in Figure (1).

This ODE
\begin{equation}\label{ODE}
\frac{\textrm{d}^2f(t)}{\textrm{d}t^2} + \beta_{0} f(t) + \beta_{1} \frac{\textrm{d}f(t)}{\textrm{d}t} + \alpha_{0} u(t)
\end{equation}
contains three parameters $\beta_{0},\beta_{1}$ and $\alpha_{0},$ and these convey the rate of the restoring force (as $t \rightarrow \infty,$ the acceleration will tend to revert back zero), the rate of the friction force (as $t \rightarrow \infty,$ the oscillations in the acceleration reduce to zero) and the rate of force from the unit pulse.

While there are several methods for estimating ODE parameters with partially observed data, they are invariably subject to several problems including high computational cost, sensitivity to initial values or large sampling variability.

We propose a method called Data2LD for data to linear dynamics that overcomes these issues and produces estimates of the ODE parameters that have less bias, a smaller sampling variance and a ten-fold improvement in computation.

The final parameter estimates with 95% confidence intervals are, $\hat{\beta_{0}} = -0.056 \pm 0.002,$ $\hat{\beta_{1}} = -0.150 \pm 0.018$ and $\hat{\alpha_{0}} = 0.395 \pm 0.032.$ indicating that the acceleration is an under-damped process; after the blow to the head, the acceleration will oscillate with a decreasing amplitude that will quickly decay to zero.

Figure (2)

Figure (2) shows the accelerometer readings of the brain tissue before and after a series of five blows to the head indicated by the circles. The fitted curve produced by Data2LD (solid line), the 95% confidence interval for this curve (dashed line) and the 95% prediction interval for this curve (grey region). We can see the fitted curve approximating the ODE solution provides an adequate description of the acceleration of the brain tissue.

Matlab code to produce the results from the above example