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) illustrates the acceleration of the brain tissue before and after a series of five blows to the cranium.

**Figure (1)**

The laws of motion tell us that the acceleration f(t) can be modeled by a second order linear differential equation (LDE) with a point impulse u(t) representing the blow to the cranium and shown in the dashed lines in Figure (1).

This LDE

\begin{equation*}

\frac{\textrm{d}^2f}{\textrm{d}t^2} = \beta_{0} f + \beta_{1} \frac{\textrm{d}f}{\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 point impulse.

While there are several methods for estimating LDE 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 LDE 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 cranium 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, the fitted curve produced by Data2LD (solid line), the numerical approximation to the solution of the LDE with the parameters identified by Data2LD (dashed line) and the impulse function $u(t)$ representing the blow to the cranium (dotted line). We can see the LDE solution with the parameters defined by Data2LD is very close to the fitted curve produced by Data2LD, which indicates that the LDE provides an adequate description of the acceleration of the brain tissue.

**Papers:**

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

http://www.sciencedirect.com/science/article/pii/S0275531912000438

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

https://doi.org/10.1080/10618600.2016.1265526

Carey, M., Ramsay J. (2018) 'Parameter Estimation and Dynamic Smoothing

with Linear Differential Equations'. *Journal of Computational and Graphical Statistics*. (in press)