|Abstract:|| Discrete observations of curves are often smoothed by attaching a penalty to the error sum of squares, and the most popular penalty is the integrated squared second derivative of the function that fits the data. But it has been known since the earliest days of smoothing splines that, if the linear differential operator D^2 is replaced by a more general differential operator L that annihilates most of the variation in the observed curves, then the resulting smooth has less bias and greatly reduced mean squared error.
This talk will show how we can use the data to estimate such an operator. The differential equation estimated in this way is already an interesting model for the data that represents the dynamics of the processes being estimated. But, in addition to the advantages in bias and MSE, it emerges that exciting new ways of representing the data emerge that use an orthogonal basis system defined by the estimated operator.