Corrections and Updates

Functional Data Analysis

The function smooth_basis in the FDA Matlab package outputs the GCV correctly when using iPDA or other linear smoothers. It is not applicable when using generalized smoothing approach to parameter estimation. The way I suggested to get GCV in class is much less numerically stable.

generalized smoothing

Generalized smoothing approach is like NLS, but with a relaxation of the differential equation solution. This makes the likelihood smoother and helps parameter optimization. But we need to use a strategy of estimating parameters by incrementally increasing the smoothing parameter. Use a strategy like:

1. 1.Make initial guesses about the ODE parameter values and choose a very small smoothing parameter.
   

2. 2.Run generalized smoothing and get parameter estimates.
   

3. 3.Increase the smoothing parameter by a factor of 10, use final ODE parameter estimates from step (2) as initial guesses and re-run the generalized smoothing method.
   

4. 4.Repeat step 3 until your estimates become stable, or use a better convergence criterion.
   

Increasing lambda and using your last ODE parameter estimates as initial guesses is vital to the success of this optimization method, because it incrementally enforces the ODE solution to a larger and larger degree.

iPDA

When we looked at the sky diver problem with iPDA, we were using the wrong model to try and model the acceleration of the person, so if we try to use our results in a different situation we will not fit the data well using the model we built from this data (and iPDA). It is important to notice that our iPDA model uses two covariates (position and velocity), where the true model uses only one covariate (velocity).

Time varying coefficients are often used when we just don't know what is happening and can't come up with a better model or better covariates. Time varying coefficients are most useful when they really capture some seasonality or help us to realize which time varying covariate might be worth considering adding into the model further or how to build a differential equation model. Think of this version of iPDA as an exploratory tool when often a differential equation is much better and explains the system behaviour better and with easier interpretation.