Spline smoothing of motorcycle data (DK §8.5)
=============================================

Head acceleration against time in a simulated motorcycle accident
(Silverman 1985): 133 observations, irregularly spaced, some at the same
time. The cubic smoothing spline is the smoothed level of a continuous-time
smooth trend plus an irregular (DK §3.8.2, §3.9.2), and its smoothing
parameter :math:`\lambda = \sigma^2_\zeta / \sigma^2_\varepsilon` is
estimated by maximum likelihood.

Results
-------

==============================  ==============  ==============
quantity                        DK              this library
==============================  ==============  ==============
:math:`\psi = \log\lambda`      -3.59 (0.22)    -2.36 (0.44)
:math:`\lambda`                 0.0275          0.0945
AIC                             9.43            9.42
==============================  ==============  ==============

The AIC agrees but :math:`\lambda` does not. The profile log likelihood
over :math:`\psi` has a single maximum, at -2.36; at DK's -3.59 it is 4.3
lower, which would give an AIC of 9.49. DK's AIC is therefore that of this
optimum, and their printed :math:`\lambda` comes from a scaling we could not
identify. The spline itself matches SciPy's cubic smoothing spline in the
tests.

Program
-------

.. literalinclude:: ../../example/dk_8_5_spline.f90
   :language: fortran

Output
------

.. literalinclude:: output/dk_8_5_spline.txt
   :language: text