Biomedical Engineering Reference
In-Depth Information
6.2
Monotone Splines
Estimation of the unknown parameters in models (6.2) through (6.4) based
on interval-censored data is challenging as each model contains an unknown
increasing function. For many real-life data sets, the endpoints of observed in-
tervals show a continuous nature, as the observed intervals are overlapping and
different for different subjects. In this case, the number of unknown parame-
ters in each semiparametric model is on the order of sample size, causing great
estimation diculty from both theoretical and computational perspectives.
To reduce the number of parameters while also allowing adequate modeling
flexibility, we propose to model these unknown functions with the monotone
splines of Ramsay (1988).
We first give a brief review of the monotone splines of Ramsay (1988).
Suppose that the interest is to estimate an increasing function within a
closed interval [L; U]. Assume that (L) = 0 temporarily. One easy form is
to take
k X
(t) =
l b l (t);
(6.5)
l=1
where the b l are the integrated spline basis functions, each of which is nonde-
creasing from 0 to 1, and l are nonnegative spline coecients to ensure that
(t) is nondecreasing.
The spline basis functions bl l are essentially piecewise polynomials, and
the construction of such basis functions requires one to specify an increasing
sequence of knots f 1 ; ; m g within (L;U) and the degree d for the simple
case (Ramsay, 1988).
The degree controls the overall smoothness of the basis functions, taking
value 1 for piecewise linear, 2 for quadratic functions, 3 for cubic functions,
etc. Once the knots and degree are specified, the spline basis functions are
deterministic and the number k of basis functions is equal to the number of
 
Search WWH ::




Custom Search