ABSTRACT
The general smoothing spline regression model (2.10) in Chapter 2 and the SS ANOVA model (4.31) in Chapter 4 assume that the unknown function is observed through some linear functionals. This chapter deals with situations when some unknown functions are observed indirectly through nonlinear functionals. We discuss some potential applications in this section. More examples can be found in Section 7.6. In some applications, the theoretical models depend on the unknown
functions nonlinearly. For example, in remote sensing, the satellite upwelling radiance measurements Rv are related to the underlying atmospheric temperature distribution f through the following equation
Rv(f) = Bv(f(xs))τv(xs)− ∫ xs x0
Bv(f(x))τ ′ v(x)dx,
where x is some monotone transformation of pressure p, for example, the kappa units x(p) = p5/8; x0 and xs are x values at the surface and top of the atmosphere; τv(x) is the transmittance of the atmosphere above x at wavenumber v; and Bv(t) is the Planck’s function, Bv(t) = c1v
3/{exp(c2v/t) − 1}, with known constants c1 and c2. The goal is to estimate f as a function of x based on noisy observations of Rv(f). Obviously, Rv(f) is nonlinear in f . Other examples involving reservoir modeling and three-dimensional atmospheric temperature distribution from satellite-observed radiances can be found in Wahba (1987) and O’Sullivan (1986). Very often there are certain constraints such as positivity and mono-
tonicity on the function of interest, and sometimes nonlinear transformations may be used to relax those constraints. For example, consider the following nonparametric regression model
Suppose g is known to be positive. The transformation g = exp(f) substitutes the original constrained estimation of g by the unconstrained estimation of f . The resulting transformed model,
yi = exp{f(xi)}+ ǫi, xi ∈ [0, 1], i = 1, . . . , n, (7.2)
depends on f nonlinearly. Monotonicity is another common constraint. Consider model (7.1) again and suppose g is known to be strictly increasing with g′(x) > 0. Write g′ as g′(x) = exp{f(x)}. Reexpressing g as g(x) = f(0) +
∫ x 0 exp{f(s)}ds leads to the following model
yi = β +
exp{f(s)}ds+ ǫi, xi ∈ [0, 1], i = 1, . . . , n. (7.3)
The function f is free of constraints and acts nonlinearly. Strictly speaking, model (7.3) is a semiparametric nonlinear regression model in Section 8.3 since it contains a parameter β. Sometimes one may want to consider empirical models that depend
on unknown functions nonlinearly. One such model, the multiplicative model, will be introduced in Section 7.6.4.
