The S function for fitting a SNR model is snr. A typical call is
snr(formula, func, params, start, data)The first four arguments are required. Arguments formula and func are the same as in nnr. params and start specify models for parameters and their initial values for the iterative procedure.
Method for selecting smoothing parameters is specified by the argument spar. spar=''v'', spar=''m'' and spar=''u'' correspond to GCV, GML and UBR methods respectively, with GCV as the default. Other options include correlation, weights, and control. They all have the same function as in ssr.
An object of snr class is returned. Generic functions summary, predict and intervals can be applied to extract further information. See help files for details.
Example 9. Projection Pursuit models assume that . They are also known as the multiple index models. Note that our estimation procedure is similar to that used in Roosen and Hastie (1994). For example, suppose that , , and both and are modeled using TPS on with . Instead of polynomial splines, we use TPS's to avoid the limitation on the domain. For identifiability, we need the side condition . We can fit such a model by
snr(y~f1(a11*t1+a12*t2+sqrt(1-a11**2-a12**2)*t3) + f2(a21*t1+a22*t2+sqrt(1-a21**2-a22**2)*t3), func=list(f1(z)+f2(z)~list(~z,rk=tp.pseudo(z)), params=list(a11+a12+a21+a22~1), start=c(a110,a120,a210,a220))
Example 10. Nonlinear partial splines assume that , where is a known function depends nonlinearly on parameters . For example, suppose that , is a function of modeled using a periodic spline, and . we can fit such a model by
snr(y~f(t1)+a*exp(b*t2), func=list(f(z)~list(~1,rk=periodic(z)), params=list(a+b~1), start=c(a0,b0))The function can also be input as an outside S function as in gnls.
Example 11. Monotone spline. Consider model ()
where is assumed to be a strictly increasing function.
Nychka and Ruppert (1995) used the following transformation
snr(y~a+h(g), func=list(g(z)~list(~z,rk=cubic(z)), params=list(a~1), start=c(a0))where is an S function calculating the integral .
For monotone (either increasing or decreasing),
Ramsay (1998) suggested the following transformation
snr(y~a+b*h(g), func=list(g(z)~list(~z,rk=cubic(z))), params=list(a~1,b~1), start=c(a0,b0))where is a S function calculating the integral .
Example 12. Positive monotone spline. Consider
model () where is assumed to be a strictly
positive and increasing function. One may use the following
snr(y~exp(a+h(g)), func=list(g(z)~list(~z,rk=cubic(z)), params=list(a~1), start=c(a0))where is a S function calculating the integral .
Example 13. Varying coefficient models.
. The varying coefficient
models in Hastie and Tibshirani (1993) assume that
snr(y~a+f1(z)*x1+f2(z)*x2, func=list(f1(z)+f2(z)~list(~z,cubic(z))), params=list(a~1), start=c(a0))
Example 14. Self-modeling nonlinear regression
(SEMOR) models were first proposed by Lawton et al. (1972)
to fit repeated measures data. They assumed that there exists
a common curve for all subjects and that a
particular subject's response curve is some parametric
transformation of the common curve. We consider a more
general SEMOR model
snr(y~a1+exp(a2)*f((t-a3)/exp(a4)), func=list(f(z)~list(~z,tp.pseudo(z))), params=list(a1+a2+a3+a4~subject-1), start=c(a10,a20,a30,a40))
We may also impose side conditions on . For example, we may remove the constant functions from the model space of to make it identifiable with 's. We may use the constraint to make identifiable with 's. Under these alternative side conditions and , model () can be fitted by
snr(y~a1+exp(a2)*f((t-a3)/exp(a4)), func=list(f(z)~list(~z-1,tp.pseudo(z))), params=list(a1+a2~subject,a3+a4~subject-1), start=c(a10,a20,a30,a40), constraint=list(maxValue=1))Note that the ``constraint'' option is not available in the current version.