Expertise:
Intermediate
Do we have any other extensions of these ideas?
Indeed we do. So far we have assumed that the breakpoints are a strictly
increasing sequence. But in fact we can actually place more than one breakpoint
at the a specific value t. The argument gets rather complicated, and de Boor
(2002) is the standard reference, but what happens is simple: For each additional
breakpoint ξ equal to t, the spline function loses one degree of smoothness, and
thus has one less smooth derivative. In the extreme, if there are m coincident
breakpoints, the resulting spline function can be discontinuous at that point. In
this way, we can accommodate even abrupt changes of level in the spline family.
The term knot is used routinely in the literature on splines to refer
to what we have called a breakpoint here. Strictly speaking, however, the
term knot should be reserved for possibly coincident breakpoints, and the term
breakpoint then means technically the strictly increasing sequence
of t-values at which knots are located.
However, there is another whole world beyond even this. Here we have confined our
attention to spline functions that are piecewise polynomials. Suppose,
however, that we work with some other set of primitive basis functions other
than powers of t. Suppose, for the sake of argument, that the functions shall
be piecewise Fourier functions instead. Or piecewise sums of exponentials with
a sequence of rate constants. The possibilities are unlimited.
But enough is enough. Or, if not, see the wonderful book by L. Schumaker(1981).
What makes spline functions easy to fit to data?
Notice that a spline basis function of order m will be positive over at most m
adjacent subintervals. In fact, the spline basis functions next to the boundaries
are positive over a decreasing number of subintervals to the point where the
extreme basis functions are positive over a only the end intervals.
This means that a matrix of values Z of spline basis functions will be band
structured in the sense that the nonzero entries in the matrix will be only found
in a band running from the top left to the lower right.
Now the equation defining the coefficient vector a has the following form:
a = (Z'Z + λR)-1Z'y
where λ is a scalar smoothing parameter and matrix R is called the
roughness penalty matrix.
The band-structure is also found in the cross-product matrix Z'Z and in
the roughness penalty matrix R. When thousands of points of observation are
used, this can result in a huge saving of computer time for algorithms making use
of this band structure over algorithms which assume that any entries can be nonzero.
Conclusion
Now look for the shift-scale-smooth operations in other basis function families, and you will
find them. The Fourier series are already infinitely differentiable and therefore no further
smoothness is possible, so only shift-scale is involved. Moreover, only a single shift is
involved, and that is the shift by πω units to convert sin(ωt) into cos(ωt).
The rescaling is the multiplication of the base frequency ω by the successive natural numbers.
Polynomials involve neither shifting or scaling, but smoothness is achieved by exactly the same
tilting operation that we used for splines. This is not surprising since splines are actually
polynomials over sub-intervals.
Wavelets are a result of playing especially clever and beautiful games with these three operations.
References
de Boor, C. (2002) A Practical Guide to Splines. Second Edition. New
York: Springer.
Schumaker, L. (1981) Spline Functions: Basic Theory. New York:
John Wiley and Sons. |
