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fda (version 6.2.0)

create.basis: Create Basis Set for Functional Data Analysis

Description

Functional data analysis proceeds by selecting a finite basis set and fitting data to it. The current fda package supports fitting via least squares penalized with lambda times the integral over the (finite) support of the basis set of the squared deviations from a linear differential operator.

Arguments

Author

J. O. Ramsay and Spencer Graves

Details

The most commonly used basis in fda is probably B-splines. For periodic phenomena, Fourier bases are quite useful. A constant basis is provided to facilitation arithmetic with functional data objects. To restrict attention to solutions of certain differential equations, it may be useful to use a corresponding basis set such as exponential, monomial or power basis sets.

Power bases support the use of negative and fractional powers, while monomial bases are restricted only to nonnegative integer exponents.

The polygonal basis is essentially a B-spline of order 2, degree 1.

The following summarizes arguments used by some or all of the current create.basis functions:

rangeval

a vector of length 2 giving the lower and upper limits of the range of permissible values for the function argument.

For bspline bases, this can be inferred from range(breaks). For polygonal bases, this can be inferred from range(argvals). In all other cases, this defaults to 0:1.

nbasis

an integer giving the number of basis functions.

This is not used for two of the create.basis functions: For constant this is 1, so there is no need to specify it. For polygonal bases, it is length(argvals), and again there is no need to specify it.

For bspline bases, if nbasis is not specified, it defaults to (length(breaks) + norder - 2) if breaks is provided. Otherwise, nbasis defaults to 20 for bspline bases.

For exponential bases, if nbasis is not specified, it defaults to length(ratevec) if ratevec is provided. Otherwise, in fda_2.0.2, ratevec defaults to 1, which makes nbasis = 1; in fda_2.0.4, ratevec will default to 0:1, so nbasis will then default to 2.

For monomial and power bases, if nbasis is not specified, it defaults to length(exponents) if exponents is provided. Otherwise, nbasis defaults to 2 for monomial and power bases. (Temporary exception: In fda_2.0.2, the default nbasis for power bases is 1. This will be increased to 2 in fda_2.0.4.)

In addition to rangeval and nbasis, all but constant bases have one or two parameters unique to that basis type or shared with one other:

Beginning with fda_2.1.0, the last 6 arguments for all the create.basis functions will be as follows; some but not all are available in the previous versions of fda:

References

Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009), Functional data analysis with R and Matlab, Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2005), Functional Data Analysis, 2nd ed., Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York.

See Also

create.bspline.basis create.constant.basis create.exponential.basis create.fourier.basis create.monomial.basis create.polygonal.basis create.power.basis