norm: Calculate the norm of functional data

Description

This function calculates the norm for each observation of a funData, irregFunData or multiFunData object.

Value

A numeric vector representing the norm of each observation.

Arguments

object: An object of class funData, irregFunData or multiFunData.
...: Further parameters (see Details).

Warning

The function is currently implemented only for functional data with one- and two-dimensional domains.

Details

For funData objects, the standard $L^2$ norm is calculated: $$||f|| = \left( \int_{\mathcal{T}} f(t)^2 dt \right)^{1/2}.$$ For irregFunData objects, each observed function is integrated only on the observed grid points (unless fullDom = TRUE).

The (weighted) norm of a multivariate functional data object $f = (f_1 , \ldots, f_p)$ is defined as $$||| f ||| := \left(\sum_{j=1}^p w_j || f_j ||^2 \right) ^{1/2}.$$

Further parameters passed to this function may include:

squared: Logical. If TRUE (default), the function calculates the squared norm, otherwise the result is not squared.
obs: A numeric vector, giving the indices of the observations, for which the norm is to be calculated. Defaults to all observations.
method: A character string, giving the integration method to be used. See integrate for details.
weight: An optional vector of weights for the scalar product; particularly useful for multivariate functional data, where each entry can be weighted in the scalar product / norm. Defaults to 1 for each element.
fullDom: Logical. If object is of class irregFunData and fullDom = TRUE, all functions are extrapolated to the same domain. Defaults to FALSE. See integrate for details.

Examples

Run this code

# Univariate
object <- funData(argvals = 1:5, X = rbind(1:5, 6:10))
norm(object)

# Univariate (irregular)
irregObject <- irregFunData(argvals = list(1:5, 2:4), X = list(2:6, 3:5))
norm(irregObject) # no extrapolation
norm(irregObject, fullDom = TRUE) # extrapolation (of second function)

# Multivariate
multiObject <- multiFunData(object, funData(argvals = 1:3, X = rbind(3:5, 6:8)))
norm(multiObject)
norm(multiObject, weight = c(2,1)) # with weight vector, giving more weight to the first element