Differentiating functional data: approximating derivative functions

Derivatives of tf-objects use finite differences of the evaluations for tfd and finite differences of the basis functions for tfb.

Usage

tf_derive(f, arg, order = 1, ...)

# S3 method for class 'matrix'
tf_derive(f, arg, order = 1, ...)

# S3 method for class 'tfd'
tf_derive(f, arg = tf_arg(f), order = 1, ...)

# S3 method for class 'tfd_irreg'
tf_derive(f, arg, order = 1, ...)

# S3 method for class 'tfb_spline'
tf_derive(f, arg = tf_arg(f), order = 1, ...)

# S3 method for class 'tfb_fpc'
tf_derive(f, arg = tf_arg(f), order = 1, ...)

Arguments

f: a tf-object
arg: grid to use for the finite differences.
order: order of differentiation. Maximal value for tfb_spline is 2. For tfb_spline-objects, order = -1 yields integrals (used internally).
...: not used

Value

a tf (with the same arg for tfd-inputs, possibly different basis for tfb-inputs, see details).

Details

The derivatives of tfd objects use second-order accurate central differences for interior points and second-order accurate one-sided differences at boundaries, following the non-uniform grid formulas from numpy.gradient with edge_order=2 (Fornberg, 1988). Domain and grid of the returned object are identical to the input. Unless the tfd has a rather fine and regular grid, representing the data in a suitable basis representation with tfb() and then computing the derivatives (or integrals) of those is usually preferable.

Note that, for spline bases like "cr" or "tp" which are constrained to begin/end linearly, computing second derivatives will produce artefacts at the outer limits of the functions' domain due to these boundary constraints. Basis "bs" does not have this problem for sufficiently high orders (but tends to yield slightly less stable fits).

Methods (by class)

tf_derive(matrix): row-wise finite differences
tf_derive(tfd): derivatives by finite differencing of function evaluations.
tf_derive(tfd_irreg): element-wise finite differencing for irregular grids. Falls back to tf_derive.tfd (interpolating to a common grid) if an explicit arg vector is supplied.
tf_derive(tfb_spline): derivatives by finite differencing of spline basis functions.
tf_derive(tfb_fpc): derivatives by finite differencing of FPC basis functions.

References

Fornberg, Bengt (1988). “Generation of Finite Difference Formulas on Arbitrarily Spaced Grids.” Mathematics of Computation, 51(184), 699–706.

Examples

arg <- seq(0, 1, length.out = 31)
x <- tfd(rbind(arg^2, sin(2 * pi * arg)), arg = arg)
dx <- tf_derive(x)
x
#> tfd[2]: [0,1] -> [-0.9945219,1] based on 31 evaluations each
#> interpolation by tf_approx_linear 
#> [1]: ▄▅▅▅▅▅▅▅▅▅▅▅▅▆▆▆▆▆▇▇▇▇████
#> [2]: ▅▆▆▇██████▆▆▅▄▃▂▁▁▁▁▁▁▂▂▃▄
dx
#> tfd[2]: [0,1] -> [-6.237351,6.373652] based on 31 evaluations each
#> interpolation by tf_approx_linear 
#> [1]: ▄▅▅▅▅▅▅▅▅▅▅▅▅▅▅▅▅▅▅▅▅▅▆▆▆▆
#> [2]: ███▇▆▅▄▄▃▂▁▁▁▁▁▁▂▃▄▄▅▆▇███
tf_arg(dx)
#>  [1] 0.00000000 0.03333333 0.06666667 0.10000000 0.13333333 0.16666667
#>  [7] 0.20000000 0.23333333 0.26666667 0.30000000 0.33333333 0.36666667
#> [13] 0.40000000 0.43333333 0.46666667 0.50000000 0.53333333 0.56666667
#> [19] 0.60000000 0.63333333 0.66666667 0.70000000 0.73333333 0.76666667
#> [25] 0.80000000 0.83333333 0.86666667 0.90000000 0.93333333 0.96666667
#> [31] 1.00000000