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Pointwise norm and inner product for functional data

Source: R/geometry-mv.R
tf_geom.Rd

Small geometric helpers defined by component-wise composition of the existing univariate Ops / Math machinery:

  • tf_norm(f) – pointwise Euclidean norm \(\lVert f(t) \rVert\);

  • tf_speed(f) – pointwise speed \(\lVert f'(t) \rVert\);

  • tf_inner(f, g) – pointwise inner product \(\langle f(t), g(t) \rangle\);

  • tf_distance(f, g) – pointwise Euclidean distance \(\lVert f(t) - g(t) \rVert\);

  • tf_tangent(f) – unit tangent \(f'(t) / \lVert f'(t) \rVert\) (undefined where the speed is zero – callers get NaNs there);

  • tf_reparam_arclength(f) – re-parametrize the curve at constant speed (i.e. by its normalized cumulative arc length).

Usage

tf_norm(f)

# Default S3 method
tf_norm(f)

# S3 method for class 'tf'
tf_norm(f)

# S3 method for class 'tf_mv'
tf_norm(f)

tf_speed(f)

tf_inner(f, g)

# Default S3 method
tf_inner(f, g)

# S3 method for class 'tf'
tf_inner(f, g)

# S3 method for class 'tf_mv'
tf_inner(f, g)

tf_distance(f, g)

tf_tangent(f)

# Default S3 method
tf_tangent(f)

# S3 method for class 'tf'
tf_tangent(f)

# S3 method for class 'tf_mv'
tf_tangent(f)

tf_reparam_arclength(f)

Arguments

f, g

tf_mv objects, or univariate tf (tfd/tfb) objects (with identical d and component names where two tf_mv arguments are required).

Value

a univariate tfd for tf_norm/tf_speed/tf_inner/tf_distance; tf_tangent returns a tf_mv (or a univariate tf for univariate input) and tf_reparam_arclength a tf_mv.

Details

These also apply to univariate tfd/tfb (treated as scalar-valued curves \(f: T \to \mathbb{R}\)), where they reduce to their one-dimensional specializations: \(\lVert f(t) \rVert = |f(t)|\), \(\langle f(t), g(t) \rangle = f(t)\,g(t)\), and the unit tangent \(f'(t) / |f'(t)| = \mathrm{sign}(f'(t))\).

See also

Other tf_mv-class: plot.tf_mv(), tf_arclength(), tf_mv_methods, tfb_mfpc(), tfb_mv(), tfd_mv()

Examples

set.seed(1)
f <- tfd_mv(list(x = tf_rgp(2), y = tf_rgp(2)))
tf_norm(f)
#> tfd[2]: [0,1] -> [0.4845396,2.208042] based on 51 evaluations each
#> interpolation by tf_approx_linear 
#> 1: ▂▃▄▄▄▄▃▂▂▁▁▁▂▃▄▆▇█████▇▇▆▅
#> 2: ▃▃▃▄▅▅▆▆▇▇▆▆▅▅▃▂▁▁▁▂▂▁▁▁▂▃
tf_speed(f)
#> tfd[2]: [0,1] -> [0.7946952,12.11029] based on 51 evaluations each
#> interpolation by tf_approx_linear 
#> 1: ▅▄▂▂▂▃▃▃▄▄▅▅▆▆▅▅▄▃▂▃▂▃▄▃▄▃
#> 2: ▇█▇▇▆▅▄▄▄▄▃▃▄▅▅▆▅▅▃▁▁▄▆▇▇▄
tf_distance(f, tfd_mv(list(x = tf_rgp(2), y = tf_rgp(2))))
#> tfd[2]: [0,1] -> [0.2677482,2.923029] based on 51 evaluations each
#> interpolation by tf_approx_linear 
#> 1: ▁▂▂▂▃▃▄▅▆▆▆▆▅▅▅▅▅▆▆▇█████▇
#> 2: ▃▄▅▆▆▆▇▆▆▆▅▅▄▄▂▁▁▁▂▂▃▃▄▅▇█
# univariate: tf_norm reduces to the pointwise absolute value
u <- tf_rgp(2)
tf_norm(u)
#> tfd[2]: [0,1] -> [0.0002797784,2.825257] based on 51 evaluations each
#> interpolation by tf_approx_linear 
#> 1: ▂▂▂▁▁▁▂▂▃▄▄▄▄▄▄▄▄▅▅▆▇█████
#> 2: ▅▅▅▅▄▄▃▃▃▂▂▂▂▃▂▂▂▂▂▁▁▁▁▁▁▁
tf_inner(u, tf_rgp(2))
#> tfd[2]: [0,1] -> [-1.494741,4.227105] based on 51 evaluations each
#> interpolation by tf_approx_linear 
#> 1: ▁▁▂▂▂▃▃▃▂▂▂▁▁▁▁▂▂▂▃▄▅▆▇███
#> 2: ▃▃▄▄▄▄▄▄▄▄▃▃▃▃▃▃▃▃▂▃▃▃▃▃▃▃