Package 'tf'

Accessing, evaluating, subsetting and subassigning tf vectors

Description

These functions access, subset, replace and evaluate tf objects. For more information on creating tf objects and converting them to/from list, data.frame or matrix, see tfd() and tfb(). See details.

Usage

## S3 method for class 'tf_mv'
x[i, j, component = NULL, interpolate = TRUE, matrix = TRUE]

## S3 replacement method for class 'tf_mv'
x[i] <- value

## S3 method for class 'tf'
x[i, j, interpolate = TRUE, matrix = TRUE]

## S3 replacement method for class 'tf'
x[i] <- value

Arguments

x	an tf.
i	index of the observations (integerish, character or logical, usual R rules apply). Can also be a two-column matrix for extracting specific (function, arg-value) pairs: the first column gives the function indices, the second column gives the arg values at which to evaluate each function. Returns a numeric vector in that case. j must not be provided when i is a matrix.
j	The arg used to evaluate the functions. A (list of) numeric vectors. NOT interpreted as a column number but as the argument value of the respective functional datum. If j is missing but matrix is explicitly given, j defaults to tf_arg(x).
component	for tf_mv objects only: optionally restrict evaluation / extraction to a subset of the output dimensions, given by name or integer index. A single name/index drops to the univariate component (a tfd or tfb); a vector of length > 1 returns a sub-tf_mv containing just those components. NULL (default) keeps all d components.
interpolate	should functions be evaluated (i.e., inter-/extrapolated) for values in arg for which no original data is available? Only relevant for the raw data class tfd, for which it defaults to TRUE. Basis-represented tfb are always "interpolated".
matrix	should the result be returned as a matrix or as a list of data.frames? If TRUE, j has to be a (list of a) single vector of arg. See return value.
value	tf object for subassignment. This is typed more strictly than concatenation: subassignment only happens if the common type of value and x is the same as the type of x, so subassignment never changes the type of x but may do a potentially lossy cast of value to the type of x (with a warning).

Details

Note that these break certain (terrible) R conventions for vector-like objects:

• no argument recycling,

• no indexing with NA,

• no indexing with names not present in x,

• no indexing with integers ⁠> length(x)⁠

All of the above will trigger errors.

Value

If i is a two-column matrix, a numeric vector of pointwise evaluations (one per row of i).
If j is missing (and i is not a matrix), a subset of the functions in x as given by i.
If j is given and matrix == TRUE, a numeric matrix of function evaluations in which each row represents one function and each column represents one argval as given in argument j, with an attribute arg=j and row- and column-names derived from x[i] and j.
If j is given and matrix == FALSE, a list of tbl_dfs with columns arg = j and value = evaluations at j for each observation in i.

Examples

x <- 1:3 * tfd(data = 0:10, arg = 0:10)
plot(x)
# this operator's 2nd argument is quite overloaded -- you can:
# 1. simply extract elements from the vector if no second arg is given:
x[1]
x[c(TRUE, FALSE, FALSE)]
x[-(2:3)]
# 2. use the second argument and optional additional arguments to
#    extract specific function evaluations in a number of formats:
x[1:2, c(4.5, 9)] # returns a matrix of function evaluations
x[1:2, c(4.5, 9), interpolate = FALSE] # NA for arg-values not in the original data
x[-3, seq(1, 9, by = 2), matrix = FALSE] # list of data.frames for each function
# 3. use a 2-column matrix to extract specific (function, arg) pairs:
x[cbind(1:3, c(0, 5, 10))] # one value per function
# 4. use matrix= with a missing j to evaluate on the default arg grid:
x[1:2, , matrix = FALSE] # same as x[1:2, tf_arg(x), matrix = FALSE]
# in order to evaluate a set of observed functions on a new grid and
# save them as a functional data vector again, use `tfd` or `tfb` instead:
tfd(x, arg = seq(0, 10, by = 0.01))

---

Convert functional data back to tabular data formats

Description

Various converters to turn tfb- or tfd-vectors into data.frames or matrices, or even an actual R function.

Usage

## S3 method for class 'tf'
as.data.frame(x, row.names = NULL, optional = FALSE, unnest = FALSE, ...)

## S3 method for class 'tf'
as.matrix(x, arg, interpolate = FALSE, ...)

## S3 method for class 'tf'
as.function(x, ...)

Arguments

x	a tf object.
row.names	NULL or a character vector giving the row names for the data frame. Missing values are not allowed.
optional	not used.
unnest	if TRUE, the function will return a data.frame with the evaluated functions.
...	additional arguments to be passed to or from methods.
arg	a vector of argument values / evaluation points for x. Defaults to tf_arg(x) (so for x on irregular grids, this will be the union of all observed arg-values by default).
interpolate	should functions be evaluated (i.e., inter-/extrapolated) for values in arg for which no original data is available? Only relevant for the raw data class tfd, for which it defaults to FALSE. Basis-represented functional data tfb are always "interpolated".

Value

for as.data.frame.tf: if unnest is FALSE (default), a one-column data.frame with a tf-column containing x. if unnest is TRUE, a 3-column data frame with columns id (containing (unique) names of x or a numeric identifier if x is unnamed), arg, and value, with each row containing one function evaluation at the original arg-values.

for as.matrix.tf: a matrix with one row per function and one column per arg.

for as.function.tf: an R function with argument arg that evaluates x on arg and returns the list of function values

See Also

Other tidyfun converters: converters-mv

Examples

f <- tfd(sin(seq(0, 2 * pi, length.out = 11)), arg = seq(0, 1, length.out = 11))
as.data.frame(f)
as.data.frame(f, unnest = TRUE)
as.matrix(f)
fun <- as.function(f)
fun(c(0, 0.5, 1))

---

Coerce a tf_mv to a matrix or data.frame

Description

as.matrix.tf_mv returns a 3-d array ⁠[curve, arg, component]⁠ – the natural shape for a vector-valued evaluation. This is deliberately different from as.matrix.tf (2-d, ⁠[curve, arg]⁠); see ⁠@seealso⁠.

Usage

## S3 method for class 'tf_mv'
as.matrix(x, arg, interpolate = FALSE, ...)

## S3 method for class 'tf_mv'
as.data.frame(
  x,
  row.names = NULL,
  optional = FALSE,
  unnest = FALSE,
  long = TRUE,
  arg = NULL,
  interpolate = TRUE,
  grids = c("union", "component"),
  ...
)

Arguments

x	a tf_mv object.
arg	optional evaluation grid (numeric vector or per-curve list). When NULL (default for as.data.frame.tf_mv; equivalent to "missing" for as.matrix.tf_mv), the per-curve union of all components' native argument grids is used.
interpolate	forwarded to the underlying tf evaluation. tfb components are always interpolated.
...	passed through.
row.names, optional	standard as.data.frame plumbing.
unnest	if TRUE, return an evaluated data.frame (see long); if FALSE (default), a one-column data.frame wrapping x.
long	when unnest = TRUE, controls the schema. long = TRUE (default) returns a 4-column data.frame ⁠(id, arg, component, value)⁠ – the multivariate analogue of the univariate ⁠(id, arg, value)⁠ contract, with component a factor over attr(x, "comp_names"). long = FALSE returns the wide ⁠(id, arg, comp1, ..., compd)⁠ schema.
grids	when unnest = TRUE, controls where components are evaluated when they live on different argument grids (for shared grids both settings agree). "union" (default) evaluates every component on each curve's union grid, so components get (interpolated) values at the other components' arg values inside their observed range. "component" evaluates each component strictly on its own grid (or on arg, if supplied): no values are fabricated at args a component was not observed at – in the long schema such rows are simply absent, in the wide schema the other components' columns are NA there. Use "union" for paired evaluations (e.g. trajectory plots), "component" for faithful tabular exports of the observed data.

Details

as.data.frame.tf_mv returns either a single-column wrapping data.frame (unnest = FALSE, for storing a tf_mv in a tibble column) or an evaluated long/wide data.frame (unnest = TRUE).

Value

a 3-d array (as.matrix.tf_mv) or a data.frame (as.data.frame.tf_mv).

See Also

as.matrix.tf() (2-d sibling), as.data.frame.tf() (univariate contract), tf_evaluate().

Other tidyfun converters: as.data.frame.tf()

Examples

arg <- seq(0, 1, length.out = 11)
xf <- tfd(t(sapply(1:3, function(i) sin(2 * pi * arg + i))), arg = arg)
yf <- tfd(t(sapply(1:3, function(i) cos(2 * pi * arg + i))), arg = arg)
mv <- tfd_mv(list(x = xf, y = yf))
dim(as.matrix(mv))
head(as.data.frame(mv, unnest = TRUE))

---

Wrap a non-list object in a list

Description

Returns x unchanged if it is already a list, otherwise wraps it in a one-element list. Used internally to normalize arg inputs that may be either a single numeric vector or a list of per-curve vectors.

Usage

ensure_list(x)

Arguments

x	any input.

Value

x if it is a list, otherwise list(x).

See Also

Other tidyfun utility functions: in_range(), tf_arg(), tf_zoom(), unique_id()

Examples

ensure_list(1:3)
ensure_list(list(1:3, 4:6))

---

Tukey's Five Number Summary for tf vectors

Description

Computes a depth-based five number summary for functional data: the observations with minimum, lower-hinge, median, upper-hinge, and maximum depth values.

Usage

fivenum(x, na.rm = FALSE, ...)

## Default S3 method:
fivenum(x, na.rm = FALSE, ...)

## S3 method for class 'tf'
fivenum(x, na.rm = FALSE, depth = "MHI", ...)

## S3 method for class 'tf_mv'
fivenum(x, na.rm = FALSE, depth = "MBD", ...)

Arguments

x	a tf vector (or numeric for the default method).
na.rm	logical; if TRUE, NA observations are removed first.
...	passed to tf_depth().
depth	depth method for ordering. See tf_depth(). Defaults to "MHI" for an up-down ordering.

Value

fivenum.tf: a named tf vector of length 5.
fivenum.default: see stats::fivenum().

See Also

Other tidyfun summary functions: functionwise, median.tf_mv(), tfsummaries

Examples

set.seed(1)
f <- tf_rgp(7)
fivenum(f)

---

Eigenfunctions via weighted, regularized SVD

Description

Compute (truncated) orthonormal eigenfunctions and scores for (partially missing) data on a common (potentially non-equidistant) grid.

Usage

fpc_wsvd(data, arg, pve = 0.995)

## S3 method for class 'matrix'
fpc_wsvd(data, arg, pve = 0.995)

## S3 method for class 'data.frame'
fpc_wsvd(data, arg, pve = 0.995)

Arguments

data	numeric matrix of function evaluations (each row is one curve, no NAs).
arg	numeric vector of argument values.
pve	percentage of variance explained.

Details

Performs a weighted SVD with trapezoidal quadrature weights s.t. returned vectors represent (evaluations of) orthonormal eigenfunctions $\phi_j(t)$, not eigenvectors $\phi_j = (\phi_j(t_1), \dots, \phi_j(t_n))$, specifically:
$\int_T \phi_j(t)^2 dt \approx \sum_i \Delta_i \phi_j(t_i)^2 = 1$ given quadrature weights $\Delta_i$, not $\phi_j'\phi_j = \sum_i \phi_j(t_i)^2 = 1$;
$\int_T \phi_j(t) \phi_k(t) dt = 0$ not $\phi_j'\phi_k = \sum_i \phi_j(t_i)\phi_k(t_i) = 0$.
For incomplete data, this uses a soft-impute iterative-SVD scheme (see references). Note that this will not work well for data on a common grid if more than a few percent of data points are missing, and it breaks down completely for truly irregular data with no/few common timepoints, even if observed very densely. For such data, either re-evaluate on a common grid first or use more advanced FPCA approaches like refund::fpca_sc(), see last example for tfb_fpc()

Value

a list with entries

• mu estimated mean function (numeric vector)

• efunctions estimated FPCs (numeric matrix, columns represent FPCs)

• scores estimated FPC scores (one row per observed curve)

• npc how many FPCs were returned for the given pve (integer)

• scoring_function a function that returns FPC scores for new data and given eigenfunctions, see tf:::.fpc_wsvd_scores for an example.

Author(s)

Fabian Scheipl

References

the soft-impute SVD algorithm for incomplete data is described in Mazumder, Rahul, Hastie, Trevor, Tibshirani, Robert (2010). “Spectral Regularization Algorithms for Learning Large Incomplete Matrices.” The Journal of Machine Learning Research, 11, 2287–2322.

See Also

Other tfb-class: tfb(), tfb_fpc(), tfb_spline()

Other tfb_fpc-class: tfb_fpc(), tfb_mfpc()

Examples

arg <- seq(0, 1, length.out = 41)
data <- t(sapply(seq(0, 2 * pi, length.out = 10),
                 function(p) sin(2 * pi * arg + p)))
fpc <- fpc_wsvd(data, arg = arg, pve = 0.99)
fpc$npc

---

Summarize each tf in a vector (function-wise)

Description

These functions extract (user-specified) function-wise summary statistics from every entry in a tf-vector. To summarize a vector of functions at each argument value, see ?tfsummaries. Note that most of these will tend to yield lots of NAs for irregular tfd unless you set a tf_evaluator()-function that does inter- and extrapolation for them beforehand.

Usage

tf_fwise(x, .f, arg = tf_arg(x), ...)

tf_fmax(x, arg = tf_arg(x), na.rm = FALSE)

## S3 method for class 'tf_mv'
tf_fmax(x, arg = tf_arg(x), na.rm = FALSE)

tf_fmin(x, arg = tf_arg(x), na.rm = FALSE)

## S3 method for class 'tf_mv'
tf_fmin(x, arg = tf_arg(x), na.rm = FALSE)

tf_fmedian(x, arg = tf_arg(x), na.rm = FALSE)

## S3 method for class 'tf_mv'
tf_fmedian(x, arg = tf_arg(x), na.rm = FALSE)

tf_frange(x, arg = tf_arg(x), na.rm = FALSE, finite = FALSE)

tf_fmean(x, arg = tf_arg(x))

## S3 method for class 'tf_mv'
tf_fmean(x, arg = tf_arg(x))

tf_fvar(x, arg = tf_arg(x))

## S3 method for class 'tf_mv'
tf_fvar(x, arg = tf_arg(x))

tf_fsd(x, arg = tf_arg(x))

## S3 method for class 'tf_mv'
tf_fsd(x, arg = tf_arg(x))

tf_crosscov(x, y, arg = tf_arg(x))

tf_crosscor(x, y, arg = tf_arg(x))

Arguments

x	a tf object.
.f	a function or formula that is applied to each entry of x, see purrr::as_mapper() and details.
arg	defaults to standard argument values of x.
...	additional arguments for purrr::as_mapper().
na.rm	a logical (TRUE or FALSE) indicating whether missing values should be removed.
finite	logical, indicating if all non-finite elements should be omitted.
y	a tf object.

Details

tf_fwise turns x into a list of data.frames with columns arg and values internally, so the function/formula in .f gets a data.frame .x with these columns, see examples below or source code for tf_fmin(), tf_fmax(), etc.

Value

a list (or vector) of the same length as x with the respective summaries.

Functions

• tf_fwise(): User-specified function-wise summary statistics

• tf_fmax(): maximal value of each function

• tf_fmax(tf_mv): component-wise maxima of each vector-valued function

• tf_fmin(): minimal value of each function

• tf_fmin(tf_mv): component-wise minima of each vector-valued function

• tf_fmedian(): median value of each function

• tf_fmedian(tf_mv): component-wise medians of each vector-valued function

• tf_frange(): range of values of each function

• tf_fmean(): mean of each function: $\tfrac{1}{|T|}\int_T x_i(t) dt$

• tf_fmean(tf_mv): component-wise means of each vector-valued function

• tf_fvar(): variance of each function: $\tfrac{1}{|T|}\int_T (x_i(t) - \bar x(t))^2 dt$

• tf_fvar(tf_mv): component-wise variances of each vector-valued function

• tf_fsd(): standard deviation of each function: $\sqrt{\tfrac{1}{|T|}\int_T (x_i(t) - \bar x(t))^2 dt}$

• tf_fsd(tf_mv): component-wise standard deviations of each vector-valued function

• tf_crosscov(): cross-covariances between two functional vectors: $\tfrac{1}{|T|}\int_T (x_i(t) - \bar x(t)) (y_i(t)-\bar y(t)) dt$

• tf_crosscor(): cross-correlation between two functional vectors: tf_crosscov(x, y) / (tf_fsd(x) * tf_fsd(y))

See Also

Other tidyfun summary functions: fivenum(), median.tf_mv(), tfsummaries

Examples

x <- tf_rgp(3)
layout(t(1:3))
plot(x, col = 1:3)
#  each function's values to [0,1]:
x_clamp <- (x - tf_fmin(x)) / (tf_fmax(x) - tf_fmin(x))
plot(x_clamp, col = 1:3)
# standardize each function to have mean / integral 0 and sd 1:
x_std <- (x - tf_fmean(x)) / tf_fsd(x)
tf_fvar(x_std) == c(1, 1, 1)
plot(x_std, col = 1:3)
# Custom functions:
# 80%tiles of each function's values:
tf_fwise(x, \(.x) quantile(.x$value, 0.8)) |> unlist()
# minimal value of each function for t > 0.5
tf_fwise(x, \(.x) min(.x$value[.x$arg > 0.5])) |> unlist()

tf_crosscor(x, -x)
tf_crosscov(x, x) == tf_fvar(x)

---

Hip and knee angle while walking data

Description

Hip and knee angle measurements in degrees through a 20-point movement cycle for 39 children. The data represents the angular positions of hip and knee joints during normal walking gait, captured at evenly spaced time points throughout the gait cycle.

Usage

gait

Format

A data frame with 39 rows and 3 variables:

subject_id

subject identifier

knee_angle

a tfd column of knee-joint angles (degrees)

hip_angle

a tfd column of hip-joint angles (degrees)

Details

Data is also included in the fda package in another format.

References

Olshen, A R, Biden, N E, Wyatt, P M, Sutherland, H D (1989). “Gait Analysis and the Bootstrap.” The Annals of Statistics, 17(4), 1419–1440.

Examples

head(gait)

---

Berkeley growth study data

Description

Heights of 39 boys and 54 girls measured from age 1 to 18 years as part of the Berkeley Growth Study. The data tracks physical development over time with measurements at 31 different ages that are not equally spaced.

Usage

growth

Format

A data frame with 93 rows and 2 variables:

gender

sex of the subject, a factor with levels female/male

height

a tfd column of height-over-age curves (cm)

Details

Data is also included in the fda package in another format.

References

Ramsay, O. J, Hooker, Giles, Graves, Spencer (2009). Functional Data Analysis with R and MATLAB, series Use R!, 1 edition. Springer New York, New York. ISBN 978-0-387-98184-0. doi:10.1007/978-0-387-98185-7.

Ramsay, O. J, Silverman, W. B (2005). Functional Data Analysis, series Springer Series in Statistics, 2nd edition. Springer, New York. ISBN 978-0-387-40080-8.

Ramsay, O. J, Silverman, W. B (2002). Applied Functional Data Analysis. Springer.

Tuddenham, D R (1954). “Physical growth of California boys and girls from birth to eighteen years.” University of California Publications in Child Development, 1, 183–364.

Examples

head(growth)

---

Find out if values are inside given bounds

Description

in_range and its infix-equivalent ⁠%inr%⁠ return TRUE for all values in the numeric vector f that are within the range of values in r.

Usage

in_range(f, r)

f %inr% r

Arguments

f	a numeric vector.
r	numeric vector used to specify a range, only the minimum and maximum of r are used.

Value

a logical vector of the same length as f.

See Also

Other tidyfun utility functions: ensure_list(), tf_arg(), tf_zoom(), unique_id()

Examples

in_range(1:10, c(3, 7))
1:10 %inr% c(3, 7)

---

Joint depth-median for vector-valued functional data

Description

The median of a tf_mv vector is the single observed curve with maximal joint depth (see tf_depth()): one which.max index selects the same observation across every component, so the result is never a "chimera" stitched together from different curves. Note the deliberate divergence from median.tf() on ties: the univariate median averages tied maximal-depth curves, but averaging components would break the observed-curve guarantee, so median.tf_mv returns the first tied curve (with a message). On tied data, median(f)$x and median(f$x) can therefore differ.

Usage

## S3 method for class 'tf_mv'
median(x, na.rm = FALSE, depth = "MBD", ...)

Arguments

x	a tf_mv vector.
na.rm	if FALSE (default), any NA observation makes the result NA; if TRUE, NA observations are dropped first.
depth	the joint depth method, see tf_depth().
...	passed to tf_depth().

Value

a length-1 tf_mv: the observed curve with maximal joint depth.

See Also

Other tidyfun summary functions: fivenum(), functionwise, tfsummaries

Examples

set.seed(1)
f <- tfd_mv(list(x = tf_rgp(5), y = tf_rgp(5)))
# the joint median is the observed curve with maximal joint depth:
median(f)
tf_depth(f)

---

Pinch force data

Description

Measurements of pinch force during 20 replications, with 151 observations recorded every 2 milliseconds over 300 milliseconds. The data captures the dynamics of finger pinch force applied during controlled motor tasks.

Usage

pinch

Format

A tfd vector of length 20 containing the pinch-force curves.

Details

Data is also included in the fda package in another format.

References

Ramsay, O. J, Hooker, Giles, Graves, Spencer (2009). Functional Data Analysis with R and MATLAB, series Use R!, 1 edition. Springer New York, New York. ISBN 978-0-387-98184-0. doi:10.1007/978-0-387-98185-7.

Ramsay, O. J, Silverman, W. B (2005). Functional Data Analysis, series Springer Series in Statistics, 2nd edition. Springer, New York. ISBN 978-0-387-40080-8.

Ramsay, O. J, Silverman, W. B (2002). Applied Functional Data Analysis. Springer.

Examples

pinch

---

base plots for tfs

Description

Some base functions for displaying functional data in spaghetti- (i.e., line plots) and lasagna- (i.e., heat map) flavors.

Usage

## S3 method for class 'tf'
plot(
  x,
  y,
  n_grid = 50,
  points = is_irreg(x),
  type = c("spaghetti", "lasagna"),
  alpha = min(1, max(0.05, 2/length(x))),
  ...
)

## S3 method for class 'tf'
lines(x, arg, n_grid = 50, alpha = min(1, max(0.05, 2/length(x))), ...)

## S3 method for class 'tf'
points(
  x,
  arg,
  n_grid = NA,
  alpha = min(1, max(0.05, 2/length(x))),
  interpolate = FALSE,
  ...
)

Arguments

x	a tf object.
y	(optional) numeric vector of evaluation points to use as arg for plotting (passed as the function's input grid, not the y-axis values).
n_grid	minimal size of equidistant grid used for plotting, defaults to 50. See details.
points	should the original evaluation points be marked by points? Defaults to TRUE for irregular tfd and FALSE for all others.
type	"spaghetti": line plots, "lasagna": heat maps.
alpha	alpha-value (see grDevices::rgb()) for noodle transparency. Defaults to 2/(no. of observations). Lower is more transparent.
...	additional arguments for graphics::matplot() ("spaghetti") or image() ("lasagna").
arg	evaluation grid (vector).
interpolate	should functions be evaluated (i.e., inter-/extrapolated) for arg for which no original data is available? Only relevant for tfd, defaults to FALSE.

Details

If no second argument y is given, evaluation points (arg) for the functions are given by the union of the tf's arg and an equidistant grid over its domain with n_grid points. If you want to only see the original data for tfd-objects without inter-/extrapolation, use n_grid < 1 or n_grid = NA.

Value

the plotted tf-object, invisibly.

References

Swihart, J B, Caffo, Brian, James, D B, Strand, Matthew, Schwartz, S B, Punjabi, M N (2010). “Lasagna plots: a saucy alternative to spaghetti plots.” Epidemiology (Cambridge, Mass.), 21(5), 621–625.

Examples

f <- tfd(sin(seq(0, 2 * pi, length.out = 51)), arg = seq(0, 1, length.out = 51))
plot(f)
plot(c(f, 2 * f), type = "lasagna")

---

Plot vector-valued functional data

Description

Two simple display modes for tf_mv objects: "facet" draws one panel per output dimension (delegating to the univariate plot.tf()); "trajectory" (only for d == 2) draws the curves in the plane, i.e. $y(t)$ against $x(t)$ – the natural view for movement data.

Usage

## S3 method for class 'tf_mv'
plot(x, y, ..., type = NULL)

## S3 method for class 'tf_mv'
lines(x, ..., type = NULL)

## S3 method for class 'tf_mv'
points(x, ..., type = NULL)

Arguments

x	a tf_mv object.
y	ignored.
...	passed to the underlying plotting calls. Per-curve graphical parameters (col, lty, lwd, ...) are recycled across curves.
type	"trajectory" or "facet". Defaults to "trajectory" for two-component (d == 2) objects and to "facet" otherwise.

Details

In "trajectory" mode the two components must be paired at common argument values to form $(x(t), y(t))$ points. When the components are sampled on different (or per-curve irregular) grids they are therefore evaluated on the union of their argument grids with interpolate = TRUE (values outside a component's observed range become NA and are skipped). For components that already share a grid this is a no-op.

Value

x, invisibly.

See Also

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

Examples

arg <- seq(0, 1, length.out = 31)
xf <- tfd(t(sapply(1:5, function(i) sin(2 * pi * arg + i / 5))), arg = arg)
yf <- tfd(t(sapply(1:5, function(i) cos(2 * pi * arg + i / 5))), arg = arg)
mv <- tfd_mv(list(x = xf, y = yf))
plot(mv, type = "trajectory")
plot(mv, type = "facet")

---

Preprocess evaluation grid for plotting

Description

(internal function exported for re-use in upstream packages)

Usage

prep_plotting_arg(f, n_grid)

Arguments

f	a tf-object.
n_grid	length of evaluation grid.

Value

a semi-regular grid rounded down to appropriate resolution.

Examples

f <- tfd(sin(seq(0, 2 * pi, length.out = 21)), arg = seq(0, 1, length.out = 21))
prep_plotting_arg(f, n_grid = 50)

---

Pretty printing and formatting for functional data

Description

Prints and formats tf-objects for display. See details / examples for options that give finer control.

Usage

## S3 method for class 'tf'
print(x, n = 6, ...)

## S3 method for class 'tfd_reg'
print(x, n = 6, ...)

## S3 method for class 'tfd_irreg'
print(x, n = 6, ...)

## S3 method for class 'tfb'
print(x, n = 5, ...)

## S3 method for class 'tf'
format(
  x,
  digits = 2,
  nsmall = 0,
  width = options()$width,
  sparkline = TRUE,
  prefix = FALSE,
  ...
)

Arguments

x	any R object (conceptually); typically numeric.
n	how many elements of x to print out at most, defaults to 6.
...	handed over to format.tf().
digits	a positive integer indicating how many significant digits are to be used for numeric and complex x. The default, NULL, uses getOption("digits"). This is a suggestion: enough decimal places will be used so that the smallest (in magnitude) number has this many significant digits, and also to satisfy nsmall. (For more, notably the interpretation for complex numbers see signif.)
nsmall	the minimum number of digits to the right of the decimal point in formatting real/complex numbers in non-scientific formats. Allowed values are 0 <= nsmall <= 20.
width	default method: the minimum field width or NULL or 0 for no restriction. AsIs method: the maximum field width for non-character objects. NULL corresponds to the default 12.
sparkline	use a sparkline representation? defaults to TRUE (not available for irregular data).
prefix	prefix with names / index positions? defaults to FALSE.

Details

By default, tf objects on regular grids are shown as "sparklines" (cli::spark_bar()), set sparkline = FALSE for a text representation.

Sparklines are based on running mean values of the function values, but these don't check for non-equidistant grids, so the visual impression will be misleading for very unequal grid distances.

Sparklines use options()$width/3 bins for printing/formatting by default, use bins argument to set the number of bins explicitly. For pillar::glimpse(), we use 8 bins by default for compact display.

Value

print: prints out x and returns it invisibly.

a character representation of x.

Examples

t <- seq(0, 1, l = 201)
cosine <- lapply(1:4, \(i) cos(i * pi * t)) |> tfd(arg = t)
cosine
tf_sparsify(cosine, dropout = .8)

format(cosine, sparkline = FALSE)
format(cosine, bins = 5)
format(cosine, bins = 40)

#! very non-equidistant grids --> sparklines can mislead about actual shapes:
tfd(cosine, arg = t^3)

---

Apply warping functions to align functional data

Description

tf_align() applies the inverse warping function to unregistered data to obtain aligned (registered) functions.

Usage

tf_align(x, warp, ...)

## S3 method for class 'tfd'
tf_align(x, warp, ..., keep_new_arg = FALSE)

## S3 method for class 'tfb'
tf_align(x, warp, ...)

Arguments

x	tf vector of functions. For tf_warp(), these should be registered/aligned functions and unaligned functions for tf_align().
warp	tf vector of warping functions used for transformation. See Details.
...	additional arguments passed to tfd().
keep_new_arg	keep new arg values after (un)warping or return tfd vector on arg values of the input (default FALSE is the latter)? See Details.

Value

the aligned tf vector (registered functions)

See Also

Other registration functions: tf_estimate_warps(), tf_landmarks_extrema(), tf_register(), tf_register_shape(), tf_registration, tf_warp()

Examples

# Estimate warps, then align manually:
t <- seq(0, 2 * pi, length.out = 101)
x <- tfd(t(sapply(c(-0.3, 0, 0.3), function(s) sin(t + s))), arg = t)
warps <- tf_estimate_warps(x, method = "affine", type = "shift")
aligned <- tf_align(x, warps)
plot(aligned, col = 1:3)

---

Inter- and extrapolation functions for tfd-objects

Description

These are exported evaluator callbacks for tfd objects. They control how function values are inter-/extrapolated to previously unseen arg values and are used by tf_evaluate().

In typical use, set an evaluator when constructing a tfd (tfd(..., evaluator = tf_approx_linear)) or replace it later via tf_evaluator(x) <- tf_approx_none.

These helpers are wrappers around zoo::na.fill(), zoo::na.approx(), etc. and all share the same signature (x, arg, evaluations), so they can also be called directly.

The list:

• tf_approx_linear for linear interpolation without extrapolation (i.e., zoo::na.approx() with na.rm = FALSE) – this is the default,

• tf_approx_spline for cubic spline interpolation, (i.e., zoo::na.spline() with na.rm = FALSE),

• tf_approx_none in order to not inter-/extrapolate ever (i.e., zoo::na.fill() with fill = NA)

• tf_approx_fill_extend for linear interpolation and constant extrapolation (i.e., zoo::na.fill() with fill = "extend")

• tf_approx_locf for "last observation carried forward" (i.e., zoo::na.locf() with na.rm = FALSE)

• tf_approx_nocb for "next observation carried backward" (i.e., zoo::na.locf() with ⁠na.rm = FALSE, fromLast = TRUE⁠).

For implementing your own, see source code of tf:::zoo_wrapper.

Usage

tf_approx_linear(x, arg, evaluations)

tf_approx_spline(x, arg, evaluations)

tf_approx_none(x, arg, evaluations)

tf_approx_fill_extend(x, arg, evaluations)

tf_approx_locf(x, arg, evaluations)

tf_approx_nocb(x, arg, evaluations)

Arguments

x	new arg values to approximate/interpolate/extrapolate the function for.
arg	the arg values of the evaluations.
evaluations	the function values at arg.

Value

a vector of values of the function defined by the given $(x_i, f(x_i))$=⁠(arg, evaluations)⁠-tuples at new argument values x.

See Also

tfd()

Other tidyfun inter/extrapolation functions: tf_evaluate(), tf_interpolate()

Examples

x <- tfd(matrix(c(0, 1), nrow = 1), arg = c(0, 1))
tf_evaluate(x, c(0, 0.5, 1))
tf_evaluator(x) <- tf_approx_none
tf_evaluate(x, c(0, 0.5, 1))

tf_approx_linear(
  x = c(0, 0.5, 1),
  arg = c(0, 1),
  evaluations = c(0, 1)
)

---

Arc length of vector-valued functional data

Description

For a vector-valued curve ⁠f: [a, b] -> R^d⁠, the arc length is $\int_a^b \lVert f'(t) \rVert\, dt$ – the length traced out by f in R^d.

Usage

tf_arclength(f, ...)

## Default S3 method:
tf_arclength(f, ...)

## S3 method for class 'tf_mv'
tf_arclength(
  f,
  arg = NULL,
  lower = tf_domain(f)[1],
  upper = tf_domain(f)[2],
  definite = TRUE,
  method = c("polyline", "derive"),
  ...
)

Arguments

f	a tf_mv object.
...	forwarded to tf_integrate() when method = "derive".
arg, lower, upper	optional evaluation/integration grid and limits.
definite	TRUE (default) returns a numeric vector of total arc lengths per curve; FALSE returns the cumulative arc length $s(t) = \int_a^t \lVert f'(u) \rVert\, du$ as a univariate tfd.
method	"polyline" (default) or "derive".

Details

Two methods are supported:

• "polyline" (default): sum of the Euclidean lengths of the line segments between consecutive sample points (in R^d). Each curve is evaluated on the union of its components' argument grids (or a supplied arg) and the segment-sum is computed in closed form. For raw tfd_mv data this is more accurate than "derive" because it avoids the compounding error of numerical differentiation followed by quadrature.

• "derive": composes the existing verbs – per-component differentiation (tf_derive()), pointwise speed tf_speed(), then tf_integrate(). Best for tfb_mv (analytical derivatives) or when a custom tf_integrate(...) argument is needed.

Value

a numeric vector (definite) or a univariate tfd (indefinite).

See Also

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

Examples

# unit circle parameterised on [0, 1] -- arc length is 2*pi
t <- seq(0, 1, length.out = 401)
circ <- tfd_mv(list(
  x = tfd(matrix(cos(2 * pi * t), nrow = 1), arg = t),
  y = tfd(matrix(sin(2 * pi * t), nrow = 1), arg = t)
))
tf_arclength(circ)
tf_arclength(circ, lower = 0, upper = 0.25) # quarter -> pi/2
tf_arclength(circ, definite = FALSE)        # cumulative s(t)

---

Utility functions for tf-objects

Description

A bunch of methods & utilities that do what they say: get or set the respective attributes of a tf-object.

is_tf_1d() distinguishes univariate tf vectors (length-n samples of f: R -> R) from any tf vector. Returns TRUE for tfd / tfb and FALSE for tfd_mv / tfb_mv. Useful as a dispatch / guard predicate inside helpers that assume scalar per-arg evaluations.

Usage

tf_arg(f)

tf_evaluations(f)

tf_count(f)

tf_domain(f)

tf_domain(x) <- value

tf_evaluator(f)

tf_evaluator(x) <- value

tf_basis(f, as_tfd = FALSE)

tf_arg(x) <- value

## S3 replacement method for class 'tfd_irreg'
tf_arg(x) <- value

## S3 replacement method for class 'tfd_reg'
tf_arg(x) <- value

## S3 replacement method for class 'tfb'
tf_arg(x) <- value

## S3 method for class 'tfb'
coef(object, ...)

## S3 method for class 'tf'
rev(x)

## S3 method for class 'tf_mv'
rev(x)

## S3 method for class 'tf'
is.na(x)

## S3 method for class 'tfd_irreg'
is.na(x)

is_tf(x)

is_tf_1d(x)

is_tfd(x)

is_reg(x)

is_tfd_reg(x)

is_irreg(x)

is_tfd_irreg(x)

is_tfb(x)

is_tfb_spline(x)

is_tfb_fpc(x)

is_tf_mv(x)

is_tfd_mv(x)

is_tfb_mv(x)

Arguments

f	an tf object.
x	an tf object.
value	for ⁠tf_evaluator<-⁠: (bare or quoted) name of a function that can be used to interpolate an tfd. Needs to accept vector arguments x, arg, evaluations and return evaluations of the function defined by arg, evaluations at x. for ⁠tf_arg<-⁠: (list of) new arg-values. for ⁠tf_domain<-⁠: sorted numeric vector with the 2 new endpoints of the domain.
as_tfd	should the basis be returned as a tfd-vector evaluated on tf_arg(f)? Defaults to FALSE, which returns the matrix of basis functions (columns) evaluated on tf_arg(f) (rows).
object	a tfb object.
...	not used.

Value

either the respective attribute or, for setters (assignment functions), the input object with modified properties.

See Also

Other tidyfun utility functions: ensure_list(), in_range(), tf_zoom(), unique_id()

Examples

x <- tf_rgp(3)
tf_arg(x)
tf_evaluations(x)
tf_count(x)
tf_domain(x)
tf_evaluator(x)
tf_evaluate(x, 0.25)
tf_evaluator(x) <- tf_approx_none
tf_evaluate(x, 0.25)
c(is_tf(x), is_tfd(x), is_reg(x), is_irreg(x))

xb <- tfb(x, k = 4, penalized = FALSE, verbose = FALSE)
tf_basis(xb)
tf_basis(xb)(c(0, .1, .2))
c(is_tfb(xb), is_tfb_spline(xb), is_tfb_fpc(xb))

---

Functional Data Depth

Description

Data depths for functional data. All depths are scaled so that 1 means most central and 0 means most extreme. Available methods:

Usage

tf_depth(x, arg, depth = "MBD", na.rm = TRUE, ...)

## S3 method for class 'matrix'
tf_depth(
  x,
  arg,
  depth = c("MBD", "MHI", "FM", "FSD", "RPD"),
  na.rm = TRUE,
  ...
)

## S3 method for class 'tf'
tf_depth(x, arg, depth = "MBD", na.rm = TRUE, ...)

## S3 method for class 'tf_mv'
tf_depth(
  x,
  arg,
  depth = c("MBD", "MHI", "FM", "FSD", "RPD"),
  na.rm = TRUE,
  weights = "equal",
  ...
)

Arguments

x	tf (or a matrix of evaluations).
arg	grid of evaluation points.
depth	one of "MBD", "MHI", "FM", "FSD", or "RPD".
na.rm	remove missing observations? Defaults to TRUE.
...	for "RPD": u (regularization quantile, default 0.01), n_projections (M, default 5000), n_projections_beta (L, default 500).
weights	(tf_mv only) component weighting scheme for the weighted componentwise aggregation. Either "equal" (default), "inverse_variance" (weights proportional to the inverse of each component's mean pointwise variance), or a numeric vector of d strictly positive weights. Weights are normalized to sum to 1.

Details

• "MBD": Modified Band-2 Depth (default). Scale of 0 (most extreme) to 1 (most central).

• "MHI": Modified Hypograph Index. Ranks functions from lowest (0) to highest (1) instead of most extreme to most central! For functions that never cross: $MBD = -2(MHI - 0.5)^2 + .5$.

• "FM": Fraiman-Muniz depth. Integrates pointwise univariate halfspace depths over the domain. Scale of 0 (most extreme) to 1 (most central).

• "FSD": Functional Spatial Depth. Based on spatial signs; robust to outliers. Scale of 0 (most extreme) to 1 (most central).

• "RPD": Regularized Projection Depth. Projects curves onto random directions and computes outlyingness. Especially useful for detecting shape outliers. Scale of 0 (most extreme) to 1 (most central). Note: results depend on the RNG state; set a seed (e.g. set.seed(...)) before calling for reproducibility. Accepts additional arguments via ...: u (quantile level for regularization, default 0.01), n_projections (M, number of projection directions, default 5000), n_projections_beta (L, directions for estimating regularization parameter, default 500).

Value

vector of depth values

References

Sun, Ying, Genton, G M, Nychka, W D (2012). “Exact fast computation of band depth for large functional datasets: How quickly can one million curves be ranked?” Stat, 1(1), 68–74.

López-Pintado, Sara, Romo, Juan (2009). “On the concept of depth for functional data.” Journal of the American Statistical Association, 104(486), 718–734.

López-Pintado, Sara, Romo, Juan (2011). “A half-region depth for functional data.” Computational Statistics & Data Analysis, 55(4), 1679–1695.

Fraiman, Ricardo, Muniz, Graciela (2001). “Trimmed means for functional data.” Test, 10(2), 419–440.

Chakraborty, Anirvan, Chaudhuri, Probal (2014). “The spatial distribution in infinite dimensional spaces and related quantiles and depths.” The Annals of Statistics, 42(3), 1203–1231.

Bočinec, Filip, Nagy, Stanislav, Yeon, Hyemin (2026). “Projection depth for functional data: Practical issues, computation and applications.” arXiv preprint arXiv:2602.22877.

See Also

Other tidyfun ordering and ranking functions: tf_minmax, tf_order()

Examples

x <- tf_rgp(3)/3 + 1:3
tf_depth(x, depth = "MBD")
tf_depth(x, depth = "MHI")
tf_depth(x, depth = "FM")
tf_depth(x, depth = "FSD")

---

Differentiating functional data: approximating derivative functions

Description

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

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).

Value

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

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.

See Also

Other tidyfun calculus functions: tf_integrate()

Examples

arg <- seq(0, 1, length.out = 31)
x <- tfd(rbind(arg^2, sin(2 * pi * arg)), arg = arg)
dx <- tf_derive(x)
x
dx
tf_arg(dx)

---

Estimate warping functions for registration

Description

tf_estimate_warps() is the low-level workhorse for functional data registration. It estimates warping functions that align a set of functions to a template, but does not apply them. For a one-shot interface that also aligns the data, see tf_register().

Usage

tf_estimate_warps(
  x,
  ...,
  template = NULL,
  method = c("srvf", "srvf_mv", "cc", "affine", "landmark"),
  max_iter = 3L,
  tol = 0.01
)

Arguments

x	a tf vector of functions to register.
...	additional method-specific arguments passed to backend routines (for example crit for method = "cc").
template	an optional tf vector of length 1 to use as the template. If NULL, a default template is computed (method-dependent). Not used for method = "landmark".
method	the registration method to use: "srvf": Square Root Velocity Framework (elastic registration). For details, see fdasrvf::time_warping(). Default template is the Karcher mean. "srvf_mv": true multivariate SRVF time registration for tf_mv objects on a regular shared grid. This method estimates one shared warp from all components jointly and does not rotate or rescale curves. "cc": continuous-criterion registration via a tf-native dense-grid optimizer with monotone spline warps. Default template is the arithmetic mean. "affine": affine (linear) registration with warps of the form $h(t) = a \cdot t + b$. Simpler than elastic registration, appropriate when phase variability consists only of shifts and/or uniform speed-up/slow-down. Default template is the arithmetic mean. "landmark": piecewise-linear warps that align user-specified landmark features. Requires landmarks argument.
max_iter	integer: maximum number of Procrustes-style template refinement iterations when template = NULL. The iteration cycle is: (1) estimate template as mean of (aligned) curves, (2) register all curves to current template, (3) update template as mean of newly aligned curves, (4) repeat until convergence or max_iter reached. Ignored when template is provided (no refinement needed) or for method = "landmark" (template not used). For method = "srvf" with template = NULL, the outer Procrustes loop is skipped regardless of max_iter because fdasrvf::time_warping() already computes the Karcher mean internally. Default is 3L.
tol	numeric: convergence tolerance for template refinement. For method = "cc", iteration stops when the relative improvement in the registration criterion becomes negligible; for the other iterative methods, iteration stops when the relative change in the template (L2 norm) falls below tol. Default is 1e-2.

Details

For method = "cc", tf uses a tf-native dense-grid optimizer with monotone spline warps. Each warp is represented as the normalized cumulative integral of exp(eta(t)), where eta(t) is a spline with nbasis coefficients. Registration is then carried out curve-by-curve by minimizing either an integrated squared-error criterion (crit = 1) or the first-eigenfunction variance criterion (crit = 2) plus an optional spline roughness penalty (lambda). The outer max_iter loop, when template = NULL, still performs the same Procrustes-style template refinement as the other methods.

Value

tfd vector of (forward) warping functions $h_i(s) = t$ with the same length as x. Apply with tf_align() to obtain registered functions, or use tf_invert() to obtain inverse warps $h_i^{-1}(t) = s$. The returned warps carry an attr(, "template") with the template used (NULL for landmark registration, which has no template).

Important method-specific arguments (passed via ...)

For method = "srvf":

lambda

non-negative number: penalty controlling the flexibility of warpings (default is 0 for unrestricted warps).

penalty_method

cost function used to penalize warping functions. Defaults to "roughness" (norm of their second derivative), "geodesic" uses the geodesic distance to the identity and "norm" uses Euclidean distance to the identity.

For method = "cc":

nbasis

integer: number of B-spline basis functions for the monotone warp basis (default 6L, minimum 2).

lambda

non-negative number: roughness penalty for the warp basis (default 0 for unpenalized warping).

crit

registration criterion. Defaults to 2 for the first-eigenfunction variance criterion; alternative is 1 for integrated squared error.

conv

non-negative convergence tolerance for the inner optimizer. Default is 1e-4.

iterlim

maximum number of inner optimization iterations per curve. Default is 20L.

For method = "affine":

type

character: "shift" (translation only), "scale" (scaling only), or "shift_scale" (both). Default is "shift".

shift_range

numeric(2): bounds for shift parameter. Default is c(-range/2, range/2) where range is the domain width. Larger bounds allow greater shifts but may result in more NA values.

scale_range

numeric(2): bounds for scale parameter. Default is c(0.5, 2). Must have lower > 0.

For method = "landmark":

landmarks

(required) numeric matrix of landmark positions with one row per function and one column per landmark. Use tf_landmarks_extrema() to find peaks/valleys automatically.

template_landmarks

numeric vector of target landmark positions. Default is column-wise mean of landmarks.

Author(s)

Maximilian Muecke, Fabian Scheipl, Claude Opus 4.6

See Also

Other registration functions: tf_align(), tf_landmarks_extrema(), tf_register(), tf_register_shape(), tf_registration, tf_warp()

Examples

# see tf_register() for full registration examples
set.seed(1)
f <- tf_rgp(5)
warps <- tf_estimate_warps(f, method = "srvf")
plot(warps)

---

Evaluate tf-vectors for given argument values

Description

Also used internally by the [-operator for tf data (see ?tfbrackets) to evaluate object, see examples.

Usage

tf_evaluate(object, arg, ...)

## Default S3 method:
tf_evaluate(object, arg, ...)

## S3 method for class 'tfd'
tf_evaluate(object, arg, evaluator = tf_evaluator(object), ...)

## S3 method for class 'tfb'
tf_evaluate(object, arg, ...)

Arguments

object	a tf, or a data.frame-like object with tf columns.
arg	optional evaluation grid (vector or list of vectors). Defaults to tf_arg(object), implicitly.
...	not used.
evaluator	optional. The function to use for inter/extrapolating the tfd. Defaults to tf_evaluator(object). See e.g. tf_approx_linear() for details.

Value

A list of numeric vectors containing the function evaluations on arg.

See Also

Other tidyfun inter/extrapolation functions: tf_approx_linear(), tf_interpolate()

Examples

f <- tf_rgp(3, arg = seq(0, 1, length.out = 11))
tf_evaluate(f) |> str()
tf_evaluate(f, arg = 0.5) |> str()
# equivalent, as matrix:
f[, 0.5]
new_grid <- seq(0, 1, length.out = 6)
tf_evaluate(f, arg = new_grid) |> str()
# equivalent, as matrix:
f[, new_grid]

---

Pointwise norm and inner product for functional data

Description

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).

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 $f'(t) / |f'(t)| = \mathrm{sign}(f'(t))$.

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.

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)
tf_speed(f)
tf_distance(f, tfd_mv(list(x = tf_rgp(2), y = tf_rgp(2))))
# univariate: tf_norm reduces to the pointwise absolute value
u <- tf_rgp(2)
tf_norm(u)
tf_inner(u, tf_rgp(2))

---

Integrals and anti-derivatives of functional data

Description

Integrals of tf-objects are computed by simple quadrature (trapezoid rule). By default the scalar definite integral $\int^{upper}_{lower}f(s)ds$ is returned (option definite = TRUE), alternatively for definite = FALSE the anti-derivative on ⁠[lower, upper]⁠, e.g. a tfd or tfb object representing $F(t) \approx \int^{t}_{lower}f(s)ds$, for $t \in$⁠[lower, upper]⁠, is returned.

Usage

tf_integrate(f, arg, lower, upper, ...)

## Default S3 method:
tf_integrate(f, arg, lower, upper, ...)

## S3 method for class 'tfd'
tf_integrate(
  f,
  arg = tf_arg(f),
  lower = tf_domain(f)[1],
  upper = tf_domain(f)[2],
  definite = TRUE,
  ...
)

## S3 method for class 'tfb'
tf_integrate(
  f,
  arg = tf_arg(f),
  lower = tf_domain(f)[1],
  upper = tf_domain(f)[2],
  definite = TRUE,
  ...
)

Arguments

f	a tf-object
arg	(optional) grid to use for the quadrature.
lower	lower limits of the integration range. For definite = TRUE, this can be a vector of the same length as f.
upper	upper limits of the integration range (but see definite arg / description). For definite = TRUE, this can be a vector of the same length as f.
...	not used
definite	should the definite integral be returned (default) or the antiderivative. See description.

Details

When f is irregular and lower / upper are not supplied explicitly, they default to each curve's own observed arg range (i.e., the range of its tf_arg() values) rather than the (shared) domain endpoints; for regular tfd the defaults remain the domain endpoints. Without this per-curve default, curves that do not span the full domain would silently NA-poison the trapezoidal sum, because the default linear evaluator does not extrapolate. Pass explicit lower / upper to integrate over a fixed sub-interval, or switch to an extrapolating evaluator (e.g. tf_approx_fill_extend()) to integrate over the full domain.

Value

For definite = TRUE, the definite integrals of the functions in f. For definite = FALSE and tf-inputs, a tf object containing their anti-derivatives

See Also

Other tidyfun calculus functions: tf_derive()

Examples

arg <- seq(0, 1, length.out = 11)
x <- tfd(rbind(arg, arg^2), arg = arg)
tf_integrate(x)
anti <- tf_integrate(x, definite = FALSE)
tf_arg(anti)

---

Re-evaluate tf-objects on a new grid of argument values.

Description

Change the internal representation of a tf-object so that it uses a different grid of argument values (arg). Useful for

• thinning out dense grids to make data smaller

• filling out sparse grids to make derivatives/integrals and locating extrema or zero crossings more accurate (... if the interpolation works well ...)

• making irregular functional data into (more) regular data.

For tfd-objects, this is just syntactic sugar for tfd(object, arg = arg). To inter/extrapolate more reliably and avoid NAs, call tf_interpolate with evaluator = tf_approx_fill_extend.
For tfb-objects, this re-evaluates basis functions on the new grid which can speed up subsequent computations if they all use that grid. NB: To reliably impute very irregular data on a regular, common grid, you'll be better off doing FPCA-based imputation or other model-based approaches in most cases.

Usage

tf_interpolate(object, arg, ...)

## S3 method for class 'tfb'
tf_interpolate(object, arg, ...)

## S3 method for class 'tfd'
tf_interpolate(object, arg, ...)

## S3 method for class 'tf_mv'
tf_interpolate(object, arg, ...)

Arguments

object	an object inheriting from tf.
arg	a vector of argument values on which to evaluate the functions in object.
...	additional arguments handed over to tfd or tfb, for the construction of the returned object.

Value

a tfd or tfb object on the new grid given by arg.

See Also

tf_rebase(), which is more general.

Other tidyfun inter/extrapolation functions: tf_approx_linear(), tf_evaluate()

Examples

# thinning out a densely observed tfd
dense <- tf_rgp(10, arg = seq(0, 1, length.out = 1001))
less_dense <- tf_interpolate(dense, arg = seq(0, 1, length.out = 101))
dense
less_dense
# filling out sparse data (use a suitable evaluator-function!)
sparse <- tf_rgp(10, arg = seq(0, 5, length.out = 11))
plot(sparse, points = TRUE)
# change evaluator for better interpolation
tfd(sparse, evaluator = tf_approx_spline) |>
  tf_interpolate(arg = seq(0, 5, length.out = 201)) |>
  lines(col = 2, lty = 2)

set.seed(1860)
sparse_irregular <- tf_rgp(5) |>
  tf_sparsify(0.5) |>
  tf_jiggle()
tf_interpolate(sparse_irregular, arg = seq(0, 1, length.out = 51))

---

Invert a tf vector

Description

Computes the functional inverse of each function in the tf vector, such that if $y = f(x)$, then $x = f^{-1}(y)$.

Usage

tf_invert(x, ...)

Arguments

x	a tf vector.
...	optional arguments for the returned object, see tfd() / tfb()

Value

a tf vector of the inverted functions.

Examples

arg <- seq(0, 2, length.out = 50)
x <- tfd(rbind(2 * arg, arg^2), arg = arg)
x_inv <- tf_invert(x)
layout(t(1:2))
plot(x, main = "original functions", ylab = "")
plot(x_inv, main = "inverted functions", ylab = "", points = FALSE)

---

Make a tf (more) irregular

Description

Randomly create some irregular functional data from regular ones.

• jiggle it by randomly moving around its arg-values inside the intervals defined by its grid neighbors on the original argument grid.

• sparsify it by removing (100*dropout)% of the function values

Usage

tf_jiggle(f, amount = 0.4, ...)

## S3 method for class 'tf_mv'
tf_jiggle(f, amount = 0.4, same_arg = TRUE, ...)

tf_sparsify(f, dropout = 0.5, ...)

## S3 method for class 'tf_mv'
tf_sparsify(f, dropout = 0.5, same_arg = TRUE, ...)

Arguments

f	a tfd object.
amount	how far away from original grid points can the jiggled grid points lie, at most (relative to original distance to neighboring grid points). Defaults to at most 40% (0.4) of the original grid distances. Must be lower than 0.5.
...	additional args for the returned tfd in tf_jiggle.
same_arg	for tf_mv objects, should all components receive the same random argument-grid changes? Defaults to TRUE; use FALSE to jitter or sparsify each component independently.
dropout	what proportion of values of f to drop, on average. Defaults to half.

Value

an (irregular) tfd object.

See Also

Other tidyfun RNG functions: tf_rgp()

Examples

set.seed(1)
(x <- tf_rgp(2, arg = 21L))
(x_jig <- tf_jiggle(x, amount = 0.2))
(x_sp <- tf_sparsify(x, dropout = 0.3))
c(is_irreg(x_jig), is_irreg(x_sp))

---

Find Extrema Locations in Functional Data

Description

Find landmark locations for registration

Usage

tf_landmarks_extrema(x, which = "all", threshold = 0.5, boundary_tol = NULL)

Arguments

x	a tf vector.
which	character: which features to detect. Either "all" (maxima, minima, and zero crossings), "both" (maxima and minima), or any subset of c("max", "min", "zero").
threshold	numeric in (0, 1]: minimum proportion of curves that must contain a feature for it to be retained as a landmark. Defaults to 0.5.
boundary_tol	numeric: features within this distance of the domain boundary are dropped (they are redundant with the boundary anchors in landmark registration). Defaults to 2x the grid spacing. Set to 0 to keep all features.

Details

Detects local maxima, minima, and/or zero crossings in each function and returns a landmark matrix suitable for tf_register() with method = "landmark". Uses position-based clustering across curves to establish feature correspondence and majority-count filtering to discard unstable landmarks.

Value

A numeric matrix with one row per function and one column per landmark, sorted left-to-right on the domain. Has attribute "feature_types" (character vector of "max", "min", or "zero" for each column). Contains NA where a curve is missing a landmark.

See Also

tf_register() with method = "landmark"

Other registration functions: tf_align(), tf_estimate_warps(), tf_register(), tf_register_shape(), tf_registration, tf_warp()

Examples

t <- seq(0, 1, length.out = 101)
x <- tfd(t(sapply(c(0.3, 0.5, 0.7), function(s) dnorm(t, s, 0.1))), arg = t)
tf_landmarks_extrema(x, "max")
tf_landmarks_extrema(x, "both")

---

Depth-based minimum, maximum and range for tf vectors

Description

By default, min, max, and range compute pointwise extremes (the existing behaviour). When a depth argument is supplied, they instead return the most extreme / most central observation according to the chosen depth. For the default "MHI" depth this gives the lowest / highest function in an up-down sense.

Usage

## S3 method for class 'tf'
min(..., na.rm = FALSE, depth = NULL)

## S3 method for class 'tf'
max(..., na.rm = FALSE, depth = NULL)

## S3 method for class 'tf'
range(..., na.rm = FALSE, depth = NULL)

Arguments

...	tf objects (and na.rm for the pointwise default).
na.rm	logical; passed on to the pointwise summary or used to filter NAs before computing depth.
depth	depth method to use. NULL (default) gives the pointwise min/max/range. Supply a depth name (e.g. "MHI") or a custom depth function for depth-based selection.

Value

a tf object.

See Also

tf_depth(), rank.tf()

Other tidyfun ordering and ranking functions: tf_depth(), tf_order()

Examples

x <- tf_rgp(5) + 1:5
# pointwise (default):
min(x)
max(x)
# depth-based:
min(x, depth = "MHI")
max(x, depth = "MHI")

---

Accessors and methods for vector-valued functional data

Description

Utilities for tf_mv objects (see tfd_mv() / tfb_mv()). tf_ncomp() returns the number of output dimensions $d$, tf_components() the list of the d underlying univariate tf vectors, and tf_component() extracts or replaces a single one (also available via the $ operator, e.g. f$x).

Usage

tf_ncomp(f)

tf_components(f)

tf_component(f, which)

tf_component(f, which) <- value

Arguments

f	a tf_mv object.
which	a component name or index.
value	a univariate tf vector (replacement) of matching length and domain.

Details

Most univariate tf verbs also work on tf_mv objects by acting on each component: tf_rebase() (and hence tfd_mv/tfb_mv conversion), tf_derive(), tf_integrate() (definite integrals return an ⁠n x d⁠ matrix), tf_smooth() and tf_zoom(). Registration (tf_register() / tf_estimate_warps() / tf_warp() / tf_align()) estimates a single, shared time-warp per curve and applies it jointly to every component. The registration signal is, by default, the first component; use ref_component to pick another component (by name/index), "norm" for the pointwise Euclidean norm, or a function mapping the tf_mv to a univariate tf vector.

is.na() flags a curve as missing if any of its components is missing (the union, not the intersection), which also drives the na.rm behaviour of mean() / median() etc.

Value

tf_ncomp(): an integer. tf_components(): a named list of tf vectors. tf_component(): a single univariate tf vector.

See Also

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

Examples

f <- tfd_mv(list(x = tf_rgp(3), y = tf_rgp(3)))
tf_ncomp(f)
tf_components(f)
tf_component(f, "y")
f$y

---

Rank, order and sort tf vectors

Description

These methods use tf_depth() to rank, order, and sort functional data. By default they use the modified hypograph index ("MHI") which provides an up-down ordering (lowest to highest). You can also use any of the other depth methods available via tf_depth(), or supply a custom depth function.

Usage

rank(
  x,
  na.last = TRUE,
  ties.method = c("average", "first", "last", "random", "max", "min"),
  ...
)

## Default S3 method:
rank(
  x,
  na.last = TRUE,
  ties.method = c("average", "first", "last", "random", "max", "min"),
  ...
)

## S3 method for class 'tf'
rank(
  x,
  na.last = TRUE,
  ties.method = c("average", "first", "last", "random", "max", "min"),
  depth = "MHI",
  ...
)

## S3 method for class 'tf'
xtfrm(x)

## S3 method for class 'tf'
sort(x, decreasing = FALSE, na.last = NA, depth = "MHI", ...)

tf_order(f, ...)

## Default S3 method:
tf_order(f, ...)

## S3 method for class 'tf'
tf_order(f, depth = "MHI", ...)

## S3 method for class 'tf_mv'
tf_order(f, by = "norm", ...)

Arguments

x	a tf vector.
na.last	for handling of NAs; see base::rank() and base::sort().
ties.method	a character string for handling ties; see base::rank().
...	passed to tf_depth() (e.g. arg).
depth	the depth function to use for ranking. One of the depths available via tf_depth() (default: "MHI") or a function that takes a tf vector and returns a numeric vector of depth values.
decreasing	logical. Should the sort be decreasing?
f	a tf or tf_mv vector (for tf_order).
by	(tf_mv only) the scalar reduction to order by: "norm" (the default, uses tf_norm(f)) or the name of a component.

Details

rank assigns ranks based on depth values: lower depth values get lower ranks. For "MHI" this gives an ordering from lowest to highest function. For centrality-based depths ("MBD", "FM", "FSD", "RPD"), the most extreme function gets rank 1 and the most central gets the highest rank.

tf_order returns the permutation which rearranges x into ascending order according to depth. For vector-valued (tf_mv) data there is no canonical total order on $R^d$: tf_order.tf_mv() requires an explicit scalar reduction via by (either "norm" for tf_norm(f), or a component name), and then applies the univariate depth order to that reduction.

sort.tf returns the sorted tf vector.

xtfrm.tf returns a numeric vector of MHI depth values, enabling base::order and base::rank to work on tf vectors.

Value

rank: a numeric vector of ranks.
tf_order: an integer vector of indices.
sort.tf: a sorted tf vector.
xtfrm.tf: a numeric vector of depth values.

See Also

tf_depth(), min.tf(), max.tf()

Other tidyfun ordering and ranking functions: tf_depth(), tf_minmax

Examples

x <- tf_rgp(5) + 1:5
rank(x)
order(x)
sort(x)
# use a centrality-based depth instead:
rank(x, depth = "MBD")

---

Change (basis) representation of a tf-object

Description

Apply the representation of one tf-object to another; i.e. re-express it in the other's basis, on its grid, etc.
Useful for making different functional data objects compatible so they can be combined, compared or computed with.

Usage

tf_rebase(object, basis_from, arg = tf_arg(basis_from), ...)

## S3 method for class 'tfd'
tf_rebase(object, basis_from, arg = tf_arg(basis_from), ...)

## S3 method for class 'tfb'
tf_rebase(object, basis_from, arg = tf_arg(basis_from), ...)

Arguments

object	a tf object whose representation should be changed.
basis_from	the tf object with the desired basis, arg, evaluator, etc.
arg	optional new arg values, defaults to those of basis_from.
...	forwarded to the tfb or tfd constructors.

Details

This uses double dispatch (S3) internally, so the methods defined below are themselves generics for methods tf_rebase.tfd.tfd, tf_rebase.tfd.tfb_spline, tf_rebase.tfd.tfb_fpc, tf_rebase.tfb.tfd, tf_rebase.tfb.tfb that dispatch on object_from.

Value

a tf-vector containing the data of object in the same representation as basis_from (potentially modified by the arguments given in ...).

Methods (by class)

• tf_rebase(tfd): re-express a tfd-vector in the same representation as some other tf-vector

• tf_rebase(tfb): re-express a tfb-vector in the same representation as some other tf-vector.

Examples

x <- tf_rgp(3)
xb <- tfb(x, k = 8, penalized = FALSE, verbose = FALSE)
tf_rebase(tf_rgp(3), xb)

---

Register / align a tf vector against a template function

Description

tf_register() is the high-level entry point for functional data registration. It estimates warping functions, applies them to align the data, and returns a tf_registration result object containing the aligned curves, inverse warping functions (observed to aligned time), and template. Use tf_aligned(), tf_inv_warps(), and tf_template() to extract components.

Usage

tf_register(
  x,
  ...,
  template = NULL,
  method = c("srvf", "srvf_mv", "cc", "affine", "landmark"),
  max_iter = 3L,
  tol = 0.01,
  store_x = TRUE
)

Arguments

x	a tf vector of functions to register.
...	additional method-specific arguments passed to backend routines (for example crit for method = "cc"). See tf_estimate_warps() for method-specific argument documentation.
template	an optional tf vector of length 1 to use as the template. If NULL, a default template is computed (method-dependent). Not used for method = "landmark".
method	the registration method to use: "srvf": Square Root Velocity Framework (elastic registration). "srvf_mv": true multivariate SRVF time registration for tf_mv objects on a regular shared grid. "cc": continuous-criterion registration via a tf-native dense-grid optimizer with monotone spline warps. "affine": affine (linear) registration. "landmark": piecewise-linear warps aligning user-specified landmarks.
max_iter	integer: maximum Procrustes-style template refinement iterations. Default 3L.
tol	numeric: convergence tolerance for template refinement. Default 1e-2.
store_x	logical: store original data in the result object? Default TRUE. Set to FALSE to save memory.

Details

For a lower-level interface that returns only warping functions (without performing alignment), see tf_estimate_warps().

Value

A tf_registration object. Access components with tf_aligned(), tf_inv_warps(), tf_template().

Important method-specific arguments (passed via ...)

For method = "srvf":

lambda

non-negative number: penalty controlling the flexibility of warpings (default is 0 for unrestricted warps).

penalty_method

cost function used to penalize warping functions. Defaults to "roughness" (norm of their second derivative), "geodesic" uses the geodesic distance to the identity and "norm" uses Euclidean distance to the identity.

For method = "cc":

nbasis

integer: number of B-spline basis functions for the monotone warp basis (default 6L, minimum 2).

lambda

non-negative number: roughness penalty for the warp basis (default 0 for unpenalized warping).

crit

registration criterion. Defaults to 2 for the first-eigenfunction variance criterion; alternative is 1 for integrated squared error.

conv

non-negative convergence tolerance for the inner optimizer. Default is 1e-4.

iterlim

maximum number of inner optimization iterations per curve. Default is 20L.

For method = "affine":

type

character: "shift" (translation only), "scale" (scaling only), or "shift_scale" (both). Default is "shift".

shift_range

numeric(2): bounds for shift parameter. Default is c(-range/2, range/2) where range is the domain width. Larger bounds allow greater shifts but may result in more NA values.

scale_range

numeric(2): bounds for scale parameter. Default is c(0.5, 2). Must have lower > 0.

For method = "landmark":

landmarks

(required) numeric matrix of landmark positions with one row per function and one column per landmark. Use tf_landmarks_extrema() to find peaks/valleys automatically.

template_landmarks

numeric vector of target landmark positions. Default is column-wise mean of landmarks.

Author(s)

Maximilian Muecke, Fabian Scheipl, Claude Opus 4.6

References

Ramsay JO, Hooker G, Graves S (2009). Functional Data Analysis with R and MATLAB. Springer, New York. doi:10.1007/978-0-387-98185-7.

Srivastava A, Wu W, Kurtek S, Klassen E, Marron JS (2011). "Registration of Functional Data Using Fisher-Rao Metric." arXiv:1103.3817.

Tucker JD, Wu W, Srivastava A (2013). "Generative models for functional data using phase and amplitude separation." Computational Statistics & Data Analysis, 61, 50–66. doi:10.1016/j.csda.2012.12.001.

See Also

Other registration functions: tf_align(), tf_estimate_warps(), tf_landmarks_extrema(), tf_register_shape(), tf_registration, tf_warp()

Examples

# Elastic registration (SRVF method)
height_female <- subset(growth, gender == "female", select = height, drop = TRUE)
growth_female <- tf_derive(height_female) |> tfd(arg = seq(1.125, 17.8), l = 101)
reg <- tf_register(growth_female)
layout(t(1:3))
plot(growth_female, xlab = "Chronological Age", ylab = "Growth Rate (cm/year)")
plot(tf_inv_warps(reg), xlab = "Chronological Age", ylab = "Biological Age")
plot(tf_aligned(reg), xlab = "Biological Age", ylab = "Growth Rate (cm/year)")

# Affine registration (shift only)
t <- seq(0, 2 * pi, length.out = 101)
x <- tfd(t(sapply(c(-0.3, 0, 0.3), function(s) sin(t + s))), arg = t)
reg <- tf_register(x, method = "affine", type = "shift")
plot(tf_aligned(reg), col = 1:3)

# Landmark registration
peaks <- tf_landmarks_extrema(x, "max")
reg <- tf_register(x, method = "landmark", landmarks = peaks)
plot(tf_aligned(reg), col = 1:3)

---

Register vector-valued curves in elastic shape space

Description

tf_register_shape() aligns vector-valued tf_mv curves with SRVF curve alignment, optionally allowing rotations and rescaling in addition to time warping. Unlike tf_register(), this is a shape-registration interface: the aligned curves live in centered shape space and the result stores rotations and scale factors.

Usage

tf_register_shape(
  x,
  ...,
  template = NULL,
  max_iter = 3L,
  tol = 0.01,
  rotation = TRUE,
  scale = TRUE,
  mode = c("O", "C"),
  lambda = 0,
  store_x = TRUE
)

Arguments

x	a regular shared-grid tfd_mv object.
...	additional arguments passed to fdasrvf alignment routines, such as ncores or verbose.
template	optional length-one tf_mv template. If NULL, a template is refined from the first curve by iterative group alignment.
max_iter	integer: maximum template refinement iterations when template = NULL. Default 3L.
tol	numeric: relative template convergence tolerance.
rotation	logical: allow rotations? Default TRUE.
scale	logical: allow scale changes? Default TRUE.
mode	character: curve mode for fdasrvf. Only open curves ("O", the default) are currently supported; "C" (closed curves) is reserved for a future release (see Details).
lambda	numeric: non-negative elastic penalty passed to the fdasrvf alignment routines. Larger values penalise warping. Default 0.
store_x	logical: store original data in the result object?

Details

When scale = TRUE the aligned curves returned by tf_aligned() are renormalised to a common (mean) arc length so that congruent shapes overlay. The per-curve factors returned by tf_scales() are the sizes that were removed: multiplying an aligned curve by its scale factor rescales it back to the corresponding input curve's arc length, so a value ⁠> 1⁠ means the input curve was larger than the shared aligned size and ⁠< 1⁠ means it was smaller. With scale = FALSE warping and rotation preserve arc length, so all factors are 1. With template = NULL the returned tf_template() is the empirical mean of the aligned shape-space curves rather than any single input curve.

Only open curves (mode = "O") are supported. Closed curves (mode = "C") additionally optimise over a circular seed shift that the returned warping functions do not represent, which would make the stored warps inconsistent with the aligned curves; mode = "C" is therefore rejected for now.

Value

A tf_shape_registration object. Access aligned curves with tf_aligned(), inverse warps with tf_inv_warps(), the template with tf_template(), rotations with tf_rotations(), and scales with tf_scales().

See Also

Other registration functions: tf_align(), tf_estimate_warps(), tf_landmarks_extrema(), tf_register(), tf_registration, tf_warp()

Examples

t <- seq(0, 1, length.out = 51)
base <- rbind(t, t^2)
beta <- array(NA_real_, dim = c(3, length(t), 2))
for (i in 1:3) {
  beta[i,, 1] <- base[1, ]
  beta[i,, 2] <- base[2, ]
}
curves <- tfd_mv(beta, arg = t)
reg <- tf_register_shape(curves, max_iter = 1)
tf_rotations(reg)
tf_scales(reg)

---

Registration Result Object

Description

tf_registration objects store the result of tf_register(), including the aligned (registered) curves, estimated inverse warping functions $h_i^{-1}$ (observed $\to$ aligned time), and the template used. Use accessors tf_aligned(), tf_inv_warps(), and tf_template() to extract components. tf_shape_registration objects, returned by tf_register_shape(), extend this structure with shape-space rotations and scale factors. Use tf_rotations() and tf_scales() to extract those components.

Usage

tf_aligned(x)

tf_inv_warps(x)

tf_template(x)

tf_rotations(x)

tf_scales(x)

## S3 method for class 'tf_registration'
print(x, ...)

## S3 method for class 'tf_shape_registration'
print(x, ...)

## S3 method for class 'tf_registration'
summary(object, ...)

## S3 method for class 'summary.tf_registration'
print(x, ...)

## S3 method for class 'tf_shape_registration'
summary(object, ...)

## S3 method for class 'summary.tf_shape_registration'
print(x, ...)

## S3 method for class 'tf_registration'
plot(x, ...)

## S3 method for class 'tf_registration'
x[i]

## S3 method for class 'tf_registration'
length(x)

## S3 method for class 'tf_shape_registration'
x[i]

Arguments

x	a tf_registration or tf_shape_registration object
...	additional arguments (currently unused)
object	a tf_registration object
i	index for subsetting (integer, logical, or character)

Value

For tf_registration objects: a list with entries registered (tf-vector of aligned/registered functions from x), inv_warps (inverse warping functions aligning x to the template function), the template function, the original data x (if store_x = TRUE was used in tf_register()), and the call to tf_register() that created the object. tf_shape_registration objects additionally contain the forward warps, rotations, and scales. Accessors return the respective component.

Summary diagnostics

summary() computes per-curve diagnostics for assessing registration quality and prints their averages and/or deciles. The printed output contains four sections:

Amplitude variance reduction (only if store_x = TRUE): the proportion of pointwise variance removed by registration, computed as $1 - \bar{V}_{\mathrm{registered}} / \bar{V}_{\mathrm{original}}$ where $\bar{V}$ is the mean (across the domain) of the pointwise variance (across curves). Values near 1 indicate that registration removed most of the original variability; values near 0 indicate little change; negative values indicate that registration increased variability (a sign that something went wrong).

Warp deviation from identity (deciles across curves): each curve's inverse warping function $h_i^{-1}$ is compared to the identity via the normalized integral $2/L^2 \int |h_i^{-1}(t) - t|\, dt$, where $L$ is the domain length. The normalizing constant $L^2/2$ is the theoretical upper limit deviation for a monotone, domain-preserving warp that maps all timepoints to the first or last timepoint, so values range from 0 (identity warp, no time deformation) to 1 (maximal crazy warping). Values above $\approx 0.3$ may suggest aggressive warping that could warrant inspection.

Warp slopes (deciles of per-curve min and max $dh^{-1}/dt$): a slope of 1 of the warp corresponds to no local time deformation (identity). Slopes $> 1$ indicate local time dilation (the warped curve is "stretched" relative to the template), slopes $< 1$ indicate local time compression, so slopes near 0 or very large slopes indicate extreme local deformation. For affine shift warps, all slopes are exactly 1.

Domain coverage loss (only printed if any loss occurs): the fraction of the original domain range that is lost per curve after alignment, computed as 1 - range(aligned_arg) / range(original_arg). This is only relevant for affine (non-domain-preserving) warps where alignment can shift parts of curves outside the original domain. Domain-preserving methods (srvf, cc, landmark) always have zero domain loss.

Accessors

• tf_aligned(x): extract the registered/aligned curves (tfd vector).

• tf_inv_warps(x): extract the estimated inverse warping functions $h_i^{-1}(t)$ that map observed time to aligned time (tfd vector). Use tf_invert() on the result to obtain forward warps if needed.

• tf_template(x): extract the template function (tf vector of length 1).

• tf_rotations(x): extract the per-curve rotation matrices from a tf_shape_registration object.

• tf_scales(x): extract the per-curve scale factors from a tf_shape_registration object. Each factor is relative to the template (template SRVF norm divided by the curve's SRVF norm); see tf_register_shape().

Author(s)

Fabian Scheipl, Claude Opus 4.6

See Also

Other registration functions: tf_align(), tf_estimate_warps(), tf_landmarks_extrema(), tf_register(), tf_register_shape(), tf_warp()

Examples

reg <- tf_register(pinch[1:5], method = "affine", type = "shift_scale")
reg
summary(reg)
plot(reg)

---

Gaussian Process random generator

Description

Generates n realizations of a zero-mean Gaussian process. The function also accepts user-defined covariance functions (without "nugget" effect, see cov), The implemented defaults with scale parameter $\phi$, order $o$ and nugget effect variance $\sigma^2$ are:

• squared exponential: $Cov(x(t), x(t')) = \exp(-(t-t')^2)/\phi) + \sigma^2 \delta_{t}(t')$.

• Wiener process: $Cov(x(t), x(t')) = \min(t',t)/\phi + \sigma^2 \delta_{t}(t')$,

• Matérn process: $Cov(x(t), x(t')) = \tfrac{2^{1-o}}{\Gamma(o)} (\tfrac{\sqrt{2o}|t-t'|}{\phi})^o \text{Bessel}_o(\tfrac{\sqrt{2o}|t-t'|}{s}) + \sigma^2 \delta_{t}(t')$

• Brownian Bridge process for $t, t' \in [a, b]$: $Cov(x(t), x(t')) = \frac{(b - \max(s,t))(\min(s, t) - a)}{\phi (b - a)} + \sigma^2 \delta_{t}(t')$

Usage

tf_rgp(
  n,
  arg = 51L,
  cov = c("squareexp", "wiener", "matern", "brown_bridge"),
  scale = diff(domain)/10,
  nugget = scale/200,
  order = 1.5,
  domain = NULL
)

Arguments

n	how many realizations to draw.
arg	vector of evaluation points (arg of the return object). Defaults to (0, 0.02, 0.04, ..., 1). If given as a single integer (don't forget the L...), creates a regular grid of that length over (0,1). If given as a n-long list of vectors, irregular functional data are created.
cov	type of covariance function to use. Implemented defaults are "squareexp", "wiener", "matern", see description. Can also be any vectorized function returning $Cov(x(t), x(t'))$ without nugget effect for pairs of inputs t and t'.
scale	scale parameter (see description). Defaults to the width of the domain divided by 10.
nugget	nugget effect for additional white noise / unstructured variability. Defaults to scale/200 (so: very little white noise).
order	order of the Matérn covariance (if used, must be >0), defaults to 1.5. The higher, the smoother the process. Evaluation of the covariance function becomes numerically unstable for large (>20) order, use "squareexp".
domain	of the generated functions. If not provided, the range of the supplied arg values.

Value

an tfd-vector of length n.

See Also

Other tidyfun RNG functions: tf_jiggle()

Examples

(x1 <- tf_rgp(10, cov = "squareexp", nugget = 0))
tf_rgp(2, arg = list(sort(runif(25)), sort(runif(34))))

---

Simple smoothing of tf objects

Description

Apply running means or medians, lowess or Savitzky-Golay filtering to smooth functional data. This does nothing for tfb-objects, which should be smoothed by using a smaller basis / stronger penalty.

Usage

tf_smooth(x, ...)

## S3 method for class 'tfb'
tf_smooth(x, verbose = TRUE, ...)

## S3 method for class 'tfd'
tf_smooth(
  x,
  method = c("lowess", "rollmean", "rollmedian", "savgol"),
  verbose = TRUE,
  ...
)

Arguments

x	a tf object containing functional data.
...	arguments for the respective method. See details.
verbose	give lots of diagnostic messages? Defaults to TRUE.
method	one of "lowess" (see stats::lowess()), "rollmean", "rollmedian" (see zoo::rollmean()) or "savgol" (a Savitzky-Golay filter, see savgol()).

Details

tf_smooth.tfd overrides/automatically sets some defaults of the used methods:

• lowess uses a span parameter of f = 0.15 (instead of 0.75) by default.

• rollmean/median use a window size of k = $<$number of grid points$>$/20 (i.e., the nearest odd integer to that) and sets fill= "extend" (i.e., constant extrapolation to replace missing values at the extremes of the domain) by default. Use fill= NA for zoo's default behavior of shortening the smoothed series.

• savgol uses a window size of k = $<$number of grid points$>$/10 (i.e., the nearest odd integer to that).

Value

a smoothed version of the input. For some methods/options, the smoothed functions may be shorter than the original ones (at both ends).

Examples

library(zoo)
f <- tf_sparsify(tf_jiggle(tf_rgp(4, 201, nugget = 0.05)))
f_lowess <- tf_smooth(f, "lowess")
# these methods ignore the distances between arg-values:
f_mean <- tf_smooth(f, "rollmean")
f_median <- tf_smooth(f, "rollmedian", k = 31)
f_sg <- tf_smooth(f, "savgol", fl = 31)
layout(t(1:4))
plot(f, points = FALSE, main = "original")
plot(f_lowess,
  points = FALSE, col = "blue", main = "lowess (default,\n span 0.9 in red)"
)
lines(tf_smooth(f, "lowess", f = 0.9), col = "red", alpha = 0.2)
plot(f_mean,
  points = FALSE, col = "blue", main = "rolling means &\n medians (red)"
)
lines(f_median, col = "red", alpha = 0.2) # note constant extrapolation at both ends!
plot(f, points = FALSE, main = "original and\n savgol (red)")
lines(f_sg, col = "red")

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