Type:	Package
Title:	Elastic Functional Data Analysis
Version:	2.4.0
Date:	2025-6-27
Description:	Performs alignment, PCA, and modeling of multidimensional and unidimensional functions using the square-root velocity framework (Srivastava et al., 2011 <doi:10.48550/arXiv.1103.3817> and Tucker et al., 2014 <doi:10.1016/j.csda.2012.12.001>). This framework allows for elastic analysis of functional data through phase and amplitude separation.
License:	GPL-3
LazyData:	TRUE
Imports:	cli, coda, doParallel, fields, foreach, lpSolve, Matrix, mvtnorm, Rcpp, rlang, minpack.lm, tolerance, viridisLite
Suggests:	covr, interp, plot3D, plot3Drgl, rgl, testthat (≥ 3.0.0), withr
Depends:	R (≥ 4.3.0),
RoxygenNote:	7.3.2
Encoding:	UTF-8
Config/testthat/edition:	3
LinkingTo:	Rcpp, RcppArmadillo
URL:	https://github.com/jdtuck/fdasrvf_R
BugReports:	https://github.com/jdtuck/fdasrvf_R/issues
NeedsCompilation:	yes
Packaged:	2025-06-27 12:50:54 UTC; jdtuck
Author:	J. Derek Tucker [aut, cre], Aymeric Stamm [ctb]
Maintainer:	J. Derek Tucker <jdtuck@sandia.gov>
Repository:	CRAN
Date/Publication:	2025-06-27 13:30:02 UTC

Elastic Functional Data Analysis

Description

A library for functional data analysis using the square root velocity framework which performs pair-wise and group-wise alignment as well as modeling using functional component analysis.

Author(s)

Maintainer: J. Derek Tucker jdtuck@sandia.gov (ORCID)

Other contributors:

• Aymeric Stamm aymeric.stamm@math.cnrs.fr (ORCID) [contributor]

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using Fisher-Rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative models for functional data using phase and amplitude separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

J. D. Tucker, W. Wu, and A. Srivastava, Phase-amplitude separation of proteomics data using extended Fisher-Rao metric, Electronic Journal of Statistics, Vol 8, no. 2. pp 1724-1733, 2014.

J. D. Tucker, W. Wu, and A. Srivastava, “Analysis of signals under compositional noise with applications to SONAR data," IEEE Journal of Oceanic Engineering, Vol 29, no. 2. pp 318-330, Apr 2014.

Tucker, J. D. 2014, Functional Component Analysis and Regression using Elastic Methods. Ph.D. Thesis, Florida State University.

Robinson, D. T. 2012, Function Data Analysis and Partial Shape Matching in the Square Root Velocity Framework. Ph.D. Thesis, Florida State University.

Kurtek, S., Srivastava, A., Klassen, E., and Ding, Z. (2012), “Statistical Modeling of Curves Using Shapes and Related Features,” Journal of the American Statistical Association, 107, 1152–1165.

Huang, W. 2014, Optimization Algorithms on Riemannian Manifolds with Applications. Ph.D. Thesis, Florida State University.

Cheng, W., Dryden, I. L., and Huang, X. (2016). Bayesian registration of functions and curves. Bayesian Analysis, 11(2), 447-475.

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Cheng, W., Dryden, I. L., and Huang, X. (2016). Bayesian registration of functions and curves. Bayesian Analysis, 11(2), 447-475.

W. Xie, S. Kurtek, K. Bharath, and Y. Sun, A geometric approach to visualization of variability in functional data, Journal of American Statistical Association 112 (2017), pp. 979-993.

Lu, Y., R. Herbei, and S. Kurtek, 2017: Bayesian registration of functions with a Gaussian process prior. Journal of Computational and Graphical Statistics, 26, no. 4, 894–904.

Lee, S. and S. Jung, 2017: Combined analysis of amplitude and phase variations in functional data. arXiv:1603.01775, 1–21.

J. D. Tucker, J. R. Lewis, and A. Srivastava, “Elastic Functional Principal Component Regression,” Statistical Analysis and Data Mining, vol. 12, no. 2, pp. 101-115, 2019.

J. D. Tucker, J. R. Lewis, C. King, and S. Kurtek, “A Geometric Approach for Computing Tolerance Bounds for Elastic Functional Data,” Journal of Applied Statistics, 10.1080/02664763.2019.1645818, 2019.

T. Harris, J. D. Tucker, B. Li, and L. Shand, "Elastic depths for detecting shape anomalies in functional data," Technometrics, 10.1080/00401706.2020.1811156, 2020.

J. D. Tucker and D. Yarger, “Elastic Functional Changepoint Detection of Climate Impacts from Localized Sources”, Envirometrics, 10.1002/env.2826, 2023.

See Also

Useful links:

• https://github.com/jdtuck/fdasrvf_R

• Report bugs at https://github.com/jdtuck/fdasrvf_R/issues

Long Run Covariance Matrix Estimation for Multivariate Time Series

Description

This function estimates the long run covariance matrix of a given multivariate data sample.

Usage

LongRunCovMatrix(mdobj, h = 0, kern_type = "bartlett")

Arguments

Value

Returns long run covariance matrix

SRVF transform of warping functions

Description

This function calculates the srvf of warping functions with corresponding shooting vectors and finds the mean

Usage

SqrtMean(gam)

Arguments

Value

Returns a list containing

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

out <- SqrtMean(simu_warp$warping_functions)

SRVF transform of warping functions

Description

This function calculates the srvf of warping functions with corresponding shooting vectors and finds the inverse of mean

Usage

SqrtMeanInverse(gam)

Arguments

Value

gamI inverse of Karcher mean warping function

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

gamI <- SqrtMeanInverse(simu_warp$warping_functions)

SRVF transform of warping functions

Description

This function calculates the srvf of warping functions with corresponding shooting vectors and finds the median

Usage

SqrtMedian(gam)

Arguments

Value

Returns a list containing

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

out <- SqrtMedian(simu_warp_median$warping_functions)

Group-wise function alignment and PCA Extractions

Description

This function aligns a collection of functions while extracting principal components.

Usage

align_fPCA(
  f,
  time,
  num_comp = 3L,
  showplot = TRUE,
  smooth_data = FALSE,
  sparam = 25L,
  parallel = FALSE,
  cores = NULL,
  max_iter = 51L,
  lambda = 0
)

Arguments

Value

A list with the following components:

• f0: A numeric matrix of shape M \times N storing the original functions;

• fn: A numeric matrix of the same shape as f0 storing the aligned functions;

• qn: A numeric matrix of the same shape as f0 storing the aligned SRSFs;

• q0: A numeric matrix of the same shape as f0 storing the SRSFs of the original functions;

• mqn: A numeric vector of length M storing the mean SRSF;

• gam: A numeric matrix of the same shape as f0 storing the estimated warping functions;

• vfpca: A list storing information about the vertical PCA with the following components:

  • q_pca: A numeric matrix of shape (M + 1) \times 5 \times \mathrm{num\_comp} storing the first 3 principal directions in SRSF space; the first dimension is M + 1 because, in SRSF space, the original functions are represented by the SRSF and the initial value of the functions.

  • f_pca: A numeric matrix of shape M \times 5 \times \mathrm{num\_comp} storing the first 3 principal directions in original space;

  • latent: A numeric vector of length M + 1 storing the singular values of the SVD decomposition in SRSF space;

  • coef: A numeric matrix of shape N \times \mathrm{num\_comp} storing the scores of the N original functions on the first num_comp principal components;

  • U: A numeric matrix of shape (M + 1) \times (M + 1) storing the eigenvectors associated with the SVD decomposition in SRSF space.

• Dx: A numeric vector of length max_iter storing the value of the cost function at each iteration.

References

Tucker, J. D., Wu, W., Srivastava, A., Generative models for functional data using phase and amplitude separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

## Not run: 
  out <- align_fPCA(simu_data$f, simu_data$time)

## End(Not run)

Amplitude Boxplot Data

Description

This function constructs the amplitude boxplot.

Usage

ampbox_data(warp_median, alpha = 0.05, ka = 1)

Arguments

Value

Returns an ampbox object containing:

• median_y: median function

• Q1: First quartile

• Q3: Second quartile

• Q1a: First quantile based on alpha

• Q3a: Second quantile based on alpha

• minn: minimum extreme function

• maxx: maximum extreme function

• outlier_index: indexes of outlier functions

• fmedian: median function

References

Xie, W., S. Kurtek, K. Bharath, and Y. Sun (2016). "A geometric approach to visualization of variability in functional data." Journal of the American Statistical Association in press: 1-34.

MPEG7 Curve Dataset

Description

Contains the MPEG7 curve data set.

Usage

beta

Format

beta

An array of shape 2 \times 100 \times 65 \times 20 storing a sample of 20 curves from R to R^2 distributed in 65 different classes, evaluated on a grid of size 100.

Tolerance Bound Calculation using Bootstrap Sampling

Description

This function computes tolerance bounds for functional data containing phase and amplitude variation using bootstrap sampling

Usage

bootTB(
  f,
  time,
  a = 0.05,
  p = 0.99,
  B = 500,
  no = 5,
  Nsamp = 100,
  parallel = TRUE
)

Arguments

Value

Returns a list containing

References

J. D. Tucker, J. R. Lewis, C. King, and S. Kurtek, “A Geometric Approach for Computing Tolerance Bounds for Elastic Functional Data,” Journal of Applied Statistics, 10.1080/02664763.2019.1645818, 2019.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Jung, S. L. a. S. (2016). "Combined Analysis of Amplitude and Phase Variations in Functional Data." arXiv:1603.01775.

Examples

## Not run: 
  out1 <- bootTB(simu_data$f, simu_data$time)

## End(Not run)

Functional Boxplot

Description

This function computes the required statistics for building up a boxplot of the aligned functional data. Since the process of alignment provides separation of phase and amplitude variability, the computed boxplot can focus either on amplitude variability or phase variability.

Usage

## S3 method for class 'fdawarp'
boxplot(
  x,
  variability_type = c("amplitude", "phase"),
  alpha = 0.05,
  range = 1,
  what = c("stats", "plot", "plot+stats"),
  ...
)

## S3 method for class 'ampbox'
boxplot(x, ...)

## S3 method for class 'phbox'
boxplot(x, ...)

## S3 method for class 'curvebox'
boxplot(x, ...)

Arguments

Details

The function boxplot.fdawarp() returns optionally an object of class either ampbox if variability_type = "amplitude" or phbox if variability_type = "phase". S3 methods specialized for objects of these classes are provided as well to avoid re-computation of the boxplot statistics.

Value

If what contains stats, a list containing the computed statistics necessary for drawing the boxplot. Otherwise, the function simply draws the boxplot and no object is returned.

Examples

## Not run: 
out <- time_warping(simu_data$f, simu_data$time)
boxplot(out, what = "stats")

## End(Not run)

Elastic Shape Distance

Description

Calculates the elastic shape distance between two curves beta1 and beta2.

Usage

calc_shape_dist(
  beta1,
  beta2,
  mode = "O",
  alignment = TRUE,
  rotation = TRUE,
  scale = TRUE,
  optim_method = "DP",
  include.length = FALSE,
  lambda = 0
)

Arguments

Details

Distances are computed between the SRVFs of the original curves. Hence, they are intrinsically invariant to position. The user can then choose to make distances invariant to rotation and scaling by setting rotation and scale accordingly. Distances can also be made invariant to reparameterization by setting alignment = TRUE, in which case curves are aligned using an appropriate action of the diffeomorphism group on the space of SRVFs.

Value

A list with the following components:

• d: the amplitude (geodesic) distance;

• dx: the phase distance;

• q1: the SRVF of beta1;

• q2n: the SRVF of beta2 after alignment and possible optimal rotation and scaling;

• beta1: the input curve beta1;

• beta2n: the input curve beta2 after alignment and possible optimal rotation and scaling.

• R: the optimal rotation matrix that has been applied to the second curve;

• gam: the optimal warping function that has been applied to the second curve;

• betascale: the optimal scaling factor that has been applied to the second curve.

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Kurtek, S., Srivastava, A., Klassen, E., and Ding, Z. (2012), “Statistical Modeling of Curves Using Shapes and Related Features,” Journal of the American Statistical Association, 107, 1152–1165.

Srivastava, A., Klassen, E. P. (2016). Functional and shape data analysis, 1. New York: Springer.

Examples

out <- calc_shape_dist(beta[, , 1, 1], beta[, , 1, 4])

Converts a curve to its SRVF representation

Description

Converts a curve to its SRVF representation

Usage

curve2srvf(beta, is_derivative = FALSE)

Arguments

Value

A function that takes a numeric vector s of values in [0, 1] as input and returns the values of the SRVF of the original curve at s.

Examples

curve2srvf(beta[, , 1, 1])

Curve Boxplot

Description

This function computes the required statistics for building up a boxplot of the aligned curve data. The computed boxplot focuses on the aligned curves.

Usage

curve_boxplot(
  x,
  alpha = 0.05,
  range = 1,
  what = c("stats", "plot", "plot+stats"),
  ...
)

Arguments

Details

The function returns optionally an object of class either curvebox

Value

If what contains stats, a list containing the computed statistics necessary for drawing the boxplot. Otherwise, the function simply draws the boxplot and no object is returned.

Examples

## Not run: 
out <- multivariate_karcher_mean(beta[, , 1, ], ms="median")
curve_boxplot(out, what = "stats")

## End(Not run)

Calculates elastic depth for a set of curves

Description

This functions calculates the elastic depth between set of curves. If the curves are describing multidimensional functional data, then rotated == FALSE and mode == 'O'

Usage

curve_depth(beta, mode = "O", rotated = TRUE, scale = TRUE, parallel = FALSE)

Arguments

Value

Returns a list containing

References

T. Harris, J. D. Tucker, B. Li, and L. Shand, "Elastic depths for detecting shape anomalies in functional data," Technometrics, 10.1080/00401706.2020.1811156, 2020.

Examples

data("mpeg7")
# note: use more shapes and iterations, small for speed
out = curve_depth(beta[,,1,1:2])

Distance Matrix Computation

Description

Computes the pairwise distance matrix between a set of curves using the elastic shape distance as computed by calc_shape_dist().

Usage

curve_dist(
  beta,
  mode = "O",
  alignment = TRUE,
  rotation = FALSE,
  scale = FALSE,
  include.length = FALSE,
  lambda = 0,
  ncores = 1L
)

Arguments

Details

Distances are computed between the SRVFs of the original curves. Hence, they are intrinsically invariant to position. The user can then choose to make distances invariant to rotation and scaling by setting rotation and scale accordingly. Distances can also be made invariant to reparameterization by setting alignment = TRUE, in which case curves are aligned using an appropriate action of the diffeomorphism group on the space of SRVFs.

Value

A list of two objects, Da and Dp, each of class dist containing the amplitude and phase distances, respectively.

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Kurtek, S., Srivastava, A., Klassen, E., and Ding, Z. (2012), “Statistical Modeling of Curves Using Shapes and Related Features,” Journal of the American Statistical Association, 107, 1152–1165.

Srivastava, A., Klassen, E. P. (2016). Functional and shape data analysis, 1. New York: Springer.

Examples

out <- curve_dist(beta[, , 1, 1:4])

Form geodesic between two curves

Description

Form geodesic between two curves using Elastic Method

Usage

curve_geodesic(beta1, beta2, k = 5)

Arguments

Value

a list containing

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Examples

out <- curve_geodesic(beta[, , 1, 1], beta[, , 1, 5])

Pairwise align two curves

Description

This function aligns to curves using Elastic Framework

Usage

curve_pair_align(beta1, beta2, mode = "O", rotation = TRUE, scale = TRUE)

Arguments

Value

a list containing

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Examples

out <- curve_pair_align(beta[, , 1, 1], beta[, , 1, 5])

Curve to SRVF Space

Description

This function computes the SRVF of a given curve. The function also outputs the length of the original curve and the L^2 norm of the SRVF.

Usage

curve_to_q(beta, scale = TRUE)

Arguments

Value

A list with the following components:

• q: A numeric matrix of the same shape as the input matrix beta storing the (possibly scaled) SRVF of the original curve beta;

• len: A numeric value specifying the length of the original curve;

• lenq: A numeric value specifying the L^2 norm of the SRVF of the original curve.

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in Euclidean spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(7), 1415-1428.

Examples

q <- curve_to_q(beta[, , 1, 1])$q

Curve Boxplot Data

Description

This function constructs the amplitude boxplot.

Usage

curvebox_data(align_median, alpha = 0.05, ka = 1)

Arguments

Value

Returns an curvebox object containing:

• median_y: median curve

• Q1: First quartile

• Q3: Second quartile

• Q1a: First quantile based on alpha

• Q3a: Second quantile based on alpha

• minn: minimum extreme curve

• maxx: maximum extreme curve

• outlier_index: indexes of outlier curves

• fmedian: median curve

References

Xie, W., S. Kurtek, K. Bharath, and Y. Sun (2016). "A geometric approach to visualization of variability in functional data." Journal of the American Statistical Association in press: 1-34.

Converts a curve from matrix to functional data object

Description

Converts a curve from matrix to functional data object

Usage

discrete2curve(beta)

Arguments

Value

A function that takes a numeric vector s of values in [0, 1] as input and returns the values of the original curve at s.

Examples

discrete2curve(beta[, , 1, 1])

Converts a warping function from vector to functional data object

Description

Converts a warping function from vector to functional data object

Usage

discrete2warping(gam)

Arguments

Value

A function that takes a numeric vector s of values in [0, 1] as input and returns the values of the original warping function at s.

Examples

discrete2warping(toy_warp$gam[, 1])

Calculates elastic depth

Description

This functions calculates the elastic depth between set of functions in R^1

Usage

elastic.depth(f, time, lambda = 0, pen = "roughness", parallel = FALSE)

Arguments

Value

Returns a list containing

References

T. Harris, J. D. Tucker, B. Li, and L. Shand, "Elastic depths for detecting shape anomalies in functional data," Technometrics, 10.1080/00401706.2020.1811156, 2020.

Examples

depths <- elastic.depth(simu_data$f[, 1:4], simu_data$time)

Calculates two elastic distance

Description

This functions calculates the distances between functions in R^1 D_y and D_x, where function 2 is aligned to function 1

Usage

elastic.distance(f1, f2, time, lambda = 0, pen = "roughness")

Arguments

Value

Returns a list containing

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

distances <- elastic.distance(
  f1 = simu_data$f[, 1],
  f2 = simu_data$f[, 2],
  time = simu_data$time
)

Elastic Logistic Regression

Description

This function identifies a logistic regression model with phase-variability using elastic methods

Usage

elastic.logistic(
  f,
  y,
  time,
  B = NULL,
  df = 20,
  max_itr = 20,
  smooth_data = FALSE,
  sparam = 25,
  parallel = FALSE,
  cores = 2
)

Arguments

Value

Returns a list containing

References

Tucker, J. D., Wu, W., Srivastava, A., Elastic Functional Logistic Regression with Application to Physiological Signal Classification, Electronic Journal of Statistics (2014), submitted.

Elastic logistic Principal Component Regression

Description

This function identifies a logistic regression model with phase-variability using elastic pca

Usage

elastic.lpcr.regression(
  f,
  y,
  time,
  pca.method = "combined",
  no = 5,
  smooth_data = FALSE,
  sparam = 25
)

Arguments

Value

Returns a lpcr object containing

References

J. D. Tucker, J. R. Lewis, and A. Srivastava, “Elastic Functional Principal Component Regression,” Statistical Analysis and Data Mining, 10.1002/sam.11399, 2018.

Elastic Multinomial Logistic Regression

Description

This function identifies a multinomial logistic regression model with phase-variability using elastic methods

Usage

elastic.mlogistic(
  f,
  y,
  time,
  B = NULL,
  df = 20,
  max_itr = 20,
  smooth_data = FALSE,
  sparam = 25,
  parallel = FALSE,
  cores = 2
)

Arguments

Value

Returns a list containing

References

Tucker, J. D., Wu, W., Srivastava, A., Elastic Functional Logistic Regression with Application to Physiological Signal Classification, Electronic Journal of Statistics (2014), submitted.

Elastic Multinomial logistic Principal Component Regression

Description

This function identifies a multinomial logistic regression model with phase-variability using elastic pca

Usage

elastic.mlpcr.regression(
  f,
  y,
  time,
  pca.method = "combined",
  no = 5,
  smooth_data = FALSE,
  sparam = 25
)

Arguments

Value

Returns a mlpcr object containing

References

J. D. Tucker, J. R. Lewis, and A. Srivastava, “Elastic Functional Principal Component Regression,” Statistical Analysis and Data Mining, 10.1002/sam.11399, 2018.

Elastic Linear Principal Component Regression

Description

This function identifies a regression model with phase-variability using elastic pca

Usage

elastic.pcr.regression(
  f,
  y,
  time,
  pca.method = "combined",
  no = 5,
  smooth_data = FALSE,
  sparam = 25,
  parallel = F,
  C = NULL
)

Arguments

Value

Returns a pcr object containing

References

J. D. Tucker, J. R. Lewis, and A. Srivastava, “Elastic Functional Principal Component Regression,” Statistical Analysis and Data Mining, 10.1002/sam.11399, 2018.

Elastic Prediction from Regression Models

Description

This function performs prediction from an elastic regression model with phase-variability

Usage

elastic.prediction(f, time, model, y = NULL, smooth_data = FALSE, sparam = 25)

Arguments

Value

Returns a list containing

References

Tucker, J. D., Wu, W., Srivastava, A., Elastic Functional Logistic Regression with Application to Physiological Signal Classification, Electronic Journal of Statistics (2014), submitted.

Elastic Linear Regression

Description

This function identifies a regression model with phase-variability using elastic methods

Usage

elastic.regression(
  f,
  y,
  time,
  B = NULL,
  lam = 0,
  df = 20,
  max_itr = 20,
  smooth_data = FALSE,
  sparam = 25,
  parallel = FALSE,
  cores = 2
)

Arguments

Value

Returns a list containing

References

Tucker, J. D., Wu, W., Srivastava, A., Elastic Functional Logistic Regression with Application to Physiological Signal Classification, Electronic Journal of Statistics (2014), submitted.

Elastic Amplitude Changepoint Detection

Description

This function identifies a amplitude changepoint using a fully functional approach

Usage

elastic_amp_change_ff(
  f,
  time,
  d = 1000,
  h = 0,
  smooth_data = FALSE,
  sparam = 25,
  showplot = TRUE
)

Arguments

Value

Returns a list object containing

References

J. D. Tucker and D. Yarger, “Elastic Functional Changepoint Detection of Climate Impacts from Localized Sources”, Envirometrics, 10.1002/env.2826, 2023.

Elastic Changepoint Detection

Description

This function identifies changepoints using a functional PCA

Usage

elastic_change_fpca(
  f,
  time,
  pca.method = "combined",
  pc = 0.95,
  d = 1000,
  n_pcs = 5,
  smooth_data = FALSE,
  sparam = 25,
  showplot = TRUE
)

Arguments

Value

Returns a list object containing

References

J. D. Tucker and D. Yarger, “Elastic Functional Changepoint Detection of Climate Impacts from Localized Sources”, Envirometrics, 10.1002/env.2826, 2023.

Elastic Phase Changepoint Detection

Description

This function identifies a phase changepoint using a fully functional approach

Usage

elastic_ph_change_ff(
  f,
  time,
  d = 1000,
  h = 0,
  smooth_data = FALSE,
  sparam = 25,
  showplot = TRUE
)

Arguments

Value

Returns a list object containing

References

J. D. Tucker and D. Yarger, “Elastic Functional Changepoint Detection of Climate Impacts from Localized Sources”, Envirometrics, 10.1002/env.2826, 2023.

Plot functional data

Description

This function plots functional data on a time grid

Usage

f_plot(t, f)

Arguments

Transformation to SRVF Space

Description

This function transforms functions in R^1 from their original functional space to the SRVF space.

Usage

f_to_srvf(f, time)

Arguments

Value

A numeric array of the same shape as the input array f storing the SRVFs of the original curves.

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using Fisher-Rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative models for functional data using phase and amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

q <- f_to_srvf(simu_data$f, simu_data$time)

Bayesian Group Warping

Description

This function aligns a set of functions using Bayesian SRSF framework

Usage

function_group_warp_bayes(
  f,
  time,
  iter = 50000,
  powera = 1,
  times = 5,
  tau = ceiling(times * 0.04),
  gp = seq(dim(f)[2]),
  showplot = TRUE
)

Arguments

Value

Returns a list containing

References

Cheng, W., Dryden, I. L., and Huang, X. (2016). Bayesian registration of functions and curves. Bayesian Analysis, 11(2), 447-475.

Examples

## Not run: 
  out <- function_group_warp_bayes(simu_data$f, simu_data$time)

## End(Not run)

Bayesian Karcher Mean Calculation

Description

This function calculates karcher mean of functions using Bayesian method

Usage

function_mean_bayes(f, time, times = 5, group = 1:dim(f)[2], showplot = TRUE)

Arguments

Value

Returns a list containing

References

Cheng, W., Dryden, I. L., and Huang, X. (2016). Bayesian registration of functions and curves. Bayesian Analysis, 11(2), 447-475.

Examples

## Not run: 
  out <- function_mean_bayes(simu_data$f, simu_data$time)

## End(Not run)

map warping function to tangent space at identity

Description

map warping function to tangent space at identity

Usage

gam_to_h(gam, smooth = TRUE)

Arguments

Value

A numeric array of the same shape as the input array gamma storing the shooting vectors of gamma obtained via finite differences.

map warping function to Hilbert Sphere

Description

map warping function to Hilbert Sphere

map warping function to hilbert sphere

Usage

gam_to_psi(gam, smooth = TRUE)

gam_to_psi(gam, smooth = TRUE)

Arguments

Value

A numeric array of the same shape as the input array gamma storing the shooting vectors of gamma obtained via finite differences.

A numeric array of the same shape as the input array gamma storing the shooting vectors of gamma obtained via finite differences.

map warping function to tangent space at identity

Description

map warping function to tangent space at identity

Usage

gam_to_v(gam, smooth = TRUE)

Arguments

Value

A numeric array of the same shape as the input array gamma storing the shooting vectors of gamma obtained via finite differences.

Gaussian model of functional data

Description

This function models the functional data using a Gaussian model extracted from the principal components of the srvfs

Usage

gauss_model(warp_data, n = 1, sort_samples = FALSE)

Arguments

Value

Returns a fdawarp object containing

References

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

out1 <- gauss_model(simu_warp, n = 10)

Computes the centroid of a curve

Description

Computes the centroid of a curve

Usage

get_curve_centroid(betafun)

Arguments

Value

A numeric vector of length L storing the centroid of the curve.

Examples

betafun <- discrete2curve(beta[, , 1, 1])
get_curve_centroid(betafun)

Computes the distance matrix between a set of shapes.

Description

Computes the distance matrix between a set of shapes.

Usage

get_distance_matrix(
  qfuns,
  alignment = FALSE,
  rotation = FALSE,
  scale = FALSE,
  lambda = 0,
  M = 200L,
  parallel_setup = 1L
)

Arguments

Value

An object of class dist containing the distance matrix between the shapes.

Examples

N <- 4L
srvfs <- lapply(1:N, \(n) curve2srvf(beta[, , 1, n]))
get_distance_matrix(srvfs, parallel_setup = 1L)

Computes the geodesic distance between two SRVFs on the Hilbert sphere

Description

Computes the geodesic distance between two SRVFs on the Hilbert sphere

Usage

get_hilbert_sphere_distance(q1fun, q2fun)

Arguments

Value

A numeric value storing the geodesic distance between the two SRVFs.

Examples

q1 <- curve2srvf(beta[, , 1, 1])
q2 <- curve2srvf(beta[, , 1, 2])
q1p <- to_hilbert_sphere(q1)
q2p <- to_hilbert_sphere(q2)
get_hilbert_sphere_distance(q1p, q2p)

Computes the identity warping function

Description

Computes the identity warping function

Usage

get_identity_warping()

Value

A function that takes a numeric vector s of values in [0, 1] as input and returns the values of the identity warping function at s.

Examples

get_identity_warping()

Computes the L^2 distance between two SRVFs

Description

Computes the L^2 distance between two SRVFs

Usage

get_l2_distance(q1fun, q2fun, method = "quadrature")

Arguments

Value

A numeric value storing the L^2 distance between the two SRVFs.

Examples

q1 <- curve2srvf(beta[, , 1, 1])
q2 <- curve2srvf(beta[, , 1, 2])
get_l2_distance(q1, q2)

Computes the L^2 inner product between two SRVFs

Description

Computes the L^2 inner product between two SRVFs

Usage

get_l2_inner_product(q1fun, q2fun)

Arguments

Value

A numeric value storing the L^2 inner product between the two SRVFs.

Examples

q1 <- curve2srvf(beta[, , 1, 1])
q2 <- curve2srvf(beta[, , 1, 2])
get_l2_inner_product(q1, q2)

Computes the L^2 norm of an SRVF

Description

Computes the L^2 norm of an SRVF

Usage

get_l2_norm(qfun)

Arguments

Value

A numeric value storing the L^2 norm of the SRVF.

Examples

q <- curve2srvf(beta[, , 1, 1])
get_l2_norm(q)

Computes the distance between two shapes

Description

Computes the distance between two shapes

Usage

get_shape_distance(
  q1fun,
  q2fun,
  alignment = FALSE,
  rotation = FALSE,
  scale = FALSE,
  lambda = 0,
  M = 200L
)

Arguments

Value

A list with the following components:

• amplitude_distance: A numeric value storing the amplitude distance between the two SRVFs.

• phase_distance: A numeric value storing the phase distance between the two SRVFs.

• optimal_warping: A function that takes a numeric vector s of values in [0, 1] as input and returns the optimal warping function evaluated at s.

• q1fun_modified: A function that takes a numeric vector s of values in [0, 1] as input and returns the first SRVF modified according to the optimal alignment, rotation, and scaling.

• q2fun_modified: A function that takes a numeric vector s of values in [0, 1] as input and returns the second SRVF modified according to the optimal alignment, rotation, and scaling.

Examples

q1 <- curve2srvf(beta[, , 1, 1])
q2 <- curve2srvf(beta[, , 1, 2])
get_shape_distance(q1, q2)

Computes the distance between two warping functions

Description

Computes the distance between two warping functions

Usage

get_warping_distance(gam1fun, gam2fun)

Arguments

Value

A numeric value storing the distance between the two warping functions.

Examples

gam1 <- discrete2warping(toy_warp$gam[, 1])
gam2 <- discrete2warping(toy_warp$gam[, 2])
get_warping_distance(gam1, gam2)
get_warping_distance(gam1, get_identity_warping())

Gradient using finite differences

Description

This function computes the gradient of f using finite differences.

Usage

gradient(f, binsize, multidimensional = FALSE)

Arguments

Value

A numeric array of the same shape as the input array f storing the gradient of f obtained via finite differences.

Examples

out <- gradient(simu_data$f[, 1], mean(diff(simu_data$time)))

Berkeley Growth Velocity Dataset

Description

Combination of both boys and girls growth velocity from the Berkley dataset.

Usage

growth_vel

Format

growth_vel

A list with two components:

• f: A numeric matrix of shape 69 \times 93 storing a sample of size N = 93 of curves evaluated on a grid of size M = 69.

• time: A numeric vector of size M = 69 storing the grid on which the curves f have been evaluated.

map shooting vector to warping function at identity

Description

map shooting vector to warping function at identity

Usage

h_to_gam(h)

Arguments

Value

A numeric array of the same shape as the input array h storing the warping functions of h.

Horizontal Functional Principal Component Analysis

Description

This function calculates vertical functional principal component analysis on aligned data

Usage

horizFPCA(warp_data, no = 3, var_exp = NULL, ci = c(-1, 0, 1), showplot = TRUE)

Arguments

Value

Returns a hfpca object containing

References

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

hfpca <- horizFPCA(simu_warp, no = 3)

Example Image Data set

Description

Contains two simulated images for registration.

Usage

im

Format

im

A list with two components:

• I1: A numeric matrix of shape 64 \times 64 storing the 1st image;

• I2: A numeric matrix of shape 64 \times 64 storing the 2nd image.

map square root of warping function to tangent space

Description

map square root of warping function to tangent space

Usage

inv_exp_map(Psi, psi)

Arguments

Value

A numeric array of the same length as the input array psi storing the shooting vector of psi

Invert Warping Function

Description

This function calculates the inverse of gamma

Usage

invertGamma(gam)

Arguments

Value

Returns gamI inverted vector

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

out <- invertGamma(simu_warp$warping_functions[, 1])

Joint Vertical and Horizontal Functional Principal Component Analysis

Description

This function calculates amplitude and phase joint functional principal component analysis on aligned data

Usage

jointFPCA(
  warp_data,
  no = 3,
  var_exp = NULL,
  id = round(length(warp_data$time)/2),
  C = NULL,
  ci = c(-1, 0, 1),
  srvf = TRUE,
  showplot = TRUE
)

Arguments

Value

Returns a list containing

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Jung, S. L. a. S. (2016). "Combined Analysis of Amplitude and Phase Variations in Functional Data." arXiv:1603.01775.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

jfpca <- jointFPCA(simu_warp, no = 3)

Joint Vertical and Horizontal Functional Principal Component Analysis

Description

This function calculates amplitude and phase joint functional principal component analysis on aligned data using the SRVF framework using MFPCA and h representation

Usage

jointFPCAh(
  warp_data,
  var_exp = 0.99,
  id = round(length(warp_data$time)/2),
  C = NULL,
  ci = c(-1, 0, 1),
  srvf = TRUE,
  showplot = TRUE
)

Arguments

Value

Returns a list containing

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Jung, S. L. a. S. (2016). "Combined Analysis of Amplitude and Phase Variations in Functional Data." arXiv:1603.01775.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

jfpcah <- jointFPCAh(simu_warp)

Gaussian model of functional data using joint Model

Description

This function models the functional data using a Gaussian model extracted from the principal components of the srvfs using the joint model

Usage

joint_gauss_model(warp_data, n = 1, no = 5)

Arguments

Value

Returns a fdawarp object containing

References

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Jung, S. L. a. S. (2016). "Combined Analysis of Amplitude and Phase Variations in Functional Data." arXiv:1603.01775.

Examples

out1 <- joint_gauss_model(simu_warp, n = 10)

K-Means Clustering and Alignment

Description

This function clusters functions and aligns using the elastic square-root velocity function (SRVF) framework.

Usage

kmeans_align(
  f,
  time,
  K = 1L,
  seeds = NULL,
  centroid_type = c("mean", "medoid"),
  nonempty = 0L,
  lambda = 0,
  showplot = FALSE,
  smooth_data = FALSE,
  sparam = 25L,
  parallel = FALSE,
  alignment = TRUE,
  rotation = FALSE,
  scale = TRUE,
  omethod = c("DP", "RBFGS"),
  max_iter = 50L,
  thresh = 0.01,
  use_verbose = FALSE
)

Arguments

Value

An object of class fdakma which is a list containing:

• f0: the original functions;

• q0: the original SRSFs;

• fn: the aligned functions as matrices or a 3D arrays of the same shape than f0 by clusters in a list;

• qn: the aligned SRSFs as matrices or a 3D arrays of the same shape than f0 separated in clusters in a list;

• labels: the cluster memberships as an integer vector;

• templates: the centroids in the original functional space;

• templates.q: the centroids in SRSF space;

• distances_to_center: A numeric vector storing the distances of each observed curve to its center;

• gam: the warping functions as matrices or a 3D arrays of the same shape than f0 by clusters in a list;

• qun: cost function value.

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using Fisher-Rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative models for functional data using phase and amplitude separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Sangalli, L. M., et al. (2010). "k-mean alignment for curve clustering." Computational Statistics & Data Analysis 54(5): 1219-1233.

Examples

## Not run: 
  out <- kmeans_align(growth_vel$f, growth_vel$time, K = 2)

## End(Not run)

Group-wise function alignment to specified mean

Description

This function aligns a collection of functions using the elastic square-root slope (srsf) framework.

Usage

multiple_align_functions(
  f,
  time,
  mu,
  lambda = 0,
  pen = "roughness",
  showplot = TRUE,
  smooth_data = FALSE,
  sparam = 25,
  parallel = FALSE,
  cores = -1,
  omethod = "DP",
  MaxItr = 20,
  iter = 2000,
  verbose = TRUE
)

Arguments

Value

Returns a fdawarp object containing

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Group-wise multivariate function alignment to specified mean

Description

This function aligns a collection of functions using the elastic square-root velocity (srvf) framework.

Usage

multiple_align_multivariate(
  beta,
  mu,
  mode = "O",
  alignment = TRUE,
  rotation = FALSE,
  scale = FALSE,
  lambda = 0,
  ncores = 1L,
  verbose = TRUE
)

Arguments

Value

Returns a fdacurve object containing:

• beta: A numeric array of shape L \times M \times N storing the original input data.

• q: A numeric array of shape L \times M \times N storing the SRVFs of the input data.

• betan: A numeric array of shape L \times M \times N storing the aligned, possibly optimally rotated and optimally scaled, input data.

• qn: A numeric array of shape L \times M \times N storing the SRVFs of the aligned, possibly optimally rotated and optimally scaled, input data.

• gamma: A numeric array of shape L \times M \times N storing the warping functions of the aligned, possibly optimally rotated and optimally scaled, input data.

• R: A numeric array of shape L \times L \times N storing the rotation matrices of the aligned, possibly optimally rotated and optimally scaled, input data.

• betamean: A numeric array of shape L \times M storing the Karcher mean or median of the input data.

• qmean: A numeric array of shape L \times M storing the Karcher mean or median of the SRVFs of the input data.

• type: A character string indicating whether the Karcher mean or median has been returned.

• E: A numeric vector storing the energy of the Karcher mean or median at each iteration.

• qun: A numeric vector storing the cost function of the Karcher mean or median at each iteration.

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (7), 1415-1428.

Curve Karcher Covariance

Description

Calculate Karcher Covariance of a set of curves

Usage

multivariate_karcher_cov(align_data)

Arguments

Value

K covariance matrix

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Examples

out <- multivariate_karcher_mean(beta[, , 1, 1:2], maxit = 2)
# note: use more shapes, small for speed
K <- multivariate_karcher_cov(out)

Karcher Mean of Multivariate Functional Data

Description

Calculates the Karcher mean or median of a collection of multivariate functional data using the elastic square-root velocity (SRVF) framework. While most of the time, the setting does not require a metric that is invariant to rotation and scale, this can be achieved through the optional arguments rotation and scale.

Usage

multivariate_karcher_mean(
  beta,
  mode = "O",
  alignment = TRUE,
  rotation = FALSE,
  scale = FALSE,
  lambda = 0,
  maxit = 20L,
  ms = c("mean", "median"),
  exact_medoid = FALSE,
  ncores = 1L,
  verbose = FALSE
)

Arguments

Value

A fdacurve object with the following components:

• beta: A numeric array of shape L \times M \times N storing the original input data.

• q: A numeric array of shape L \times M \times N storing the SRVFs of the input data.

• betan: A numeric array of shape L \times M \times N storing the aligned, possibly optimally rotated and optimally scaled, input data.

• qn: A numeric array of shape L \times M \times N storing the SRVFs of the aligned, possibly optimally rotated and optimally scaled, input data.

• gamma: A numeric array of shape L \times M \times N storing the warping functions of the aligned, possibly optimally rotated and optimally scaled, input data.

• R: A numeric array of shape L \times L \times N storing the rotation matrices of the aligned, possibly optimally rotated and optimally scaled, input data.

• betamean: A numeric array of shape L \times M storing the Karcher mean or median of the input data.

• qmean: A numeric array of shape L \times M storing the Karcher mean or median of the SRVFs of the input data.

• type: A character string indicating whether the Karcher mean or median has been returned.

• E: A numeric vector storing the energy of the Karcher mean or median at each iteration.

• qun: A numeric vector storing the cost function of the Karcher mean or median at each iteration.

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33 (7), 1415-1428.

Examples

out <- multivariate_karcher_mean(beta[, , 1, 1:2], maxit = 2)
# note: use more functions, small for speed

Curve PCA

Description

Calculate principal directions of a set of curves

Usage

multivariate_pca(
  align_data,
  no = 3,
  var_exp = NULL,
  ci = c(-1, 0, 1),
  mode = "O",
  showplot = TRUE
)

Arguments

Value

Returns a curve_pca object containing

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Examples

align_data <- multivariate_karcher_mean(beta[, , 1, 1:2], maxit = 2)
out <- multivariate_pca(align_data)

Align two functions

Description

This function aligns the SRVFs of two functions in R^1 defined on an interval [t_{\min}, t_{\max}] using dynamic programming or RBFGS

Usage

optimum.reparam(
  Q1,
  T1,
  Q2,
  T2,
  lambda = 0,
  pen = "roughness",
  method = c("DP", "DPo", "SIMUL", "RBFGS"),
  f1o = 0,
  f2o = 0,
  nbhd_dim = 7
)

Arguments

Value

A numeric vector of size n_points storing discrete evaluations of the estimated boundary-preserving warping diffeomorphism on the initial grid.

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using Fisher-Rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative models for functional data using phase and amplitude separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

q <- f_to_srvf(simu_data$f, simu_data$time)
gam <- optimum.reparam(q[, 1], simu_data$time, q[, 2], simu_data$time)

Outlier Detection

Description

This function calculates outlier's using geodesic distances of the SRVFs from the median

Usage

outlier.detection(q, time, mq, k = 1.5)

Arguments

Value

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

q_outlier <- outlier.detection(
  q = toy_warp$q0,
  time = toy_data$time,
  mq = toy_warp$mqn,
  k = .1
)

Align two functions

Description

This function aligns two functions using SRSF framework. It will align f2 to f1

Usage

pair_align_functions(
  f1,
  f2,
  time,
  lambda = 0,
  pen = "roughness",
  method = "DP",
  w = 0.01,
  iter = 2000
)

Arguments

Value

Returns a list containing

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Cheng, W., Dryden, I. L., and Huang, X. (2016). Bayesian registration of functions and curves. Bayesian Analysis, 11(2), 447-475.

Lu, Y., Herbei, R., and Kurtek, S. (2017). Bayesian registration of functions with a Gaussian process prior. Journal of Computational and Graphical Statistics, DOI: 10.1080/10618600.2017.1336444.

Examples

out <- pair_align_functions(
  f1 = simu_data$f[, 1],
  f2 = simu_data$f[, 2],
  time = simu_data$time
)

Align two functions

Description

This function aligns two functions using Bayesian SRSF framework. It will align f2 to f1

Usage

pair_align_functions_bayes(
  f1,
  f2,
  timet,
  iter = 15000,
  times = 5,
  tau = ceiling(times * 0.4),
  powera = 1,
  showplot = TRUE,
  extrainfo = FALSE
)

Arguments

Value

Returns a list containing

References

Cheng, W., Dryden, I. L., and Huang, X. (2016). Bayesian registration of functions and curves. Bayesian Analysis, 11(2), 447-475.

Examples

out <- pair_align_functions_bayes(
  f1 = simu_data$f[, 1],
  f2 = simu_data$f[, 2],
  timet = simu_data$time
)

Align two functions using geometric properties of warping functions

Description

This function aligns two functions using Bayesian framework. It will align f2 to f1. It is based on mapping warping functions to a hypersphere, and a subsequent exponential mapping to a tangent space. In the tangent space, the Z-mixture pCN algorithm is used to explore both local and global structure in the posterior distribution.

Usage

pair_align_functions_expomap(
  f1,
  f2,
  timet,
  iter = 20000,
  burnin = min(5000, iter/2),
  alpha0 = 0.1,
  beta0 = 0.1,
  zpcn = list(betas = c(0.5, 0.05, 0.005, 1e-04), probs = c(0.1, 0.1, 0.7, 0.1)),
  propvar = 1,
  init.coef = rep(0, 2 * 10),
  npoints = 200,
  extrainfo = FALSE
)

Arguments

Details

The Z-mixture pCN algorithm uses a mixture distribution for the proposal distribution, controlled by input parameter zpcn. The zpcn$betas must be between 0 and 1, and are the coefficients of the mixture components, with larger coefficients corresponding to larger shifts in parameter space. The zpcn$probs give the probability of each shift size.

Value

Returns a list containing

References

Lu, Y., Herbei, R., and Kurtek, S. (2017). Bayesian registration of functions with a Gaussian process prior. Journal of Computational and Graphical Statistics, DOI: 10.1080/10618600.2017.1336444.

Examples

## Not run: 
  # This is an MCMC algorithm and takes a long time to run
  myzpcn <- list(
    betas = c(0.1, 0.01, 0.005, 0.0001),
    probs = c(0.2, 0.2, 0.4, 0.2)
  )
  out <- pair_align_functions_expomap(
    f1 = simu_data$f[, 1],
    f2 = simu_data$f[, 2],
    timet = simu_data$time,
    zpcn = myzpcn,
    extrainfo = TRUE
  )
  # overall acceptance ratio
  mean(out$accept)
  # acceptance ratio by zpcn coefficient
  with(out, tapply(accept, myzpcn$betas[betas.ind], mean))

## End(Not run)

Pairwise align two images This function aligns to images using the q-map framework

Description

Pairwise align two images This function aligns to images using the q-map framework

Usage

pair_align_image(
  I1,
  I2,
  M = 5,
  ortho = TRUE,
  basis_type = "t",
  resizei = FALSE,
  N = 64,
  stepsize = 1e-05,
  itermax = 1000
)

Arguments

Value

Returns a list containing

References

Q. Xie, S. Kurtek, E. Klassen, G. E. Christensen and A. Srivastava. Metric-based pairwise and multiple image registration. IEEE European Conference on Computer Vision (ECCV), September, 2014

Examples

## Not run: 
  # This is a gradient descent algorithm and takes a long time to run
  out <- pair_align_image(im$I1, im$I2)

## End(Not run)

Tolerance Bound Calculation using Elastic Functional PCA

Description

This function computes tolerance bounds for functional data containing phase and amplitude variation using principal component analysis

Usage

pcaTB(f, time, m = 4, B = 1e+05, a = 0.05, p = 0.99)

Arguments

Value

Returns a list containing

References

J. D. Tucker, J. R. Lewis, C. King, and S. Kurtek, “A Geometric Approach for Computing Tolerance Bounds for Elastic Functional Data,” Journal of Applied Statistics, 10.1080/02664763.2019.1645818, 2019.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Jung, S. L. a. S. (2016). "Combined Analysis of Amplitude and Phase Variations in Functional Data." arXiv:1603.01775.

Examples

## Not run: 
  out1 <- pcaTB(simu_data$f, simu_data$time)

## End(Not run)

Phase Boxplot Data

Description

This function constructs the phase boxplot.

Usage

phbox_data(warp_median, alpha = 0.05, kp = 1)

Arguments

Value

Returns a phbox object containing:

• median_x: median warping function

• Q1: First quartile

• Q3: Second quartile

• Q1a: First quantile based on alpha

• Q3a: Second quantile based on alpha

• minn: minimum extreme function

• maxx: maximum extreme function

• outlier_index: indexes of outlier functions

References

Xie, W., S. Kurtek, K. Bharath, and Y. Sun (2016). "A geometric approach to visualization of variability in functional data." Journal of the American Statistical Association in press: 1-34.

Plot Curve

Description

This function plots open or closed curves

Usage

plot_curve(beta, add = FALSE, ...)

Arguments

Value

Return shape confidence intervals

Elastic Prediction for curve PCA

Description

This function performs projection of new curves on fPCA basis

Usage

## S3 method for class 'curve_pca'
predict(object, newdata = NULL, ...)

Arguments

Value

Returns a matrix

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Elastic Prediction for functional PCA

Description

This function performs projection of new functions on fPCA basis

Usage

## S3 method for class 'hfpca'
predict(object, newdata = NULL, ...)

Arguments

Value

Returns a matrix

References

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Elastic Prediction for functional PCA

Description

This function performs projection of new functions on fPCA basis

Usage

## S3 method for class 'jfpca'
predict(object, newdata = NULL, ...)

Arguments

Value

Returns a matrix

References

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Elastic Prediction for functional PCA

Description

This function performs projection of new functions on fPCA basis

Usage

## S3 method for class 'jfpcah'
predict(object, newdata = NULL, ...)

Arguments

Value

Returns a matrix

References

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Elastic Prediction for functional logistic PCR Model

Description

This function performs prediction from an elastic logistic fPCR regression model with phase-variability

Usage

## S3 method for class 'lpcr'
predict(object, newdata = NULL, y = NULL, ...)

Arguments

Value

Returns a list containing

References

J. D. Tucker, J. R. Lewis, and A. Srivastava, “Elastic Functional Principal Component Regression,” Statistical Analysis and Data Mining, 10.1002/sam.11399, 2018.

Elastic Prediction for functional multinomial logistic PCR Model

Description

This function performs prediction from an elastic multinomial logistic fPCR regression model with phase-variability

Usage

## S3 method for class 'mlpcr'
predict(object, newdata = NULL, y = NULL, ...)

Arguments

Value

Returns a list containing

References

J. D. Tucker, J. R. Lewis, and A. Srivastava, “Elastic Functional Principal Component Regression,” Statistical Analysis and Data Mining, 10.1002/sam.11399, 2018.

Elastic Prediction for functional PCR Model

Description

This function performs prediction from an elastic pcr regression model with phase-variability

Usage

## S3 method for class 'pcr'
predict(object, newdata = NULL, y = NULL, ...)

Arguments

Value

Returns a list containing

References

J. D. Tucker, J. R. Lewis, and A. Srivastava, “Elastic Functional Principal Component Regression,” Statistical Analysis and Data Mining, 10.1002/sam.11399, 2018.

Elastic Prediction for functional PCA

Description

This function performs projection of new functions on fPCA basis

Usage

## S3 method for class 'vfpca'
predict(object, newdata = NULL, ...)

Arguments

Value

Returns a matrix

References

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

map Hilbert sphere to warping function

Description

map Hilbert sphere to warping function

map hilbert sphere to warping function

Usage

psi_to_gam(psi)

psi_to_gam(psi)

Arguments

Value

A numeric array of the same shape as the input array psi storing the warping function obtained via finite integration

A numeric array of the same shape as the input array psi storing the warping functions psi.

Convert to curve space

Description

This function converts SRVFs to curves

Usage

q_to_curve(q, scale = 1)

Arguments

Value

beta array describing curve

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Examples

q <- curve_to_q(beta[, , 1, 1])$q
beta1 <- q_to_curve(q)

Align two curves

Description

This function aligns two SRVF functions using Dynamic Programming. If the curves beta1 and beta2 are describing multidimensional functional data, then rotation == FALSE and mode == 'O'

Usage

reparam_curve(
  beta1,
  beta2,
  lambda = 0,
  method = "DP",
  w = 0.01,
  rotated = TRUE,
  isclosed = FALSE,
  mode = "O"
)

Arguments

Value

return a List containing

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Examples

gam <- reparam_curve(beta[, , 1, 1], beta[, , 1, 5])$gam

Find optimum reparameterization between two images

Description

Finds the optimal warping function between two images using the elastic framework

Usage

reparam_image(It, Im, gam, b, stepsize = 1e-05, itermax = 1000, lmark = FALSE)

Arguments

Value

Returns a list containing

References

Q. Xie, S. Kurtek, E. Klassen, G. E. Christensen and A. Srivastava. Metric-based pairwise and multiple image registration. IEEE European Conference on Computer Vision (ECCV), September, 2014

Resample Curve

Description

This function resamples a curve to a number of points

Usage

resamplecurve(x, N = 100, mode = "O")

Arguments

Value

xn matrix defining resampled curve

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Examples

xn <- resamplecurve(beta[, , 1, 1], 200)

Random Warping

Description

Generates random warping functions

Usage

rgam(N, sigma, num)

Arguments

Value

gam warping functions

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

gam = rgam(N=101, sigma=.01, num=35)

Rotation Principal Component Analysis

Description

This function calculates functional principal component analysis on rotation data from

Usage

rotation_pca(align_data, no = 3, var_exp = NULL)

Arguments

Value

Returns a rotpca object containing

Sample shapes from model

Description

Sample shapes from model

Usage

sample_shapes(x, no = 3, numSamp = 10)

Arguments

Value

Returns a fdacurve object containing

References

Srivastava, A., Klassen, E., Joshi, S., Jermyn, I., (2011). Shape analysis of elastic curves in euclidean spaces. Pattern Analysis and Machine Intelligence, IEEE Transactions on 33 (7), 1415-1428.

Examples

out <- multivariate_karcher_mean(beta[, , 1, 1:5], scale=TRUE, maxit = 2)
# note: use more shapes, small for speed
out.samples <- sample_shapes(out)

Shape Confidence Interval Calculation using Bootstrap Sampling

Description

This function computes Confidence bounds for shapes using elastic metric

Usage

shape_CI(
  beta,
  a = 0.95,
  no = 5,
  Nsamp = 100,
  mode = "O",
  rotated = TRUE,
  scale = TRUE,
  lambda = 0,
  parallel = TRUE
)

Arguments

Value

Return shape confidence intervals

References

J. D. Tucker, J. R. Lewis, C. King, and S. Kurtek, “A Geometric Approach for Computing Tolerance Bounds for Elastic Functional Data,” Journal of Applied Statistics, 10.1080/02664763.2019.1645818, 2019.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Simulated two Gaussian Dataset

Description

A functional dataset where the individual functions are given by: y_i(t) = z_{i,1} e^{-(t-1.5)^2/2} + z_{i,2}e^{-(t+1.5)^2/2}, t \in [-3, 3], ~i=1,2,\dots, 21, where z_{i,1} and z_{i,2} are i.i.d. normal with mean one and standard deviation 0.25. Each of these functions is then warped according to: \gamma_i(t) = 6({e^{a_i(t+3)/6} -1 \over e^{a_i} - 1}) - 3 if a_i \neq 0, otherwise \gamma_i = \gamma_{id} (gamma_{id}(t) = t) is the identity warping). The variables are as follows: f containing the 21 functions of 101 samples and time which describes the sampling.

Usage

simu_data

Format

simu_data

A list with 2 components:

• f: A numeric matrix of shape 101 \times 21 storing a sample of size N = 21 of curves evaluated on a grid of size M = 101.

• time: A numeric vector of size M = 101 storing the grid on which the curves f have been evaluated.

Aligned Simulated two Gaussian Dataset

Description

A functional dataset where the individual functions are given by: y_i(t) = z_{i,1} e^{-(t-1.5)^2/2} + z_{i,2}e^{-(t+1.5)^2/2}, t \in [-3, 3], ~i=1,2,\dots, 21, where z_{i,1} and z_{i,2} are i.i.d. normal with mean one and standard deviation 0.25. Each of these functions is then warped according to: \gamma_i(t) = 6({e^{a_i(t+3)/6} -1 \over e^{a_i} - 1}) - 3 if a_i \neq 0, otherwise \gamma_i = \gamma_{id} (gamma_{id}(t) = t) is the identity warping). The variables are as follows: f containing the 21 functions of 101 samples and time which describes the sampling which has been aligned.

Usage

simu_warp

Format

simu_warp

A list which contains the output of the time_warping() function applied on the data set simu_data.

Aligned Simulated two Gaussian Dataset using Median

Description

A functional dataset where the individual functions are given by: y_i(t) = z_{i,1} e^{-(t-1.5)^2/2} + z_{i,2}e^{-(t+1.5)^2/2}, t \in [-3, 3], ~i=1,2,\dots, 21, where z_{i,1} and z_{i,2} are i.i.d. normal with mean one and standard deviation 0.25. Each of these functions is then warped according to: \gamma_i(t) = 6({e^{a_i(t+3)/6} -1 \over e^{a_i} - 1}) - 3 if a_i \neq 0, otherwise \gamma_i = \gamma_{id} (gamma_{id}(t) = t) is the identity warping). The variables are as follows: f containing the 21 functions of 101 samples and time which describes the sampling which has been aligned.

Usage

simu_warp_median

Format

simu_warp_median

A list which contains the output of the time_warping() function finding the median applied on the data set simu_data.

Smooth Functions

Description

This function smooths functions using standard box filter

Usage

smooth.data(f, sparam)

Arguments

Value

fo smoothed functions

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

fo <- smooth.data(simu_data$f, 25)

Converts from SRVF to curve representation

Description

Converts from SRVF to curve representation

Usage

srvf2curve(qfun, beta0 = NULL)

Arguments

Value

A function that takes a numeric vector t of values in [0, 1] as input and returns the values of the underlying curve at t.

Examples

srvf2curve(curve2srvf(beta[, , 1, 1]))

Transformation from SRSF Space

Description

This function transforms SRVFs back to the original functional space for functions in R^1.

Usage

srvf_to_f(q, time, f0 = 0)

Arguments

Value

A numeric array of the same shape as the input q storing the transformation of the SRVFs q back to the original functional space.

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative models for functional data using amplitude and phase separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

q <- f_to_srvf(simu_data$f, simu_data$time)
f <- srvf_to_f(q, simu_data$time, simu_data$f[1, ])

Alignment of univariate functional data

Description

This function aligns a collection of 1-dimensional curves that are observed on the same grid.

Usage

time_warping(
  f,
  time,
  lambda = 0,
  penalty_method = c("roughness", "geodesic", "norm"),
  centroid_type = c("mean", "median"),
  center_warpings = TRUE,
  smooth_data = FALSE,
  sparam = 25L,
  parallel = FALSE,
  cores = -1,
  optim_method = c("DP", "DPo", "DP2", "RBFGS"),
  max_iter = 20L
)

Arguments

Value

An object of class fdawarp which is a list with the following components:

• time: a numeric vector of length M storing the original grid;

• f0: a numeric matrix of shape M \times N storing the original sample of N functions observed on a grid of size M;

• q0: a numeric matrix of the same shape as f0 storing the original SRSFs;

• fn: a numeric matrix of the same shape as f0 storing the aligned functions;

• qn: a numeric matrix of the same shape as f0 storing the aligned SRSFs;

• fmean: a numeric vector of length M storing the mean or median curve;

• mqn: a numeric vector of length M storing the mean or median SRSF;

• warping_functions: a numeric matrix of the same shape as f0 storing the estimated warping functions;

• original_variance: a numeric value storing the variance of the original sample;

• amplitude_variance: a numeric value storing the variance in amplitude of the aligned sample;

• phase_variance: a numeric value storing the variance in phase of the aligned sample;

• qun: a numeric vector of maximum length max_iter + 2 storing the values of the cost function after each iteration;

• lambda: the input parameter lambda which specifies the elasticity;

• centroid_type: the input centroid type;

• optim_method: the input optimization method;

• inverse_average_warping_function: the inverse of the mean estimated warping function;

• rsamps: TO DO.

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using Fisher-Rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative models for functional data using phase and amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

## Not run: 
  out <- time_warping(simu_data$f, simu_data$time)

## End(Not run)

Projects an SRVF onto the Hilbert sphere

Description

Projects an SRVF onto the Hilbert sphere

Usage

to_hilbert_sphere(qfun, qnorm = get_l2_norm(qfun))

Arguments

Value

A function that takes a numeric vector s of values in [0, 1] as input and returns the values of the SRVF projected onto the Hilbert sphere.

Examples

q <- curve2srvf(beta[, , 1, 1])
to_hilbert_sphere(q)

Distributed Gaussian Peak Dataset

Description

A functional dataset where the individual functions are given by a Gaussian peak with locations along the x-axis. The variables are as follows: f containing the 29 functions of 101 samples and time which describes the sampling.

Usage

toy_data

Format

toy_data

A list with two components:

• f: A numeric matrix of shape 101 \times 29 storing a sample of size N = 29 of curves evaluated on a grid of size M = 101.

• time: A numeric vector of size M = 101 storing the grid on which the curves f have been evaluated.

Aligned Distributed Gaussian Peak Dataset

Description

A functional dataset where the individual functions are given by a Gaussian peak with locations along the x-axis. The variables are as follows: f containing the 29 functions of 101 samples and time which describes the sampling which as been aligned.

Usage

toy_warp

Format

toy_warp

A list which contains the output of the time_warping() function applied on the data set toy_data.

map shooting vector to curve at mean

Description

map shooting vector to curve at mean

Usage

v_to_curve(v, mu, mode = "O", scale = 1)

Arguments

Value

A numeric array of the same shape as the input array v storing the curves of v.

map shooting vector to warping function at identity

Description

map shooting vector to warping function at identity

Usage

v_to_gam(v)

Arguments

Value

A numeric array of the same shape as the input array v storing the warping functions v.

Vertical Functional Principal Component Analysis

Description

This function calculates vertical functional principal component analysis on aligned data

Usage

vertFPCA(
  warp_data,
  no = 3,
  var_exp = NULL,
  id = round(length(warp_data$time)/2),
  ci = c(-1, 0, 1),
  showplot = TRUE
)

Arguments

Value

Returns a vfpca object containing

References

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

vfpca <- vertFPCA(simu_warp, no = 3)

Applies a warping function to a given curve

Description

Applies a warping function to a given curve

Usage

warp_curve(betafun, gamfun)

Arguments

Value

A function that takes a numeric vector s of values in [0, 1] as input and returns the values of the warped curve at s.

Examples

curv <- discrete2curve(beta[, , 1, 1])
gamf <- discrete2warping(seq(0, 1, length = 100)^2)
warp_curve(curv, gamf)

Warp Function

Description

This function warps function f by \gamma

Usage

warp_f_gamma(f, time, gamma, spl.int = FALSE)

Arguments

Value

fnew warped function

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

fnew <- warp_f_gamma(
  f = simu_data$f[, 1],
  time = simu_data$time,
  gamma = seq(0, 1, length.out = 101)
)

Warp SRSF

Description

This function warps srsf q by \gamma

Usage

warp_q_gamma(q, time, gamma, spl.int = FALSE)

Arguments

Value

qnew warped function

References

Srivastava, A., Wu, W., Kurtek, S., Klassen, E., Marron, J. S., May 2011. Registration of functional data using fisher-rao metric, arXiv:1103.3817v2.

Tucker, J. D., Wu, W., Srivastava, A., Generative Models for Function Data using Phase and Amplitude Separation, Computational Statistics and Data Analysis (2012), 10.1016/j.csda.2012.12.001.

Examples

q <- f_to_srvf(simu_data$f, simu_data$time)
qnew <- warp_q_gamma(q[, 1], simu_data$time, seq(0, 1, length.out = 101))

Applies a warping function to a given SRVF

Description

Applies a warping function to a given SRVF

Usage

warp_srvf(qfun, gamfun, betafun = NULL)

Arguments

Value

A function that takes a numeric vector s of values in [0, 1] as input and returns the values of the warped SRVF.

Examples

q <- curve2srvf(beta[, , 1, 1])
warp_srvf(q, get_identity_warping())