Tolerance Bands for Functional Data — tolerance.band • fdars

Compute tolerance bands that are expected to contain a given fraction of individual curves in the population. Functional Tolerance Band

Usage

tolerance.band(
  fdataobj,
  method = c("fpca", "conformal", "scb", "exponential", "elastic", "phase",
    "elastic.config"),
  coverage = 0.95,
  ncomp = 3,
  nb = 500,
  band.type = c("pointwise", "simultaneous"),
  cal.fraction = 0.2,
  score.type = c("supnorm", "l2"),
  bandwidth = NULL,
  confidence = NULL,
  multiplier = c("gaussian", "rademacher"),
  family = c("gaussian", "binomial", "poisson"),
  max.iter = 10,
  ncomp.phase = 3,
  tol = 1e-04,
  seed = NULL
)

Arguments

fdataobj: An object of class 'fdata'.
method: Method to use. One of "fpca" (default), "conformal", "scb", "exponential", "elastic", "phase", or "elastic.config".
coverage: Target coverage probability (default 0.95).
ncomp: Number of FPCA components (default 3). Used by "fpca", "exponential", "elastic", "phase", and "elastic.config" (as amplitude components for "elastic.config").
nb: Number of bootstrap replicates (default 500). Used by "fpca", "scb", "exponential", "elastic", "phase", and "elastic.config".
band.type: "pointwise" (default) or "simultaneous". Used by "fpca", "elastic", "phase", and "elastic.config".
cal.fraction: Calibration fraction for conformal method (default 0.2).
score.type: Nonconformity score: "supnorm" (default) or "l2". Used by "conformal".
bandwidth: Kernel bandwidth for SCB Degras method. If NULL, a default is computed.
confidence: Confidence level for SCB method (default is coverage).
multiplier: Multiplier distribution: "gaussian" (default) or "rademacher". Used by "scb".
family: Exponential family: "gaussian" (default), "binomial", or "poisson". Used by "exponential".
max.iter: Maximum iterations for elastic method Karcher mean (default 10).
ncomp.phase: Number of FPCA components for the phase band (default 3). Used by "elastic.config".
tol: Convergence tolerance for Karcher mean (default 1e-4). Used by "elastic.config".
seed: Random seed for reproducibility (default NULL).

Value

An object of class 'tolerance.band' with components:

lower: numeric vector of lower bounds
upper: numeric vector of upper bounds
center: numeric vector of center function
half_width: numeric vector of half-widths
method: the method used
coverage: the target coverage
argvals: evaluation points
fdataobj: the original fdata input

The "phase" method additionally returns gamma.lower, gamma.upper, and gamma.center (warping function bounds), with lower/upper/center/half_width from the tangent-space band.

The "elastic.config" method additionally returns phase.lower, phase.upper, and phase.center (warping function bounds), with the primary lower/upper/center/half_width from the amplitude band.

Returns NULL with a warning if computation fails.

Details

Compute a tolerance band for functional data using one of several methods.

Available methods:

fpca: FPCA + bootstrap tolerance band. Reconstructs curves from PC scores and uses bootstrap to estimate pointwise or simultaneous quantiles.
conformal: Distribution-free conformal prediction band. Splits data into training and calibration sets.
scb: Simultaneous confidence band for the mean (Degras method). Uses multiplier bootstrap for critical values.
exponential: Tolerance band for exponential family functional data. Applies link function transformation.
elastic: Tolerance band in elastic (aligned) space. First computes Karcher mean, then applies FPCA band on aligned data.
phase: Phase tolerance band for warping function variation. Computes Karcher mean, extracts warping functions, and builds a tolerance band in the tangent (shooting vector) space. Returns both the warping-function bounds and the tangent-space band.
elastic.config: Joint amplitude and phase tolerance band with full configuration control. Separately controls the number of FPCA components for amplitude (ncomp) and phase (ncomp.phase), plus convergence tolerance (tol).

References

Rathnayake, L.N. and Cuevas, A. (2016). Tolerance bands for functional data. Technometrics, 58(3):326–334.

Lei, J. and Wasserman, L. (2014). Distribution-free prediction bands for non-parametric regression. Journal of the Royal Statistical Society: Series B, 76(1):71–96.

Degras, D. (2011). Simultaneous confidence bands for nonparametric regression with functional data. Statistica Sinica, 21(4):1735–1765.

Examples

fd <- fdata(matrix(rnorm(500), 50, 10), argvals = seq(0, 1, length.out = 10))
# \donttest{
band <- tolerance.band(fd, method = "fpca", nb = 100)
# }