Fit a FARFIMA model

Fits a functional autoregressive fractionally integrated moving-average (FARFIMA) model to the training sample and calibrates conformal prediction bands on the validation sample.

Usage

farfima(
  train,
  valid = NULL,
  d = c("bc_lpwn2", "bc_lpwn1", "lpwn2", "lpwn1", "lw", "dfa"),
  order = c("ic_full", "ic_auto", "bj", "cv"),
  p = 0:5,
  q = 0:5,
  method = c("farma", "far"),
  h = 10,
  level = c(0.8, 0.95),
  band = c("SR", "NS", "AC"),
  window = c("expanding", "fixed"),
  valid_eval = NULL,
  Npc = 1,
  trim = NULL,
  far_pve = 0.95,
  memory_args = list(),
  order_args = list(),
  eta_grid = c(0, 0.005, 0.01, 0.02, 0.05, 0.1),
  zeta_grid = seq(0.1, 50, length.out = 2496),
  symmetric = TRUE,
  integrate = TRUE,
  KI = NULL,
  scorecast = c("none", "naive", "theta"),
  acmcp_regression = TRUE,
  seed = NULL,
  verbose = TRUE
)

Arguments

train: Training sample: a matrix of curves (grid points by time) or an fts object.
valid: Validation sample, following train. It is needed for the conformal prediction bands and for order = "cv".
d: Memory parameter estimator for estimate_memory() ("lw", "dfa", "lpwn1", "lpwn2", "bc_lpwn1" or "bc_lpwn2"), or a known value.
order: Order selection method for select_order() ("ic_full", "ic_auto", "bj" or "cv"), or known orders c(p, q). With "cv", the forecasts are compared with the validation curves on their original scale.
p, q: Candidate AR and MA orders searched by the selection method. Not used when order gives the orders. (0, 0) is excluded.
method: "farma" for the functional Hannan-Rissanen estimator, or "far" for the FAR(1) benchmark fitted by far::far(). The benchmark ignores order, and with d = 1 it is the FARFIMA(1, 1, 0) benchmark.
h: Maximum forecast horizon of the conformal prediction bands and of order = "cv". valid needs at least h + 1 curves.
level: Nominal coverage levels of the conformal prediction bands.
band: Conformal prediction bands to calibrate: "SR" (scaled residual), "NS" (non-conformity score), "AC" (adaptive conformal), "PID" (conformal PID control, proportional-integral-derivative) and "AcMCP" (autocorrelated multistep conformal). See predict().
window: "expanding" or "fixed" window at each forecast origin. The fixed window has the length of the training sample.
valid_eval: Optional curves to check the validation forecasts against, in place of valid. The forecasts are still made from valid. Use it when the model is fitted to smoothed curves but the bands should be calibrated on the raw curves. It must have the same size as valid.
Npc: Number of principal components in the operator estimator.
trim: Number of initial fractionally differenced curves dropped before order selection and estimation. Defaults to \(\min(100, \lfloor T/3 \rfloor)\), where \(T\) is the number of training curves.
far_pve: Proportion of variance explained by the components of the FAR(1) benchmark.
memory_args: List of other arguments to estimate_memory(), such as m_const or B.
order_args: List of other arguments to select_order(), such as n_lags or alpha.
eta_grid: Candidate learning rates of the AC, PID and AcMCP conformal prediction bands. One rate is chosen for each band and level, by the mean interval score on the second half of the validation sample.
zeta_grid: Candidate multipliers of the SR band.
symmetric: If TRUE, the AC and PID bands are symmetric about the point forecast. If FALSE, their lower and upper edges are calibrated separately, each at level \(\alpha/2\). The SR and NS bands are always symmetric and the AcMCP band is always asymmetric.
integrate: Whether the PID and AcMCP bands include the integral term, which widens the band after repeated under-coverage and narrows it after repeated over-coverage. In the extreme, the band becomes infinite or empty.
KI: Scale of the integral term. Defaults to the largest absolute standardised validation residual at each horizon.
scorecast: Scorecaster of the PID band: "none", "naive" (the latest observed score curve) or "theta" (the theta method: exponential smoothing of the past scores plus half their linear trend).
acmcp_regression: Whether the centre of the AcMCP band also uses the regression on the scores at shorter horizons. If FALSE, only the MA(h - 1) forecast is used.
seed: Optional seed for the BC-LPWN bootstrap and the white-noise test of order = "bj".
verbose: Whether to show progress messages and a progress bar.

Value

An object of class farfima, a list with

d and order: the memory parameter and c(p = , q = );
phi, theta: the AR and MA operators (grid by grid by lag);
fitted, fitted_differenced, residuals: the in-sample fit on the scale of the data and on the fractionally differenced scale, and the residuals, also given by fitted() and residuals() (NULL for the FAR(1) benchmark);
memory, order_selection: the estimate_memory() and select_order() results, or NULL when d or the orders are given;
operators: the estimate_operators() result, or NULL for the FAR(1) benchmark;
calibration: a table of the calibrated conformal prediction bands by level and h: SR_zeta (SR multiplier), NS_quantile (NS quantile), AC_eta, PID_eta and AcMCP_eta (learning rates, for the bands chosen) and n_origin (number of validation origins). NULL without a validation sample.

print() and summary() show the fit, coef() gives d, phi and theta, and plot() draws the fitted or residual curves.

Details

The fit has three steps. The memory parameter is estimated by estimate_memory(). The training curves are then centred and fractionally differenced, and the first trim differenced curves are dropped, which gives short-memory curves. Their FARMA orders are selected as in select_order(), and their operators are estimated by the functional Hannan-Rissanen method, as in estimate_operators().

With a validation sample, forecasts up to h steps ahead are made from every validation origin, and their errors calibrate the conformal prediction bands for each horizon and level. A validation origin whose fit fails is skipped with a warning. A warning is also given when the fitted operators are unstable.

References

Kuenzer, T. (2024). Estimation of functional ARMA models. Bernoulli.

Li, D., Robinson, P. M. and Shang, H. L. (2020). Long-range dependent curve time series. Journal of the American Statistical Association.

Examples

sim = sim_farfima(n = 200, d = 0.3, p = 1, q = 0, no_grid = 51, seed = 1)
blocks = split_curves(sim)
fit = farfima(blocks$train, blocks$valid, d = "lpwn2", order = "ic_full",
               p = 0:2, q = 0:2, h = 5, trim = 30)
#> Selecting (p, q) by functional IC over 8 candidates
#> Calibrating the SR, NS, AC bands on 40 validation origins (36 at h = 5)
fit
#> FARFIMA(p = 1, d = 0.2924, q = 0)
#>   Data:      51 grid points x 120 training curves, 40 validation curves
#>   Memory:    LPWN2 on the leading score process of the empirical covariance operator (Covar), bandwidth exponent 0.75
#>   Orders:    functional IC (exhaustive search), 8 candidates
#>   Operators: functional Hannan-Rissanen estimator with 1 principal component, first 30 fractionally differenced curves trimmed
#>   Bands:     SR, NS, AC at 80%, 95% for h = 1..5, calibrated on 40 validation curves (expanding window)