Interval regression: a coefficient identified only up to bounds

tda_ivreg, tda_ivreg1 and tda_ivreg2 are TDA's three variants of interval regression – both response and regressor are interval-valued – and tda_ivls solves the same kind of problem as a system of linear interval equations. All four return bounds on beta rather than a point estimate, so coef() would be misleading; use tda_bounds.

Usage

tda_ivreg(
  formula,
  data,
  method = c("two_step", "heuristic", "exact"),
  max_boxes = NULL,
  max_iter = NULL,
  tol_width = NULL,
  tol_beta = NULL,
  options = list(),
  dir = tempfile("tda"),
  ...
)

tda_ivreg1(
  formula,
  data,
  search_box = list(alpha = c(-10, 10), alpha_radius = c(0, 5), beta = c(-10, 10),
    beta_radius = c(0, 5)),
  max_boxes = 4000,
  max_iter = 4000,
  tol_width = NULL,
  tol_beta = NULL,
  tol_min = 0.001,
  options = list(),
  dir = tempfile("tda"),
  ...
)

tda_ivreg2(
  formula,
  data,
  method = c("direct", "search", "minimizer", "contour"),
  options = list(),
  dir = tempfile("tda"),
  ...
)

tda_ivls(formula, data, options = list(), dir = tempfile("tda"), ...)

Arguments

formula: a formula using iv() for interval-valued variables.
data: a data frame.
method: for tda_ivreg: "two_step" (default, only the first two steps), "heuristic", or "exact" optimization – ivreg's opt=. For tda_ivreg2: "direct" (default, direct calculation), "search" (direct search), "minimizer" (TDA's general minimiser), or "contour" – the same opt= option with a different meaning for each command. The choice changes the fit: on the same data, "exact" converges to a different beta than "two_step" and can exhaust the box search where the default does not.
max_boxes, max_iter, tol_width, tol_beta: for tda_ivreg and tda_ivreg1, the box search's iteration limits and tolerances (nbox=/mxit=/tolbw=/tolf=); tda_control does not apply here, these are specific to the interval-regression box search, not TDA's general minimiser. For tda_ivreg1 the limits default to 4000 each – TDA's 100/100 rarely completes a real search.
options: a named list of further TDA options, passed through.
dir: working directory.
...: passed to tda_run.
search_box: for tda_ivreg1, the domain the box search covers: a named list of c(lower, upper) ranges for alpha, alpha_radius, beta and beta_radius (equivalently a 4x2 matrix, or 8 numbers, four lower then four upper). The solution must lie inside it; widen it if the fitted bounds hug an edge. The radii cannot go below 0.
tol_min: for tda_ivreg1, the certification tolerance (tolfe=): a box is accepted when its inclusion lower bound is within this of the best function value found. See Details for why TDA's default is unusable.

Value

An object of class tda_fit.

Details

Interval-valued data is a lower and an upper bound per case, a different thing from interval censoring, and has no equivalent in base R or in survival. iv(lo, hi) marks such a variable in a formula, the way Surv() marks an episode.

ivreg1 and ivreg2 are global-optimization (branch-and-bound) searches, and each reports its bounds in its own format – ivreg2 a centre and a half-width (“Beta (center-radius): C R”, read as [C - R, C + R]), and ivls a bare “Final bounds” with no label at all.

ivreg1 needs a word of its own. As shipped, the command's search domain is hard-coded – alpha in [1, 2.5], beta in [0, 0.1], evidently the dataset it was being developed against – so for almost any other data the answer is simply not inside the searched box and TDA reports “0 finally accepted boxes”. The command was left mid-study, not broken: on data whose solution lies inside the box, the search certifies cleanly. tda_ivreg1 therefore passes the domain in through the package's sbox= option (a guarded addition to the C; the standalone program is unchanged), always requests the prot= protocol file, and reads the accepted boxes back: boxes holds every box the protocol reports, parameters the hull of the near-optimal ones over (alpha, alpha radius, beta, beta radius), and beta/alpha – what tda_bounds returns – the centre rows of that hull. Certification follows TDA's rule: a box counts when its inclusion lower bound is within tol_min of the best value found. The shipped default of 1e-10 for that tolerance is unreachable (interval arithmetic over-estimates a box's range), which is the second reason the command always seemed to fail; tol_min defaults to 1e-3 here instead.

Examples

set.seed(1)
d <- data.frame(xlo = 1:20)
d$xhi <- d$xlo + 2
d$ylo <- 2 + 0.5 * d$xlo + rnorm(20)
d$yhi <- d$ylo + 1.5
tda_ivreg(iv(ylo, yhi) ~ iv(xlo, xhi), d)
#> Call: tda_ivreg(iv(ylo, yhi) ~ iv(xlo, xhi), d)
#> 
#> Cases: 20 
#> Mean of the response: [8.040524, 8.340524] 
#> Mean of the regressor: [10.5, 12.5] 
#> Variance of the regressor: [24.225, 44.25] 
#> 
#> Bounds on the slope: [0.1868326, 0.8283934]  (outer interval)

# ivreg2: converges readily even where ivreg1 often does not
d2 <- data.frame(xlo = c(1, 2, 3, 4))
d2$xhi <- d2$xlo + 1
d2$ylo <- 2 + 0.5 * d2$xlo
d2$yhi <- d2$ylo + 1
tda_ivreg2(iv(ylo, yhi) ~ iv(xlo, xhi), d2)
#> Call: tda_ivreg2(iv(ylo, yhi) ~ iv(xlo, xhi), d2)
#> 
#> Cases: 4 
#> 
#> Bounds on the slope: [0.5, 0.5]
#> Bounds on the intercept: [0, 0] 

# ivls: the same interval-equations idea, its bound format
tda_ivls(iv(ylo, yhi) ~ iv(xlo, xhi), d2)
#> Call: tda_ivls(iv(ylo, yhi) ~ iv(xlo, xhi), d2)
#> 
#> Cases: 4 
#> 
#> Bounds on the slope: [1.25, 1.25]