R: BACON for Regression or Multivariate Covariance Estimation

BACON {robustX}

R Documentation

BACON for Regression or Multivariate Covariance Estimation

Description

BACON, short for ‘Blocked Adaptive Computationally-Efficient Outlier Nominators’, is a somewhat robust algorithm (set), with an implementation for regression or multivariate covariance estimation.

BACON() applies the multivariate (covariance estimation) algorithm, using mvBACON(x) in any case, and when y is not NULL adds a regression iteration phase, using the auxiliary .lmBACON() function.

Usage

BACON(x, y = NULL, intercept = TRUE,
      m = min(collect * p, n * 0.5),
      init.sel = c("Mahalanobis", "dUniMedian", "random", "manual", "V2"),
      man.sel, init.fraction = 0, collect = 4,
      alpha = 0.05, alphaLM = alpha, maxsteps = 100, verbose = TRUE)

## *Auxiliary* function:
.lmBACON(x, y, intercept = TRUE,
         init.dis, init.fraction = 0, collect = 4,
         alpha = 0.05, maxsteps = 100, verbose = TRUE)

Arguments

`x`	a multivariate matrix of dimension [n x p] considered as containing no missing values.
`y`	the response (n vector) in the case of regression, or `NULL` for the multivariate case, where just `mvBACON()` is returned.
`intercept`	logical indicating if an intercept has to be used for the regression.
`m`	integer in `1:n` specifying the size of the initial basic subset; used only when `init.sel` is not `"manual"`; see `mvBACON`.
`init.sel`	character string, specifying the initial selection mode; see `mvBACON`.
`man.sel`	only when `init.sel == "manual"`, the indices of observations determining the initial basic subset (and `m <- length(man.sel)`).
`init.dis`	the distances of the x matrix used for the initial subset determined by `mvBACON`.
`init.fraction`	if this parameter is > 0 then the tedious steps of selecting the initial subset are skipped and an initial subset of size n * init.fraction is chosen (with smallest dis)
`collect`	numeric factor chosen by the user to define the size of the initial subset (p * collect)
`alpha`	number in `(0, 1)` determining the cutoff value for the Mahalanobis distances (multivariate outlier nomination in `mvBACON()`), or the discrepancies for regression, see `alphaLM`.
`alphaLM`	number in `(0, 1)` where a `1-alphaM` t-quantile is the cutoff for the discrepancies (for regression, `.lmBACON()`); see details.
`maxsteps`	the maximal number of iteration steps (to prevent infinite loops)
`verbose`	logical indicating if messages are printed which trace progress of the algorithm.

Details

Notably about the initial selection mode, init.sel, see its description in the mvBACON arguments list.

The choice of alpha and alphaLM:

Multivariate outlier nomination: see the Details section of mvBACON.
Regression: Let t_r(\alpha) denote the 1-\alpha quantile of the Student t-distribution with r degrees of freedom, where r is the number of elements in the current subset; e.g., t_r(0.05) is the 0.95 quantile. Following Billor et al. (2000), the cutoff value for the discrepancies is defined as t_r(\alpha/(2r + 2)), and they use \alpha=0.05. Note that this is argument alphaLM (defualting to alpha) for BACON().

Value

BACON(x,y,..) (for regression) returns a list with components

`subset`	the observation indices (in `1:n`) denoting a subset of “good” supposedly outlier-free observations.
`tis`	the `t_i(y_m, X_m)` of eq (6) in the reference; the clean “basic subset” in the algorithm is defined the observations `i` with the smallest `\|t_i\|`, and the `t_i` can be regarded as scaled predicted errors.
`mv.dis`	the (final) discrepancies or distances of `mvBACON()`.
`mv.subset`	the “good” subset from `mvBACON()`, used to start the regression iterations.

Note

“BACON” was also chosen in honor of Francis Bacon:

Whoever knows the ways of Nature will more easily notice her deviations; and, on the other hand, whoever knows her deviations will more accurately describe her ways.
Francis Bacon (1620), Novum Organum II 29.

Author(s)

Ueli Oetliker, Swiss Federal Statistical Office, for S-plus 5.1; 25.05.2001; modified six times till 17.6.2001.

Port to R, testing etc, by Martin Maechler. Daniel Weeks (at pitt.edu) proposed a fix to a long standing buglet in GiveTis() computing the t_i, which was further improved Maechler, for robustX version 1.2-3 (Feb. 2019).

Correction of alpha default, from 0.95 to 0.05, by Tobias Schoch, see mvBACON.

References

Billor, N., Hadi, A. S., and Velleman , P. F. (2000). BACON: Blocked Adaptive Computationally-Efficient Outlier Nominators; Computational Statistics and Data Analysis 34, 279–298. doi:10.1016/S0167-9473(99)00101-2

Examples

data(starsCYG, package = "robustbase")
## Plot simple data and fitted lines
plot(starsCYG)
lmST <- lm(log.light ~ log.Te, data = starsCYG)
abline(lmST, col = "gray") # least squares line
str(B.ST <- with(starsCYG,  BACON(x = log.Te, y = log.light)))
## 'subset': A good set of of points (to determine regression):
colB <- adjustcolor(2, 1/2)
points(log.light ~ log.Te, data = starsCYG, subset = B.ST$subset,
       pch = 19, cex = 1.5, col = colB)
## A BACON-derived line:
lmB <- lm(log.light ~ log.Te, data = starsCYG, subset = B.ST$subset)
abline(lmB, col = colB, lwd = 2)

require(robustbase)
(RlmST <- lmrob(log.light ~ log.Te, data = starsCYG))
abline(RlmST, col = "blue")

[Package robustX version 1.2-7 Index]