# BACON: Blocked Adaptive Computationally-Efficient Outlier Nominators This function performs an outlier identification algorithm to the data in the x array [n x p] and y vector [n] following the lines described by Hadi et al. for their BACON outlier procedure. 1.1 ```r mvBACON(x, collect = 4, m = min(collect * p, n * 0.5), alpha = 0.05, init.sel = c("Mahalanobis", "dUniMedian", "random", "manual", "V2"), man.sel, maxsteps = 100, allowSingular = FALSE, verbose = TRUE) ``` ## Arguments - `x`: numeric matrix (of dimension $[n x p]$), not supposed to contain missing values. - `collect`: a multiplication factor $c$, when `init.sel` is not `"manual"`, to define $m$, the size of the initial basic subset, as $c * p$, in practice, `m <- min(p * collect, n/2)`. - `m`: integer in `1:n` specifying the **size** of the initial basic subset; used only when `init.sel` is not `"manual"`. - `alpha`: determines the cutoff value for the Mahalanobis distances (see details). - `init.sel`: character string, specifying the initial selection mode; implemented modes are: - **"Mahalanobis"**: based on Mahalanobis distances (default); the version $V1$ of the reference; affine invariant but not robust. - **"dUniMedian"**: based on the distances from the uni variate medians; similar to the version $V2$ of the reference; robust but not affine invariant. - **"random"**: based on a random selection, i.e., reproducible only via `set.seed()`. - **"manual"**: based on manual selection; in this case, a vector `man.sel` containing the indices of the selected observations must be specified. - **"V2"**: based on the Euclidean norm from the uni variate medians; this is the version $V2$ of the reference; robust but not affine invariant. `"Mahalanobis"` and `"V2"` where proposed by Hadi and the other authors in the reference as versions V_1 and V_2 , as well as `"manual"`, while `"random"` is provided in order to study the behaviour of BACON. Option `"dUniMedian"` is similar to `"V2"` and is due to U. Oetliker. - `man.sel`: only when `init.sel == "manual"`, the indices of observations determining the initial basic subset (and `m <- length(man.sel)`). - `maxsteps`: maximal number of iteration steps. - `allowSingular`: logical indicating a solution should be sought also when no matrix of rank $p$ is found. - `verbose`: logical indicating if messages are printed which trace progress of the algorithm. ## Details Remarks on the tuning parameter `alpha`: Let $\chi^2_p$ be a chi-square distributed random variable with $p$ degrees of freedom ($p$ is the number of variables; $n$ is the number of observations). Denote the $(1-\alpha)$ quantile by $\chi^2_p(\alpha)$, e.g., $\chi^2_p(0.05)$ is the 0.95 quantile. Following Billor et al. (2000), the cutoff value for the Mahalanobis distances is defined as $\chi_p(\alpha/n)$ (the square root of $chi^2_p$) times a correction factor $c(n,p)$, $n$ and $p$, and they use $\alpha=0.05$. ## Returns a `list` with components - **subset**: logical vector of length `n` where the `i`-th entry is true iff the i-th observation is part of the final selection. - **dis**: numeric vector of length `n` with the (Mahalanobis) distances. - **cov**: $p x p$ matrix, the corresponding robust estimate of covariance. ## 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. tools:::Rd_expr_doi("10.1016/S0167-9473(99)00101-2") ## Author(s) Ueli Oetliker, Swiss Federal Statistical Office, for S-plus 5.1. Port to , testing etc, by Martin Maechler; Init selection `"V2"` and correction of default `alpha` from 0.95 to 0.05, by Tobias Schoch, FHNW Olten, Switzerland. ## See Also `covMcd` for a high-breakdown (but more computer intensive) method; `BACON` for a generalization , notably to **regression**. ## Examples ```r require(robustbase) # for example data and covMcd(): ## simple 2D example : plot(starsCYG, main = "starsCYG data (n=47)") B.st <- mvBACON(starsCYG) points(starsCYG[ ! B.st$subset,], pch = 4, col = 2, cex = 1.5) stopifnot(identical(which(!B.st$subset), c(7L,11L,20L,30L,34L))) ## finds the 4 clear outliers (and 1 "borderline"); ## it does not find obs. 14 which is an outlier according to covMcd(.) iniS <- setNames(, eval(formals(mvBACON)$init.sel)) # all initialization methods, incl "random" set.seed(123) Bs.st <- lapply(iniS[iniS != "manual"], function(s) mvBACON(as.matrix(starsCYG), init.sel = s, verbose=FALSE)) ii <- - match("steps", names(Bs.st[[1]])) Bs.s1 <- lapply(Bs.st, `[`, ii) stopifnot(exprs = { length(Bs.s1) >= 4 length(unique(Bs.s1)) == 1 # all 4 methods give the same }) ## Example where "dUniMedian" and "V2" differ : data(pulpfiber, package="robustbase") dU.plp <- mvBACON(as.matrix(pulpfiber), init.sel = "dUniMedian") V2.plp <- mvBACON(as.matrix(pulpfiber), init.sel = "V2") (oU <- which(! dU.plp$subset)) (o2 <- which(! V2.plp$subset)) stopifnot(setdiff(o2, oU) %in% c(57L,58L,59L,62L)) ## and 57, 58, 59, and 62 *are* outliers according to covMcd(.) ## 'coleman' from pkg 'robustbase' coleman.x <- data.matrix(coleman[, 1:6]) Cc <- covMcd (coleman.x) # truly robust summary(Cc) # -> 6 outliers (1,3,10,12,17,18) Cb1 <- mvBACON(coleman.x) ##-> subset is all TRUE hmm?? Cb2 <- mvBACON(coleman.x, init.sel = "dUniMedian") stopifnot(all.equal(Cb1, Cb2)) ## try 20 different random starts: Cb.r <- lapply(1:20, function(i) { set.seed(i) mvBACON(coleman.x, init.sel="random", verbose=FALSE) }) nm <- names(Cb.r[[1]]); nm <- nm[nm != "steps"] all(eqC <- sapply(Cb.r[-1], function(CC) all.equal(CC[nm], Cb.r[[1]][nm]))) # TRUE ## --> BACON always breaks down, i.e., does not see the outliers here ## breaks down even when manually starting with all the non-outliers: Cb.man <- mvBACON(coleman.x, init.sel = "manual", man.sel = setdiff(1:20, c(1,3,10,12,17,18))) which( ! Cb.man$subset) # the outliers according to mvBACON : _none_ ```

mvBACON function