# BoxPlot based outlier detection Identifies univariate outliers by using methods based on BoxPlots ```r boxB(x, k=1.5, method='asymmetric', weights=NULL, id=NULL, exclude=NA, logt=FALSE) ``` ## Arguments - `x`: Numeric vector that will be searched for outliers. - `k`: Nonnegative constant that determines the extension of the 'whiskers'. Commonly used values are 1.5 (default), 2, or 3. Note that when `method="adjbox"` then `k` is set automatically equal to 1.5 - `method`: Character, identifies the method to be used: `method="resistant"` provides the `standard' boxplot fences; `method="asymmetric"` is a modification of standard method to deal with (moderately) skewed data; `method="adjbox"` uses Hubert and Vandervieren (2008) adjusted boxplot for skewed distributions. - `weights`: Optional numeric vector with units' weights associated to the observations in `x`. Only nonnegative weights are allowed. Weights are used in estimating the quartiles (see Details). - `id`: Optional vector with identifiers of units in `x`. If missing (`id=NULL`, default) the identifiers will be set equal to the positions in the vector (i.e. `id=1:length(x)`). - `exclude`: Values of `x` that will be excluded by the analysis. By default missing values are excluded (`exclude=NA`). - `logt`: Logical, if `TRUE`, before searching outliers the `x` variable is log-transformed (`log(x+1)` is considered). In this case the summary outputs (bounds, etc.) will refer to the log-transformed `x` ## Details When `method="resistant"` the outlying observations are those outside the interval: $$ [Q_1 - k \times IQR;\quad Q_3 + k \times IQR][Q1 - k*IQR; Q3 + k*IQR] $$ where $Q1$ and $Q3$ are respectively the 1st and the 3rd quartile of `x`, while $IQR=(Q3-Q1)$ is the Inter-Quartile Range. The value $k=1.5$ (said 'inner fences') is commonly used when drawing a boxplot. Values $k=2$ and $k=3$ provide middle and outer fences, respectively. When `method="asymmetric"` the outlying observations are those outside the interval: $$ [Q_1 - 2k \times (Q_2-Q_1);\quad Q_3 + 2k \times (Q_3-Q_2)][Q1 - 2k*(Q2-Q1); Q3 + 2k*(Q3-Q2)] $$ being $Q2$ the median; such a modification allows to account for slight skewness of the distribution. Finally, when `method="adjbox"` the outlying observations are identified using the method proposed by Hubert and Vandervieren (2008) and based on the Medcouple measure of skewness; in practice the bounds are: $$ [Q_1-1.5 \times e^{aM} \times IQR;\quad Q_3+1.5 \times e^{bM}\times IQR ][Q1 - 1.5*exp(a*M)*IQR; Q3 + 1.5*exp(b*M)*IQR] $$ Where M is the medcouple; when $M>0$ (positive skewness) then $a=-4$ and $b=3$; on the contrary $a = -3$ and $b = 4$ for negative skewness ($M < 0$). This adjustment of the boxplot, according to Hubert and Vandervieren (2008), works with moderate skewness ($-0.6 <= M <= 0.6$). The bounds of the adjusted boxplot are derived by applying the function `adjboxStats` in the package `robustbase`. When weights are available (passed via the argument `weights`) then they are used in the computation of the quartiles. In particular, the quartiles are derived using the function `wtd.quantile` in the package `Hmisc`. Remember that when asking a log transformation (argument `logt=TRUE`) all the estimates (quartiles, etc.) will refer to $log(x+1)$. ## Returns The output is a list containing the following components: - **quartiles**: The quartiles of `x` after discarding the values in the `exclude` argument. When weights are provided they are used in quartiles estimation trough the function `wtd.quantile` in the package `Hmisc`. - **fences**: The bounds of the interval, values outside the interval are detected as outliers. - **excluded**: The identifiers or positions (when `id=NULL`) of units in `x` excluded by the computations, according to the argument `exclude`. - **outliers**: The identifiers or positions (when `id=NULL`) of units in `x` detected as outliers. - **lowOutl**: The identifiers or positions (when `id=NULL`) of units in `x` detected as outliers in the lower tail of the distribution. - **upOutl**: The identifiers or positions (when `id=NULL`) of units in `x` detected as outliers in the upper tail of the distribution. ## References McGill, R., Tukey, J. W. and Larsen, W. A. (1978) `Variations of box plots'. **The American Statistician**, 32, pp. 12-16. Hubert, M., and Vandervieren, E. (2008) `An Adjusted Boxplot for Skewed Distributions', **Computational Statistics and Data Analysis**, 52, pp. 5186-5201. ## Author(s) Marcello D'Orazio mdo.statmatch@gmail.com ## See Also `adjboxStats`, `wtd.quantile` ## Examples ```r set.seed(321) x <- rnorm(30, 50, 10) x[10] <- 1 x[20] <- 100 out <- boxB(x = x, k = 1.5, method = 'asymmetric') out$fences out$outliers x[out$outliers] out <- boxB(x = x, k = 1.5, method = 'adjbox') out$fences out$outliers x[out$outliers] x[24] <- NA x.ids <- paste0('obs',1:30) out <- boxB(x = x, k = 1.5, method = 'adjbox', id = x.ids) out$excluded out$fences out$outliers set.seed(111) w <- round(runif(n = 30, min=1, max=10)) out <- boxB(x = x, k = 1.5, method = 'adjbox', id = x.ids, weights = w) out$excluded out$fences out$outliers ```

boxB function