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FinchModern methods for the detection of multivariate outliersTable 1 | Summary of outlier detection methods.Method Equation Reference Strengths WeaknessesD 2 i (x i − ¯x) ′ S −1 (x i − ¯x) Mahalanobis (1936) Intuitively easy to understand;MVEMCDMGVP1P2P3Identify subset of data contained within theellipsoid that has minimized volumeIdentify subset of data that minimizes thedeterminant of the covariance matrixCalculate MAD version of D 2 as∑ nj=1√∑pl=1(xjl −x ilMAD l) 2to identify most centralpoints; calculate variance of this central set asadditional observations are added one by one;examine this generalized variance and retainthose with√values less than the adjusted medianM G + χ 2 0.975.p (q 3 − q 1 )Identify the multivariate center of data usingMCD or MVE and then determine its relativedistance from this center (depth); use the MGVcriteria based on this depth to identify outliersIdentify all possible lines between all pairs ofobservations in order to determine depth ofeach pointRousseeuw andLeroy (1987)Rousseeuw and vanDriessen (1999)Wilcox (2005)Donoho and Gasko(1992)Donoho and Gasko(1992)Same approach as P1 except√that the criteria for Donoho and Gaskoidentifying outliers is M D + χ 2 0.975.p ( MAD i0.6745 ) (1992)easy to calculate; familiar toother researchersYields mean with maximumpossible breakdown pointYields mean with maximumpossible breakdown pointTypically removes fewerobservations than either MVE orMCDApproximates an affineequivariant outlier detectionmethod; may not exclude asmany cases as MVE or MCDSome evidence that this methodis more accurate than P1 interms of identifying outliersMay yield a mean with a higherbreakdown point than otherprojection methodsSensitive to outliers; assumesdata are continuousMay remove as much as 50%of sampleMay remove as much as 50%of sampleGenerally does not have ashigh a breakdown point asMVE or MCDWill not typically lead to amean with the maximumpossible breakdown pointExtensive computationaltime, particularly for largedatasetsWill likely lead to exclusion ofmore observations as outliersthan will other projectionapproacheswherex i = Vector of scores on the set of p variables for subject i¯x = Vector of sample means on the set of p variablesS = Covariance matrix for the p variablesA number of recommendations exist in the literature for identifyingwhen this value is large; i.e., when an observation might be anoutlier. The approach used here will be to compare Di 2 to the χ 2distribution with p degrees of freedom and declare an observationto be an outlier if its value exceeds the quantile for some inverseprobability; i.e., χ 2 p (0.005) (Mahalanobis).D 2 is easy to compute using existing software and allows fordirect hypothesis testing regarding outlier status (Wilcox, 2005).Despite these advantages, D 2 is sensitive to outliers because it isbased on the sample covariance matrix, S, which is itself sensitiveto outliers (Wilcox, 2005). In addition, D 2 assumes that the dataare continuous and not categorical so that when data are ordinal,for example, it may be inappropriate for outlier detection (Zijlstraet al., 2007). Given these problems, researchers have developedalternatives to multivariate outlier detection that are more robustand more flexible than D 2 .MINIMUM VOLUME ELLIPSOIDOne of the earliest of alternative approach to outlier detection wasthe Minimum Volume Ellipsoid (MVE), developed by Rousseeuwand Leroy (1987). In concept, the goal behind this method is toidentify a subsample of observations of size h (where h < n) thatcreates the smallest volume ellipsoid of data points, based on thevalues of the variables. By definition, this ellipsoid should be free ofoutliers, and estimates of central tendency and dispersion would beobtained using just this subset of observations. The MVE approachto dealing with outliers can, in practice, be all but intractable tocarry out as the number of possible ellipsoids to investigate willtypically be quite large. Therefore, an alternative approach is totake a large number of random samples of size h with replacement,whereh = n 2+ 1, (2)and calculate the volume of the ellipsoids created by each. Thefinal sample to be used in further analyses is that which yieldsthe smallest ellipsoid. An example of such an ellipsoid based onMVE can be seen in Figure 1. The circles represent observationsthat have been retained, while those marked with a star representoutliers that will be removed from the sample for future analyses.MINIMUM COVARIANCE DETERMINANTThe minimum covariance determinant (MCD) approach to outlierdetection is similar to the MVE in that it searches for a portionof the data that eliminates the presence and impact of outliers.However, whereas MVE seeks to do this by minimizing the volumeof an ellipsoid created by the retained points, MCD does it bywww.frontiersin.org July 2012 | Volume 3 | Article 211 | 61
FinchModern methods for the detection of multivariate outliersFIGURE 1 | Scatterplot of observations identified as outliers based onthe MVE method.FIGURE 2 | Scatterplot of observations identified as outliers based onthe MCD method.minimizing the determinant of the covariance matrix, which is anestimate of the generalized variance in a multivariate set of data(Rousseeuw and van Driessen, 1999). The dataset with the smallestdeterminant will be the one least influenced by outliers and whichcan then be used for future statistical analyses. Statistics calculatedon data to which MCD and MVE have been applied will typicallyhave high breakdown points (Rousseeuw and van Driessen, 1999).As with MVE, the logistics of searching every possible subsetof the data of size h to find the one that yields the smallestdeterminant are not practical in the vast majority of situations. Asa consequence Rousseeuw and van Driessen (1999) developed amultiple step algorithm to approximate the MCD, obviating theneed to examine all possible subsets of the data. This approach,known as Fast MCD involves the random selection of an initialsubsample from the data of size h, for which the values of D 2 iarecalculated and ordered from smallest to largest. The h smallest D 2 ivalues (and thus the data points associated with them) are thenretained into a new subset of the data, after which individuals fromthe full dataset are randomly added and the value of the determinantcalculated. The algorithm stops when it attains a subsample(size h) of the full data that yields the smallest determinant. Variantsof this algorithm involve the selection of multiple subsamplesin the initial step, and with several minimization procedures runningparallel to one another simultaneously (Hardin and Rocke,2004). Figure 2 includes a scatterplot identifying individuals asoutliers based on the MCD method. Again, outliers are markedwith a star.MINIMUM GENERALIZED VARIANCEOne potential difficulty with both MVE and MCD is that theytend to identify a relatively large number of outliers when thevariables under examination are not independent of one another(Wilcox, 2005). A third approach for outlier detection that wasdesigned to avoid this problem is the Minimum Generalized Variance(MGV). MGV is based on a similar principle to MCD in thatthe set of data with the smallest overall variance is identified. However,rather than relying on the random addition of observations tothe core data set to be retained, it includes those individuals whoseinclusion increases the generalized variance as little as possible.As with MVE and MCD, MGV is an iterative procedure. In thefirst step the p most centrally located points are identified using anon-parametric estimate of D i which is calculated asD i =wheren∑∑√ p (xjl)−x il 2,MAD (3)lj=1l=1MAD l = MED{[x i − M]}. (4)In other words, MAD, the median absolute deviation, is themedian of the deviations between each individual data point andthe median of the data set, M. The most centrally located observationsare those with the smallest value of D i as calculated above.These points are then placed in a new data set, after which thegeneralized variance associated with adding each of the remainingobservations not originally placed in this new data is calculated.The observation with the smallest generalized variance is thenadded to the new data set. For each data point remaining outside ofthe new data set, the generalized variance is recalculated, accountingfor the new observation that was just added. Once again, theobservation with the lowest generalized variance is then added tothe new data set. This process is repeated until all of the originaldata points are included in the new data set; i.e., the new data setis identical in terms of membership to the old one. However, nowFrontiers in Psychology | Quantitative Psychology and Measurement July 2012 | Volume 3 | Article 211 | 62
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