COMPUTERS GEOSCIENCES ELSEVIER Computers Geosciences 31 (2005)579-587 www.elsevier.com/locate/cageo Multivariate outlier detection in exploration geochemistry* Peter Filzmoser*,Robert G.Garrett,Clemens Reimann "Institute of Statistics and Probability Theory.Vienna University of Technology.Wiedner Hauptstr.8-10.A-1040 Wien.Austria bGeological Survey of Canada,Natural Resources Canada,601 Booth Street,Ottawa,Ontario,Canada,KIA 0E8 Geological Survey of Norway.N-7491 Trondheim.Norway Received 16 November 2004:accepted 16 November 2004 Abstract A new method for multivariate outlier detection able to distinguish between extreme values of a normal distribution and values originating from a different distribution (outliers)is presented.To facilitate visualising multivariate outliers spatially on a map,the multivariate outlier plot,is introduced.In this plot different symbols refer to a distance measure from the centre of the distribution,taking into account the shape of the distribution,and different colours are used to signify the magnitude of the values for each variable.The method is illustrated using a real geochemical data set from far-northern Europe.It is demonstrated that important processes such as the input of metals from contamination sources and the contribution of sea-salts via marine aerosols to the soil can be identified and separated. C 2004 Elsevier Ltd.All rights reserved. Keywords:Multivariate outliers:Robust statistics:Exploration geochemistry:Background 1.Introduction tances.The definition of an outlier limit or threshold, dividing background data from outliers.has found The detection of data outliers and unusual data much attention in the geochemical literature and to date structures is one of the main tasks in the statistical no universally applicable method of identifying outliers analysis of geochemical data.Traditionally,despite the has been proposed (see discussion in Reimann et al., fact that geochemistry data sets are almost always 2005).In this context,background is defined by the multivariate,outliers are most frequently sought for properties,location and spread,of geochemical samples each single variable in a given data set(Reimann et al., that represent the natural variation of the material being 2005).The search for outliers is usually based on studied in a specific area that are uninfluenced by location and spread of the data.The higher (lower)the extraneous and exotic processes such as those related to analytical result of a sample,the greater is the distance rare rock types,mineral deposit forming processes,or of the observation from the central location of all anthropogenic contamination.In geochemistry,outliers observations;outliers thus,typically,have large dis- are generally observations resulting from a secondary process and not extreme values from the background *Code available from server at http://cran.r-project.org/ distribution.Samples where the analytical values are *Corresponding author.Tel.:+431 58801 10733; derived from a secondary process-be it mineralisation fax:+4315880110799. or contamination-do not need to be especially high E-mail addresses:p.filzmoser@tuwien.ac.at (P.Filzmoser). (or low)in relation to all values of a variable in a data garrett@gsc.NRCan.gc.ca(R.G.Garrett), set,and thus attempts to identify these samples with Clemens.Reimann@ngu.no(C.Reimann). classical univariate methods commonly fail.However, 0098-3004/S-see front matter 2004 Elsevier Ltd.All rights reserved. doi:10.1016j.cageo.2004.11.013
Computers & Geosciences 31 (2005) 579–587 Multivariate outlier detection in exploration geochemistry$ Peter Filzmosera,, Robert G. Garrettb , Clemens Reimannc a Institute of Statistics and Probability Theory, Vienna University of Technology, Wiedner Hauptstr. 8-10, A-1040 Wien, Austria b Geological Survey of Canada, Natural Resources Canada, 601 Booth Street, Ottawa, Ontario, Canada, K1A 0E8 c Geological Survey of Norway, N-7491 Trondheim, Norway Received 16 November 2004; accepted 16 November 2004 Abstract A new method for multivariate outlier detection able to distinguish between extreme values of a normal distribution and values originating from a different distribution (outliers) is presented. To facilitate visualising multivariate outliers spatially on a map, the multivariate outlier plot, is introduced. In this plot different symbols refer to a distance measure from the centre of the distribution, takinginto account the shape of the distribution, and different colours are used to signify the magnitude of the values for each variable. The method is illustrated using a real geochemical data set from far-northern Europe. It is demonstrated that important processes such as the input of metals from contamination sources and the contribution of sea-salts via marine aerosols to the soil can be identified and separated. r 2004 Elsevier Ltd. All rights reserved. Keywords: Multivariate outliers; Robust statistics; Exploration geochemistry; Background 1. Introduction The detection of data outliers and unusual data structures is one of the main tasks in the statistical analysis of geochemical data. Traditionally, despite the fact that geochemistry data sets are almost always multivariate, outliers are most frequently sought for each single variable in a given data set (Reimann et al., 2005). The search for outliers is usually based on location and spread of the data. The higher (lower) the analytical result of a sample, the greater is the distance of the observation from the central location of all observations; outliers thus, typically, have large distances. The definition of an outlier limit or threshold, dividingbackground data from outliers, has found much attention in the geochemical literature and to date no universally applicable method of identifyingoutliers has been proposed (see discussion in Reimann et al., 2005). In this context, background is defined by the properties, location and spread, of geochemical samples that represent the natural variation of the material being studied in a specific area that are uninfluenced by extraneous and exotic processes such as those related to rare rock types, mineral deposit formingprocesses, or anthropogenic contamination. In geochemistry, outliers are generally observations resulting from a secondary process and not extreme values from the background distribution. Samples where the analytical values are derived from a secondary process—be it mineralisation or contamination—do not need to be especially high (or low) in relation to all values of a variable in a data set, and thus attempts to identify these samples with classical univariate methods commonly fail. However, ARTICLE IN PRESS www.elsevier.com/locate/cageo 0098-3004/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2004.11.013 $Code available from server at http://cran.r-project.org/. Correspondingauthor. Tel.: +43 1 58801 10733; fax: +43 1 58801 10799. E-mail addresses: p.filzmoser@tuwien.ac.at (P. Filzmoser), garrett@gsc.NRCan.gc.ca (R.G. Garrett), Clemens.Reimann@ngu.no (C. Reimann)
580 P.Filzmoser et al.Computers Geosciences 31 (2005)579-587 this problem often may be overcome by utilising the measure which takes into account the covariance matrix multivariate nature of most geochemical data sets. is the Mahalanobis distance.For a p-dimensional In the multivariate case not only the distance of an multivariate sample x,...,the Mahalanobis distance observation from the centroid of the data but also the is defined as: shape of the data have to be considered.To illustrate this,two variables with normal distributions having a MD=((i-1)TC-l(1))2 for i=1,....n,(1) defined correlation (Fig.1)are simulated.The estimated where t is the estimated multivariate location and C the central location of each variable is indicated by dashed estimated covariance matrix.Usually,t is the multivariate lines (their intersection marks the multivariate centre or arithmetic mean.the centroid.and C is the sample centroid of the data). covariance matrix.For multivariate normally distributed In the absence of a prior threshold (Rose et al..1979) data the values MD are approximately chi-square a common practice of geochemists is to identify some distributed with p degrees of freedom ()By setting fraction,often 2%,of the data at the upper and lower the (squared)Mahalanobis distance equal to a certain extremes for further investigation.Today this is constant,ie.to a certain quantile of it is possible to achieved by direct estimation of the percentiles and define ellipsoids having the same Mahalanobis distance visual (EDA)inspection of the data.When computers from the centroid (e.g,Gnanadesikan,1977). were not widely available an approximation of the Fig.I illustrates this for the bivariate normally 97.5th percentile was obtained by estimating the mean distributed data.The ellipses correspond to the quantiles and standard deviation (SD)for each variate and 0.25.0.50,0.75 and 0.98 of Points lying on an ellipse computing the value of mean +2SD.The 2%limits thus have the same distance from the centroid.This are indicated by dotted lines on Fig.1.If candidates for distance measure takes the shape of the data cloud into outliers are defined to be observations falling in the account and has potential for more reliably identifying extreme 2%fractions of the univariate data for each extreme values. variable,the rectangle visualised with bold dots sepa- Multivariate outliers can now simply be defined as rates potential outliers from non-outliers.This proce- observations having a large (squared)Mahalanobis dure ignores the elliptical shape of the bivariate data and distance.As noted above for the univariate case.when therefore it is not effective. no prior threshold is available a certain proportion of the The shape and size of multivariate data are quantified data or quantile of the normal distribution is selected for by the covariance matrix.A well-known distance identifying extreme samples for further study.Similarly, in the multivariate case a quantile of the chi-squared 0 distribution (e.g.,the 98%quantile 2:9s)could be considered for this purpose.However,this approach has 90 several shortcomings that will be investigated in this 0 6◆·意归 $°。g paper.The Mahalanobis distances need to be estimated by a robust procedure in order to provide reliable measures for the recognition of outliers.In the geochem- ical context what is required is a reliable estimate of the 0 statistical properties of natural background.Using robust estimates that remove (trim)or downweight extreme 0 values in a population is an effective,if conservative, 8 8 solution.It is conservative to the extent that if there are in o:8 8808 fact no outliers the only consequence is that the true variability (variance-covariance)of the data will be 4 8 o underestimated.Furthermore,by selecting a fixed quan- tile for outlier identification there is no adjustment for ● different sample sizes.To address this situation an adaptive outlier identification method has been devel- -2 0 oped.Finally,the multivariate outlier plot is introduced as Fig.1.Simulated standard normally distributed data with a a helpful tool for the interpretation of multivariate data. predetermined correlation.Dashed lines mark locations (means)of variates,ellipses correspond to 0.25,0.50,0.75 and 0.98 quantiles of chi-squared distribution,and bold dotted lines to 2nd and 98th empirical percentiles for individual variables. 2.The robust distance(RD) Hence,inner rectangular(bold dotted lines)can be considered for univariate outlier recognition,outer ellipse for multivariate The Mahalanobis distance is very sensitive to the outlier identification. presence of outliers (Rousseeuw and Van Zomeren
this problem often may be overcome by utilisingthe multivariate nature of most geochemical data sets. In the multivariate case not only the distance of an observation from the centroid of the data but also the shape of the data have to be considered. To illustrate this, two variables with normal distributions havinga defined correlation (Fig. 1) are simulated. The estimated central location of each variable is indicated by dashed lines (their intersection marks the multivariate centre or centroid of the data). In the absence of a prior threshold (Rose et al., 1979) a common practice of geochemists is to identify some fraction, often 2%, of the data at the upper and lower extremes for further investigation. Today this is achieved by direct estimation of the percentiles and visual (EDA) inspection of the data. When computers were not widely available an approximation of the 97.5th percentile was obtained by estimatingthe mean and standard deviation (SD) for each variate and computingthe value of mean 72SD. The 2% limits are indicated by dotted lines on Fig. 1. If candidates for outliers are defined to be observations fallingin the extreme 2% fractions of the univariate data for each variable, the rectangle visualised with bold dots separates potential outliers from non-outliers. This procedure ignores the elliptical shape of the bivariate data and therefore it is not effective. The shape and size of multivariate data are quantified by the covariance matrix. A well-known distance measure which takes into account the covariance matrix is the Mahalanobis distance. For a p-dimensional multivariate sample x1; ... ; xn the Mahalanobis distance is defined as: MDi :¼ ððxi tÞ TC1 ðxi tÞÞ1=2 for i ¼ 1; ... ; n, (1) where t is the estimated multivariate location and C the estimated covariance matrix. Usually, t is the multivariate arithmetic mean, the centroid, and C is the sample covariance matrix. For multivariate normally distributed data the values MD2 i are approximately chi-square distributed with p degrees of freedom ðw2 pÞ: By setting the (squared) Mahalanobis distance equal to a certain constant, i.e. to a certain quantile of w2 p; it is possible to define ellipsoids havingthe same Mahalanobis distance from the centroid (e.g, Gnanadesikan, 1977). Fig. 1 illustrates this for the bivariate normally distributed data. The ellipses correspond to the quantiles 0.25, 0.50, 0.75 and 0.98 of w2 2: Points lyingon an ellipse thus have the same distance from the centroid. This distance measure takes the shape of the data cloud into account and has potential for more reliably identifying extreme values. Multivariate outliers can now simply be defined as observations havinga large (squared) Mahalanobis distance. As noted above for the univariate case, when no prior threshold is available a certain proportion of the data or quantile of the normal distribution is selected for identifyingextreme samples for further study. Similarly, in the multivariate case a quantile of the chi-squared distribution (e.g., the 98% quantile w2 p;0:98) could be considered for this purpose. However, this approach has several shortcomings that will be investigated in this paper. The Mahalanobis distances need to be estimated by a robust procedure in order to provide reliable measures for the recognition of outliers. In the geochemical context what is required is a reliable estimate of the statistical properties of natural background. Using robust estimates that remove (trim) or downweight extreme values in a population is an effective, if conservative, solution. It is conservative to the extent that if there are in fact no outliers the only consequence is that the true variability (variance–covariance) of the data will be underestimated. Furthermore, by selectinga fixed quantile for outlier identification there is no adjustment for different sample sizes. To address this situation an adaptive outlier identification method has been developed. Finally, the multivariate outlier plot is introduced as a helpful tool for the interpretation of multivariate data. 2. The robust distance (RD) The Mahalanobis distance is very sensitive to the presence of outliers (Rousseeuw and Van Zomeren, ARTICLE IN PRESS -3 -2 -1 0 1 -3 -2 -1 0 1 2 3 2 3 Fig. 1. Simulated standard normally distributed data with a predetermined correlation. Dashed lines mark locations (means) of variates, ellipses correspond to 0.25, 0.50, 0.75 and 0.98 quantiles of chi-squared distribution, and bold dotted lines to 2nd and 98th empirical percentiles for individual variables. Hence, inner rectangular (bold dotted lines) can be considered for univariate outlier recognition, outer ellipse for multivariate outlier identification. 580 P. Filzmoser et al. / Computers & Geosciences 31 (2005) 579–587
P.Filzmoser et al.Computers Geosciences 31 (2005)579-587 581 1990).Single extreme observations,or groups of Classical cor 0.66 Robust cor=0.18 observations,departing from the main data structure can have a severe influence on this distance measure. 000 This is somewhat obscure because the Mahalanobis distance should be able to detect outliers,but the same o60 0°。 outliers can heavily affect the Mahalanobis distance. 0 The reason is the sensitivity of arithmetic mean and 00 sample covariance matrix to outliers (Hampel et al., 4 1986).A solution to this problem is well-known in robust statistics:t and Cin Eq.(1)have to be estimated N in a robust manner.where the expression 'robust'means resistance against the influence of outlying observations. Many robust estimators for location and covariance have been introduced in the literature,for a review see Maronna and Yohai (1998).The minimum covariance determinant (MCD)estimator (Rousseeuw,1985)is probably most frequently used in practice,partly because it is a computationally fast algorithm (Rous- -3 -2 7 seeuw and Van Driessen,1999). log (Be)in Chorizon The MCD estimator is determined by that subset of Fig.2.Scatterplot of log (Be)and log (Sr).Covariance is observations of size h which minimises the determinant visualised by tolerance ellipses.Non-robust estimation (dotted of the sample covariance matrix,computed from only ellipse)leads to a Pearson correlation coefficient of 0.66,robust these h points.The location estimator is the average of procedure (solid ellipse)estimates a Pearson correlation of 0.18 these h points,whereas the scatter estimator is propor- for core population,i.e.weight of 1.identified by MCD tional to their covariance matrix.As a compromise procedure. between robustness and efficiency,a value of h0.75n (n is the sample size)will be employed in this study. through their influence on the classical non-robust The choice of h also determines the robustness of the computation.This influence is also reflected in the estimator.The breakdown value of the MCD estimator resulting correlation coefficients.Whereas the Pearson is approximately (n-h)/n,with h0.75n the break- correlation based on the classical estimates is 0.66.the down is approximately 25%.The breakdown value is robust correlation based on the MCD estimator is only the fraction of outliers that when exceeded will lead to 0.18.The next step would be an appropriate visualisa- completely biased estimates(Hampel et al.,1986). tion of the outliers in a map in order to support the Using robust estimators of location and scatter in the geochemical interpretation of the observations.This will formula for the Mahalanobis distance Eq.(1)leads to be demonstrated later for other examples.The high the so-called robust distances (RDs).Rousseeuw and correlation of Be and Sr in Fig.2 is due to a few samples Van Zomeren (1990)used these RDs for multivariate of soil developed on alkaline rocks that display outlier detection.If the squared RD for an observation unusually high concentrations of both these elements. is larger than,say,it can be declared a candidate The high non-robust correlation coefficient is thus an outlier. inappropriate estimate for the majority of the data as it This procedure is illustrated using real data from the is unduly influenced by true outliers (due to completely Kola project (Reimann et al.,1998).Fig.2 shows the different geology)】 plot of Be and Sr determined in C-horizon soils.Using the arithmetic mean and the sample covariance matrix in Eq.(1)it is possible to construct the ellipse correspond- ing to the squared Mahalanobis distance equal to 72.0.9s. 3.Multivariate outliers or extremes? This ellipse (often called a tolerance ellipse)is visualised as a dotted line in Fig.2.It identifies the extreme In the univariate case,Reimann et al.(2005)pointed members of the bivariate population and its shape out the difference between extremes of a distribution reflects the structure of the covariance matrix.By and true outliers.Outliers are thought to be observa- computing the RDs with the MCD estimator another tions coming from one or more different distributions, tolerance ellipse (solid line in Fig.2)can be constructed and extremes are values that are far away from the using the same quantile,29s.It is clearly apparent that centre but which belong to the same distribution.In an many more points in the upper right of Fig.2 are exploratory univariate data analysis it is convenient to identified as candidate outliers.These outliers cause the start with simply identifying all extreme observations as elongated orientation and shape of the dotted ellipse extreme.It is an important aim of data interpretation to
1990). Single extreme observations, or groups of observations, departingfrom the main data structure can have a severe influence on this distance measure. This is somewhat obscure because the Mahalanobis distance should be able to detect outliers, but the same outliers can heavily affect the Mahalanobis distance. The reason is the sensitivity of arithmetic mean and sample covariance matrix to outliers (Hampel et al., 1986). A solution to this problem is well-known in robust statistics: t and C in Eq. (1) have to be estimated in a robust manner, where the expression ‘robust’ means resistance against the influence of outlying observations. Many robust estimators for location and covariance have been introduced in the literature, for a review see Maronna and Yohai (1998). The minimum covariance determinant (MCD) estimator (Rousseeuw, 1985) is probably most frequently used in practice, partly because it is a computationally fast algorithm (Rousseeuw and Van Driessen, 1999). The MCD estimator is determined by that subset of observations of size h which minimises the determinant of the sample covariance matrix, computed from only these h points. The location estimator is the average of these h points, whereas the scatter estimator is proportional to their covariance matrix. As a compromise between robustness and efficiency, a value of h 0:75n (n is the sample size) will be employed in this study. The choice of h also determines the robustness of the estimator. The breakdown value of the MCD estimator is approximately ðn hÞ=n; with h 0:75n the breakdown is approximately 25%. The breakdown value is the fraction of outliers that when exceeded will lead to completely biased estimates (Hampel et al., 1986). Usingrobust estimators of location and scatter in the formula for the Mahalanobis distance Eq. (1) leads to the so-called robust distances (RDs). Rousseeuw and Van Zomeren (1990) used these RDs for multivariate outlier detection. If the squared RD for an observation is larger than, say, w2 2;0:98; it can be declared a candidate outlier. This procedure is illustrated usingreal data from the Kola project (Reimann et al., 1998). Fig. 2 shows the plot of Be and Sr determined in C-horizon soils. Using the arithmetic mean and the sample covariance matrix in Eq. (1) it is possible to construct the ellipse correspondingto the squared Mahalanobis distance equal to w2 2;0:98: This ellipse (often called a tolerance ellipse) is visualised as a dotted line in Fig. 2. It identifies the extreme members of the bivariate population and its shape reflects the structure of the covariance matrix. By computingthe RDs with the MCD estimator another tolerance ellipse (solid line in Fig. 2) can be constructed usingthe same quantile, w2 2;0:98: It is clearly apparent that many more points in the upper right of Fig. 2 are identified as candidate outliers. These outliers cause the elongated orientation and shape of the dotted ellipse through their influence on the classical non-robust computation. This influence is also reflected in the resultingcorrelation coefficients. Whereas the Pearson correlation based on the classical estimates is 0.66, the robust correlation based on the MCD estimator is only 0.18. The next step would be an appropriate visualisation of the outliers in a map in order to support the geochemical interpretation of the observations. This will be demonstrated later for other examples. The high correlation of Be and Sr in Fig. 2 is due to a few samples of soil developed on alkaline rocks that display unusually high concentrations of both these elements. The high non-robust correlation coefficient is thus an inappropriate estimate for the majority of the data as it is unduly influenced by true outliers (due to completely different geology). 3. Multivariate outliers or extremes? In the univariate case, Reimann et al. (2005) pointed out the difference between extremes of a distribution and true outliers. Outliers are thought to be observations comingfrom one or more different distributions, and extremes are values that are far away from the centre but which belongto the same distribution. In an exploratory univariate data analysis it is convenient to start with simply identifyingall extreme observations as extreme. It is an important aim of data interpretation to ARTICLE IN PRESS -3 -2 -1 0 2 0 2 log (Be) in Chorizon log (Sr) in Chorizon Classical cor = 0.66 Robust cor = 0.18 4 6 1 Fig. 2. Scatterplot of loge(Be) and loge(Sr). Covariance is visualised by tolerance ellipses. Non-robust estimation (dotted ellipse) leads to a Pearson correlation coefficient of 0.66, robust procedure (solid ellipse) estimates a Pearson correlation of 0.18 for core population, i.e. weight of 1, identified by MCD procedure. P. Filzmoser et al. / Computers & Geosciences 31 (2005) 579–587 581
582 P.Filzmoser et al.Computers Geosciences 31 (2005)579-587 identify the different geochemical processes that influence detect outliers.The tails will be defined by =for the data.Only in doing so can the true outliers be a certain small a(e.g.,=0.02),and identified and differentiated from extreme members of the one or more background populations in the data.This pn(=sup(G()-Gn()计 (2) u≥6 distinction should also be made in the multivariate case. In the previous section the assumption of multivariate is considered,where +indicates the positive differences. normality was implicitly used because this led to chi- In this way,p()measures the departure of the square distributed Mahalanobis distances.Also for the empirical from the theoretical distribution only in the RD this assumption was used,at least for the majority tails,defined by the value of p()can be considered as of data (depending on the choice of h for the MCD a measure of outliers in the sample.Gervini (2003)used estimator).Defining outliers by using a fixed threshold this idea as a reweighting step for the robust estimation value (e.g.,:9s)is rather subjective because of multivariate location and scatter.In this way,the efficiency (in terms of statistical precision)of the estimator could be improved considerably. (1)If the data should indeed come from a single P()will not be directly used as a measure of outliers. multivariate normal distribution.the threshold As mentioned in the previous section,the threshold would be infinity because there are no observations should be infinity in case of multivariate normally from a different distribution (only extremes); distributed background data.This means,that if the (2)There is no reason why this fixed threshold should be data are coming from a multivariate normal distribu- appropriate for every data set;and tion,no observation should be declared as an outlier. (3)The threshold has to be adjusted to the sample size Instead.observations with a large RD should be seen as (see Reimann et al.,2005;and simulations below). extremes of the distribution.Therefore a critical value Pr is introduced,which helps to distinguish between A better procedure than using a fixed threshold is to outliers and extremes.The measure of outliers in the adjust the threshold to the data set at hand.Garrett sample is then defined as (1989)used the chi-square plot for this purpose,by plotting the squared Mahalanobis distances(which have 0 ifPm()≤Peit(⑥，n,p, 以a()= Pn(5)if pn(5)>Peru(6.n.p). (3) to be computed on the basis of robust estimations of location and scatter)against the quantiles of the most extreme points are deleted until the remaining points The threshold value is then determined as c()= follow a straight line.The deleted points are the G(1-xn(). identified outliers,the multivariate threshold corre- The critical value porir for distinguishing between sponds to the distance of the closest outlier,the farthest outliers and extremes can be derived by simulation.For background individual,or some intermediate distance. different sample sizes n and different dimensions(num- Alternately,the cube root of the squared Mahalanobis bers of variables)p data from a multivariate normal distances may be plotted against normal quantiles (e.g., distribution are simulated.Then Eq.(2)is applied for Chork.1990).This procedure (Garrett.1989)is not computing the value p()for a fixed value (in the automatic,it needs user interaction and experience on simulations =72:0.9s is used).The procedure is repeated the part of the analyst.Moreover,especially for large 1000 times for every considered value of n and p. data sets,it can be time consuming,and also to some To directly compute the limiting distribution of the extent it is subjective.In the next section a procedure statistic defined by Eq.(2)would be a more elegant way that does not require analyst intervention,is reprodu- for determining the critical value.However,even for cible and therefore objective,and takes the above points, related simpler problems Csorgo and Revesz (1981, (1)-(3),into consideration is introduced. Chapter 5)note that this is analytically extremely difficult and they recommend simulation. The resulting values give an indication of the differences between the theoretical and the empirical 4.Adaptive outlier detection distributions,G(u)-G(u),if the data are sampled from multivariate normal distributions.To be on the safe side. The chi-square plot is useful for visualising the the 95%percentile of the 1000 simulated values can be deviation of the data distribution from multivariate used for every n and p,and these percentiles are shown normality in the tails.This principle is used in the for p=2,4,6,8,10 by different symbols in Fig.3.By following.Let G(u)denote the empirical distribution transforming the x-axis by the inverse of n it can be function of the squared robust distances RD,and let seen that-at least for larger sample size-the points lie G(u)be the distribution function of For multivariate on a line(see Fig.3).The lines in Fig.3 are estimated by normally distributed samples,Gn converges to G. least trimmed sum of squares (LTS)regression (Rous- Therefore the tails of G and G can be compared to seeuw.1984).Using LTS regression the less precise
identify the different geochemical processes that influence the data. Only in doingso can the true outliers be identified and differentiated from extreme members of the one or more background populations in the data. This distinction should also be made in the multivariate case. In the previous section the assumption of multivariate normality was implicitly used because this led to chisquare distributed Mahalanobis distances. Also for the RD this assumption was used, at least for the majority of data (dependingon the choice of h for the MCD estimator). Definingoutliers by usinga fixed threshold value (e.g., w2 p;0:98) is rather subjective because (1) If the data should indeed come from a single multivariate normal distribution, the threshold would be infinity because there are no observations from a different distribution (only extremes); (2) There is no reason why this fixed threshold should be appropriate for every data set; and (3) The threshold has to be adjusted to the sample size (see Reimann et al., 2005; and simulations below). A better procedure than usinga fixed threshold is to adjust the threshold to the data set at hand. Garrett (1989) used the chi-square plot for this purpose, by plottingthe squared Mahalanobis distances (which have to be computed on the basis of robust estimations of location and scatter) against the quantiles of w2 p; the most extreme points are deleted until the remainingpoints follow a straight line. The deleted points are the identified outliers, the multivariate threshold corresponds to the distance of the closest outlier, the farthest background individual, or some intermediate distance. Alternately, the cube root of the squared Mahalanobis distances may be plotted against normal quantiles (e.g., Chork, 1990). This procedure (Garrett, 1989) is not automatic, it needs user interaction and experience on the part of the analyst. Moreover, especially for large data sets, it can be time consuming, and also to some extent it is subjective. In the next section a procedure that does not require analyst intervention, is reproducible and therefore objective, and takes the above points, (1)–(3), into consideration is introduced. 4. Adaptive outlier detection The chi-square plot is useful for visualisingthe deviation of the data distribution from multivariate normality in the tails. This principle is used in the following. Let Gnð Þ u denote the empirical distribution function of the squared robust distances RD2 i ; and let G uð Þ be the distribution function of w2 p: For multivariate normally distributed samples, Gn converges to G. Therefore the tails of Gn and G can be compared to detect outliers. The tails will be defined by d ¼ w2 p;1a for a certain small a (e.g., a ¼ 0:02), and pnð Þ¼ d sup uXd ð Þ G uð Þ Gnð Þ u þ (2) is considered, where + indicates the positive differences. In this way, pnð Þ d measures the departure of the empirical from the theoretical distribution only in the tails, defined by the value of d: pnð Þ d can be considered as a measure of outliers in the sample. Gervini (2003) used this idea as a reweighting step for the robust estimation of multivariate location and scatter. In this way, the efficiency (in terms of statistical precision) of the estimator could be improved considerably. pnð Þ d will not be directly used as a measure of outliers. As mentioned in the previous section, the threshold should be infinity in case of multivariate normally distributed background data. This means, that if the data are comingfrom a multivariate normal distribution, no observation should be declared as an outlier. Instead, observations with a large RD should be seen as extremes of the distribution. Therefore a critical value pcrit is introduced, which helps to distinguish between outliers and extremes. The measure of outliers in the sample is then defined as anðdÞ ¼ 0 if pnðdÞppcritðd; n; pÞ; pnðdÞ if pnðdÞ4pcritðd; n; pÞ: ( (3) The threshold value is then determined as cnð Þ¼ d G1 n ð Þ 1 anð Þ d : The critical value pcrit for distinguishing between outliers and extremes can be derived by simulation. For different sample sizes n and different dimensions (numbers of variables) p data from a multivariate normal distribution are simulated. Then Eq. (2) is applied for computingthe value pnð Þ d for a fixed value d (in the simulations d ¼ w2 p;0:98 is used). The procedure is repeated 1000 times for every considered value of n and p. To directly compute the limitingdistribution of the statistic defined by Eq. (2) would be a more elegant way for determiningthe critical value. However, even for related simpler problems Cso¨rgo+ and Re´ve´sz (1981, Chapter 5) note that this is analytically extremely difficult and they recommend simulation. The resultingvalues give an indication of the differences between the theoretical and the empirical distributions, G uð Þ Gnð Þ u ; if the data are sampled from multivariate normal distributions. To be on the safe side, the 95% percentile of the 1000 simulated values can be used for every n and p, and these percentiles are shown for p ¼ 2; 4, 6, 8, 10 by different symbols in Fig. 3. By transformingthe x-axis by the inverse of ffiffiffi n p it can be seen that—at least for larger sample size—the points lie on a line (see Fig. 3). The lines in Fig. 3 are estimated by least trimmed sum of squares (LTS) regression (Rousseeuw, 1984). UsingLTS regression the less precise ARTICLE IN PRESS 582 P. Filzmoser et al. / Computers & Geosciences 31 (2005) 579–587
P.Filzmoser et al.Computers Geosciences 31(2005)579-587 583 simulation results for smaller sample sizes have less dimension and low sample size.The estimated slopes influence.The slopes of the different lines(the intercept form a linear trend(Fig.6)and the resulting approx- is 0 because for n tending to infinity the difference imative formula is between empirical and theoretical distribution is 0)are shown in Fig.4.The resulting points can again be P(6,np)= 0.252-0.0018p for p>10. (5) approximated by a straight line,which allows definition √m of the critical value as a function of n and p: Peru(o.n,p)= 0.24-0.003p forp≤10. (4) √i 5.Example For larger dimension(p>10)the same procedure can be To test the procedure.data from the Kola project applied.The 95%percentiles of 1000 simulated values for different sample sizes and dimensions are shown in (Reimann et al.,1998)are again used.The objective is to Fig.5.The linear dependency becomes worse for high identify outliers in the O-horizon (organic surface soil) 0.020 p=2 0.012 p=15 -- P=4 p=20 90.015 p=6 0.010 p=30 p=8 -X- p=50 d 0.008 p=70 p=10 0.010 0.006 0.005 0.004 0.002 0.000 0.000 100001000500300200150 100 Sample size 10000 1000 500 300 200 Sample size Fig.3.Simulated critical values according to Eq.(2)for multivariate normal distributions with different sample sizes(x- Fig.5.Simulated critical values analogous to Fig.3.but for axis)and dimensions p.Linear trends for dimensions plotted. higher dimensions (p>10) and increasing sample size,are indicated by lines. 0.230 0.22 0.225 0.20 f0230 复018 0.215 0.16 0.210 0.14 0 2 4 6 8 10 20 30 4050 60 70 Dimension p Dimension p Fig.4.Slopes of lines from Fig.3 plotted against dimension p. Fig.6.Slopes of lines from Fig.5 plotted against dimension p. Line is an estimation of linear trend,and leads to Eg.(4). Line is an estimation of linear trend,and leads to Eq.(5)
simulation results for smaller sample sizes have less influence. The slopes of the different lines (the intercept is 0 because for n tendingto infinity the difference between empirical and theoretical distribution is 0) are shown in Fig. 4. The resultingpoints can again be approximated by a straight line, which allows definition of the critical value as a function of n and p: pcritðd; n; pÞ ¼ 0:24 0:003p ffiffiffi n p for pp10. (4) For larger dimension (p410) the same procedure can be applied. The 95% percentiles of 1000 simulated values for different sample sizes and dimensions are shown in Fig. 5. The linear dependency becomes worse for high dimension and low sample size. The estimated slopes form a linear trend (Fig. 6) and the resultingapproximative formula is pcritðd; n; pÞ ¼ 0:252 0:0018p ffiffiffi n p for p410. (5) 5. Example To test the procedure, data from the Kola project (Reimann et al., 1998) are again used. The objective is to identify outliers in the O-horizon (organic surface soil) ARTICLE IN PRESS Max. difference between distribution functions 10000 1000 500 300 200 150 100 Sample size 0.000 0.005 0.010 0.015 0.020 p=2 p=4 p=6 p=8 p=10 Fig. 3. Simulated critical values according to Eq. (2) for multivariate normal distributions with different sample sizes (xaxis) and dimensions p. Linear trends for dimensions plotted, and increasingsample size, are indicated by lines. 2 8 10 0.210 0.215 0.220 0.225 0.230 Dimension p Slopes from simulation 4 6 Fig. 4. Slopes of lines from Fig. 3 plotted against dimension p. Line is an estimation of linear trend, and leads to Eq. (4). Max. difference between distribution functions 10000 1000 500 300 200 Sample size 0.000 0.002 0.004 0.006 0.008 0.010 0.012 p=15 p=20 p=30 p=50 p=70 Fig. 5. Simulated critical values analogous to Fig. 3, but for higher dimensions (p410). 20 30 40 50 60 70 0.14 0.16 0.18 0.20 0.22 Dimension p Slopes from simulation Fig. 6. Slopes of lines from Fig. 5 plotted against dimension p. Line is an estimation of linear trend, and leads to Eq. (5). P. Filzmoser et al. / Computers & Geosciences 31 (2005) 579–587 583
584 P.Filzmoser et al.Computers Geosciences 31 (2005)579-587 data caused by industrial contamination from Ni- 6.Visualisation of multivariate outliers smelters.A combination of two typical contaminant elements (Co and Cu).three minor contaminants (As. An important issue is the visualisation of multivariate Cd and Pb)and two elements that are not part of the outliers,in the simplest case it is possible to plot them on emission spectrum of the Ni-smelters (Mg and Zn)are a map.On a map,clusters of outliers would indicate that used as a test data set.Magnesium is influenced by a some regions have a completely different data structure second major process in the study area,the steady input than others.Fig.8 shows the multivariate outliers for of marine aerosols near the Arctic coast.This leads to a the above example on such a map,using the symbol build-up of Mg in the O-horizon,and this process can be for outliers.Two clusters of outliers occur in Russia.As detected for more than 100 km inland (Reimann et al., expected,they mark the two large industrial centres at 2000).Thus the test-task is to detect outliers in the Monchegorsk and Nikel with neighbouring Zapoljarnij. seven-dimensional space at the basis of 617 observa- There are a number of outliers in the northwestern, tions.The procedure for adaptive outlier detection is Norwegian part of the region.This is an almost pristine illustrated in Fig.7.The solid line is the distribution area with little industry and a low population density function of Robust squared distances RD on the (see Reimann et al.,1998).At a first glance it is perhaps basis of the MCD estimator are computed,and their surprising to find outliers in this area.The detection of empirical distribution function,G,is represented by outliers due to contamination was the prime objective of small circles.According to Eq.(2)the task is to find the the investigation.However.multivariate outliers are not supremum of the difference between these two functions only observations with high values for every variable. in the tails.With==16.62(dotted line in Fig. more importantly they are observations departing from 7)a supremum of p(5)=0.1026 is obtained.Eq.(4) the dominant data structure.In the case of a data set of gives a critical value per(,n,p)=0.0088,which is contamination-related variables,outliers also could be clearly lower than the above supremum.For this reason observations with very low values for the contamina- it can be assumed that large RD come from at least one tion-related elements,indicating extremely clean (less- different distribution.From Eq.(3)the measure of contaminated)regions.The reality is that Mg is highly outliers is 10.26%.corresponding to 65 outliers.The enriched in marine aerosols and thus enriched in the O- resulting threshold value cn()=18.64 is slightly larger horizon of podzols along the Norwegian coast,and in than 6,and presented in Fig.7 as a dashed line.This new this remote near-pristine area the levels of the contam- threshold value is called the adjusted quantile ination related elements are within normal background ranges or low.Thus the reason for the Norwegian coast outliers is apparent,but Fig.8 makes no distinction between contamination and pristine coastal multivariate 1.0 outliers 0.8 7900000 8o 0.6 00 7800000 anneinwno 0.4 0 88.9 7700000 88 0.2 98%quantile 8g88 点58890。 %80°。 88088 88 Adjusted quantile 7600000 0 0 0.0 839t8wg 86 %g+ 89 0 100 200 300 7500000 。多之色会一 0 o8 09 8o88o0 Ordered squared robust distances 00 888898 06 Fig.7.Adaptive outlier detection rule for Kola O-horizon 7400000 data:In tails of distribution(chosen asand indicated by a 88 88°006e +0 0 dotted line)we search for supremum of positive differences between distribution function ofy(solid line)and empirical 40000 5000060000 70000 80000 distribution function of RD?(small circles).Resulting value is adjusted quantile(dashed line)that separates outliers from non- Fig.8.Map showing regular observations (circles)and outliers. identified multivariate outliers (+)
data caused by industrial contamination from Nismelters. A combination of two typical contaminant elements (Co and Cu), three minor contaminants (As, Cd and Pb) and two elements that are not part of the emission spectrum of the Ni-smelters (Mgand Zn) are used as a test data set. Magnesium is influenced by a second major process in the study area, the steady input of marine aerosols near the Arctic coast. This leads to a build-up of Mgin the O-horizon, and this process can be detected for more than 100 km inland (Reimann et al., 2000). Thus the test-task is to detect outliers in the seven-dimensional space at the basis of 617 observations. The procedure for adaptive outlier detection is illustrated in Fig. 7. The solid line is the distribution function of w2 7: Robust squared distances RD2 i on the basis of the MCD estimator are computed, and their empirical distribution function, Gn; is represented by small circles. Accordingto Eq. (2) the task is to find the supremum of the difference between these two functions in the tails. With d ¼ w2 7;0:98 ¼ 16:62 (dotted line in Fig. 7) a supremum of pnðdÞ ¼ 0:1026 is obtained. Eq. (4) gives a critical value pcritð Þ¼ d; n; p 0:0088; which is clearly lower than the above supremum. For this reason it can be assumed that large RD come from at least one different distribution. From Eq. (3) the measure of outliers is 10.26%, correspondingto 65 outliers. The resultingthreshold value cnð Þ d ¼ 18:64 is slightly larger than d; and presented in Fig. 7 as a dashed line. This new threshold value is called the adjusted quantile. 6. Visualisation of multivariate outliers An important issue is the visualisation of multivariate outliers, in the simplest case it is possible to plot them on a map. On a map, clusters of outliers would indicate that some regions have a completely different data structure than others. Fig. 8 shows the multivariate outliers for the above example on such a map, usingthe symbol + for outliers. Two clusters of outliers occur in Russia. As expected, they mark the two large industrial centres at Monchegorsk and Nikel with neighbouring Zapoljarnij. There are a number of outliers in the northwestern, Norwegian part of the region. This is an almost pristine area with little industry and a low population density (see Reimann et al., 1998). At a first glance it is perhaps surprisingto find outliers in this area. The detection of outliers due to contamination was the prime objective of the investigation. However, multivariate outliers are not only observations with high values for every variable, more importantly they are observations departingfrom the dominant data structure. In the case of a data set of contamination-related variables, outliers also could be observations with very low values for the contamination-related elements, indicatingextremely clean (lesscontaminated) regions. The reality is that Mg is highly enriched in marine aerosols and thus enriched in the Ohorizon of podzols alongthe Norwegian coast, and in this remote near-pristine area the levels of the contamination related elements are within normal background ranges or low. Thus the reason for the Norwegian coast outliers is apparent, but Fig. 8 makes no distinction between contamination and pristine coastal multivariate outliers. ARTICLE IN PRESS 0 100 200 300 0.0 0.2 0.4 0.6 0.8 1.0 Ordered squared robust distances Cumulative probability 98% quantile Adjusted quantile Fig. 7. Adaptive outlier detection rule for Kola O-horizon data: In tails of distribution (chosen as w2 7;0:98 and indicated by a dotted line) we search for supremum of positive differences between distribution function of w2 7 (solid line) and empirical distribution function of RD2 i (small circles). Resultingvalue is adjusted quantile (dashed line) that separates outliers from nonoutliers. 7400000 7500000 7600000 7700000 7800000 7900000 40000 50000 60000 70000 80000 Fig. 8. Map showing regular observations (circles) and identified multivariate outliers (+). 584 P. Filzmoser et al. / Computers & Geosciences 31 (2005) 579–587
P.Filzmoser et al.Computers Geosciences 31 (2005)579-587 585 The above demonstrates the necessity for developing a 7900000 more effective way of visualising multivariate outliers. Firstly,it should be possible to provide a better visualisation of the distribution of the RDs,and 7800000 secondly,it is desirable to distinguish between outliers with extremely low values and outliers having very high values of the variables. 7700000 Both features are fulfilled with the visualisation in Fig.9,the multivariate outlier plot.The simulated two- dimensional data set in Fig.9 represents a background 7600000 and an outlying population.The RDs were computed and-similar to Fig.1-three inner tolerance ellipses (dotted lines)are shown for 0.25,0.5,and 0.75 quantiles 7500000 of 7.The outer ellipse corresponds to the threshold cn()with 6=720.9s of the adaptive outlier detection method.Values in the inner ellipse,which are at the 7400000 centre of the main mass of the data,are represented by a small dot.Observations between the 0.25 and 0.5 40000 50000 60000 70000 80000 tolerance ellipses are shown by a larger dot.Going further outwards,a small circle is used as a symbol,and Fig.10.Multivariate outlier plot with symbols according to the most distant non-outliers are plotted as a small plus. Fig.9 provides an alternative presentation to Fig.8. Finally,multivariate outliers that are outside the outer tolerance ellipse are represented by a large plus. according to the Euclidean distances (dashed lines)of For the second feature,i.e.distinguishing between the scaled observations from the coordinate-wise mini- different types of outliers,a colour (heat)scale that mum,such that all coordinates have the same influence depends on the magnitude of the values for each variable on the symbol colour.This procedure is illustrated in is used.Low values are depicted in blue,and high values Fig.9 for the Euclidean distances of the simulated data. in red.More specifically,the colour scale is chosen Applying the above visualisation technique to the O- horizon soil data gives the multivariate outlier plot in Fig.10.Indeed,the spatial distribution of the RDs becomes much clearer with the different symbols,and the colour scale is very helpful in distinguishing the different types of multivariate outliers.Two outlier clusters are proximal to the industrial centres at Monchegorsk and Nikel.Obviously,high values for most of the variables occur there,and hence give an indication of heavy contamination.The northern region of the investigated area also includes many multivariate outliers,but the symbols are in blue or green.This region is not at all contaminated and exhibits low values of the contaminant elements,and this combined with the input of sea spray (Mg)as a locally important process results in the outliers.The proposed visualisation permits discrimination between these very different families of outliers. -3 2 1 01 2 7.From multivariate back to univariate Fig.9.Preparation for multivariate outlier plot:five different With the help of good visualisation for multivariate symbols are plotted depending on value of RD.Five classes are defined by tolerance ellipses (dotted lines)for chi-squared outliers it is easier to explain their structure and quantiles 0.25,0.5,and 0.75,and outlier threshold of adaptive interpret the geochemical data.To support interpreta- outlier detection method.Colour of symbols varies continu- tion it is useful to visualise the multivariate outliers for ously from smallest to largest values for every variable.Thus. every single variable.Highlighting the multivariate observations lying on one dashed curve have the same colour. outliers on the maps for every single element could
The above demonstrates the necessity for developinga more effective way of visualisingmultivariate outliers. Firstly, it should be possible to provide a better visualisation of the distribution of the RDs, and secondly, it is desirable to distinguish between outliers with extremely low values and outliers havingvery high values of the variables. Both features are fulfilled with the visualisation in Fig. 9, the multivariate outlier plot. The simulated twodimensional data set in Fig. 9 represents a background and an outlyingpopulation. The RDs were computed and—similar to Fig. 1—three inner tolerance ellipses (dotted lines) are shown for 0.25, 0.5, and 0.75 quantiles of w2 2: The outer ellipse corresponds to the threshold cnð Þ d with d ¼ w2 2;0:98 of the adaptive outlier detection method. Values in the inner ellipse, which are at the centre of the main mass of the data, are represented by a small dot. Observations between the 0.25 and 0.5 tolerance ellipses are shown by a larger dot. Going further outwards, a small circle is used as a symbol, and the most distant non-outliers are plotted as a small plus. Finally, multivariate outliers that are outside the outer tolerance ellipse are represented by a large plus. For the second feature, i.e. distinguishing between different types of outliers, a colour (heat) scale that depends on the magnitude of the values for each variable is used. Low values are depicted in blue, and high values in red. More specifically, the colour scale is chosen accordingto the Euclidean distances (dashed lines) of the scaled observations from the coordinate-wise minimum, such that all coordinates have the same influence on the symbol colour. This procedure is illustrated in Fig. 9 for the Euclidean distances of the simulated data. Applyingthe above visualisation technique to the Ohorizon soil data gives the multivariate outlier plot in Fig. 10. Indeed, the spatial distribution of the RDs becomes much clearer with the different symbols, and the colour scale is very helpful in distinguishing the different types of multivariate outliers. Two outlier clusters are proximal to the industrial centres at Monchegorsk and Nikel. Obviously, high values for most of the variables occur there, and hence give an indication of heavy contamination. The northern region of the investigated area also includes many multivariate outliers, but the symbols are in blue or green. This region is not at all contaminated and exhibits low values of the contaminant elements, and this combined with the input of sea spray (Mg) as a locally important process results in the outliers. The proposed visualisation permits discrimination between these very different families of outliers. 7. From multivariate back to univariate With the help of good visualisation for multivariate outliers it is easier to explain their structure and interpret the geochemical data. To support interpretation it is useful to visualise the multivariate outliers for every single variable. Highlighting the multivariate outliers on the maps for every single element could ARTICLE IN PRESS -3 -2 -1 0 -3 -2 -1 0 123 1 2 3 Fig. 9. Preparation for multivariate outlier plot: five different symbols are plotted dependingon value of RD. Five classes are defined by tolerance ellipses (dotted lines) for chi-squared quantiles 0.25, 0.5, and 0.75, and outlier threshold of adaptive outlier detection method. Colour of symbols varies continuously from smallest to largest values for every variable. Thus, observations lyingon one dashed curve have the same colour. 7400000 7500000 7600000 7700000 7800000 7900000 40000 50000 60000 70000 80000 Fig. 10. Multivariate outlier plot with symbols according to Fig. 9 provides an alternative presentation to Fig. 8. P. Filzmoser et al. / Computers & Geosciences 31 (2005) 579–587 585
586 P.Filzmoser et al.Computers Geosciences 31 (2005)579-587 10 data.In the univariate case it is often very difficult to identify data outliers originating from a second or other rare process,rather than extreme values in relation to the underlying data of the more common process(es). Extreme values can be easily detected due to their distance from the core of the data.If they originate from the underlying data they are of little interest to the exploration or environmental geochemist because they will neither identify mineralisation nor contamination. In contrast,in the multivariate case it is necessary also to consider the shape of the data,its structure,in the multivariate space and all the dependencies between the variables.Thus the really interesting data outliers, caused by additional,rare processes,can be easily identified. As Cd Co Cu Mg Pb Zn Not surprisingly the identified multivariate outliers in the test data set consisting of seven variables and 617 Fig.11.Plot of single elements for Kola O-horizon data,with samples are often not the univariate extreme values.In same symbols as used in Fig.10. the context of Fig.1,they are equivalent to the distant off-axis individuals in the middle of the data range,e.g., the individual at (-1,1).The map of the multivariate achieve this.It is possible to use the same symbols as in outliers clearly identifies contaminated sites and those the multivariate outlier plot to provide important information about the structure of these outliers. affected by the input of marine aerosols near the coast as For exploratory investigations,however,it is infor- regionally important processes causing different data mative to have an overview of the position of the outlier populations. multivariate outliers within the distribution of the single Although multivariate outlier identification is impor- elements.To achieve this we can simply plot the values tant for thorough data analysis,the task of interpreta- of the elements and use the same symbols and colours as tion goes beyond that first step as the researcher is also in the multivariate outlier plot.See Fig.11 for the Kola interested in identifying the geochemical processes O-horizon data.All variables are presented as a series of leading to the data structure.A crucial point,however, vertically scaled parallel bars,where the values are is that multivariate outliers are not simply excluded from scattered randomly in the horizontal direction (one- further analysis,but that after applying robust proce- dimensional scatter plot).Since the original values of the dures which reduce the impact of the outliers the outliers variables have very different data ranges,the data were are actually left in the data set.Working in this way first centred and scaled for this presentation by using the permits the outliers to be viewed in the context of the robust multivariate estimates of location and scatter.In main mass of the data,which facilitates an appreciation of their relationship to the core data.In this context,the this way the different variables can be easily compared. This visualisation provides insight into the data struc- data analyst should use a variety of procedures,often ture and quality.As in the multivariate outlier plot,the graphical,to gain as great an insight as possible into the multivariate outliers are presented by large symbols data structure and the controlling processes behind the for every variable.Not unsurprisingly in the light of the observations.For example,since factor analysis (like many other multivariate methods)is based on the previous discussion,the multivariate outliers occur over covariance matrix,a robust estimation of the covariance the complete univariate data ranges.and not only at the extremes.Moreover,extremely low values,e.g.,for Pb, matrix will reduce the effect of (multivariate)outlying which seem to be univariate outliers are not necessarily observations (Chork and Salminen,1993:Reimann multivariate outliers.The explanation can be found by et al..2002)and lead to a data interpretation centred looking at the simulation example.Fig.9.again,where on the dominant process(es).Furthermore,when a the lowest values for the x-axis are not multivariate single dominant process is present the factor loadings outliers but members of the main data structure. may be interpretable in the context of that process. When non-robust procedures are used in the presence of multiple processes factor analysis often behaves more like a cluster analysis procedure.In such cases the factor 8.Conclusions loadings provide little or no information on the internal structure of the processes,but define a framework for An automated method to identify outliers in multi- differentiating between them.Both applications have variate space was developed and demonstrated with real merit,the latter in exploratory data analysis,and the
achieve this. It is possible to use the same symbols as in the multivariate outlier plot to provide important information about the structure of these outliers. For exploratory investigations, however, it is informative to have an overview of the position of the multivariate outliers within the distribution of the single elements. To achieve this we can simply plot the values of the elements and use the same symbols and colours as in the multivariate outlier plot. See Fig. 11 for the Kola O-horizon data. All variables are presented as a series of vertically scaled parallel bars, where the values are scattered randomly in the horizontal direction (onedimensional scatter plot). Since the original values of the variables have very different data ranges, the data were first centred and scaled for this presentation by usingthe robust multivariate estimates of location and scatter. In this way the different variables can be easily compared. This visualisation provides insight into the data structure and quality. As in the multivariate outlier plot, the multivariate outliers are presented by large symbols + for every variable. Not unsurprisingly in the light of the previous discussion, the multivariate outliers occur over the complete univariate data ranges, and not only at the extremes. Moreover, extremely low values, e.g., for Pb, which seem to be univariate outliers are not necessarily multivariate outliers. The explanation can be found by lookingat the simulation example, Fig. 9, again, where the lowest values for the x-axis are not multivariate outliers but members of the main data structure. 8. Conclusions An automated method to identify outliers in multivariate space was developed and demonstrated with real data. In the univariate case it is often very difficult to identify data outliers originating from a second or other rare process, rather than extreme values in relation to the underlyingdata of the more common process(es). Extreme values can be easily detected due to their distance from the core of the data. If they originate from the underlyingdata they are of little interest to the exploration or environmental geochemist because they will neither identify mineralisation nor contamination. In contrast, in the multivariate case it is necessary also to consider the shape of the data, its structure, in the multivariate space and all the dependencies between the variables. Thus the really interestingdata outliers, caused by additional, rare processes, can be easily identified. Not surprisingly the identified multivariate outliers in the test data set consistingof seven variables and 617 samples are often not the univariate extreme values. In the context of Fig. 1, they are equivalent to the distant off-axis individuals in the middle of the data range, e.g., the individual at (1,1). The map of the multivariate outliers clearly identifies contaminated sites and those affected by the input of marine aerosols near the coast as regionally important processes causing different data outlier populations. Although multivariate outlier identification is important for thorough data analysis, the task of interpretation goes beyond that first step as the researcher is also interested in identifyingthe geochemical processes leadingto the data structure. A crucial point, however, is that multivariate outliers are not simply excluded from further analysis, but that after applyingrobust procedures which reduce the impact of the outliers the outliers are actually left in the data set. Workingin this way permits the outliers to be viewed in the context of the main mass of the data, which facilitates an appreciation of their relationship to the core data. In this context, the data analyst should use a variety of procedures, often graphical, to gain as great an insight as possible into the data structure and the controllingprocesses behind the observations. For example, since factor analysis (like many other multivariate methods) is based on the covariance matrix, a robust estimation of the covariance matrix will reduce the effect of (multivariate) outlying observations (Chork and Salminen, 1993; Reimann et al., 2002) and lead to a data interpretation centred on the dominant process(es). Furthermore, when a single dominant process is present the factor loadings may be interpretable in the context of that process. When non-robust procedures are used in the presence of multiple processes factor analysis often behaves more like a cluster analysis procedure. In such cases the factor loadings provide little or no information on the internal structure of the processes, but define a framework for differentiatingbetween them. Both applications have merit, the latter in exploratory data analysis, and the ARTICLE IN PRESS -4 -2 0 10 Centered and scaled data As Cd Co Cu Mg Pb Zn 2 4 6 8 Fig. 11. Plot of single elements for Kola O-horizon data, with same symbols as used in Fig. 10. 586 P. Filzmoser et al. / Computers & Geosciences 31 (2005) 579–587
P.Filzmoser et al.Computers Geosciences 31 (2005)579-587 587 former in more detailed studies.Unfortunately.the Banks.D.(Eds.).Encyclopedia of Statistical Sciences EDA approach is often misused for a detailed process Update,Vol.2.Wiley,New York.NY.pp.589-596. study,leading to questionable conclusions. Reimann,C.Ayras,M.,Chekushin,V.,Bogatyrev,I.,Boyd, We conclude that proper exploratory data analysis R.,Caritat,P.De.,Dutter,R.,Finne,T.E..Halleraker,J.H.. and outlier recognition plays an essential part in the Jager,.Kashulina,G.,Lehto,O.,Niskavaara.H.. interpretation of geochemical data,and we suggest,data Pavlov,V.,Raisanen,M.L.,Strand,T.,Volden,T.,1998 Environmental Geochemical Atlas of the Central Barents from other geoscience and physical science studies. Region.NGU-GTK-CKE Special Publication,Geological The method has been implemented in the free statistical Survey of Norway.Trondheim,Norway 745pp. software package R(see http://cran.r-project.org/).It is Reimann,C.,Banks,D.,Kashulina.G.,2000.Processes available as a contributed package called "mvoutlier", influencing the chemical composition of the O-horizon of and it contains all the programs to the proposed methods podzols along a 500 km north-south profile from the coast and additionally valuable data sets from geochemistry, of the Barents Sea to the Arctic Circle.Geoderma 95. like the Kola data (Reimann et al.,1998)and data from 113-139. Northern Europe(Reimann et al.,2003). Reimann,C..Filzmoser,P..Garrett,R.G.,2002.Factor analysis applied to regional geochemical data:problems and possibilities.Applied Geochemistry 17(2). 185-206. References Reimann,C..Filzmoser,P..Garrett.R.G..2005.Background and threshold:critical comparison of methods of determi- Chork.C.Y..1990.Unmasking multivariate anomalous ob- nation.Science of the Total Environment,in press. servations in exploration geochemical data from sheeted- Reimann,C..Siewers.U.,Tarvainen,T.,Bityukova,L.. vein tin mineralisation near Emmaville,N.S.W.,Australia. Eriksson,J.,Gilucis,A.,Gregorauskiene,V.,Lukashev, Journal of Geochemical Exploration 37(2),205-223. V.K.,Matinian,N.N.,Pasieczna,A.,2003.Agricultural Chork.C.Y.,Salminen,R..1993.Interpreting exploration soils in Northern Europe:a geochemical atlas.Geologisches geochemical data from Outukumpu,Finland:a Jahrbuch,Sonderhefte,Reihe D,Heft SD 5.2003. MVE-robust factor analysis.Journal of Geochemical Schweizerbart'sche Verlagsbuchhandlung. Stuttgart, Exploration 48(1),1-20. Germany,279pp. Csorgo,M.,Revesz,P.,1981.Strong Approximations in Rose.A.W..Hawkes.H.E..Webb.J.S..1979.Geochemistry in Probability and Statistics.Academic Press,New York. Mineral Exploration,second ed.Academic Press,London NY 284pP. 657pp. Garrett,R.G.,1989.The chi-square plot:a tool for multivariate Rousseeuw,P.J.,1984.Least median of squares regression. outlier recognition.Journal of Geochemical Exploration 32 Journal of the American Statistical Association 79 (388). (1/3),319-341. 871-880. Gervini.D..2003.A robust and efficient adaptive reweighted Rousseeuw,P.J.,1985.Multivariate estimation with high estimator of multivariate location and scatter.Journal of breakdown point.In:Grossmann,W.,Pflug,G.,Vincze, Multivariate Analysis 84,116-144. I..Wertz,W.(Eds.),Mathematical Statistics and Applica- Gnanadesikan,R..1977.Methods for the Statistical Data tions,vol.B.Akademiai Kiado,Budapest,Hungary, Analysis of Multivariate Observations.Wiley,New York, pp.283-297. NY 311PP. Rousseeuw,P.J..Van Driessen,K..1999.A fast algorithm for Hampel,F.R.,Ronchetti,E.M..Rousseeuw,P.J.,Stahel,W.. the minimum covariance determinant estimator.Techno- 1986.Robust Statistics.The Approach Based on Influence metrics 41,212-223. Functions.Wiley,New York,NY 502pp. Rousseeuw,P.J.,Van Zomeren,B.C.,1990.Unmasking multi- Maronna,R.A.,Yohai,V.J.,1998.Robust estimation of variate outliers and leverage points.Journal of the Amer- multivariate location and scatter.In:Kotz,S.,Read,C.. ican Statistical Association 85(411),633-651
former in more detailed studies. Unfortunately, the EDA approach is often misused for a detailed process study, leadingto questionable conclusions. We conclude that proper exploratory data analysis and outlier recognition plays an essential part in the interpretation of geochemical data, and we suggest, data from other geoscience and physical science studies. The method has been implemented in the free statistical software package R (see http://cran.r-project.org/). It is available as a contributed package called ‘‘mvoutlier’’, and it contains all the programs to the proposed methods and additionally valuable data sets from geochemistry, like the Kola data (Reimann et al., 1998) and data from Northern Europe (Reimann et al., 2003). References Chork, C.Y., 1990. Unmaskingmultivariate anomalous observations in exploration geochemical data from sheetedvein tin mineralisation near Emmaville, N.S.W., Australia. Journal of Geochemical Exploration 37 (2), 205–223. Chork, C.Y., Salminen, R., 1993. Interpretingexploration geochemical data from Outukumpu, Finland: a MVE-robust factor analysis. Journal of Geochemical Exploration 48 (1), 1–20. Cso¨rgo+, M., Re´ve´sz, P., 1981. StrongApproximations in Probability and Statistics. Academic Press, New York, NY 284pp. Garrett, R.G., 1989. The chi-square plot: a tool for multivariate outlier recognition. Journal of Geochemical Exploration 32 (1/3), 319–341. Gervini, D., 2003. A robust and efficient adaptive reweighted estimator of multivariate location and scatter. Journal of Multivariate Analysis 84, 116–144. Gnanadesikan, R., 1977. Methods for the Statistical Data Analysis of Multivariate Observations. Wiley, New York, NY 311pp. Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W., 1986. Robust Statistics. The Approach Based on Influence Functions. Wiley, New York, NY 502pp. Maronna, R.A., Yohai, V.J., 1998. Robust estimation of multivariate location and scatter. In: Kotz, S., Read, C., Banks, D. (Eds.), Encyclopedia of Statistical Sciences Update, Vol. 2. Wiley, New York, NY, pp. 589–596. Reimann, C., A¨yra¨s, M., Chekushin, V., Bogatyrev, I., Boyd, R., Caritat, P.De., Dutter, R., Finne, T.E., Halleraker, J.H., Jæger, Ø., Kashulina, G., Lehto, O., Niskavaara, H., Pavlov, V., Ra¨isa¨nen, M.L., Strand, T., Volden, T., 1998. Environmental Geochemical Atlas of the Central Barents Region. NGU-GTK-CKE Special Publication, Geological Survey of Norway, Trondheim, Norway 745pp. Reimann, C., Banks, D., Kashulina, G., 2000. Processes influencingthe chemical composition of the O-horizon of podzols alonga 500 km north–south profile from the coast of the Barents Sea to the Arctic Circle. Geoderma 95, 113–139. Reimann, C., Filzmoser, P., Garrett, R.G., 2002. Factor analysis applied to regional geochemical data: problems and possibilities. Applied Geochemistry 17 (2), 185–206. Reimann, C., Filzmoser, P., Garrett, R.G., 2005. Background and threshold: critical comparison of methods of determination. Science of the Total Environment, in press. Reimann, C., Siewers, U., Tarvainen, T., Bityukova, L., Eriksson, J., Gilucis, A., Gregorauskiene, V., Lukashev, V.K., Matinian, N.N., Pasieczna, A., 2003. Agricultural soils in Northern Europe: a geochemical atlas. Geologisches Jahrbuch, Sonderhefte, Reihe D, Heft SD 5, 2003, Schweizerbart’sche Verlagsbuchhandlung, Stuttgart, Germany, 279pp. Rose, A.W., Hawkes, H.E., Webb, J.S., 1979. Geochemistry in Mineral Exploration, second ed. Academic Press, London 657pp. Rousseeuw, P.J., 1984. Least median of squares regression. Journal of the American Statistical Association 79 (388), 871–880. Rousseeuw, P.J., 1985. Multivariate estimation with high breakdown point. In: Grossmann, W., Pflug, G., Vincze, I., Wertz, W. (Eds.), Mathematical Statistics and Applications, vol. B. Akade´miai Kiado´, Budapest, Hungary, pp. 283–297. Rousseeuw, P.J., Van Driessen, K., 1999. A fast algorithm for the minimum covariance determinant estimator. Technometrics 41, 212–223. Rousseeuw, P.J., Van Zomeren, B.C., 1990. Unmaskingmultivariate outliers and leverage points. Journal of the American Statistical Association 85 (411), 633–651. ARTICLE IN PRESS P. Filzmoser et al. / Computers & Geosciences 31 (2005) 579–587 587