Databases often show outliers observations, which present a different behavior from the majority. It is important to detect these observations since they affect the data analysis in different ways. In this respect, Uriel Jiménez and Aldás Manzano (2005) point out:

They could mask the data pattern and distort the results, so the conclusions would be completely different without their presence.

Outliers have different causes:

Measurement errors, collection or transcription.

Sometimes the detection of outliers is the first step in statistical analysis. Other times, the outliers need to be removed or downweighted; different causes motivate different procedures. The detection of outliers depends on the type of error (or cause) in the data. In the case of errors in measurement or data entry to the base, it is relatively simple to correct them and it is convenient to eliminate the obvious mistakes. However, a controversial question is: What should we do when the outliers derive from the intrinsic heterogeneity of the data?

In this paper we discuss some multivariate methods for detecting outliers. In multivariate methods, there exist two approaches to identify outliers: those based on the distances of the observations in the data center and those projecting the original data. The projection pursuit methods easily identify atypical observations, and they have the advantage that it is not necessary to know the data distribution. However, the disadvantage of the projection pursuit methods is that there are high requirements in terms of computational load, which increments significantly when there is an increase in the variables considered.

The document is organized as follows. The first section describes two algorithms used to detect outliers (Filzmoser et al., 2008; Peña & Prieto, 2001) based on projection pursuit. The second section compares the methods developed applying them in a group of Argentine companies that listed their shares publicly in the period 2004-2012. In final section, we developed the main conclusions, and we describe research limitations and future research work related to this topic.

Mahalanobis’ distance from the center of the data is the classical multivariate way of identifying outliers observations far from most others.

Let x ₁, x ₂...x _n be a random sample from a normal multivariate distribution N_p (µ,Σ) where µ is the multivariate location vector and Σ the p x p covariance matrix. The distance between the i-th observation x_i and the location µ, weighted by the covariance Σ is using to detect if the observation x_i is an outlier, it is (1).

(1)

The square Mahalanobis distance has Chi-square distribution with p degrees of freedom and the observation . is consider outlier if (by setting the squared Mahalanobis distance equal to certain quantile of Chi-squared distribution it is possible define ellipsoids having the same Mahalanobis distance from the data centre).

When both µ and Σ are unknown it be used estimators, i.e. the vector mean and sample covariance matrix S, to estimate the Mahalanobis distance ,see (2):

(2)

The vector mean and the covariance matrix are affected by outliers, besides the Mahalanobis distance relies on the assumption of normality. Therefore, it is affected by outliers and it does not allow identify sets of outliers (Peña, 2002).

An alternative approach is to use robust location and scale estimators, measures resistance against the influence of outlying observations. Maronna (1976) studied affinely equivariant M-estimators for covariance matrices, and Campbell (1980) proposed using the Mahalanobis distance computed using M-estimators for the mean and covariance matrix.

Nevertheless, the distance method approach presents two difficulties: (i) obtaining a reliable robust location estimator, and (ii) determining and classifying the outliers. It is important to find metric separating outliers from regular observations. Rousseeuw (1985) proposed other distance-based algorithm that computes the ellipsoid with the smallest volume or with the smallest covariance determinant that would include at least half of the data points (minimum covariance determinant, MCD). Because these procedures are based on the minimization of certain nonconvex and nondifferentiable criteria, these estimators are computed by resampling.

Rousseeuw and Driessen (1999) get faster algorithm splitting the problem into smaller subproblems (FAST-MCD algorithm).

Others outlier-detection procedures are basing on projections to identify outliers. The underlying motive of these methods is to find suitable projection of the data in which the outliers are readily apparent and can thus be downweighted to yield a robust estimator.

Gnanadesikan and Kettenring (1972) proposed to search for outliers in the direction of the first principal components: the direction of the maximum variability of the data. Although this method provides a correct solution when the outliers are located close to the directions of the principal components, it may fail to identify outliers in the general case.

From that point of view, Stahel (1981) carry on projection pursuit on the data using random directions. They proposed to compute the weight for the robust estimators from the projections of the data onto some directions. These directions were chosen maximizing distances based on robust location and scale estimators, and the optimal values for the distances could also be used to weight each point in the computation of the robust covariance matrix. Stahel (1981) developed a computer approximation based on direction from random subsamples.

Rousseeuw (1993) proposed selecting . observations from the original sample and computing the orthogonal direction of the hyperplane defined by these observations. The maximum over this finite set of directions is used as an approximation to the exact solution.

The disadvantage of the projection pursuit methods is the form to increase the computational burden with the variable number.

Peña and Prieto (2001) proposed an improve examining only the set of 2. directions that maximize or minimize the kurtosis. A small number of outliers would cause heavy tails and lead to a larger kurtosis coefficient while a large number of outliers would start introducing bimodality and decrease the kurtosis coefficient (Filzmoser et al., 2008).

Projection pursuit methods have a computational time that increase very rapidly in higher dimensions.

Principal components are those orthogonality directions that maximize the variance along each component. It is well-known the method of dimension reduction that seems intuitive to identifying outliers since outliers increase the variance along their respective directions. The outliers appear more visible in principal components space, at least in some direction of maximum variance, than the original data space. Principal components select a small quantity of highly informative components, discarding those are not contribute significant additional information. In this way, the dataset become more computationally tractable without losing a lot of information. Filzmoser et al. (2008) proposed a method based on the principal components properties useful to detect outliers in high dimensions.

In the following sections, we describe the Peña and Prieto (2001) and Filzmoser et al. (2008) methods.

In this paper we applied the detection outliers methods to a sample composed by a set of Argentine companies that quote their shares in the Buenos Aires stock exchange in the period 2004-2012. The database was prepared relying on the data publicly available in the Buenos Aires stock exchange web site (Bolsar, s.f.) including only companies that presented a positive operative ordinary income defined as net sales larger that costs of sales and selling and administrative expenses, and excluding those that belong to the financial and insurance sectors. A total of 744 observations (firms per year) belonging to 111 firms were considered.

The variables used to detect outliers are the following financial reporting indicators to analyse cost-effectiveness and cost behaviour (Anderson, Banker, & Janakiraman, 2003; Banker & Byzalov, 2014).

Market to book value. The ratio of indicates investors’ expectations of future abnormal earnings relative to assets in place. It reflects both the magnitude and persistence of sales growth expectations.

The Table 2 and Figure 1 summarize a preliminary descriptive analysis of original sample. The univariate statistical analysis (descriptive measures and boxplots) shows the skewness and outliers.

Table 2 Source: Own elaboration.

Figure 1 Figure 1 Boxplot of original variables Source: Own elaboration.

We detected outliers using Mahalanobis distance and the both projection pursuit methods presented by Peña and Prieto (2001) and Filzmoser et al. (2008) denominated Kurt and PCOut, respectively. In this work we compare the results defining finally as outliers all the outliers defined using both methods proposed.

We used the Matlab for the Kurt method and the R package mvoutlier for the PCOut method (Filzmoser, 2015).

The Table 3 shows the outliers detected by differences algorithms. There were detected 48 outliers (6.5% of the data) by the distance methods (Mahalanobis). By using projection pursuit methods a large outliers were detected. 212 (28.5%) by Kurt method and 203 (27.3%) by PCOut of outliers. The algorithms proposed detected a similar quantity. Nevertheless the 69.8% of Kurt outliers were PCOut outliers (see Table 4).

Table 3 Table 3 Outliers detected Source: Own elaboration.

Table 4 Table 4 Outliers detected comparing methods Source: Own elaboration.

The biplots (Figure 2) shows the detected outliers using different methods in the space of the first and second principal component.

Figure 2 Figure 2 Biplot showing outliers identified by Methods Source: Own elaboration.

For each method it has been calculated the first and second principal components of the non-outliers data and graphic it on biplots which shows different structures of the data (See figure 3).

Figure 3 Figure 3 Biplot without outliers by Method Source: Own elaboration.

Table 5 shows descriptive measures of the variables considered in non-outliers database, the mean vector difference, the multivariate variability (total and generalized variance) by method. Multivariate variances are significantly lower in all cases because of excluding outliers distorting the estimates.

Table 5 Table 5 Multivariate descriptive without outliers Source: Own elaboration.

A plausible criterion for determining multivariate outliers is to use different methods and consider as those who are simultaneously identified as atypical by them. Particularly, in this application 19.9% of the data were identified as outliers at Kurt and PCOut methods.

The different results and different performance of each method leads us to consider all the outliers detected for these methods. We aggregated the outliers identified by all the methods taking advantage of their performance. I this empirical application, we detected 225 outliers (30.24%). Figure 4 shows in the space of the two first principal components of the sample all the outliers detected for the algorithms.

Figure 4 Figure 4 Outliers detected by all algorithms Source: Own elaboration.

The biplot without the outliers detected by all the methods (Figure 5) shows a better performance. Besides we pointed with a circle a set of data that we could study especially because they presented a different behaviour.

Figure 5 Figure 5 Biplot without all the Outliers detected Source: Own elaboration.

Table 6 and Figure 6 show a descriptive analysis of data without all the Outliers detected. The variables exhibited less skewness and heterogeneity, resulting a data sample more homogeneous.

Table 6 Table 6 Multivariate descriptive without all the Outliers detected Source: Own elaboration.

Figure 6 Figure 6 Boxplott without all the Outliers detected Source: Own elaboration.

Outliers distort the results and mask the real data structure, so to detect them is an important task in the multivariate data analysis.

This work presented two pursuit algorithms to detect outliers, they are an example of the different algorithms available. But is not possible to point one algorithm as the better, it depends of the data sample and the algorithms could show similar performance. Each one method has some disadvantage, so we proposed aggregate the outliers detected by different methods (in this paper Kurt and PCOut methods) and use their different performance to improve the outliers detection. Specially the projection pursuit methods that they only search the useful projections, they are not affecting by non-normality and can be widely applied in diverse data situations.

A multivariate outliers detection is important for thorough data analysis, however, the researchers have to decide to exclude the outliers from further analysis or apply robust procedures to reduce the impact of them.