Outlier-Resistant L 1 Orthogonal Regression via the Reformulation-Linearization Technique

Assessing the linear relationship between a set of continuous predictors and a continuous response is a well-studied problem in statistics and data mining. L2-based methods such as ordinary least squares and orthogonal regression can be used to determine this relationship. However, both of these methods become impaired when influential values are present. This problem becomes compounded when outliers confound standard diagnostics. This work proposes an L1-norm orthogonal regression method L1OR formulated as a nonconvex optimization problem. Solution strategies for finding globally optimal solutions are presented. Simulation studies are conducted to assess the resistance of the method to outliers and the consistency of the method. The method is also applied to real-world data arising from an environmental science application.


Introduction and Background
Data analysts are often posed with the problem of determining the relationship between several variables and a response variable.The standard technique when all variables are defined on a continuous domain is ordinary least squares regression OLS .When outliers, or unusual observations, are present in data, traditional regression techniques become impaired.Methods such as M-regression M-R use M estimates to reduce the impact of outliers.These methods are not designed for developing errors-in-variables models in which both the predictors and the response have measurement error or are considered random components.An example of such a situation is studying the relationship between pH and alkalinity in freshwater habitats, where both measurements are subject to error.
Orthogonal regression L 2 OR is used when uncertainty is known to be present in both independent and dependent variables.This assumption is in contrast to OLS, where the predictors are assumed to be known with no measurement error.Furthermore, orthogonal regression measures the distances orthogonal to the fitted hyperplane whereas in OLS residuals are measured as the vertical distance of observations to the fitted surface.In traditional orthogonal regression, the sum of distances of points x, y to b T x, y b is maximized, the sum of distances of points x, y to a T x, y a is minimized, and the sum of the magnitudes of a T x, y a is maximized.As noted in the text, each of these distance measures can be modified to incorporate the L 1 norm to derive different results.In this paper, the approach is to maximize the sum of L 1 distances of points x, y to b T x, y b which is illustrated by d 1 d 2 .

Previous Work on Robust Orthogonal Regression
The sensitivity of L 2 OR to outliers has been noted, and other investigators have worked to develop robust methods 1-3 .The work of Zamar 3 includes the use of S and M estimates for orthogonal regression.Späth and Watson 4 introduce a method for incorporating the L 1 norm for measuring distances in orthogonal regression.L 2 OR can be formulated as equivalent to finding the last principal component, or the direction of minimum variation, in principal component analysis PCA .Hence, any robust PCA method can be used for robust orthogonal regression.Two main approaches for robust PCA are 1 to find robust estimates of the covariance matrix in traditional PCA, the principal components are eigenvectors of the covariance matrix and 2 to use a robust measure of dispersion.Research in the former area includes 5-11 .Robust estimates of dispersion in PCA have been investigated in [12][13][14][15][16] ; each of these works is based on a projection pursuit approach.
Our approach is closely related to that developed by Späth and Watson 4 and Kwak 16 by the manner in which we incorporate the L 1 norm into an orthogonal regression procedure.Späth and Watson 4 measure the error of an observation as the L 1 distance to its orthogonal projection onto a fitted hyperplane.Kwak 16 finds successive directions of maximum variation by maximizing the L 1 distance to the L 2 projection of points onto a line.In contrast to these methods, our approach is to directly find the direction of least variation by maximizing the L 1 distance between points and their L 2 projections onto a vector see Figure 1 .Also, the methods presented in 4, 16 guarantee only local minima to the respective optimization problems, while we present a method for deriving globally optimal solutions.These three methods can be viewed as approximating a maximum likelihood estimator MLE for the linear errors-in-variables model with independent errors with a Laplace distribution see 23, 24 .The MLE for such a model corresponds to a hyperplane that minimizes the sum of L 1 projections.Zwanzig 25 considers an L 1 estimator for a nonlinear generalization of the error-in-variables model and shows that under certain assumptions on the error distribution, the estimator is consistent.When applied to the setting of L 1 orthogonal linear regression, the estimator is similar to the approach of Späth and Watson 4 .

Traditional Orthogonal Regression
Suppose we are given observations with continuous predictors and responses x i , y i ∈ R d × R, i 1, . . ., n. L 2 OR seeks to find an orthogonal projection of the data onto a hyperplane such that the sum of the orthogonal distances of the points x i , y i to the hyperplane is minimized.We assume throughout this work that the medians have been subtracted from samples and that the fitted hyperplane passes through the origin.We note that for large values of d, the coordinate-wise median may not be a good estimate of the center of a data cloud see 26 .In L 2 OR, the sum of squared orthogonal distances of x i , y i to the hyperplane defined by b T x, y 0 is minimized.The vector b is normal to the best-fitting hyperplane and is the direction of least variation of the data.Because b is the direction of least variation, the sum of squared distances of observations to their projections along b is maximized.Therefore, we can find b by solving the following optimization problem: The variables are in the vector b ∈ R d 1 .The term b T x i , y i b represents the orthogonal projection of observation i along b in terms of the original coordinates of the data.
In this paper, we present a new outlier-resistant method for orthogonal regression called L 1 OR.The direction of least variation in data is found by maximizing the L 1 distances of observations to their projection points along a vector.The fitted hyperplane is orthogonal to the direction of least variation.The problem is formulated as a nonconvex optimization problem.We describe how globally optimal solutions can be derived based on a reformulation-linearization technique RLT developed by Sherali and Tuncbilek 27 .We present results of applying L 1 OR to simulated data that is contaminated with outliers and compare the results to robust methods for orthogonal regression.The consistency of L 1 OR is assessed using simulated data.L 1 OR is applied to data collected for the evaluation of marine habitats, where uncertainty resides in both the dependent and independent variables.

Finding the Best-Fit Hyperplane
Suppose that instead of maximizing the sum of the squared perpendicular distances of observations to their projection along the direction of least variation, we maximize the sum of the L 1 distances.Using the L 1 metric reduces the impact of outlier observations.
In Figure 1, we illustrate different methods of incorporating the L 1 norm into an orthogonal regression procedure for a two-dimensional example.The fitted hyperplane is Advances in Operations Research defined by its normal vector b, representing an approximation of the direction of least variation in data.The vector a spans the space defined by the fitted hyperplane.Our approach is to maximize the sum of L 1 distances of points onto their projections on b.The L 1 distance of x, y to its L 2 projection on b is given by d 1 d 2 in the figure.The procedure proposed by Späth and Watson 4 minimizes the sum of L 1 distances of points to their L 2 projections in a fitted hyperplane.The distance of x, y to its projection in the fitted subspace is indicated by d 3 d 4 .The procedure introduced by Kwak 16 maximizes the sum of L 1 magnitudes of the projections of points onto the fitted hyperplane.In Figure 1, this magnitude is given by d 5 d 6 .When these three distances are measured using the L 2 norm, the same regression plane is optimal 28 ; however, because the distances in each case are measured using the L 1 norm, the resulting regression planes will not always coincide.The L 1 projection of x, y on to the fitted hyperplane is given by x 1 , y 1 ; an MLE approach would minimize the sum of L 1 distances of points to their L 1 projections.
Maximizing the sum of the L 1 distances of points to a line passing through the origin is written as

2.1
The objective function is nonlinear and nonconvex.As with L 2 OR , the optimal hyperplane is defined by b T x, y 0. Let r ij be the L 1 residual for component j of observation i.Also, let a b 1, where 1 is a vector of 1's, so that all a j variables are nonnegative.This substitution is necessary for our solution method which is explained below.The math program can then be formulated as The quantities 0, 1, and 2 are vectors with each coordinate having the value 0, 1, and 2, respectively.The objective function is now linear, and the first three sets of constraints are defined by nonconvex functions.
To derive globally optimal solutions for L 1 OR , we combine the use of branch-andbound for integer programming with branch-and-bound for the reformulation-linearization technique RLT as described in 27 .Subproblem will refer to a linear mixed-integer program MIP that corresponds to a node in a branch-and-bound tree for the RLT.Each subproblem can be converted to a linear MIP by expressing the conditional constraints as for a sufficiently large constant M.
The following is a summary of RLT applied to L 1 OR .
i Subproblem optimization.Select a subproblem to solve.Each subproblem is a linear MIP that relaxes the nonconvex constraints.If all subproblems are solved, then the incumbent solution is optimal.
ii Check for new bound.If the solution satisfies the original nonconvex constraints, the current solution is feasible.Update the incumbent solution and objective value if appropriate.
iii Fathom.Fathom if 1 the solution satisfies the original constraints, 2 the subproblem is infeasible, or 3 the objective value for the subproblem is less than the incumbent objective value.
iv Branch.Select a variable for branching, creating two subproblems.
A flow-chart detailing the steps in the RLT branch-and-bound process is included in Figure 2.
We now describe the construction of the root subproblem for RLT.For each occurrence of a j a k in the constraints, substitute a new variable A jk into the formulation.Also, add constraints of the form but again replace occurrences of a j a k with A jk .The presence of 0 in the constraints is to reflect the lower bounds on the a j variables; these lower bounds will be changed during the optimization algorithm as described below.The result is a linear MIP that is a relaxation of L 1 OR 27 .
We now describe the branching procedure.The optimal solution to the relaxation is feasible for L 1 OR if A jk a j a k for all j, k.If this condition is not satisfied, then choose a variable a j with A jk / a j a k for some k with current value a j and create two new subproblems.One of the new subproblems will have constraints of the form a j ≤ a j .

2.7
Again, replace all occurrences of a j a k with A jk to create linear constraints.The other new subproblem will have the linearized form of the constraints

2.8
As nodes in the branch-and-bound tree are traversed, the bounds for the a j variables are successively tightened.Sherali and Tuncbilek 27 prove that either the search for optimal solutions terminates with a globally optimal solution in finite steps or else any accumulation point of solutions along an infinite branch of the branch-and-bound tree is a globally optimal solution.

Simulation Studies
In this section, the ability of L 1 OR to resist the effects of two types of outliers is assessed using simulation studies.The approach is compared to L 2 OR and several robust procedures.The consistency of L 1 OR is also assessed using a simulation study.
L 1 OR MIP subproblems are solved using CPLEX 12.1.If provable optimality is not achieved for MIP subproblems after 2 minutes, the best-known integer feasible solution is used.We implemented our own branch-and-bound algorithm for applying RLT in a C program, with a time limit of 7200 CPU seconds for each instance.Problems are solved on machines with 2 × 2.6 GHz Opteron processors and 2 GB RAM.
L 1 OR is compared to a robust approach based on projection pursuit 12 , a τ scalebased orthogonalized Gnanadesikan-Kettenring estimate 29 hereafter τ-OGK , and a method based on PCA-L 1 16 .The projection pursuit approach is applied by using the method for principal component analysis described in 15 .The method is modified for orthogonal regression by taking the last robust principal component as the coefficients of the orthogonal regression hyperplane.We denote this method by ppOR-mad or ppOR-qn, with the suffix indicating the scale function used.The other methods are denoted by τ-OGK and PCA-L 1 .For PCA-L 1 , the initial vector is set to w 0 arg max x i x i 2 see 16 .
L 2 OR and ppOR models are derived using prcomp() and PCAgrid() functions, respectively, called in the R environment for statistical computing 30 .The function PCAgrid() is in the pcaPP 31 library.R code for the τ-OGK estimator was provided by an anonymous referee.We implemented the PCA-L 1 method 16 in a C program.

Vertical Outliers
A simulation study is conducted to assess the ability of L 1 OR to detect linear relationships in bivariate data in the presence of vertical outliers.Vertical outliers have significant variation only in their response-variable values.A simulation design is utilized by varying the number of contaminated observations C and contamination magnitude m .Each method is run on 30 datasets with 100 observations under each treatment condition.For this study, C is varied in the following manner: no contamination, C 0, moderate contamination, C 10, and high contamination, C 25.The magnitude of contamination m is varied as m 1: low contamination, m 10: moderate magnitude, m 50: large magnitude.
The data are sampled in the following manner.
i Generate the uncontaminated data: ii Generate the contaminated data: An example dataset with fitted models generated using m 10 and C 25 is given in Figure 3 a .
To evaluate each method's ability to accurately fit the known underlying model, the following model discrepancy, D, is used: where f is the known model and f is the estimated model.Note that D corresponds to the area between f and f.If the estimated model is close to the true model, then D will be small.For each of the simulations D is computed and recorded.Using these results the average model discrepancy, D, and standard error are computed.
To analyze the simulation, the means and standard deviations of D are computed for each setting of m and C and can be found in Table 1.For all configurations with m ≤ 10, L 1 OR has lower means and standard deviations than all other methods tested, indicating superior performance in resisting outlier contamination for such conditions.For m 50, L 1 OR performs worse than the robust methods with the exception of PCA-L 1 but better than the outlier-sensitive L 2 OR.In the case of extreme contamination C 25, m 50 , L 2 OR and PCA-L 1 are extremely sensitive to outliers as indicated by large values for D. The bestperforming method for this configuration is ppOR-qn.L 1 OR has mean discrepancy that is only 0.34 more than that of ppOR-qn but is at least 1.28 less than the outlier-sensitive methods.Overall, this suggests that L 1 OR performs well when no contamination is present and in the presence of larger levels of contamination, but performance degrades relative to some of the robust methods when the contamination magnitude is very large.

Clustered Leverage Outliers
The ability of L 1 OR to detect linear relationships in bivariate data with outliers is further analyzed with a simulation using datasets with clustered leverage outliers.Clustered leverage outliers in a dataset have very similar values but are far from the rest of the data set.The simulation design varies the number of observations n and the contamination level .For each treatment condition and replication, a dataset is generated without contamination and a companion dataset is generated replacing the first n observations with contaminated data.There are 50 replications for each treatment condition.For this experiment, is varied in the following manner: low contamination: 0.05, moderate contamination: 0.10, and high contamination: 0.25.

Advances in Operations Research
The data are sampled as follows.
ii Generate the contaminated data: x i , y i ∼ N m, 10 −2 I , for i 1, . . ., n .
The covariance matrix Σ is varied across replications.First, a 2 × 2 matrix A is generated such that each entry is sampled from a N 0, 1 distribution.The QR decomposition A QR is calculated.Let B QI sgn R , where • indicates taking the diagonal elements as a vector and sgn • is the vector with the signs of the corresponding elements of a vector.Then Σ is sampled from a Wishart B, 5 .The means m for the contaminated data are generated such that 1 the Mahalanobis distance of m from the distribution N 0, Σ is at least 2χ 2 0.99,2 , 2 min{x i : i 1, . . ., n} ≤ m 1 ≤ max{x i : i 1, . . ., n}, and 3 min{y i : i 1, . . ., n} ≤ m 2 ≤ max{y i : i 1, . . ., n}.
An example dataset with 100 observations and fitted models generated using 0.25 is given in Figure 3 b .
Each method is evaluated based on the similarity of the models fit on the companion uncontaminated and contaminated datasets.The similarity measure S is defined as the absolute value of the inner product where b 1 and b 2 are the vectors of coefficients derived for the uncontaminated and contaminated datasets.The values of S can be between 0 and 1, with larger values indicating that the models are in agreement and that outliers do not affect the estimate.The means across replications and the percentage of instances with S ≥ 0.90 for each value of n and are contained in Table 2.For 0.05, 0.10, the performance of L 1 OR is nearly constant as n is increased.There is a slight degradation in performance for larger values of n, which is likely due to the increased computational complexity of instances see Section 5 .For 0.05, all methods have high mean values for S and high percentages of instances with S ≥ 0.9, including the outlier-sensitive L 2 OR.For 0.10, L 1 OR and all of the robust methods have larger mean values of S than L 2 OR.The ppOR-qn estimator has the most consistent performance across different values of n for 0.10, with mean values of S above 0.94 for each.The L 1 OR estimator has mean values above 0.93 for n ≤ 100, but performance degrades for n 200.The τ-OGK estimator has the highest or second-highest mean values of S for n ≤ 100.For 0.25, the performance of L 1 OR lags the robust methods.For n ≤ 50, the performance is similar to that of L 2 OR.For n ≥ 100, the mean values of S are less than those for L 2 OR.For 0.25, the preferred estimator appears to be ppOR-mad, as it has the highest or second-highest value of S for each n.

Consistency
The consistency of L 1 OR is assessed by performing tests on instances with various sample sizes.Bivariate data x i , y i , i 1, . . ., n are generated such that x i ν i i , where ν i ∼ U −1, 1 and i ∼ Laplace 0, 0.5 , and y i x i ξ i , where ξ i ∼ Laplace 0, 0.5 .The sample sizes tested are  3 c . Figure 4 depicts the standard error of the absolute value of the slope as a function of sample size.As sample size increases, the standard error rapidly approaches zero, indicating that the procedure is consistent.For large sample sizes, L 1 OR should provide good estimates.

An Environmental Example
The pH and alkalinity of the water in which the fish live are known to impact their overall health.Alkalinity is a measure of the ability of a solution to neutralize acids.Researchers expect pH and alkalinity be highly correlated.However, the relationship of the two variables is difficult to estimate in many datasets due to low variation in pH across streams and due to the presence of outliers.The dataset for this example is a subset of values collected across the state of Ohio resulting in 312 observations.Various subsets of this dataset have been considered previously by Norton 17 ,Lipkovich et   20 with varying degrees of success at estimating the relationship between pH and alkalinity.
For the purposes of this work both pH and alkalinity have been normalized.Note that in this data both pH and alkalinity have measurement error, and hence an orthogonal regression method should be used.The same computational settings as in the simulation studies are used for this analysis.
Figure 5 shows the scatter plot of pH versus alkalinity.There appears to be a linear relationship between alkalinity and pH.Also notice the vertical and leverage outliers present in the data.The Pearson correlation coefficient for the relationship between pH and alkalinity is r 0.3366 which is biased down due to the outliers in the data.Furthermore, since the correlation is biased down, extracting a pH-alkalinity component and using that as a predictor would not be prudent.Hence, a regression method is needed that is insensitive to outliers/influential points.With the exception of ppOR-mad, the outlier-insensitive methods all demonstrate resistance to the outliers in the measurements of pH when compared to the outlier-sensitive L 2 OR.The method based on PCA-L 1 produces a model that appears to be least affected by outliers, followed by the τ-OGK estimator and then L 1 OR.
Table 3 shows the summary of regression models for each method.Here the standard errors are bootstrap standard errors based on 100 bootstrap samples.The bootstrap standard errors vary widely across the methods, with L 1 OR having the most stable estimates, as evidenced by smaller standard errors, followed by τ-OGK and PCA-L 1 .With the exception of ppOR-mad, the P values indicate that the relationship between pH and alkalinity is statistically significant.Notice that the P value for the relationship is statistically significant using L 1 OR, τ-OGK, and PCA-L 1 with a P value of less than .00001.The L 1 OR, τ-OGK, and PCA-L 1 estimators appear to be the best choice for this data, with PCA-L 1 producing the best estimate and L 1 OR providing the most stable estimate.
An expansion of this problem is to consider how alkalinity, pH, and habitat measure qualitative habitat evaluation index QHEI impact the index of biotic integrity IBI .QHEI measures the quality of the habitat in which the fish reside 21 .QHEI is determined from the following six measures: stream substrate, in stream cover, channel morphology, riparian and bank condition, pool and riffle quality, and gradient.Here higher values correspond to better habitat quality, and lower values correspond to poorer habitat quality.IBI measures the health of the fish community.Lower values of IBI correspond only to tolerant species present, low community organization, and high proportion of fish with physical anomalies.High values correspond to highly organized fish communities, many intolerant species present, and high diversity among species 22 .The data consist of 312 observations from the same sites.
Orthogonal regression models are fit to the data with IBI as the response and QHEI, pH, and alkalinity as predictors.The method presented by Croux and Haesbroeck 10 hereafter CH is used instead of τ-OGK because of the increased number of variables.The CH method is a robust PCA based on finding eigenvalues of a robust estimate of the covariance matrix.
Table 4 shows the coefficients, bootstrap standard errors, t value, and P values for the regressions using each method.Notice that the L 1 OR estimates for the coefficients for pH and alkalinity are the most stable, as indicated by lower standard errors.The standard errors for the QHEI coefficient estimates are the smallest for each method.The estimate for the QHEI coefficient by CH has the lowest standard error and is the largest positive estimate which seems to agree best with biological expectations.The better the habitat the fish have to live in, the better the health of the fish community.For all methods except for L 2 OR and PCA-L 1 , the coefficients indicate a positive correlation between IBI and pH and a negative correlation between IBI and alkalinity.While none of the variables in any of the regressions are statistically significant, this dataset provides an example of how the regression coefficients from orthogonal regression with outliers may be suspect.

Computation Time
The solution method proposed for L 1 OR is more computationally intensive than the other methods used for comparison in this paper.The alternative methods solve all of the instances used here in less than a few seconds.In this section, we evaluate the computational performance of our implementation of L 1 OR.
Tables 5-8 contain data on the computational performance of L 1 OR in each of the experiments conducted.In each table, the first column s indicates the configuration of the data: for Table 5 the contamination level C and contamination magnitude m; for Table 6 the sample size n, contamination level , and whether the data has outliers; for Table 7 the sample size n; for Table 8 the number of variables d.The second column % Optimal indicates the percentage of instances solved to optimality, meaning that all MIP subproblems solved to optimality and the RLT branch and bound tree are fully explored.The third column Avg.MIPs Solved contains the average number of MIPs solved for each configuration.The fourth column Avg.MIPs Suboptimal contains the average number of MIPs that were not solved to optimality within the 120 CPU second time limit.The fifth column Avg.Time-to-Term.(s)  8 , the RLT branch-and-bound tree is explored in every instance.However, for many of these instances, at least one of the MIP subproblems is not solved to optimality.The solution taken in these instances is therefore not "provably" optimal.All instances with n ≤ 25 are solved to optimality.As n is increased to 50 and larger, fewer instances are solved to optimality.For n ≤ 100, the number of MIP subproblems that are not solved to optimality is less than 5% of the subproblems solved in those instances on average.For n 200 in the simulation for clustered leverage outliers and the consistency experiment, about 10% of the MIPs are not solved to optimality.For the bootstrap simulation with d 4, more than half of the MIPs were not solved to optimality.In the simulation with vertical outliers Table 5 , more instances are solved to optimality when the outlier contamination is larger.In contrast, in the simulation with clustered leverage outliers Table 6 , fewer instances with contamination are solved to optimality, than the companion datasets without contamination.Also, the number of instances solved to optimality seems to decrease as the contamination level increases.
In the simulation with vertical outliers Table 5 , at least one MIP is not solved to optimality in most instances.Except in the case of extreme outlier contamination, L 1 OR performed competitively when compared to robust methods.Also, the standard error for the slope in the consistency experiment Table 7 , the percentage of instances solved to optimality decreases dramatically as n increases, but the standard error for the estimates continues to decrease.For these instances then, the time limit for the MIP subproblems does not appear to hamper the ability to find good solutions.
Only one experiment, the bootstrap simulation with d 4, used data with more than 2 variables.The degradation in computational performance is more dramatic in the shift from d 2 to d 4 in the bootstrap simulation than the degradation observed when n is increased in the bivariate experiments.This phenomenon is likely due to the increase in nonlinear constraints needed to produce the RLT relaxation.

Discussion
This work introduces a new L 1 orthogonal regression technique that is designed to be resistant to outliers.We develop a method for deriving globally optimal solutions for problem instances.Via simulation, the method shows promise for being resistant to outliers.An application to an environmental example further demonstrates that the method produces results which are more resistant to outliers than traditional orthogonal regression and competes with other robust methods.Hence, this method gives data analysts that deal with errors-in-variables data contaminated with outliers a resistant alternative to orthogonal regression.
The computational studies presented here indicate that different robust or outlierresistant methods are suitable in different situations, and there is no clearly superior method.The pcaPP-mad method is among the best performers in the presence of vertical and clustered leverage outliers in simulated data but has perhaps the poorest estimate in the real-world example that contains both types of outliers.PCA-L 1 is among the poorest performers in the presence of vertical and clustered leverage outliers in simulated data but produces some of the best estimates in the real-world analysis.The inconsistency of the results for PCA-L 1 may be due to the dependence of the method on having a good starting point for finding a good local optimal solution.The L 1 OR method presented here performs best with respect to the other methods in the presence of moderate contamination by vertical outliers but suffers in cases of extreme contamination.
Traditional orthogonal regression L 2 OR can be formulated as a special case of PCA.The approach presented in this work for formulation and optimization can potentially be adapted to develop an outlier-resistant method for PCA.An outlier-resistant PCA algorithm would be useful for data analysts that work with contaminated data.Another possible extension is for an outlier-resistant factor analysis procedure for analyzing categorical data.

Figure 1 :
Figure1: Two-dimensional illustration of different methods for incorporating the L 1 norm into orthogonal regression.In traditional orthogonal regression, the sum of distances of points x, y to b T x, y b is maximized, the sum of distances of points x, y to a T x, y a is minimized, and the sum of the magnitudes of a T x, y a is maximized.As noted in the text, each of these distance measures can be modified to incorporate the L 1 norm to derive different results.In this paper, the approach is to maximize the sum of L 1 distances of points x, y to b T x, y b which is illustrated by d 1 d 2 .

Figure 3 :
Figure 3: Examples of data sets used in simulation experiments and fitted models: a a data set with vertical outliers generated using parameters m 10 and C 25, b a data set with clustered leverage outliers with n 100 and generated using 0.25, and c a data set with errors in both variables sampled from a Laplace distribution with n 200.

Figure 4 : 1 Figure 5 :
Figure 4: Plot of the standard error of the absolute value of the slope in a bivariate experiment as a function of sample size n .
25,0,25, 50, 100, 200; data are generated for 100 datasets for each value of n.The rlaplace() function in the R package rmutil 32 is used to sample from the Laplace distribution.An example dataset with 200 observations and the fitted L 1 OR model is given in Figure al. 18 , Noble et al. 19 , and Boone et al.
* Standard errors are bootstrap standard errors based on 100 bootstrap samples.

Table 5 :
Computational performance of L 1 OR implementation for simulation with vertical outliers.

Table 6 :
Computational performance of L 1 OR implementation for simulation with clustered leverage outliers. of the lesser of the CPU seconds before the RLT branch and bound tree is explored and 7200 seconds.The last column Avg.Time to Best Soln.(s) contains the average time to find the best feasible solution.With the exception of the bootstrap samples for the environmental data with d 4 Table

Table 7 :
Computational performance of L 1 OR implementation for consistency experiment.

Table 8 :
Computational performance of L 1 OR implementation for bootstrap simulations.