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Neutral Data Fitting 95For each data point (X i , Y i ), the deviation is the x-direction is given by∣ X i − c − bY ∣ i∣∣∣ a ∣ = aX i + bY i − ca ∣ ,whereas the deviation in the y-direction is∣ Y i − c − aX ∣ i∣∣∣ b ∣ = aX i + bY i − cb ∣ .Hence the product of the deviations associated to data point (X i , Y i ) equals ∣ (aX i+bY i −c) 2 ∣ab ∣.Summing over all data points yields the objective function, and we are faced with the followingoptimisation problem:n∑min(aX i + bY i − c) 2∣ ab ∣(3)i=1s.t. a, b, c ∈ R.In the higher dimensional case, i.e. in the presence of variables X i ∈ R k (i = 1, . . . , n), themodel generalizes in the obvious way, cf. also [10] and [6]: The hyperplane we want to fit isthen of the formk∑a j x j = c,j=1deviation of a data point X i from the hyperplane in coordinate direction j amounts to∑ k j=1 a ∣j(X i ) j − c ∣∣∣∣,∣ a jand the optimisation problem becomes( ∑k kn∑j=1 a j(X i ) j − c) min∏ ki=1 ∣j=1 a j∣s.t. a ∈ R k , c ∈ R.(4)Note that the dimension of the optimisation problem is k + 1, not the number n of datapoints. Usually, k + 1 ≪ n in practical applications. In some cases the hyperplane is knownnot to pass through the origin, such that it is possible to set c = 1, thus reducing the dimensionby one.Problem (3) falls into a class of nonlinear optimisation problems called fractional programmingproblems which have received a considerable amount of interest in the last decades(for a survey cf. [8]). While problems involving ratios of linear functions are well understood(see [5] or [9] and references therein), and certain problems involving ratios of linear functionswith absolute values have been treated in [2], to our knowledge the special structure of (3) hasnot yet been studied.References[1] Barker, F., Soh, Y.C., and Evans, R.J. (1988). Properties of the Geometric Mean Functional Relationship. Biometrics44, 279–281.[2] Chadha, S. S. (2002). Fractional Programming with Absolute-value Functions. European Journal of OperationalResearch 141, 233–238.