Algorithms for Robustified Error-in-Variables Problems

Abstract

We consider the problem of fitting a model of the form y = f (x, β) to a set of points (x _i , y _i ), i = 1,..., n. If there are measurement or observation errors in x as well as in y, we have the so called errors-in-variables-problem with model equation

$$ {y_i} = f\left( {{x_i} + {\delta _i},\beta } \right) + {\varepsilon _i},\left( {i = 1, \ldots ,n} \right) $$

((1))

where δ _i ∈ ℝ^m, i = 1,..., n are the errors in x _i ∈ ℝ^m. Then the problem is to find a vector of parameters β ∈ ℝ ^p that minimizes the errors ε _i and δ _i in some loss function subject to (1). We will present algorithms using more robust alternatives to the least squares criterion. Figure 1 gives examples where the least squares (L2), the least absolute deviation (L1) and the Huber criteria are used.