that the number of measurements contained in the zone [−TLMS, TLMS] is equal to the parameter ν. The measurement residual yν(x) is thus at one edge of the interval [−TLMS, TLMS].
In formulation (2.39), note that the minimization of the median of squared errors corresponds to the minimization of the median of absolute errors.
The proposed mathematical formulation requires the addition of the following optimization variables: (i) a binary variable vector b whose values identify those absolute errors ∣yi( x)∣, which are smaller than or equal to ∣yν( x)∣, and (ii) a variable TLMS whose value is equal to ∣yν(x)∣. Three sets of constraints must also be included. Observe that the symbol TLMS represents a variable to be optimized, not a predefined parameter.
2.5.6 Least Trimmed of Squares
An alternative to the LMS estimator is provided by the estimator that minimizes the sum of the smallest ordered squared errors up to the position ν, the so‐called least trimmed of squares (LTS) estimator [3, 27].
2.5.6.1 LTS General Formulation
The general formulation of the LTS estimator is
(2.41a)
subject to
(2.41b)
(2.41c)
(2.41d)
(2.41e)
Note that the main difference between the LMS and LTS estimators is that the former considers only one squared measurement error in the objective function, whereas the latter takes into consideration about half of the squared measurement errors.
2.5.6.2 LTS Mathematical Programming Formulation
The mathematical programming formulation of the LTS method is
(2.42a)
subject to
(2.42b)
(2.42c)
(2.42d)
(2.42e)
(2.42f)
Again, parameter ν identifies the median and can be computed as [26]
(2.43)
Problem (2.42) minimizes the sum of all squared variables si. Note that the number of optimization variables for the proposed LTS formulation is larger than that of any of the previous formulations.
2.5.7 Least Measurements Rejected
References [28, 29] propose a mathematical programming formulation for the least measurements rejected (LMR) estimator. The underlying idea is to find the largest set of measurements whose errors are within a given tolerance T, i.e. to minimize the number of measurement errors that are out of tolerance. Hereafter, this estimator is denominated as LMR.
2.5.7.1 LMR General Formulation
The general formulation of the LMR estimator is
(2.44a)
subject to
(2.44b)
(2.44c)
where card(Ω) represents the cardinality of set Ω and ΩBM is the set of those measurement errors that are out of tolerance, i.e. with ∣yi(x) ∣ ≥ T.
2.5.7.2 LMR Mathematical Programming Formulation
The mathematical programming formulation for the LMR estimator proposed in [28, 29] is
subject to
(2.45c)
(2.45d)
(2.45e)
In (2.45b), note that each binary variable bi indicates whether or not the ith weighted measurement error (yi(x)) is within the range [−T, T ]. Specifically, the value bi = 0 implies that ∣yi(x ) ∣ ≤ T. On the other hand, the value bi = 1 implies that ∣yi(x) ∣ > T. Since the objective function (2.45a) minimizes the sum of all binary variables bi, the LMR procedure searches the largest set of measurement errors
