Least Trimmed Squares

Least trimmed squares (LTS), or least trimmed sum of squares, is a robust statistical method that fits a function to a set of data whilst not being unduly affected by the presence of outliers. It is one of a number of methods for robust regression.

Instead of the standard least squares method, which minimises the sum of squared residuals over n points, the LTS method attempts to minimise the sum of squared residuals over a subset, ${\displaystyle k}$, of those points. The ${\displaystyle n-k}$ points which are not used do not influence the fit.

In a standard least squares problem, the estimated parameter values, β, are defined to be those values that minimise the objective function, S(β), of squared residuals

where the residuals are defined as the differences between the values of the dependent variables (observations) and the model values

and where n is the overall number of data points. For a least trimmed squares analysis, this objective function is replaced by one constructed in the following way. For a fixed value of β, let ${\displaystyle r_{(j)}(\beta )}$ denote the set of ordered absolute values of the residuals (in increasing order of absolute value). In this notation, the standard sum of squares function is

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