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. 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, k {displaystyle k} , of those points. The unused n − k {displaystyle n-k} points 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 r ( j ) ( β ) {displaystyle r_{(j)}(eta )} 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