# 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.

## Description of method

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, of those points. The 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

${\displaystyle S=\sum _{i=1}^{n}{r_{i}(\beta )}^{2}}$,

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

${\displaystyle r_{i}(\beta )=y_{i}-f(x_{i},\beta ),}$

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

${\displaystyle S(\beta )=\sum _{j=1}^{n}(r_{(j)}(\beta ))^{2},}$

while the objective function for LTS is

${\displaystyle S_{k}(\beta )=\sum _{j=1}^{k}(r_{(j)}(\beta ))^{2}.}$

## Computational considerations

Because this method is binary, in that points are either included or excluded, no closed form solution exists. As a result, methods which try to find a LTS solution through a problem sift through combinations of the data, attempting to find the k subset which yields the lowest sum of squared residuals. Methods exist for low n which will find the exact solution, however as n rises, the number of combinations grows rapidly, thus yielding methods which attempt to find approximate (but generally sufficient) solutions.