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[1] . 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
n-k
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:
S=
| n | |
| \sum | |
| i=1 |
| 2, | |
| r | |
| i(\beta) |
ri(\beta)=yi-f(xi,\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
r(j)(\beta)
S(\beta)=
| n | |
| \sum | |
| j=1 |
r(j)(\beta)2,
Sk(\beta)=
| k | |
| \sum | |
| j=1 |
r(j)(\beta)2.
Because this method is binary, in that points are either included or excluded, no closed-form solution exists. As a result, methods for finding the LTS solution sift through combinations of the data, attempting to find the k subset that yields the lowest sum of squared residuals. Methods exist for low n that will find the exact solution; however, as n rises, the number of combinations grows rapidly, thus yielding methods that attempt to find approximate (but generally sufficient) solutions.