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

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

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 $r_{(j)}(\beta)$ 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

S(\beta)=\sum_{j=1}^n (r_{(j)}(\beta))^2,

while the objective function for LTS is

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.

References

Rousseeuw, P. J. (1984). "Least Median of Squares Regression". Journal of the American Statistical Association. 79: 871–880. doi:10.1080/01621459.1984.10477105. JSTOR 2288718.
Rousseeuw, P. J.; Leroy, A. M. (2005) [1987]. Robust Regression and Outlier Detection. Wiley. doi:10.1002/0471725382. ISBN 978-0-471-85233-9.
Li, L. M. (2005). "An algorithm for computing exact least-trimmed squares estimate of simple linear regression with constraints". Computational Statistics & Data Analysis. 48 (4): 717–734. doi:10.1016/j.csda.2004.04.003.
Atkinson, A. C.; Cheng, T.-C. (1999). "Computing least trimmed squares regression with the forward search". Statistics and Computing. 9 (4): 251–263. doi:10.1023/A:1008942604045.
Jung, Kang-Mo (2007). "Least Trimmed Squares Estimator in the Errors-in-Variables Model". Journal of Applied Statistics. 34 (3): 331–338. doi:10.1080/02664760601004973.

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