Least trimmed squares or least trimmed sum of squares, is a robust statistical method

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

 that attempts to fit a function to a set of data whilst not being unduly affected by the presence of outliers. It is one of a number of possible applications of the ideas of robust statistics

Robust statistics

Robust statistics provides an alternative approach to classical statistical methods. The motivation is to produce estimators that are not unduly affected by small departures from model assumptions.- Introduction :...

 to the application of regression analysis

Regression analysis

In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...

.

Description of method

Instead of the standard least squares

Least squares

The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...

 method, which minimises the sum of squared residuals

Errors and residuals in statistics

In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...

 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 object function, S(β), of squared residuals,
where the residuals

Errors and residuals in statistics

In statistics and optimization, statistical errors and residuals are two closely related and easily confused measures of the deviation of a sample from its "theoretical value"...

 are defined as the differences between the values of the dependent variables

Dependent and independent variables

The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects...

(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)|} 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
while the objective function for LTS is

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.