Least Trimmed Squares (LTS) Regression
Also known as: LTS, least trimmed squares regression, trimmed least squares, robust regression, En Küçük Kırpılmış Kareler (LTS)
Least Trimmed Squares is a robust linear regression method introduced by Peter J. Rousseeuw in 1984. Instead of fitting all residuals, it estimates the coefficients by minimising the sum of only the h smallest squared residuals, which gives it a breakdown point of up to 50% and reliable estimates on data heavily contaminated by outliers.
Key highlights
- Very high robustness: a breakdown point of up to 50%, so estimates stay reliable even when nearly half the data are outliers.
- Resistant to extreme observations that would dominate an ordinary least squares fit.
- Deterministic, well-studied estimator with fast algorithms available for large data sets.
Intuition
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How it works
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When to use it
Use LTS when fitting a linear model to a continuous outcome on cross-sectional or longitudinal data that may contain a substantial share of outliers — up to roughly 50% contamination can be tolerated. It assumes the underlying linear model is valid for the clean majority of the data. A reasonable sample size is needed (at least about 30 observations); below 20 the subset selection becomes unreliable.
Strengths & limitations
- Very high robustness: a breakdown point of up to 50%, so estimates stay reliable even when nearly half the data are outliers.
- Resistant to extreme observations that would dominate an ordinary least squares fit.
- Deterministic, well-studied estimator with fast algorithms available for large data sets.
- Needs a reasonable sample size; with fewer than about 20 observations the subset selection becomes unreliable.
- Less statistically efficient than ordinary least squares when the data are actually clean and well behaved.
- The trimming parameter (h, via alpha) must be chosen with care, especially when the outlier share is very high.
Common pitfalls
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Applications
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Frequently asked
What does the breakdown point of 50% mean?
The breakdown point is the largest fraction of the data that can be replaced by arbitrary outliers before the estimator can be forced to give a meaningless answer. LTS tolerates up to about 50% contamination, which is the highest possible for a regression estimator.
How is LTS different from ordinary least squares?
OLS minimises the sum of all squared residuals, so a single extreme point can dominate the fit. LTS minimises only the sum of the h smallest squared residuals, ignoring the worst-fitting observations, which makes it resistant to outliers at the cost of some efficiency on clean data.
What is the trimming parameter h (alpha)?
h is the number of observations whose squared residuals enter the objective; alpha is the corresponding fraction of the data kept. Keeping roughly 75% (alpha ≈ 0.75) is common, while keeping about half gives the maximum breakdown point. The choice trades robustness against efficiency.
When should I prefer a different robust method?
With very small samples (n < 20) the Theil-Sen estimator is more reliable, and when the outlier ratio is very high RANSAC regression may be a better fit. LTS is strongest on moderate-to-large samples with a substantial but bounded share of outliers.
Sources
- 1.Rousseeuw, P. J. (1984). Least Median of Squares Regression. Journal of the American Statistical Association, 79(388), 871-880.
- 2.Rousseeuw, P. J., & Van Driessen, K. (2006). Computing LTS Regression for Large Data Sets. Data Mining and Knowledge Discovery, 12, 29-45.
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Cite this page
ScholarGate. (2026, June 1). Least Trimmed Squares. ScholarGate. https://scholargate.app/statistics/least-trimmed-squares