Least trimmed squares

Key takeaways

Least trimmed squares belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Least trimmed squares to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Least trimmed squares from memory before moving on to harder problems.

Reference excerpt

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 {\displaystyle k} , of those points. The unused n − k {\displaystyle n-k} points 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 = ∑ i = 1 n r i ( β ) 2 , {\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:

r i ( β ) = y i − f ( x i , β ) , {\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 r ( j ) ( β ) {\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

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

while the objective function for LTS is

S k ( β ) = ∑ j = 1 k r ( j ) ( β ) 2 . {\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 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.

References

Rousseeuw, P. J. (1984). "Least Median of Squares Regression". Journal of the American Statistical Association. 79 (388): 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. Bibcode:2007JApSt..34..331J. doi:10.1080/02664760601004973.

Worked examples

Example 1 — a first encounter with Least trimmed squares

Start with the simplest possible case. Write down what Least trimmed squares claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Least trimmed squares before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Least trimmed squares ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Least trimmed squares

In research: Least trimmed squares appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Least trimmed squares in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Least trimmed squares is common in secondary-school and first-year university syllabi. It links to neighbouring topics Robust regression, Robust statistics, so understanding it makes those chapters shorter.
In everyday life: Look for Least trimmed squares outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Least trimmed squares in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Least trimmed squares means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Least trimmed squares out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Least trimmed squares in simple terms?

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.

Why does Least trimmed squares matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Least trimmed squares?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Least trimmed squares.