Page 438 - Contributed Paper Session (CPS) - Volume 4

CPS2526 Holger Cevallos-Valdiviezo et al.
trimmed squares (MVLTS) estimator and a coordinatewise least trimmed
squares (CooLTS) estimator. The MVLTS estimator was introduced in Maronna
(2005) for multivariate data. We extend this estimator to the functional case.
Moreover, we also introduce the CooLTS estimator for both multivariate data
and functional data. For both methods we propose an algorithm based on
estimating equations to find a local minimum of their objective function.
For multivariate data we use the following notation. Consider
observations ∈ ℝ , = 1, … , . with corresponding sample mean and sample

covariance matrix . Let ∈ ℝ × be an orthogonal matrix, i.e. = with

rows , = 1, … , . Let ∈ ℝ × be a matrix with rows , = 1, … , , and ∈

ℝ . The corresponding approximations of the observations are given by

̂ ( , , ) ≡ ̂ = + , or elementwise ̂ = + . The associated

multivariate residuals are given by = − ̂ ∈ ℝ with components = −

̂ . Its Euclidean norm is denoted by ( , , ) ≡ = ‖ ‖ .

ℝ

2. Methodology
a. Multivariate least trimmed squares estimator in ℝ

It is easy to see that the classical PCA solution is found by minimizing a scale
estimate ̂ (( , , )) of the Euclidean distances of the residuals
2

( , , ) = ( , … , ), given by

1

1
2
2
̂ (( , , )) = ∑ ( , , ) (1)

=1
This classical scale estimator based on a quadratic loss function is clearly not
robust against outliers. Maronna (2005) robustified the classical approach by
replacing ̂ by a least trimmed squares (LTS) scale, defined by
2
ℎ
1 1
2
̂ 2 (( , , )) = ∑ 2 ( , , ) = ∑ ( , , ) (2)

ℎ : ℎ
=1 =1
where (1:) ( , , ) ≤ ⋯ ≤ (:) ( , , ) is the ordered sequence of

Euclidean distances and ℎ = − [] for some 0 ≤ ≤ 1 . Note that the
weights are:

1, (1:) ≤ ⋯ ≤ (ℎ:)
= { (3)

0, ℎ
The multivariate LTS-stimulator (MVLTS) is now defined as the solution
̂
( , ̂ , ̂ ) of the minimization problem
min ̂ 2 (( , , )) (4)
, ,
Where ∈ ℝ × is an orthogonal matrix.

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