Page 439 - Contributed Paper Session (CPS) - Volume 4

CPS2526 Holger Cevallos-Valdiviezo et al.
b. Coordinatewise least trimmed squares estomatpr in ℝ

Boente & Salibian-Barrera (2015) noted that the classical PCA problem can be
rewritten as

min ∑ ( , , ) (5)
2
, ,
=1
A robust alternative can thus be obtained by replacing the nonrobust sample
variance by a robust estiator of scale. We use the univariate LTS scale-
estimator, which is defined as
ℎ
1 1 2
2
2

̂ , = ℎ ∑( ) = ℎ ∑ ( − − ) (6)

:

=1 =1

For the th variable, and where the weights are given by

2
2
1, ≤ ( )
= { ℎ: (7)

2
2
0, > ( )
ℎ:

The corresponding coordinatewise LTS-estimator (CooLTS) is now defined as
̂
the solution ( , ̂ , ̂ ) of the minimization problem

min ∑ 2 , ( , , ) (8)

, ,
=1
with an orthogonal matrix.

c. Algorithm

Explicit first-order conditions can be obtained by differentiating (4) or (8) with
respect to , and . After setting them to zero and rearranging terms we

obtain
(9)

∑ ( − ) = (∑ ) , 1 ≤ ≤

=1 =1 (10)

∑ ( − ) = (∑ ) , 1 ≤ ≤

=1 =1
(11)

∑ ( − ) = ∑

=1 =1

Note that for CooLTS the weights are given by (7) while for MVLTS the

weights = are given by (3). These equations suggest an iterative re-

weighted least squares procedure to find a local minimum of (4) or (8). As
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