A fast method to compute orthogonal loadings partial least squares
✍ Scribed by Constantinos Goutis
- Publisher
- John Wiley and Sons
- Year
- 1997
- Tongue
- English
- Weight
- 176 KB
- Volume
- 11
- Category
- Article
- ISSN
- 0886-9383
No coin nor oath required. For personal study only.
✦ Synopsis
There are two commonly used partial least squares (PLS) regression algorithms. 1 -3 The first, the orthogonal scores algorithm, is the older one, 4 whereas the second, the orthogonal loadings algorithm, is more recent. 5 Their difference lies in the decomposition of the data matrix X = ∑ r i =1 t i p i T . The first one requires orthogonality of the vectors t i , whereas the second one requires orthogonality of the vectors p i . As it turns out, the two methods are equivalent for prediction purposes. 1 In the usual formulation the first algorithm is easier computationally, though not always faster, 6 and is typically used in computer packages. 7,8 This is because it does not require any multiple regression and the orthogonality of the scores simplifies their further use. The second algorithm is easier to interpret and study theoretically. 5 The purpose of this paper is to give a fast method to compute the scores and loadings of the second algorithm. Our method avoids the multiple regression step and, as we will see, requires significantly fewer computations than both the orthogonal scores and orthogonal loadings algorithms. Furthermore, it is trivial to program.
ORTHOGONAL LOADINGS PLS METHODS
Consider data in the form of a columnwise centered n × k matrix X and a centered n × 1 vector y. In the usual formulation 2 the PLS steps are as follows.