Abstract
NIPALS and SIMPLS algorithms are the most commonly used algorithms for partial least squares analysis. When the number of objects, N, is much larger than the number of explanatory, K, and/or response variables, M, the NIPALS algorithm can be time consuming. Even though the SIMPLS is not as time consuming as the NIPALS and can be preferred over the NIPALS, there are kernel algorithms developed especially for the cases where N is much larger than number of variables. In this study, the NIPALS, SIMPLS and some kernel algorithms have been used to built partial least squares regression model. Their performances have been compared in terms of the total CPU time spent for the calculations of latent variables, leave-one-out cross validation and bootstrap methods. According to the numerical results, one of the kernel algorithms suggested by Dayal and MacGregor (J Chemom 11:73–85, 1997) is the fastest algorithm.
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Abbreviations
- X :
-
N × K matrix of explanatory variables
- Y :
-
N × M matrix of response variables
- F :
-
N × M matrix of residuals
- B PLS :
-
K × M matrix of PLS regression coefficients
- T :
-
N × A matrix of PLS latent variables for X
- U :
-
N × A matrix of PLS latent variables for Y
- W :
-
K × A matrix of weights of deflated X matrix on latent variables T
- R :
-
K × A matrix of weights of original X matrix on latent variables T
- C :
-
M × A matrix of weights of Y on latent variables U
- P :
-
K × A matrix of loadings for X
- t a :
-
A column vector of T
- u a :
-
A column vector of U
- w a :
-
A column vector of W
- r a :
-
A column vector of R
- c a :
-
A column vector of C
- p a :
-
A column vector of P
- :
-
Uppercase bold variables will represent matrices and lower case bold variables will represent column vectors in the paper. The transpose of a matrix will be given with “ ′ ”. N, K, M and A are the number of objects, the number of explanatory variables, the number of response variables and the number of latent variables, respectively. The notations used in the paper are given above. It is assumed that the columns of X and Y are mean-centered and scaled prior to PLS model estimation to have mean zero and standard deviation one.
References
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Alin, A. Comparison of PLS algorithms when number of objects is much larger than number of variables. Stat Papers 50, 711–720 (2009). https://doi.org/10.1007/s00362-009-0251-7
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DOI: https://doi.org/10.1007/s00362-009-0251-7