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PLS Criterion for Unordered Lists

IP.com Disclosure Number: IPCOM000038349D
Original Publication Date: 1987-Jan-01
Included in the Prior Art Database: 2005-Jan-31
Document File: 2 page(s) / 21K

Publishing Venue

IBM

Related People

Rissanen, JJ: AUTHOR

Abstract

This is a description of the algorithm needed to implement the predictive least squares (PLS) criterion for data which is modeled as an unordered list. We give the description for the case of determining the optimal polynomial degree when polynomials are fitted to data using the least squares error measure; the general case is analogous. Let the data be indexed as follows: x = xi, ... , xn, where the order does not matter. The first data item is predicted as zero no matter what the degree of the fitted polynomial is, and we pick it as that item which can be predicted with smallest error. Therefore, let i (1) be the index of the smallest data item xi(1), or any one of them if several data items have the smallest squared value. Let Sk(1) = xi2(1) be the square of this data item for each degree k.

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PLS Criterion for Unordered Lists

This is a description of the algorithm needed to implement the predictive least squares (PLS) criterion for data which is modeled as an unordered list. We give the description for the case of determining the optimal polynomial degree when polynomials are fitted to data using the least squares error measure; the general case is analogous. Let the data be indexed as follows: x = xi, ... , xn, where the order does not matter. The first data item is predicted as zero no matter what the degree of the fitted polynomial is, and we pick it as that item which can be predicted with smallest error. Therefore, let i (1) be the index of the smallest data item xi(1), or any one of them if several data items have the smallest squared value. Let Sk(1) = xi2(1) be the square of this data item for each degree k. Working recursively, let for a selected degree k, where j runs through the indices that remain, i.e., those not in the list i(1), ... , i(t - 1) that depend on the value k. Further, where µt-1(k) denote the least squares estimates obtained when fitting a kth degree polynomial to the so-far-processed data xi(1), ... , xi(t - 1) for this value of k. Next, define i(t) as the index of the data item for which the second member in (1) is smallest. The recursion is complete, and Sk(n) defines the PLS criterion to be minimized over k. Numerical examples suggest that this criterion is considerably more "pointed" than the old criterion, defined for...