Computation of Least Square Estimates Without Matrix Manipulation

Abstract

The least square approach is undoubtedly one of the well known methods in the fields of statistics and related disciplines such as optimization, artificial intelligence, and data mining. The core of the traditional least square approach is to find the inverse of the product of the design matrix and its transpose. Therefore, it requires storing at least two matrixes – the design matrix and the inverse matrix of the product. In some applications, for example, high frequency financial data in the capital market and transactional data in the credit card market, the design matrix is huge and on line update is desirable. Such cases present a difficulty to the traditional matrix version of the least square approach. The reasons are from the following two aspects: (1) it is still a cumbersome task to manipulate the huge matrix; (2) it is difficult to utilize the latest information and update the estimates on the fly. Therefore, a new method is demanded. In this paper, authors applied the idea of CIO-component-wise iterative optimization and propose an algorithm to solve a least square estimate without manipulating matrix, i.e. it requires no storage for the design matrix and the inverse of the product, and furthermore it can update the estimates on the fly. Also, it is rigorously shown that the solution obtained by the algorithm is truly a least square estimate.