Comprehensive studies of Grassmann manifold optimization and sequential candidate set algorithm in a principal fitted component model

Abstract

In this paper we compare parameter estimation by Grassmann manifold optimization and sequential candidate set algorithm in a structured principal fitted component(PFC) model. The structured PFC model extends the form of the covariance matrix of a random error to relieve the limits that occur due to too simple form of the matrix. However, unlike other PFC models, structured PFC model does not have a closed form for parameter estimation in dimension reduction which signals the need of numerical computation. The numerical computation can be done through Grassmann manifold optimization and sequential candidate set algorithm. We conducted numerical studies to compare the two methods by computing the results of sequential dimension testing and trace correlation values where we can compare the performance in determining dimension and estimating the basis. We could conclude that Grassmann manifold optimization outperforms sequential candidate set algorithm in dimension determination, while sequential candidate set algorithm is better in basis estimation when conducting dimension reduction. We also applied the methods in real data which derived the same result;이 논문에서는 Principal fitted component(PFC) 모델에서 Grassmann manifold optimization과 sequential candidate set algorithm에 의한 모수추정을 비교한다. 다른 PFC 모델들과 달리 structured PFC 모델의 경우 차원축소를 위한 모수추정을 하기 위해서는 추가적으로 수치적 계산과정이 필요하다. Grassmann manifold optimization과 sequential candidate set algorithm을 통해 이 수치적 계산이 가능하다. 이 논문에서는 두 방법을 비교하기 위해 순차적 차원 검정 (sequential dimension testing) 결과와 trace correlation 값을 계산한 후 비교하는 시뮬레이션 연구를 진행했다. 그 결과 차원을 결정할 때는 Grassmann manifold optimization이 더 좋은 성능을 보이고, 차원 축소를 위한 기저추정 (basis estimation)을 할 때에는 sequential candidate set algorithm이 더 나은 성능을 보였으며 실제 데이터를 적용했을 때에도 같은 결과가 나왔다.