﻿ 一种收敛且解唯一的SEM算法 An Algorithm of Structural Equation Model with Convergence and Unique Solution

Pure Mathematics
Vol.06 No.01(2016), Article ID:16790,9 pages
10.12677/PM.2016.61001

An Algorithm of Structural Equation Model with Convergence and Unique Solution

Qian Liu, Shengshuang Chen*, Hengqing Tong

Department of Mathematics, College of Science, Wuhan University of Technology, Wuhan Hubei

Received: Nov. 2nd, 2015; accepted: Jan. 14th, 2016; published: Jan. 20th, 2016

ABSTRACT

The algorithm of structural equation model, partial least squares (PLS), has been widely applied to solve indefinite equations. But the iterative algorithm may have the problem of non-convergent and non-unique. In this paper, we propose an optimized algorithm based on the unit modular length constraint of latent variables and the prescription constraint of path coefficients. Simultaneously, we prove the convergence of the algorithm and take a set of data to validate the uniqueness of the solution.

Keywords:Unit Modular Length Constraint, Prescription Constraint, Convergence, Uniqueness

1. 引言

2. 结构方程模型及其PLS算法

(1)

Figure 1. Ancient building fire risk assessment model

(2)

(3)

(4)

，则(3) (4)可表示为：

(5)

(6)

(7)

(8)

，则观测方程组(7) (8)可表示为：

(9)

(10)

3. 单位模长约束下的最优迭代初值

(11)

(12)

(13)

(12) (13)联立可得：

(14)

(15)

(16)

(17)

(18)

，而中已经包含了的影响，即(18)可写成

(19)

(20)

(21)

4. 最优迭代初值下的PLS收敛

(22)

(23)

(23)将(2)的不定方程组转换成一般的回归分析，便于讨论PLS算法的收敛性。但是在这里不能直接利用(23)，因为未必能符合路径系数(0元素)的条件。在这里还得根据(1) (2)来考虑PLS的收敛。

(24)

(25)

(26)

5. 配方约束下的优化算法的解唯一

(27)

(28)

(29)

Table 1. Original data

Table 2. Prescription constraint of summary coefficients

Table 3. Path coefficient of latent variables

Table 4. Estimates of latent variables

6. 总结

An Algorithm of Structural Equation Model with Convergence and Unique Solution[J]. 理论数学, 2016, 06(01): 1-9. http://dx.doi.org/10.12677/PM.2016.61001

1. 1. Fornell, C. and Bookstein, F.L. (1982) Two Structural Equation Models: LISREL and PLS Applied to Consumer Ex-it-Voice Theory. Journal of Marketing Research, 19, 440-452. http://dx.doi.org/10.2307/3151718

2. 2. 王惠文. 偏最小二乘回归方法及其应用[M]. 北京: 国防工业出版社, 1999.

3. 3. 王惠文, 刘强. 偏最小二乘回归模型内涵分析方法研究[J]. 北京航空航天大学学报, 2000, 26(4): 473-476.

4. 4. 罗批, 郭继昌, 李锵, 滕建辅. 基于偏最小二乘回归建模的探讨[J]. 天津大学学报, 2002, 35(6): 783-786.

5. 5. Tong, H.Q. (1993) Evaluation Model and Its It-erative Algorithm by Alternating Protecting. Mathematical and Computer Modelling, 18, 55-60. http://dx.doi.org/10.1016/0895-7177(93)90162-R

6. 6. Tong, Q.L., Zou, X.C., Wang, C.M. and Tong, H.Q. (2010) A Definite Linear Algorithm for Structural Equation Model. Mathematical and Computer Modelling, 52, 744-751. http://dx.doi.org/10.1016/j.mcm.2010.05.002

7. 7. Wang, C.M. and Tong, H.Q. (2007) Best Iterative Initial Values for PLS in a CSI Model. Mathematical and Computer Modelling, 46, 439-444. http://dx.doi.org/10.1016/j.mcm.2006.10.009

8. 8. 董汉忠, 张小薇. 配方回归[J]. 计算机应用通讯, 1983(4): 36-42.

9. 9. 方开泰, 王东谦, 吴国富. 一类带约束的回归——配方回归[J]. 计算数学, 1982(1): 57-69.

*通讯作者。