一种基于Rayleigh商的复共线性的诊断指标

doi:10.11947/j.AGCS.2025.20240338

摘要/Abstract

摘要：

复共线性广泛存在于测量数据处理中，其危害性十分严重。对复共线性问题有效处理的前提是对设计阵的复共线性关系进行有效的诊断和度量。针对条件数、矩阵向量正交度等复共线性诊断指标存在的不足，本文从复共线性的本质出发，采用法矩阵的Rayleigh商作为设计阵列向量之间线性相关性的度量，并根据Rayleigh商和镶边矩阵特征值的性质，提出一种复共线性的诊断指标，该指标可以诊断多个复共线性关系。通过试验说明本文提出的复共线性诊断指标的有效性和优良性。当复共线性关系较强时，复共线性关系诊断的准确率可达100%。当复共线性关系较弱时，诊断准确率为84.9%；通过调整阈值，诊断准确率可提升至99.8%。

关键词: 复共线性, Rayleigh商, 条件数, 最小特征值

Abstract:

Multicollinearity widely exists in measurement data processing, and its harms are very serious. The premise of effective treatment of multicollinearity problem is to effectively diagnose and measure the multicollinearity of design matrix. In view of the shortcomings of multicollinearity indicators such as the condition number and orthogonal degree of matrix, based on the nature of multicollinearity, the paper adopts the Rayleigh quotient of normal matrix as the measure of linear correlation between the array vectors of the design matrix, and proposes a multicollinearity indicator according to the properties of the Rayleigh quotient and the eigenvalues of the edge-matrix. This indicator can be used to diagnose multiple multicollinearity relationships. The effectiveness and superiority of the proposed indicator for multicollinearity diagnosis are demonstrated by experiments. When the multicollinearity relationships are strong, the rate of accuracy in diagnosing the multicollinearitycan reach 100%. When the multicollinearity relationships are weak, the diagnostic accuracy is 84.9%, and by adjusting the threshold, the diagnostic accuracy can be increased to 99.8%.

Key words: multicollinearity, Rayleigh quotient, condition number, minimum eigenvalue

中图分类号:

P207

李新娜, 韩松辉, 陈轲, 郭杰. 一种基于Rayleigh商的复共线性的诊断指标[J]. 测绘学报, 2025, 54(3): 422-431.

Xinna LI, Songhui HAN, Ke CHEN, Jie GUO. An index of multicollinearity diagnosis based on Rayleigh quotient[J]. Acta Geodaetica et Cartographica Sinica, 2025, 54(3): 422-431.

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扰动	诊断结果（1000次试验）
扰动	两个强复共线性关系	一强一弱复共线性关系	两个弱复共线性关系	一个复共线性关系
	1000	0	0	0
	1000	0	0	0
	222	778	0	0
	222	778	0	0
	3	412	453	132
	3	412	453	132

参数	主对角线元素	特征值
6参数	6.5×10³、334.9、9.38、4.6×10¹¹、1.4×10¹²、4.95×10⁸	0.32、0.67、278.4、7.8×10⁷、8.3×10¹⁰、1.8×10¹²
12参数	6.5×10³、334.9、9.38、4.6×10¹¹、1.4×10¹²、4.95×10⁸、1.66×10⁶、1.63×10⁴、720、720、1.71×10²⁰、1.64×10²⁰	7.7×10^-5、0.51、2.97、4.67、59.12、160.5、570.4、7.6×10⁷、1.0×10⁹、5.6×10¹⁰、7.2×10¹⁸、3.3×10²⁰

设计阵	条件数	矩阵列向量正交度	本文提出的指标LI_i	线性相关状态
6参数	5.54×10¹²	0.000 326	0.034	良态
12参数	4.25×10²⁴	1.08×10^-13	8.22×10^-6、0.054	一个复共线性关系

列行	a₁q_1,1	a₂q_2,1	a₃q_3,1	a₄q_4,1	a₉q_9,1	a₁₀q_10,1
1	0.253	-0.247	-0.003	0.000	0.000	0.000
2	0.256	-0.216	-0.001	0.000	-0.042	0.000
3	0.255	-0.221	-0.002	0.000	0.000	-0.029
4	0.253	-0.243	-0.003	-0.004	0.000	0.000
5	0.256	-0.212	-0.001	-0.004	-0.042	0.000

列行	a₁q_1,1	a₂q_2,1	a₃q_3,1	a₄q_4,1	a₅q_5,1	a₆q_6,1	a₇q_7,1	a₈q_8,1	a₉q_9,1	a₁₀q_10,1	a₁₁q_11,1	a₁₂q_12,1
1	-0.300	0.530	-0.002	0.713	-0.726	-0.001	-6.000	0.030	-0.042	0.000	6.458	-0.659
2	-0.272	0.487	-0.003	0.675	-0.688	-0.001	-6.019	0.030	0.000	-0.029	6.450	-0.625
3	-0.140	0.287	-0.004	0.498	-0.515	-0.001	-6.146	0.033	0.000	0.000	6.453	-0.474
4	-0.300	0.534	-0.002	0.709	-0.726	-0.001	-5.988	0.030	-0.042	0.000	6.451	-0.664
5	-0.272	0.491	-0.003	0.671	-0.689	-0.001	-6.007	0.030	0.000	-0.029	6.443	-0.630