In this paper, we analyze the general errors-in-variables (EIV) model, allowing both the uncertain coefficient matrix and the dispersion matrix to be rank-deficient. We derive the weighted total least-squares (WTLS) solution in the general case and find that with the model consistency condition: (1) If the coefficient matrix is of full column rank, the parameter vector and the residual vector can be uniquely determined independently of the singularity of the dispersion matrix, which naturally extends the Neitzel/Schaffrin rank condition (NSC) in previous work. (2) In the rank-deficient case, the estimable functions and the residual vector can be uniquely determined. As a result, a unified approach for WTLS is provided by using generalized inverse matrices (g-inverses) as a principal tool. This method is unified because it fully considers the generality of the model setup, such as singularity of the dispersion matrix and multicollinearity of the coefficient matrix. It is flexible because it does not require to distinguish different cases before the adjustment. We analyze two examples, including the adjustment of the translation elimination model, where the centralized coordinates for the symmetric transformation are applied, and the unified adjustment, where the higher-dimensional transformation model is explicitly compatible with the lower-dimensional transformation problem.

在本文中，我们分析了一般变量误差（EIV）模型，允许不确定系数矩阵和色散矩阵都是秩亏的。我们推导了一般情况下的加权总最小二乘（WTLS）解，发现在模型一致性条件下：（1）如果系数矩阵是满列秩的，则参数向量和残差向量可以独立唯一地确定色散矩阵的奇异性，这自然地扩展了先前工作中的 Neitzel/Schaffrin 秩条件（NSC）。 (2)在秩亏的情况下，可估计函数和残差向量可以唯一确定。因此，通过使用广义逆矩阵（g-逆）作为主要工具，为 WTLS 提供了统一的方法。该方法是统一的，因为它充分考虑了模型设置的通用性，例如色散矩阵的奇异性和系数矩阵的多重共线性。它是灵活的，因为它不需要在调整之前区分不同的情况。我们分析了两个例子，包括平移消除模型的调整（应用对称变换的集中坐标）和统一调整（其中高维变换模型与低维变换问题显式兼容）。