发表机构
Indian Institute of Technology Madras（印度马德拉斯理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出QZ-IPCA迭代广义特征值算法，无需先验假设即可从含噪数据同时恢复多元线性系统的噪声划分、方差、关系数和回归系数，实验证明其优于传统最小二乘方法。
AI 中文摘要
从含噪测量中恢复控制系统线性关系是物理和工程科学中的一项基本任务。由于每个测量变量可能携带未知量的噪声，经典回归必须事先承诺一组结构假设：普通最小二乘要求声明输入-输出划分且输入变量无噪声，总体最小二乘假设所有变量噪声方差相等，而广义总体最小二乘还要求事先已知噪声变量划分和方差。Kalman~\cite{Kalman:1982} 表明，任何从不精确数据返回唯一线性模型的程序都必须依赖于此类无法用数据本身验证的先验假设——即“偏见”——而消除这些偏见会使辨识问题从根本上变得不确定。这些偏见是否可以直接从数据中解决，至今仍未解决。在此我们证明，一种迭代广义特征值算法 QZ-IPCA 仅使用原始数据即可同时恢复多元线性系统的噪声变量划分、噪声方差、线性关系数量和回归系数。在五变量基准网络的所有可能穷举噪声配置中，QZ-IPCA 正确识别模型结构并以低于 6.4% 的误差恢复系数。即使给定最佳划分，它也优于普通最小二乘，并且在噪声方差跨变量不同时，恰好在标准总体最小二乘失效的秩辨识中取得成功。这些结果表明，传统多元回归所需的假设并非必要，将模型辨识重新定义为仅从数据几何即可解决的问题。
英文摘要
Recovering the linear relationships that govern a system from noisy measurements is a basic task across the physical and engineering sciences. Because every measured variable may carry an unknown amount of noise, classical regression must commit in advance to a set of structural assumptions: ordinary least squares requires a declared input-output partition with input variables being noise-free, total least squares assumes equal noise variance across all variables, and generalized total least squares additionally requires the noisy-variable partition and variances to be known beforehand. Kalman~\cite{Kalman:1982} showed that any procedure returning a unique linear model from inexact data must rest on such unverifiable a priori assumptions -- ``prejudices'' -- that cannot be checked against the data itself, and that removing them leaves the identification problem fundamentally indeterminate. Whether these prejudices can instead be resolved directly from the data has remain unresolved. Here we show that an iterative generalized-eigenvalue algorithm, QZ-IPCA, recovers the noisy-variable partition, noise variances, number of linear relations, and regression coefficients of a multivariate linear system simultaneously, using only the raw data. Across all possible exhaustive noise configurations of a five-variable benchmark network, QZ-IPCA correctly identifies model structure and recovers coefficients with error below 6.4\%. It outperforms ordinary least squares even when given the best partition, and succeeds in rank identification precisely where standard total least squares falls once noise variances differ across variables. These results show that the assumptions conventionally required for multivariate regression are not necessary, recasting model identification as a problem solvable from data geometry alone.
CommentsThe manuscript is under further processing. Important modifications might take place in further versions. There are 32 pages containing 10 figures and 3 tables
