Mixed linear regression (MLR) is a powerful model for characterizing nonlinear relationships by utilizing a mixture of linear regression sub-models. The identification of MLR is a fundamental problem, where most of the existing results focus on offline algorithms, rely on independent and identically distributed (i.i.d) data assumptions, and provide local convergence results only. This paper investigates the online identification and data clustering problems for two basic classes of MLRs, by introducing two corresponding new online identification algorithms based on the expectation-maximization (EM) principle. It is shown that both algorithms will converge globally without resorting to the traditional i.i.d data assumptions. The main challenge in our investigation lies in the fact that the gradient of the maximum likelihood function does not have a unique zero, and a key step in our analysis is to establish the stability of the corresponding differential equation in order to apply the celebrated Ljung's ODE method. It is also shown that the within-cluster error and the probability that the new data is categorized into the correct cluster are asymptotically the same as those in the case of known parameters. Finally, numerical simulations are provided to verify the effectiveness of our online algorithms.

翻译：混合线性回归（MLR）是一种通过混合线性回归子模型刻画非线性关系的强大模型。MLR的辨识是一个基本问题，现有结果大多集中于离线算法、依赖独立同分布（i.i.d.）数据假设，且仅提供局部收敛结论。本文针对两类基本MLR模型，基于期望最大化（EM）原理提出两种新的在线辨识算法，研究其在线辨识与数据聚类问题。证明表明，两种算法无需传统i.i.d.数据假设即可实现全局收敛。研究的主要挑战在于极大似然函数的梯度不存在唯一零点，而分析的关键步骤是建立相应微分方程的稳定性，以便应用经典的Ljung ODE方法。同时证明，新数据被正确聚类的概率及簇内误差渐近等价于已知参数时的情形。最后通过数值仿真验证了所提在线算法的有效性。