隐马尔可夫模型的参数估计：EM 方法与拟牛顿方法的比较及一种新的混合算法
Parameter estimation of hidden Markov models: comparison of EM and quasi-Newton methods with a new hybrid algorithm
AI总结：
本文提出一种结合 Baum-Welch 与拟牛顿方法的混合算法 QNEM，通过似然函数凸性切换策略提升隐马尔可夫模型参数估计效率，并在多个模型上与 EM、SQUAREM 和 L-BFGS-B 进行比较，验证其整体性能优势。
摘要：
隐马尔可夫模型（HMM）用于对依赖于遵循马尔可夫链的隐藏（或潜在）状态的观测序列进行建模。这些模型广泛应用于生态学、语音识别和遗传学等多个领域。HMM 中的参数估计通常使用 Baum-Welch 算法进行，该算法是期望最大化（EM）算法的一种特例。虽然该方法保证收敛到局部最大值，但其收敛速度通常较慢。替代方法，例如使用拟牛顿方法（如 L-BFGS-B）直接最大化似然，可以提供更快的收敛速度，但由于需要处理参数空间上的边界约束，实现起来可能更加复杂。我们提出了一种新的混合算法 QNEM，它结合了 Baum-Welch 算法和拟牛顿算法。QNEM 旨在通过根据似然函数的凸性在两种方法之间切换，从而利用两种算法的优势。我们通过将这些算法应用于基于不同模型构建的四个示例，对 QNEM、Baum-Welch 算法、一种称为 SQUAREM（Varadhan, 2008, Scand J Statist）的 EM 加速算法以及 L-BFGS-B 拟牛顿方法进行了比较分析。我们使用不同算法估计了每个模型的参数，并评估了它们的性能。我们的结果表明，表现最佳的算法取决于所考虑的模型。QNEM 总体上表现良好，始终比 L-BFGS-B 更快或与之相当。在存在多个最优值的某些场景中，Baum-Welch 和 SQUAREM 算法比拟牛顿方法和 QNEM 算法更快。总之，QNEM 为现有算法提供了一种有前景的替代方案。
英文摘要：
Hidden Markov Models (HMM) model a sequence of observations that are dependent on a hidden (or latent) state that follow a Markov chain. These models are widely used in diverse fields including ecology, speech recognition, and genetics.Parameter estimation in HMM is typically performed using the Baum-Welch algorithm, a special case of the Expectation-Maximisation (EM) algorithm. While this method guarantee the convergence to a local maximum, its convergence rates is usually slow.Alternative methods, such as the direct maximisation of the likelihood using quasi-Newton methods (such as L-BFGS-B) can offer faster convergence but can be more complicated to implement due to challenges to deal with the presence of bounds on the space of parameters.We propose a novel hybrid algorithm, QNEM, that combines the Baum-Welch and the quasi-Newton algorithms. QNEM aims to leverage the strength of both algorithms by switching from one method to the other based on the convexity of the likelihood function.We conducted a comparative analysis between QNEM, the Baum-Welch algorithm, an EM acceleration algorithm called SQUAREM (Varadhan, 2008, Scand J Statist), and the L-BFGS-B quasi-Newton method by applying these algorithms to four examples built on different models. We estimated the parameters of each model using the different algorithms and evaluated their performances.Our results show that the best-performing algorithm depends on the model considered. QNEM performs well overall, always being faster or equivalent to L-BFGS-B. The Baum-Welch and SQUAREM algorithms are faster than the quasi-Newton and QNEM algorithms in certain scenarios with multiple optimum. In conclusion, QNEM offers a promising alternative to existing algorithms.
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