A multiple kernel-based kernel density estimator for multimodal probability density functions

The performance of the single kernel-based kernel density estimator (SK-KDE) in fitting a unimodal probability density function (PDF) depends on the choice of kernel function and the corresponding selection of kernel bandwidth. Unlike unimodal PDFs, a multimodal PDF has several distinct features. First, it has multiple local maxima. Second, it is composed of various unimodal PDFs. Each of these unimodal PDFs corresponds to a different collection of random variables. Importantly, these variables are not independent and identically distributed. Because of the difficulty in quantifying multimodality among different modes, it is extremely difficult to select an appropriate kernel function and optimal kernel bandwidth for the multimodal PDF. Multimodal PDFs are frequently encountered in real-world applications. To address this, this paper proposes a novel multiple kernel-based kernel density estimator (MK-KDE). It constructs a flexible KDE by using the weighted average of multiple kernels with consideration of their kernel efficiencies. By integrating multiple kernels, MK-KDE leverages their complementary strengths to enhance the estimation of complex and multimodal PDFs. To achieve this, a novel efficient objective function is designed to obtain the optimized kernel weights and kernel bandwidths by minimizing both the global estimation error of MK-KDE and the local estimation errors of SK-KDEs. Moreover, a sophisticated k-nearest neighbor strategy is devised as a heuristic method to determine the unknown PDF values of given data points, thereby optimizing the aforementioned objective function. A series of extensive experiments was conducted to validate the feasibility, rationality, and effectiveness of MK-KDE for 10 multimodal PDFs. The experimental results show that (1) the kernel weights and bandwidths of MK-KDE converge as the iteration number of the optimization algorithm increases; (2) MK-KDE can fit multimodal PDFs by automatically selecting the kernel functions and bandwidths; and (3) MK-KDE obtains lower estimation errors on 10 multimodal PDFs in comparison to 10 existing PDF estimation methods, demonstrating that MK-KDE is a viable approach to estimate multimodal PDFs.