A diffusion-based kernel density estimator (diffKDE, version 1) with optimal bandwidth approximation for the analysis of data in geoscience and ecological research

Probability density functions (PDFs) provide information about the probability of a random variable taking on a specific value. In geoscience, data distributions are often expressed by a parametric estimation of their PDF, such as, for example, a Gaussian distribution. At present there is growing attention towards the analysis of non-parametric estimation of PDFs, where no prior assumptions about the type of PDF are required. A common tool for such non-parametric estimation is a kernel density estimator (KDE). Existing KDEs are valuable but problematic because of the difficulty of objectively specifying optimal bandwidths for the individual kernels. In this study, we designed and developed a new implementation of a diffusion-based KDE as an open source Python tool to make diffusion-based KDE accessible for general use. Our new diffusion-based KDE provides (1) consistency at the boundaries, (2) better resolution of multimodal data, and (3) a family of KDEs with different smoothing intensities. We demonstrate our tool on artificial data with multiple and boundary-close modes and on real marine biogeochemical data, and compare our results against other popular KDE methods. We also provide an example for how our approach can be efficiently utilized for the derivation of plankton size spectra in ecological research. Our estimator is able to detect relevant multiple modes and it resolves modes that are located closely to a boundary of the observed data interval. Furthermore, our approach produces a smooth graph that is robust to noise and outliers. The convergence rate is comparable to that of the Gaussian estimator, but with a generally smaller error. This is most notable for small data sets with up to around 5000 data points. We discuss the general applicability and advantages of such KDEs for data-model comparison in geoscience.