Yuxuan | Searcharxiv

arXiv2024

Unit interval parking functions and the $r$-Fubini numbers

We recall that unit interval parking functions of length $n$ are a subset of parking functions in which every car parks in its preference or in the spot after its preference, and Fubini rankings of length $n$ are rankings of $n$ competitors allowing for ties. We present an independent proof of a result of Hadaway, which establishes that unit interval parking functions and Fubini rankings are in bijection. We also prove that the cardinality of these sets are given by Fubini numbers. In addition, we give a complete characterization of unit interval parking functions by determining when a rearrangement of a unit interval parking function is again a unit interval parking function. This yields an identity for the Fubini numbers as a sum of multinomials over compositions. Moreover, we introduce a generalization of Fubini rankings, which we call the $r$-Fubini rankings of length $n+r$. We show that this set is in bijection with unit interval parking functions of length $n+r$ where the first $r$ cars have distinct preferences. We conclude by establishing that these sets are enumerated by the $r$-Fubini numbers.

math.CO↗

arXiv2020

On the Importance of Diversity in Re-Sampling for Imbalanced Data and Rare Events in Mortality Risk Models

Surgical risk increases significantly when patients present with comorbid conditions. This has resulted in the creation of numerous risk stratification tools with the objective of formulating associated surgical risk to assist both surgeons and patients in decision-making. The Surgical Outcome Risk Tool (SORT) is one of the tools developed to predict mortality risk throughout the entire perioperative period for major elective in-patient surgeries in the UK. In this study, we enhance the original SORT prediction model (UK SORT) by addressing the class imbalance within the dataset. Our proposed method investigates the application of diversity-based selection on top of common re-sampling techniques to enhance the classifier's capability in detecting minority (mortality) events. Diversity amongst training datasets is an essential factor in ensuring re-sampled data keeps an accurate depiction of the minority/majority class region, thereby solving the generalization problem of mainstream sampling approaches. We incorporate the use of the Solow-Polasky measure as a drop-in functionality to evaluate diversity, with the addition of greedy algorithms to identify and discard subsets that share the most similarity. Additionally, through empirical experiments, we prove that the performance of the classifier trained over diversity-based dataset outperforms the original classifier over ten external datasets. Our diversity-based re-sampling method elevates the performance of the UK SORT algorithm by 1.4$.

cs.LG↗