arXiv subjects
Publications and source records attributed to Mai Zhou.
In this article we generalise the structure of Connes-Kreimer Hpof algebra consisting of Feynmam diagrams to the situations of abstract finite sets, matrices and star product of scalar field, where the construction for the case of finite sets is essential.
In this article we discuss the operations of partitions (sequence of disjoint finite subsets) which are quotient, insertion, composition and Lie bracket. Moreover, we discuss applications of those operations for Feymman diagrams and Kontesvich's graphs.
In this paper the deformation quantization is constructed in the case of scalar fields on Minkowski space-time. We construct the star products at three level concerning fields, Hamiltonian functionals and their underlying structure called Hamiltonian functions in a special sense. Which mean the star products of fields, functionals, Hamiltonian functions, and ones between the fields and functionals. As bases of star products the Poisson brackets at different level are generalized, constructed and discussed in a systematical way, where the Poisson brackets like canonical and time-equal ones. For both of the star products and Poisson brackets the construction at level of Hamiltonian functions plays the essential role. Actually, the discussion for the case of Hamiltonian functions includes the key information about Poisson brackets and the star products in our approach. All of other Poisson brackets and star products in this paper are based on ones of Hamiltonian functions, and the Poisson brackets and the star products at three level are compatible. To discuss the Poisson brackets and the star products in the case of scalar fields, we introduce the notion of algebra of Euler-Lagrange operators which related to variation calculus closely.
Random survival forest and survival trees are popular models in statistics and machine learning. However, there is a lack of general understanding regarding consistency, splitting rules and influence of the censoring mechanism. In this paper, we investigate the statistical properties of existing methods from several interesting perspectives. First, we show that traditional splitting rules with censored outcomes rely on a biased estimation of the within-node failure distribution. To exactly quantify this bias, we develop a concentration bound of the within-node estimation based on non i.i.d. samples and apply it to the entire forest. Second, we analyze the entanglement between the failure and censoring distributions caused by univariate splits, and show that without correcting the bias at an internal node, survival tree and forest models can still enjoy consistency under suitable conditions. In particular, we demonstrate this property under two cases: a finite-dimensional case where the splitting variables and cutting points are chosen randomly, and a high-dimensional case where the covariates are weakly correlated. Our results can also degenerate into an independent covariate setting, which is commonly used in the random forest literature for high-dimensional sparse models. However, it may not be avoidable that the convergence rate depends on the total number of variables in the failure and censoring distributions. Third, we propose a new splitting rule that compares bias-corrected cumulative hazard functions at each internal node. We show that the rate of consistency of this new model depends only on the number of failure variables, which improves from non-bias-corrected versions. We perform simulation studies to confirm that this can substantially benefit the prediction error.