基于一维数据逐段均匀条件下的参考分布研究

聚类分析中一个重要的问题是估计聚类数,GS方法通过参考分布对聚类数进行合理的估计,解决了其他聚类方法无法对应分为一类的数据进行分类的问题,具有更好的分类效果;目前已通过理论证明得出,在分布为对数凹且一维情况下,GS方法的参考分布为均匀分布,而有关其在其他条件下的参考分布研究较少;针对这一情况,提出解决一维且逐段均匀条件下GS方法的参考分布问题,以类内平方和为评价标准,通过拉格朗日乘数法求解与问题等价的优化问题,理论证出条件下总体均匀分布是类内平方和最大的情况,进而得出对于一维数据,逐段均匀条件下的参考分布仍为均匀分布的结论。

Study of Reference Distribution of One-dimensional Data under the Condition of Piecewise Uniform

An important issue in clustering analysis is to estimate the number of clusters. GS method makes reasonable estimation of the number of clusters by reference distribution, which solves the problem that other clustering methods cannot classify the data into one category, so it has better classification effect. Tibshirani R and others theatrically got the conclusion that under logarithmic concave and one dimensional case, GS method reference is uniform distribution. In view of the present situation that only a few reference distributions under different conditions had been studied, this paper put forward the best distribution of GS method under the condition of one dimensional and piecewise uniform, used the sum of squares in the class as the evaluation standard, solved the equivalent optimization problem through the Lagrange multiplier method, and obtained the result that under the above condition, uniform distribution is the maximum of the sum of squares within the class. So we get the conclusion that the reference distribution under the condition of segmental uniform distribution is still uniform for the one-dimensional data.