论文标题
随机过程高维观测的多项式表示
Polynomial Representations of High-Dimensional Observations of Random Processes
论文作者
论文摘要
本文研究了观察次数很大时进行相关分析的问题。在这种情况下，通常有必要将随机观察结果结合起来，以减少问题的维度。通过近似于原始随机变量的简单总和，通过单变量多项式近似单变量多项式来获得一类新的统计量度。然后，多项式的平均值是统计中心总和的加权总和，权重依赖。如果可以获得随机变量之和的分布，则计算总和在计算上是有效的，并且可以对数学分析进行。在还获得了几个辅助结果中，与样品平均值相对应的一阶总和来通过将数据分配到分离性的分离子集中来降低线性回归的数值复杂性。假设Markov过程的第一阶和二阶过程提供了说明性示例。
The paper investigates the problem of performing correlation analysis when the number of observations is very large. In such a case, it is often necessary to combine the random observations to achieve dimensionality reduction of the problem. A novel class of statistical measures is obtained by approximating the Taylor expansion of a general multivariate scalar function by a univariate polynomial in the variable given as a simple sum of the original random variables. The mean value of the polynomial is then a weighted sum of statistical central sum-moments with the weights being application dependent. Computing the sum-moments is computationally efficient and amenable to mathematical analysis, provided that the distribution of the sum of random variables can be obtained. Among several auxiliary results also obtained, the first order sum-moments corresponding to sample means are used to reduce the numerical complexity of linear regression by partitioning the data into disjoint subsets. Illustrative examples are provided assuming the first and the second order Markov processes.
