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Study of statistical properties for level crossings in discrete time series

离散时间序列中平交道口统计特性的研究

基本信息

批准号：
15500190
负责人：
SHIMIZU Kunio
金额：
$ 1.22万
依托单位：
Keio University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15500190/
关键词：
strictly stationary ellipsoidal process scale mixtures of normals level crossings binary time series trinary time series number of extremes number of inflection points length of excursions

项目摘要

In the paper by Tanaka and Shimizu (2001), the expectation formula of level crossings for discrete time series is studied in the case where the underlying process is a strictly stationary ellipsoidal process with zero-mean, unit-variance and autocorrelation function ρ(h), which is actually governed by a scale mixture of normal distributions. The present study assumes the same process and aims to get expectation and variance formulae of the numbers of one-step and two-step level-crossings for an observed discrete sample Y_1,...,Y_N. Transformation into trinary series 1, 0, -1 is used to get the results. Here the number of one-step level-crossings is defined by the number t (2≦t≦N) satisfying that [Y_t≧u and v≦Y_<t-1><u], or [v≦Y_t<u and Y_<t-1>≧u], or [v≦Y_t<u and Y_<t-1><v], or finally [Y_t<v and v≦Y_<t-1><u] for u, v(u>v). Similarly the number of two-step level-crossings is defined by the number t (2≦t≦N) satisfying that [Y_t≧u and Y_<t-1><u] or [ Y_t<v and Y_<t-1>≧u]. The expectation formulae are obtained in the paper by Shimizu and Tanaka (2003) and the paper also provides applications to the expectation formula for the number of two-step level-crossings after taking the first- and second-difference time series in a stationary Gaussian zero-mean, unit-variance and autocorrelation function ρ(h). The numbers generalize the number of extremes (local minima and maxima) and that of inflection points. Asymptotic behavior of the length of excursions is studied in Tanaka and Shimizu (2004). A paper by Shikama and Shimizu which studies variance formulae is submitted to a journal for possible publication and is now under review.

在Tanaka和Shimizu(2001)的文章中，研究了离散时间序列的能级穿越的期望公式，当基本过程是具有零均值、单位方差和自相关函数ρ(H)的严格平稳的椭球过程时，它实际上是服从正态分布的尺度混合的。本研究假定同样的过程，目的是得到观测离散样本Y_1，…，Y_N的一步和两步水平穿越次数的期望和方差公式。通过变换到三元序列1，0，-1得到结果。这里，一步道口的数目由满足[Y_t≦u和v≦Y_&lt；t-1&gt；&lt；u]或[v≧Y_t&lt；t-1&gt；u]或[v≦Y_t&lt；u和Y_&lt；t-1&gt；t；t；v]或满足[Y_t&lt；v和v≦Y_t&lt；t-1&gt；v]的数t(2_t&lt；n)来定义。&lt；u]表示u，v(u&gt；v)。类似地，两步道口的数目由满足[Y_t≦u和Y_&lt；t-1&gt；&lt；u]或[Y_t&lt；v和Y_&lt；t-1&gt；≦u]的数t(2≧t；t-1&gt；≧u)定义。文中给出了Shimizu和Tanaka(2003)的期望公式，并给出了取平稳高斯零均值、单位方差和自相关函数ρ(H)的一阶和二阶差时间序列后两步水平穿越次数的期望公式的应用。这些数字概括了极值(局部极小值和极大值)和拐点的个数。Tanaka和Shimizu(2004)研究了游程长度的渐近行为。Shikama和Shimizu的一篇研究方差公式的论文已提交给一家期刊，可能会发表，目前正在接受审查。