透過馬可夫鏈之下的 copula 模型對二項式時間序列資料線上改變點的偵測

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林宜興(Yi-Xing Lin) 查詢紙本館藏

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統計研究所

論文名稱

透過馬可夫鏈之下的 copula 模型對二項式時間序列資料線上改變點的偵測
(Online Changepoint Detection under a Copula-based Markov Chain Model for Binomial Time Series Data)

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摘要(中)

當生成資料序列的參數發生異常的改變，我們稱之為改變點，而改變點偵測就是為了偵測參數不規則變化的發生。改變點偵測又分即時偵測與線下偵測，這篇論文主要聚焦在即時改變點偵測，而即時改變點偵測是當我們隨著時間得到觀察值的同時判斷該觀察值是否為改變點。這個方法最主要的目標就是在改變點出現之後，能盡快發現改變點，並做出相對應的策略，在理想的狀況下，我們希望能在改變點出現的同時就能偵測到改變點的訊號。在此工作中，我們利用分析步長來決定可能的改變點。此外，在實務中，由於獨立的假設在大部分序列型資料是不成立的，我們考慮馬可夫鏈模型去描述資料之間的相關性。並且使用有左尾相關性的Clayton copula與右尾相關性
的Joe copula來描述聯合分配，邊際分配的部分為二項分配。在實證分析中，珠寶製造的數據為分析資料，目的是盡早找出改變點的發生，並檢查製造珠寶的過程，以降低製造珠寶的不良率。

摘要(英)

When the parameters of the marginal distribution for the sequential data are abnormally changed, we call it the changepoint, and changepoint detection is to detect
the occurrence of irregular changes in parameters. Changepoint detection is divided into online detection and offline detection. This paper mainly focuses on online changepoint detection. Online changepoint detection is to monitor whether the observation is a changepoint or not when we obtain data over time. The purpose is to detect the changepoint as soon as possible after the changepoint appears, and make corresponding strategies. We
focus on fraction nonconforming change by using the most possible run length to identify changepoint in this paper. We compute the probability of the run length distribution to
identify the most possible run length at each time. In addition, since it is not practical to assume the data is independent in most cases, we describe the dependence structure by using the copula-based Markov model. In this work, we consider the Clayton copula and
the Joe copula and the marginal distribution is the binomial distribution. In empirical analysis, jewelry manufacturing data analysis is for the purpose of finding out when the
changepoint occurs as early as possible and checking the process of manufacturing the jewelry, so as to reduce the probability of broken jewelry.

關鍵字(中)

★ 改變點
★ Copula
★ 尾部相關
★ 馬可夫鏈模型
★ 步長

關鍵字(英)

★ changepoint
★ Copula
★ tail dependence
★ Markov model
★ run length

論文目次

Contents

Abstract ii
Contents iii
1 Introduction 1
2 Background Knowledge 4

2.1 Settings and notations for the proposed methods . . . . . . . . . . . . . . . 4
2.2 The Bayesian Online Changepoint Detection Method . . . . . . . . . . . . 6

3 Copula-based Markov model 10

3.1 Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 The Chi-plot and tail dependence . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 First-order Copula-based Markov model . . . . . . . . . . . . . . . . . . . 15

4 Proposed the method of online changepoint detection 19

4.1 Model assumption and notations . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Proposed method for online changepoint detection . . . . . . . . . . . . . . 20

5 Simulation Study 24

5.1 Data generation and performance measure . . . . . . . . . . . . . . . . . . 24
5.2 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.2.1 Performance on the proposed method . . . . . . . . . . . . . . . . . 26
5.2.2 Sensitivity analysis for Binomial distribution parameters . . . . . . 35
5.3 Results for comparing with the independent model . . . . . . . . . . . . . 57

6 Empirical Study 66

6.1 Jewelry data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2 Empirical result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7 Conclusions 71

References 72

Appendices 75

A R codes for our proposed method under the Clayton copula 75
B R codes for our proposed method under the Joe copula 79

參考文獻

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指導教授

孫立憲(Li-Hsien Sun)

審核日期

2022-9-16

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