平均積分平方誤差與最大概似估計法在非齊性卜瓦松過程下估計分段線性函數參數之比較

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鄭雯馨(Wen-Hsin Cheng) 查詢紙本館藏

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工業管理研究所

論文名稱

平均積分平方誤差與最大概似估計法在非齊性卜瓦松過程下估計分段線性函數參數之比較
(Comparison of Mean Integral Squared Error and the Maximum Likelihood Estimation for Parameter Estimation of Piecewise Linear Functions under Non-homogeneous Poisson Process)

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至系統瀏覽論文 (2025-7-25以後開放)

摘要(中)

非齊性卜瓦松過程(Non-homogeneous poisson process)是類似一般的卜瓦松過程，不同的是到達的平均速率會隨時間變化而改變。在給定一組具有不同分散度的點過程事件時間，可以將時間分段、事件分組得出各時間區間的速率函數，而此速率函數在每段區間上皆為線性並且在區間邊界上是連續的，我們稱這種函數為分段線性（Piecewise Linear）函數。本論文可以透過離散的事件時間來擬合未知的連續速率函數。
因此，擬合的第一步是選擇區間的數量，在過往的文獻中往往將區間長度限制為等長，而本論文將此限制移除，將區間上的邊界改成任意於時間軸上，希望能更廣義更準確的找出切割區間的方式。根據 Chen 和 Schmeiser (2019)中的平均積分平方誤差（MeanIntegrated Square Error）創建估計量，利用最佳化過程找出最佳的區間數使 MISE 最小。本文使用兩種方法的得到擬合速率函數，第一種方法是透過 Chen 和 Schmeiser (2014)速率積分特性和速率連續性，得出各區間的一次項係數及常數；第二種方法是 Glynn (2017)的最大概似估計法 (Maximum Likelihood)求出時間區間上的速率值，將各點速率值相連以得到擬合的分段線性函數。
本論文將設定十二組隨機真實分段線性函數，每組真實函數之下再隨機模擬出四組觀測值，觀察以上兩種方法並比較結論，最後將評估兩種方法的準確度並得出結論。並還原過往文獻中區間等長的設定，比較同一種方法之下，“區間等長”的限制是否影響估計的表現。

摘要(英)

The non-homogeneous poisson process (NHPP) is similar to the general poisson process, the difference is that the average rate of arrival will change with time. At a given set of point process events with different degrees of dispersion, time segments and events can be grouped to obtain a rate function for each time interval, and this rate function is linear in each interval and is on the boundary of the interval. Continuously, we call this function a piecewise linear (PL) function. In this paper, the discrete continuous event time can be used to fit the unknown continuous rate function.
Therefore, the first step of fitting is to choose the number of intervals. In the past literature,
the length of intervals is often limited to equal length. In this paper, this restriction is removed
and the boundary on the interval is changed to be arbitrary on the time axis. I hope to find a more general and accurate way to find the cutting interval. An estimate is created based on the Mean Integrated Square Error in Chen and Schmeiser (2019), and the optimization process is used to find the optimal number of intervals to minimize MISE. This article uses two methods to get the fitting rate function. The first method is through Chen and Schmeiser (2014) rate integration characteristics and rate continuity to obtain the first-order coefficients and constants of each interval; the second method is Glynn (2017) ′s maximum likelihood estimation method (MLE) Find the velocity value in the time interval, and connect the velocity values at each point to get the piecewise linear function of the fit.
In this paper, twelve sets of random real piecewise linear functions will be set. Under each set of real functions, four sets of observations will be randomly simulated to observe the above two methods and compare the conclusions. Finally, the accuracy of the two methods will be evaluated and the conclusions will be drawn . And restore the setting of interval equal length in the previous literature, and compare whether the limitation of "interval equal length" affects the estimated performance under different methods.

關鍵字(中)

★ 非齊性卜瓦松過程
★ 分段線性函數
★ 平均積分平方誤差
★ 最佳化過程
★ 最大概似估計法

關鍵字(英)

★ Non-homogeneous poisson process
★ piecewise linear (PL) function、
★ Mean Integrated Square Error、
★ Optimization
★ Maximum Likelihood

論文目次

目錄
中文摘要 i
ABSTRACT ii
目錄 iii
圖目錄 v
表目錄 vi
第一章、緒論 1
1-1 研究背景與動機 1
1-2 研究目的 2
1-3 研究架構 4
第二章、文獻探討 5
2-1 非齊性卜瓦松過程 5
2-2 分段函數 6
2-2-1 分段常數函數 6
2-2-2 分段二次函數 7
2-3 Iterated-SMOOTH 8
2-4 生成非齊性卜瓦松過程的方法Thinning Method 9
2-5 Block-Circulant Matrix 10
第三章、研究方法 11
3-1 平均積分平方誤差 11
3-1-1 gMISE準則 11
3-1-2 積分特性和區間連續性 14
3-1-3 全局最小值 15
3-2 最大概似估計法 15
3-3 Thinning Method 16
第四章、實驗環境結果 17
4-1 根據需求調整模型 18
4-1-1 區間不等長調整gMISE準則 18
4-1-2 區間連續性 19
4-1-3 速率不為負值 19
4-2 最佳化過程 20
4-3 比較MISE與MLE模擬結果 20
4-4 比較區間等長與區間不等長模擬結果 25
第五章、結論與建議 31
5-1 結論 31
5-2 未來研究方向與建議 32
參考文獻 33
附錄 36
附錄一 36
附錄二 36

參考文獻

參考文獻
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指導教授

葉英傑(Ying-Chieh Yeh)

審核日期

2020-7-29

推文