臺灣博碩士論文加值系統

氣膠的粒徑對氣膠動力與捕集具有密切的關係，使用統計方法求得粒徑分佈與粒徑分佈參數是基本的氣膠分析工作。目前普遍使用的微軟 Excel 試算表程式含有優化計算的規劃求解功能進行非線性迴歸，可以求取與給定粒徑分佈型態具最小方差和或最大概似性的氣膠粒徑分佈，並得到分佈函數中各不同成分的比例與分佈參數。在以往的研究中，曾與實驗室所產生的氣膠比較，得到相當準確的結果。
然而，在使用於現場狀況下，預期仍會遭遇到各種問題，其中最基本的問題就是若兩氣膠粒徑分佈相當接近，上述方法是否仍可以得到正確的結果？如果要使用上述方法得到正確的結果，氣膠分佈函數必須要滿足何種條件。
本研究組合兩組粒徑成對數常態分佈的氣膠，經過各種成分比例與分佈參數組合，得到可以使用規劃求解得到正確結果的範圍。所使用的評估方式包括最小方差和與最大概似性，其中最小方差和尚包括比對分佈函數、各粒徑區間分率與累積分率。當兩組氣膠具相同的幾何標準差，經過 ANOVA 分析，若 ANOVA 參數小於某特定值時，所得到的結果與預期值會相當的差異（以計算結果與預期值的方差和做為指標）。當兩組氣膠具相同的中數粒徑時，使用 Bartlett's 檢驗，發現 Bartlett 參數非常接近 1 才會得到明顯誤差，也就是當兩組氣膠粒徑幾何標準差相當接近，或份量差異甚大時。
根據既有研究成果以及本研究所進行的大量運算結果經整理後，發現各種最小方差和法對氣膠成分及其分佈參數的解析度大略相同。然而概似性函數因本身計算即有較大的誤差，再加上對參數的敏感度較小，因此較不適用於需要較精確成分分析的狀況。但是若實務上容許較大的誤差時，使用最大概似性法反而具較快的運算速度與較大的適用範圍。
關鍵字：非線性迴歸、最小方差和、最大概似性、氣膠粒徑分佈、變異數分析

The size of an aerosol particle significantly affects the dynamic and capture properties of the particle. Therefore, analysis of particle size distribution is an essential task in the study of an aerosol sample. Currently, the Solver tool in Microsoft's Excel spreadsheet software can be employed to perform a nonlinear regression and determine a size distribution function which has the sum of least square error to a given distribution pattern. Hence, the fraction and distribution parameter of each component in an aerosol mixture can be determined. This method has been assessed successfully with the aerosol generated in the laboratory.
However, there is still one major concern. No study has shown the limit of above method to be employed to determine the fractions with similar distributions. The sum of square error and maximum likelihood method were employed. The former included the comparisons between distribution functions, fractions in each size range and cumulative fraction.
This study mixed two components of aerosol with given size distribution of lognormal and fractions. The Solver under the Excel was used to ‘un-mix’ the mixture and solve the fraction and distribution parameters of each components by optimization computation. The solved parameters and fractions were compared with given fractions and parameters. The optimization model was established based on least sum of square error and maximum likelihood. The least sum of square error model used three kinds of distributions to build the objective function: distribution function, bin fraction, and cumulative fraction.
With extensive computation, this study found while two components have the same geometrical standard deviation, a parameter given by ANOVA smaller than a certain number will obtain a significant error. While two components have the same count median diameter, above nonlinear regression will also fail when Bartlett’s parameter is close to 1.
By compiling the results of a previous study and present study, it was show that all least sum of square error gave a similar trend. Therefore the valid envelope for applying a least sum of square error was similar with least sum of square error method no matter what graph was used to build the objective function. Due to the inaccuracy of evaluating a likelihood function and the relatively less insensitivity of the parameters to the objective function, the maximum likelihood method was showed less desirable for an application with requirement of accurate component fraction. However, the maximum likelihood method is still suitable for an analysis with less accurate requirement, since it showed faster computation speed and larger envelope for valid computation.

Keywords：nonlinear regression, least sum of square error, maximum likelihood, aerosol size distribution, ANOVA

致謝 i
中文摘要 ii
Abstract iii
目錄 v
表目錄 vi
圖目錄 vii
符號說明 x
第一章 緒論 1
1-1 研究背景 1
1-2 氣膠粒徑分佈參數 5
1-3 接近對數常態分佈的粒徑分佈 10
1-4 氣膠混合物 14
1-5 非線性迴歸分析 18
1-5-1 最小方差和法 18
1-5-2 最大概似性迴歸 23
1-5-3 最小方差和與最大概似性 24
1-6 規劃求解 26
1-7 文獻回顧 31
1-8 研究目的 41
第二章 研究方法與步驟 43
2-1 每次分析運算流程 43
2-2 基本假設與粒徑範圍 44
2-3 混合–解混誤差分析 44
2-4 Excel 工作表 46
2-5 規劃求解參數設定 49
第三章 結果與討論 51
3-1 以粒徑區間分率進行最小方差和法迴歸（LSQ-ΔF） 51
3-2 以累積分率進行最小方差和法迴歸（LSQ- F） 52
3-3 以最大概似性法進行迴歸（ML） 54
第四章 結論與建議 60
4-1 結論 60
4-2 建議 61
附錄 最大概似性法與對數常態分配 62
參考文獻 63

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