基于馬氏距離的自適應高斯混合模型的設計與實現路徑目錄1引言．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．11.1選題背景及意義．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．11.2設計基本架構．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．2馬氏距離．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．2.1馬氏距離概述．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．2.2馬氏距離的幾何意義．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．2.3公式推導．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．2.4馬氏距離的優缺點．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3EM算法．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3.1極大似然估計．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3.2EM算法．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3.2.1推導．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3.2.2EM算法收斂性證明．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3.2.3EM算法性質．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．3.2.4EM算法應用．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．4高斯混合模型．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．4.1無監督學習．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．4.2模型結合及概述．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．．結束語參考文獻附錄1引言1.1選題背景及意義高斯密度函數屬于參數模型，包括單高斯模型（SingleGaussianModel,SGM）和高斯混合模型（Gaussianmixturemodel，GMM）兩種情況，SMG無法適應復雜的背景狀態只能進行微小的漸變，當混合具有不同分布的多個樣本時，單個高斯模型就沒辦法準確的顯示樣本特征，也無法清楚地分類樣本。為更精確的表示樣本所具有的不同統計規律，從而能更精確的描述樣本的統計特性，彌補SMG這方面的缺陷，因此引出高斯混合模型（陳浩宇，楊靜萱，2022）。如果有足夠多的高斯模型可以融合，并且它們之間的權重設置得當這個合理的高斯混合模型可以擬合任何分布的樣本，生成任何形狀的非線性函數，在這種理論框架的指引下可推導出并通過多次優化迭代來抵消隱藏的變量錯誤，從而生成更好的參數。為祛除數據之間的相關性，利用取自高斯分布部分參數所表示的馬氏距離來更好的描述具有不同統計概率的重疊率關系。無監督學習已經成為機器學習的趨勢，為實現自適應選擇，本文在查找大量文獻，進行對比分析算法性能，選擇貝葉斯相關學習方法。提出采用基于馬氏距離的自適應高斯混合模型確定最優數量的高斯混合模型并生成最優自適應高斯混合模型（邱天佑，秦文軒，2023）。1.2設計基本架構本研究針對高斯混合模型擬合訓練樣本成分數量難以確定問題、提出一種基于馬氏距離的增量高斯混合模型自適應確定成分數量區間、擬采用貝葉斯最優化準則通過百次運算法則確定最終自適應成分數量，實現高斯混合模型最優擬合給定的數據集，進行多次迭代優化，直至最優結果。最后通過對仿真數據集和實測數據集對所提算法進行性能評估。算法概述如下：（1）通過對數據的成分進行自適應分類來確定樣本間距。（2）根據貝葉斯信息準則（BIC）對數據進行分類，顯而易見的是并通過協方差獲得不同數據之間的位置關系，并確定類別。（3）分類的數據最適合于高斯混合模型，并經過持續迭代優化。（4）通過比較模擬和實際測量數據集來評估所提出算法的性能。2馬氏距離2.1馬氏距離概述馬氏距離（MD）由印度統計學家P.C.Mahalanobis提出，基于變量之間的相關性，通過該相關性可以識別和分析不同的模式，衡量未知樣本集與已知樣本集的相似性，是樣本點與分布之間的距離。它表示數據的協方差距離，且在總體樣本的基礎上進行計算（鄧嘉偉，李秀敏，2021）。也就是說如果拿同樣的兩個樣本，從這些反應可以推斷出放入兩個不同的總體中，最后計算得出的兩個樣本間的馬氏距離通常是不相同的，除非這兩個總體的協方差矩陣相同。對于一個均值μ=μ1，D同樣的當兩個數據點時，馬氏距離表示為：D其中∑是多維隨機變量所組成的協方差矩陣，μ為樣本均值，如果協方差矩陣是單位矩陣，也就是各維度獨立同分布，馬氏距離就變成了歐氏距離。本文在數據分析策略上，既運用了諸如描述性統計、回歸分析等傳統統計方法，也納入了現代數據挖掘技術及其算法。像利用聚類分析識別數據內部結構，或是通過決策樹進行趨勢預測。這些先進技術增強了對復雜現象的理解能力，并有助于挖掘大數據中的潛在關系。同時，本文注重融合定量與定性研究方法，力求提供一個全方位的研究視角。2.2馬氏距離的幾何意義用主成分分析將變量進行旋轉，使維度之間相互獨立；進行標準化，使維度同分布。由PCA可知，主成分就是特征向量的方向，每個方向的方差即對應的特征值，故按照特征向量的方向旋轉，再縮放特征值倍數就可得所求（尤智淵，吳芳菲，2021）。在多維高斯向量中，A圖1.卡方分布（A）和生成的數據與擬合的A圖1.卡方分布（A）和生成的數據與擬合的GMM的混合分量之間的馬氏距離（B）B2.3公式推導通過上述分析結果看首先將樣本點旋轉至主成分，使其維度間線性無關，假定此時坐標為：Aμ（其次，變換之后維度的特征值為方差，且其維度間線性無關，則有：A?==將馬氏距離進行規范化，旋轉縮放之后即為歐式距離，故馬氏距離的計算公式為（侯俊杰，寧曉紅，2022）：D===2.4馬氏距離優缺點在這個大前提下歐氏距離（Euclideandistance），是一個通常采用的距離定義，是在k維空間中兩個點之間的直線距離。在二維和三維空間中的歐氏距離的就是兩點之間的距離。但是在大部分的統計類問題中，坐標往往會有不同程度的波動。此時馬氏距離的提出，在這樣的前提之下有效的解決的這類問題（余睿德，穆俊馳，2023）。它展示了兩個服從于同一分布的且協方差矩陣為∑的隨機變量的差異程度。但是由于協方差矩陣的影響，馬氏距離的計算并不穩定。基于當前階段性的研究成果總結，對后續工作產生了指導意義。尤其是在研究方法上，本文識別出多個方面可以進行優化的空間。前一階段的經驗教訓為本文展示了哪些做法有效，哪些則需改進或淘汰。比如，在數據采集時，本文需要更加關注樣本的廣泛性和代表性，確保選取的樣本能夠全面代表目標人群的特點。另外，面對不同研究議題時，采用多種數據搜集技術有助于增強數據的覆蓋面和可信度。因為它有求逆矩陣的過程，協方差矩陣必須滿秩，所以要求數據要有原維度等量的特征值。在計算馬氏距離過程中，要求總體樣本數大于樣本的維數，否則得到的總體樣本協方差矩陣逆矩陣不存在。雖然滿足條件總體樣數大于樣本維數，但是協方差矩陣仍不存在也需要采用歐氏距離計算(邱雨昕,唐羽澄,2022)。同時，馬氏距離不被量綱所影響，且與原始數據的測量單位無關，原始數據和均值的差計算出的兩個樣本點的馬氏距離是一樣的，從這能見其概此外還可以排除變量之間的相關性的干擾(許睿羽,黃澤謙,2022)。因此，本文提出采用基于馬氏距離的自適應高斯混合模型。在經典歐氏距離的求解中，我們很直觀可以看出其分布均為球型分布，也就是說歐氏距離只適用于理想情況下的，歐幾里得空間內的直線距離；若有若干復雜模型，因互相之間的干擾構成橢球型，通過上述事實能知曉歐氏距離明顯無法完美的解決。馬氏距離就應運而生，在各變量符合正態分布的情況下，排除各成分之間的量綱干擾，無單位影響，擴大一些微小的量，排除離群值(孫羽航,周佳慧,2021)。在圖2.中，四個點到觀測中心的歐氏距離都相等，然而卻并不是都屬于該類。此時的藍色點的馬氏距離明顯要比黃色點的馬氏距離小，我們有理由判斷藍色點更有可能屬于這個類（郭浩，韓雨萱，2022）。圖2.馬氏距離圖解圖2.馬氏距離圖解3自動模型的選擇這在一定層面上表露大量的無監督學習中，聚類和降維的方法成為最基本的問題。高斯混合模型因為其包含的協方差矩陣，需要有充足的運行數據才能夠保證模型參數的準確性；但是當將其協方差矩陣換為對角協方差矩陣時，則需要足夠多的高斯模型才能提供較高的識別能力。在高斯混合模型中的降維方法中，在這般的環境中局部因子分析成為主流方法。通過該方法，降低協方差矩陣的自由度，提高準確性(鄧澤洋,吳彤彤,2021)。數據分析期間，本文借助多種統計方法來核實數據的可靠性，并探尋潛在的異常數值。通過深入探究數據分布特性，本文精確地移除了異常數據點，同時確保核心樣本信息得到保留。除此之外，本文還利用敏感性實驗來測定參數波動對研究結論的穩定性及通用性的影響。為選擇局部因子分析的成分數量和局部的維度時，通常借助的統計準則之一是極大似然學習。但是其計算較為復雜，在1994年提出BayesianYing-Yang(BYY)，經過不斷地發展完善，成為通用的學習框架。BYYharmonylearning由BYY系統和基本的harmonylearning原則組成，與局部因子分析組合，成為一種正則化的方法，執行參數學習和自動模型選擇(何炳福,周志時,2022)ADDINCSL_CITATION{"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1007/11840930_27","ISBN":"3540388710","ISSN":"16113349","abstract":"AfurtherinvestigationismadeonanadaptivelocalfactoranalysisalgorithmfromBayesianYing-Yang(BYY)harmonylearning,whichmakesparameterlearningwithautomaticdeterminationofboththecomponentnumberandthefactornumberineachcomponent.Acomparativestudyhasbeenconductedonsimulateddatasetsandseveralrealproblemdatasets.ThealgorithmhasbeencomparedwithnotonlyarecentapproachcalledIncrementalMixtureofFactorAnalysers(IMoFA)butalsotheconventionaltwo-stageimplementationofmaximumlikelihood(ML)plusmodelselection,namely,usingtheEMalgorithmforparameterlearningonaseriescandidatemodels,andselectingonebestcandidatebyAIC,CAIC,andBIC.ExperimentshaveshownthatIMoFAandML-BICoutperformML-AICorML-CAICwhiletheBYYharmonylearningconsiderablyoutperformsIMoFAandML-BIC.Furthermore,thisBYYlearningalgorithmhasbeenappliedtothepopularMNISTdatabasefordigitsrecognitionwithapromisingperformance.?Springer-VerlagBerlinHeidelberg2006.","author":[{"dropping-particle":"","family":"Shi","given":"Lei","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Xu","given":"Lei","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"LectureNotesinComputerScience(includingsubseriesLectureNotesinArtificialIntelligenceandLectureNotesinBioinformatics)","id":"ITEM-1","issue":"Ml","issued":{"date-parts":[["2006"]]},"page":"260-269","title":"Localfactoranalysiswithautomaticmodelselection:Acomparativestudyanddigitsrecognitionapplication","type":"article-journal","volume":"4132LNCS"},"uris":["/documents/?uuid=d0548008-1e94-46a5-a4c6-1ae0e7632c17"]}],"mendeley":{"formattedCitation":"[2]","plainTextFormattedCitation":"[2]","previouslyFormattedCitation":"[2]"},"properties":{"noteIndex":0},"schema":"/citation-style-language/schema/raw/master/csl-citation.json"}[2]。在文獻ADDINCSL_CITATION{"citationItems":[{"id":"ITEM-1","itemData":{"DOI":"10.1007/11840930_27","ISBN":"3540388710","ISSN":"16113349","abstract":"AfurtherinvestigationismadeonanadaptivelocalfactoranalysisalgorithmfromBayesianYing-Yang(BYY)harmonylearning,whichmakesparameterlearningwithautomaticdeterminationofboththecomponentnumberandthefactornumberineachcomponent.Acomparativestudyhasbeenconductedonsimulateddatasetsandseveralrealproblemdatasets.ThealgorithmhasbeencomparedwithnotonlyarecentapproachcalledIncrementalMixtureofFactorAnalysers(IMoFA)butalsotheconventionaltwo-stageimplementationofmaximumlikelihood(ML)plusmodelselection,namely,usingtheEMalgorithmforparameterlearningonaseriescandidatemodels,andselectingonebestcandidatebyAIC,CAIC,andBIC.ExperimentshaveshownthatIMoFAandML-BICoutperformML-AICorML-CAICwhiletheBYYharmonylearningconsiderablyoutperformsIMoFAandML-BIC.Furthermore,thisBYYlearningalgorithmhasbeenappliedtothepopularMNISTdatabasefordigitsrecognitionwithapromisingperformance.?Springer-VerlagBerlinHeidelberg2006.","author":[{"dropping-particle":"","family":"Shi","given":"Lei","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Xu","given":"Lei","non-dropping-particle":"","parse-names":false,"suffix":""}],"container-title":"LectureNotesinComputerScience(includingsubseriesLectureNotesinArtificialIntelligenceandLectureNotesinBioinformatics)","id":"ITEM-1","issue":"Ml","issued":{"date-parts":[["2006"]]},"page":"260-269","title":"Localfactoranalysiswithautomaticmodelselection:Acomparativestudyanddigitsrecognitionapplication","type":"article-journal","volume":"4132LNCS"},"uris":["/documents/?uuid=d0548008-1e94-46a5-a4c6-1ae0e7632c17"]}],"mendeley":{"formattedCitation":"[2]","plainTextFormattedCitation":"[2]","previouslyFormattedCitation":"[2]"},"properties":{"noteIndex":0},"schema":"/citation-style-language/schema/raw/master/csl-citation.json"}[2]中，比較了局部因子分析的性能，測試的范圍為極大似然估計與AIC，CAIC，BIC三者的分別組合，為了避免初始化導致的局部最優，分別進行十次BYYharmony算法并且對應實施EM算法(許珂茜,付明哲,2019)。進行100次模擬得到平均值。結果表明BYY-LFA在性能和計算時間上的表現是最好的。為了驗證和優化理論結構，本文積累了豐富的數據材料。這些數據不僅涉及多樣的研究對象，還橫跨不同的歷史時期和社會背景，為理論框架的全方位驗證提供了重要依據。借助統計軟件對數據進行解析，有助于確認理論中的各項假設是否成立，并發現其不足。后續的研究工作考慮加入更多因素或者采用更大量的樣本，以提升理論框架的有效性和前瞻性。4極大似然估計極大似然估計（MLE）就是在知道結果的情況下，通過概率的計算比較，找到最有可能導致這個結果的參數值，就需要考慮到先驗概率和后驗概率ADDINCSL_CITATION{"citationItems":[{"id":"ITEM-1","itemData":{"ISBN":"9781467347143","author":[{"dropping-particle":"","family":"Wang","given":"Pingbo","non-dropping-particle":"","parse-names":false,"suffix":""},{"dropping-particle":"","family":"Wang","given":"Yu","non-dropping-particle":"","parse-names":false,"suffix":""}],"id":"ITEM-1","issued":{"date-parts":[["2013"]]},"page":"1454-1458","publisher":"IEEE","title":"TwoIterativeAlgorithmsforMaximumLikelihoodEsitimationofGaussianMixtureParameter","type":"article-journal"},"uris":["/documents/?uuid=ebf02573-8fec-40a3-a5ce-9a2aae377728"]}],"mendeley":{"formattedCitation":"[3]","plainTextFormattedCitation":"[3]","previouslyFormattedCitation":"[3]"},"properties":{"noteIndex":0},"schema":"/citation-style-language/schema/raw/master/csl-citation.json"}[3]，在此，先介紹一下較為簡單的貝葉斯準則。經典的貝葉斯公式如下（林志博，何夢琪，2021）：p其中，pa為先驗概率，即模型內不同類別的分布概率；pab為類條件概率;相應的pba在統計學的實際應用中，似然函數因結構復雜導致難以計算最大化時，通常會有參數估計的問題ADDINCSL_CITATION{"citationItems":[{"id":"ITEM-1","itemData":{"ISBN":"9788578110796","ISSN":"1098-6596","PMID":"25246403","abstract":"Hasilpenelitian,sistemkomisimerupakansalahsatusistempenjualan(sistemoperasi)jasaangkutantaksiyangberlaku/diterapkandiDKIJakarta.Penelitianinimencobamengungkapduahalyangberkaitandengansistemkomisiditinjaudariaspekpengemudi,yaitu(1)bagaimanasikappengemuditaksiterhadapsistemkomisijasaangkutantaksidan(2)bagaimanatingkatkepuasanpengemuditaksiterhadapsistemkomisijasaangkutantaksi.Hasilkajianmenunjukkansikappengemuditaksibluebirdterhadapsistemkomisijasaangkutantaksibluebirdumumnyarelatif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準確無誤。借助嚴格的數據甄選機制和規范化的處理流程，提升了信息品質，同時也強調了信息流的公開透明和可追溯性。在指定混合模型的初始值時，有以下方法(徐星宇,李若彤,2020)：在g個成分的混合模型的獨立數據，E-step的作用在更新該成分的后驗概率，可通過以下方法找到第一個E-step：指定τj(0)（j=1，…,n）來執行第一步，τZ將整體分為x個組件，比如在g=1的正態分量和對應的協方差矩陣混合的情況下，采用初步劃分數據的方法，在這般的環境中提供兩個p變量。對于高維