【教材專區】學習AI有困難？ 讓Cupoy助教來幫你！
學習資料科學時遇到問題，希望能有專業人士線上即時為您解答嗎？ 想提高模型準確率，卻不知從何問起嗎？ 如果您有以上的需求，歡迎來參加我們每週一次的線上助教時間！
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2022/05/14（六）20:00-21:00 助教時間
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【QA】有哪些文字表示模型？它們各有什麼優缺點？
文字本身是一種非數值化的資料，無法直接被計算。 我們需要將文字轉換成機器看得懂的形式。我們將文字訊息轉成結構化的電腦可讀資訊，就稱作：單詞表示 (word representation) 。 文字本身是一種非數值化的資料，無法直接被計算。 我們需要將文字轉換成機器看得懂的形式。我們將文字訊息轉成結構化的電腦可讀資訊，就稱作：單詞表示 (word representation) 。 獨熱編碼(One-hot encoding)： 整數編碼(Integer encoding) 整數編碼英文是Integer encoding： 形式跟獨熱編碼(One-hot encoding)相同，但我們會用數字直接代表單詞，以此來減少使用空間。 但是以上的方式因為都是用數字來表達單詞，不同文字之間，沒有差異，所以無法清楚了解詞之間的差異，因此後續就有詞嵌入向量 (Word Embedding)的概念， 詞嵌入向量 (Word Embedding) 詞嵌入向量的概念是：“用語言模型表示詞的特徵，把我們所知道的單詞投射到連續的向量空間中，每個單詞都用向量來表達”。 詞袋 (Bag-of-words) 模型 詞袋 (Bag-of-words)模型的概念是，為了用最簡單的方式表達出一句話或一篇文章，我們會直接將這句話或這篇文章中出現過的文字全部加起來，可以想像成我們把所有的單詞放進一個袋子當中(詞袋)。這些文字就會被視作一系列詞的集合，我們就稱作詞袋 (Bag-of-words)。 ，這裡有一篇英文的文章，我們將所有出現的單詞找出之後將其建造一個字典，也就是所有的單詞需要有對應的索引 (index)，其中沒有順序的限制，但固定後就不可改變。 只要是那個文本內有出現的單詞取出，找到對應的 index，將向量中這個位置的值 +1。 例如我們在一個文本中找到兩個 good，good 對應到的 index 為 5，所以我們就在向量 [5] 的位置 +2，變為 [0, 0, 0, 0 , 0, 2,.....]。 Bag-of-words的優點是它直觀，操作容易，並且不需要任何預訓練模型，可套用在任何需要將文字轉向量的任務上。 但也有顯而易見的缺點，就是無法表達前後語意關係，詞袋 (Bag-of-words)模型就是一個像袋子，而袋子裡的東西是同時裝在一起的，並沒有一個排列的順序關係。無法呈現單字含義：許多單字有多種不同含義，以此兩個句子為例："購買蘋果手機"&"去菜市場購買蘋果"，這兩句話中的蘋果意義不相同，但在詞袋 (Bag-of-words)模型中無法呈現其含意的不同。當文字數量太多時，會形成稀疏矩陣，不利於部分模型訓練。假設我們訓練的 corpus 內有 100000 個單字，那要表達每一個單字就是(1,100000) 的 vector，其中絕大部分都是0的數值。 儘管詞袋 (Bag-of-words)模型有諸多不足的地方，然而在較為簡單的情境下，其效果仍然相當不錯。 詞嵌入向量 (Word Embedding)是自然語言處理當中相當重要的領域，在詞袋 (Bag-of-words)模型之後有許多改進的方式陸續被提出，如 TFIDF、Word2Vec、GloVe到近期的基於上下文的詞向量( BERT, ERNIE, RoBerta )。 TF-IDF 詞頻(Term Frequency, TF)：某一特定單詞在整個文件中出現的頻率 計算公式：特定單詞出現在整個文件的次數/該文件中所有單詞的數量 逆向檔案頻率(Inverse Document Frequency, IDF) : 整個文件的數量與某一特定單詞文件的數量的比率 計算公式：Log (所有文件的數目/包含這個單詞的文件數目) TF-IDF就是將TF乘上IDF：TF * IDF，我們用這個數值來表現出一個單詞在文件中的重要程度。 綜合上述，當一個單詞的 TF * IDF 值越大時，代表這個單詞對整段文章的重要性也越大，我們可以歸納出： 不同單詞在同一個文章中獲得的 TF-IDF 值可能不相同，值的高低代表了單詞對整段文章的重要性。同一個單詞在不同文章所得到的 TF-IDF 值也可能不同。 Word2Vec word2vec為 Google的研究團隊於2013在 ”Efficient Estimation of Word Representations in Vector Space” 所提出，是一個用來表示字詞的方法。 他們提出了兩種訓練詞嵌入的模型： 連續詞袋模型 (Continuous Bag Of Words, CBOW) 跟 Skip-Gram 連續詞袋模型(CBOW)是一個類神經網路模型，是由”相鄰”的上下文來推測目標單詞(Target word)的模型架構，是一個讓機器選填空題的概念(一句話挖掉中間的字，請你填空)。 Skip-gram的概念與連續詞袋模型(CBOW)類似，但Skip-gram模型是使用目標單詞(中間的單詞)來推測周圍單詞(上下文)。是一個讓機器給定一個詞，去無限猜測前後文的概念(一句話只給中間的字，請你發揮想像力拼湊出上下文)。 GloVe GloVe (Global Vector for Word Representation)是由史丹佛(Stanford)研究 團隊所創建的,他是對 Word2Vec 方法進行擴展。 是一種基於全局詞頻統計 (count- based & overall statistics) 作為單詞表示 (word representation) 的工具,它以實數 組成的向量標示每一個單詞。如右圖所示,這些向量捕獲了單詞之間的一些語 義特徵,例如相似性,類比 等。 我們可以通過對向量的操作來計算兩個單詞之間的語義相似性,例如歐幾里德距離或餘弦相似度 (cosine similarity)。 GloVe綜合了潛在語義分析 (Latent semantic analysis note, LSA)、 CBOW 的優點,模型訓練速度更快,對大型語料庫具有良好的可擴展性、在小型 語料庫或小向量表現亦良好。
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【QA】Batch Size 是甚麼？
「Epoch」、「Batch size」、「Iteration」、「Learning rate」這些字眼，在人工智慧這領域很常見，訓練 AI 模型的過程，時常需要調整這些參數來影響模型的準確度。 Epoch 中文直翻的話是「時代」、「時期」的意思，但在神經網路訓練上，比較接近中文「期」的含意，指的是訓練模型過程裡，演算法完整使用過資料集每筆資料的狀態。 假設我們有一個資料集，有 1000 筆資料，把它們全丟進神經網路運算完，那就是完成了 1 個 Epoch 的訓練。 Batch size Batch 中文直翻的話意思是「批次」。 假設剛剛的 1000 筆資料，每筆數據量都很大，電腦因為記憶體不夠用所以無法一次跑 1000 筆。這時我們只好把 1000 筆資料分批送進神經網路，這裡分批的數量就是 Batch size，今天我一次放 500 筆資料進神經網路運算，那 Batch size 就等於 500。 Iteration Iteration 意思是「迭代」，我們將資料分批送進神經網路的時候，會透過迴圈來執行。 延續剛剛的例子，我們如果一次送 500 筆資料進神經網路，那就要跑 兩 次迴圈才算 1 個 Epoch，這時就稱訓練過程迭代了 兩 次。 Data set size = Iteration * Batch size （1 Epoch） 這時換個角度來說，如果開發者決定要用 Batch size = 100 的條件，來訓練 5 個 Epoch，那總共要迭代幾次？ Iteration = （Data set size / Batch size）* Epoch 代入數字就是 1000 / 100 * 5 = 50，總共要迭代 50 次才能完成開發者的期望。
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【QA】Cross entropy 作為損失函數帶來哪些好處？
在分類任務中，我們希望模型可以分辨出不同類別的資料，希望分類出來的「錯誤率」越小越好，或是說正確率越大越好。 右邊上面第一個數學式子錯誤率( error rate )數學表式為： y表示實際值、ŷ表示預測值、n為資料大小。 但是真正進行機器學習的時候，我們不會拿模型分類結果的錯誤率( error rate )直接當作損失函數， 但我們設計損失函數時，當中要有錯誤率的隱含意義。 這裡介紹交叉熵(cross-entropy)之前，我們需要了解熵(Entropy)是什麼。 熵(Entropy)：是從熱力學引用過來的一個名詞，在資訊理論中，代表接收的每個消息之中包含之資訊的平均量。 熵可以直接理解成衡量不確定性的度量衡，越隨機發生的事件熵越大。 例如以下兩個例子。 訊息量非常大，是一件不太尋常、不太可能發生的事件： 昨天投資虛擬貨幣不小心賺了100萬，於是去買了一台汽車。 常理或是不可能改變的真理，訊息量非常低： 水從高處往低處流。 熵(Entropy)的數學表示為這樣 交叉熵(Cross-entropy)：測量"預測機率分布"與"實際機率分布"的誤差範圍，將各類別的熵(Entropy)算出來之後，再相加算總和。 當預測結果與實際結果差距很多，代表此結果所含有的資訊量非常大，不確定性很多，交叉熵(Cross-entropy)就會非常高。 交叉熵(Cross-entropy)的數學表示為： C是類別數、n是資料總數、 yc,i是第i筆資料屬於第c類的正確類別、 pc,i是第i筆資料屬於第c類預測出來的機率。 那Cross entropy 作為損失函數帶來哪些好處呢？ 可以看到數學式中，交叉熵會考慮每個類別機率值跟正確值的差異，因此可以修正每次分類錯誤的結果，一次一次的優化模型
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【QA】常用的激活函數有哪些？
一般機器學習使用之神經元如下圖： 輸入有三個x output有一個，這時我們需要一個啟動的機制來讓資訊從輸入傳遞給輸出 這時我們需要一個閥值機制(threshold ) 我們稱這個函數叫激活函數Activation functions。 使用的激活函數、激勵函數為下： 常用的激活函數有： 第一個是Sigmoid： Sigmoid 函數（Sigmoid Function）又稱為 Logistic 函數，是一個在生物學中常見的 S 型函數，也稱為 S 型生長曲線，它將負無窮到正無窮的區間映射到了 0 到 1 中。很容易理解和應用，但他有以下幾個缺點： 1梯度消失問題2它的輸出不是以0為中心。表示它的梯度更新在不同的方向上且走得太遠。 0 <output <1，使優化更加困難。而且收斂比較慢 再來是雙曲正切函數——Tanh：它的輸出是以0中心的，因為它的值區間在-1到1之間，即-1 <output <1。 因此，使用此激活函數模優化型更容易一點，從而其在實踐應用中總是優於Sigmoid函數。 但它依然存在著梯度消失問題。 那麼我們該如何處理和糾正梯度消失問題呢？ 我們使用ReLu -Rectified linear units（線性修正單元）：在過去幾年中此激活函數就已經非常受歡迎了。 最近證明，相較於Tanh函數，它的收斂性提高了6倍。如果輸入<0，輸出= 0，如果輸入> = 0，則輸出=輸入。 因此，只看這個函數的數學形式，我們就可以看到它非常簡單、有效。它可以避免梯度消失。現在幾乎所有深度學習模型現在都使用ReLu函數。但它的局限性在於它只能在神經網絡模型的隱藏層中使用。 ReLu函數的另一個問題是，一些梯度在訓練過程中可能很脆弱，甚至可能會死亡。它可以導致權重更新，這將使其永遠不會在任何數據點上激活。簡單地說ReLu可能會導致死亡神經元。 為了解決這個問題，我們引進了另一個被稱為Leaky ReLu的修改函數，讓它來解決死亡神經元的問題。它引入了一個小斜坡從而保持更新值具有活力。 然後，我們還有另一個變體，它形成於ReLu函數和Leaky ReLu函數的結合，我們稱之為Maxout函數。
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【QA】過擬合、欠擬合與誤差之間的關係？
損失又有分作兩種：偏差(Bias)與方差(Variance) 偏差(Bias)：打得準，離靶心越遠偏差越大。 方差(Variance)：打得平均、打得明確，越隨機方差越大。 模型訓練的過程中，偏差(Bias)與方差(Variance)的變化基本上會如同圖所示： 在機器學習訓練模型的過程中，驗證資料集的誤差曲線如下： 而當偏差過大或方差過大時會產生兩種問題： 過擬合(Overfitting)：模式複雜度過高，Bias平方降低，Variance增加。 欠擬合(Underfitting)：模式複雜度過低，Variance降低，Bias平方增加。 下方圖形為模型切合實際資料的狀態，由左至右，分別是Underfitting, Good fitting, Overfitting： 我們希望最小化損失函數的值，讓模型的預測與正確答案越接近越好。
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【免費直播】學習AI有困難？ 讓Cupoy助教來幫你！ - 第一堂分享會 - YouTube
筆記連結：https://www.cupoy.com/collection/0000018008CD5D70000000046375706F795F72656C656173654355?layoutType=content提問表單：https://forms.gle/P7XELxgJncn7kL6y6Zoom 會...
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【QA】有哪些文字表示模型？它們各有什麼優缺點？
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【QA】Batch Size 是甚麼？
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【QA】Cross entropy 作為損失函數帶來哪些好處？
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【QA】常用的激活函數有哪些？
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【QA】過擬合、欠擬合與誤差之間的關係？
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2022/05/17（二）20:00-21:00 助教時間
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【QA】什麼是降維（Dimension Reduction)？
https://drive.google.com/drive/folders/1WoTFt_kFeMQN9H0bB-t_FWVTfOXyDnE8?usp=sharing 廣義問題1,2,3 1.什麼是降維? 降維廣義上來說是指通過保留一些比較重要的特徵，去除一些冗餘的特徵，減少數據特徵的維度。 而特徵的重要性取決於該特徵能夠表達多少數據集的資訊，也取決於使用什麼方法進行降維。 一般情況會先使用線性的降維方法再使用非線性的降維方法，通過結果去判斷哪種方法比較合適。 2.甚麼情況下用? 1）特徵維度過大，可能會導致過擬合時 2）某些樣本數據不足的情況（缺失值很多） 3）特徵間的相關性比較大時 3.降維的好處? （1）節省存儲空間; （2）加速計算速度，維度越少，計算量越少，並且能夠使用那些不適合於高維度的演算法; （3）去除一些冗餘的特徵（原數據中既有平方米和平方英里的特徵--即相關性大的特徵） （4）便於觀察和挖掘資訊（如將數據維度降到2維或者3維使之能可視化） （5）特徵太多或者太複雜會使得模型過擬合。 降維方法兩大類: 1.特徵選擇(Feature Selection): Backward Elimination 反向淘汰 Forward Selection 順向選擇 Bidirectional 雙向淘汰 2.特徵提取(Feature Extration): 主成分分析(PCA) 核函數主成分分析 (KernelPCA) LDA Note: 通過提取之後產生的自變量跟原來的自變量完全不一樣了。 主要降低自變量個數 當數據裡面有m個自變量時，主成分分析PCA可以從所有自變量中提取出p個新的自變量(p<=m)， 這p個自變量可以較好的解釋數據自變量的差異性(方差)，無論因變量是多少。 PCA是一個非監督式學習，因為它的因變量不被考慮在內。 通常在數據預處理當中 PCA:尋找最大方差所屬方向 後來的自變量為原本自變量重組後的結果。 總結一些降維方法
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【QA】如何使用PCA(主成份分析)做資料降維？
step1 : 將X記做包含了所有m個自變量的矩陣，對X進行特徵縮放 step2 : 計算X的方差矩陣， 並將這個方差矩陣記做A; step3 : 計算矩陣A的特徵值(eigenvalue)和特徵向量(eigenvectors); step4 : 選擇需解釋方差的百分比P， 並選取前p個特徵值使得: stpe5 : 對應前p個特徵值對應的特徵向量就是最終選取的主要成分。 運用投射矩陣(這個矩陣由選取的p個特徵向量組成)，最初的m維特徵空間被投射到p維空上。 直接看code! Kernel PCA: 線性不可分--> 透過kernel PCA將資料投射到高維空間。 資料在高維的空間中就變成線性可分， 接著再進行一般的主成分分析PCA，以達到降維效果。
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【QA】LDA和PCA有什麼相同和相異之處？
先導數學知識準備 在正式深入理解之前還是需要鋪墊一些線性代數的基本概念及其計算公式，以方便大家理解之後的數學符號以及矩陣運算的推導過程。 1.1.1 內積與投影 兩個向量A,B內積的計算公式爲， 1.1.2 方差 用於描述數據的離散程度，計算公式爲， 一般爲了簡便計算，我們會對數值進行去中心化處理，即mu爲0，則此時計算公式可簡便爲， 1.1.3 協方差 用於描述倆個變量之間的相關性，數值去中心化處理以後，其計算公式爲， 利用該公式可以推廣出矩陣X的協方差矩陣的計算公式爲， 1.1.4 矩陣求導的性質 矩陣的求導，有如下性質， 矩陣的跡（即矩陣的對角線元素之和）有如下性質， 1.2 PCA基本概念理解 PCA（Principal Component Analysis），中文名爲“主成成分分析”。顧名思義，其目的就是找到高維數據中的主成分，並利用“主成分”數據，來表示原始數據，從而達到降維的目的。借鑑一個簡單的例子，假設有一組數據存在於三維空間的一個平面上（此時需要3個維度來表徵數據向量），若我們選擇旋轉座標軸使得數據所在平面與x,y平面重合，則此時我們只需要2個緯度即可表徵數據，且沒有丟失任何數據信息，這就是最簡單的數據降維。但是現實生活中的情況，往往數據特徵高達上百甚至上千維，我們很難直觀去找出一組基平面來完成對數據的降維，此時PCA就有其用武之地了。先直接上一張PCA分解後的圖來幫助大家理解。 如圖可見，綠色線爲第一主成成分方向，黑色線爲第二主成成分方向。仔細觀察一下圖，大家是否能得出如下倆個結論： （1） 樣本點在綠色線上的投影其離散程度要大於其在黑色線上的投影程度。（2） 樣本點到綠色線的平均距離都要小於其到黑色線的距離。 （2） 樣本點到綠色線的平均距離都要小於其到黑色線的距離。 兩個結論歸納爲最大可分析性和最近重構性。其中最大可分性可以理解爲我們希望降維過後的數據不影響後續我們對其的分類處理，其數據特徵的差異性仍然足夠強，也即方差最大；最近重構性可以理解爲我們希望降維過後的數據仍然保留有其主要的特徵，也即數據樣本點到這個超平面的距離和最小。 最大可分性理論的目標函數就是最大化數據在主軸上的投影的方差。假設現有一組已去中心化的數據 X 投影后的方差就是協方差矩陣的特徵值，而我們想要的最大方差，顯然就是協方差矩陣最大的特徵值，最佳投影方向就是最大特徵值所對應的特徵向量。 PCA降維的流程，如下: （1）樣本去中心化 （2）計算樣本的協方差矩陣 （3）對協方差矩陣做特徵值分解 （4）取最大的 個特徵值所對應的特徵向量 （5）計算投影矩陣 2.1 LDA基本概念理解 LDA（Linear Discriminant Analysis），中文名爲“線性判別分析”，是目前數據挖掘領域中比較經典且熱門的一種有監督的算法。從降維的層面考慮，其也是在尋找一個投影矩陣，使得投影之後數據樣本，同類的接近，而不同類的遠離。（其作爲分類器的時候，就可以對新的數據也進行投影，依據與哪一個類別接近來確定類別）。 仔細觀察投影後的結果，不難發現，LDA的中心思想就是最大化類間距離以及最小化類內距離。 LDA用於降維的流程，歸納如下， （1）計算每個類別的均值 ，全局樣本均值 （2）計算類內散度矩陣 ，全局散度矩陣 ，類間散度矩陣 （3）對矩陣 做特徵值分解 （4）取最大的 個特徵值所對應的特徵向量 （5）計算投影矩陣 PCA和LDA的相同點和不同點: LDA用於降維，和PCA有很多相同，也有很多不同的地方，因此值得好好的比較一下兩者的降維異同點。 首先我們看看相同點： 1）兩者均可以對數據進行降維。 2）兩者在降維時均使用了矩陣特徵分解的思想。 3）兩者都假設數據符合高斯分佈。 我們接著看看不同點： 1）LDA是有監督的降維方法，而PCA是無監督的降維方法 2）LDA降維最多降到類別數k-1的維數，而PCA沒有這個限制。 3）LDA除了可以用於降維，還可以用於分類。 4）LDA選擇分類性能最好的投影方向，而PCA選擇樣本點投影具有最大方差的方向。
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【QA】在對資料進行預處理時，應該怎樣處理類別型資料？
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請問有甚麼圖表可以直接展示出所有特徵值對於y的相關程度呢？
請問有甚麼圖表可以直接展示出所有特徵值對於y的相關程度呢? - Cupoy [特徵選擇]使用皮爾遜積差相關係數來做特徵選擇 - iT 邦幫忙::一起幫忙解決難題，拯救 IT 人的一天 (ithome.com.tw)
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【免費直播】學習AI有困難？ 讓Cupoy助教來幫你！ - 第二堂分享會 - YouTube
筆記連結：https://www.cupoy.com/collection/0000018008CD5D70000000046375706F795F72656C656173654355?layoutType=content 範例程式碼連結：https://drive.google.com/drive/u/2/fol... 備註：影片前三十分鐘為設備測試，學員觀看時可直接跳過測試環節。
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2022/05/28（六）20:00-21:00 助教時間
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【QA】如何給二分類模型打分數？
給模型打分數為何需要這麼複雜，我們可以知道在學習的過程中不會都順利，不論是訓練的資料或是模型本身，如果評估的方式很簡單，那麼就無法兼顧每個面向，意思就是不能只看到表面的數值。 Part 1 Accuracy 先以大家常見、常用的指標出發，Accuracy中文是準確率，從小到大如何得知準確率，例:考試題目有10題，答對5題，答對率就是50%，文字說明就是正確的比例，那為甚麼不用這個當作評估模型的指標就好，下面舉出另一個例子 今天訓練出兩個動物分類器，分類對象為90隻狗跟10隻貓: 1號分類器: 分類結果為85隻狗、15隻貓，代表有5隻狗和5隻貓被分類錯誤，準確率為95% 2號分類器: 分類結果為100隻狗、0隻貓，代表有10隻貓被分類錯誤，準確率為90% 可以發現兩個分類器的Accuracy差不多，以數值來看兩者效能接近，但就分類結果來看也知道1號分類器能夠對兩種動物分類，而2號分類器則完全無法對2號分類器分類，顯然2號分類器效能更差。 引出了Accuracy指標的問題，當兩個類別數量不平衡時，Accuracy就有可能會無法呈現出真正的意義。 Part 2 Confuse Matrix 為了避免因為類別不平衡導致的衡量問題，下面介紹混淆矩陣，如下表 其中有四種情況，分別是TP、TN、FP和FN，各自代表的意義為 1. TP(True Positive):正確預測成功的正樣本 2. TN(True Negative):正確預測成功的負樣本 3. FP(False Positive):錯誤預測成正樣本，實際為負樣本 4. FN(False Negative):錯誤預測成負樣本，實際為正樣本 舉以下例子，假設有8張圖片，5隻狗跟3隻貓，狗為0(正樣本)貓為1(負樣本)，拿一個訓練好的模型對這8張圖預測，實際情況為:[0,0,0,0,0,1,1,1]，預測結果為:[1,1,0,0,0,1,0,1]，此時將結果放到混淆矩陣內為下表 以左上角TP說明，TP為正樣本被預測為正的數量，正的為前五筆，前五筆被預測結果有三筆一樣為正，因此TP為3，其他TN、FP和FN同理，接著要介紹的評估指標會透過混淆矩陣的這四個指標去計算得出。 Precision Recall F1-score Accuracy Part 3 ROC/ AUC 首先ROC曲線會依照下面兩個指標繪製，AUC則是計算ROC曲線下的面積(界在0~1之間) 下面先看到TPR、FPR，一樣利用混淆矩陣計算。可以看到TPR分子為TP，因此TPR越大越好，FPR分子是FP，因此FPR越小越好。 有了TPR、FPR後，就可以繪製ROC曲線，但要繪製比較完整的ROC曲線需要多個閥值下的TPR、FPR，就會根據不同閥值下的TPR、FPR繪製ROC曲線。 這裡範例就取一個閥值下的TPR和FPR，如下面左圖，只有一條斜線，代表在這一個閥值底下的TPR和FPR相同，右圖的紅線為左圖紅線，新加入的綠線可以看到左上的點，那就是在設定的閥值底下，(FPR, TPR)的點。 此時計算兩張圖曲線下面積，左圖為0.5，因為曲線下面積只有右下的一半(完整為1)。右圖則是大略估計為0.8左右。AUC越高代表該模型效能越好。
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【QA】正規化(Normalization)為什麼能夠提升機器學習的成效？
直接舉例來說明，有兩筆資料，分別如下表 用網路流量當範例，平常在傳輸的都是正常流量(Normal)，少數時候會遇到攻擊流量(Attack)，上表顯示的封包大小和傳輸速度是分別在正常和攻擊下的數值，可以發現第一個現象為封包大小的最大值跟最小值差距較大，而傳輸速度中的最大值和最小值差距較小。 第二個現象為當今天要分辨是否為正常或攻擊時，兩個特徵都是可以觀察的，以封包大小為例，正常流量傳輸時封包都會有一定內容，而攻擊者只是為了傳輸大量封包，因此攻擊的封包大小較小。傳輸速度則是攻擊者希望用越快的速度去傳輸，以癱瘓受害者頻寬。簡單來說就是兩個特徵都是很重要的。 了解以上兩個現象後，從機器學習的角度切入，機器在學習分類正常和攻擊時不是像我們用上述文字來推敲，而是學習上表中的數值，而這些數值也能畫成如下的座標 以最直接的KNN來說，該模型會需要計算每一個點和其他點的距離來做分群，計算距離就會有問題，第一個特徵為封包大小，差距為980，而傳輸速度差距僅有50，看起來就像是這兩筆數據的傳輸速度是接近的，而能真正比較出兩者差距的是封包大小，但事實上兩筆數據的兩個特徵都有很大的差距，只是在於特徵值的大小或標準不同。 如何讓不同的特徵都有相同的標準去給機器學習，就會用到正規化(Normalization)，將數值縮放到[0, 1]，公式如下列所示，這個方法基本上是所有的數據在訓練前都會經過的預處理，常常看到範例程式碼中會有這個動作就是為了讓後續學習能更好。接著看經過正規化後的數據長怎樣 因為現在每個特徵只有兩個數值，因此最大值會變1，最小值會變0，當每個特徵的數值更多時會看到所有數值在轉換後都介在0~1之間，可以發現兩個特徵的差值都變成1，也代表機器在看這兩個特徵時會是相同的距離，不會像前面一樣A特徵的差值較大因此較能夠分類兩者。 總結 正規化後的數據分布並不會改變， 一樣保持相同的分布，從範例也能看到，因此並不影響數據在空間上的分布情況。 能夠有效地讓所有特徵都有相同的單位標準，不會因為數值範圍不同而有所影響。 範例 from sklearn import preprocessing from sklearn.datasets import make_classification import matplotlib.pyplot as plt ## 生成具有兩個特徵，共100筆數據 X, y = make_classification( n_samples=300, n_features=2, n_redundant=0, n_informative=2, random_state=20, n_clusters_per_class=1, scale=100) ## 可視化數據 plt.scatter(X[:,0], X[:,1], c=y) plt.show() #################################### # 將數據標準化 X = preprocessing.minmax_scale(X, feature_range=(0,1)) ## 可視化數據 plt.scatter(X[:,0], X[:,1], c=y) plt.show() https://drive.google.com/drive/folders/1qktFFBon0BzfmNuqIHEVq5V3p9TJR2xp?usp=sharing
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【QA】為何用CNN來做影像處理？
Part 1 在最開始先提到簡單的深度學習架構-DNN，接著就會介紹耳熟能詳的CNN，讓大家先了解DNN後再去看CNN，就會發現為甚麼CNN為大多數常用的深度學習方法。 DNN的樣子 圖片來源:https://jason-chen-1992.weebly.com/uploads/1/0/8/5/108557741/dnn_orig.png 上圖是一個簡單的DNN架構，分為input layer(輸入層)、hidden layer(隱藏層)和output layer(輸出層)，輸入層和輸出層是固定的，隱藏層這東西是包在輸入層以及輸出層中間的，也可以說是隱藏在中間的那幾層就稱為隱藏層。 DNN從字面上來看就是深層神經網路，但要有多少數量的隱藏層才算深沒有明確定義，可以是5層、20層或是說要50層才算，理論上來說整個架構越深可以讓模型學到更多東西，但實際上越深不代表越好，要依實際應用而定。 回到上面的圖，input layer中的每一點都會連接到hidden layer 1的每一個點，像這樣子的話有另一個專有名詞叫做「全連接層, Fully Connected Layer」。這一段記住一個重點，就是**每一點都會連接到下一層的每一點**。這裡所說到的「點」都是指「神經元」，也就是說輸入層有8個神經元，所有的hidden layer和output layer也依此類推。 --- Part 2 剛剛看到input layer的每一個神經元會連接到下一層中的每一個神經元，連接起來是口頭說法，實際上電腦做的事情是將兩個神經元相乘，以下圖舉例，一個簡單的深層網路，特別將輸入層中的一個神經元連接情況用不同顏色標註，紅色線條做的事情就是(2*2)，紫色線條為(2*5)而綠色線條則是(2*1)，這是一個最簡單的說明，想像一下假如每一層的神經元都幾百幾千個，電腦一次要計算這麼多的數字，想必是要計算的東西越多會讓整體時間越長。 目前了解了每一點都會連接到下一層的每一點以及電腦做的事情是將兩個神經元相乘這兩個重點，就會發現為了讓層數更深，勢必會有更多的神經元，這樣會讓電腦做的事情增加，建構好一個架構後可以計算一下參數量，參數量簡單來講就是指整個網路的複雜度，參數量越多複雜度越大，計算時間也越長。這樣就進一步想到真的有需要在隱藏層用全連接層嗎，以及CNN為甚麼可以達到更好的效果並且參數量比DNN來的更少。 有需要在隱藏層用全連接層嗎: 1. 參數量多 : 全連接層的參數量藉由剛剛的說明了解到，在深層網路裡的參數量只會隨著深度而增加，雖然是為了達到更好的效果，但更大的問題在於計算時間和複雜度很高，這裡可以先想像全連接層能夠讓時間花越多，並且帶來比其它方法還要好的效果。 2. 真正的需求 : 當今來看會用到深度學習的應用有目標檢測、影像辨識等等，舉例來說今天我們要訓練一個能夠分類不同種類的狗狗時，以人來說，人類是靠毛的顏色、狗的外觀(例:鼻子嘴巴眼睛樣貌)或是體型來分辨是哪種狗狗，機器也一樣，只要讓機器學習到每一種狗狗毛的顏色、外觀體型等等的就能讓機器認出不同的品種。 以下圖說明，一張圖為20 * 30個pixel，假設只有一層隱藏層，這樣參數量就有(20*30)*100=60000，另外這600個點就代表圖中的任一點，隱藏層的100個神經元將這輸入的600個點都訓練，這樣對機器來說是多餘的。 圖片來源:https://www.google.com/url?sa=i&url=https://digiphoto.techbang.com/posts/10182-why-do-you-take-black-and-white-photos&psig=AOvVaw35rDiTLhrcPquyGCf7cdxo&ust=1644911180183000&source=images&cd=vfe&ved=0CAsQjRxqFwoTCKCVxufZ_vUCFQAAAAAdAAAAABAD --- Part 3 CNN最重要的精神就是卷積這個東西，為甚麼常常聽到CNN對圖像有很好的分類效果，對應到前面說到的，機器真正要學到的東西就只是一張圖片中關鍵的地方，DNN中隱藏層中每個神經元都會連接輸入的每一點，等於每個神經元都會訓練到完整的圖，是多餘的，另外就是參數量過多的問題。 在CNN裡隱藏層不使用全連接層，而是指在輸出層前才使用，為了避免DNN的問題，使用卷積能夠解決在DNN中參數量過多和不必要的訓練問題，可以把CNN的一個filter想像成DNN中一個神經元，但這裡不同的是filter可以根據設定的大小去抓取一張圖中特定範圍的像素，而filter會對整張圖用「滑動窗口」方式做計算。用下圖進一步說明為甚麼filter可以避免在DNN中有不必要訓練的問題。 圖中有兩個filter，大小為3*3，fulter中每個權重值如圖，並且兩個filter值不同，接著用這兩個filter對同一張圖做卷積，可以發現filter1權重值為1的地方是斜的，代表filter1和其經過的區塊相乘後，只有為1的點會有值，filter2同理，將重點放在filter2，當filter2經過中間樹幹的地方時，跟filter相乘會發現只有樹幹的地方會有值，其他部分相乘後都為0，代表filter是負責樹幹的特徵，而filter1就可能是負責樹葉，因為兩個filter的權重值不同，就算兩個filter都對整張圖處理，但出來的結果會相同。每個filter負責不同部分，避免DNN的不必要訓練問題。
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【QA】在圖像分類任務中，訓練資料量不足會帶來什麼問題？如何緩解資料量不足帶來的問題？
前述 訓練時會有各種不同應用，而不同的應用中不會是所有數據都很好取得，以要談的圖像分類任務為例，要訓練一個動物分類器需要很多種類的動物，常見的貓咪和狗狗等等的很好蒐集，但如果是獅子、大象等等的相比貓狗而言非常難獲取。 如果這些難取得的動物類別只有少量數據，拿這些少量數據肯定無法讓分類器好的效果，此時需要有補充數據量的方法，藉此緩解資料量不足的問題。 Part 1 訓練資料量不足會帶來甚麼問題? 1、機器和人相似，對一個東西要學習足夠才有相當的能力做分類，因此資料量不足會帶來的最直接問題是模型缺乏訓練，分類效果不佳。 2、假如模型要訓練兩個類別，類別比為1000:1，當模型訓練這兩類後會發現後續對少數的類別分類效能較差，因為模型訓練時會不斷更新權重，而訓練時的損失函數或優化函數都偏向於多數類，等同多數類會將少數類蓋過。 如何緩解資料量不足帶來的問題? 1、第一個問題提到CNN可以有效訓練圖像並分類，因此這裡都用CNN來說明，在圖像分類中無非是要獲得更好的圖像辨識器，第一個解決方法就是「Data Augmentation」，藉由對原始圖像旋轉、裁切、增加噪點、白化等技術，將原本的圖片做了一些「變化」，就能獲得和原先圖像不同的資料。 圖片來源:https://chtseng.files.wordpress.com/2017/11/4674_hxh_mbr4kq.png 2、對圖像進行採樣，這裡介紹SMOTE，是一種採樣方式，透過對需平衡的樣本計算離其最近的數據，並利用自身和最近數據得到新增的數據點。此方法可以得到近似於自身的數據，但又不完全相同，和Data Augmentation生成的也不太相同。 3、抓取原始數據特徵並生成，前面都是利用原始數據本身來生成，但可以利用每個類別都有不同特徵的觀點出發，例如貓跟狗一定有不同特徵，就可以利用抓取特徵的方式來生成，VAE就是一個很好的方法，利用原始數據訓練編碼/解碼器，編碼器能抓取特徵，而解碼器就根據此特徵就能生成想要數量的資料。 Part 2 介紹了三種不同的方法，分別是Data Augmentation、SMOTE和VAE，整理三者的優缺點 Data Augmentation 優點: 透過對原始圖像做翻轉、裁切等等的簡單操作可以很快速的獲得資料量。 缺點: 經由圖像處理後的圖片有可能失去本身擁有的特性，例如模型訓練翻轉後的圖像後，變得無法分類原始圖像。 SMOTE 優點: 能夠生成跟自身較相近的圖像。 缺點: 需大量計算時間、假如周圍鄰居相隔較遠則無法有效生成相近數據。 VAE 優點: 抓取特徵後，透過VAE中的解碼器(生成器)，能快速生成所需數量的數據，且一定包含原始圖像所擁有的特徵。 缺點: 假如原始數據量過少，會發生訓練編碼/解碼器時缺乏訓練，後果是編碼或解碼時不會是預想中的數據。 VAE、SMOTE參考程式碼:https://drive.google.com/drive/folders/1kfjZmKC53PjKB3yxyyPpOH0PiigVSWUY?usp=sharing
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2022/05/28 - 助教時間完整影片
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【QA】如何給二分類模型打分數？
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【QA】正規化(Normalization)為什麼能夠提升機器學習的成效？
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【QA】為何用CNN來做影像處理？
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【QA】在圖像分類任務中，訓練資料量不足會帶來什麼問題？如何緩解資料量不足帶來的問題？
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2022/05/31（二）20:00-21:00 助教時間
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【QA】什麼是轉置卷積（Transposed Convolution）？
目錄 Convolution 簡介 Transposed Convolution 簡介 用數學方式理解 Convolution、Transpose Convolution 的運作 實際案例 卷積（Convolution）簡介 Convolution 是一個「將原始資料縮透過遮罩（Mask）縮減成少量關鍵特徵」的技術，用數學角度想就是將原始資料與 Mask 進行「線性轉換」。 圖像辨識：透過 Convolution 將原始圖片搭配 Mask 運算後，得出有用的特徵圖（Feature Map）。 Output Size = 1 + \frac{size\ of\ image\ – kernel\ size\ + 2<em>padding}{stride}1+size of image –kernel size +2∗paddingstride1 + \frac{size\ of\ image\ – kernel\ size\ + 2padding}{stride}1+stridesize of image –kernel size +2∗padding 訊號處理：透過 1D Convolution 將原始訊號轉換成「特徵訊號」。 Transposed Convolution 簡介 剛好跟 Convolution 的功能「完全相反」。 將某個 Pixel 或資料點以「線性轉換」方式轉換成多個資料 用數學方式理解 Convolution、Transpose Convolution 的運作 已知有一張 44 的 Image、一張 33 的 Kernel，以及透過 Kernel 將 Image 線性轉換後的 Output，Image 中每個 Pixel 為 $I={I_{ij} | i = 1, 2, 3, 4; j = 1, 2, 3, 4}$，Kernel 中每個 Pixel 為 K={K_{ij} | i = 1, 2, 3; j = 1, 2, 3}K={Kij∣i=1,2,3;j=1,2,3}K={K_{ij} | i = 1, 2, 3; j = 1, 2, 3}K={Kij∣i=1,2,3;j=1,2,3}，Output 中每個 Pixel 為 O={O_{ij} | i = 1, 2; j = 1, 2}O={Oij∣i=1,2;j=1,2}O={O_{ij} | i = 1, 2; j = 1, 2}O={Oij∣i=1,2;j=1,2}。O_{11}=I_{11}K_{11}+I_{12}K_{12}+I_{13}K_{13}+I_{21}K_{21}+I_{22}K_{22}+I_{23}K_{23}+I_{31}K_{31}+I_{32}K_{32}+I_{33}K_{33} \<br>O_{12}= I_{12}K_{11}+I_{13}K_{12}+I_{14}K_{13}+I_{22}K_{21}+I_{23}K_{22}+I_{24}K_{23}+I_{32}K_{31}+I_{33}K_{32}+I_{34}K_{33}\<br>O_{21}= I_{21}K_{11}+I_{22}K_{12}+I_{23}K_{13}+I_{31}K_{21}+I_{32}K_{22}+I_{33}K_{23}+I_{41}K_{31}+I_{42}K_{32}+I_{43}K_{33}\<br>O_{22}= I_{22}K_{11}+I_{23}K_{12}+I_{24}K_{13}+I_{32}K_{21}+I_{33}K_{22}+I_{34}K_{23}+I_{42}K_{31}+I_{43}K_{32}+I_{44}K_{33}\O11=I11K11+I12K12+I13K13+I21K21+I22K22+I23K23+I31K31+I32K32+I33K33O12=I12K11+I13K12+I14K13+I22K21+I23K22+I24K23+I32K31+I33K32+I34K33O21=I21K11+I22K12+I23K13+I31K21+I32K22+I33K23+I41K31+I42K32+I43K33O22=I22K11+I23K12+I24K13+I32K21+I33K22+I34K23+I42K31+I43K32+I44K33O_{11}=I_{11}K_{11}+I_{12}K_{12}+I_{13}K_{13}+I_{21}K_{21}+I_{22}K_{22}+I_{23}K_{23}+I_{31}K_{31}+I_{32}K_{32}+I_{33}K_{33} \O_{12}= I_{12}K_{11}+I_{13}K_{12}+I_{14}K_{13}+I_{22}K_{21}+I_{23}K_{22}+I_{24}K_{23}+I_{32}K_{31}+I_{33}K_{32}+I_{34}K_{33}\O_{21}= I_{21}K_{11}+I_{22}K_{12}+I_{23}K_{13}+I_{31}K_{21}+I_{32}K_{22}+I_{33}K_{23}+I_{41}K_{31}+I_{42}K_{32}+I_{43}K_{33}\O_{22}= I_{22}K_{11}+I_{23}K_{12}+I_{24}K_{13}+I_{32}K_{21}+I_{33}K_{22}+I_{34}K_{23}+I_{42}K_{31}+I_{43}K_{32}+I_{44}K_{33}\O11=I11K11+I12K12+I13K13+I21K21+I22K22+I23K23+I31K31+I32K32+I33K33O12=I12K11+I13K12+I14K13+I22K21+I23K22+I24K23+I32K31+I33K32+I34K33O21=I21K11+I22K12+I23K13+I31K21+I32K22+I33K23+I41K31+I42K32+I43K33O22=I22K11+I23K12+I24K13+I32K21+I33K22+I34K23+I42K31+I43K32+I44K33 若將 Convolution 以 Matrix Multiplication 計算，則：I= \begin{bmatrix}<br> I_{11} \<br> I_{12} \<br> I_{13} \<br> I_{14} \<br> I_{21} \<br> I_{22} \<br> I_{23} \<br> I_{24} \<br> I_{31} \<br> I_{32} \<br> I_{33} \<br> I_{34} \<br> I_{41} \<br> I_{42} \<br> I_{43} \<br> I_{44}<br> \end{bmatrix}<em>{1*16}<br>\<br>K= \begin{bmatrix}<br> K</em>{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 & 0 & 0 & 0 & 0 \<br> 0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 & 0 & 0 & 0 \<br> 0 & 0 & 0 & 0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 \<br> 0 & 0 & 0 & 0 & 0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33}<br> \end{bmatrix}<em>{4*16}<br>\<br>O= K</em>{4<em>16} \cdot I_{16</em>1}=\begin{bmatrix}<pre><code><span class="xml"> O_</span><span class="hljs-template-variable">{11}</span><span class="xml"> \\ O_</span><span class="hljs-template-variable">{12}</span><span class="xml"> \\ O_</span><span class="hljs-template-variable">{21}</span><span class="xml"> \\ O_</span><span class="hljs-template-variable">{22}</span><span class="xml"> \\ \end</span><span class="hljs-template-variable">{bmatrix}</span><span class="xml"> \\<span class="hljs-tag"></<span 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class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>13<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>14<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>21<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>22<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>23<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>24<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>31<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>32<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>33<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>34<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>41<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>42<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>43<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>44<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"></<span class="hljs-name">mtable</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span> <span class="hljs-attr">fence</span>=<span class="hljs-string">"true"</span>></span>]<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"></<span class="hljs-name">mrow</span>></span><span class="hljs-tag"><<span class="hljs-name">mrow</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>1<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span>></span>∗<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>16<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mrow</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mspace</span> <span class="hljs-attr">linebreak</span>=<span class="hljs-string">"newline"</span>></span><span class="hljs-tag"></<span class="hljs-name">mspace</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span>></span>=<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mrow</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span> <span class="hljs-attr">fence</span>=<span class="hljs-string">"true"</span>></span>[<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"><<span class="hljs-name">mtable</span> <span class="hljs-attr">rowspacing</span>=<span class="hljs-string">"0.15999999999999992em"</span> <span class="hljs-attr">columnspacing</span>=<span class="hljs-string">"1em"</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>11<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>12<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>13<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>21<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>22<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>23<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>31<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>32<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>33<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>11<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>12<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>13<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>21<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>22<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>23<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>31<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>32<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>33<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>11<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>12<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>13<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>21<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>22<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>23<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>31<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>32<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>33<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>11<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>12<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>13<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>21<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>22<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>23<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>0<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>31<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>32<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>33<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"></<span class="hljs-name">mtable</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span> <span class="hljs-attr">fence</span>=<span class="hljs-string">"true"</span>></span>]<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"></<span class="hljs-name">mrow</span>></span><span class="hljs-tag"><<span class="hljs-name">mrow</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>4<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span>></span>∗<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>16<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mrow</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mspace</span> <span class="hljs-attr">linebreak</span>=<span class="hljs-string">"newline"</span>></span><span class="hljs-tag"></<span class="hljs-name">mspace</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>O<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span>></span>=<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>K<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mrow</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>4<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span>></span>∗<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>16<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mrow</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span>></span>⋅<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>I<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mrow</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>16<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span>></span>∗<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>1<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">mrow</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span>></span>=<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"><<span class="hljs-name">mrow</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span> <span class="hljs-attr">fence</span>=<span class="hljs-string">"true"</span>></span>[<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"><<span class="hljs-name">mtable</span> <span class="hljs-attr">rowspacing</span>=<span class="hljs-string">"0.15999999999999992em"</span> <span class="hljs-attr">columnspacing</span>=<span class="hljs-string">"1em"</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>O<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>11<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>O<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>12<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>O<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>21<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtr</span>></span><span class="hljs-tag"><<span class="hljs-name">mtd</span>></span><span class="hljs-tag"><<span class="hljs-name">mstyle</span> <span class="hljs-attr">scriptlevel</span>=<span class="hljs-string">"0"</span> <span class="hljs-attr">displaystyle</span>=<span class="hljs-string">"false"</span>></span><span class="hljs-tag"><<span class="hljs-name">msub</span>></span><span class="hljs-tag"><<span class="hljs-name">mi</span>></span>O<span class="hljs-tag"></<span class="hljs-name">mi</span>></span><span class="hljs-tag"><<span class="hljs-name">mn</span>></span>22<span class="hljs-tag"></<span class="hljs-name">mn</span>></span><span class="hljs-tag"></<span class="hljs-name">msub</span>></span><span class="hljs-tag"></<span class="hljs-name">mstyle</span>></span><span class="hljs-tag"></<span class="hljs-name">mtd</span>></span><span class="hljs-tag"></<span class="hljs-name">mtr</span>></span><span class="hljs-tag"></<span class="hljs-name">mtable</span>></span><span class="hljs-tag"><<span class="hljs-name">mo</span> <span class="hljs-attr">fence</span>=<span class="hljs-string">"true"</span>></span>]<span class="hljs-tag"></<span class="hljs-name">mo</span>></span><span class="hljs-tag"></<span class="hljs-name">mrow</span>></span><span class="hljs-tag"><<span class="hljs-name">mspace</span> <span class="hljs-attr">linebreak</span>=<span class="hljs-string">"newline"</span>></span><span class="hljs-tag"></<span class="hljs-name">mspace</span>></span><span class="hljs-tag"></<span class="hljs-name">mrow</span>></span><span class="hljs-tag"><<span class="hljs-name">annotation</span> <span class="hljs-attr">encoding</span>=<span class="hljs-string">"application/x-tex"</span>></span>I= \begin</span><span class="hljs-template-variable">{bmatrix}</span> </code></pre> I_{11} \<br> I_{12} \<br> I_{13} \<br> I_{14} \<br> I_{21} \<br> I_{22} \<br> I_{23} \<br> I_{24} \<br> I_{31} \<br> I_{32} \<br> I_{33} \<br> I_{34} \<br> I_{41} \<br> I_{42} \<br> I_{43} \<br> I_{44}<br> \end{bmatrix}<em>{1*16}<br>\<br>K= \begin{bmatrix}<br> K</em>{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 & 0 & 0 & 0 & 0 \<br> 0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 & 0 & 0 & 0 \<br> 0 & 0 & 0 & 0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33} & 0 \<br> 0 & 0 & 0 & 0 & 0 & K_{11} & K_{12} & K_{13} & 0 & K_{21} & K_{22} & K_{23} & 0 & K_{31} & K_{32} & K_{33}<br> \end{bmatrix}<em>{4*16}<br>\<br>O= K</em>{4<em>16} \cdot I_{16</em>1}=\begin{bmatrix}<pre><code><span class="xml"> O_</span><span class="hljs-template-variable">{11}</span><span class="xml"> \\ O_</span><span class="hljs-template-variable">{12}</span><span class="xml"> \\ O_</span><span class="hljs-template-variable">{21}</span><span class="xml"> \\ O_</span><span class="hljs-template-variable">{22}</span><span class="xml"> \\ \end</span><span class="hljs-template-variable">{bmatrix}</span><span class="xml"> \\<span class="hljs-tag"></<span class="hljs-name">annotation</span>></span><span class="hljs-tag"></<span class="hljs-name">semantics</span>></span><span class="hljs-tag"></<span class="hljs-name">math</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"katex-html"</span> <span class="hljs-attr">aria-hidden</span>=<span class="hljs-string">"true"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"base"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"strut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:0.68333em;vertical-align:0em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord mathnormal"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"margin-right:0.07847em;"</span>></span>I<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mspace"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"margin-right:0.2777777777777778em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mrel"</span>></span>=<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mspace"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"margin-right:0.2777777777777778em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"base"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"strut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:19.27673em;vertical-align:-9.399849999999999em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"minner"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"minner"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mopen"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizing mult"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-t vlist-t2"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-r"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:9.876880000000002em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:5.550129999999999em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎣<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:4.400129999999999em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎢<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:3.804129999999999em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎢<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:3.208129999999999em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎢<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:2.6121299999999987em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎢<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:2.0161299999999986em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎢<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:1.4201299999999986em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎢<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:0.8241299999999985em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎢<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:0.2281299999999984em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎢<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:-0.3678700000000017em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span 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class="hljs-attr">class</span>=<span class="hljs-string">"mord"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord mathnormal"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"margin-right:0.02778em;"</span>></span>O<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"msupsub"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-t vlist-t2"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-r"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:0.30110799999999993em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:2.7em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"sizing reset-size6 size3 mtight"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord mtight"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord mtight"</span>></span>1<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord mtight"</span>></span>2<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-s"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-r"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:0.15em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:-2.4099999999999997em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord 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class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord mtight"</span>></span>1<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-s"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-r"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:0.15em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:-1.2099999999999997em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord mathnormal"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"margin-right:0.02778em;"</span>></span>O<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"msupsub"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-t vlist-t2"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-r"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:0.30110799999999993em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:2.7em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"sizing reset-size6 size3 mtight"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord mtight"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord mtight"</span>></span>2<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mord mtight"</span>></span>2<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-s"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-r"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:0.15em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-s"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-r"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:2.1500000000000004em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mclose"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizing mult"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-t vlist-t2"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-r"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:2.6529999999999996em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:-1.6499900000000003em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎦<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:-2.79999em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎥<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:-3.3959900000000003em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎥<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:-3.4119800000000002em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎥<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"top:-4.653em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"pstrut"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:3.1550000000000002em;"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"delimsizinginner delim-size4"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span>⎤<span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-s"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist-r"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"vlist"</span> <span class="hljs-attr">style</span>=<span class="hljs-string">"height:2.15003em;"</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"><<span class="hljs-name">span</span> <span class="hljs-attr">class</span>=<span class="hljs-string">"mspace newline"</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span><span class="hljs-tag"></<span class="hljs-name">span</span>></span></span> </code></pre> </li> <li>Transposed Convolution 就是從轉成矩陣的 Convolution 進行轉換的</li> </ul> <p><span class="cupoymd-latex-code-block" widget-latex-code><textarea style=" position: absolute; left: 0; top: 0; width: 0px; height: 0px; opacity: 0; display: none; " readonly lessthen-regex="###md_code_symbol_lessthen###">K^{T}=\begin{bmatrix}<br> K_{11} & 0 & 0 & 0 \<br> K_{12} & K_{11} & 0 & 0 \<br> K_{13} & K_{12} & 0 & 0 \<br> 0 & K_{13} & 0 & 0 \<br> K_{21} & 0 & K_{11} & 0 \<br> K_{22} & K_{21} & K_{12} & K_{11} \<br> K_{23} & K_{22} & K_{13} & K_{12} \<br> 0 & K_{23} & 0 & K_{13} \<br> K_{31} & 0 & K_{21} & 0 \<br> K_{32} & K_{31} & K_{22} & K_{21} \<br> K_{33} & K_{32} & K_{23} & K_{22} \<br> 0 & K_{33} & 0 & K_{23} \<br> 0 & 0 & K_{31} & 0 \<br> 0 & 0 & K_{32} & K_{31} \<br> 0 & 0 & K_{33} & K_{32} \<br> 0 & 0 & 0 & K_{33}<br> \end{bmatrix}<em>{4*16}<br>O=\begin{bmatrix}<br> O</em>{11} \<br> O_{12} \<br> O_{21} \<br> O_{22}<br> \end{bmatrix}<em>{1*4}<br>I= O</em>{1<em>4}\cdot K_{4</em>16} \KT=[K11000K12K1100K13K12000K1300K210K110K22K21K12K11K23K22K13K120K230K13K310K210K32K31K22K21K33K32K23K220K330K2300K31000K32K3100K33K32000K33]4∗16O=[O11O12O21O22]1∗4I=O1∗4⋅K4∗16K^{T}=\begin{bmatrix} K_{11} & 0 & 0 & 0 \ K_{12} & K_{11} & 0 & 0 \ K_{13} & K_{12} & 0 & 0 \ 0 & K_{13} & 0 & 0 \ K_{21} & 0 & K_{11} & 0 \ K_{22} & K_{21} & K_{12} & K_{11} \ K_{23} & K_{22} & K_{13} & K_{12} \ 0 & K_{23} & 0 & K_{13} \ K_{31} & 0 & K_{21} & 0 \ K_{32} & K_{31} & K_{22} & K_{21} \ K_{33} & K_{32} & K_{23} & K_{22} \ 0 & K_{33} & 0 & K_{23} \ 0 & 0 & K_{31} & 0 \ 0 & 0 & K_{32} & K_{31} \ 0 & 0 & K_{33} & K_{32} \ 0 & 0 & 0 & K_{33} \end{bmatrix}{4*16}O=\begin{bmatrix} O{11} \ O_{12} \ O_{21} \ O_{22} \end{bmatrix}{1*4}I= O{14}\cdot K_{416} \KT=⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎡K11K12K130K21K22K230K31K32K33000000K11K12K130K21K22K230K31K32K3300000000K11K12K130K21K22K230K31K32K33000000K11K12K130K21K22K230K31K32K33⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎤4∗16O=⎣⎢⎢⎢⎡O11O12O21O22⎦⎥⎥⎥⎤1∗4I=O1∗4⋅K4∗16 實際案例 Generative Adversarial Network 一種影像生成的深度學習演算法 Discriminator Network：判斷生成出來的圖片與真實圖片是否相同或相似。 Generator Network：藉由 D-dimensional noise vector 產生出一張偽圖片，其中就會大量使用到 Transposed Convolution。 U-Net 影像切割案例中最常使用的深度學習架構 在 Encoder 階段，將圖片以 CNN 方式萃取出關鍵特徵 在 Decoder 階段，除了以 Transposed Convolution 將 Feature Map 中每個特徵轉換成多個特徵外，部分 Encoder 階段得到的 Feature Map 也會在此階段搭配運算。 Reference Mozaffari M.H. and Tay L.L. (2020). A Review of 1D Convolutional Neural Networks toward Unknown Substance Identification in Portable Raman Spectrometer. arXiv:2006.10575
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【QA】循序向後選擇演算法(Sequential Backward Selection)？
目錄 為何要挑選有用的特徵？ Sequential Backward Selection 其他有關選擇演算法的變形 實際案例 為何要挑選有用的特徵？ 不是所有特徵對目標的預測都有幫助 沒有影響力的特徵容易對預測目標造成負面影響 特徵之間的交互影響過多導致不容易解釋 Sequential Backward Selection 運用模型好壞判斷哪些特徵需要從特徵集中剔除掉 Objective Function：用來判斷模型好壞藉以決定哪個特徵需要剔除的函數 演算法 Step1. 決定 Objective Function、模型種類、所有想要納入挑選的特徵，以及設定最終想要有幾個特徵 Step2. 使用所有挑選的特徵建立模型 Step3. 輪流扣除一個特徵後，再次建立模型，並運用 Objective Function 計算模型效能 Step4. 根據模型評估結果，挑選最佳特徵組合 Step5. 重複 Step3、Step4，直到剩下指定特徵數的特徵 優點：觀念簡單、操作容易 缺點：容易忽略掉特定特徵組合、變異多 其他有關選擇演算法的變形 Sequential Forward Selection 與 Sequential Backward Selection 完全相反，從一開始只有一個特徵，到最後擁有指定數量的特徵 優點、缺點與 Sequential Forward Selection 相似。 Sequential Floating Forward Selection 在選入特徵的過程中，插入一次或數次移除已被選入的特徵的判斷，增加特徵組合的數量以達到更大的模型效能 Sequential Floating Backward Selection 與 Sequential Floating Forward Selection 剛好相反，在某一次排除特徵的過程中，插入一次或數次加入已被排除的特徵的判斷。 Reference Raschka S. [Sebastian Raschka] (2022, January 28). 13.4.4 Sequential Feature Selection (L13: Feature Selection) [Video]. Youtube. https://www.youtube.com/watch?v=0vCXcGJg5BoJoe Bemister-Buffington, Alex J. Wolf, Sebastian Raschka, and Leslie A. Kuhn (2020)Machine Learning to Identify Flexibility Signatures of Class A GPCR InhibitionBiomolecules 2020, 10, 454. (https://www.mdpi.com/2218-273X/10/3/454#)
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【QA】甚麼方法能用深度神經網路當作是降維的工具？
目錄 什麼是「降維」？ AutoEncoder 作為降維的工具 PCA v.s. AutoEncoder 實作範例 Reference 什麼是「降維」？ 「降維」本身是個「非監督式學習」的技術 將原始資料內的特徵以「有用的技巧」萃取出有用的資訊，除了能更減少特徵數，更能強調出關鍵特徵 常見降維方法：Principle Components Analysis (PCA)、Factorial Analysis (FA)、AutoEncoder AutoEncoder 作為降維的工具 輸入：某一筆資料 輸出：與輸入相同的資料 架構 作為降維結果的地方：架構中正中間維度最小處（如上圖中 Z） PCA v.s. AutoEncoder PCA 是個「線性轉換」的降維方法；AutoEncoder是個「線性轉換」也是個「非線性轉換」的降維方法 PCA 本身屬於「公式解」，變異低；AutoEncoder 本身屬於「數值解」，會受到模型架構、神經元數量、模型訓練狀況影響產生出來的特徵，變異高 PCA 是個「可解釋」的方法，每個主成分中會含有各個特徵的權重，透過權重即可知道該主成分的含意；AutoEncoder 產生出來的低維度特徵由於已經經過複雜計算且是以「Gradient Descent」得出的結果，不容易解釋 實際案例 Fournier and Aloise (2021) 針對 PCA 與 AutoEncoder 在實際使用時做了實驗 Datasets: MNIST、Fashion-MNIST、CIFAR10 X軸：降維後取多少 percent 的特徵；Y軸：Accuracy 或 Training Time Reference Mungoli A. (2022 Jun. 19). Dimensionality Reduction: PCA versus Autoencoders. https://towardsdatascience.com/dimensionality-reduction-pca-versus-autoencoders-338fcaf3297dFournier Q. and Aloise D. (2021). Empirical comparison between autoencoders and traditional dimensionality reduction methods. arXiv:2103.04874
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【QA】為什麼要對數值作特徵正規化？
目錄 什麼是特徵正規化 特徵正規化有哪些方法 若不做特徵正規化，將會對模型造成什麼影響 有哪些模型就算沒有特徵正規化，也不會對模型效能造成影響 Reference 什麼是特徵正規化 將不同範圍的特徵轉換成同一個範圍的特徵 例如：將數值轉換成 [0, 1] 的範圍內 特徵正規化有哪些方法 Standardization（Z-Score Normalization） 將資料統一轉換成平均數為 0、標準差為 1 的資料 $X_{Standardization}=\frac{X-\mu}{\sigma}$ Normalization（Min-Max Scaler） 將資料統計轉換成 [0, 1] 的資料 $X_{Normalization}=\frac{X-X_{min}}{X_{max}-X_{min}}$ 若不做特徵正規化，將會對模型造成什麼影響？做了正規化又有什麼好處？ 先備知識：「電腦」沒有單位的概念，只知道數字大小 反面案例：假設有兩個特徵，一個特徵是年齡，另一個特徵是每月薪資。有某筆資料的年齡、每月薪資都是 100 元，請問以「人」角度跟以「電腦」角度思考面向有相同嗎？ 透過正規化，不僅讓每個特徵的範圍具體被定義，每個特徵的單位也都被轉換一致。 有哪些模型就算沒有特徵正規化，也不會對模型造成影響 Tree-based Model Decision Tree Random Forest eXtreme Gradient Boosting (XGBoost) Light Gradient Boosting Machine (LightGBM) Reference TOWARD AI 2020. Machine Learning Standardization (Z-Score Normalization) with Mathematics. https://towardsai.net/p/machine-learning/machine-learning-standardization-z-score-normalization-with-mathematicsJaitley U. 2018. Wht Data Normalization is necessary for Machine Learning Models. https://medium.com/@urvashilluniya/why-data-normalization-is-necessary-for-machine-learning-models-681b65a05029
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助教時間_轉置卷積
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助教時間_SequentialBackwardSelection
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助教時間_什麼方法能用深度神經網路當作是降維的工具
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2022/06/11（六）20:00-21:00 助教時間
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【QA】甚麼是 Bootstrapping (拔靴法)？
Statistics is the science of learning from experience. 翻譯成中文，統計就是從經驗中學習的科學。 本質上統計是用來嘗試回答3個基本的問題： Data Collection: How should I collect my data? 舉例來說：如何做實驗收集資料 Summary: How should I analyze and summarize the data that I’ve collected? 舉例來說：如何從資料中分析並得到一個結果，也許是圖表或是在一定信賴區間(95%)下的數值分布區間 Statistical Inference: How accurate are my data summaries? 舉例來說：我的結果有多可信或多準確 第3點，所謂的統計推論(statistical inferences)就是我們的Bootstrap所發揮作用的地方。 我們接下來看看Bootstrap的效果與目的。 根據一些名間傳言，Bootstrap的命名由來是來自於Adventures of Baron Munchausen這部作品，有一幕是主角被困在泥沼時脫下靴子用來幫助自己脫離泥沼，因此稱為拔靴法或拔靴自助法(參考連結)，自助是這裡很重要的概念，Bootstrap就是不斷從同一個資料集中學習並實現減少variance的效果。 首先我們看看Bradley Efron的An Introduction to the Bootstrap(連結)中的範例，給定兩個資料集，分別是給予aspirin(阿斯匹靈)和(placebo)安慰劑兩個實驗組， 給予aspirin的組別中共有11037個人，其中有119個發生中風 給予placebo的組別中共有11034個人，其中有98個人發生中風 我們用python來模擬這個實驗條件，code如下： import numpy as np import random # 1代表發生中風的樣本，0代表沒有發生的樣本 lis1 = [1]*119 + [0]*(11037-119) lis2 = [1]*98 + [0]*(11034-98) 我們可以估計一個theta，用來計算所有病患中，給予阿斯匹靈的患者發生中風的比例，對比給予安慰劑的患者發生中風的比例，這裡可以得到1.21這個數值，單從結果上來看給予阿斯匹靈是很危險的。 ratio = lambda lis: sum(lis)/len(lis) theta = ratio(lis1)/ratio(lis2) print('患者中風比值：給予阿斯匹靈/給予安慰劑:', theta) 接下來我們用bootstrap來估計一個theta在95%信心水準下的信賴區間。 首先定義用來計算theta^的函式 def props(lis, k): a = random.choices(lis, k=k) # 從原始資料中抽出長度為k的子集合 return sum(a)/len(a) # 下面的k可以選擇任意小於list總長的數 theta = props(lis1, k=6666) / (props(lis2, k=8787)+1e-12) # 加上一個微小常數避免分母0 print(theta) Bootstrap的本質是基於大數法則(**Law of large numbers)**的重複抽樣，而每一次抽樣並不會把所有資料抽出來，也就是上面的k值不會等於總資料長度，而k值越大，能夠更快的收斂，但k值足夠小降低variance的效果才會明顯。 那麼上面只有抽樣一次而已，接下來我們用一個for迴圈進行重複抽樣。 theta_lis = [] replica = 1000 for loop in range(replica): theta = props(lis1, k=6666) / (props(lis2, k=8787)+1e-12) theta_lis.append(theta) theta_lis = sorted(theta_lis) # 排序theta_lis 首先設定一個空集合用來收集每次實驗得到的theta^，然後設定要重複的次數replica=1000，接下來我們可以觀察theta_lis的一些統計量。 你會發現，theta_lis的平均值與前面我們從整個母群體估計出來的theta是非常靠近的，就只差一個標準差而已。這反映了Bootstrap的精神：我們不需要直接看到整個母群體(實務上，大部分的情況我們也無法觀察真正的母群體)，而是透過不斷抽樣子集合，這些子集合的平均應該能夠趨近母群體的平均值，並且這些子集合的分布能夠很好的反應母群體的分布。 Bootstrap還有一個很好用的地方在於信賴區間的估計，我們手上的theta_lis是經過排序的一個histrogram，當我們要計算95%的信賴區間時，可以直接找2.5百分位的數值和97.5百分位的數值，他就是信賴區間的近似解。如下面所示的[0.86, 1.65] 以上就是Bootstrap的介紹，完整程式碼如下： import numpy as np import random # 1代表發生中風的樣本，0代表沒有發生的樣本 lis1 = [1]*119 + [0]*(11037-119) lis2 = [1]*98 + [0]*(11034-98) ratio = lambda lis: sum(lis)/len(lis) theta = ratio(lis1)/ratio(lis2) print('患者中風比值：給予阿斯匹靈/給予安慰劑:', theta) # Bootstrap方法 def props(lis, k): a = random.choices(lis, k=k) return sum(a)/len(a) # 下面的k可以選擇任意小於list總長的數 theta = props(lis1, k=6666) / (props(lis2, k=8787)+1e-12) print(theta) theta_lis = [] replica = 1000 for loop in range(replica): theta = props(lis1, k=6666) / (props(lis2, k=8787)+1e-12) theta_lis.append(theta) theta_lis = sorted(theta_lis) print(np.mean(theta_lis), np.std(theta_lis)) # 近似的母體平均值和標準差 print(np.quantile(theta_lis, (0.025, 0.975))) # 95%下的信賴區間
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【QA】集成學習（Ensemble Learning)中的boosting方法是什麼？
Bagging and Boosting 在集成學習中，這兩者是相提並論的主流方法。 Bagging(Bootstrap Aggregation)分成兩個部分，Bootstrap和Aggregation。Bootstrap是基於機率論上的大數法則(Law of large numbers)發展而來，也就是實驗重複越多次，累積結果的統計值越逼近母群體的參數。我們都知道丟骰子任何一面出現的機率都是1/6，但你實際上丟6次的結果又如何呢?這個就是我們實現次數太少導致的結果，依循大數法則，你可以預期當你投600萬次，每一個點數出現的次數應該都是100萬次左右，這就是大數法則的意義。 所以在Bootstrap階段我們會不斷的重複抽樣，並用抽出來的子集合訓練一個弱的模型，然後放回，重複實驗，不斷產出新的弱模型。最後Aggregation的階段就是讓所有弱進行投票，得到預測結果。如果把每個弱模型分開來看可能varience很大，但把不同的varience很大的model集合起來看以後，varience就被縮小了。 使用Bagging的時機 當model的複雜度很高，擔心會過擬合(overfitting)時才做bagging 減低varience Decision tree很容易overfit所以特別需要，用Decision tree做bagging的結果就是Random Forest Boosting方法 A Decision-Theoretic Generalization of On-Line Learning and an Application to Boosting, 1997 如剛才所提到，Bagging是用在很強的model，也就是複雜度高的模型，與之相對Boosting是用在很弱的model。 當你有一個很弱的model，你沒辦法讓他充分的fit你的data時，你就會想要用Boosting。 特色 Boosting有一個數學上，很有力的Guarantee(保證) 假設你有一個ML的演算法，它可以給你一個錯誤率略低過50%的分類器。(這樣是一個很爛的模型) Boosting的方法可以保證讓這些錯誤率略低於50%的分類器，最後的錯誤率達到0%。(不過可能是過擬合) Boosting的framework(框架) 找一個f1(x)分類器 找另一個f2(x)來輔助f1(x) 注意: f1和f2不可以相似, 他們的特性需要是互補的 再找一個f3跟f2是互補的, 再一個f4跟f3互補........ 以此類推，找到n個，舉例來說：100個分類器，再把100個集合起來，就可以得到很低的error rate. 就算每一個都很弱也沒關係 要注意的是： Boosting的classifier(分類器)的訓練是有順序性的, 要先找f1, 才有f2, 才有f3...... Bagging的分類器是沒有順序問題的, 100顆決策樹可以平行製作 用AdaBoost產出互補的分類器 互補可以很直觀的理解為「解決你沒能解決的問題」也就是處理殘差，最簡單的方法就是針對前一個模型分類錯誤的資料點進行補償。這裡我們用經典的AdaBoost(Adaptive Boosting)來做簡單說明。 首先，我們還是做Bootstrap進行重複抽樣，設定對所有資料點的抽樣權重是相同的1。 訓練f1(x)，並記錄抽樣到的x子集合中包含哪些資料點，在這裡將f1稱為stump1。 評估f1(x)，針對分類錯誤的資料點，調整進行抽樣時的抽樣權重，並降低分類正確者的抽樣權重 評估f1(x)，計算Total Error，並基於該值計算f1在最後投票時的話語權(最後投票時並不是每一票都等值的) 以此類推訓練f2(也就是stump2)，重複執行，直到達到停止條件。 小結 Boosting方法能夠有效率地找到並克服不好訓練的資料點，適用於模型複雜度不高，或者資料點複雜度高的情況。如果覺得AdaBoost不太有效率，你也可以採用Gradient Boosting方法來執行，受限於篇幅這邊就不做太深入的討論，大致上Gradient Boosting是先初始化第一個分類器的目標函數g0(x)，第二個分類器的目標函數則是g0(x)+α1f1(x)=g1(x)，以此類推，第三個分類器的目標函數就是g0(x)+g1(x)+α2f2(x)=g2(x)。透過不斷迭代來找到優化方向。 以上就是Boosting方法的分享。
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【QA】甚麼是聯邦式學習(Federated Learning)？
【QA】甚麼是聯邦式學習(Federated Learning)？ 聯邦學習(Federated learning)應該被歸納為分散式學習(Distributed learning)的分支，不同的是，聯邦學習通常是一種不得不的分散式學習。 分散式學習透過將資料分配到更多的運算單元，有了更高的平行運算能力。而聯邦學習則是因為個別運算單元(e.g. 手機)中收集到的資料，不允許離開該單元，或者單一單元能收集到的資料有限，不足以訓練複雜的模型，為了追求更好的模型訓練效果與預測能力只好透過分散式學習的方式來進行。 舉例來說，Google為了聯合不同用戶的使用習慣優化智慧輸入法(Gboard)，同時不違反歐盟GDPR對用戶隱私的保護，於是透過聯邦學習來進行模型訓練，這個過程中資料能一直保持在本地端，只有梯度會被共享出去，所以是相對安全的選擇。 而聯邦學習應用在經典機器學習問題、例如說訂單預測時，有一種經典的方法是利用GBDT(梯度提升決策樹)來進行，GBDT利用了Bootstrap方法來對母體進行估計，而在這個場景下我們假設3家企業的資料雖然有不同的收集來源，但有相似、相同特徵，並相信他們服從於相同的母體分布(真實值)。 於是我們將分別對家企業的資料抽樣，產出多個基於抽樣資料的決策樹(弱分類器)，並將其聚合在一起共同決策；或者產出多個gini table，並共享這些gini table以及估計的殘差。 這種做法會有一個前提是三方企業必須事先溝通，找到各自資料中的共同特徵(by column)或者共同實例(by row)，基於特徵或實例來進行聯邦學習的方法由大陸楊強教授定義為橫向或綜向聯邦學習。 共同特徵：舉例來說，時間、日期、產品特性、供應商、某項醫療檢驗數值 共同實例：舉例來說，在醫療紀錄中的某個人、跨電商販售的某個產品 但聯邦學習也並非萬能，除了安全性是否真的可靠的疑慮外，聯邦學習由於是多個企業合作進行，某些情況下每家企業都只擁有一個半模型，你可以想像這是半把(或N分之1)鑰匙，結合所有半模型的系統才能有好的預測能力，但如果某家企業突然不參與，這些半模型都會跟著失效，這也是聯邦學習在應用上經常討論的議題。
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【QA】高維資料的處理
高維資料的處理 當我們說到二維資料，很直觀就能想到表示身高和體重的XY散佈圖，或者是加入一天當中的溫度變化折線，他們都是很常見的二維資料表示法，這種用來表示兩個變數之間關係的就是二維資料。 而高維資料就是資料中包含兩個以上的變數，舉例來說，所有學生的性向測驗上各項指標的分數，或者簡單的各科目分數成績(但可能無意義)，實務上可能會是一個工廠生產線上某個流程上數百個感測器在各個時間點的數值(我們這邊先不討論變數類型(自變數和應變數等等)，在同時存在大量變數時，就是高維度資料，高維度資料會大幅影響呈現、分析或是建模的各個部分。這個章節就是討論高維資料的處理。 如果只考慮機器學習上的做法，這也是我唯一熟悉的領域，大致上高維資料有兩個有效手段可以幫助你：降維以及分群。 降維 降維的方法可以簡單分為線性與非線性方法： 線性方法包括 PCA(主成分分析）、LDA(線性判斷分析)等等，目的在於尋找最佳解釋數據的變數的線性組合，如何定義最佳解釋這件事是其中重要的課題，以PCA為例，他會嘗試找到變數的最大變異量（或者我們說資料在某個空間中最大的分佈）。 非線性的方法則包括流型學習或t-SNE等方法，t-SNE也是用來做可視化時非常可靠的方法，這個方法將高維資料用高斯分佈近似，低維資料則視同樣本較少的抽樣資料、我們使用t分佈近似。接著用KL距離評估相似度，使用梯度下降方法求最佳解。 ref: https://medium.com/d-d-mag/淺談兩種降維方法-pca-與-t-sne-d4254916925b 另外值得一提的是深度學習也可以用在降維，而且有相當好的效果。比較常見的包括： 時序型資料：1D-conv，LSTM/attention Auto encoder Metric learning methods(siamese network...etc.) 分群 而關於分群，我們可以舉一個直觀的例子，當我們經營電商平台時，使用者的足跡和購買紀錄就是我們的高維資料。如果要將所有資料的所有變數納入考量並計算出在所有使用者都通用的準則（函數）是很困難的，這時候我們傾向於從使用者的屬性（足跡、紀錄）歸納出相似的使用者，並為他們推薦類似的產品。 這種作法可以用在非監督或自監督模型 並不是找到函數來精確描述類別，而是找到類別的中心點，或者基於類別之間的共性找到一個函數讓同類相近（或異類相斥） 而用在高維資料時，有一個顯著的問題是高維度的資料描述了一個非常大的空間（你可以想像低維空間是一張a4紙的大小，高維可能是一個銀河系）在這個空間中資料點是相當稀疏的，因此需要更多的資料，評估距離（聚類）的演算法也需要調整，或者，你可以先用維度縮減的方法減少維度，然後應用聚類模型進行描述。
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2022/06/14（二）
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如何使用梯度下降法（Gradient Descent）優化技術找出最優模型？
目錄 涉及梯度下降法的要素 微分技巧：鎖鏈率 以白話文方式深度學習優化過程 以數學公式方式表達深度學習優化過程 與梯度下降法有關的優化演算法 涉及梯度下降法的要素 最佳化概念 - Convex function 與 Non-convex function Convex function：透過二次、四次函數或限制集合，可以找到一個最佳解。 Non-convex function：該函數很多個微分為零的點，會有很多的局部最佳解。在深度學習的最佳化計算就是最典型的 Non-convex function。 損失函數（Loss()Loss()Loss()Loss()）：計算出真實結果與預測結果的差值，例如：真實的溫度與預測的溫度差多少min\ Loss()min Loss()min\ Loss()min Loss() 優化的關鍵：當微分為零時，代表該點在該曲線的相對高點或低點，因此可透過「微分」方式計算出往最小值的方向與幅度。 學習率： 微分技巧：鎖鏈率 已知有個函數 g(x)g(x)g(x)g(x) 是另一個函數 f(x)f(x)f(x)f(x) 的 xxxx，則：\frac{d}{dx}f(g(x))=f'(g(x))\times g'(x)ddxf(g(x))=f′(g(x))×g′(x)\frac{d}{dx}f(g(x))=f'(g(x))\times g'(x)dxdf(g(x))=f′(g(x))×g′(x) 舉例：Given\ g(x) = x^2+3,\ f(x)=x^2, then \<br>f(g(x))= (x^2+3)^2 \<br>\frac{d}{dx} f(g(x))= 2\times (x^2+3)\times (2\times x)Given g(x)=x2+3, f(x)=x2,thenf(g(x))=(x2+3)2ddxf(g(x))=2×(x2+3)×(2×x)Given\ g(x) = x^2+3,\ f(x)=x^2, then \f(g(x))= (x^2+3)^2 \\frac{d}{dx} f(g(x))= 2\times (x^2+3)\times (2\times x)Given g(x)=x2+3, f(x)=x2,thenf(g(x))=(x2+3)2dxdf(g(x))=2×(x2+3)×(2×x) 以白話文方式深度學習優化過程 已知有個深度學習模型、輸入資料、真實結果、損失函數 Step1. 模型中的權重值隨機產生出來 Step2. 將輸入資料放入模型中，得到預測結果 Step3. 使用損失函數評斷預測結果與真實結果的差距 Step4. 計算梯度（Gradient） Step5. 優化參數 以數學公式方式表達深度學習優化過程 已知有個兩層神經網路模型（L_1, L_2L1,L2L_1, L_2L1,L2）、輸入資料（X_0X0X_0X0）、真實結果（Y）、損失函數（Loss()Loss()Loss()Loss()）、需要優化的參數（W_1,\ W_2,\ W_OW1, W2, WOW_1,\ W_2,\ W_OW1, W2, WO） 經過第一層後輸出特徵（X_1X1X_1X1）為：X_1=X_0\times W_1X1=X0×W1X_1=X_0\times W_1X1=X0×W1 經過第二層後輸出特徵（X_2X2X_2X2）為：X_2= X_1\times W_2 = (X_0\times W_1)\times W_2X2=X1×W2=(X0×W1)×W2X_2= X_1\times W_2 = (X_0\times W_1)\times W_2X2=X1×W2=(X0×W1)×W2 經過輸出層後預測結果（O_{pred}OpredO_{pred}Opred）為：O_{pred}=X_2\times W_O = ((X_0\times W_1)\times W_O)Opred=X2×WO=((X0×W1)×WO)O_{pred}=X_2\times W_O = ((X_0\times W_1)\times W_O)Opred=X2×WO=((X0×W1)×WO) 使用損失函數（Loss()Loss()Loss()Loss()）計算出預測值與實際值的差距（llll）l=\frac{1}{N} (Y-O_{pred})^2 \<br>= \frac{1}{N} (Y-X_2\times W_O)^2l=1N(Y−Opred)2=1N(Y−X2×WO)2l=\frac{1}{N} (Y-O_{pred})^2 \= \frac{1}{N} (Y-X_2\times W_O)^2l=N1(Y−Opred)2=N1(Y−X2×WO)2 使用損失函數分別對 W_1,\ W_2,\ W_OW1, W2, WOW_1,\ W_2,\ W_OW1, W2, WO 微分\frac{d}{dW_O}l=2\times (Y-X_2\times W_O)\times X_2=gradient(W_O) \<br>\frac{d}{dW_2}l=2\times (Y-X_2\times W_O)\times (X_2\times W_2)\times X_1=gradient(W_2) \<br>\frac{d}{dW_1}l=2\times (Y-X_2\times W_O)\times (X_2\times W_2)\times (X_0\times W_1)\times X_0=gradient(W_1)ddWOl=2×(Y−X2×WO)×X2=gradient(WO)ddW2l=2×(Y−X2×WO)×(X2×W2)×X1=gradient(W2)ddW1l=2×(Y−X2×WO)×(X2×W2)×(X0×W1)×X0=gradient(W1)\frac{d}{dW_O}l=2\times (Y-X_2\times W_O)\times X_2=gradient(W_O) \\frac{d}{dW_2}l=2\times (Y-X_2\times W_O)\times (X_2\times W_2)\times X_1=gradient(W_2) \\frac{d}{dW_1}l=2\times (Y-X_2\times W_O)\times (X_2\times W_2)\times (X_0\times W_1)\times X_0=gradient(W_1)dWOdl=2×(Y−X2×WO)×X2=gradient(WO)dW2dl=2×(Y−X2×WO)×(X2×W2)×X1=gradient(W2)dW1dl=2×(Y−X2×WO)×(X2×W2)×(X0×W1)×X0=gradient(W1) 與梯度下降法有關的優化演算法 Stochastic Gradient Descent Adam AdamW Amsgrad ADMM-Adam
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如何處理不平衡資料
目錄 何謂「不平衡資料」？ 不平衡資料對模型訓練、預測有何影響？ 如何處理表格資料中不平衡資料的現象？ 如何處理影像資料中不平衡資料的現象？ 何謂「不平衡資料」？ 簡介：當資料中某類別變數的類別比例差異大，造成某類別特別多或特別少，就稱為「不平衡資料」 實際案例 想預測成骨不全症（俗稱：玻璃娃娃）等罕見疾病 想預測某個人信用卡信用風險 不平衡資料對模型訓練、預測有何影響？ 對模型訓練的影響 對評估的影響 如使用 Accuracy：會得到很高的準確度，但並沒有達到模型本身的意義 實際案例顯示問題：假設有 100 筆資料以預測是否不信任某信用卡用戶，其中 5 筆為不信任、其餘 95 筆為信任，如使用 Accuracy ，光直接猜測全部用戶皆為「信任」也有 95% 的準確度，但並不符合模型建立的初衷，因此應針對少量類別的預測準確狀況評估模型好壞。 如何處理表格資料中不平衡資料的現象？ Undersampling 將多類別資料以取後不放回的隨機抽樣、分層隨機抽樣或配對抽樣等方式抽出近似於少類別數量的資料 程式碼實現 import numpy as np import pandas as pd from IPython.display import display from imblearn.under_sampling import RandomUnderSampler from imblearn.over_sampling import RandomOverSampler, SMOTE from sklearn.model_selection import train_test_split # Step1. Load Data # OneHot Encoding and Label Encoding row_data = pd.read_csv("dataset.csv") print(row_data.info()) # Step2. EDA and preprocessing work_type_dict = { class_name: [0 if i != class_index else 1 for i in range(row_data["work_type"].unique().tolist().__len__())]\ for class_index, class_name in enumerate(row_data["work_type"].unique()) } smoking_status_dict = { class_name: [0 if i != class_index else 1 for i in range(row_data["smoking_status"].unique().tolist().__len__())]\ for class_index, class_name in enumerate(row_data["smoking_status"].unique()) } row_data["gender"] = row_data["gender"].apply(lambda x: 0 if x == "Female" else 1) row_data["ever_married"] = row_data["ever_married"].apply(lambda x: 0 if x == "No" else 1) row_data["Residence_type"] = row_data["Residence_type"].apply(lambda x: 0 if x == "Rural" else 1) row_data = pd.concat([ row_data.drop(columns = ["work_type", "smoking_status"]), pd.DataFrame(np.array( [work_type_dict[i] for i in row_data["work_type"]] ), columns = [f"work_type_{class_name}" for class_name in work_type_dict.keys()]), pd.DataFrame(np.array( [smoking_status_dict[i] for i in row_data["smoking_status"]] ), columns = [f"smoking_status_{class_name}" for class_name in smoking_status_dict.keys()]) ], axis = 1) # Missing value row_data = row_data.dropna() print(display(row_data["stroke"].value_counts())) # Step3. Define Input and Outcome Data xdata, ydata = row_data.drop(columns = ["id", "stroke"]), row_data["stroke"] # Step4. Split data into xtrain, xtest, ytrain, ytest xtrain, xtest, ytrain, ytest = train_test_split(xdata, ydata, test_size = 0.2) # Step5. Balanced train data undersample = RandomUnderSampler() xtrain_undersample, ytrain_undersample = undersample.fit_resample(xtrain, ytrain) print(display(ytrain_undersample.value_counts())) Oversampling 將少類別資料以取後放回的隨機抽樣、分層隨機抽樣或配對抽樣等方式抽至近似於多類別數量的資料 程式碼實現# Step5. Balanced train data oversample = RandomOverSampler() xtrain_oversample, ytrain_oversample = oversample.fit_resample(xtrain, ytrain) print(display(ytrain_oversample.value_counts())) Synthetic Minority Oversampling Technique (SMOTE) SMOTE 是依照少類別數量資料的特質生成出近似於少類別數量資料的方法 程式碼實現# Step5. Balanced train data smote = SMOTE() xtrain_smote, ytrain_smote = smote.fit_resample(xtrain, ytrain) print(display(ytrain_smote.value_counts())) 如何處理影像資料中不平衡資料的現象？ Rotation Flip Crop Reference Tan, J. (2020). How to deal with imbalanced data in Python. (https://towardsdatascience.com/how-to-deal-with-imbalanced-data-in-python-f9b71aba53eb)Liu, Tianyu; Fan, Wenhui; Wu, Cheng (2019), “Data for A hybrid machine learning approach to cerebral stroke prediction based on imbalanced medical-datasets”, Mendeley Data, V1, doi: 10.17632/x8ygrw87jw.1
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餘弦相似度的應用
目錄 餘弦相似度基本觀念 向量內積數學公式 皮爾森相關係數公式 餘弦相似度與相關係數的關係 實際案例 餘弦相似度基本觀念 餘弦定理是一個計算兩個向量相似度的方式，透過 \cos{\theta}cosθ\cos{\theta}cosθ 的計算方式，確認該兩向量的向量是否相似，兩向量越相似，方向會越趨近於同一方向。 實際案例：詞向量 向量內積數學公式 已知 \mathbf{x}={x_1, x_2, ..., x_n}x={x1,x2,...,xn}\mathbf{x}={x_1, x_2, ..., x_n}x={x1,x2,...,xn} 為 n 維的向量，則該向量的長度為：||\mathbf{x}||=\sqrt{x_1^2+x_2^2+...+x_n^2}∣∣x∣∣=x12+x22+...+xn2||\mathbf{x}||=\sqrt{x_1^2+x_2^2+...+x_n^2}∣∣x∣∣=x12+x22+...+xn2 已知有 \mathbf{x}={x_1, x_2, ..., x_n}, \mathbf{y}={y_1, y_2, ..., y_n}x={x1,x2,...,xn},y={y1,y2,...,yn}\mathbf{x}={x_1, x_2, ..., x_n}, \mathbf{y}={y_1, y_2, ..., y_n}x={x1,x2,...,xn},y={y1,y2,...,yn} 為兩個各為 n 維的向量，則兩向量的內積為：\mathbf{x}\cdot \mathbf{y}=||x||\times ||y||\times \cos{\theta} \<br>=\displaystyle \sum_{i=1}^n x_i\times y_ix⋅y=∣∣x∣∣×∣∣y∣∣×cosθ=∑i=1nxi×yi\mathbf{x}\cdot \mathbf{y}=||x||\times ||y||\times \cos{\theta} \=\displaystyle \sum_{i=1}^n x_i\times y_ix⋅y=∣∣x∣∣×∣∣y∣∣×cosθ=i=1∑nxi×yi 皮爾森相關係數公式 已知 x, yx,yx, yx,y 為兩各為 N 筆資料的連續型變數，則該兩變數的相關係數為：\rho_{x, y}=\frac{ \sum_{i=1}^N (x_i-\bar{x})\times (y_i-\bar{y})}{ \sqrt{\ \sum_{i=1}^N (x_i-\bar{x})^2}\times \sqrt{ \sum_{i=1}^N (y_i-\bar{y})^2} }ρx,y=∑i=1N(xi−xˉ)×(yi−yˉ) ∑i=1N(xi−xˉ)2×∑i=1N(yi−yˉ)2\rho_{x, y}=\frac{ \sum_{i=1}^N (x_i-\bar{x})\times (y_i-\bar{y})}{ \sqrt{\ \sum_{i=1}^N (x_i-\bar{x})^2}\times \sqrt{ \sum_{i=1}^N (y_i-\bar{y})^2} }ρx,y= ∑i=1N(xi−xˉ)2×∑i=1N(yi−yˉ)2∑i=1N(xi−xˉ)×(yi−yˉ) 餘弦定理與相關係數的關係 目標函數：\rho_{x, y}=\cos{\theta}ρx,y=cosθ\rho_{x, y}=\cos{\theta}ρx,y=cosθ\because \mathbf{x}\cdot \mathbf{y}=||\mathbf{x}||\times ||\mathbf{y}||\times \cos{\theta} \<br>\therefore \cos{\theta}=\frac{\mathbf{x}\cdot \mathbf{y}}{||\mathbf{x}||\times ||\mathbf{y}||} \<br>Let\ \mathbf{x} = {(x_1-\bar{x}), (x_2-\bar{x}), ..., (x_N-\bar{x})},\ y=\mathbf{y}={(y_1-\bar{y}), (y_2-\bar{y}), ..., (y_N-\bar{y})} \<br>Then,\ \cos{\theta}= \frac{\sum_{i=1}^n(x_i-\bar{x})\times (y_i-\bar{y})}{\sqrt{\sum_{i=1}^n(x_i-\bar{x})^2}\sqrt{\sum_{i=1}^n(y_i-\bar{y})^2}} = \rho_{x, y}∵x⋅y=∣∣x∣∣×∣∣y∣∣×cosθ∴cosθ=x⋅y∣∣x∣∣×∣∣y∣∣Let x={(x1−xˉ),(x2−xˉ),...,(xN−xˉ)}, y=y={(y1−yˉ),(y2−yˉ),...,(yN−yˉ)}Then, cosθ=∑i=1n(xi−xˉ)×(yi−yˉ)∑i=1n(xi−xˉ)2∑i=1n(yi−yˉ)2=ρx,y\because \mathbf{x}\cdot \mathbf{y}=||\mathbf{x}||\times ||\mathbf{y}||\times \cos{\theta} \\therefore \cos{\theta}=\frac{\mathbf{x}\cdot \mathbf{y}}{||\mathbf{x}||\times ||\mathbf{y}||} \Let\ \mathbf{x} = {(x_1-\bar{x}), (x_2-\bar{x}), ..., (x_N-\bar{x})},\ y=\mathbf{y}={(y_1-\bar{y}), (y_2-\bar{y}), ..., (y_N-\bar{y})} \Then,\ \cos{\theta}= \frac{\sum_{i=1}^n(x_i-\bar{x})\times (y_i-\bar{y})}{\sqrt{\sum_{i=1}^n(x_i-\bar{x})^2}\sqrt{\sum_{i=1}^n(y_i-\bar{y})^2}} = \rho_{x, y}∵x⋅y=∣∣x∣∣×∣∣y∣∣×cosθ∴cosθ=∣∣x∣∣×∣∣y∣∣x⋅yLet x={(x1−xˉ),(x2−xˉ),...,(xN−xˉ)}, y=y={(y1−yˉ),(y2−yˉ),...,(yN−yˉ)}Then, cosθ=∑i=1n(xi−xˉ)2∑i=1n(yi−yˉ)2∑i=1n(xi−xˉ)×(yi−yˉ)=ρx,y 實際案例 Reference Han, J., Kamber, M. and Pei, J. (2012). Data Mining: Concepts and Techniques. doi: https://doi.org/10.1016/C2009-0-61819-5
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第六次助教時間-如何使用梯度下降法找到最優模型
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第六次助教時間-不平衡資料的應用
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第六次助教時間-餘弦相似度的應用
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2022/06/25（六）20:00-21:00 助教時間
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2022/06/28（二）助教時間
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圖像分析時做subsampling 帶來哪些好處？
Subsampling也可以叫做Pooling Layer，是CNN在做圖像分析時一個非常重要的步驟。 Subsampling的好處有以下兩點 1.模型參數減少但仍能保留重要的特徵 我們知道當電腦要辨識一張圖時會把圖像轉成矩陣的形式，當模型學習的參數越多，模型也會更容易產生overfitting的現象，所以我們會希望模型可以減少一些參數但又不能刪減到重要的特徵，這時候 Subsampling就可以起到很大的作用 2.保持參數間的不變性 什麼是不變性呢？就是圖片再經過Subsampling之後會成為一個較小的圖片，但不會因為圖片變小了就失去了原圖的樣貌 那Subsampling要如何使用呢？ 目前常見的方法有三種 1.Max pooling 2.Mean pooling 3.Min pooling 下面將為三個方法做詳細的解釋 Max pooling 假設現在我們有個4X4的矩陣，那我們希望透過Subsampling將其轉換成2X2的矩陣 一開始我們會先把4X4的矩陣分割成4個2X2的矩陣 Max pooling顧名思義就是將4個2X2的矩陣中數值最大的提出來當成新矩陣的數值 Mean pooling 我們以一個新的4X4矩陣為例子，一樣是切成4個2X2的矩陣 Mean pooling也不難懂，就是從4個2X2的矩陣取出各矩陣中的平均數做為新矩陣的數值 Min pooling Min pooling通常比較少會被使用到，因為相對比較無法體現出圖片的重要特徵，只有希望對圖片達到比較特別的效果時才會使用到，那Min pooling的方式就是把矩陣中最小的數值提取出來成為一個新的矩陣 那Max pooling 、Mean pooling 、 Min pooling 要怎麼選擇呢？ 通常這會取決於你想讓圖片最後呈現什麼效果而決定 我們以下圖為例 這是一張鳶尾花的照片，我們可以看到根據三種pooling方法得到的效果都有所不同 Max pooling讓照片呈現較亮、鮮豔的效果 Mean pooling讓照片呈現較平滑的效果 Min pooling讓照片呈現較暗沈的效果
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常見的相關係數有哪些？
目前生活中常見的相關係數有以下四種 1.皮爾森相關係數（Pearson correlation coefficient） 2.斯皮爾曼相關係數（Spearsan correlation coefficient） 3.重複測量相關係數（Repeat measure correlation） 4.xi correlation 以下針對四種的相關係數做更加詳細的介紹 皮爾森相關係數（Pearson correlation coefficient） 皮爾森相關係數主要是探討兩個連續性特徵的線性相關程度，他是兩個變數的共變異數與標準差乘積之比，那共變異數是什麼呢？共變異數主要是表現兩個變數一同變化的程度，就是當A變數離均值越遠時，B變數是否也會跟著變遠，接著讓我們來看看皮爾森相關係數的公式： 式子中的CovXY代表X跟Y的共變異數，SX、SY則代表X跟Y的標準差，那得到的數值r會介於-1~1之間，r如果是正的代表正相關，負的就代表負相關，數值越高代表相關性也越高，不過皮爾森相關係數也有個缺點，就是在判斷中會容易有偏誤，當r很低有時並不代表兩個變數之間沒有相關性，可能是因為兩變數呈現非線性相關，這用皮爾森相關係數就無法察覺，於是需要多方的查證與參考來確認。 斯皮爾曼相關係數（Spearsan correlation coefficient） 斯皮爾曼相關係數主要是探討兩序列特徵（可以依照等級做排序的特徵，如排名、年紀）之間的相關性，他與皮爾森相關係數的差別是不容易受線性相關或離群值影響，接著讓我們來看斯皮爾曼相關係數的公式： 其中di指的是第i個配對等級的差值，也就是第i個第一個變項的等級減去第i個第二個變項的等級，n則為樣本總個數 與皮爾森相關係數相同的是，斯皮爾曼係數得到的數值p也是介於-1~1之間，且正號為正相關，負號為負相關。 重複測量相關係數（Repeat measure correlation） Repeat measure correlation主要是應用在時間序列或是需要長期追蹤資料分析的相關係數，可以去探討兩個特徵在相同樣本下隨著時間變化的線性相關性，藉由不斷的重複測量，來探索兩特徵是否會隨著時間而有相同的變化。如果想在python中做使用，可以透過pingouin套件來操作。 xi correlation xi correlation是一個非常新的相關係數，在2020才被提出，主要是探討兩特徵之間”可以形成一個function"的程度，即使兩個特徵相關性再多麼高，也不一定能透過一個funtion來表示，那xi correlation的優點就是不管兩個特徵相關性是線性、非線性、二次項等等，都可以使用xi correlation來做檢測。如果想在python中使用，可以透過xicor套件來操作xi correlation。
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歸一化、標準化、中心化分別是什麼？
這三種方法都有一個共同的目標：為了要讓數據在訓練上能變得更有意義 在此三種方法中，歸一化及標準化的目標較相似，他們主要針對的問題都是把數據範圍縮小卻又不影響原始數據的分佈。我們在比較兩組數據時，可能會遭遇因單位的不同(例如：身高與體重)，或數字大小的代表性不同(例如：房價1000萬與坪數30坪)，那麼若把這兩個特徵丟入模型，在收斂上會產生問題，所以這時候就會需要使用歸一化跟標準化來處理數據，讓模型也能夠更加的完整與準確。 至於中心化主要針對的問題是當數據的中心非原點時，我們會無法簡單的以向量的方式來表示數據，所以要把數據的中心移到原點。 以下會對三種方法進行更詳細的介紹與應用 標準化(Standardization) 標準化主要是應用在機器學習的演算法當中，目的是為了要讓數據變成同一個單位，讓原本不同單位的數據可以進行比較，並提升模型的精確度。 常見的標準化方法就是Z分數標準化(Z-Score Standardization) 我們會使用到下列公式： 式子中的X代表數值、μ代表平均數、σ代表標準差 再經過Z分數標準化後，數據會呈現常態分佈（如下圖） 常態分佈就是讓數據變成平均值=0、標準差=1 Z分數標準化適用於分佈大致對稱的資料，因為在非常不對稱的分佈中，標準差的意義並不明確，此時若標準化資料，可能會對結果做出錯誤的解讀，另外，當我們未知資料的最大值與最小值，或存在超出觀察範圍的離群值時，也可透過 Z分數標準化來降低離群值對整個模型的影響。 我們來看一個實際應用的例子，請觀察下圖 經過觀察我們發現兩組數據的平均值與標準差並不相同且差異巨大，所以我們利用Z分數標準化把兩組數據皆轉換成平均值近似於0且標準差近似於1的資料 並我們把標準化前後的數據放在同個圖表上後發現，兩組數據在標準化後確實都成常態分佈 （紅色為標準化前，藍色為標準化後） 歸一化（又稱正規化 Normalization）： 歸一化的目標在於要把原始數據按比例縮放成[0,1]之間且不改變其原本分佈。舉例來說，若我們現有兩組數據資料（如下圖）分別表示 500 項商品的銷售量及銷售額。 如左圖所示，可以看到很明顯的，此兩組資料的單位不同，且在數字上有著懸殊的差異，不過透過資料歸一化後，如右圖所示，兩組資料將同時轉換成純量並縮放於 [0,1] 區間中，且沒有影響到資料本來的分佈狀況，這樣的資料轉換，能排除資料單位的限制，提供我們一個相同的基準來進行後續比較分析。 常見的歸一化方法是使用最小值最大值歸一化法(Min-Max Normalization) 我們會使用到下列公式 其中Xmax代表最大值、Xmin代表最小值，要注意的地方是當有新資料進來時，可能會有最大值或最小值的改變，這時候我們就要重新定義式子中的Xmax跟Xmin才不會導致錯誤。 這裡我們準備兩組資料，這兩組資料的分佈情況及各別平均值、標準差，請詳下面圖表： 我們經過觀察能發現兩組數據的最大值、最小值、平均值跟標準差都有十分明顯的差異 透過使用最小值最大值歸一化法，將兩組資料皆縮放於 [0, 1] 區間中，新的最大值、最小值、平均值跟標準差如下圖所示 接下來我們可以觀察兩組資料各自歸一化前後的分佈，請詳下圖(紅色為歸一化前，藍色為歸一化後)，我們瞭解到最小值最大值歸一化法僅將資料等比例縮放進 [0, 1] 區間中，但並不會改變原本的分佈情況。 中心化(Zero-centered) : 中心化實際應用的層面較少，主要的目的只是讓數據能回到以中心為起點的數據 中心化的式子為：數值-平均數 因為只是平移數據，所以對數據本身的影響不大 參考文獻：https://aifreeblog.herokuapp.com/posts/54/data_science_203/
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如何有效地找到組合特徵？
我們在建立模型的時候，會加入許多的參數，但是若加入的參數過多，也會容易造成模型過於複雜或是overfitting的現象，而且也並不是所有的特徵組合都是有意義的，所以能運用什麼方法，來讓我們找出正確且有意義的組合特徵呢？ 這次想介紹給大家的方法是基於決策樹來尋找有意義的特徵組合 先簡單介紹一下決策樹是什麼？ 決策樹是一種監督式學習的模型，會根據我們事先所輸入的資料去做訓練，而訓練的成果會是一個樹狀結構，當我們輸入新資料時，模型就會根據所訓練出的規則來對其進行預測。 舉一個常見的例子：當一個人去商場購買東西，要如何知道他的忠誠度？ 這裡面會牽扯到非常多的因素例如：購買商品的數量、購物頻率、購買的商品是否重疊率高等等 當我們將這個人過往的購買記錄放進決策樹中，決策樹就會根據所有顧客的購買紀錄中去預測他是否為商場的忠實客群。 接下來我們以一個實際的例子展示給大家看 有一位Y女士她到了適婚年齡想要結婚，於是她請了相親公司來幫他尋找適合的對象，女方開出了幾個她會參考的條件（年齡、長相、薪水高低、是否會寫程式）如果以上條件都有符合她內心的要求她就願意跟對方見面。 以下是相親公司給這位Y女士介紹五位對象的資料，以及Y女士最後是否有答應見面 其實當我們再根據條件做選擇時就是一個很典型的決策樹分類，我們可以將上述五位的資料放入決策樹模型做訓練，那就會得到以下結果 從這個決策樹我們能看到Y女士的條件順序由上至下是 1.年齡是否小於30歲 2.長相是否可接受 3.薪水是否大於40000元 4.是否會寫程式 如果這四個條件都滿足就會得到100%願意見面的結果 回到正題，所以到底要怎麼用決策樹有效的找到組合特徵呢？ 這時候就要介紹決策樹中的演算法（ID3、C4.5、CART） 藉由這些演算法能讓模型知道哪個特徵適合當作根節點、哪個特徵對模型的影響較大等等 那因為篇幅關係所以我們就先介紹ID3 ID3(最大資訊增益) ID3中有個很重要的名詞叫做熵（entropy） 熵越大代表不確定性就越大 ID3背後的公式原理如下 1.計算數據及D的經驗熵H(D) 其中Ck表示在集合D中屬於第k類樣本的樣本子集合 2.利用條件機率計算特徵A對數據集D的條件熵H(D|A) 其中Di表示在D中特徵A取i個值的樣本子集合，Dik表示Di中屬於第k類樣本的樣本子集合 3.計算訊息增益 經驗熵H(D) - 條件熵H(D|A) 我們在回到剛剛Y女士相親的例子用ID3測試 1.計算H(D) 我們知道Y女士在最後見面有兩個選擇：見面與不見面 並選擇了兩位見面，三位不見面 所以我們要將兩種分類分開計算 H(D) = (-3/5 x log2 x 3/5 ) + (-2/5 x log2 x 2/5) = 0.971 2.計算H(D|A) H(D|年齡）= 1/5(H|>30）＋ 4/5(H|<30) 在四位年齡<30歲的人中，Y女士選擇了兩位見面兩位不見面 =1/5 x (0) + 4/5 x [(-2/4 x log2 x 2/4 )-( 2/4 x log2 x 2/4)] = 0.8 依照此公式計算其他特徵（公式省略） H(D|長相）= 0.551 H(D|薪水）= 0.551 H(D|寫程式） = 0 3.計算g(D,A) g(D,年齡) = 0.971 - 0.8 = 0.171 g(D,長相) = 0.42 g(D,薪水) = 0.42 g(D,寫程式) = 0.971 因此我們可以看出寫程式的資訊增益最大，也就是這個特徵對模型的影響力較大，長相跟薪水是其次，年齡則是最小，依照這個結果我們就能找出有效的組合特徵了。 參考文獻：https://roger010620.medium.com/決策樹-decision-tree-常見的三種算法-id3-c4-5-cart-54091ca85044
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