非侵入性檢查 (non-invasive examination) 為疾病診斷中對身體造成傷害最低的醫療輔助行為。心臟超音波即為一種非侵入性之檢查，且能作為心臟結構異常診斷的輔助工具。而隨著對心臟疾病的認識，可了解到心臟衰竭與左心室的收縮舒張功能密切相關。本文延續 Chen (2011) 和 Kao (2011) 所提之問題，藉由心臟超音波影像中，左心室收縮及舒張時的變化，分析圖片灰階值之改變，進一步探討與心臟疾病的關聯。由於資料屬於大維度矩陣之類型，本文延續 Chen (2011) 和 Kao (2011) 所提出之方法，使用因素分析後所得到之因素分數作為分類的依據。在此使用正常受試者之因素分數建立準則，再計算每位異常受試者與正常群受試者數據所建立之模型的馬氏距離 (Mahalanobis Distance)，利用此距離來區分心臟功能之正常與否。由於計算馬氏距離時需要估計正常群受試者之共變異數矩陣，為了提高對分類之精確度，希望從正常群受試者之樣本共變異矩陣中找出影響點；此處引用實驗設計裡所討論之最適設計的概念，建立一套方法作為篩選出影響點之準則。

Non-invasive physical examination helps to make disease diagnosis with minimum injury to the body. Cardiology ultrasound is a non-invasive examination which can be used as a auxiliary tool for diagnose cardiac structure abnormalities. With more understanding of heart diseases, it has been recognized that heart failures are closely related to left ventricular systolic and diastolic function. Following Chen (2011) and Kao (2011), we study association of heart diseases with the change of gray-scale values in the cardiology ultrasound images of left ventricular systolic and diastolic.
Since data obtained from ultrasound image is of matrix type with high dimensions, following the method proposed by Chen (2011) and Kao (2011), factor scores obtained from factor analysis are used as a basis for classification. We take the factor scores of normal subjects to establish the bench mark and calculate the Mahalanobis distance of each abnormal subject with the model established by the data from normal group. Later based on this distance to the normal group, cardiac function of the subject is distinguished as normal or not. In order to improve the accuracy of the classification, influential points which may cause inaccurate covariance matrix estimate on the subjects in normal group are identified. Based on concepts from optimal designs theory, some criteria are established for screening out the influential points.

論文審定書...i
誌謝...ii
摘要...iii
Abstract...iv
1 前言...1
2 資料描述...1
2.1 資料蒐集...2
2.2 資料處理...2
2.2.1 旋轉...3
2.2.2 截取與壓縮...3
2.2.3 灰階值校正...4
2.2.4 外圍淡化...5
3 研究方法...6
3.1 距離指標模型...6
3.1.1 異常群分類...7
3.1.2 正常群組內交叉驗證...8
3.1.3 臨界點分析...8
3.2 共變異矩陣估計之敏感性分析...9
3.2.1 矩陣分析-最適準則...9
3.3 因素篩選...10
4 研究結果...11
4.1 距離指標分類結果...12
4.2 影響點...13
4.2.1 D 最適準則下之影響點...13
4.2.2 A 最適準則下之影響點...14
4.2.3 E 最適準則下之影響點...15
4.2.4 影響點之摘要...16
4.3 因素篩選...18
4.4 臨界點分析...22
5 討論與結語...24
參考文獻...25
附錄...26

[1] Chen, J.-M. (2011). Discriminant analysis for cardiology ultrasound in left ventricle. Master thesis in Chinese, Department of Applied Mathematics, National Sun Yat-sen University.
[2] Friedman, J. H. (1989). Regularized discriminant analysis, Journal of the American Statistical Association, 84, 165-175.
[3] Kao, L.-W. (2011). Comparison of discriminant between logistic model with distance indicator and regularized function for cardiology ultrasound in left ventricle. Master thesis in Chinese, Department of Applied Mathematics, National Sun Yat-sen University.
[4] Lee, T.-L., Li, T.-Y. and Zeng, Z.-G. (2009). A rank-revealing method with updating downdating, and applications. Part II. SIAM Journal on Matrix Analysis and Applications, 31, 503-525.
[5] Rencher, A. C. (2002). Methods of Multivariate Analysis, 2nd ed. John Wiley, New York.
[6] Yang, H.-Y. (2012). Comparison of Classification Effects of Principal Component and Sparse Principal Component Analysis for Cardiology Ultrasound in Left Ventricle. Master thesis in Chinese, Department of Applied Mathematics, National Sun Yat-sen University.