臺灣博碩士論文加值系統

摘　要

獨立成份分析是近年來在多變量上發展的統計方法。它被設計在多維度的類神經網路(neural networks)、訊號傳輸(signal processing)資料中尋找隱藏的獨立因子。ICA在其它應用上，也有不錯的結果。如：腦波資料(brain image)、聲頻區別(audio separation)、經濟計量(economics)、生醫工程(biomedical engineering)和生物資訊(bioinformatics)。然而在一份資料中，我們所得到的資訊除了有興趣的混合變數(mixing variables)外，還有可能得到會影響到混合變數的相關因子。在目前做法上，處理ICA資料時，是直接將混合變數放至ICA線性模型，分析結果可得到估計的獨立因子及其對應的係數。在本文的內容考慮到的模型中，發現直接以混合變數進行ICA分析時獲得的係數是有偏的。然而，本文建議先將混合變數在ICA分析前先對因子進行迴歸分析，將其殘差放入ICA分析，獲得的結果較為正確。

Independent component analysis (ICA) is a recently developed statistical and computational technique for discovering mutually independent nongaussian latent variables from observed multivariate data in the fields of neural networks and signal processing. It can potentially be applied to many application fields such as brain imaging, audio separation, telecommunication, feature extraction, economics, psychology, physiology, biomedical engineering, and bioinformatics, whenever the assumptions of statistical independence and nongaussianity are substantively justifiable. In current practice, one applies the standard procedure(s) of ICA directly to the observed multivariate variables, even though they may be affected by some known covariates, to identify the independent components and estimate the mixing coefficients. In this study, we find that ignoring those relevant covariates may lead to a biased result of ICA, and then suggest a covariate-adjusted ICA to minimize such biases by applying the standard procedure(s) of ICA to the residuals from the regressions of the observed multivariate variables on those relevant covariates in a linear ICA model. A simulation study is conducted using the FastICA algorithm to examine the statistical properties of our covariate-adjusted ICA and to derive numerically the sampling distributions of the estimated mixing coefficients as an interesting by-product. Finally, two examples are given for illustration.

Contents
1 Introduction 1
2 The Under-Fitting Problem 3
3 A Two-Step Solution 5
4 Simulations 7
4.1 Data 7
4.2 Approaches 9
4.3 Results 10
4.3.1 Approach 1: The Direct ICA Method 10
4.3.2 Approach 2: The Covariate-Adjusted ICA Method 11
5 Two Examples 11
5.1 Example 1: The Crab Data 12
5.2 Example 2: The Electromyography (EMG) Data 14
6 Discussions 17
7 Appendix: The fastICA Package in S 18
8 References 19
9 Tables 22
10 Figures 33

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