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研究生:李婉菁
論文名稱:主成分選取與因子選取在費雪區別分析上的探討
論文名稱(外文):Discussion of the Fisher's Discriminant Analysis Based on Choices of Principal Components and Factors
指導教授:姜志銘姜志銘引用關係
學位類別:碩士
校院名稱:國立政治大學
系所名稱:應用數學研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2007
畢業學年度:95
語文別:英文
論文頁數:26
中文關鍵詞:主成分
外文關鍵詞:Principal Component
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當我們的資料變數很多時,我們通常會使用主成分
或因子來降低資料變數;
在選取主成分與因子時,我們通常會以特徵值來做選擇,
然而變異數大(亦即特徵值大)的主成分或因子雖然解釋了大部分變異,
但卻不一定保留了最多後續要分析的資訊,
例如利用由特徵值所選取出來最好的主成分或因子
來當做區別資料之變數,所得結果不一定理想。
在此我們假設資料是來自於兩個多維常態母體,
我們將分別利用由Mardia等人 (1979) 和Chang (1983) 所提出的兩種方法
來選取出具區別能力的主成分,將其區別結果與由特徵值所選取出最好的主成分
之區別結果作一比較;並且將此二方法應用在選取因子上。
同時我們也證明Mardia等人 (1979) 和Chang (1983)的方法對於
主成分及因子(利用主成分方法轉換)有相同的選取順序。
本文更進一步地將Mardia等人
所提出之方法運用至三群資料上,探討當資料來自於三個
多維常態母體時,我們該如何利用此方法來選取具區別能力之變數。
Principal component analysis or factor analysis are often used
to reduce the dimensionality of the original variables.
However, the principal component or factor, which has
larger variance (i.e eigenvalue) explaining larger proportion of total sample
variance, may not retain the most information for other analyses later.
For example, using the first few principal components or factors
having the largest corresponding eigenvalues as
discriminant variables, the discriminant result
may not be good or even appropriate.

\hspace{2.05em}We first discuss two methods, given by Mardia et al. (1979) and Chang (1983)
for choosing discriminant variables when data are randomly obtained from
a mixture of two multivariate normal distributions.
We then use the discriminant result (or classification error rates)
to compare these two methods and the traditional method of using the
principal components, which have the larger corresponding eigenvalues,
as discriminant variables. We also prove that the both the two methods
have the same selection order on principal components and factor (obtained
by the principal component method).
Furthermore, we use the method of
Mardia et al. to select appropriate discriminators when data is from
three populations.
[1] Mardia K.V., Kent J.T. and Bibby J.M., Multivariate Analysis, Academic
Press, (1979), 322–324.
[2] Chang W.C., On using principal components before separating a mixture of two
multivariate normal distributions, Appl. Statist., 32 (1983), 267–275.
[3] Jolliffe I.T., Morgan B.J.T. and Young P.J., A simulation study of the use of
principal components in linear discriminant analysis, J. Stat. Comput. Simul.,
55 (1996), 353–366.
[4] Jolliffe I.T., Morgan B.J.T. and Young P.J., A note on using principal components
in linear discriminant analysis, (1995). Submitted for publication.
http://citeseer.ist.psu.edu/jolliffe95note.html
[5] Murry G.D., A cautionary note on selection of variables in discriminant analysis,
Appl. Statist., 3 (1977), 246–250.
[6] Namkoon G., Statistical analysis of introgression, Biomtrics, 22 (1966), 488–
502.
[7] Wolfe J.H., Computational methods for estimating the parameters of multivariate
normal mixtures of distribution, U.S. Naval Personnel Research Activity,
San Diego (1967), SRM 68–2.
[8] Dillon W.R., Mulani N. and Frederick D.G., On the use of component scores
in the presence of group structures, J. Consumer Research, 16 (1989), 106–112.
[9] Kemsley E.K., Discriminant analysis of high-dimensional data: a comparsion
of principal components analysis and least squares data reduction methods,
Journal of Statistical Computitation and Simulation, 55 (1996), 353–366.
[10] Song C.C., Jiang T.J. and Kuo K.L., On the Fisher’s discriminant analysis,
Technical Report # NCCU 701-05-T04-01, Department of Mathematical Sciences,
National Chengchi University.
20
[11] Jackson J.E., A user’s guide to principal components, Wiley, New York (1991).
[12] Flury B.D., Developments In Principal Component Analysis, (1995), 14–23.
[13] Johnson R.A., Wichern D.W., Alllied Multivariate Statistical Analysis, Prentice
Hall, (2002).
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