2014 3 Å 28 1 Ð Mar. 2014 Communication on Applied Mathematics and Computation Vol.28 No.1 DOI 10.3969/j.issn.1006-6330.2014.01.012 Ç ÖÇ Ú ¾Ä Î Þ Ý ( 200433) Ç È ß ³ Õº º ÅÂÞÐÆÈÛ CAC40 Õ Û ËÛ ¾ ÆÄ (ICA) ½ (PCA) ¾ È Ü Í ÆÄ Û 2010 Å Ý 62P05 Ó Ý O213.9 Ñ A Ý 1006-6330(2014)01-0086-10 Application of ICA and PCA to extracting structure from stock return ZHANG Jie (School of Mathematical Sciences, Fudan University, Shanghai 200433, China) Abstract The paper exhibits a new methodology to analyze and recognize the financial market structure. The method is applied to the underlying stocks of the index CAC40. The trend of the stock price is analyzed, and finally a comparison of the independent component analysis (ICA) and the principal component analysis (PCA) is given. Key words ICA; PCA; stock index 2010 Mathematics Subject Classification 62P05 Chinese Library Classification O213.9 1 Ì Æ 1.1 ICA Ò ¹ ICA(Åà ) Ô¹¼ ( µ ) ½ ¾ ¹ º Ô¹ ÝÍ Ä ¹ Ö Ý ¹ ¾ ICA ØÙ ÅÃÛ Û¾ ICA É ÙÃÛ² Back Weigend [1] Ð»Ç ICA Ú ² Ý Å [1], Πл ICA Ú ² Ý Ó 2013-05-17; Ä 2013-07-20 ÁÜ µ Î ¼Đ E-mail: 062018048@fudan.edu.cn
1 Ð ÆÄ ÆÛ ³ Ð 87 1.2 ICA à ÂÔ Þ Æ x(t) = (x 1 (t), x 2 (t),, x n (t)) T, Ñ T x i (t) = n a ij s j (t). s(t) = (s 1 (t), s 2 (t),, s n (t)) T ÍÍÆ j=1 s 1, s 2,, s n ÅÅÅà A = (a ij ) n i=1,j=1 ICA Ê Þ»Ï n Åà y(t) = (y 1 (t), y 2 (t),, y n (t)) T. Ò Ñ Ô y(t) = Wx(t) = WAs(t). Ç ÐÔ W,»  i) Þ ÈÅà ii) ²ÌÔ iii) Ä Ó«Ì ICA  ¹ [2] : Algorithme HJ Maximisation de contraste (CoM) JADE Fast-ICA Infomax Ý Matlab Fast-ICA [3] CAC40 40 ±Ú 1.3 PCA PCA ( ) ² ÖÅÛ PCA ¹ Æ ¹ ± Û Å± Ê ± Û ÈÝÁ Ê»Û Åà ÍÊÔ«Û Ð Î Ã Û¾ Û Ö Â ÑÜ½Ë Ê ²½ ÅÃ Ê Æ Ô¹ Æ ± ØÙ ÅÃ. ICA Ü½Ç Î±Ñ ± À л ICA PCA Ä Å ³º½» ¾ 2 ICA Û Ð 2.1 Ç ICA ÊºÐ É ICA Ý ºÐ ² À Ø CAC40 40 ±Ú ( 2008 7 22 ² 2009 3 16 1). È Û GDF Suez Ô Ú ÓÄ Suez GDF Ô GDF Suez Ú 2008 7 22 Æ»Ä CAC40 2» 2008 9 Ì Ú ÂÁ ºÐ ÓÔ
88 28 1 Societe Generale 2008 7 22 µò 2009 3 16 µ Ù Ú Ü Á BNP Societe Generale, Ñ Ü ºÐ ßË 2 CAC40 Ó Ô±º Ù À ² 2.2 Ë ÌÑ É ÔÐ ICA ÐÑØ ¾¼ ÊÚ p(t) Ê Ô± x(t) = ln(p(t)) ln(p(t 1)). Â Ê 2 10 ±Ú Ð ßÔ¼ Ð ¹ Ø Đ ºÐ 9 ² 12 ÚÍ ( 3).
1 Ð ÆÄ ÆÛ ³ Ð 89 3 10 ÐÕÇÙ Þ É (2008 7 Ò 2009 3 ) 2.3 ÊØ«Ö Ê Ú ÚÍÝ Fast-ICA(Hyvarinen et Oja) [3] ½ Ë Ú Åà À È Ô 40 Åà À ¾¼ ±Ê» 10 Åà À ½ Þ 10 Û Ö ÇÊ ± ßË ÙÌ À Þ ÅÃÛ Ê ² ßË Ü Í Æ ( 4). 4 ± 10 ÕÕÇ ËÙ Â SG ±Ú ICA PCA ÅÃÛ ¹ (Û» ) A 32 (¾ ÊÝ SG). 5 SG 6 Ñ Å ÃÛ Î Ê SG Ú ÅÆÍ 5, Back Weigend È
90 28 5 Ï Ò ÈÈ Ô A Ð 6 ÕÛÔÇ ËÙ ± ÅÃÛ Ç Ú ÅÆÍ Ã ÑÅÃÛ Ã Ú ÅÆÍ Ã È ÅÃÛ Ï Ã ³ÅÆÍ Ç Ï Ã «Ö ßË À 6 ÑÅÃÛ 6. 6 ÔÀßÆ SG Ù» ÔÆà 6 SG Ú À 6 ÑÅÃÛ Ú Ë 34 ÅÃÛ ÊÚ ßË 34 ÑÅÃÛ ÊÚ ßË Ï Ò Þ 6 ÑÅÃÛ ÊÚ Ò Â 2.4 ÊØ ÉØ Ó Ô³ºÎÇ ÑÅÃÛ Ú Ú Ä Ê ÆµÂ ÑÅÃÛ Ä Ð»Ô Â À Ì ± Ì ÀÖ ²É ÅÃÛ. ¹ ²É ÅÃÛ Ú
1 Ð ÆÄ ÆÛ ³ Ð 91 ¾¼ Ú 3, 7 лÇĐ Ð 6 ²É ÑÅÃÛ Ú SG ÅÆÍ 7 Ï 6 ÕÅ Ð ÕÇ ËÙßÆ Đ 8 ±¼ Ú SG Ú È± ¹Õ лÇÚ Ã Ð ½ÓÇ À À 8 Ï 7 Ê ÂÙ Ú¼ 2.5 PCA PCA ʺР² Ô¹ Ò» ÊÚ ² Û¾ Ê Ô ICA PCA Ä Ê CAC40 40 ±Ú Ý PCA  6 ÑÛ ¹ Ç 85% ( 9). 9 Ç»Ñ Û ¹ Ç 60% À 6 ÑÛ» SG Ú ( 10). Ð 6 ÑÛ ½ ²É ( 11). Đ 12» Đ 6 ÑÛ ²É Ú ²É ¹ ( 10). Û PCA ± ÇÛ Î±ÈÝÁ 0, Û¾ ¹Ö  ICA PCA
92 28 9 6 ÕÛÔÇ ÕÇËÙ 10 ßÆ Ô½Ï 6 ÕÕÇËÙÔÀßÆ Ô»Ù Ô Å (Âà 34 ÕËÙ Ô Í ßÆ Ô» Ô Ã) 11 ÏÅ Ð 6 ÕÕÇËÙßÆ Ù Đ ÎÌ ³ 17 12 Ï 11 ßÆ Ô
1 Ð ÆÄ ÆÛ ³ Ð 93 3 ICA Ç CAC40 Đ 40 ±Ú ÅÆÍ Ç 40 Ø Ù ÅÃÛ ÅÃÛ Ñ ÂŹ i) Í Ã ¹«ÜÒÚ Ã ii) Ï Í Ã ¹ Ü «³ Ã Þ ICA ÊÚ Ã PCA Ñ A A1 CAC 40 Æ (2009 3 ) Õ ¼ /% Total SA 17.79 Sanofi-Aventis SA 8.70 France Telecom SA 6.79 Gaz de France SA 6.66 BNP Paribas 5.30 Vivendi 4.46 Groupe Danone 3.35 Carrefour SA 3.28 Air Liquide 3.19 Societe Generale 2.91 L Oreal SA 2.76 Vinci SA 2.66 AXA SA 2.64 LVMH Moet Hennessy Louis Vuitton SA 2.60 Arcelor Mittal 2.33 Schneider Electric SA 2.12 Electricite de France 2.04 Credit Agricole SA 1.69 Unibail-Rodamco 1.64 Pernod-Ricard SA 1.62 Ñ B Matlab л ICA Æ Number of signals: 40 Number of samples: 169 Õ ¼ /% Alstom 1.50 Bouygues 1.47 Cie de Saint-Gobain 1.26 Veolia Environnement 1.23 Essilor International SA 1.15 Lafarge SA 0.93 Accor SA 0.93 PPR 0.76 EADS 0.70 Cap Gemini SA 0.67 Michelin 0.67 Vallourec 0.62 Suez SA 0.60 Renault SA 0.53 Peugeot SA 0.51 Alcatel-Lucent 0.47 Lagardere SCA 0.41 STMicroelectronics NV 0.41 Air France-KLM 0.36 Dexia SA 0.29
94 28 Calculating covariance Dimension not reduced. Selected [40] dimensions. Smallest remaining (non-zero) eigenvalue [0.004 388 3]. Largest remaining (non-zero) eigenvalue [71.020 9]. Sum of removed eigenvalues [0] [100] % of (non-zero) eigenvalues retained. Whitening Check: covariance differs from identity by [1.341 15E 013]. Used approach [defl]. Used nonlinearity [pow3]. Starting ICA calculation IC 1...computed (10 steps) IC 2...computed (9 steps) IC 3...computed (11 steps) IC 4...computed (10 steps) IC 5...computed (13 steps) IC 6...computed (18 steps) IC 7...computed (6 steps) IC 8...computed (8 steps) IC 9...computed (11 steps) IC 10...computed (12 steps) IC 11...computed (10 steps) IC 12...computed (8 steps) IC 13...computed (11 steps) IC 14...computed (9 steps) IC 15...computed (9 steps) IC 16...computed (8 steps) IC 17...computed (10 steps) IC 18...computed (9 steps) IC 19...computed (31 steps) IC 20...computed (9 steps) IC 21...computed (13 steps) IC 22...computed (14 steps) IC 23...computed (13 steps) IC 24...computed (10 steps) IC 25...computed (13 steps) IC 26...computed (10 steps) IC 27...computed (25 steps)
1 Ð ÆÄ ÆÛ ³ Ð 95 IC 28...computed (71 steps) IC 29...computed (14 steps) IC 30...computed (15 steps) IC 31...computed (17 steps) IC 32...computed (33 steps) IC 33...computed (94 steps) IC 34...computed (9 steps) IC 35...computed (37 steps) IC 36...computed (14 steps) IC37...computed (293 steps) IC 38...computed (12 steps) IC 39...computed (5 steps) IC 40...computed (2 steps) Done. Adding the mean back to the data. Ñ C ³Åà ¹ FAST-ICA (mixedsig) ÐÂ Ì ½ÚÅÃÛ mixedsig Ô ÊÝ Ä Ô Þ FAST-ICA À Ç Hyvarinen ÜÂÀ ÉÖ Å [3]. Á [1] Back A D, Weigend A S. A first application of independent component analysis to extracting structure from stock returns [J]. Int J Neural Syst, 1997, 8(4): 473-484. [2] Wikipédia. Analyse en composantes indépendantes [EB/OL]. [2009-03-01]. http://fr.wikipedia.org /wiki/. [3] Matlab Fast-ICA µ Ï [EB/OL]. [2009-03-01]. http://www.cis.hut.fi/projects/ica/fastica.