MSG Multivariate Analysis [Analisis Multivariat]

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UNIVERSITI SAINS MALAYSIA Secod Semester Examiatio 016/017 Academic Sessio Jue 017 MSG 466 - Multivariate Aalysis [Aalisis Multivariat] Duratio : 3 hours [Masa : 3 jam] Please check that this

1 UNIVERSITI SAINS MALAYSIA Secod Semester Examiatio 016/017 Academic Sessio Jue 017 MSG Multivariate Aalysis [Aalisis Multivariat] Duratio : 3 hours [Masa : 3 jam] Please check that this examiatio paper cosists of TWENTY-ONE pages of prited material before you begi the examiatio. [Sila pastika bahawa kertas peperiksaa ii megadugi DUA PULUH SATU muka surat yag bercetak sebelum ada memulaka peperiksaa ii.] Istructios: [Araha: Aswer all eight [8] questios. Jawab semua lapa [8] soala.] I the evet of ay discrepacies, the Eglish versio shall be used. [Sekiraya terdapat sebarag percaggaha pada soala peperiksaa, versi Bahasa Iggeris hedaklah digua pakai]. /-

2 - - Questio 1 (a) What are the three properties that must be satisfied by a distace measure, d (P, Q) betwee two poits P ( x1, x, K, x p ) ad Q (y 1, y, K, y p )? (b) The distace from the poit P = ( x1, x ) to the poit Q = ( y1, y ) is defied as d( P, Q) max x y, x y. 1 1 (i) Does this measure of distace satisfy the three coditios that have bee stated i (a) above? Justify your aswer. Soala 1 (a) (ii) Compute the distace from the poit P = (1, 3) to the poit Q = (, 1). (iii) Plot the locus of poits whose distace from P = (1, 3) is. [ 0 marks ] Apakah tiga sifat yag mesti dipeuhi bagi suatu ukura jarak, d (P, Q) atara dua titik P ( x1, x, K, x p ) da Q (y 1, y, K, y p )? (b) Jarak dari titik P = ( x1, x ) ke titik Q = ( y1, y ) ditakrifka sebagai d( P, Q) max x y, x y. 1 1 (i) Adakah ukura jarak ii memeuhi ketiga-tiga syarat yag diyataka dalam (a) di atas? Tetusahka jawapa ada. (ii) Kira jarak atara titik P = (1, 3) ke titik Q = (, 1). (iii) Lakar lokus titik-titik yag jarakya dari P = (1, 3) adalah. Questio [ 0 markah ] (a) Suppose X : N ( μσ, ). Based o a radom sample of size 30, the sample mea vector ad the sample variace-covariace matrix are obtaied as below: x ad 4 S 1 (i) Is S a positive defiite matrix? 3/-

3 - 3 - (ii) Determie the geeralized sample variace ad total sample variace. (iii) Fid the maximum likelihood estimates of the 1 mea vector μ ad the variace-covariace matrix Σ. (b) Let X be N3 ( μσ, ) where X X X, μ 3 ad Σ X (i) Fid the distributios of X1 X. 3 (ii) Determie the coditioal distributio of X give that X1 x ad X3 x3. [ 30 marks ] Soala (a) Adaika X : N ( μσ, ). Berdasarka suatu sampel rawak bersaiz 30, vektor mi sampel da matriks varias-kovarias sampel diperoleh seperti di bawah: x da 4 S 1 (i) Adakah S matriks tetu positif? (ii) Tetuka varias sampel teritlak da varias sampel keseluruha. (iii) Cari aggara kebolehjadia maksimum bagi vektor mi 1, μ da matriks varias-kovarias, Σ. (b) Biar X sebagai N3 ( μσ, ) yag maa X X X, μ 3 da Σ X (i) Cari tabura bagi X1 X. 3 (ii) Tetuka tabura bersyarat bagi X diberi bahawa X1 x da X3 x3. [ 30 markah ] 4/-

4 - 4 - Questio 3 Let X1, X ad X 3 be mutually idepedet ad idetically distributed 3 1 radom vectors from N3 ( μσ,, ) where μ (3,1, )' ad 1 0 Σ (a) Fid the margial distributio for each of the radom vectors ad V 1 X 1 X 1 X V X X X. 1 3 (b) Are V 1 ad V idepedet? Justify your aswer. (c) Specify the joit distributio of V 1 ad V. [ 5 marks ] Soala 3 Biar X1, X da X 3 sebagai vektor rawak 3 1 secama daripada N3 ( μσ,, ) yag maa μ (3,1, )' da 1 0 Σ bertabura salig tak bersadar da (a) Cari tabura sut bagi setiap vektor rawak da V 1 X 1 X 1 X V X X X. 1 3 (b) Adakah V 1 da V tak bersadar? Tetusahka jawapa ada. (c) Pericika tabura tercatum V 1 da V. [ 5 markah ]...5/-

5 - 5 - Questio 4 Madible measuremets were made o eight male moder dogs from Thailad. Two of the measuremets are legth of first molar (i mm) ad breadth of first molar (i mm), which are assumed to have a bivariate ormal distributio. The sample mea vector ad the sample variace-covariace matrix are as follows: x m ad m 7.8 S (a) Test Ho: μ m (18.5, 8.5)' at State ay assumptios that you have made ad give your coclusio from this test. (b) Similar measuremets are also made for eight female dogs of the same species. The sample mea vector ad the sample variace-covariace matrix are give by x f ad f 7.5 S Is there evidece of differeces betwee the madible size of the male ad the female dogs? Perform a appropriate test ad state your assumptios. Soala 4 [ 0 marks ] Ukura-ukura rahag dibuat pada lapa ekor ajig jata mode daripada Thailad. Dua daripada ukura adalah pajag molar pertama (dalam mm) da lebar molar pertama (dalam mm), yag diaggap mempuyai tabura ormal bivariat. Vektor mi sampel da matriks varias-kovarias sampel adalah seperti berikut: x m da m 7.8 S (a) Uji Ho: μ m (18.5, 8.5)' pada Nyataka sebarag adaia yag telah ada buat da berika kesimpula daripada ujia ii. (b) Ukura yag sama juga telah dibuat pada lapa ekor ajig betia daripada spesies yag sama. Vektor mi sampel da matriks variaskovarias sampelya diberi oleh x f da f 7.5 S Adakah terdapat bukti perbezaa saiz rahag atara ajig jata da ajig betia? Lakuka ujia yag bersesuaia da yataka adaia-adaia ada. [ 0 markah ] 6/-

6 - 6 - Questio 5 The table below displays the edited data of four measuremets made o 40 male skulls from the area of Thebes i Egypt. There are two samples of 0 skulls each from the early predyastic period ad the Roma period. For each skull, the measuremets (i mm) are: X1 = maximum breadth X = basibregmatic height X3 = basialveolar legth X4 = asal height Skull X1 X X3 X M M M M M M M M M M Skulls 1 to 0 are from the early predyastic period while skulls 1 to 40 are from the Roma period. Statistical aalyses usig MINITAB have bee performed for this dataset ad the output is displayed i Appedix A. (a) (b) (c) (d) Based o the output, i your opiio, what aalyses have bee performed? How are the four measuremets related? Do the skulls from the early predyastic ad Roma periods have statistically sigificat differece for their mea values of the variables? Discuss your aswer. Suppose, a skull is just foud from the area ad the four measuremets o the skull are (16, 133, 10, 51). Usig the iformatio i the output, determie whether the skull is most likely from the early predyastic or Roma period. Discuss your aswer. [ 5 marks ] 7/-

7 - 7 - Soala 5 Jadual di bawah mempamerka data yag telah disutig bagi empat ukura yag dibuat pada 40 tegkorak lelaki daripada kawasa Thebes di Mesir. Terdapat sampel dega 0 tegkorak setiap satuya daripada zama awal pradiastik da zama Roma. Bagi setiap tegkorak, ukura-ukuraya (dalam mm) adalah: X1 = lebar maksimum X = tiggi basibregmatik X3 = pajag basialveolar X4 = tiggi hidug Tegkorak X1 X X3 X M M M M M M M M M M Tegkorak 1 ke 0 adalah daripada zama awal pradiastik semetara tegkorak 1 ke 40 adalah daripada zama Roma. Aalisis statistik megguaka MINITAB telah dijalaka pada set data ii da outputya dipamerka dalam Lampira A. (a) (b) (c) (d) Berdasarka output, pada pedapat ada, aalisis apa yag telah dijalaka? Bagaimaa keempat-empat ukura ii berkait? Adakah tegkorak pada zama awal pradiastik da tegkorak pada zama Roma mempuyai perbezaa bererti secara statistik bagi ilai mi pembolehubah-pembolehubah mereka? Bicagka jawapa ada. Adaika suatu tegkorak baru sahaja dijumpai daripada kawasa itu da empat ukura pada tegkorak adalah (16, 133, 10, 51). Megguaka maklumat daripada output, tetuka sama ada tegkorak itu berkemugkia besar daripada zama awal pradiastik atau daripada zama Roma. Bicagka jawapa ada. [ 5 markah ] Questio 6 (a) Aswer the followig questios o Factor Aalysis: (i) (ii) (iii) (iv) What is the purpose of the aalysis? How does it differ from Pricipal Compoet Aalysis? How are the variables grouped? What is the purpose of factor rotatio? 8/-

8 - 8 - (b) A set of data cosists of 130 observatios geerated by scores o a psychological test admiistered to Peruvia teeagers (see table below). The scores were accumulated ito five subscale scores labeled idepedece (idep), support (supp), beevolece (beev), coformity (coform) ad leadership (leader). Idep Supp Beev Coform Leader M M M M M Soala 6 Factor aalysis was performed o this data set usig the pricipal compoet method ad the maximum likelihood method. The output of the aalysis is displayed i Appedix B. Iterpret ad discuss the results. [ 5 marks ] (a) Jawab soala-soala berkeaa Aalisis Faktor berikut: (i) (ii) (iii) (iv) Apakah tujua aalisis tersebut? Bagaimaa ia berbeza daripada Aalisis Kompoe Utama? Bagaimaa pembolehubah-pembolehubah dikelaska? Apakah tujua putara faktor? (b) Suatu set data megadugi 130 cerapa yag dijaa daripada skor suatu ujia psikologi terhadap remaja-remaja Peru (lihat jadual di bawah). Skor-skor dikumpul dalam lima skor subskala berlabel berdikari (idep), sokoga (supp), kebajika (beev), pematuha (coform) da kepimpia (leader). Idep Supp Beev Coform Leader M M M M M Aalisis faktor telah dijalaka pada set data ii megguaka kaedah kompoe utama da kaedah kebolehjadia maksimum. Output aalisis dipamerka dalam Lampira B. Tafsir da bicag keputusaya. [ 5 markah ] 9/-

9 - 9 - Questio 7 The percetages of the labour force i three differet types of idustry (agriculture, miig ad maufacturig) for five Europea coutries are show i the followig table Coutry AGR MIN MAN Belgium Romaia Switzerlad Frace Bulgaria (a) (b) Determie the distace matrix for the data usig the city-block metric. Cluster the five coutries usig the complete likage hierarchical procedure. Draw a dedogram ad discuss your results. [ 30 marks ] Soala 7 Peratusa teaga kerja dalam tiga jeis idustri berbeza (pertaia, perlomboga da pembuata) bagi lima egara Eropah ditujukka dalam jadual di bawah : Negara AGR MIN MAN Belgium Romaia Switzerlad Peracis Bulgaria (a) (b) Tetuka matriks jarak bagi data megguaka metrik blok-badar. Klusterka kelima-lima egara tersebut megguaka tatacara berhierarki pauta legkap. Lukis dedogram da bicag keputusa ada. [ 30 markah ] 10/-

10 Questio 8 Observatios o two resposes are collected for four treatmets. The summary statistics of the two variables for Treatmet 1, Treatmet, Treatmet 3 ad Treatmet 4 are as follows: Treatmet 1: x 5 3 1, 1, S Treatmet : x,, 13 1 S 1 Treatmet 3: x , 3, S 1 1 Treatmet 4: x 3 4, 4, S 1 Costruct a oe-way MANOVA table ad test for treatmet effects usig Give your coclusio ad state ay assumptios that you have made. [ 5 marks ] Soala 8 Cerapa daripada dua respo dikumpul bagi empat rawata. Statistik rigkas daripada dua pembolehubah bagi Rawata 1, Rawata, Rawata 3 da Rawata 4 adalah seperti berikut: Rawata 1: x 5 3 1, 1, S Rawata : x,, 13 1 S 1 Rawata 3: x , 3, S 1 1 Rawata 4: x 3 4, 4, S 1 Bia jadual MANOVA satu-hala da uji kesa rawata megguaka Berika kesimpula ada da yataka sebarag adaia yag ada telah buat. [ 5 markah ] 11/-

11 Appedix/Lampira A *Period: Early Predyastic (1), Roma () Descriptive Statistics: X1, X, X3, X4 Total Variable Period Cout Mea Variace Miimum Maximum X X X X Correlatios: X1, X, X3, X4 X1 X X3 X X X Cell Cotets: Pearso correlatio P-Value ANOVA: X1, X, X3, X4 versus Period Factor Type Levels Values Period fixed 1, Aalysis of Variace for X1 Source DF SS MS F P Period Error Total S = R-Sq = 1.05% R-Sq(adj) = 9.74% Aalysis of Variace for X Source DF SS MS F P Period Error Total S = R-Sq = 9.66% R-Sq(adj) = 7.8% 1/-

12 - 1 - Aalysis of Variace for X3 Source DF SS MS F P Period Error Total S = R-Sq = 1.9% R-Sq(adj) = 9.98% Aalysis of Variace for X4 Source DF SS MS F P Period Error Total S = R-Sq = 3.70% R-Sq(adj) = 1.16% MANOVA for Period s = 1 m = 1.0 = 16.5 Test DF Criterio Statistic F Num Deom P Wilks' Lawley-Hotellig Pillai's Roy's SSCP Matrix for Period X1 X X3 X4 X X X X SSCP Matrix for Error X1 X X3 X4 X X X X Partial Correlatios for the Error SSCP Matrix X1 X X3 X4 X X X X Discrimiat Aalysis: Period versus X1, X, X3, X4 Liear Method for Respose: Period Predictors: X1, X, X3, X4 13/-

13 Group 1 Cout 0 0 Summary of classificatio True Group Put ito Group Total N 0 0 N correct Proportio N = 40 N Correct = 34 Proportio Correct = Squared Distace Betwee Groups Liear Discrimiat Fuctio for Groups 1 Costat X X X X Summary of Misclassified Observatios True Pred Squared Observatio Group Group Group Distace Probability 6** ** ** ** ** ** /-

14 Appedix/Lampira B Descriptive Statistics: Idep, Supp, Beev, Coform, Leader Total Variable Cout Mea StDev Variace Idep Supp Beev Coform Leader Correlatios: Idep, Supp, Beev, Coform, Leader Idep Supp Beev Coform Supp Beev Coform Leader Cell Cotets: Pearso correlatio P-Value.5 Scree Plot of Idep,..., Leader.0 Eigevalue Factor Number /-

15 Factor Aalysis: Idep, Supp, Beev, Coform, Leader Pricipal Compoet Factor Aalysis of the Correlatio Matrix Urotated Factor Loadigs ad Commualities Variable Factor1 Factor Commuality Idep Supp Beev Coform Leader Variace % Var Rotated Factor Loadigs ad Commualities Varimax Rotatio Variable Factor1 Factor Commuality Idep Supp Beev Coform Leader Variace % Var Factor Score Coefficiets Variable Factor1 Factor Idep Supp Beev Coform Leader Factor Aalysis: Idep, Supp, Beev, Coform, Leader Pricipal Compoet Factor Aalysis of the Correlatio Matrix Urotated Factor Loadigs ad Commualities Variable Factor1 Factor Factor3 Commuality Idep Supp Beev Coform Leader Variace % Var Rotated Factor Loadigs ad Commualities Varimax Rotatio Variable Factor1 Factor Factor3 Commuality Idep Supp Beev Coform Leader /-

16 Variace % Var Factor Score Coefficiets Variable Factor1 Factor Factor3 Idep Supp Beev Coform Leader Factor Aalysis: Idep, Supp, Beev, Coform, Leader Maximum Likelihood Factor Aalysis of the Correlatio Matrix * NOTE * Heywood case Urotated Factor Loadigs ad Commualities Variable Factor1 Factor Commuality Idep Supp Beev Coform Leader Variace % Var Rotated Factor Loadigs ad Commualities Varimax Rotatio Variable Factor1 Factor Commuality Idep Supp Beev Coform Leader Variace % Var Factor Score Coefficiets Variable Factor1 Factor Idep Supp Beev Coform Leader * WARNING * Too may factors, solutio is ot uique 17/-

17 Factor Aalysis: Idep, Supp, Beev, Coform, Leader Maximum Likelihood Factor Aalysis of the Correlatio Matrix * NOTE * Heywood case Urotated Factor Loadigs ad Commualities Variable Factor1 Factor Factor3 Commuality Idep Supp Beev Coform Leader Variace % Var Rotated Factor Loadigs ad Commualities Varimax Rotatio Variable Factor1 Factor Factor3 Commuality Idep Supp Beev Coform Leader Variace % Var Factor Score Coefficiets Variable Factor1 Factor Factor3 Idep Supp Beev Coform Leader /-

18 Appedix/Lampira C FORMULAE SHEET 1. Suppose X has E( X) c' Σc. μ ad Cov( X) Σ. Thus cx ' has mea. Bivariate ormal p.d.f.: 1 1 f ( x, x ) exp x1 1 x x1 1 x c' μ ad variace 3. Multivariate ormal p.d.f.: 4. If X : N ( μσ, ), the p f( x) 1 p/ (a) a' X : N a' μ, a' Σa (b) AX : N q Aμ, AΣA ' (c) Xd : N p μ d, Σ (d) AX d : N q Aμ d, AΣA ' (e) 1 X μ' Σ X μ : p Σ 1/ e 1 ( xμ)' Σ ( xμ)/ 5. Let X j : N p( μj, Σ), j1,..., be mutually idepedet. The V1 c jxj : N p c j j, c j μ Σ. Moreover, V 1 ad V bjx j are j1 j1 j1 j1 joitly multivariate ormal with covariace matrix j1 c j Σ bc ' Σ b ' c Σ. b j Σ j1 X1 μ1 6. Let X be distributed as X : N p( μσ, ) with μ X μ, Σ11 Σ1 Σ ad Σ1 Σ Σ 0. The the coditioal distributio of X 1, give that X x, is ormal ad 1 has mea = μ Σ Σ x μ ad covariace = Σ Σ Σ Σ. 19/-

19 Oe-sample results: 1 (a) T X μ' S X μ X 1 1, ' X S X X X X j j j j1 1 j1 p T : Fp, p 1 (b) 100(1 )% simultaeous cofidece itervals for a' μ : p p ( 1) a' X Fp, p( ) a' Sa ( p) (c) 100(1 )% Boferroi cofidece iterval for i : x i p t1 (d) 100(1 )% large sample cofidece iterval for i : sii x i sii p( ) 8. Two-sample results (Paired comparisos): 1 (a) T Dδ' δd Dδ D 1 1, ' D S D D D D j d j j j1 1 j1 p T : Fp, p 1 (b) 100(1 )% simultaeous cofidece itervals for i : p d i p ( 1) sd F, ( ) i p p ( p) 9. Two-sample results (Idepedet samples): p (a) T X X μ μ ' S X X μ μ 1 0/-

20 - 0-1 p 1 T : F 1 p 1 p, p1 S p S i 1S 1 i j S ij i ij i x x x x i 1 ' (b) 100(1 )% simultaeous cofidece iterval for ' a μ μ : a' X1 X c a' S pa 1 c 1 p F 1 p 1 1 p, p1 ( ) (c) For large 1 p ad p, a approximate 100(1 )% simultaeous cofidece iterval for ' a μ μ : a' X1 X c a' S1 S a 1 c ( ) p 10. Oe-way MANOVA g l1 B x x x x l l l ' g l 1 1 W x x x x ' ( 1) S ( 1) S... ( 1) S l1 j1 lj l lj l g g * W B + W Distributio of * : g1 * For p 1, g : : Fg 1, g 1 * g g11 * p, g : F g 1 * : For ( g1),( g1) 1/-

21 - 1 - p11 * For p 1, g : : Fp, p1 p * p 1 * p 1, g 3: F p * : For p, ( p) l 11. The Estimated Miimum ECM Rule for two ormal populatios: Allocate x 0 to populatio 1 if c(1 ) p x1 x ' Sp x0 x1 x ' Sp x1 x l c( 1) p1 Allocate x 0 to populatio otherwise. - ooo O ooo -

MSG 366 Multivariate Analysis [Analisis Multivariat]

MSG 366 Multivariate Analysis [Analisis Multivariat] UNIVERSITI SAINS MALAYSIA First Semester Examiatio Academic Sessio 015/016 Jauary 016 MSG 366 Multivariate Aalysis [Aalisis Multivariat] Duratio : 3 hours [Masa : 3 jam] Please check that this examiatio