MSG 366 Multivariate Analysis [Analisis Multivariat]

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1 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 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: Aswer EIGHT (8) questios. [Araha: Jawab LAPAN (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 Give the data matrix, 6 X (a) (b) (c) Determie the deviatio vectors ad its legths. Obtai the ier product ad the agle betwee the deviatio vectors. Calculate the geeralized sample variace ad total sample variace. [ 5 marks ] 1. Diberi matriks data, 6 X (a) (b) (c) Tetuka vektor-vektor sisiha serta pajagya. Dapatka hasil darab terkedalam da sudut atara vektor-vektor sisiha. Kira varias sampel teritlak da varias sampel keseluruha. [ 5 markah ]. Let X be N3 ( μσ, ) where X X X, μ 5, ad Σ 1 9. X (a) Fid the distributios of (i) X X 3, X X1 (ii). 3/-

3 - 3 - (b) Are X X 3 X X1 ad idepedet? [ 0 marks ]. Biar X sebagai N3 ( μσ, ) yag maa X X X, μ 5, ad Σ 1 9. X (a) Cari tabura bagi (i) X X 3, X X1 (ii). (b) Adakah X X 3 X X1 da tidak bersadar? [ 0 markah ] 3. Readigs at a particular day for two meteorological variables at 0 moitorig statios are observed. The values observed for ozoe cocetratio (i part per millio) ad air temperature (i degree Celsius), which are assumed to have a multivariate ormal distributio, gave x , 7.55 S (a) Obtai the maximum likelihood estimates of the mea vector, μ ad the variace-covariace matrix, Σ. (b) Test at the 5% level the ull hypothesis that μ ' (0.080, 6.45). State ay assumptios you make before performig the test. (c) Obtai the 95% simultaeous cofidece itervals for the two populatio meas. [ 5 marks ] 4/-

4 Bacaa pada hari tertetu bagi dua pembolehubah meteorologi di 0 stese pemataua dicerap. Nilai yag dicerap bagi kepekata ozo (dalam bahagia per juta) da suhu udara (dalam Celsius), yag diadaika mempuyai tabura tercatum multivariat ormal, memberika x , 7.55 S (a) Dapatka aggara kebolehjadia maksimum bagi vektor mi μ da matriks varias-kovarias, Σ. (b) Uji pada aras 5% hipotesis ol μ ' (0.080, 6.45). Nyataka sebarag adaia yag telah ada dibuat sebelum melaksaaka ujia tersebut. (c) Dapatka selag keyakia seretak 95% bagi dua mi populasi tersebut. [ 5 markah ] 4. The followig table displays the edited data of five measuremets o 0 moribud female sparrows brought to a biological laboratory i Rhode Islad after a severe storm i For each bird, the measuremets (i mm) are: X1 = total legth X = alar extet X3 = legth of beak ad head X4 = legth of humerus X5 = legth of keel of sterum Birds X1 X X3 X4 X /-

5 - 5 - Birds 1 to 1 survived while birds 13 to 0 died. Statistical aalyses usig MINITAB have bee performed for this dataset ad the output is displayed i Appedix A. (a) (b) (c) Based o the output, i your opiio what aalyses have bee performed? How are the five measuremets related? Do the survivors ad osurvivors have statistically sigificat differece for their mea values of the variables? Discuss your aswer. (d) Suppose, the five measuremets o a bird are (155, 35, 30.7, 17.7, 19.6), but the iformatio regardig the survival of this bird is missig. Usig the iformatio i the output, determie whether the bird most likely survived or died. Discuss your aswer. [ 30 marks ] 4. Jadual berikut memaparka data yag telah disutig bagi lima ukura pada 0 ekor burug pipit betia yag hampir meemui ajal yag telah dibawa ke sebuah makmal biologi di Rhode Islad selepas suatu ribut yag teruk. Bagi setiap burug, ukura-ukura (dalam mm) adalah: X1 = pajag keseluruha X= pajag alar X3 = pajag paruh da kepala X4 = pajag humerus X5 = pamjag luas tulag dada Burug X1 X X3 X4 X /-

6 - 6 - Burug 1 ke 1 terselamat semetara burug 13 ke 0 meemui ajal. Aalisisaalisis statistik megguaka MINITAB telah dijalaka terhadap set data ii da outputya dipamerka di Lampira A. (a) (b) (c) Berdasarka output, pada pedapat ada apakah aalisis-aalisis yag telah dijalaka? Bagaimaa lima ukura ii berkaita? Adakah yag terselamat da yag meemui ajal mempuyai perbezaa ilai-ilai mi pembolehubah yag bererti secara statistik? Bicagka jawapa ada. (d) Adaika lima ukura seekor burug adalah (155, 35, 30.7, 17.7, 19.6), tetapi maklumat berkeaa kemadiria burug ii hilag. Dega megguaka maklumat-maklumat yag terdapat pada output, tetuka sama ada burug tersebut berkemugkia besar terselamat atau meemui ajal. Bicagka jawapa ada. [ 30 markah ] 5. Observatios o two resposes are collected for four treatmets. The summary statistics of the two resposes for Treatmet 1, Treatmet, Treatmet 3 ad Treatmet 4 are as follows: Treatmet 1: , 1, Treatmet : ,, Treatmet 3: , 3, Treatmet 4: , 4, Costruct the oe-way MANOVA table ad test for treatmet effects usig Give your coclusio ad state ay assumptios you make. [ 30 marks ] 7/-

7 Cerapa utuk dua respo dikumpul bagi empat rawata. Statistik rigkasa bagi dua respo tersebut bagi Rawata 1, Rawata, Rawata 3 da Rawata 4 adalah seperti berikut: Rawata 1: , 1, Rawata : ,, Rawata 3: , 3, Rawata 4: , 4, Bia jadual MANOVA satu-hala da uji kesa rawata megguaka Beri kesimpula ada da yataka sebarag adaia yag dibuat. [ 30 markah ] 6. Suppose X ( X1, X)' N ( 0, Σ ), where 9 4 Σ. 4 3 (a) Obtai the pricipal compoets of Σ. (b) Determie the joit distributio of the pricipal compoets. Specify the mea vector ad variace-covariace matrix. [ 5 marks ] 6. Adaika X ( X1, X)' N ( 0, Σ ), yag maa 9 4 Σ. 4 3 (a) Dapatka kompoe-kompoe prisipal bagi Σ. (b) Tetuka tabura tercatum bagi kompoe-kompoe prisipal. Pericika vektor mi da matriks varias-kovarias. [ 5 markah ] 8/-

8 The followig data measure the amout of protei cosumed for ie food groups i 5 Europea coutries. The ie food groups are red meat (X1), white meat (X), eggs (X3), milk (X4), fish (X5), cereal (X6), starch (X7), uts (X8), ad fruits ad vegetables (X9). Couty X1 X X3 X4 X5 X6 X7 X8 X9 Albaia Austria Belgium Bulgaria Czechoslovakia Demark East Germay Filad Frace Greece Hugary Irelad Italy Netherlads Norway Polad Portugal Romaia Spai Swede Switzerlad UK USSR West Germay Yugoslavia /-

9 - 9 - Factor aalysis o this data set is performed usig Miitab ad the output is displayed i Appedix B. Discuss the results ad idetify the importat factors uderlyig the observed variables ad examie the relatioship betwee the coutries with respect to these factors. [ 15 marks ] 7. Data berikut megukur jumlah protei yag dimaka bagi sembila kumpula makaa di 5 buah egara Eropah. Sembila kumpula makaa tersebut adalah dagig merah (X1), dagig putih (X), telur (X3), susu (X4), ika (X5), bijiri (X6), kaji (X7), kekacag (X8), da buah-buaha da sayur-sayura (X9). Negara X1 X X3 X4 X5 X6 X7 X8 X9 Albaia Austria Belgium Bulgaria Czechoslovakia Demark East Germay Filad Frace Greece Hugary Irelad Italy Netherlads Norway Polad Portugal Romaia Spai Swede Switzerlad UK USSR West Germay Yugoslavia /-

10 Aalisis faktor terhadap set data dijalaka megguaka perisia Miitab da outputya dipamerka di Lampira B. Bicagka keputusa da kealpasti faktor-faktor petig yag medasari pembolehubah-pembolehubah yag dicerap da kaji perhubuga atara egara-egara dega faktor-faktor ii. [ 15 markah ] 8. The followig data displays the mea values for four madible measuremets for five caie groups. The four measuremets are breadth of madible (X1), height of madible below the first molar (X), legth of the first molar (X3) ad breadth of the first molar (X4). Measuremets (i mm) Group X1 X X3 X4 Moder Dog Chiese Wolf Idia Wolf Cuo Prehistoric Dog (a) (b) Determie the distace matrix D for the data, usig the city block metric. Cluster the five groups usig the complete likage hierarchical procedure. Draw a dedogram ad discuss the results. [ 30 marks ] 11/-

11 Data berikut mempamerka ilai-ilai mi bagi empat ukura rahag bawah bagi lima kumpula ajig. Empat ukura tersebut adalah lebar rahag (X1), tiggi rahag bawah molar pertama (X), pajag molar pertama (X3) da lebar molar pertama (X4). Ukura (dalam mm) Kumpula X1 X X3 X4 Ajig Mode Serigala Chia Serigala Idia Cuo Ajig Prasejarah (a) (b) Tetuka matriks jarak D bagi data, megguaka metrik blok badar. Kelompokka lima kumpula tersebut megguaka tatacara berhierarki pauta legkap. Lukis suatu dedogram da bicagka keputusa. [ 30 markah ] 1/-

12 - 1 - APPENDICES/LAMPIRAN Appedix A/Lampira A *1: Survived, : Died (Nosurvived) Descriptive Statistics: X1, X, X3, X4, X5 Total Variable Survival Cout N N* Mea Variace Miimum Maximum X X X X X Correlatios: X1, X, X3, X4, X5 X1 X X3 X4 X X X X Cell Cotets: Pearso correlatio P-Value Geeral Liear Model: X1, X, X3, X4, X5 versus Factor Type Levels Values Aalysis of Variace for X1, usig Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Survival Error Total S = R-Sq = 0.0% R-Sq(adj) = 0.00% Term Coef SE Coef T P Costat Survival /-

13 Uusual Observatios for X1 Obs X1 Fit SE Fit Residual St Resid R R deotes a observatio with a large stadardized residual. Aalysis of Variace for X, usig Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Survival Error Total S = R-Sq = 0.0% R-Sq(adj) = 0.00% Term Coef SE Coef T P Costat Survival Uusual Observatios for X Obs X Fit SE Fit Residual St Resid R R R deotes a observatio with a large stadardized residual. Aalysis of Variace for X3, usig Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Survival Error Total S = R-Sq = 3.41% R-Sq(adj) = 0.00% Term Coef SE Coef T P Costat Survival Aalysis of Variace for X4, usig Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Survival Error Total S = R-Sq = 0.50% R-Sq(adj) = 0.00% Term Coef SE Coef T P Costat Survival Uusual Observatios for X4 Obs X4 Fit SE Fit Residual St Resid R R 14/-

14 R deotes a observatio with a large stadardized residual. Aalysis of Variace for X5, usig Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Survival Error Total S = R-Sq = 5.6% R-Sq(adj) = 0.38% Term Coef SE Coef T P Costat Survival MANOVA for Survival s = 1 m = 1.5 = 6.0 Test DF Criterio Statistic F Num Deom P Wilks' Lawley-Hotellig Pillai's Roy's SSCP Matrix (adjusted) for Survival X1 X X3 X4 X5 X X X X X SSCP Matrix (adjusted) for Error X1 X X3 X4 X5 X X X X X Partial Correlatios for the Error SSCP Matrix X1 X X3 X4 X5 X X X X X /-

15 Discrimiat Aalysis: Survival versus X1, X, X3, X4, X5 Liear Method for Respose: Survival Predictors: X1, X, X3, X4, X5 Group 1 Cout 1 8 Summary of classificatio True Group Put ito Group Total N 1 8 N correct 8 6 Proportio N = 0 N Correct = 14 Proportio Correct = Squared Distace Betwee Groups Liear Discrimiat Fuctio for Groups 1 Costat X X X X X Summary of Misclassified Observatios True Pred Squared Observatio Group Group Group Distace Probability 4** ** ** ** ** ** /-

16 Appedix B/Lampira B Descriptive Statistics: X1, X, X3, X4, X5, X6, X7, X8, X9 Variable Mea Variace X X X X X X X X X Correlatios: X1, X, X3, X4, X5, X6, X7, X8, X9 X1 X X3 X4 X5 X6 X7 X8 X X X X X X X X Cell Cotets: Pearso correlatio P-Value Factor Aalysis: X1, X, X3, X4, X5, X6, X7, X8, X9 Pricipal Compoet Factor Aalysis of the Correlatio Matrix Urotated Factor Loadigs ad Commualities Variable Factor1 Factor Factor3 Factor4 Factor5 Commuality X X X X X X X X X Variace % Var /-

17 Rotated Factor Loadigs ad Commualities Varimax Rotatio Variable Factor1 Factor Factor3 Factor4 Factor5 Commuality X X X X X X X X X Variace % Var Factor Score Coefficiets Variable Factor1 Factor Factor3 Factor4 Factor5 X X X X X X X X X Score Plot of X1,..., X9 17 Secod Factor First Factor 1 18/-

18 Appedix C/Lampira C FORMULAE SHEET 1. Suppose X has E( X) c' Σc. μ ad Cov( X) Σ. Thus cx ' has mea c' μ ad variace. Bivariate ormal p.d.f.: 1 1 f ( x, x ) exp x1 1 x x1 1 x Multivariate ormal p.d.f.: f( x) 1 p/ Σ 1/ e 1 ( xμ)' Σ ( xμ)/ 4. If X N ( μσ,, ) the 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 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 j1 j1 j1 j1 are joitly multivariate ormal with covariace matrix j1 c j Σ bc ' Σ b ' c Σ. b j Σ j1 19/- j

19 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 1 ad has mea = 1 μ Σ Σ x μ ad covariace = Σ Σ Σ Σ 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) 0/-

20 Two-sample results (Idepedet samples): p (a) T X X μ μ ' S X X μ μ 1 p 1 T F 1 p 1 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 a' μ μ : cofidece iterval for 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 1/-

21 - 1 - Distributio of * : g1 * For p 1, g : Fg 1, g 1 * g g11 * p, g : F g 1 * For ( g1),( g1) 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 Multivariate Analysis [Analisis Multivariat]

MSG Multivariate Analysis [Analisis Multivariat] 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 examiatio