MST 565 Linear Models [Model Linear]

Transcription

1 UNIVERSITI SAINS MALAYSIA Secod Semester Examato 04/05 Academc Sesso Jue 05 MST 565 Lear Models [Model Lear] Durato : 3 hours [Masa : 3 jam] Please check that ths examato paper cossts of TEN pages of prted materal before you beg the examato. [Sla pastka bahawa kertas peperksaa megadug SEPULUH muka surat yag bercetak sebelum ada memulaka peperksaa.] Istructos : Aswer EIGHT (8) questos. [Araha : Jawab LAPAN (8) soala.] I the evet of ay dscrepaces, the Eglsh verso shall be used. [Sekraya terdapat sebarag percaggaha pada soala peperksaa, vers Bahasa Iggers hedaklah dgua paka]. /-

2 - -. Show that matrx A below s postve defte ad fd a matrx P such that P P=A. 0 A = 0 [ 8 marks ]. Tujukka matrk A d bawah adalah tetu postf da car matrks P supaya P P=A. 0 A = 0 [ 8 markah ]. Show that the system below s cosstet. x + x + x = 3 3 x x + x = 3 3 3x x + 5x = 3 x + x + x = 4 3 [ 8 marks ]. Tujukka sstem d bawah adalah tak kosste. x + x + x = 3 3 x x + x = 3 3 3x x + 5x = 3 x + x + x = 4 3 [ 8 markah ] 3/-

3 Let Y,, Y be ucorrelated radom varable wth equal meas µ ad Var Y = σ ( ). For Y Y, Fd a costat of Var ( Y ). = what s ( ) Var Y? K such that Q = K ( Y Y) s a ubased estmator [ 8 marks ] 3. Barka Y,, Y pemboleh ubah rawak tak berkoleras dega m yag sama µ da Var ( Y ) = σ. Utuk Y Y, Car pemalar = apakah ( ) pcag bag Var ( Y ). Var Y? K supaya Q = K ( Y Y) adalah pegaggar tak [ 8 markah ] 4. Cosder the radom vector X ~ N,, Fd the jot dstrbuto of X ad ( X X ). Are X ad ( X X ) depedet? Expla your aswer. [ 0 marks ] 4. Pertmbagka vektor rawak X ~ N,, Car tabura tercatum Adakah X ad ( X X ) X da ( X X ). tak bersadar? Teragka jawapa ada. [ 0 markah ] 4/-

4 If y s ( Xβ, I) ( ) ( ) ( ) ( ) yxβ y Xβ L β e σ σ = πσ N σ ad the lkelhood fucto s gve by,, Dfferetate l ( β, ) ˆβ= X X Xy. L σ wth respect to β to obta ( ) Dfferetate l ( β, ) ( ) ( ) σ = yβ X ˆ y Xβ ˆ. L σ wth respect to σ to obta 5. Jka y adalah ( Xβ, I) ( ) ( ) ( ) ( ) yxβ y Xβ L β e σ σ = πσ Bezaka l ( β, ) N σ da fugs kebolehjada dberka sebaga,, L σ terhadap β utuk medapatka ( ) [ 0 marks ] ˆβ= X X Xy. Bezaka l ( β, ) ( ) ( ) σ = yβ X ˆ y Xβ ˆ. L σ terhadap σ utuk medapatka [ 0 markah ] 5/-

5 Table below shows hotel room rates for several hotels. The varables are defed as follows: y= daly rate of the room, x = populato ( hudreds) of the cty whch the hotel s located, x = ratg of the hotel (, or 3) ad x 3 = umber of rooms (uts) the hotel. Table x x x 3 y /-

6 - 6 - Calculate R for each the followg model; y = β0 + βx+ ε, y = β0 + βx + ε, ad y = β0 + βx3 + ε. Whch model explas the hghest proporto of the total varato the observed room rates? (d) Whch of the above models best fts the data? Justfy your aswers. Usg the best ftted model, compute a 95% predcto terval for a dvdual room rate for a hotel havg 00 uts of room, gve a rate 3 ad located a cty wth a populato Usg the best ftted model, compute a 95% cofdece terval for the average room rate for a hotel havg 00 uts of room, gve a rate 3 ad located a cty wth a populato [ 0 marks ] 6. Jadual d bawah meujukka kadar sewa blk bag beberapa hotel. Pembolehubah adalah dtakrfka sepert berkut: y = kadar sewa blk, x = jumlah peduduk (dalam ratusa) badar d maa hotel tu terletak, x = pearafa hotel (, atau 3) da x 3 = blaga blk (ut) d hotel. Jadual x x x 3 y /-

7 Kra R bag setap model berkut y = β0 + βx+ ε, y = β0 + βx + ε, da y = β0 + βx3 + ε. Model yag maakah meeragka kadar tertgg darpada jumlah varas dalam cerapa sewa blk? (d) Model yag maakah d atas meyua data palg bak? Tetusahka jawapa ada. Dega megguaka model peyuaa terbak, kraka selag ramala 95% utuk sewa blk dvdu utuk sebuah hotel yag mempuya 00 ut, dega taraf 3 da terletak d badar dega jumlah peduduk 50,000. Dega megguaka model peyuaa terbak, kraka selag keyaka 95% bag purata sewa blk utuk hotel yag mempuya 00 ut, dega taraf 3 da terletak d badar dega jumlah peduduk 50, For the geeral lear model, y βx ε = + where ε ~ ( 0, I) [ 0 markah ] N σ, the least square = For a estmate of β s gve as soluto(s) of the ormal equato, ( XXβ) Xy. gve stuato wth β ( β β β3 β4 β5 β6) β ˆ = ( ) ad ˆ ( ) solutos of the ormal equatos. = t s foud that β = are two Are there lkely to be more solutos of the ormal equatos ad, f so, how may? The rak of the desg matrx, X, s 5, 6 or 7. State, wth reasos, whch rak should t be. Exactly oe of the followg three parametrc fuctos s estmable. Whch s t, ad why? () β 3 () β3+ β4 () β3 β4. [ 4 marks ]...8/-

8 Utuk suatu model lear am, y βx ε = + d maa ε ~ ( 0, I) N σ, pegaggar kuasa dua terkecl utuk β dberka sebaga peyelesaa persamaa ormal, ( XXβ ) = Xy. Dalam keadaa tertetu dega β ( β β β3 β4 β5 β6) ddapat bahawa β ˆ = ( ) da β ˆ = ( ) adalah dua peyelesaa persamaa ormal tersebut. =, Adakah kemugka persama ormal mempuya peyelesaa yag lebh da jka demka, berapa bayak? Pagkat bag matrk rekabetuk, X adalah 5, 6 atau 7. Terag dega sebab pagkat yag maa sepatutya. Dega tepat salah satu fugs berparameter berkut adalah teraggarka. Yag maakah da keapa? () β 3 () β3+ β4 () β3 β4. [ 4 markah ] 8. Petroas wshes to exame the cotet of two dfferet RON 95 petrol addtves ther sx gas statos. They radomly select sx gas statos Peag ad radomly assg three gas statos to each addtve type (Type ad Type ). The results of the expermet are gve the table below Table RON 95 Petrol addtve (ml per lter petrol) Type Type The lear model that was used s yj = µ + τ + εj, =, j =,,3 where τ s the addtve Type ad ε j s the error for the jth sample from addtve Type. Wrte dow the lear model a matrx form. Fd two dfferet codtoal verses for XX, where X s the desg matrx. Uder what codtos does a matrx have 0,, or a fte umber of codtoal verses? (d) Estmate τ τ. (e) Fd a 95% cofdece terval for τ τ. [ marks ] 9/-

9 Petroas g memerksa kaduga dua pemagk berbeza utuk petrol RON 95 d eam stese myak mereka. Mereka memlh secara rawak eam stese myak d Pulau Pag da membahag secara rawak tga stese myak utuk setap jes baha pemagk (Jes da Jes ). Keputusa eksperme dberka dalam jadual d bawah Pemagk petrol RON 95 (ml per lter petrol) Jes Jes Model lear yag dguaka adalah yj = µ + τ + εj, =, j =,,3 yag maa τ adalah pemagk Jes ad ε j adalah ralat bag jth sampel dar pemagk Jes. Tulska model lear dalam betuk matrks. Car dua sogsaga bersyarat berbeza utuk XX, yag maa X adalah matrks reka betuk. Dalam keadaa apakah matrks mempuya 0,, atau blaga tak terhgga sogsaga bersyarat? (d) Aggarka τ τ. (e) Car selag keyaka 95% utuk τ τ. [ markah ] 0/-

10 - 0 - FORMULAE AA A = A c = P ρ = ΣDDσ σ, Σ= DPD σ ρ σ ( ) cov Ay + b = A, E r b bb z = Ay = A Ax = λx ( ) ΣA ( ) ( ) ΣB ( ) cov cov, ΣA ( ) ( ty M ) y t E e cov z, w = cov Ay, By = A, = E ( y xμ) = Σ + Σ x μ( ), y yx xx ( y Ay) Σ= tr ( μa Aμ) + cov ( By, y AyΣAμ ) = B ( y xσ) = Σ Σ Σ = ( yˆ y) ( ) SSR = = = SST y y SSR k F = SSE k ( ) β ˆ c c c XX Xy = = Sxxs * ( βxy ˆ βˆ X y) F = yx ( yyβxy ˆ ) ( k ) h cov yy yx xx xy ( xσ) = Σ + x μ ( ) E y µ y yx xx x, ( y xσ) = Σ σ σ var y yx xx xy ( k ) s ( ) k s ˆ ( ) 0, xβ ± tα ks + x 0 XX x ( y Ay) Σ tr ( A ) s χα, k χ α, k ˆ t j α, k s g jj β ± ( ) ( ) ˆ t s ˆ var = μ AΣAμ + 4 Cβ C X X C Cβˆ q SSE k xβ 0 ± tα, ks x 0 XX x 0 ( ˆ) ( ) F = αβ ± α, k α XX α ( ) ( ) ( k+ ) sf α, +, ( ) ˆ β β XX βˆ β k k A = CD C, σ = Xy c s yx = DΣ dag ( ) Σ = E ( )( ) = E( ) y μ y μ yy μμ XX = c c Sxx R = ˆβXy y yy y t j ˆ j β = s g jj ˆ ˆ β + β x ± t s + + ( x0 x) ( ) 0 0 α, = x0 x F = ( R Rr ) h ( R ) ( k ) ˆ ˆ β + β x ± t s + ( x0 x) ( ) 0 0 α, = x x - ooo0ooo -