Please use R studio to solve this question. THX!! Question 1: Load the \"gala\"
ID: 3075126 • Letter: P
Question
Please use R studio to solve this question. THX!!
Question 1: Load the "gala" data from the "faraway" package. The goal is to predict the num- ber of plant species found on the island (Species) using five other predictors (Area, Elevation, Nearest, Scruz, and Adjacent). You can read either the R document of the "faraway" package or the JF text book page 18 to get more information. For this question, you are NOT allowed to use the lm() function. Make sure that your R output is simplified so that you don't produce an extremely long report. In R markdown, To prevent a chunk of R code from printing the result you can specify 'r, eval-FALSE) as demonstrated in the Rintro file a). (1 point) Construct the design matrix X (with intercept) and verify that XTX is positive definite b). (1 point) Calculate the regression coefficients c). (1 point) Construct the "hat matrix" H and utilize it in the calculation of the residuals From the residuals, calculate SSE and 2 d). (1 point) Calculate the R2 e). (1 point) A researcher claims that a new variable an averaged distance measure (Nearest+ Scruz)/2should be added to the existing model and the R2 should be improved. Is this true? Explain your reason f). (1 point) Verify that the sum of errors is 0 g)(1 point) Now exclude the intercept in your MLR model, check whether the sum of errors is still 0. What do you find?Explanation / Answer
Answer:
rm(list=ls(all=TRUE))
> library(faraway)
> gala
Species Endemics Area Elevation Nearest Scruz Adjacent
Baltra 58 23 25.09 346 0.6 0.6 1.84
Bartolome 31 21 1.24 109 0.6 26.3 572.33
Caldwell 3 3 0.21 114 2.8 58.7 0.78
Champion 25 9 0.10 46 1.9 47.4 0.18
Coamano 2 1 0.05 77 1.9 1.9 903.82
Daphne.Major 18 11 0.34 119 8.0 8.0 1.84
Daphne.Minor 24 0 0.08 93 6.0 12.0 0.34
Darwin 10 7 2.33 168 34.1 290.2 2.85
Eden 8 4 0.03 71 0.4 0.4 17.95
Enderby 2 2 0.18 112 2.6 50.2 0.10
Espanola 97 26 58.27 198 1.1 88.3 0.57
Fernandina 93 35 634.49 1494 4.3 95.3 4669.32
Gardner1 58 17 0.57 49 1.1 93.1 58.27
Gardner2 5 4 0.78 227 4.6 62.2 0.21
Genovesa 40 19 17.35 76 47.4 92.2 129.49
Isabela 347 89 4669.32 1707 0.7 28.1 634.49
Marchena 51 23 129.49 343 29.1 85.9 59.56
Onslow 2 2 0.01 25 3.3 45.9 0.10
Pinta 104 37 59.56 777 29.1 119.6 129.49
Pinzon 108 33 17.95 458 10.7 10.7 0.03
Las.Plazas 12 9 0.23 94 0.5 0.6 25.09
Rabida 70 30 4.89 367 4.4 24.4 572.33
SanCristobal 280 65 551.62 716 45.2 66.6 0.57
SanSalvador 237 81 572.33 906 0.2 19.8 4.89
SantaCruz 444 95 903.82 864 0.6 0.0 0.52
SantaFe 62 28 24.08 259 16.5 16.5 0.52
SantaMaria 285 73 170.92 640 2.6 49.2 0.10
Seymour 44 16 1.84 147 0.6 9.6 25.09
Tortuga 16 8 1.24 186 6.8 50.9 17.95
Wolf 21 12 2.85 253 34.1 254.7 2.33
> y=gala$Species;y #response variable
[1] 58 31 3 25 2 18 24 10 8 2 97 93 58 5 40 347 51 2 104 108 12 70
[23] 280 237 444 62 285 44 16 21
> n=length(y);n
[1] 30
> x0=rep(1,30);x0
[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
> x1=gala$Area
> x2=gala$Elevation
> x3=gala$Nearest
> x4=gala$Scruz
> x5=gala$Adjacent
> #design matrix X with intercept
> x=matrix(c(x0,x1,x2,x3,x4,x5),30,6);x
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 1 25.09 346 0.6 0.6 1.84
[2,] 1 1.24 109 0.6 26.3 572.33
[3,] 1 0.21 114 2.8 58.7 0.78
[4,] 1 0.10 46 1.9 47.4 0.18
[5,] 1 0.05 77 1.9 1.9 903.82
[6,] 1 0.34 119 8.0 8.0 1.84
[7,] 1 0.08 93 6.0 12.0 0.34
[8,] 1 2.33 168 34.1 290.2 2.85
[9,] 1 0.03 71 0.4 0.4 17.95
[10,] 1 0.18 112 2.6 50.2 0.10
[11,] 1 58.27 198 1.1 88.3 0.57
[12,] 1 634.49 1494 4.3 95.3 4669.32
[13,] 1 0.57 49 1.1 93.1 58.27
[14,] 1 0.78 227 4.6 62.2 0.21
[15,] 1 17.35 76 47.4 92.2 129.49
[16,] 1 4669.32 1707 0.7 28.1 634.49
[17,] 1 129.49 343 29.1 85.9 59.56
[18,] 1 0.01 25 3.3 45.9 0.10
[19,] 1 59.56 777 29.1 119.6 129.49
[20,] 1 17.95 458 10.7 10.7 0.03
[21,] 1 0.23 94 0.5 0.6 25.09
[22,] 1 4.89 367 4.4 24.4 572.33
[23,] 1 551.62 716 45.2 66.6 0.57
[24,] 1 572.33 906 0.2 19.8 4.89
[25,] 1 903.82 864 0.6 0.0 0.52
[26,] 1 24.08 259 16.5 16.5 0.52
[27,] 1 170.92 640 2.6 49.2 0.10
[28,] 1 1.84 147 0.6 9.6 25.09
[29,] 1 1.24 186 6.8 50.9 17.95
[30,] 1 2.85 253 34.1 254.7 2.33
> p=6
> pd=t(x)%*%x;pd
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 30.00 7851.26 11041.0 301.80 1709.30 7832.95
[2,] 7851.26 23708665.46 10852798.5 39240.84 275516.84 5950313.65
[3,] 11041.00 10852798.53 9218227.0 109139.20 616237.80 8553187.95
[4,] 301.80 39240.84 109139.2 8945.30 34527.34 37196.67
[5,] 1709.30 275516.84 616237.8 34527.34 231613.77 534409.98
[6,] 7832.95 5950313.65 8553187.9 37196.67 534409.98 23719568.46
> eigen(pd)
$values
[1] 3.659801e+07 1.787397e+07 2.243836e+06 1.679281e+05 3.293730e+03 1.013371e+01
$vectors
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] -0.0004070598 -6.565549e-06 -0.002253151 -0.003779895 0.001827127 0.9999885657
[2,] -0.6578883570 6.498814e-01 0.377036883 -0.051574948 0.004486495 0.0003828472
[3,] -0.4498348101 6.597096e-02 -0.878358151 0.147036209 -0.012699699 -0.0015827832
[4,] -0.0026812736 2.339028e-04 -0.034471516 -0.128656732 0.991083382 -0.0023759306
[5,] -0.0214797443 -1.047954e-02 -0.151781563 -0.979199426 -0.132458010 -0.0038100959
[6,] -0.6036247336 -7.570945e-01 0.249196996 -0.017944910 0.004884175 0.0002340453
> #Here XtX is a symmetric matrix and the eigne values are postive .So XtX is positive definite
> xty=t(x)%*%y;xty
[,1]
[1,] 2557.0
[2,] 2444013.2
[3,] 1976099.0
[4,] 25054.6
[5,] 106983.1
[6,] 742829.8
> xtxinverse=solve(pd);xtxinverse
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 9.867829e-02 3.778242e-05 -1.561976e-04 -2.339027e-04 -3.760293e-04 2.309832e-05
[2,] 3.778242e-05 1.352247e-07 -2.593617e-07 1.294003e-06 -4.913149e-08 4.620303e-08
[3,] -1.561976e-04 -2.593617e-07 7.745339e-07 -3.549366e-06 3.080831e-07 -1.640241e-07
[4,] -2.339027e-04 1.294003e-06 -3.549366e-06 2.988732e-04 -3.821077e-05 1.424729e-06
[5,] -3.760293e-04 -4.913149e-08 3.080831e-07 -3.821077e-05 1.247941e-05 -1.958356e-07
[6,] 2.309832e-05 4.620303e-08 -1.640241e-07 1.424729e-06 -1.958356e-07 8.426543e-08
> b=xtxinverse%*%xty;b
[,1]
[1,] 7.068220709
[2,] -0.023938338
[3,] 0.319464761
[4,] 0.009143961
[5,] -0.240524230
[6,] -0.074804832
> h=x%*%xtxinverse%*%t(x);h #hat matrix
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 0.07871937 0.030046980 0.036597135 0.033995316 2.166880e-02 0.044989777
[2,] 0.03004698 0.091353235 0.056168898 0.069268961 1.155680e-01 0.059722906
[3,] 0.03659714 0.056168898 0.062314427 0.065015280 4.747372e-02 0.036964513
[4,] 0.03399532 0.069268961 0.065015280 0.072376765 6.986545e-02 0.047974532
[5,] 0.02166880 0.115568041 0.047473720 0.069865454 1.687837e-01 0.082270132
[6,] 0.04498978 0.059722906 0.036964513 0.047974532 8.227013e-02 0.071637899
[7,] 0.04376576 0.064461309 0.043749546 0.054794511 8.378158e-02 0.068915612
[8,] -0.06408858 -0.006558534 0.060993396 0.044015503 -5.968361e-02 -0.049331245
[9,] 0.05138259 0.074102594 0.053156692 0.064795655 9.042923e-02 0.069450832
[10,] 0.03910303 0.057849373 0.060426662 0.064227214 5.257285e-02 0.041486608
[11,] 0.03616807 0.041436482 0.069698186 0.064487048 1.068569e-02 0.011775742
[12,] -0.01141441 0.081282022 -0.021482704 -0.028815595 1.365081e-01 -0.031343661
[13,] 0.02037118 0.065344307 0.079298996 0.081061534 4.686534e-02 0.022421046
[14,] 0.04494162 0.036425903 0.052866214 0.049879887 1.937056e-02 0.027425274
[15,] -0.03846472 0.036120440 -0.017243516 0.003015155 1.070553e-01 0.088030804
[16,] -0.03072350 0.004621363 0.002609610 0.011134084 8.275937e-03 -0.005252051
[17,] 0.01095834 0.006411174 0.003576389 0.004218613 2.106413e-02 0.042095499
[18,] 0.03048936 0.071964605 0.062986080 0.072210067 7.777675e-02 0.052692539
[19,] 0.04715056 -0.067108507 -0.013691893 -0.040618917 -1.054559e-01 -0.015741882
[20,] 0.07493222 0.003919105 0.012061472 0.006184427 1.894630e-06 0.043678273
[21,] 0.05333291 0.070688314 0.051610872 0.062138892 8.531628e-02 0.067507282
[22,] 0.05217274 0.048095762 0.032433143 0.033739661 5.698846e-02 0.042953670
[23,] 0.01618007 -0.048391388 -0.053181340 -0.056833256 -1.995102e-02 0.048360679
[24,] 0.10521102 -0.042531039 0.014238024 -0.011224683 -9.632178e-02 -0.007304476
[25,] 0.09036268 -0.021518127 0.013308244 -0.001673163 -5.367739e-02 0.013444520
[26,] 0.04508936 0.031484662 0.013841704 0.020186561 5.441604e-02 0.066910134
[27,] 0.08564336 -0.020400646 0.030035379 0.008970007 -6.878930e-02 -0.002767701
[28,] 0.05601181 0.060859025 0.050804734 0.057453426 6.703796e-02 0.057979055
[29,] 0.04057647 0.044177721 0.047564166 0.049292177 4.060787e-02 0.040694123
[30,] -0.04516931 -0.014864940 0.045805971 0.028864884 -6.050474e-02 -0.037640434
[,7] [,8] [,9] [,10] [,11] [,12]
[1,] 0.0437657622 -0.064088576 0.051382589 0.039103030 0.036168073 -0.011414414
[2,] 0.0644613091 -0.006558534 0.074102594 0.057849373 0.041436482 0.081282022
[3,] 0.0437495459 0.060993396 0.053156692 0.060426662 0.069698186 -0.021482704
[4,] 0.0547945114 0.044015503 0.064795655 0.064227214 0.064487048 -0.028815595
[5,] 0.0837815769 -0.059683614 0.090429228 0.052572849 0.010685689 0.136508065
[6,] 0.0689156123 -0.049331245 0.069450832 0.041486608 0.011775742 -0.031343661
[7,] 0.0682865118 -0.036375924 0.071407274 0.047417930 0.022453955 -0.032320249
[8,] -0.0363759242 0.465517265 -0.048793668 0.044517125 0.119417643 0.008129048
[9,] 0.0714072738 -0.048793668 0.080225388 0.056556734 0.035210996 -0.030412892
[10,] 0.0474179302 0.044517125 0.056556734 0.059330985 0.064178163 -0.023085561
[11,] 0.0224539548 0.119417643 0.035210996 0.064178163 0.098245833 -0.011603719
[12,] -0.0323202486 0.008129048 -0.030412892 -0.023085561 -0.011603719 0.949673245
[13,] 0.0351872887 0.135616328 0.048739640 0.073859734 0.100228954 -0.009832660
[14,] 0.0321392953 0.058253822 0.039315762 0.050742410 0.065865990 -0.012238238
[15,] 0.0698856825 0.052116090 0.031045170 -0.010003378 -0.082868184 -0.014499224
[16,] -0.0007091869 0.016646452 0.002350385 0.001903810 0.009332430 0.007709218
[17,] 0.0326242441 0.058959450 0.013102492 0.005543695 -0.014063322 -0.003732930
[18,] 0.0586504478 0.039849739 0.067060759 0.062795858 0.057798450 -0.031435899
[19,] -0.0264382650 0.098055907 -0.045571409 -0.016436530 0.012254581 0.054976326
[20,] 0.0356629367 -0.068063995 0.032087219 0.015925032 0.004903598 -0.004524818
[21,] 0.0691613205 -0.050002146 0.077790976 0.054953357 0.034928235 -0.027141150
[22,] 0.0420044888 -0.031958079 0.046317177 0.034472755 0.026509132 0.101034205
[23,] 0.0232721448 -0.018055847 -0.016918483 -0.046011679 -0.091607308 -0.013152432
[24,] -0.0103118578 -0.044654849 -0.003225864 0.013223107 0.050152838 0.013974377
[25,] 0.0095821040 -0.074164127 0.015149232 0.014706127 0.030933834 -0.009484868
[26,] 0.0580614080 -0.054638383 0.049006918 0.019373109 -0.014555455 -0.023043624
[27,] -0.0023654001 0.015677359 0.004779036 0.027385471 0.064744889 0.013351559
[28,] 0.0600317936 -0.035307439 0.068887943 0.053108189 0.041479333 -0.021302132
[29,] 0.0430956013 0.034143358 0.047262389 0.047320963 0.048273937 -0.015160674
[30,] -0.0298718620 0.389767939 -0.044690763 0.032556858 0.093533979 0.009389376
[,13] [,14] [,15] [,16] [,17] [,18]
[1,] 0.020371183 0.044941620 -0.038464716 -0.0307235041 0.0109583370 0.030489359
[2,] 0.065344307 0.036425903 0.036120440 0.0046213627 0.0064111741 0.071964605
[3,] 0.079298996 0.052866214 -0.017243516 0.0026096097 0.0035763890 0.062986080
[4,] 0.081061534 0.049879887 0.003015155 0.0111340844 0.0042186133 0.072210067
[5,] 0.046865341 0.019370561 0.107055263 0.0082759367 0.0210641306 0.077776746
[6,] 0.022421046 0.027425274 0.088030804 -0.0052520505 0.0420954994 0.052692539
[7,] 0.035187289 0.032139295 0.069885683 -0.0007091869 0.0326242441 0.058650448
[8,] 0.135616328 0.058253822 0.052116090 0.0166464515 0.0589594500 0.039849739
[9,] 0.048739640 0.039315762 0.031045170 0.0023503854 0.0131024922 0.067060759
[10,] 0.073859734 0.050742410 -0.010003378 0.0019038097 0.0055436948 0.062795858
[11,] 0.100228954 0.065865990 -0.082868184 0.0093324296 -0.0140633216 0.057798450
[12,] -0.009832660 -0.012238238 -0.014499224 0.0077092180 -0.0037329301 -0.031435899
[13,] 0.117956891 0.063873745 -0.049341071 0.0169371219 -0.0140461242 0.077130710
[14,] 0.063873745 0.052735679 -0.031959172 -0.0133689781 0.0088059319 0.046118775
[15,] -0.049341071 -0.031959172 0.430065941 0.0080340168 0.1813790769 0.024567669
[16,] 0.016937122 -0.013368978 0.008034017 0.9685320732 0.0010400066 0.013637975
[17,] -0.014046124 0.008805932 0.181379077 0.0010400066 0.1035450235 0.010837954
[18,] 0.077130710 0.046118775 0.024567669 0.0136379750 0.0108379540 0.073385171
[19,] -0.035128420 0.028252930 0.031453021 -0.0742853614 0.0878296621 -0.046162454
[20,] -0.019700015 0.029961322 0.025370593 -0.0494963594 0.0490017742 0.004950906
[21,] 0.046189613 0.039537335 0.026354555 -0.0008291358 0.0132303136 0.063987406
[22,] 0.023895392 0.033780944 0.009886985 -0.0323737732 0.0203021036 0.033113364
[23,] -0.113927074 -0.027902406 0.300846422 0.0492895260 0.1746819446 -0.045168378
[24,] -0.008111044 0.053896500 -0.164035649 0.0313158656 -0.0016431570 -0.024569519
[25,] -0.011827290 0.040412928 -0.101006393 0.1194546107 0.0078180799 -0.009930531
[26,] -0.015286379 0.015292618 0.132217419 -0.0199961544 0.0736978372 0.026045038
[27,] 0.022853069 0.059093414 -0.133572376 -0.0301357084 0.0006478486 -0.002570019
[28,] 0.047132800 0.043130377 0.004234388 -0.0069376095 0.0101333925 0.057512770
[29,] 0.051912356 0.043652943 0.014640849 -0.0090714814 0.0222754431 0.048622997
[30,] 0.100324028 0.049696616 0.066674137 0.0003548197 0.0697051160 0.025651417
[,19] [,20] [,21] [,22] [,23] [,24]
[1,] 0.04715056 7.493222e-02 0.0533329139 0.052172740 0.016180066 0.105211022
[2,] -0.06710851 3.919105e-03 0.0706883141 0.048095762 -0.048391388 -0.042531039
[3,] -0.01369189 1.206147e-02 0.0516108724 0.032433143 -0.053181340 0.014238024
[4,] -0.04061892 6.184427e-03 0.0621388917 0.033739661 -0.056833256 -0.011224683
[5,] -0.10545593 1.894630e-06 0.0853162820 0.056988456 -0.019951022 -0.096321780
[6,] -0.01574188 4.367827e-02 0.0675072825 0.042953670 0.048360679 -0.007304476
[7,] -0.02643826 3.566294e-02 0.0691613205 0.042004489 0.023272145 -0.010311858
[8,] 0.09805591 -6.806399e-02 -0.0500021458 -0.031958079 -0.018055847 -0.044654849
[9,] -0.04557141 3.208722e-02 0.0777909760 0.046317177 -0.016918483 -0.003225864
[10,] -0.01643653 1.592503e-02 0.0549533573 0.034472755 -0.046011679 0.013223107
[11,] 0.01225458 4.903598e-03 0.0349282349 0.026509132 -0.091607308 0.050152838
[12,] 0.05497633 -4.524818e-03 -0.0271411498 0.101034205 -0.013152432 0.013974377
[13,] -0.03512842 -1.970001e-02 0.0461896128 0.023895392 -0.113927074 -0.008111044
[14,] 0.02825293 2.996132e-02 0.0395373346 0.033780944 -0.027902406 0.053896500
[15,] 0.03145302 2.537059e-02 0.0263545553 0.009886985 0.300846422 -0.164035649
[16,] -0.07428536 -4.949636e-02 -0.0008291358 -0.032373773 0.049289526 0.031315866
[17,] 0.08782966 4.900177e-02 0.0132303136 0.020302104 0.174681945 -0.001643157
[18,] -0.04616245 4.950906e-03 0.0639874058 0.033113364 -0.045168378 -0.024569519
[19,] 0.24690400 1.027707e-01 -0.0380536295 0.025721887 0.188412676 0.177771022
[20,] 0.10277071 9.775826e-02 0.0354571172 0.049345657 0.108285479 0.118060387
[21,] -0.03805363 3.545712e-02 0.0757044571 0.046893422 -0.013720347 0.004941445
[22,] 0.02572189 4.934566e-02 0.0468934217 0.053595160 0.017171303 0.047531472
[23,] 0.18841268 1.082855e-01 -0.0137203469 0.017171303 0.375247525 0.046522642
[24,] 0.17777102 1.180604e-01 0.0049414451 0.047531472 0.046522642 0.267547021
[25,] 0.12090140 1.004102e-01 0.0207201405 0.042477190 0.060680449 0.207640849
[26,] 0.04321053 6.822134e-02 0.0488396615 0.040944387 0.130607386 0.018283715
[27,] 0.14404706 8.935017e-02 0.0108394678 0.043406742 0.012853915 0.204656450
[28,] -0.01801087 3.928450e-02 0.0676906562 0.045983476 -0.016456453 0.026718009
[29,] 0.01350422 3.060149e-02 0.0466020707 0.034948825 -0.001170677 0.026798880
[30,] 0.11948757 -3.640093e-02 -0.0446696972 -0.021387645 0.030035933 -0.014549709
[,25] [,26] [,27] [,28] [,29] [,30]
[1,] 0.090362679 0.04508936 0.0856433607 0.056011806 0.040576466 -0.0451693148
[2,] -0.021518127 0.03148466 -0.0204006459 0.060859025 0.044177721 -0.0148649395
[3,] 0.013308244 0.01384170 0.0300353790 0.050804734 0.047564166 0.0458059713
[4,] -0.001673163 0.02018656 0.0089700067 0.057453426 0.049292177 0.0288648837
[5,] -0.053677390 0.05441604 -0.0687892968 0.067037958 0.040607872 -0.0605047409
[6,] 0.013444520 0.06691013 -0.0027677013 0.057979055 0.040694123 -0.0376404337
[7,] 0.009582104 0.05806141 -0.0023654001 0.060031794 0.043095601 -0.0298718620
[8,] -0.074164127 -0.05463838 0.0156773594 -0.035307439 0.034143358 0.3897679391
[9,] 0.015149232 0.04900692 0.0047790361 0.068887943 0.047262389 -0.0446907634
[10,] 0.014706127 0.01937311 0.0273854706 0.053108189 0.047320963 0.0325568578
[11,] 0.030933834 -0.01455545 0.0647448886 0.041479333 0.048273937 0.0935339792
[12,] -0.009484868 -0.02304362 0.0133515588 -0.021302132 -0.015160674 0.0093893763
[13,] -0.011827290 -0.01528638 0.0228530689 0.047132800 0.051912356 0.1003240280
[14,] 0.040412928 0.01529262 0.0590934141 0.043130377 0.043652943 0.0496966158
[15,] -0.101006393 0.13221742 -0.1335723760 0.004234388 0.014640849 0.0666741369
[16,] 0.119454611 -0.01999615 -0.0301357084 -0.006937609 -0.009071481 0.0003548197
[17,] 0.007818080 0.07369784 0.0006478486 0.010133393 0.022275443 0.0697051160
[18,] -0.009930531 0.02604504 -0.0025700190 0.057512770 0.048622997 0.0256514171
[19,] 0.120901399 0.04321053 0.1440470629 -0.018010872 0.013504221 0.1194875712
[20,] 0.100410240 0.06822134 0.0893501657 0.039284500 0.030601492 -0.0364009299
[21,] 0.020720140 0.04883966 0.0108394678 0.067690656 0.046602071 -0.0446696972
[22,] 0.042477190 0.04094439 0.0434067419 0.045983476 0.034948825 -0.0213876450
[23,] 0.060680449 0.13060739 0.0128539151 -0.016456453 -0.001170677 0.0300359327
[24,] 0.207640849 0.01828372 0.2046564501 0.026718009 0.026798880 -0.0145497086
[25,] 0.178116117 0.03264404 0.1500003703 0.033475165 0.025416351 -0.0443727781
[26,] 0.032644038 0.08550763 0.0110099454 0.041833386 0.031112830 -0.0303176570
[27,] 0.150000370 0.01100995 0.1682503376 0.029236986 0.034132657 0.0296356567
[28,] 0.033475165 0.04183339 0.0292369859 0.063008931 0.045732346 -0.0307459422
[29,] 0.025416351 0.03111283 0.0341326568 0.045732346 0.042206398 0.0302334005
[30,] -0.044372778 -0.03031766 0.0296356567 -0.030745942 0.030233400 0.3334687104
> i=diag(30) #identity matrix
> e=(i-h)%*%y;e #residuals
[,1]
[1,] -58.725946
[2,] 38.273154
[3,] -26.330659
[4,] 14.635734
[5,] 38.383916
[6,] -25.087705
[7,] -9.919668
[8,] 19.018992
[9,] -20.314202
[10,] -28.785943
[11,] 49.343513
[12,] -3.989598
[13,] 62.033276
[14,] -59.633796
[15,] 40.497176
[16,] -39.403558
[17,] -37.694540
[18,] -2.037233
[19,] -111.679486
[20,] -42.475375
[21,] -23.075807
[22,] -5.553122
[23,] 73.048122
[24,] -40.676318
[25,] 182.583587
[26,] -23.376486
[27,] 89.383371
[28,] -5.805095
[29,] -36.935732
[30,] -5.700573
> sse=sum(e^2);sse #error sum of squares
[1] 89231.37
> s2=sse/(n-p);s2 #estimate of variance
[1] 3717.974
> sst=t(y)%*%y-(sum(y)^2)/n ;sst #total sum of squares
[,1]
[1,] 381081.4
> r2=1-(sse/sst);r2 #coefficient of deternmination
[,1]
[1,] 0.7658469