TP2 Linear Models
A.V Hurtado Quiceno
2/5/2023
- Introduction Ozone Data
- Modèle
linear simple \(\mathcal{M}_{1}\) avec
la varialble
T12 - Modèle
linear multiple \(\mathcal{M}_{2}\)
avec les variables
T12,Vx,N12andmax03v - Modèle lineaire avec tout les variables
- Analysis
- Intervalles de Confiance
- Propriété de sans biais des estimateurs de coefficients \(\beta_{j}\)
- Echantionage
- Question 1
- Question 2
- Question 3
- Bootstrap
install.packages(c(“ggplot2”, “dplyr”, “ISwR”, “MASS”, “Sleuth3”, “rmarkdown”, “tidyr”))
Introduction Ozone Data
\(\textbf{Objective:}\) L’objective
est d’expliquer la variable max03 par rapport a
T12 dans le modele lineare et par rapport a dans le modele
multiple et faire un comparison avec le deux.
| date | maxO3 | T6 | T9 | T12 | T15 | T18 | Ne6 | Ne9 | Ne12 | Ne15 | Ne18 | Vx | maxO3v |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19940401 | 56.0 | 8.6 | 9.5 | 6.8 | 9.1 | 7.7 | 6 | 3 | 6 | 7 | 4 | -10.8329 | 59.6 |
| 19940402 | 39.2 | 3.6 | 5.6 | 9.2 | 8.4 | 4.9 | 3 | 4 | 6 | 7 | 7 | -10.3366 | 56.0 |
| 19940403 | 36.0 | 2.7 | 7.3 | 6.3 | 7.0 | 7.9 | 6 | 8 | 8 | 8 | 8 | -1.0419 | 39.2 |
| 19940404 | 41.2 | 11.8 | 11.8 | 11.0 | 7.0 | 7.7 | 8 | 7 | 6 | 7 | 3 | -10.3366 | 36.0 |
| 19940405 | 27.6 | 3.7 | 8.3 | 11.6 | 10.7 | 7.9 | 6 | 3 | 6 | 6 | 6 | -8.8633 | 41.2 |
ou les varaibles sont:
obs: mois-jourmaxO3: teneur maximale en ozone observée sur la journée (en \(\mu g r / m 3)\);T6, .. ,T18: température observée à \(6 \mathrm{~h}, \ldots 18 \mathrm{~h}\);Ne6, ..,Ne18: nébulosité observée à \(6 \mathrm{~h}, \ldots, 18 \mathrm{~h}\);$\mathrm{Vx}$: composante est-ouest du vent ;maxO3v: teneur maximale en ozone observée la veille ;vent: orientation du vent à \(12 \mathrm{~h}\);pluie: occurrence ou non de précipitations.
Modèle linear simple \(\mathcal{M}_{1}\) avec la varialble
T12
On considére el modèle lineaire simple \(\mathcal{M}_{1}\) avec la variable
explicative T12.
\[\begin{equation} (\mathcal{M}_{1}): max03= \beta_{0} + \beta_{1} T12 + \epsilon_{1} \end{equation}\]
Résumé
Plot
Scatter plot
## `geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = "cs")'
Correlation
Ici on etude la correlation entre les deux variables T12
et maxo3, on a que le valuer est \(0.492\) i.e que on peut expliquer la
variable maxo3 par la variable T12
## [1] 0.4922408Summary
##
## Call:
## lm(formula = maxO3 ~ T12)
##
## Residuals:
## Min 1Q Median 3Q Max
## -75.675 -14.746 -0.297 17.459 66.312
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 28.2761 2.5242 11.20 <2e-16 ***
## T12 2.6254 0.1257 20.89 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 23.59 on 1364 degrees of freedom
## Multiple R-squared: 0.2423, Adjusted R-squared: 0.2417
## F-statistic: 436.2 on 1 and 1364 DF, p-value: < 2.2e-16Coefficients
## (Intercept) T12
## 28.276122 2.625433Les estimations des coefficients sont \(\beta_0\) et \(\beta_1\) sont respectivement \(\beta_0=28.2761\) et \(\beta_1=2.6254\). L’équation de la regression lineaire est:
\[y=28.2761+2.6254 \,\, T12. \]
Residuals
#Le residuals sont:
residuals_simple <- res_simple$residuals
residuals_simple## 1 2 3 4 5
## 9.870933177 -13.230106149 -8.816350297 -15.955885643 -31.131145474
## 6 7 8 9 10
## -21.318775390 -8.617736064 -0.730452591 -5.979932928 1.495673346
## 11 12 13 14 15
## -5.843169117 -10.168948611 -5.304326654 -26.429413265 -3.228720381
## 16 17 18 19 20
## -15.743515559 9.032437157 3.607696988 -14.942129791 -24.467909285
## 21 22 23 24 25
## -39.319468274 -21.758657178 -40.870680820 -38.220507599 -41.369641495
## 26 27 28 29 30
## -46.932531242 -52.920854041 -59.197846082 -43.361428713 -37.244554884
## 31 32 33 34 35
## -8.497499640 -18.784783114 -33.296113873 -32.782704463 -48.607444631
## 36 37 38 39 40
## -50.869987937 -31.831491916 -29.957271410 -33.608137515 -37.058310736
## 41 42 43 44 45
## -13.609523282 -7.870680820 -3.434956335 -17.920854041 -19.897499640
## 46 47 48 49 50
## -44.269987937 -30.608137515 -38.118775390 -20.244554884 -41.158657178
## 51 52 53 54 55
## -22.819814716 -33.909176841 -31.159350062 -38.821200483 -25.921546925
## 56 57 58 59 60
## -15.745594210 -15.395074547 -18.045940652 -11.157964294 -9.708483957
## 61 62 63 64 65
## 18.026894087 -14.749751512 -21.833570567 -25.395767431 -17.207444631
## 66 67 68 69 70
## -24.745594210 -54.809523282 -40.610216166 -27.821200483 -32.183397346
## 71 72 73 74 75
## -25.832877684 2.654405790 -0.997153198 9.177413750 28.364350781
## 76 77 78 79 80
## 2.538917729 -20.173105913 -17.061775155 12.301114593 2.901114593
## 81 82 83 84 85
## -37.121546925 -56.885822439 -67.060389387 -30.837034986 16.547823395
## 86 87 88 89 90
## -45.047326419 -26.358657178 -23.861775155 -9.500964059 -42.672759471
## 91 92 93 94 95
## -22.536688544 -16.212987701 -4.122932692 -13.425357785 -62.060389387
## 96 97 98 99 100
## -37.334609893 -47.309176841 -53.783397346 -54.408830399 -43.810216166
## 101 102 103 104 105
## -23.562814480 15.011059584 34.010366700 -36.672066588 -30.048712187
## 106 107 108 109 110
## -56.286515323 9.736839078 -22.125011344 -31.086515323 -37.934609893
## 111 112 113 114 115
## -5.973798797 -11.487208207 -29.098885407 7.699035941 9.211059584
## 116 117 118 119 120
## -46.361428713 -59.224664902 -36.887901091 -59.109869724 -47.824664902
## 121 122 123 124 125
## -46.112641259 -75.674838123 -66.661082271 -56.386168881 -39.276916774
## 126 127 128 129 130
## -65.026050669 -70.211601934 -67.374491681 -55.487901091 -26.725704227
## 131 132 133 134 135
## -43.987554649 -59.810216166 -56.097499640 -46.947672861 -42.522239809
## 136 137 138 139 140
## -33.272759471 -33.474145239 -35.499578291 -55.233570567 -51.610216166
## 141 142 143 144 145
## -67.173105913 -52.922932692 -64.074838123 -49.500271175 -62.709869724
## 146 147 148 149 150
## -59.696806756 -66.910562608 -62.810216166 -65.097499640 -61.146979977
## 151 152 153 154 155
## -59.534609893 -39.311255492 -55.472066588 -46.071373704 -40.947672861
## 156 157 158 159 160
## -43.734609893 -56.971720146 -47.022586250 -48.070680820 -30.334609893
## 161 162 163 164 165
## -35.444554884 -57.021893367 -58.272066588 -55.921546925 -51.058310736
## 166 167 168 169 170
## -45.445247768 -40.619814716 -44.507098189 -50.120161157 -44.506405306
## 171 172 173 174 175
## -57.383397346 -46.845940652 -41.708483957 -29.808830399 -49.071373704
## 176 177 178 179 180
## -49.507791073 -46.371027262 -24.883743788 -15.594381663 -8.307098189
## 181 182 183 184 185
## -10.583397346 -7.406058864 -4.707098189 -7.956578527 -13.794381663
## 186 187 188 189 190
## 13.567122316 16.231051389 27.305271895 17.691516043 29.066082991
## 191 192 193 194 195
## 7.805618337 1.305271895 29.817988421 29.256484441 11.818681305
## 196 197 198 199 200
## 11.831051389 14.645500125 7.144460799 14.393248252 20.908043430
## 201 202 203 204 205
## 20.406311220 14.969200968 22.469547409 -0.668602169 4.431744273
## 206 207 208 209 210
## -4.368948611 13.193248252 15.430358505 24.553366465 25.389090950
## 211 212 213 214 215
## 22.738224845 19.687358741 25.749209162 43.074295773 58.624815436
## 216 217 218 219 220
## 14.478453075 4.642035706 12.366429433 -4.469295053 11.707350546
## 221 222 223 224 225
## 15.268854526 20.305271895 10.067468758 -17.243862000 6.554752232
## 226 227 228 229 230
## 16.543075032 5.630358505 5.543075032 11.016602654 13.415216886
## 231 232 233 234 235
## 3.138917729 -8.283050905 -20.197153198 -14.709176841 -24.785475998
## 236 237 238 239 240
## -20.260389387 -4.371027262 -1.482358021 5.729319180 -6.120854041
## 241 242 243 244 245
## -12.146287093 -21.195074547 -5.083743788 1.129319180 -1.272066588
## 246 247 248 249 250
## -29.283743788 5.079145959 2.404232569 -6.745594210 -24.483050905
## 251 252 253 254 255
## -3.582704463 -7.132531242 8.879145959 -11.246633535 -29.860389387
## 256 257 258 259 260
## -23.308483957 -39.971027262 -18.385475998 -26.023279134 -32.261082271
## 261 262 263 264 265
## -6.985475998 2.128626296 -9.984783114 6.251634255 -1.688593974
## 266 267 268 269 270
## 9.323776110 13.248169837 23.984933648 25.110020258 28.773256447
## 271 272 273 274 275
## -1.873452355 5.715563328 -5.022586250 -27.684436672 -31.034956335
## 276 277 278 279 280
## -16.274145239 31.161579247 15.472217122 4.362965014 -17.397846082
## 281 282 283 284 285
## -21.461775155 -17.023279134 -29.586168881 -8.410909050 -23.485822439
## 286 287 288 289 290
## -33.223279134 -24.049405070 4.512098909 8.222043901 -11.474145239
## 291 292 293 294 295
## -13.860389387 15.888744508 15.149209162 18.335453310 3.538224845
## 296 297 298 299 300
## -36.197846082 -30.285822439 -3.588247533 5.723083226 32.811059584
## 301 302 303 304 305
## 28.510020258 27.823429668 17.474988656 16.526547645 -4.447326419
## 306 307 308 309 310
## -6.047326419 23.988398066 44.312098909 35.785626531 26.102500360
## 311 312 313 314 315
## -16.261082271 1.776720866 19.663311456 31.774642215 28.236492636
## 316 317 318 319 320
## 23.811752467 49.811059584 58.561579247 23.086665857 11.712098909
## 321 322 323 324 325
## 2.738917729 -21.186168881 -17.761428713 -2.135995660 -15.386168881
## 326 327 328 329 330
## -15.384783114 -18.797846082 2.778106633 -21.497499640 -20.409523282
## 331 332 333 334 335
## -16.460389387 -10.197153198 -20.422586250 -11.959350062 -9.421200483
## 336 337 338 339 340
## -10.896806756 -2.560042945 -16.760042945 -19.159350062 -2.259696503
## 341 342 343 344 345
## -20.384090230 -4.408830399 -18.382011579 -24.708483957 2.028972738
## 346 347 348 349 350
## 4.390476718 -5.059696503 13.115563328 -2.758657178 -3.721546925
## 351 352 353 354 355
## -2.108483957 -22.333917009 -0.069987937 -12.545594210 -21.608137515
## 356 357 358 359 360
## -9.620507599 -20.157964294 -23.445247768 -13.019814716 23.820067072
## 361 362 363 364 365
## 30.583649703 34.683303261 39.956830883 11.307350546 2.145153683
## 366 367 368 369 370
## 14.881917494 22.568508084 18.817295537 15.880531726 15.180185284
## 371 372 373 374 375
## 4.528626296 -11.881665137 26.756138000 -1.595767431 28.690823159
## 376 377 378 379 380
## 13.742382148 0.017295537 12.253712907 17.803539686 12.714870444
## 381 382 383 384 385
## 0.143075032 6.980878168 15.780185284 22.518334863 16.629665622
## 386 387 388 389 390
## 30.827586970 22.828279854 21.942382148 6.679838843 26.481917494
## 391 392 393 394 395
## 16.593248252 13.107350546 24.431051389 38.992555369 29.982956819
## 396 397 398 399 400
## 7.417988421 19.219374189 23.180878168 18.268854526 13.393941136
## 401 402 403 404 405
## 32.730704947 29.117641979 18.530704947 8.330704947 8.532783598
## 406 407 408 409 410
## -11.592995896 0.404232569 23.855791558 18.292901811 22.705271895
## 411 412 413 414 415
## 3.743075032 -22.120854041 -16.157964294 -0.919468274 -0.507098189
## 416 417 418 419 420
## 2.167122316 -3.720854041 7.977413750 15.012445351 -14.283743788
## 421 422 423 424 425
## 13.628972738 12.229665622 12.828279854 4.226894087 40.362272130
## 426 427 428 429 430
## 30.460539921 21.385626531 33.243421473 2.339610613 -6.298192524
## 431 432 433 434 435
## -30.498885407 1.615909770 33.389090950 22.276374424 34.700421709
## 436 437 438 439 440
## 25.799382383 24.524468994 26.899728825 14.112098909 -19.597846082
## 441 442 443 444 445
## 21.366429433 20.828279854 33.990476718 15.615216886 10.614524002
## 446 447 448 449 450
## -10.571027262 -40.560042945 -12.648019303 -3.796460314 -18.659003620
## 451 452 453 454 455
## -5.146287093 -22.333917009 -39.896806756 -18.410216166 -26.396460314
## 456 457 458 459 460
## -18.646633535 -0.883743788 -17.734609893 -22.684436672 -11.297499640
## 461 462 463 464 465
## -0.958657178 -22.309176841 -49.810216166 -31.723625576 4.063311456
## 466 467 468 469 470
## 13.000768151 18.125161877 20.112098909 27.299728825 46.461232805
## 471 472 473 474 475
## 32.747823395 21.584933648 22.050248488 -17.298192524 10.738224845
## 476 477 478 479 480
## 32.575335098 -19.885822439 -41.673452355 -26.211601934 -16.610216166
## 481 482 483 484 485
## -27.010216166 -15.785475998 -17.609523282 16.926547645 33.412445351
## 486 487 488 489 490
## 9.699728825 -7.385475998 -8.234956335 -3.724318460 -29.371720146
## 491 492 493 494 495
## -19.285129556 -17.259696503 -4.309869724 19.976720866 16.363657898
## 496 497 498 499 500
## 24.302500360 18.912098909 28.473602889 37.011059584 19.960193479
## 501 502 503 504 505
## 7.039264171 -20.448019303 -17.123625576 -24.460389387 -18.048019303
## 506 507 508 509 510
## -2.109869724 -20.372413030 -15.921546925 12.128626296 -2.896806756
## 511 512 513 514 515
## -11.521546925 -7.697499640 7.802846802 12.327240529 20.177413750
## 516 517 518 519 520
## 19.475681540 12.776720866 19.438571287 18.289437392 18.640649938
## 521 522 523 524 525
## 0.378106633 -16.496113873 -2.971720146 -7.221200483 -5.845940652
## 526 527 528 529 530
## 2.215909770 6.977413750 11.052327139 15.014524002 15.128626296
## 531 532 533 534 535
## -20.219814716 -23.768948611 14.191169601 7.167815200 -9.133917009
## 536 537 538 539 540
## 4.053020023 4.378799517 -28.508483957 -16.934609893 -13.634263451
## 541 542 543 544 545
## -27.647326419 -13.958657178 16.777413750 30.814524002 47.140303497
## 546 547 548 549 550
## 31.031051389 26.855791558 19.729319180 25.204232569 21.330704947
## 551 552 553 554 555
## 24.242728590 38.842035706 30.218681305 46.906657662 38.543767915
## 556 557 558 559 560
## 40.217988421 27.817295537 7.328626296 0.643421473 -4.745594210
## 561 562 563 564 565
## -3.520161157 -11.508483957 0.242035706 6.927933412 -6.084436672
## 566 567 568 569 570
## -8.410909050 -3.410216166 -19.724318460 -25.647326419 -18.384090230
## 571 572 573 574 575
## -13.735302777 1.077760191 -1.483743788 13.114870444 23.316256212
## 576 577 578 579 580
## -10.821893367 -20.459696503 -17.509869724 -10.021893367 -11.660389387
## 581 582 583 584 585
## -8.733224125 -9.795074547 -7.632184800 -7.032877684 5.954752232
## 586 587 588 589 590
## -4.546287093 8.792555369 -15.046633535 -4.694728105 3.491516043
## 591 592 593 594 595
## 12.665390107 15.264697223 -5.986861765 -0.337381428 -4.511948376
## 596 597 598 599 600
## 11.013831119 6.464004340 -16.210909050 -14.486515323 -12.459696503
## 601 602 603 604 605
## -24.373798797 -14.960042945 -13.197153198 -1.372413030 11.064004340
## 606 607 608 609 610
## 7.976027982 6.788398066 11.874988656 37.224815436 32.200768151
## 611 612 613 614 615
## -43.472759471 -49.261082271 -13.861082271 -2.924318460 9.161579247
## 616 617 618 619 620
## -8.310562608 -15.885129556 -10.122932692 -47.848019303 -11.938074312
## 621 622 623 624 625
## -17.986168881 -39.123625576 -5.986861765 -8.949058628 15.962272130
## 626 627 628 629 630
## 17.123776110 32.085280090 9.889437392 35.787705183 -10.437034986
## 631 632 633 634 635
## 4.749902046 31.299035941 15.773949331 4.189783834 13.825508319
## 636 637 638 639 640
## 14.837185520 8.336146194 -9.861775155 24.937531962 -1.486515323
## 641 642 643 644 645
## -3.999924733 -17.609523282 -8.547672861 -18.421200483 -12.783397346
## 646 647 648 649 650
## -26.747672861 -36.772413030 -19.373798797 -17.534609893 -22.122932692
## 651 652 653 654 655
## -21.810909050 -21.171027262 -20.109176841 -10.209523282 -17.684436672
## 656 657 658 659 660
## -17.159350062 -14.546287093 -0.660389387 3.839264171 -14.733917009
## 661 662 663 664 665
## -9.783397346 -1.271373704 0.427586970 9.476374424 22.013138235
## 666 667 668 669 670
## 9.086665857 24.848862720 23.637878404 17.676374424 24.251634255
## 671 672 673 674 675
## 37.963657898 28.738224845 28.626201203 32.688744508 52.976720866
## 676 677 678 679 680
## 53.053020023 44.525161877 29.725854761 21.892901811 21.743075032
## 681 682 683 684 685
## 26.593248252 23.842728590 28.305964779 41.833130040 16.558909534
## 686 687 688 689 690
## 22.857870209 39.396019787 36.857870209 30.132783598 27.045500125
## 691 692 693 694 695
## 37.719720630 34.431744273 31.268161642 8.279145959 -9.959350062
## 696 697 698 699 700
## 11.218681305 20.642728590 32.655791558 27.281224610 21.617988421
## 701 702 703 704 705
## 18.255098674 17.780878168 26.844114357 44.193248252 26.767122316
## 706 707 708 709 710
## 8.380185284 30.516256212 24.638571287 26.140303497 17.988398066
## 711 712 713 714 715
## 25.262618572 21.225508319 15.150594930 35.802153918 38.575335098
## 716 717 718 719 720
## 52.063311456 52.301114593 22.550594930 41.900421709 57.138917729
## 721 722 723 724 725
## 27.626201203 23.366429433 30.940996380 19.891516043 6.578799517
## 726 727 728 729 730
## 4.815909770 9.979492401 11.266082991 26.905271895 12.091516043
## 731 732 733 734 735
## 11.666082991 -6.732531242 11.228279854 18.028279854 17.464697223
## 736 737 738 739 740
## -3.285129556 16.114870444 6.965736549 2.165736549 -18.834956335
## 741 742 743 744 745
## -7.384090230 6.681224610 -0.782704463 0.004925453 -3.496113873
## 746 747 748 749 750
## 18.142382148 8.278453075 24.265390107 7.464697223 15.249555604
## 751 752 753 754 755
## 37.072217122 25.227586970 2.714870444 7.089437392 -0.786168881
## 756 757 758 759 760
## -8.972413030 -10.797153198 -4.745594210 -8.971720146 -15.296113873
## 761 762 763 764 765
## -39.534609893 3.540303497 -3.483050905 14.391169601 4.490823159
## 766 767 768 769 770
## -14.571027262 -29.845940652 4.253712907 -4.296806756 -21.496806756
## 771 772 773 774 775
## -12.621893367 -5.197846082 -47.197846082 -12.496806756 -11.221893367
## 776 777 778 779 780
## -16.958657178 -29.597153198 -5.672759471 -15.734609893 -5.199924733
## 781 782 783 784 785
## -19.799231849 -5.172413030 -8.785475998 -19.546287093 -4.472759471
## 786 787 788 789 790
## 21.864004340 40.815216886 -9.021893367 -24.171027262 -17.584090230
## 791 792 793 794 795
## -25.109869724 -23.584783114 -10.146979977 -4.522932692 0.589090950
## 796 797 798 799 800
## 29.359500595 -22.175184564 -11.098885407 37.402153918 -12.361428713
## 801 802 803 804 805
## 9.238571287 -13.723625576 -15.685822439 21.753366465 -8.210909050
## 806 807 808 809 810
## -26.297499640 -39.648019303 -16.722932692 -14.472759471 10.402153918
## 811 812 813 814 815
## 17.727933412 22.653020023 19.452327139 27.138917729 -0.808830399
## 816 817 818 819 820
## -9.210909050 -10.586168881 -13.323625576 -20.861082271 -3.696113873
## 821 822 823 824 825
## 2.180185284 17.005618337 20.667468758 -0.169641495 -12.984090230
## 826 827 828 829 830
## -1.459003620 -8.484436672 5.639264171 4.776720866 -3.586168881
## 831 832 833 834 835
## 12.764350781 10.664697223 28.589090950 6.865390107 -12.220507599
## 836 837 838 839 840
## -4.847326419 -5.109176841 -8.946287093 1.092208927 10.879838843
## 841 842 843 844 845
## 16.703886127 20.855098674 22.566429433 -1.933224125 6.540996380
## 846 847 848 849 850
## 14.754752232 28.143075032 25.592555369 -22.119468274 31.906657662
## 851 852 853 854 855
## 38.493594694 23.555445116 33.257177325 36.108736314 35.407696988
## 856 857 858 859 860
## 36.957523767 36.718334863 39.194634020 33.657870209 33.780878168
## 861 862 863 864 865
## 36.092901811 34.117641979 34.807004104 30.479838843 7.481224610
## 866 867 868 869 870
## 25.242035706 23.103193244 23.428279854 14.153366465 17.678453075
## 871 872 873 874 875
## 25.514177561 22.413831119 19.450941372 20.553366465 34.004232569
## 876 877 878 879 880
## 12.453020023 -0.859003620 16.340996380 21.789090950 14.066082991
## 881 882 883 884 885
## 18.567815200 11.154752232 18.291516043 13.929319180 17.178799517
## 886 887 888 889 890
## 29.415909770 24.429665622 9.266775875 23.354059348 7.916256212
## 891 892 893 894 895
## -0.720854041 19.579492401 11.964350781 19.750594930 5.401461034
## 896 897 898 899 900
## 29.700421709 5.739610613 6.176720866 -0.812294817 15.954059348
## 901 902 903 904 905
## 7.254405790 1.292208927 15.242728590 6.228972738 5.442728590
## 906 907 908 909 910
## 12.642035706 5.841342822 25.803539686 17.490823159 27.602846802
## 911 912 913 914 915
## 25.678453075 40.102500360 48.488744508 60.075681540 26.889437392
## 916 917 918 919 920
## 23.827586970 24.802153918 -5.971720146 4.579492401 4.666082991
## 921 922 923 924 925
## -4.284436672 -7.295420989 -24.223279134 -6.683743788 -33.748365745
## 926 927 928 929 930
## 2.348516279 -15.347672861 -15.946979977 -7.946979977 -12.097499640
## 931 932 933 934 935
## -26.496806756 12.750594930 22.250248488 22.712791793 30.137531962
## 936 937 938 939 940
## -6.248019303 -11.810909050 -3.872066588 11.262618572 28.549209162
## 941 942 943 944 945
## -21.648019303 -23.098192524 -7.946979977 -14.645940652 -0.785475998
## 946 947 948 949 950
## 15.650248488 12.348516279 12.272910005 37.624815436 42.762272130
## 951 952 953 954 955
## 45.812445351 31.136839078 -0.375184564 -5.561428713 -38.597846082
## 956 957 958 959 960
## -29.349058628 -19.098192524 -10.599231849 18.241342822 -11.834956335
## 961 962 963 964 965
## -6.259696503 -0.797153198 -1.910562608 -21.348365745 -28.085822439
## 966 967 968 969 970
## -28.797153198 -22.847326419 -14.371027262 -11.072759471 -9.622586250
## 971 972 973 974 975
## -5.822586250 17.650941372 15.125854761 -21.735995660 -1.212987701
## 976 977 978 979 980
## 10.038571287 -16.761428713 -12.410909050 -4.498885407 35.413138235
## 981 982 983 984 985
## 4.950594930 -14.498192524 -1.160735829 13.724468994 11.700421709
## 986 987 988 989 990
## 42.925161877 38.524468994 36.150594930 16.738917729 -5.098885407
## 991 992 993 994 995
## -27.072066588 11.436492636 24.523083226 -12.009523282 -11.696806756
## 996 997 998 999 1000
## -9.645247768 -0.334609893 -7.246633535 -11.283743788 -13.845940652
## 1001 1002 1003 1004 1005
## -9.521546925 -22.759350062 -24.420507599 -26.998538966 -14.560042945
## 1006 1007 1008 1009 1010
## -19.360735829 -2.333917009 -2.434263451 2.091516043 -11.083743788
## 1011 1012 1013 1014 1015
## -8.921546925 -9.471373704 21.732090715 -2.055539201 -10.657617852
## 1016 1017 1018 1019 1020
## 11.457870209 17.496366229 23.782263936 2.543075032 6.566429433
## 1021 1022 1023 1024 1025
## 12.905964779 -0.893342338 14.469547409 11.882610377 0.755445116
## 1026 1027 1028 1029 1030
## 5.508043430 11.245500125 19.982956819 1.068161642 -2.731145474
## 1031 1032 1033 1034 1035
## -2.719468274 -9.657617852 -1.908483957 0.817295537 8.492901811
## 1036 1037 1038 1039 1040
## -1.731145474 -4.294728105 -0.344901326 -7.456924968 5.917641979
## 1041 1042 1043 1044 1045
## 0.242035706 -3.946979977 -2.309869724 -12.645940652 -9.132531242
## 1046 1047 1048 1049 1050
## -16.283050905 4.489437392 4.176720866 -4.734609893 -5.846633535
## 1051 1052 1053 1054 1055
## -4.159350062 0.678453075 -7.271373704 4.454405790 18.728626296
## 1056 1057 1058 1059 1060
## 3.076374424 16.462618572 9.064697223 16.191862485 17.442728590
## 1061 1062 1063 1064 1065
## 12.130012063 11.454405790 9.442728590 3.504579011 4.678453075
## 1066 1067 1068 1069 1070
## -14.333224125 7.566429433 9.017988421 10.130012063 10.079838843
## 1071 1072 1073 1074 1075
## 7.242035706 -8.920161157 -4.333224125 -8.271373704 2.327240529
## 1076 1077 1078 1079 1080
## -6.159350062 3.840649938 -10.333224125 -4.483743788 7.890823159
## 1081 1082 1083 1084 1085
## 17.813831119 6.153366465 -2.634263451 -22.047326419 -22.510562608
## 1086 1087 1088 1089 1090
## -30.360042945 6.250248488 7.524468994 11.061232805 28.424122552
## 1091 1092 1093 1094 1095
## 9.551287813 -8.985475998 -3.159350062 -14.483743788 -3.171027262
## 1096 1097 1098 1099 1100
## 6.253712907 20.226894087 23.338917729 36.450941372 10.813831119
## 1101 1102 1103 1104 1105
## 7.701807476 -16.286515323 -12.248019303 -10.109176841 -14.935302777
## 1106 1107 1108 1109 1110
## -2.360042945 -1.711255492 5.041342822 -6.109176841 -24.059003620
## 1111 1112 1113 1114 1115
## -2.120854041 -9.171027262 -15.997153198 -15.259696503 -8.584090230
## 1116 1117 1118 1119 1120
## -3.533917009 4.103193244 7.226894087 28.964350781 34.450941372
## 1121 1122 1123 1124 1125
## 33.350594930 35.200075267 14.037878404 14.613138235 -3.309869724
## 1126 1127 1128 1129 1130
## -8.722932692 -13.572413030 8.415909770 -21.197846082 -19.784783114
## 1131 1132 1133 1134 1135
## -10.460389387 8.937531962 -1.761428713 -19.711255492 -9.572413030
## 1136 1137 1138 1139 1140
## -2.622586250 -9.410216166 -22.722932692 -24.811601934 -21.085822439
## 1141 1142 1143 1144 1145
## -1.950444396 -29.885129556 3.775335098 2.825508319 0.338917729
## 1146 1147 1148 1149 1150
## -25.186168881 -13.398538966 -29.348365745 -7.873452355 -29.097499640
## 1151 1152 1153 1154 1155
## -20.309869724 -15.020507599 -10.784783114 -1.035649219 21.099728825
## 1156 1157 1158 1159 1160
## 38.736839078 15.810366700 21.740303497 -19.248019303 -13.047326419
## 1161 1162 1163 1164 1165
## -14.823279134 1.288744508 21.188398066 -3.097499640 -14.097499640
## 1166 1167 1168 1169 1170
## 1.628279854 -1.259696503 16.226894087 -20.460389387 -8.097499640
## 1171 1172 1173 1174 1175
## -36.448712187 -13.274838123 -8.100964059 14.385626531 7.890823159
## 1176 1177 1178 1179 1180
## -13.896806756 -18.923625576 -19.348365745 -14.147672861 -9.371720146
## 1181 1182 1183 1184 1185
## -12.946979977 -9.507098189 -7.707791073 -12.696113873 -25.711255492
## 1186 1187 1188 1189 1190
## 4.350594930 -4.109176841 -9.834956335 -13.248019303 -27.684436672
## 1191 1192 1193 1194 1195
## -6.433570567 -5.109176841 10.242035706 4.991169601 19.130012063
## 1196 1197 1198 1199 1200
## 30.631744273 16.056484441 15.330704947 23.168508084 26.380878168
## 1201 1202 1203 1204 1205
## 7.917641979 32.944460799 27.118334863 21.743767915 22.531397831
## 1206 1207 1208 1209 1210
## 43.419374189 12.492901811 15.218681305 29.531397831 34.207004104
## 1211 1212 1213 1214 1215
## 41.457870209 39.508043430 38.681917494 29.994634020 30.855791558
## 1216 1217 1218 1219 1220
## 27.342382148 18.180185284 19.593248252 19.867468758 31.118334863
## 1221 1222 1223 1224 1225
## 33.894287578 34.631744273 3.543075032 5.369200968 26.257177325
## 1226 1227 1228 1229 1230
## 27.956138000 28.994634020 8.130012063 26.079838843 -9.020507599
## 1231 1232 1233 1234 1235
## -10.696113873 23.964350781 42.462618572 21.616602654 14.265390107
## 1236 1237 1238 1239 1240
## 20.604925453 26.504579011 19.755445116 9.454405790 17.191862485
## 1241 1242 1243 1244 1245
## 20.477760191 30.701807476 21.651634255 33.864004340 50.138224845
## 1246 1247 1248 1249 1250
## 45.539610613 28.427586970 -1.309869724 7.099728825 -23.460389387
## 1251 1252 1253 1254 1255
## -13.911948376 5.578106633 10.153366465 5.415909770 17.516256212
## 1256 1257 1258 1259 1260
## 34.002846802 11.902500360 -0.259696503 9.767122316 5.492901811
## 1261 1262 1263 1264 1265
## 24.516256212 4.778799517 21.265390107 20.226894087 22.053020023
## 1266 1267 1268 1269 1270
## 8.427586970 -7.734609893 -6.059003620 9.041342822 9.578106633
## 1271 1272 1273 1274 1275
## 52.138224845 -6.085822439 29.188398066 42.636492636 26.562965014
## 1276 1277 1278 1279 1280
## 31.872217122 18.026201203 -5.398538966 -30.085822439 -13.174491681
## 1281 1282 1283 1284 1285
## -22.711255492 -17.873452355 3.161579247 31.698343058 1.365736549
## 1286 1287 1288 1289 1290
## 16.940996380 6.501114593 36.940996380 -11.935302777 1.964350781
## 1291 1292 1293 1294 1295
## -5.622586250 -12.410216166 -17.696113873 -21.344901326 -10.109176841
## 1296 1297 1298 1299 1300
## -11.421893367 2.165043665 -19.097499640 -0.232877684 13.578106633
## 1301 1302 1303 1304 1305
## 6.427586970 -18.985475998 -13.773105913 -6.348365745 -4.348365745
## 1306 1307 1308 1309 1310
## 50.099728825 51.999382383 56.424122552 48.261925688 54.011059584
## 1311 1312 1313 1314 1315
## 47.648169837 -13.325011344 20.273602889 25.238571287 0.940996380
## 1316 1317 1318 1319 1320
## -19.360042945 -23.371720146 -23.661082271 -12.985475998 -24.460389387
## 1321 1322 1323 1324 1325
## -3.171027262 -3.784783114 9.864004340 16.562965014 -7.097499640
## 1326 1327 1328 1329 1330
## 9.485972973 18.501114593 -16.672759471 -8.197846082 19.613138235
## 1331 1332 1333 1334 1335
## 18.099728825 28.161579247 66.312098909 42.771870680 11.601461034
## 1336 1337 1338 1339 1340
## 16.674988656 19.663311456 25.991169601 -1.109176841 -22.572413030
## 1341 1342 1343 1344 1345
## -3.309869724 -11.873452355 -17.383397346 -7.684436672 -11.896806756
## 1346 1347 1348 1349 1350
## -17.321546925 -14.885129556 -17.259696503 -8.560735829 -3.371720146
## 1351 1352 1353 1354 1355
## -4.271373704 -5.746287093 -3.746287093 -5.032184800 -23.819814716
## 1356 1357 1358 1359 1360
## -6.070680820 -4.533917009 16.790476718 18.002846802 17.516256212
## 1361 1362 1363 1364 1365
## 1.253712907 9.253712907 -5.885129556 10.338917729 2.740303497
## 1366
## -7.109176841Fitted Values
#Le fitted values sont:
fitted_simple <- res_simple$fitted
fitted_simple## 1 2 3 4 5 6 7 8
## 46.12907 52.43011 44.81635 57.15589 58.73115 59.51878 53.21774 54.53045
## 9 10 11 12 13 14 15 16
## 51.37993 47.70433 55.84317 60.56895 47.70433 48.22941 44.02872 57.94352
## 17 18 19 20 21 22 23 24
## 52.16756 50.59230 49.54213 54.26791 63.71947 73.95866 71.07068 70.02051
## 25 26 27 28 29 30 31 32
## 64.76964 67.13253 72.12085 84.19785 90.76143 64.24455 82.09750 80.78478
## 33 34 35 36 37 38 39 40
## 73.69611 68.18270 66.60744 66.86999 60.83149 65.55727 70.80814 71.85831
## 41 42 43 44 45 46 47 48
## 79.20952 71.07068 81.83496 72.12085 82.09750 66.86999 70.80814 59.51878
## 49 50 51 52 53 54 55 56
## 64.24455 73.95866 65.81981 77.10918 78.15935 74.22120 76.32155 70.54559
## 57 58 59 60 61 62 63 64
## 67.39507 72.64594 69.75796 72.90848 85.77311 95.74975 73.43357 71.59577
## 65 66 67 68 69 70 71 72
## 66.60744 70.54559 79.20952 83.41022 74.22120 72.38340 69.23288 70.54559
## 73 74 75 76 77 78 79 80
## 79.99715 82.62259 86.03565 88.66108 85.77311 92.86178 90.49889 90.49889
## 81 82 83 84 85 86 87 88
## 76.32155 87.08582 84.46039 94.43703 110.45218 81.04733 73.95866 92.86178
## 89 90 91 92 93 94 95 96
## 103.10096 83.67276 92.33669 100.21299 84.72293 99.42536 84.46039 79.73461
## 97 98 99 100 101 102 103 104
## 77.10918 72.38340 75.00883 83.41022 99.16281 105.98894 110.18963 79.47207
## 105 106 107 108 109 110 111 112
## 89.44871 91.28652 101.26316 97.32501 91.28652 79.73461 89.97380 95.48721
## 113 114 115 116 117 118 119 120
## 90.49889 103.10096 105.98894 90.76143 95.22466 99.68790 81.30987 95.22466
## 121 122 123 124 125 126 127 128
## 98.11264 96.27484 88.66108 89.18617 108.87692 103.62605 91.81160 94.17449
## 129 130 131 132 133 134 135 136
## 99.68790 101.52570 97.58755 83.41022 82.09750 83.14767 80.52224 83.67276
## 137 138 139 140 141 142 143 144
## 92.07415 94.69958 73.43357 83.41022 85.77311 84.72293 96.27484 98.90027
## 145 146 147 148 149 150 151 152
## 81.30987 77.89681 85.51056 83.41022 82.09750 78.94698 79.73461 89.71126
## 153 154 155 156 157 158 159 160
## 79.47207 75.27137 83.14767 79.73461 77.37172 82.62259 71.07068 79.73461
## 161 162 163 164 165 166 167 168
## 64.24455 78.42189 79.47207 76.32155 71.85831 68.44525 65.81981 64.50710
## 169 170 171 172 173 174 175 176
## 67.92016 60.30641 72.38340 72.64594 72.90848 75.00883 75.27137 68.70779
## 177 178 179 180 181 182 183 184
## 73.17103 74.48374 63.19438 64.50710 72.38340 58.20606 64.50710 61.35658
## 185 186 187 188 189 190 191 192
## 63.19438 69.23288 60.56895 65.29473 72.90848 75.53392 63.19438 65.29473
## 193 194 195 196 197 198 199 200
## 63.98201 57.94352 59.78132 60.56895 48.75450 55.05554 62.40675 48.49196
## 201 202 203 204 205 206 207 208
## 58.99369 56.63080 54.53045 58.46860 56.36826 60.56895 62.40675 64.76964
## 209 210 211 212 213 214 215 216
## 76.84663 87.61091 92.86178 98.11264 102.05079 101.52570 98.37518 76.32155
## 217 218 219 220 221 222 223 224
## 69.75796 73.43357 62.66930 52.69265 58.73115 65.29473 67.13253 60.04386
## 225 226 227 228 229 230 231 232
## 68.44525 63.45692 64.76964 63.45692 72.38340 80.78478 88.66108 70.28305
## 233 234 235 236 237 238 239 240
## 79.99715 77.10918 84.98548 84.46039 73.17103 66.08236 71.07068 72.12085
## 241 242 243 244 245 246 247 248
## 74.74629 67.39507 74.48374 71.07068 79.47207 74.48374 72.12085 71.59577
## 249 250 251 252 253 254 255 256
## 70.54559 70.28305 68.18270 67.13253 72.12085 76.84663 84.46039 72.90848
## 257 258 259 260 261 262 263 264
## 73.17103 84.98548 86.82328 88.66108 84.98548 75.27137 80.78478 87.34837
## 265 266 267 268 269 270 271 272
## 103.88859 104.67622 108.35183 112.81507 112.28998 107.82674 87.87345 78.68444
## 273 274 275 276 277 278 279 280
## 82.62259 78.68444 81.83496 92.07415 102.83842 114.12778 94.43703 84.19785
## 281 282 283 284 285 286 287 288
## 92.86178 86.82328 89.18617 87.61091 87.08582 86.82328 93.64941 99.68790
## 289 290 291 292 293 294 295 296
## 115.17796 92.07415 84.46039 89.71126 102.05079 109.66455 92.86178 84.19785
## 297 298 299 300 301 302 303 304
## 87.08582 101.78825 108.87692 105.98894 112.28998 106.77657 97.32501 87.87345
## 305 306 307 308 309 310 311 312
## 81.04733 81.04733 91.81160 99.68790 108.61437 82.09750 88.66108 86.82328
## 313 314 315 316 317 318 319 320
## 92.33669 99.42536 103.36351 101.78825 105.98894 102.83842 102.31333 99.68790
## 321 322 323 324 325 326 327 328
## 88.66108 89.18617 90.76143 88.13600 89.18617 80.78478 84.19785 78.42189
## 329 330 331 332 333 334 335 336
## 82.09750 79.20952 84.46039 79.99715 82.62259 78.15935 74.22120 77.89681
## 337 338 339 340 341 342 343 344
## 82.36004 82.36004 78.15935 80.25970 76.58409 75.00883 63.98201 72.90848
## 345 346 347 348 349 350 351 352
## 73.17103 79.20952 80.25970 78.68444 73.95866 76.32155 72.90848 75.53392
## 353 354 355 356 357 358 359 360
## 66.86999 70.54559 70.80814 70.02051 69.75796 68.44525 65.81981 51.37993
## 361 362 363 364 365 366 367 368
## 44.81635 46.91670 55.84317 52.69265 50.85485 55.31808 60.83149 68.18270
## 369 370 371 372 373 374 375 376
## 63.71947 65.81981 75.27137 61.88167 60.04386 71.59577 77.10918 67.65762
## 377 378 379 380 381 382 383 384
## 68.18270 74.74629 75.79646 82.88513 63.45692 61.61912 65.81981 61.88167
## 385 386 387 388 389 390 391 392
## 68.97033 81.57241 77.37172 67.65762 67.92016 55.31808 62.40675 52.69265
## 393 394 395 396 397 398 399 400
## 60.56895 66.60744 49.01704 63.98201 55.58063 61.61912 58.73115 58.20606
## 401 402 403 404 405 406 407 408
## 62.66930 66.08236 62.66930 62.66930 50.06722 54.79300 71.59577 62.14421
## 409 410 411 412 413 414 415 416
## 64.50710 65.29473 63.45692 72.12085 69.75796 63.71947 64.50710 69.23288
## 417 418 419 420 421 422 423 424
## 72.12085 82.62259 97.58755 74.48374 73.17103 68.97033 77.37172 85.77311
## 425 426 427 428 429 430 431 432
## 98.63773 109.13946 108.61437 61.35658 84.46039 86.29819 90.49889 76.58409
## 433 434 435 436 437 438 439 440
## 87.61091 88.92363 94.69958 101.00062 100.47553 98.90027 99.68790 84.19785
## 441 442 443 444 445 446 447 448
## 73.43357 77.37172 79.20952 80.78478 84.98548 73.17103 82.36004 85.24802
## 449 450 451 452 453 454 455 456
## 75.79646 76.05900 74.74629 75.53392 77.89681 83.41022 75.79646 76.84663
## 457 458 459 460 461 462 463 464
## 74.48374 79.73461 78.68444 82.09750 73.95866 77.10918 83.41022 88.92363
## 465 466 467 468 469 470 471 472
## 92.33669 92.59923 96.27484 99.68790 98.90027 104.93877 110.45218 112.81507
## 473 474 475 476 477 478 479 480
## 95.74975 86.29819 92.86178 95.22466 87.08582 87.87345 91.81160 83.41022
## 481 482 483 484 485 486 487 488
## 83.41022 84.98548 79.20952 87.87345 97.58755 98.90027 84.98548 81.83496
## 489 490 491 492 493 494 495 496
## 93.12432 77.37172 82.88513 80.25970 81.30987 86.82328 90.23634 82.09750
## 497 498 499 500 501 502 503 504
## 99.68790 105.72640 105.98894 111.23981 86.56074 85.24802 88.92363 84.46039
## 505 506 507 508 509 510 511 512
## 85.24802 81.30987 81.57241 76.32155 75.27137 77.89681 76.32155 82.09750
## 513 514 515 516 517 518 519 520
## 79.99715 83.67276 82.62259 93.12432 86.82328 90.76143 85.51056 78.15935
## 521 522 523 524 525 526 527 528
## 78.42189 73.69611 77.37172 74.22120 72.64594 76.58409 82.62259 83.14767
## 529 530 531 532 533 534 535 536
## 84.98548 75.27137 65.81981 60.56895 75.00883 65.03218 75.53392 78.94698
## 537 538 539 540 541 542 543 544
## 74.22120 72.90848 79.73461 77.63426 81.04733 73.95866 82.62259 84.98548
## 545 546 547 548 549 550 551 552
## 80.25970 60.56895 62.14421 71.07068 71.59577 62.66930 65.55727 69.75796
## 553 554 555 556 557 558 559 560
## 59.78132 56.89334 59.25623 63.98201 68.18270 75.27137 61.35658 70.54559
## 561 562 563 564 565 566 567 568
## 67.92016 72.90848 69.75796 79.47207 78.68444 87.61091 83.41022 93.12432
## 569 570 571 572 573 574 575 576
## 81.04733 76.58409 83.93530 80.52224 74.48374 82.88513 74.48374 78.42189
## 577 578 579 580 581 582 583 584
## 80.25970 81.30987 78.42189 84.46039 71.33322 67.39507 65.03218 69.23288
## 585 586 587 588 589 590 591 592
## 68.44525 74.74629 66.60744 76.84663 65.29473 72.90848 79.73461 83.93530
## 593 594 595 596 597 598 599 600
## 93.38686 96.53738 93.91195 89.18617 88.13600 87.61091 91.28652 80.25970
## 601 602 603 604 605 606 607 608
## 89.97380 82.36004 79.99715 81.57241 88.13600 91.02397 91.81160 97.32501
## 609 610 611 612 613 614 615 616
## 98.37518 92.59923 83.67276 88.66108 88.66108 93.12432 102.83842 85.51056
## 617 618 619 620 621 622 623 624
## 82.88513 84.72293 85.24802 100.73807 89.18617 88.92363 93.38686 91.54906
## 625 626 627 628 629 630 631 632
## 98.63773 104.67622 110.71472 85.51056 96.01229 94.43703 97.85010 103.10096
## 633 634 635 636 637 638 639 640
## 103.62605 83.41022 94.17449 99.16281 105.46385 92.86178 97.06247 91.28652
## 641 642 643 644 645 646 647 648
## 96.79992 79.20952 83.14767 74.22120 72.38340 83.14767 81.57241 89.97380
## 649 650 651 652 653 654 655 656
## 79.73461 84.72293 87.61091 73.17103 77.10918 79.20952 78.68444 78.15935
## 657 658 659 660 661 662 663 664
## 74.74629 84.46039 86.56074 75.53392 72.38340 75.27137 81.57241 88.92363
## 665 666 667 668 669 670 671 672
## 93.38686 102.31333 104.15114 94.96212 88.92363 87.34837 90.23634 92.86178
## 673 674 675 676 677 678 679 680
## 89.97380 89.71126 86.82328 78.94698 96.27484 92.07415 64.50710 63.45692
## 681 682 683 684 685 686 687 688
## 62.40675 65.55727 61.09404 47.96687 43.24109 49.54213 45.60398 49.54213
## 689 690 691 692 693 694 695 696
## 50.06722 48.75450 53.48028 56.36826 62.93184 72.12085 78.15935 59.78132
## 697 698 699 700 701 702 703 704
## 65.55727 62.14421 59.51878 63.98201 66.34490 61.61912 57.15589 62.40675
## 705 706 707 708 709 710 711 712
## 69.23288 65.81981 74.48374 90.76143 80.25970 91.81160 96.53738 94.17449
## 713 714 715 716 717 718 719 720
## 93.64941 84.19785 95.22466 92.33669 90.49889 93.64941 94.69958 88.66108
## 721 722 723 724 725 726 727 728
## 89.97380 73.43357 76.05900 72.90848 74.22120 76.58409 70.02051 75.53392
## 729 730 731 732 733 734 735 736
## 65.29473 72.90848 75.53392 67.13253 77.37172 77.37172 83.93530 82.88513
## 737 738 739 740 741 742 743 744
## 82.88513 77.63426 77.63426 81.83496 76.58409 59.51878 68.18270 67.39507
## 745 746 747 748 749 750 751 752
## 73.69611 67.65762 76.32155 79.73461 83.93530 99.95044 114.12778 81.57241
## 753 754 755 756 757 758 759 760
## 82.88513 85.51056 89.18617 81.57241 79.99715 70.54559 77.37172 73.69611
## 761 762 763 764 765 766 767 768
## 79.73461 80.25970 70.28305 75.00883 77.10918 73.17103 72.64594 74.74629
## 769 770 771 772 773 774 775 776
## 77.89681 77.89681 78.42189 84.19785 84.19785 77.89681 78.42189 73.95866
## 777 778 779 780 781 782 783 784
## 79.99715 83.67276 79.73461 96.79992 92.59923 81.57241 84.98548 74.74629
## 785 786 787 788 789 790 791 792
## 83.67276 88.13600 80.78478 78.42189 73.17103 76.58409 81.30987 80.78478
## 793 794 795 796 797 798 799 800
## 78.94698 84.72293 87.61091 115.44050 98.37518 90.49889 84.19785 90.76143
## 801 802 803 804 805 806 807 808
## 90.76143 88.92363 87.08582 76.84663 87.61091 82.09750 85.24802 84.72293
## 809 810 811 812 813 814 815 816
## 83.67276 84.19785 79.47207 78.94698 83.14767 88.66108 75.00883 87.61091
## 817 818 819 820 821 822 823 824
## 89.18617 88.92363 88.66108 73.69611 65.81981 63.19438 67.13253 64.76964
## 825 826 827 828 829 830 831 832
## 76.58409 76.05900 78.68444 86.56074 86.82328 89.18617 86.03565 83.93530
## 833 834 835 836 837 838 839 840
## 87.61091 79.73461 70.02051 81.04733 77.10918 74.74629 68.70779 67.92016
## 841 842 843 844 845 846 847 848
## 73.69611 66.34490 73.43357 71.33322 76.05900 68.44525 63.45692 66.60744
## 849 850 851 852 853 854 855 856
## 63.71947 56.89334 60.30641 64.24455 53.74282 44.29126 50.59230 51.64248
## 857 858 859 860 861 862 863 864
## 61.88167 54.00537 49.54213 61.61912 64.50710 66.08236 54.79300 67.92016
## 865 866 867 868 869 870 871 872
## 59.51878 69.75796 77.89681 77.37172 76.84663 76.32155 87.08582 89.18617
## 873 874 875 876 877 878 879 880
## 91.54906 76.84663 71.59577 78.94698 76.05900 76.05900 87.61091 75.53392
## 881 882 883 884 885 886 887 888
## 65.03218 68.44525 72.90848 71.07068 74.22120 76.58409 68.97033 71.33322
## 889 890 891 892 893 894 895 896
## 72.64594 74.48374 72.12085 70.02051 86.03565 93.64941 88.39854 94.69958
## 897 898 899 900 901 902 903 904
## 84.46039 86.82328 96.01229 72.64594 70.54559 68.70779 65.55727 73.17103
## 905 906 907 908 909 910 911 912
## 65.55727 69.75796 73.95866 75.79646 77.10918 79.99715 76.32155 82.09750
## 913 914 915 916 917 918 919 920
## 89.71126 93.12432 85.51056 81.57241 84.19785 77.37172 70.02051 75.53392
## 921 922 923 924 925 926 927 928
## 78.68444 69.49542 86.82328 74.48374 87.34837 106.25148 83.14767 78.94698
## 929 930 931 932 933 934 935 936
## 78.94698 82.09750 77.89681 93.64941 95.74975 95.48721 97.06247 85.24802
## 937 938 939 940 941 942 943 944
## 87.61091 79.47207 96.53738 102.05079 85.24802 86.29819 78.94698 72.64594
## 945 946 947 948 949 950 951 952
## 84.98548 95.74975 106.25148 109.92709 98.37518 98.63773 97.58755 101.26316
## 953 954 955 956 957 958 959 960
## 98.37518 90.76143 84.19785 91.54906 86.29819 92.59923 73.95866 81.83496
## 961 962 963 964 965 966 967 968
## 80.25970 79.99715 85.51056 87.34837 87.08582 79.99715 81.04733 73.17103
## 969 970 971 972 973 974 975 976
## 83.67276 82.62259 82.62259 91.54906 92.07415 88.13600 100.21299 90.76143
## 977 978 979 980 981 982 983 984
## 90.76143 87.61091 90.49889 93.38686 93.64941 86.29819 86.56074 100.47553
## 985 986 987 988 989 990 991 992
## 94.69958 96.27484 100.47553 93.64941 88.66108 90.49889 79.47207 103.36351
## 993 994 995 996 997 998 999 1000
## 108.87692 79.20952 77.89681 68.44525 79.73461 76.84663 74.48374 72.64594
## 1001 1002 1003 1004 1005 1006 1007 1008
## 76.32155 78.15935 70.02051 88.39854 82.36004 86.56074 75.53392 77.63426
## 1009 1010 1011 1012 1013 1014 1015 1016
## 72.90848 74.48374 76.32155 75.27137 54.26791 55.05554 67.65762 49.54213
## 1017 1018 1019 1020 1021 1022 1023 1024
## 43.50363 53.21774 63.45692 73.43357 61.09404 56.89334 54.53045 51.11739
## 1025 1026 1027 1028 1029 1030 1031 1032
## 64.24455 48.49196 48.75450 49.01704 62.93184 58.73115 63.71947 67.65762
## 1033 1034 1035 1036 1037 1038 1039 1040
## 72.90848 68.18270 64.50710 58.73115 65.29473 66.34490 63.45692 66.08236
## 1041 1042 1043 1044 1045 1046 1047 1048
## 69.75796 78.94698 81.30987 72.64594 67.13253 70.28305 85.51056 86.82328
## 1049 1050 1051 1052 1053 1054 1055 1056
## 79.73461 76.84663 78.15935 76.32155 75.27137 70.54559 75.27137 88.92363
## 1057 1058 1059 1060 1061 1062 1063 1064
## 96.53738 83.93530 70.80814 65.55727 66.86999 70.54559 65.55727 69.49542
## 1065 1066 1067 1068 1069 1070 1071 1072
## 76.32155 71.33322 73.43357 63.98201 66.86999 67.92016 69.75796 67.92016
## 1073 1074 1075 1076 1077 1078 1079 1080
## 71.33322 75.27137 83.67276 78.15935 78.15935 71.33322 74.48374 77.10918
## 1081 1082 1083 1084 1085 1086 1087 1088
## 89.18617 76.84663 77.63426 81.04733 85.51056 82.36004 95.74975 100.47553
## 1089 1090 1091 1092 1093 1094 1095 1096
## 104.93877 102.57588 89.44871 84.98548 78.15935 74.48374 73.17103 74.74629
## 1097 1098 1099 1100 1101 1102 1103 1104
## 85.77311 88.66108 91.54906 89.18617 86.29819 91.28652 85.24802 77.10918
## 1105 1106 1107 1108 1109 1110 1111 1112
## 83.93530 82.36004 89.71126 73.95866 77.10918 76.05900 72.12085 73.17103
## 1113 1114 1115 1116 1117 1118 1119 1120
## 79.99715 80.25970 76.58409 75.53392 77.89681 85.77311 86.03565 91.54906
## 1121 1122 1123 1124 1125 1126 1127 1128
## 93.64941 96.79992 94.96212 93.38686 81.30987 84.72293 81.57241 76.58409
## 1129 1130 1131 1132 1133 1134 1135 1136
## 84.19785 80.78478 84.46039 97.06247 90.76143 89.71126 81.57241 82.62259
## 1137 1138 1139 1140 1141 1142 1143 1144
## 83.41022 84.72293 91.81160 87.08582 99.95044 82.88513 95.22466 94.17449
## 1145 1146 1147 1148 1149 1150 1151 1152
## 88.66108 89.18617 88.39854 87.34837 87.87345 82.09750 81.30987 70.02051
## 1153 1154 1155 1156 1157 1158 1159 1160
## 80.78478 86.03565 98.90027 101.26316 110.18963 80.25970 85.24802 81.04733
## 1161 1162 1163 1164 1165 1166 1167 1168
## 86.82328 89.71126 91.81160 82.09750 82.09750 77.37172 80.25970 85.77311
## 1169 1170 1171 1172 1173 1174 1175 1176
## 84.46039 82.09750 89.44871 96.27484 103.10096 108.61437 77.10918 77.89681
## 1177 1178 1179 1180 1181 1182 1183 1184
## 88.92363 87.34837 83.14767 77.37172 78.94698 64.50710 68.70779 73.69611
## 1185 1186 1187 1188 1189 1190 1191 1192
## 89.71126 93.64941 77.10918 81.83496 85.24802 78.68444 73.43357 77.10918
## 1193 1194 1195 1196 1197 1198 1199 1200
## 69.75796 75.00883 66.86999 56.36826 57.94352 62.66930 60.83149 61.61912
## 1201 1202 1203 1204 1205 1206 1207 1208
## 66.08236 55.05554 61.88167 59.25623 58.46860 55.58063 64.50710 59.78132
## 1209 1210 1211 1212 1213 1214 1215 1216
## 58.46860 54.79300 49.54213 48.49196 55.31808 54.00537 62.14421 67.65762
## 1217 1218 1219 1220 1221 1222 1223 1224
## 65.81981 62.40675 67.13253 61.88167 56.10571 56.36826 63.45692 56.63080
## 1225 1226 1227 1228 1229 1230 1231 1232
## 53.74282 60.04386 54.00537 66.86999 67.92016 70.02051 73.69611 86.03565
## 1233 1234 1235 1236 1237 1238 1239 1240
## 96.53738 72.38340 79.73461 67.39507 69.49542 64.24455 70.54559 70.80814
## 1241 1242 1243 1244 1245 1246 1247 1248
## 80.52224 86.29819 87.34837 88.13600 92.86178 84.46039 81.57241 81.30987
## 1249 1250 1251 1252 1253 1254 1255 1256
## 98.90027 84.46039 93.91195 78.42189 76.84663 76.58409 74.48374 79.99715
## 1257 1258 1259 1260 1261 1262 1263 1264
## 82.09750 80.25970 69.23288 64.50710 74.48374 74.22120 79.73461 85.77311
## 1265 1266 1267 1268 1269 1270 1271 1272
## 78.94698 81.57241 79.73461 76.05900 73.95866 78.42189 92.86178 87.08582
## 1273 1274 1275 1276 1277 1278 1279 1280
## 91.81160 103.36351 94.43703 114.12778 89.97380 88.39854 87.08582 94.17449
## 1281 1282 1283 1284 1285 1286 1287 1288
## 89.71126 87.87345 102.83842 107.30166 77.63426 76.05900 90.49889 76.05900
## 1289 1290 1291 1292 1293 1294 1295 1296
## 83.93530 86.03565 82.62259 83.41022 73.69611 66.34490 77.10918 78.42189
## 1297 1298 1299 1300 1301 1302 1303 1304
## 81.83496 82.09750 69.23288 78.42189 81.57241 84.98548 85.77311 87.34837
## 1305 1306 1307 1308 1309 1310 1311 1312
## 87.34837 98.90027 101.00062 102.57588 100.73807 105.98894 108.35183 97.32501
## 1313 1314 1315 1316 1317 1318 1319 1320
## 105.72640 90.76143 76.05900 82.36004 77.37172 88.66108 84.98548 84.46039
## 1321 1322 1323 1324 1325 1326 1327 1328
## 73.17103 80.78478 88.13600 94.43703 82.09750 106.51403 90.49889 83.67276
## 1329 1330 1331 1332 1333 1334 1335 1336
## 84.19785 93.38686 98.90027 102.83842 99.68790 116.22813 88.39854 97.32501
## 1337 1338 1339 1340 1341 1342 1343 1344
## 92.33669 75.00883 77.10918 81.57241 81.30987 87.87345 72.38340 78.68444
## 1345 1346 1347 1348 1349 1350 1351 1352
## 77.89681 76.32155 82.88513 80.25970 86.56074 77.37172 75.27137 74.74629
## 1353 1354 1355 1356 1357 1358 1359 1360
## 74.74629 65.03218 65.81981 71.07068 75.53392 79.20952 79.99715 74.48374
## 1361 1362 1363 1364 1365 1366
## 74.74629 74.74629 82.88513 88.66108 80.25970 77.10918The means
## The mean of the residuals is -6.454147e-15## The mean of the fitted values is 79.28141\(R^{2}\) squared et \(R^{2}_{a}\) adjusted
## Le R2 est value est 0.242301## Le R2 adjusted value est 0.2417455Modèle linear multiple \(\mathcal{M}_{2}\) avec les variables
T12, Vx, N12 and
max03v
On considére el modèle lineaire simple \(\mathcal{M}_{2}\) avec la variable
explicative T12, Vx, N12 and
max03v.
\[\begin{equation} (\mathcal{M}_{2}): max03= \beta_{0} + \beta_{1} T12 + \beta_{1} Vx + \beta_{1} N12 + \beta_{1} T12max03V + \epsilon_{2} \end{equation}\]
Résumé
Plot
Summary
##
## Call:
## lm(formula = maxO3 ~ T12 + Vx + Ne12 + maxO3v, data = ozone)
##
## Residuals:
## Min 1Q Median 3Q Max
## -70.313 -9.129 0.935 9.953 58.950
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 32.76618 2.87129 11.412 < 2e-16 ***
## T12 0.70387 0.10074 6.987 4.38e-12 ***
## Vx 0.93921 0.13449 6.983 4.50e-12 ***
## Ne12 -2.55848 0.22418 -11.413 < 2e-16 ***
## maxO3v 0.59597 0.01701 35.046 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.68 on 1361 degrees of freedom
## Multiple R-squared: 0.6659, Adjusted R-squared: 0.6649
## F-statistic: 678.1 on 4 and 1361 DF, p-value: < 2.2e-16Coefficients
## (Intercept) T12 Vx Ne12 maxO3v
## 32.7661837 0.7038747 0.9392082 -2.5584780 0.5959748Les estimations des coefficients sont \(\beta_0, \beta_1, \beta_2, \beta_3\) et \(\beta_4\) sont respectivement \(\beta_0=32.76618, \beta_1=0.70387\), \(\beta_2=0.93921, \beta_3=-2.55848\) et \(\beta_4=0.59597\). Le modele lineaire \(\mathcal{M}_{2}\) est:
\[y=32.76618+0.70387 \,\, T12 + 0.93921 \,\, Vx -2.55848 \,\, Ne12 +0.59597 \,\, maxO3v.\]
Residuals
#Le residuals sont:
residuals_multiple <- res_multiple$residuals
#residuals_multipleFitted Values
#Le fitted values sont:
fitted_multiple <- res_multiple$fitted
#fitted_multipleThe means
## The mean of the fitted values is 79.28141## The mean of the residuals is 2.397328e-15\(R^{2}\) squared et \(R^{2}_{a}\) adjusted
## Le R2 est value est 0.6658794## Le R2 adjusted value est 0.6648975Modèle lineaire avec tout les variables
On considere le modèle lineaire avec tout les variables explicatives
Resume
Summary
##
## Call:
## lm(formula = maxO3 ~ ., data = ozone)
##
## Residuals:
## Min 1Q Median 3Q Max
## -66.664 -9.063 0.277 9.660 49.455
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 41.20014 11.31132 3.642 0.000280 ***
## T6 -1.54951 0.28029 -5.528 3.88e-08 ***
## T9 0.10141 0.43054 0.236 0.813825
## T12 1.35109 0.40963 3.298 0.000998 ***
## T15 0.18992 0.42341 0.449 0.653831
## T18 0.15683 0.33125 0.473 0.635972
## Ne6 -0.09294 0.21773 -0.427 0.669545
## Ne9 -0.69359 0.31252 -2.219 0.026632 *
## Ne12 -0.84100 0.36079 -2.331 0.019900 *
## Ne15 -0.38348 0.34556 -1.110 0.267311
## Ne180 -15.79476 10.95285 -1.442 0.149516
## Ne181 -13.17350 10.83916 -1.215 0.224441
## Ne182 -12.00948 10.87340 -1.104 0.269581
## Ne183 -10.84205 10.86677 -0.998 0.318592
## Ne184 -12.95055 10.88617 -1.190 0.234400
## Ne185 -11.82442 10.84382 -1.090 0.275719
## Ne186 -13.48212 10.81210 -1.247 0.212634
## Ne187 -13.23046 10.78884 -1.226 0.220297
## Ne188 -13.92541 10.82302 -1.287 0.198439
## Vx 0.59868 0.14277 4.193 2.93e-05 ***
## maxO3v 0.58786 0.01667 35.266 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.17 on 1345 degrees of freedom
## Multiple R-squared: 0.6909, Adjusted R-squared: 0.6863
## F-statistic: 150.3 on 20 and 1345 DF, p-value: < 2.2e-16## Le R2 adjusted value est 0.6863159Analysis
\(\textbf{Resume Modèle}\) \(\mathcal{M}_{1}:\)
- Est-ce le modele \(\mathcal{M}_{1}\) vous semple-t-il adapte et \(\mathcal{M}_{2}\):
- Dans le modele de regression simple on a que \(R^{2}_{\mathcal{M}_{1}}= 24\%\), on peut
concluire que les modele n’est pas adapte, mais dans le modele de
regression multiple on a que \(R^{2}_{\mathcal{M}_{2}} = 66\%\) alors on
peut concluire que le modele est meiller adapte, en plus, on n’observe
pas de forme particulière avec le nuage des points. On montre que les
variables ajoutées qui sont
V x,Ne12,maxO3vcontribue à mieux expliquer le modele.
\(\textbf{Observation}:\) \(R^2\) augmente avec chaque prédicteur ajouté à un modèle. Comme \(R^2\) augmente toujours et ne diminue jamais, il peut sembler être un meilleur ajustement avec le plus de termes que vous ajoutez au modèle. Mais ce n’est pas toujours le meilleur choix.
- Est-ce qu’on valide a chaque fois le modele global ? les variables explicatives
- On voit que seul les variables que sont significatives dans le
modele sont \(\beta_0\),
T6,T12,Ne9,Ne12````,VxetmaxO3v```` sont significative dans le modèle. On voit aussi que le valeur de \(R^2\) a augmenté de \(0.0204365\) i.e \(2.4 \%\) seulement.
Intervalles de Confiance
Considérer maintenant le modèle \(\mathrm{M}_2\). Calculer les intervalles de confiance pour chaque coefficient (créer une fonction si possible). Vous pouvez utiliser la fonction coef de la façon suivante pour récupérer directement les estimations des coefficients avec les statistiques et $df.residual pour récupérer directement les degrés de liberté nécéssaires dans le calcul des intervalles de confiance:
Coefficiets
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 32.7661837 2.87128787 11.411668 7.164360e-29
## T12 0.7038747 0.10073849 6.987147 4.377991e-12
## Vx 0.9392082 0.13449333 6.983307 4.495141e-12
## Ne12 -2.5584780 0.22417730 -11.412743 7.083953e-29
## maxO3v 0.5959748 0.01700557 35.045870 2.668762e-192Degres de liberte
## Les degres de liberte 1361Intervalles de Confiance avec confit
## 2.5 % 97.5 %
## (Intercept) 27.1335537 38.3988136
## T12 0.5062551 0.9014943
## Vx 0.6753715 1.2030449
## Ne12 -2.9982486 -2.1187075
## maxO3v 0.5626149 0.6293348Intervalles de Confiance a la main
Cette fonction calcule l’intervalle de confiance
#Hacer una funcion que calcule los intervalos de confianza.
#the estimae +-std.error *qt(0.96,degrefreedom )
# With alpha the significance level
conf_int <- function(model, alpha=0.05) {
# Extract the coefficients and their standard errors from the model summary
coef_summary <- summary(model)$coefficients
coef_values <- coef_summary[, 1]
coef_se <- coef_summary[, 2]
# Compute the t-values for the given confidence level
t_values <- qt(1 - alpha / 2, df = model$df.residual)
# Compute the confidence intervals for each coefficient
lower_bounds <- coef_values - t_values * coef_se
upper_bounds <- coef_values + t_values * coef_se
# Combine the lower and upper bounds into a matrix and add column names
conf_int <- cbind(lower_bounds, upper_bounds)
colnames(conf_int) <- c("Lower Bound", "Upper Bound")
# Return the confidence intervals
return(conf_int)
}
conf_int(res_multiple)## Lower Bound Upper Bound
## (Intercept) 27.1335537 38.3988136
## T12 0.5062551 0.9014943
## Vx 0.6753715 1.2030449
## Ne12 -2.9982486 -2.1187075
## maxO3v 0.5626149 0.6293348Cette focntion ici, c’est l’intervalle de confiance avec R
IC<-confint(res_multiple,level = 0.95)
IC## 2.5 % 97.5 %
## (Intercept) 27.1335537 38.3988136
## T12 0.5062551 0.9014943
## Vx 0.6753715 1.2030449
## Ne12 -2.9982486 -2.1187075
## maxO3v 0.5626149 0.6293348L’ellipse de confiance de 95%
Dessinez les ellipses de confiance (à 95%) des paramètres \(\beta\) considérés deux à deux et sur chaque ellipse le rectangle de confiance construit à partir de chaque intervalle de confiance pris séparement.
Propriété de sans biais des estimateurs de coefficients \(\beta_{j}\)
Vérification de la propriété de sans biais des estimateurs de coefficients \(\beta_j\) d’un modèle linéaire et calcul des taux de couverture
Nous allons tester dans cette section que les estimateurs de coefficients \(\beta\) d’un modèle linéaire sont sans biais. En pratique, on ne peut jamais réaliser cela. On va simuler cela lors du TP. On considère pour cela que la table ozone est notre population d’étude, on connait donc les vraies valeurs des coefficients \(\beta\) (cela n’est jamais possible en pratique car on n’a jamais la population d’étude). Dans cette population, nous allons tirer des échantillons aléatoires de taille \(n=200\) et on va s’interesser à l’estimation du coefficient de la variable T12 qui est égal à:
Echantionage
Le valeur de T12
## Le vrai valeur de T12 est: 0.7038747Echantionage avec \(n=200\) observations
## [1] 1135 791 905 1341 807 1246 1311 292 1250 297 860 605 637 1063 1261
## [16] 165 619 1057 83 866 277 1233 1023 76 1118 1054 1241 946 1199 374
## [31] 323 115 850 608 682 938 1120 397 1172 989 392 593 744 243 106
## [46] 11 625 386 403 461 141 31 1139 94 16 178 177 524 924 204
## [61] 1338 373 646 384 1146 315 259 494 1072 1124 1016 1132 10 170 402
## [76] 1345 108 8 626 261 541 326 1098 282 1307 1243 696 667 990 1143
## [91] 452 856 622 1304 1060 1079 1264 891 1034 665 1129 793 463 1204 278
## [106] 241 24 679 37 686 566 19 378 549 48 1212 464 393 1163 670
## [121] 1213 311 189 38 1108 319 1257 846 120 712 441 1223 599 72 714
## [136] 677 81 1271 134 424 756 6 1152 879 668 49 193 709 459 303
## [151] 1328 898 190 191 446 119 1083 817 61 1295 1335 930 950 698 983
## [166] 758 993 947 690 251 560 643 545 1278 162 576 168 788 78 1267
## [181] 445 995 95 379 221 1280 620 448 242 927 814 926 407 229 785
## [196] 699 1047 218 648 79Valeur de T12
##
## Call:
## lm(formula = maxO3 ~ T12 + Vx + Ne12 + maxO3v, data = ozone.s)
##
## Coefficients:
## (Intercept) T12 Vx Ne12 maxO3v
## 40.4068 0.4622 1.3314 -2.6449 0.5804## T12
## 0.4621573Question 1
Considérer 10000 échantillons comme ci-dessus et calculer pour chaque échantillon l’estimation du coefficient de T12. Mettez ces estimations dans un vecteur de taille 10000.
\(\textbf{Answer:}\)
Ici nous montrons juste la première ligne du \(10000\) et \(20000\) echantillons.
10000 Echantillon avec \(n=200\) observations
nsim <- 10000
echantillon <- numeric(10000)
confinterval <- c()
for (i in 1:10000)
{
#sample with 200 observations
N <- nrow(ozone)
n <- 200
s <- sample(N,n)
ozone.s <- ozone[s,]
ozone.s
res2.s <- lm(maxO3 ~T12 + Vx + Ne12 +maxO3v, data=ozone.s)
rescof <- res2.s$coefficients[2]
## Interval de confiance
confinterval <- append(confinterval,confint(res2.s,level=0.95)[2,])
echantillon[i] <- rescof
}
head(echantillon)## [1] 0.5049869 0.3758093 0.7963478 0.9651165 0.8457399 0.9183359#echantillon20000 Echantillon avec \(n=200\) observations
## [1] 0.7562587 0.7299679 0.4765873 0.5390277 0.5694816 1.074340610000 Echantillon avec \(n=400\) observations
## [1] 0.6902926 0.7332076 0.8283719 0.9676027 0.4767603 0.8830131Function que depend de m nombre de simulation et r nombre des observations
## n is the number of simulations:
echantillonvector <- function(m,r){
nsim <- m
echantillon <- numeric(m)
for (i in 1:m)
{
#sample with 200 observations
N <- nrow(ozone)
n <- r
s <- sample(N,n)
ozone.s <- ozone[s,]
ozone.s
res2.s <- lm(maxO3 ~T12 + Vx + Ne12 +maxO3v, data=ozone.s)
rescof <- res2.s$coefficients[2]
echantillon[i] <- rescof
}
return(echantillon)
}
echantillon3 <- echantillonvector(10000,400)
head(echantillon3)## [1] 1.0135206 0.8473761 0.7595781 0.6837330 0.6804568 0.7997701Question 2
Calculer la moyenne de ces 10000 estimations et comparer avec la vrai valeur du coeffcient. Si ces deux valeurs sont très proches, on peut décider que l’estimateur est sans biais. Pour nous aider, on peut calculer le biais relatif de \(\hat{\beta}\) :
\[ RB(\hat{\beta})=\frac{\frac{1}{10000} \sum_{i=1}^{10000} \hat{\beta}^{(i)}-\beta}{\beta}, \] où \(\hat{\beta}^{(i)}\) est une estimation du paramètre \(\beta\) à partir de ième échantillon et \(\beta\) est la vraie valeur du paramètre.
On peut aussi représenter graphiquement la moyenne des estimations (voir figure plus bas). Attention, le calcul du biais relatif nous dit que la courbe est assez proche de la ligne en rouge qui est le vrai beta.
\(\textbf{Answer:}\)
a). La moyenne de ces 10000 et 20000 estimations:
Le valeur de T12
## Le vrai valeur de T12
cat("Le vrai valuer del coefficient correspondant a T12 est: ", T12)## Le vrai valuer del coefficient correspondant a T12 est: 0.7038747Le means
#10000
mean1 <- mean(echantillon)
cat("Le moyenne de 10000 echantillonage est ", mean1)## Le moyenne de 10000 echantillonage est 0.6973428## 20000
mean2 <- mean(echantillon2)
cat("Le moyenne de 20000 est ", mean2)## Le moyenne de 20000 est 0.6974637## 10000 avec n=400
mean3 <- mean(echantillon3)
cat("Le moyenne de 20000 echantillonage avec n=400 pbservations est ", mean3)## Le moyenne de 20000 echantillonage avec n=400 pbservations est 0.7040497\(\textbf{Conclusion}\) Comme le vrai valeur et les valeurs de 10000 et 20000 echantionage sont tres proche, on peut concluire que le estimateur est sans biais.
Le bias avec la focntion de R
b). Le biais relatif de \(\hat{\beta}\) est different chaque fois:
#biais del echantillon 10000
bi1 <- bias(echantillon, T12)
cat("Le bias de 10000 echantillonage avec n=200 avec R:",bi1 )## Le bias de 10000 echantillonage avec n=200 avec R: -0.00653193#biais del echantillon 20000
bi2 <- bias(echantillon2, T12)
cat("Le bias de 20000 echantillonage avec n=200 avec R:",bi2 )## Le bias de 20000 echantillonage avec n=200 avec R: -0.006410988#biais del echantillon 20000 avec n=400 observations
bi2 <- bias(echantillon3, T12)
cat("Le bias de 10000 echantillonage avec n=400 observations avec R:",bi2 )## Le bias de 10000 echantillonage avec n=400 observations avec R: 0.0001750234Le bias a la main
#EL VALOR MENOS EL DE ECHANTILLON
bia1 <- mean(echantillon) - T12
cat("Le bias de 10000 echantillonage avec n=200 est", bia1)## Le bias de 10000 echantillonage avec n=200 est -0.00653193bia2 <- mean(echantillon2) - T12
cat("Le bias de 20000 echantillonage avec n=200", bia2)## Le bias de 20000 echantillonage avec n=200 -0.006410988bia3 <- mean(echantillon3) - T12
cat("Le bias de 10000 echantillonage avec n=400 observations est", bia3)## Le bias de 10000 echantillonage avec n=400 observations est 0.0001750234Graphe
c). Graphe moyenne en terms de nombre de simulations
# 10000 nombre de simulations avec n=200
plot( cumsum(echantillon)/(1:10000), type="l" ,xlab = "Nombre Simulations ", ylab = " Moyennes des estimateurs de beta de T12" )
abline(h=T12, col="red")
# 20000 nombre de simulations avec n=200
plot( cumsum(echantillon2)/(1:20000), type="l" ,xlab = "Nombre Simulations ", ylab = " Moyennes des estimateurs de beta de T12" )
abline(h=T12, col="red")
# 10000 nombre de simulations avec n=400
plot( cumsum(echantillon3)/(1:10000), type="l" ,xlab = "Nombre Simulations ", ylab = " Moyennes des estimateurs de beta de T12" )
abline(h=T12, col="red")
Question 3
- On veut calculer maintenant le taux de couverture des intervalles de confiance de \(\beta_{T 12}\). Il faut donc, pour chanque échantillon, regarder si \(\beta_{T 12}\) appartient à l’intervalle de confiance construit à partir de cet échantillon et obtenir ensuite le taux de couverture (nombre d’échantillons qui contiennent \(\beta_{T 12}\) divisé par 10000). Inspirez vous du point précédent pour calculer ce taux de couverture. Le mieux est de faire les deux calculs sur les mêmes échantillons (biais et taux de couverture).
\(\textbf{Answer:}\)
beta <- T12
#echantillon
vecteurinter <- c()
#vecteurinter <- append(vecteurinter, confinterval[1] > beta && confinterval[2]>beta)
for (i in 1:10000){
k=i-1
vecteurinter <- append(vecteurinter, isTRUE(beta>confinterval[i+k] && beta<confinterval[i+k+1]))
}
taux <- mean(vecteurinter)
cat("Le taux de couverture des intervalles de confiance de beta_{T 12} est:", taux)## Le taux de couverture des intervalles de confiance de beta_{T 12} est: 0.9755Bootstrap
Le modèle utilisé est \(Y=X \beta+\varepsilon\) où \(\varepsilon\) est une variable aléatoire de loi \(F\) inconnue et d’espérance nulle. L’idée du bootstrap est d’estimer cette loi par ré-échantillonnage.
On considere le modele suivante:
\[ maxo3 = \beta_{0} + \beta_{1} T12 + \beta_{2} Vx + \beta_{3} Ne12\]
##
## Call:
## lm(formula = maxO3 ~ T12 + Vx + Ne12, data = ozoneb)
##
## Residuals:
## Min 1Q Median 3Q Max
## -75.461 -11.868 0.175 14.667 68.954
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 67.4299 3.7166 18.143 < 2e-16 ***
## T12 1.6163 0.1342 12.046 < 2e-16 ***
## Vx 1.3979 0.1846 7.574 6.63e-14 ***
## Ne12 -3.4475 0.3071 -11.226 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 21.62 on 1362 degrees of freedom
## Multiple R-squared: 0.3644, Adjusted R-squared: 0.363
## F-statistic: 260.2 on 3 and 1362 DF, p-value: < 2.2e-16Les residus estimés
On calcule les residus estimes \(\hat{\epsilon} = \hat{Y} -Y\) et ajustements \(\hat{Y}\).
res<-residuals(modele3)
y_chapeau <-predict(modele3)
COEFF <-matrix(0, ncol = 4,nrow = 10000)
colnames(COEFF) <-names(coef(modele3))
ozone.boot<-ozonebOn applique la methode de bootstrap avec 10000 (\(dim(COEF)\)) echantillons bootstrapes:
for (i in 1:nrow(COEFF)){
Regetoile <- sample(res,length(res),replace=T)
mod_etoile<-y_chapeau + Regetoile
ozone.boot[,"T12"] <-mod_etoile
modele3boot<-lm(formula(modele3),data = ozone.boot)
COEFF[i,] <-coef(modele3boot) # On obtient une matrice de 1000 coefficients estimés
}Confidence Interval
Nous avons obtenu un matrice de 10000 coefficients estimés
(COEF) et nous choisir les quantiles empiriques a \(2.5\%\) et \(97.5\%\).
## (Intercept) T12 Vx Ne12
## 2.5% 92.48188 0.04563539 1.231091 -4.977863
## 97.5% 103.37990 0.14635306 1.394780 -4.437713Un IC de \(95\%\) pour le
coefficient T12 est donc donné pas \([0.045,0.146]\).
Histogram
Cette histogramme semble indiquer que la loi est proche d’une loi normale.
hist(COEFF[,"T12"],main="",xlab="Coefficient de T12")