class: center, middle # Functions, Attributes, and Class ## Data Analysis with R and Python ### Deepayan Sarkar <h1 onclick='document.documentElement.requestFullscreen();' style='cursor: pointer;'> <svg xmlns='http://www.w3.org/2000/svg' width='16' height='16' fill='currentColor' class='bi bi-arrows-fullscreen' viewBox='0 0 16 16'> <path fill-rule='evenodd' d='M5.828 10.172a.5.5 0 0 0-.707 0l-4.096 4.096V11.5a.5.5 0 0 0-1 0v3.975a.5.5 0 0 0 .5.5H4.5a.5.5 0 0 0 0-1H1.732l4.096-4.096a.5.5 0 0 0 0-.707zm4.344 0a.5.5 0 0 1 .707 0l4.096 4.096V11.5a.5.5 0 1 1 1 0v3.975a.5.5 0 0 1-.5.5H11.5a.5.5 0 0 1 0-1h2.768l-4.096-4.096a.5.5 0 0 1 0-.707zm0-4.344a.5.5 0 0 0 .707 0l4.096-4.096V4.5a.5.5 0 1 0 1 0V.525a.5.5 0 0 0-.5-.5H11.5a.5.5 0 0 0 0 1h2.768l-4.096 4.096a.5.5 0 0 0 0 .707zm-4.344 0a.5.5 0 0 1-.707 0L1.025 1.732V4.5a.5.5 0 0 1-1 0V.525a.5.5 0 0 1 .5-.5H4.5a.5.5 0 0 1 0 1H1.732l4.096 4.096a.5.5 0 0 1 0 .707z'/> </svg> </h1> --- # Functions * Most useful things in R happen by calling functions * Functions have one or more arguments * All arguments have names (available as variables inside the function) * Arguments may be compulsory or optional * Optional arguments usually have "default" values -- * Arguments may or may not be named when the function is called * Unnamed arguments are matched by position * Optional arguments are usually named * Functions normally also have a useful "return" value <div> $$ \newcommand{\sub}{_} \newcommand{\nseq}[1]{ {#1}\sub{1}, {#1}\sub{2}, \dotsc, {#1}\sub{n} } $$ </div> --- layout: true # Example: Delhi Air Quality data --- ```r aqi <- read.csv("https://deepayan.github.io/BSDS/2024-01-DE/data/rkpuram-aqi.csv") str(aqi) ``` ``` 'data.frame': 3930 obs. of 7 variables: $ date: chr "2024/11/1" "2024/11/2" "2024/11/3" "2024/11/4" ... $ pm25: int 300 306 308 300 282 267 307 275 269 260 ... $ pm10: int 260 249 298 246 227 251 205 198 195 264 ... $ o3 : int 40 57 53 56 52 47 40 46 53 46 ... $ no2 : int 19 16 21 12 15 16 13 9 8 9 ... $ so2 : int 8 5 3 9 8 5 9 10 11 18 ... $ co : int 17 19 17 24 22 19 21 22 19 32 ... ``` --- * How is AQI (PM10) related to `no2` (Nitrogen dioxide)? -- * Basic tools we are familiar with * Scatter plot * Linear Regression --- ```r plot(pm10 ~ no2, data = aqi) ```  --- ```r plot(sqrt(pm10) ~ sqrt(no2), data = aqi) ```  --- * Formula for linear regression line <div> $$ \begin{eqnarray*} \hat{b} &=& \frac{\sum (x\sub{i} - \bar{x}) (y\sub{i} - \bar{y})}{\sum (x\sub{i} - \bar{x})^2} \cr \hat{a} &=& \bar{y} - \hat{b} \bar{x} \end{eqnarray*} $$ </div> --- * Calculation of linear regression line ```r x <- sqrt(aqi$no2) y <- sqrt(aqi$pm10) xbar <- mean(x) ybar <- mean(y) sxy <- sum( (x - xbar) * (y - ybar) ) sxx <- sum( (x - xbar)^2 ) ``` -- ```r c(xbar, ybar, sxx, sxy) ``` ``` [1] NA NA NA NA ``` --- * Regression coefficients removing missing values ```r x <- sqrt(aqi$no2) y <- sqrt(aqi$pm10) xbar <- mean(x, na.rm = TRUE) ybar <- mean(y, na.rm = TRUE) sxy <- sum( (x - xbar) * (y - ybar), na.rm = TRUE) sxx <- sum( (x - xbar)^2, na.rm = TRUE) ``` ```r c(xbar, ybar, sxx, sxy) ``` ``` [1] 4.703466 11.808792 7506.779092 11173.744082 ``` --- * Even this may not be correct: should remove both if only one missing! ```r x <- sqrt(aqi$no2) y <- sqrt(aqi$pm10) ok <- is.finite(x) & is.finite(y) xbar <- mean(x[ok]); ybar <- mean(y[ok]) sxy <- sum( (x[ok] - xbar) * (y[ok] - ybar)) sxx <- sum( (x[ok] - xbar)^2) ``` ```r c(xbar, ybar, sxx, sxy) ``` ``` [1] 4.700322 11.827206 7480.368670 11173.964771 ``` --- * Coefficients of linear regression line ```r bhat <- sxy / sxx ahat <- ybar - bhat * xbar c(intercept = ahat, slope = bhat) ``` ``` intercept slope 4.805996 1.493772 ``` -- * Standard R function to do this ```r lm(sqrt(pm10) ~ sqrt(no2), data = aqi) ``` ``` Call: lm(formula = sqrt(pm10) ~ sqrt(no2), data = aqi) Coefficients: (Intercept) sqrt(no2) 4.806 1.494 ``` --- * Add regression line to scatter plot ```r plot(sqrt(pm10) ~ sqrt(no2), data = aqi) abline(ahat, bhat, col = 2) ```  --- layout: true # Functions --- * We have used several functions above * `sqrt()`, `mean()`, `sum()` — basic mathematical / summary functions * `plot()` — "high level" plotting function * `abline()` — "low level" plotting function * `lm()` — "high level" modeling function --- * We will discuss graphics functions in more detail later -- * `lm()` is a good example to study the behaviour of modeling functions in R -- * Recall: * Functions have one or more arguments * All arguments have names (available as variables inside the function) * Arguments may be compulsory or optional * Optional arguments usually have "default" values --- * `lm()` fits a more general class of models known as _linear models_ ```r str(lm) ``` ``` function (formula, data, subset, weights, na.action, method = "qr", model = TRUE, x = FALSE, y = FALSE, qr = TRUE, singular.ok = TRUE, contrasts = NULL, offset, ...) ``` -- * These calls are equivalent ```r fm1 <- lm(pm10 ~ no2, aqi, (no2 < 100)) fm2 <- lm(pm10 ~ no2, data = aqi, method = "qr", subset = (no2 < 100)) coef(fm1) ``` ``` (Intercept) no2 52.011396 4.228562 ``` ```r coef(fm2) ``` ``` (Intercept) no2 52.011396 4.228562 ``` --- * Rule: named arguments are matched by name, remaining by position * Convention: - First few "standard" arguments are usually unnamed (matched by position) - Usually unnamed arguments are _not_ used after named arguments -- * The following call is equivalent to previous two, but not recommended ```r fm3 <- lm(pm10 ~ no2, data = aqi, no2 < 100) coef(fm3) ``` ``` (Intercept) no2 52.011396 4.228562 ``` --- * The return value of `lm()` is a list .scrollable400[ ```r str(fm2) ``` ``` List of 13 $ coefficients : Named num [1:2] 52.01 4.23 ..- attr(*, "names")= chr [1:2] "(Intercept)" "no2" $ residuals : Named num [1:3810] 128 129 157 143 112 ... ..- attr(*, "names")= chr [1:3810] "1" "2" "3" "4" ... $ effects : Named num [1:3810] -9466 3472 155 139 108 ... ..- attr(*, "names")= chr [1:3810] "(Intercept)" "no2" "" "" ... $ rank : int 2 $ fitted.values: Named num [1:3810] 132 120 141 103 115 ... ..- attr(*, "names")= chr [1:3810] "1" "2" "3" "4" ... $ assign : int [1:2] 0 1 $ qr :List of 5 ..$ qr : num [1:3810, 1:2] -61.7252 0.0162 0.0162 0.0162 0.0162 ... .. ..- attr(*, "dimnames")=List of 2 .. .. ..$ : chr [1:3810] "1" "2" "3" "4" ... .. .. ..$ : chr [1:2] "(Intercept)" "no2" .. ..- attr(*, "assign")= int [1:2] 0 1 ..$ qraux: num [1:2] 1.02 1.01 ..$ pivot: int [1:2] 1 2 ..$ tol : num 1e-07 ..$ rank : int 2 ..- attr(*, "class")= chr "qr" $ df.residual : int 3808 $ na.action : 'omit' Named int [1:118] 14 27 166 237 238 406 434 455 456 458 ... ..- attr(*, "names")= chr [1:118] "NA" "NA.1" "NA.2" "NA.3" ... $ xlevels : Named list() $ call : language lm(formula = pm10 ~ no2, data = aqi, subset = (no2 < 100), method = "qr") $ terms :Classes 'terms', 'formula' language pm10 ~ no2 .. ..- attr(*, "variables")= language list(pm10, no2) .. ..- attr(*, "factors")= int [1:2, 1] 0 1 .. .. ..- attr(*, "dimnames")=List of 2 .. .. .. ..$ : chr [1:2] "pm10" "no2" .. .. .. ..$ : chr "no2" .. ..- attr(*, "term.labels")= chr "no2" .. ..- attr(*, "order")= int 1 .. ..- attr(*, "intercept")= int 1 .. ..- attr(*, "response")= int 1 .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv> .. ..- attr(*, "predvars")= language list(pm10, no2) .. ..- attr(*, "dataClasses")= Named chr [1:2] "numeric" "numeric" .. .. ..- attr(*, "names")= chr [1:2] "pm10" "no2" $ model :'data.frame': 3810 obs. of 2 variables: ..$ pm10: int [1:3810] 260 249 298 246 227 251 205 198 195 264 ... ..$ no2 : int [1:3810] 19 16 21 12 15 16 13 9 8 9 ... ..- attr(*, "terms")=Classes 'terms', 'formula' language pm10 ~ no2 .. .. ..- attr(*, "variables")= language list(pm10, no2) .. .. ..- attr(*, "factors")= int [1:2, 1] 0 1 .. .. .. ..- attr(*, "dimnames")=List of 2 .. .. .. .. ..$ : chr [1:2] "pm10" "no2" .. .. .. .. ..$ : chr "no2" .. .. ..- attr(*, "term.labels")= chr "no2" .. .. ..- attr(*, "order")= int 1 .. .. ..- attr(*, "intercept")= int 1 .. .. ..- attr(*, "response")= int 1 .. .. ..- attr(*, ".Environment")=<environment: R_GlobalEnv> .. .. ..- attr(*, "predvars")= language list(pm10, no2) .. .. ..- attr(*, "dataClasses")= Named chr [1:2] "numeric" "numeric" .. .. .. ..- attr(*, "names")= chr [1:2] "pm10" "no2" ..- attr(*, "na.action")= 'omit' Named int [1:118] 14 27 166 237 238 406 434 455 456 458 ... .. ..- attr(*, "names")= chr [1:118] "NA" "NA.1" "NA.2" "NA.3" ... - attr(*, "class")= chr "lm" ``` ] --- * Individual elements can be extracted using list indexing ```r fm2$coefficients ``` ``` (Intercept) no2 52.011396 4.228562 ``` ```r fm2$residuals ``` ``` 1 2 3 4 5 6 127.64591729 129.33160471 157.18879234 143.24585460 111.56016718 131.33160471 7 8 9 10 11 12 98.01729213 107.93154203 109.16010450 173.93154203 98.61722945 414.61722945 13 15 16 17 18 19 255.07435439 337.41735482 653.33160471 55.58885503 113.64591729 36.27454245 20 21 22 23 24 25 95.90316761 104.04597998 -2.49689508 73.58885503 -16.72545755 -3.95402002 26 28 29 30 31 32 9.27454245 17.70297955 19.47441708 27.93154203 26.16010450 11.38866697 33 34 35 36 37 38 13.93154203 9.93154203 3.70297955 4.93154203 1.70297955 4.56016718 39 40 41 42 43 44 14.56016718 22.33160471 7.10304224 32.41735482 60.87447976 54.56016718 45 46 47 48 49 50 46.33160471 76.01729213 82.78872966 51.33160471 78.33160471 -16.06814455 51 52 53 54 55 56 12.44604266 54.76035524 31.13173008 19.27454245 11.73166740 60.58885503 57 58 59 60 61 62 134.64591729 171.47441708 -30.21127034 -51.66839529 -67.43983282 -63.12552024 63 64 65 66 67 68 -49.75414540 -57.29702045 -54.98270787 -61.12552024 -51.35408271 -31.43983282 69 70 71 72 73 74 -45.21127034 -66.58264518 -60.35408271 -49.66839529 -24.06845797 -22.06845797 75 76 77 78 79 80 -60.12552024 -19.21127034 -20.06845797 -22.06845797 -34.98270787 -35.98270787 81 82 83 84 85 86 -34.52558292 -44.43983282 -35.98270787 -41.29702045 -31.06845797 -32.52558292 87 88 89 90 91 92 -41.52558292 -53.83989550 -52.98270787 -37.83989550 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44.18879234 214 215 216 217 218 219 -35.58264518 133.10304224 68.56016718 -42.26833260 -60.95402002 -56.46820723 220 221 222 223 224 225 -51.32539487 -23.26833260 73.93154203 -2.66839529 -39.41114497 -11.81120766 226 227 228 229 230 231 93.78872966 48.24585460 6.01729213 -11.12552024 -13.35408271 33.53147934 232 233 234 235 236 239 -14.83989550 17.84579192 15.07435439 -9.69708313 -4.69708313 51.73166740 240 241 242 243 244 245 51.96022987 150.47441708 71.33160471 78.93154203 94.53147934 199.47441708 246 247 248 249 250 251 82.93154203 45.56016718 79.41735482 35.18879234 315.44604266 378.27454245 252 253 254 255 256 257 123.73166740 119.87447976 72.64591729 15.41735482 22.10304224 50.87447976 258 259 260 261 262 263 51.10304224 67.87447976 90.96022987 205.50310492 113.73166740 300.04597998 264 265 266 267 268 269 120.41735482 114.50310492 106.50310492 115.50310492 148.33160471 -25.52558292 270 271 272 273 274 275 -27.35408271 -28.72545755 -14.26833260 -25.43983282 -37.58264518 -32.26833260 276 277 278 279 280 281 -28.12552024 -46.49689508 17.36029255 -9.63970745 -7.55395734 45.90316761 282 283 284 285 286 287 27.50310492 -1.81120766 25.64591729 37.10304224 75.70297955 59.38866697 288 289 290 291 292 293 41.93154203 45.16010450 0.70297955 -2.29702045 27.24585460 23.70297955 294 295 296 297 298 299 23.47441708 7.93154203 11.38866697 9.93154203 25.70297955 0.07435439 300 301 302 303 304 305 22.93154203 -12.52558292 -7.52558292 -16.98270787 -13.89695776 -15.43983282 306 307 308 309 310 311 -17.21127034 -2.66839529 25.47441708 3.01729213 -3.75414540 1.78872966 312 313 314 315 316 317 -14.52558292 -5.43983282 3.33160471 21.78872966 26.78872966 40.78872966 318 319 320 321 322 323 22.16010450 46.61722945 84.53147934 4.70297955 45.61722945 3.38866697 324 325 326 327 328 329 13.16010450 11.16010450 9.47441708 33.70297955 -4.03977013 -76.18258250 330 331 332 333 334 335 -31.81120766 -39.26833260 -36.72545755 -15.72545755 -23.18258250 -8.58264518 336 337 338 339 340 341 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-32.38245713 3397 3398 3399 3400 3401 3402 49.53179277 -12.06814455 30.84610535 -34.72514412 -19.43951939 577.38898040 3404 3405 3406 3407 3408 3409 69.96054330 188.98891771 152.36029255 84.98891771 162.04597998 174.18879234 3410 3411 3412 3413 3414 3415 108.04597998 99.73166740 121.73166740 129.78872966 169.04597998 111.58885503 3416 3417 3418 3419 3420 3421 151.90316761 176.81741750 173.96022987 231.90316761 214.36029255 83.50310492 3422 3423 3424 3425 3426 3427 123.96022987 -11.86826992 196.58885503 194.27454245 238.36029255 108.64591729 3428 3429 3430 3431 3432 3433 143.90316761 129.73166740 81.81741750 108.98891771 10.53179277 73.21748019 3434 3435 3436 3437 3438 3439 96.56048061 84.16041793 111.73198082 58.30323029 53.96022987 132.50310492 3440 3441 3442 3443 3444 3445 -15.86826992 42.58885503 74.21748019 22.36029255 16.50310492 95.81741750 3446 3447 3448 3449 3452 3453 188.44604266 71.67460513 118.50310492 256.27454245 -103.09683239 17.78872966 3454 3455 3456 3457 3458 3459 34.01729213 67.24585460 353.33160471 56.24585460 21.18879234 241.78872966 3460 3461 3462 3463 3464 3465 170.96022987 307.18879234 112.24585460 42.96022987 -4.95402002 294.13173008 3467 3468 3469 3470 3471 3472 -97.55395734 -63.72545755 -41.72545755 -24.41114497 -45.41114497 -23.95402002 3473 3474 3475 3476 3477 3478 102.27454245 96.87447976 220.33160471 79.18879234 -12.35408271 -19.66839529 3479 3480 3481 3482 3483 3484 -49.26833260 -36.12552024 -83.49689508 -72.12552024 -100.81120766 -73.58264518 3485 3486 3487 3488 3489 3490 -47.12552024 -52.26833260 -39.95402002 -43.12552024 -80.35408271 -59.81120766 3493 3494 3495 3497 3498 3500 -52.81120766 -67.35408271 -75.12552024 -31.09683239 -32.72545755 -37.12552024 3501 3502 3503 3504 3505 3506 -55.35408271 -80.26833260 -42.43983282 -3.52558292 -0.06845797 -44.06845797 3507 3509 3510 3511 3513 3514 -53.29702045 6.13173008 20.18879234 -11.03977013 -47.03977013 -52.81120766 3515 3516 3518 3519 3520 3521 -97.32539487 -86.86826992 -59.21127034 -55.98270787 -53.29702045 -74.43983282 3522 3523 3524 3525 3526 3527 -48.89695776 21.47441708 -43.21127034 -36.66839529 -52.89695776 -55.58264518 3528 3530 3531 3532 3533 3534 -21.75414540 -45.75414540 -32.03977013 -20.98270787 -15.21127034 -35.12552024 3535 3537 3538 3539 3540 3542 -18.21127034 19.47441708 17.56016718 -13.52558292 11.70297955 -21.43983282 3543 3544 3545 3546 3547 3548 -1.52558292 20.73166740 1.01729213 47.24585460 -22.81120766 81.10304224 3549 3550 3551 3552 3553 3554 13.56016718 1.16010450 6.56016718 -1.43983282 9.64591729 6.58885503 3555 3556 3557 3558 3559 3560 6.18879234 -17.35408271 1.47473050 24.36029255 -11.03977013 -34.21127034 3561 3563 3564 3565 3566 3567 -11.75414540 35.78872966 14.10304224 119.73166740 97.50310492 71.64591729 3568 3569 3570 3571 3572 3573 43.64591729 111.18879234 116.50310492 86.81741750 133.73166740 137.33160471 3574 3575 3576 3577 3578 3579 43.47441708 99.33160471 3.33160471 -33.81120766 0.50310492 95.41735482 3580 3581 3582 3583 3584 3585 25.41735482 108.41735482 -17.43983282 40.10304224 -38.98270787 -31.75414540 3586 3587 3588 3589 3590 3591 7.78872966 18.01729213 39.10304224 20.24585460 5.47441708 5.78872966 3592 3593 3594 3595 3596 3597 7.56016718 36.64591729 55.01729213 -13.98270787 -8.75414540 -41.98270787 3598 3599 3600 3601 3602 3603 -16.49689508 23.96022987 4.04597998 221.18879234 49.16010450 15.38866697 3604 3605 3606 3607 3608 3610 51.24585460 45.33160471 26.41735482 37.10304224 30.78872966 58.78872966 3611 3612 3613 3614 3616 3617 64.47441708 78.70297955 32.16010450 16.16010450 12.01729213 27.38866697 3619 3621 3622 3623 3624 3625 0.96022987 -32.86826992 -65.21095692 -65.41083155 144.67491856 296.07498125 3626 3627 3628 3629 3630 3631 43.87479319 69.38898040 78.27485588 228.36060598 275.13204351 -14.92533218 3632 3633 3635 3636 3637 3639 37.21748019 254.30291687 19.18941920 275.53179277 137.56016718 -5.98270787 3640 3641 3642 3643 3644 3645 -9.29702045 -0.75414540 21.47441708 28.47441708 32.24585460 26.41735482 3646 3647 3648 3649 3650 3651 56.87447976 100.18879234 168.27454245 143.27454245 212.33160471 98.18879234 3652 3653 3654 3655 3656 3657 226.18879234 57.64591729 46.93154203 31.58885503 39.16010450 39.84579192 3658 3659 3660 3661 3662 3663 2.87447976 133.41735482 95.64591729 110.10304224 71.10304224 -23.98270787 3664 3665 3666 3667 3668 3669 51.10304224 71.47441708 27.70297955 101.93154203 189.04597998 216.13173008 3670 3671 3672 3673 3674 3675 105.33160471 226.73166740 561.96022987 -37.18258250 50.90316761 36.07466782 3676 3677 3678 3679 3680 3681 -11.35408271 -76.03977013 -47.35408271 -24.66839529 -5.26833260 -25.41114497 3682 3683 3684 3685 3686 3687 -11.72545755 -38.26833260 -26.49689508 -31.26833260 -26.26833260 -92.49689508 3688 3689 3690 3691 3692 3693 -31.21127034 -80.26833260 -31.89695776 -2.12552024 14.58885503 -4.03977013 3694 3695 3696 3697 3698 3699 26.33160471 -7.52558292 -25.89695776 -43.72545755 -29.86826992 78.13173008 3700 3701 3702 3703 3704 3705 34.73166740 -54.58264518 -16.26833260 -117.52526950 -69.32539487 -60.55395734 3706 3707 3708 3709 3710 3711 -37.58264518 -20.98270787 -23.12552024 14.27454245 198.58885503 -33.98270787 3712 3713 3714 3715 3716 3717 -61.12552024 -4.75414540 -34.98270787 -10.26833260 443.84579192 -123.03977013 3718 3719 3720 3721 3722 3723 -57.29702045 -56.06845797 -60.81120766 -19.92564561 -12.15420808 -25.83989550 3724 3725 3728 3729 3730 3731 -38.58264518 593.38866697 -49.81120766 -62.95402002 -58.72545755 -52.03977013 3732 3733 3734 3735 3736 3737 -41.12552024 -44.72545755 -65.09683239 -74.78251981 -29.63970745 0.81741750 3738 3739 3740 3741 3742 3743 -14.09683239 -36.21127034 -44.43983282 -77.26833260 -60.43983282 -58.89695776 3744 3745 3746 3747 3748 3749 -42.58264518 90.56016718 142.10304224 165.56016718 137.56016718 73.56016718 3750 3751 3752 3753 3754 3755 2.64591729 -89.72545755 -76.12552024 -76.35408271 -52.03977013 -64.26833260 3756 3757 3758 3759 3760 3761 -61.12552024 -62.03977013 -45.26833260 -77.89695776 -82.12552024 -60.75414540 3762 3763 3764 3765 3766 3767 -66.43983282 -68.12552024 -61.66839529 -58.43983282 -67.12552024 -61.12552024 3768 3769 3770 3771 3772 3773 -56.12552024 -83.35408271 -73.35408271 -76.89695776 -59.98270787 -62.52558292 3774 3775 3776 3777 3778 3779 -53.52558292 -70.21127034 -76.43983282 -80.35408271 -65.66839529 -81.21127034 3780 3781 3782 3783 3784 3785 -70.83989550 -67.38277055 -63.12552024 -44.66839529 -35.12552024 -55.58264518 3786 3787 3788 3789 3790 3791 -41.89695776 -47.58264518 -57.21127034 -70.43983282 -65.12552024 -57.66839529 3792 3793 3794 3795 3796 3797 -50.21127034 -52.66839529 -66.26833260 -74.58264518 -51.72545755 -50.72545755 3798 3799 3800 3801 3802 3803 -51.43983282 -50.43983282 -51.21127034 -51.89695776 -37.58264518 -31.95402002 3804 3805 3806 3807 3808 3809 -34.12552024 -51.21127034 -50.75414540 -46.75414540 -55.03977013 -67.72545755 3810 3811 3812 3813 3814 3815 -52.43983282 -59.98270787 -57.66839529 -35.21127034 -72.89695776 -63.66839529 3816 3817 3818 3819 3820 3824 -60.12552024 -73.12552024 -59.98270787 -27.69708313 -26.38277055 -68.86826992 3825 3826 3827 3828 3829 3830 -64.92533218 -64.15389465 -31.69676971 -4.55395734 -1.09683239 -67.55395734 3831 3832 3833 3834 3835 3836 -36.32539487 -62.41114497 -31.09683239 -50.49689508 -67.21127034 -91.72545755 3837 3838 3839 3840 3841 3842 -75.86826992 -68.23964476 -89.46820723 -53.38245713 -55.38245713 -36.01108229 3843 3844 3845 3846 3847 3848 1.36029255 -48.46820723 -2.86826992 -32.55395734 -109.46789381 -50.29670702 3849 3850 3851 3852 3853 3854 -51.01108229 -81.63970745 -68.78251981 -45.32539487 -47.46820723 1.76035524 3855 3856 3857 3858 3859 3860 93.44604266 17.73166740 -9.32539487 63.36029255 29.21748019 -60.03977013 3861 3862 3863 3864 3865 3866 -72.81120766 -16.29702045 -53.12552024 253.96022987 -71.69676971 -81.63970745 3867 3868 3869 3870 3871 3872 -69.86826992 -235.21064349 -45.18258250 -46.58264518 44.24585460 32.10304224 3873 3874 3875 3876 3877 3878 38.98891771 2.41735482 19.67460513 -36.09683239 9.07466782 89.16041793 3879 3880 3881 3882 3883 3884 140.44604266 26.27454245 21.36029255 54.81741750 -68.35408271 -76.89695776 3885 3886 3887 3888 3889 3890 -79.03977013 -66.35408271 -95.72545755 -74.35408271 -77.12552024 -79.49689508 3891 3892 3893 3894 3895 3896 -84.81120766 -72.41114497 -31.18258250 -14.21095692 190.10335566 26.07466782 3897 3898 3899 3900 3901 3902 84.53179277 -141.09683239 72.93154203 18.24585460 -38.09683239 -84.78251981 3903 3904 3905 3906 3907 3908 -7.21127034 93.90348103 -17.23964476 -137.63970745 -53.43983282 -91.98239444 3909 3910 3911 3912 3913 3914 -22.89695776 186.56048061 -67.26833260 -52.89695776 -72.58264518 58.27454245 3915 3916 3917 3918 3919 3920 -73.18258250 -58.66839529 -4.29702045 15.01729213 -23.21127034 -19.26833260 3921 3922 3923 3924 3925 3926 103.81741750 8.56016718 8.56016718 192.36029255 -43.29670702 -0.03977013 ``` --- layout: true # Attributes --- * The names (derived from row names of the data) can be used as index ```r fm2$residuals["25"] ``` ``` 25 -3.95402 ``` * The names associated with a vector can be obtained using `names()` ```r names(fm2$residuals) ``` ``` [1] "1" "2" "3" "4" "5" "6" "7" "8" "9" "10" "11" "12" [13] "13" "15" "16" "17" "18" "19" "20" "21" "22" "23" "24" "25" [25] "26" "28" "29" "30" "31" "32" "33" "34" "35" "36" "37" "38" [37] "39" "40" "41" "42" "43" "44" "45" "46" "47" "48" "49" "50" [49] "51" "52" "53" "54" "55" "56" "57" "58" "59" "60" "61" "62" [61] "63" "64" "65" "66" "67" "68" "69" "70" "71" "72" "73" "74" [73] "75" "76" "77" "78" "79" "80" "81" "82" "83" "84" "85" "86" [85] "87" "88" "89" "90" "91" "92" "93" "94" "95" "96" "97" "98" [97] "99" "100" "101" "102" "103" "104" "105" "106" "107" "108" "109" "110" [109] "111" "112" "113" "114" "115" "116" "117" "118" "119" "120" "121" "122" [121] "123" "124" "125" "126" "127" "128" "129" "130" "131" "132" "133" "134" [133] "135" "136" "137" "138" "139" "140" "141" "142" "143" "144" "145" "146" [145] "147" "148" "149" "150" "151" "152" "153" "154" "155" "156" "157" "158" [157] "159" "160" "161" "162" "163" "164" "165" "167" "168" "169" "170" "171" [169] "172" "173" "174" "175" "176" "177" "178" "179" "180" "181" "182" "183" [181] "184" "185" "186" "187" "188" "189" "190" "191" "192" "193" "194" "195" [193] "196" "197" "198" "199" "200" "201" "202" "203" "204" "205" "206" "207" [205] "208" "209" "210" "211" "212" "213" "214" "215" "216" "217" "218" "219" [217] "220" "221" "222" "223" "224" "225" "226" "227" "228" "229" "230" "231" [229] "232" "233" "234" "235" "236" "239" "240" "241" "242" "243" "244" "245" [241] "246" "247" "248" "249" "250" "251" "252" "253" "254" "255" "256" "257" [253] "258" "259" "260" "261" "262" "263" "264" "265" "266" "267" "268" "269" [265] "270" "271" "272" "273" "274" "275" "276" "277" "278" "279" "280" "281" [277] "282" "283" "284" "285" "286" "287" "288" "289" "290" "291" "292" "293" [289] "294" "295" "296" "297" "298" "299" "300" "301" "302" "303" "304" "305" [301] "306" "307" "308" "309" "310" "311" "312" "313" "314" "315" "316" "317" [313] "318" "319" "320" "321" "322" "323" "324" "325" "326" "327" "328" "329" [325] "330" "331" "332" "333" "334" "335" "336" "337" "338" "339" "340" "341" [337] "342" "343" "344" "345" "346" "347" "348" "349" "350" "351" "352" "353" [349] "354" "355" "356" "357" "358" "359" "360" "361" "362" "363" "364" "365" [361] "366" "367" "368" "369" "370" "371" "372" "373" "374" "375" "376" "377" [373] "378" "379" "380" "381" "382" "383" "384" "385" "386" "387" "388" "389" [385] "390" "391" "392" "393" "394" "395" "396" "397" "398" "399" "400" "401" [397] "402" "403" "404" "405" "407" "408" "409" "410" "411" "412" "413" "414" [409] "415" "416" "417" "418" "419" "420" "421" "422" "423" "424" "425" "426" [421] "427" "428" "429" "430" "431" "432" "433" "435" "436" "437" "438" "439" [433] "440" "441" "442" "443" "444" "445" "446" "447" "448" "449" "450" "451" [445] "452" "453" "454" "458" "460" "461" "462" "463" "465" "466" "467" "468" [457] "469" "470" "471" "472" "473" "474" "475" "476" "477" "478" "479" "480" [469] "481" "482" "483" "484" "485" "486" "487" "488" "489" "490" "491" "492" [481] "493" "494" "495" "496" "497" "498" "499" "500" "501" "502" "503" "504" [493] "505" "506" "507" "508" "509" "510" "511" "512" "513" "514" "515" "516" [505] "517" "518" "519" "520" "521" "522" "523" "524" "525" "526" "527" "528" [517] "529" "530" "531" "532" "533" "534" "535" "536" "537" "538" "539" "540" [529] "541" "542" "543" "544" "545" "546" "547" "548" "549" "550" "551" "552" [541] "553" "554" "555" "556" "557" "558" "559" "560" "561" "562" "563" "564" [553] "565" "566" "567" "568" "569" "570" "571" "572" "573" "574" "575" "576" [565] "577" "578" "579" "580" "581" "582" "585" "586" "587" "588" "589" "590" [577] "591" "592" "593" "594" "595" "596" "597" "598" "599" "600" "601" "602" [589] "603" "604" "605" "606" "607" "608" "609" "610" "611" "612" "613" "614" [601] "615" "616" "617" "618" "619" "620" "621" "622" "623" "624" "625" "626" [613] "627" "628" "629" "630" "631" "632" "633" "634" "635" "636" "637" "638" [625] "639" "640" "641" "642" "643" "644" "645" "646" "647" "648" "649" "650" [637] "651" "652" "653" "654" "655" "656" "657" "658" "659" "660" "661" "662" [649] "663" "664" "665" "666" "667" "668" "669" "670" "671" "672" "673" "674" [661] "675" "676" "677" "678" "679" "680" "681" "682" "683" "684" "685" "686" [673] "687" "688" "689" "690" "691" "692" "693" "694" "695" "696" "697" "698" [685] "699" "700" "701" "702" "703" "704" "705" "706" "707" "708" "709" "710" [697] "711" "712" "713" "714" "715" "716" "717" "718" "719" "720" "721" "722" [709] "723" "724" "725" "726" "727" "728" "729" "730" "731" "732" "733" "734" [721] "735" "736" "737" "738" "739" "740" "741" "742" "743" "744" "745" "746" [733] "747" "748" "749" "750" "751" "752" "753" "754" "755" "756" "757" "758" [745] "759" "760" "761" "762" "763" "764" "765" "766" "767" "768" "769" "770" [757] "771" "772" "773" "774" "775" "776" "777" "778" "779" "780" "781" "782" [769] "783" "784" "785" "786" "787" "788" "789" "790" "791" "792" "793" "794" [781] "795" "796" "797" "798" "799" "800" "801" "802" "803" "804" "805" "806" [793] "807" "808" "809" "810" "811" "812" "813" "814" "815" "816" "817" "818" [805] "819" "820" "821" "822" "823" "824" "825" "826" "827" "828" "829" "830" [817] "831" "832" "833" "834" "835" "836" "837" "838" "839" "840" "841" "842" [829] "843" "844" "845" "846" "847" "848" "849" "850" "851" "852" "853" "854" [841] "855" "856" "857" "858" "859" "860" "861" "862" "863" "864" "865" "866" [853] "867" "868" "869" "870" "871" "872" "873" "874" "875" "876" "877" "878" [865] "879" "880" "881" "882" "883" "884" "885" "886" "887" "888" "889" "890" [877] "891" "892" "893" "894" "895" "896" "897" "898" "899" "900" "901" "902" [889] "903" "904" "905" "906" "907" "908" "909" "910" "911" "912" "913" "914" [901] "915" "916" "917" "918" "919" "920" "921" 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"2308" "2309" "2310" "2311" "2312" "2313" [2245] "2314" "2315" "2316" "2317" "2318" "2319" "2320" "2321" "2322" "2323" "2324" "2325" [2257] "2326" "2327" "2328" "2329" "2330" "2331" "2332" "2333" "2334" "2335" "2336" "2337" [2269] "2338" "2339" "2340" "2341" "2342" "2343" "2344" "2345" "2346" "2347" "2348" "2349" [2281] "2350" "2351" "2352" "2353" "2354" "2355" "2356" "2357" "2358" "2359" "2360" "2361" [2293] "2362" "2363" "2364" "2365" "2366" "2367" "2368" "2369" "2370" "2371" "2372" "2373" [2305] "2374" "2375" "2376" "2377" "2378" "2379" "2380" "2381" "2382" "2383" "2384" "2385" [2317] "2386" "2387" "2388" "2389" "2390" "2391" "2392" "2393" "2394" "2395" "2396" "2397" [2329] "2398" "2399" "2400" "2401" "2402" "2403" "2404" "2405" "2406" "2407" "2408" "2409" [2341] "2410" "2411" "2412" "2413" "2414" "2415" "2416" "2417" "2418" "2419" "2420" "2421" [2353] "2422" "2423" "2424" "2425" "2426" "2427" "2428" "2429" "2430" "2431" "2432" "2433" [2365] "2434" "2435" "2436" "2437" "2438" "2439" "2440" "2441" "2442" "2443" "2444" "2445" [2377] "2446" "2447" "2448" "2449" "2450" "2451" "2452" "2453" "2454" "2455" "2456" "2457" [2389] "2458" "2459" "2460" "2461" "2462" "2463" "2464" "2465" "2466" "2467" "2468" "2469" [2401] "2470" "2471" "2472" "2473" "2474" "2475" "2476" "2477" "2478" "2479" "2480" "2481" [2413] "2482" "2483" "2484" "2485" "2486" "2487" "2488" "2489" "2490" "2491" "2492" "2493" [2425] "2494" "2495" "2496" "2497" "2498" "2499" "2500" "2501" "2502" "2503" "2505" "2506" [2437] "2507" "2508" "2510" "2511" "2512" "2513" "2514" "2515" "2516" "2517" "2518" "2519" [2449] "2520" "2521" "2522" "2523" "2524" "2525" "2526" "2527" "2528" "2529" "2530" "2531" [2461] "2532" "2533" "2534" "2535" "2536" "2537" "2538" "2539" "2540" "2541" "2543" "2544" [2473] "2545" "2546" "2547" "2548" "2549" "2550" "2551" "2552" "2553" "2554" "2555" "2556" [2485] "2557" "2558" "2559" "2560" "2561" "2562" "2563" "2564" "2565" "2566" "2567" "2568" [2497] "2569" "2570" "2571" "2572" "2573" "2574" "2575" "2576" "2577" "2578" "2579" "2580" [2509] "2581" "2582" "2583" "2584" "2585" "2586" "2587" "2588" "2589" "2590" "2591" "2592" [2521] "2593" "2594" "2595" "2596" "2597" "2598" "2599" "2600" "2601" "2602" "2603" "2604" [2533] "2605" "2606" "2607" "2608" "2609" "2610" "2611" "2612" "2613" "2614" "2615" "2616" [2545] "2617" "2618" "2619" "2620" "2621" "2622" "2623" "2624" "2625" "2626" "2627" "2628" [2557] "2629" "2630" "2631" "2632" "2633" "2634" "2635" "2636" "2637" "2638" "2639" "2640" [2569] "2641" "2642" "2643" "2644" "2645" "2646" "2647" "2648" "2649" "2650" "2651" "2652" [2581] "2653" "2654" "2655" "2656" "2657" "2658" "2659" "2660" "2661" "2662" "2663" "2664" [2593] "2665" "2666" "2667" "2668" "2669" "2670" "2671" "2672" "2673" "2674" "2675" "2676" [2605] "2677" "2678" "2679" "2680" "2681" "2682" "2683" "2684" "2685" "2686" "2687" "2688" [2617] "2689" "2690" "2691" "2692" "2693" "2694" "2695" "2696" "2697" "2698" "2699" "2700" [2629] "2701" "2702" "2703" "2704" "2705" "2706" "2707" "2708" "2709" "2710" "2711" "2712" [2641] "2713" "2714" "2715" "2716" "2717" "2718" "2719" "2720" "2721" "2722" "2723" "2724" [2653] "2725" "2726" "2727" "2728" "2729" "2730" "2731" "2732" "2733" "2734" "2735" "2736" [2665] "2737" "2738" "2739" "2740" "2741" "2742" "2743" "2744" "2745" "2746" "2747" "2748" [2677] "2749" "2750" "2751" "2752" "2753" "2754" "2755" "2756" "2757" "2758" "2759" "2760" [2689] "2761" "2762" "2763" "2764" "2765" "2766" "2767" "2768" "2769" "2770" "2771" "2772" [2701] "2773" "2774" "2775" "2776" "2777" "2778" "2779" "2780" "2781" "2782" "2783" "2784" [2713] "2785" "2786" "2787" "2788" "2789" "2790" "2791" "2792" "2793" "2794" "2795" "2796" [2725] "2797" "2798" "2799" "2800" "2801" "2802" "2803" "2804" "2805" "2806" "2807" "2808" [2737] "2809" "2810" "2811" "2812" "2813" "2814" "2815" "2816" "2817" "2818" "2819" "2820" [2749] "2821" "2822" "2823" "2824" "2825" "2826" "2827" "2828" "2829" "2830" "2831" "2832" [2761] "2833" "2834" "2835" "2836" "2837" "2838" "2839" "2840" "2841" "2842" "2843" "2844" [2773] "2845" "2846" "2847" "2848" "2849" "2850" "2851" "2852" "2853" "2854" "2855" "2856" [2785] "2857" "2858" "2859" "2860" "2861" "2862" "2863" "2864" "2865" "2866" "2867" "2868" [2797] "2869" "2870" "2871" "2872" "2873" "2874" "2875" "2876" "2877" "2878" "2879" "2880" [2809] "2881" "2882" "2883" "2884" "2885" "2886" "2887" "2888" "2889" "2890" "2891" "2892" [2821] "2893" "2894" "2895" "2896" "2897" "2898" "2899" "2900" "2901" "2902" "2903" "2904" [2833] "2905" "2906" "2907" "2908" "2909" "2910" "2911" "2912" "2913" "2914" "2915" "2916" [2845] "2917" "2918" "2919" "2920" "2921" "2922" "2923" "2924" "2925" "2926" "2927" "2928" [2857] "2929" "2930" "2931" "2932" "2933" "2934" "2935" "2936" "2937" "2938" "2939" "2940" [2869] "2941" "2943" "2944" "2945" "2946" "2947" "2948" "2949" "2950" "2951" "2952" "2953" [2881] "2954" "2955" "2956" "2957" "2958" "2959" "2960" "2961" "2962" "2963" "2964" "2965" [2893] "2966" "2967" "2968" "2969" "2970" "2971" "2972" "2973" "2974" "2975" "2976" "2977" [2905] "2978" "2979" "2980" "2981" "2982" "2983" "2984" "2985" "2986" "2987" "2988" "2989" [2917] "2990" "2991" "2992" "2993" "2994" "2995" "2996" "2997" "2998" "2999" "3000" "3001" [2929] "3002" "3003" "3004" "3005" "3006" "3007" "3008" "3009" "3010" "3011" "3012" "3013" [2941] "3014" "3015" "3016" "3017" "3018" "3019" "3020" "3021" "3022" "3023" "3024" "3025" [2953] "3026" "3027" "3028" "3029" "3030" "3031" "3032" "3033" "3034" "3035" "3036" "3037" [2965] "3038" "3039" "3040" "3041" "3042" "3043" "3044" "3045" "3046" "3047" "3048" "3049" [2977] "3051" "3052" "3054" "3055" "3056" "3057" "3058" "3059" "3060" "3061" "3062" "3063" [2989] "3064" "3065" "3066" "3067" "3068" "3069" "3070" "3071" "3072" "3073" "3074" "3075" [3001] "3076" "3077" "3079" "3080" "3081" "3082" "3083" "3084" "3085" "3086" "3087" "3088" [3013] "3089" "3090" "3091" "3092" "3093" "3094" "3095" "3096" "3097" "3098" "3099" "3100" [3025] "3101" "3102" "3103" "3104" "3105" "3106" "3107" "3108" "3109" "3110" "3111" "3112" [3037] "3113" "3114" "3115" "3116" "3117" "3118" "3119" "3121" "3122" "3123" "3124" "3126" [3049] "3127" "3128" "3129" "3130" "3131" "3132" "3133" "3134" "3135" "3136" "3137" "3138" [3061] "3139" "3140" "3141" "3142" "3143" "3144" "3145" "3146" "3147" "3148" "3149" "3150" [3073] "3151" "3152" "3153" "3154" "3155" "3156" "3157" "3158" "3159" "3160" "3161" "3162" [3085] "3163" "3164" "3165" "3166" "3167" "3168" "3169" "3170" "3171" "3172" "3173" "3174" [3097] "3175" "3176" "3177" "3178" "3180" "3181" "3182" "3183" "3184" "3186" "3187" "3188" [3109] "3189" "3191" "3192" "3193" "3194" "3195" "3196" "3197" "3198" "3199" "3200" "3201" [3121] "3202" "3204" "3205" "3206" "3207" "3208" "3209" "3210" "3211" "3212" "3213" "3214" [3133] "3215" "3216" "3217" "3218" "3219" "3220" "3221" "3222" "3223" "3224" "3225" "3226" [3145] "3227" "3228" "3229" "3230" "3231" "3232" "3233" "3234" "3235" "3236" "3237" "3238" [3157] "3239" "3240" "3241" "3242" "3243" "3244" "3245" "3246" "3247" "3250" "3251" "3252" [3169] "3253" "3254" "3255" "3256" "3257" "3258" "3259" "3260" "3261" "3262" "3263" "3264" [3181] "3265" "3266" "3267" "3268" "3269" "3270" "3271" "3272" "3273" "3274" "3276" "3277" [3193] "3278" "3279" "3280" "3281" "3282" "3283" "3284" "3285" "3286" "3287" "3288" "3289" [3205] "3290" "3291" "3292" "3293" "3294" "3295" "3296" "3297" "3298" "3299" "3300" "3301" [3217] "3302" "3303" "3304" "3305" "3306" "3307" "3308" "3309" "3310" "3311" "3312" "3313" [3229] "3314" "3315" "3316" "3317" "3318" "3319" "3320" "3321" "3322" "3323" "3324" "3325" [3241] "3326" "3327" "3328" "3329" "3330" "3331" "3332" "3333" "3334" "3335" "3337" "3338" [3253] "3339" "3340" "3341" "3342" "3343" "3344" "3345" "3346" "3347" "3348" "3349" "3350" [3265] "3351" "3352" "3353" "3355" "3356" "3357" "3358" "3359" "3360" "3361" "3362" "3363" [3277] "3364" "3365" "3366" "3367" "3368" "3369" "3370" "3371" "3372" "3373" "3374" "3375" [3289] "3376" "3377" "3378" "3379" "3380" "3382" "3383" "3384" "3386" "3387" "3388" "3389" [3301] "3390" "3392" "3393" "3394" "3395" "3396" "3397" "3398" "3399" "3400" "3401" "3402" [3313] "3404" "3405" "3406" "3407" "3408" "3409" "3410" "3411" "3412" "3413" "3414" "3415" [3325] "3416" "3417" "3418" "3419" "3420" "3421" "3422" "3423" "3424" "3425" "3426" "3427" [3337] "3428" "3429" "3430" "3431" "3432" "3433" "3434" "3435" "3436" "3437" "3438" "3439" [3349] "3440" "3441" "3442" "3443" "3444" "3445" "3446" "3447" "3448" "3449" "3452" "3453" [3361] "3454" "3455" "3456" "3457" "3458" "3459" "3460" "3461" "3462" "3463" "3464" "3465" [3373] "3467" "3468" "3469" "3470" "3471" "3472" "3473" "3474" "3475" "3476" "3477" "3478" [3385] "3479" "3480" "3481" "3482" "3483" "3484" "3485" "3486" "3487" "3488" "3489" "3490" [3397] "3493" "3494" "3495" "3497" "3498" "3500" "3501" "3502" "3503" "3504" "3505" "3506" [3409] "3507" "3509" "3510" "3511" "3513" "3514" "3515" "3516" "3518" "3519" "3520" "3521" [3421] "3522" "3523" "3524" "3525" "3526" "3527" "3528" "3530" "3531" "3532" "3533" "3534" [3433] "3535" "3537" "3538" "3539" "3540" "3542" "3543" "3544" "3545" "3546" "3547" "3548" [3445] "3549" "3550" "3551" "3552" "3553" "3554" "3555" "3556" "3557" "3558" "3559" "3560" [3457] "3561" "3563" "3564" "3565" "3566" "3567" "3568" "3569" "3570" "3571" "3572" "3573" [3469] "3574" "3575" "3576" "3577" "3578" "3579" "3580" "3581" "3582" "3583" "3584" "3585" [3481] "3586" "3587" "3588" "3589" "3590" "3591" "3592" "3593" "3594" "3595" "3596" "3597" [3493] "3598" "3599" "3600" "3601" "3602" "3603" "3604" "3605" "3606" "3607" "3608" "3610" [3505] "3611" "3612" "3613" "3614" "3616" "3617" "3619" "3621" "3622" "3623" "3624" "3625" [3517] "3626" "3627" "3628" "3629" "3630" "3631" "3632" "3633" "3635" "3636" "3637" "3639" [3529] "3640" "3641" "3642" "3643" "3644" "3645" "3646" "3647" "3648" "3649" "3650" "3651" [3541] "3652" "3653" "3654" "3655" "3656" "3657" "3658" "3659" "3660" "3661" "3662" "3663" [3553] "3664" "3665" "3666" "3667" "3668" "3669" "3670" "3671" "3672" "3673" "3674" "3675" [3565] "3676" "3677" "3678" "3679" "3680" "3681" "3682" "3683" "3684" "3685" "3686" "3687" [3577] "3688" "3689" "3690" "3691" "3692" "3693" "3694" "3695" "3696" "3697" "3698" "3699" [3589] "3700" "3701" "3702" "3703" "3704" "3705" "3706" "3707" "3708" "3709" "3710" "3711" [3601] "3712" "3713" "3714" "3715" "3716" "3717" "3718" "3719" "3720" "3721" "3722" "3723" [3613] "3724" "3725" "3728" "3729" "3730" "3731" "3732" "3733" "3734" "3735" "3736" "3737" [3625] "3738" "3739" "3740" "3741" "3742" "3743" "3744" "3745" "3746" "3747" "3748" "3749" [3637] "3750" "3751" "3752" "3753" "3754" "3755" "3756" "3757" "3758" "3759" "3760" "3761" [3649] "3762" "3763" "3764" "3765" "3766" "3767" "3768" "3769" "3770" "3771" "3772" "3773" [3661] "3774" "3775" "3776" "3777" "3778" "3779" "3780" "3781" "3782" "3783" "3784" "3785" [3673] "3786" "3787" "3788" "3789" "3790" "3791" "3792" "3793" "3794" "3795" "3796" "3797" [3685] "3798" "3799" "3800" "3801" "3802" "3803" "3804" "3805" "3806" "3807" "3808" "3809" [3697] "3810" "3811" "3812" "3813" "3814" "3815" "3816" "3817" "3818" "3819" "3820" "3824" [3709] "3825" "3826" "3827" "3828" "3829" "3830" "3831" "3832" "3833" "3834" "3835" "3836" [3721] "3837" "3838" "3839" "3840" "3841" "3842" "3843" "3844" "3845" "3846" "3847" "3848" [3733] "3849" "3850" "3851" "3852" "3853" "3854" "3855" "3856" "3857" "3858" "3859" "3860" [3745] "3861" "3862" "3863" "3864" "3865" "3866" "3867" "3868" "3869" "3870" "3871" "3872" [3757] "3873" "3874" "3875" "3876" "3877" "3878" "3879" "3880" "3881" "3882" "3883" "3884" [3769] "3885" "3886" "3887" "3888" "3889" "3890" "3891" "3892" "3893" "3894" "3895" "3896" [3781] "3897" "3898" "3899" "3900" "3901" "3902" "3903" "3904" "3905" "3906" "3907" "3908" [3793] "3909" "3910" "3911" "3912" "3913" "3914" "3915" "3916" "3917" "3918" "3919" "3920" [3805] "3921" "3922" "3923" "3924" "3925" "3926" ``` --- * These names are actually stored as an _attribute_ called "names" ```r attr(fm2$residuals, "names") ``` ``` [1] "1" "2" "3" "4" "5" "6" "7" "8" "9" "10" "11" "12" [13] "13" "15" "16" "17" "18" "19" "20" "21" "22" "23" "24" "25" [25] "26" "28" "29" "30" "31" "32" "33" "34" "35" "36" "37" "38" [37] "39" "40" "41" "42" "43" "44" "45" "46" "47" "48" "49" "50" [49] "51" "52" "53" "54" "55" "56" "57" "58" "59" "60" "61" "62" [61] "63" "64" "65" "66" "67" "68" "69" "70" "71" "72" "73" "74" [73] "75" "76" "77" "78" "79" "80" "81" "82" "83" "84" "85" "86" [85] "87" "88" "89" "90" "91" "92" "93" "94" "95" "96" "97" "98" [97] "99" "100" "101" "102" "103" "104" "105" "106" "107" "108" "109" "110" [109] "111" "112" "113" "114" "115" "116" "117" "118" "119" "120" "121" "122" [121] "123" "124" "125" "126" "127" "128" "129" "130" "131" "132" "133" "134" [133] "135" "136" "137" "138" "139" "140" "141" "142" "143" "144" "145" "146" [145] "147" "148" "149" "150" "151" "152" "153" "154" "155" "156" "157" "158" [157] "159" "160" "161" "162" "163" "164" "165" "167" "168" "169" "170" "171" [169] "172" "173" "174" "175" "176" "177" "178" "179" "180" "181" "182" "183" [181] "184" "185" "186" "187" "188" "189" "190" "191" "192" "193" "194" "195" [193] "196" "197" "198" "199" "200" "201" "202" "203" "204" "205" "206" "207" [205] "208" "209" "210" "211" "212" "213" "214" "215" "216" "217" "218" "219" [217] "220" "221" "222" "223" "224" "225" "226" "227" "228" "229" "230" "231" [229] "232" "233" "234" "235" "236" "239" "240" "241" "242" "243" "244" "245" [241] "246" "247" "248" "249" "250" "251" "252" "253" "254" "255" "256" "257" [253] "258" "259" "260" "261" "262" "263" "264" "265" "266" "267" "268" "269" [265] "270" "271" "272" "273" "274" "275" "276" "277" "278" "279" "280" "281" [277] "282" "283" "284" "285" "286" "287" "288" "289" "290" "291" "292" "293" [289] "294" "295" "296" "297" "298" "299" "300" "301" "302" "303" "304" "305" [301] "306" "307" "308" "309" "310" "311" "312" "313" "314" "315" "316" "317" [313] "318" "319" "320" "321" "322" "323" "324" "325" "326" "327" "328" "329" [325] "330" "331" "332" "333" "334" "335" "336" "337" "338" "339" "340" "341" [337] "342" "343" "344" "345" "346" "347" "348" "349" "350" "351" "352" "353" [349] "354" "355" "356" "357" "358" "359" "360" "361" "362" "363" "364" "365" [361] "366" "367" "368" "369" "370" "371" "372" "373" "374" "375" "376" "377" [373] "378" "379" "380" "381" "382" "383" "384" "385" "386" "387" "388" "389" [385] "390" "391" "392" "393" "394" "395" "396" "397" "398" "399" "400" "401" [397] "402" "403" "404" "405" "407" "408" "409" "410" "411" "412" "413" "414" [409] "415" "416" "417" "418" "419" "420" "421" "422" "423" "424" "425" "426" [421] "427" "428" "429" "430" "431" "432" "433" "435" "436" "437" "438" "439" [433] "440" "441" "442" "443" "444" "445" "446" "447" "448" "449" "450" "451" [445] "452" "453" "454" "458" "460" "461" "462" "463" "465" "466" "467" "468" [457] "469" "470" "471" "472" "473" "474" "475" "476" "477" "478" "479" "480" [469] "481" "482" "483" "484" "485" "486" "487" "488" "489" "490" "491" "492" [481] "493" "494" "495" "496" "497" "498" "499" "500" "501" "502" "503" "504" [493] "505" "506" "507" "508" "509" "510" "511" "512" "513" "514" "515" "516" [505] "517" "518" "519" "520" "521" "522" "523" "524" "525" "526" "527" "528" [517] "529" "530" "531" "532" "533" "534" "535" "536" "537" "538" "539" "540" [529] "541" "542" "543" "544" "545" "546" "547" "548" "549" "550" "551" "552" [541] "553" "554" "555" "556" "557" "558" "559" "560" "561" "562" "563" "564" [553] "565" "566" "567" "568" "569" "570" "571" "572" "573" "574" "575" "576" [565] "577" "578" "579" "580" "581" "582" "585" "586" "587" "588" "589" "590" [577] "591" "592" "593" "594" "595" "596" "597" "598" "599" "600" "601" "602" [589] "603" "604" "605" "606" "607" "608" "609" "610" "611" "612" "613" "614" [601] "615" "616" "617" "618" "619" "620" "621" "622" "623" "624" "625" "626" [613] "627" "628" "629" "630" "631" "632" "633" "634" "635" "636" "637" "638" [625] "639" "640" "641" "642" "643" "644" "645" "646" "647" "648" "649" "650" [637] "651" "652" "653" "654" "655" "656" "657" "658" "659" "660" "661" "662" [649] "663" "664" "665" "666" "667" "668" "669" "670" "671" "672" "673" "674" [661] "675" "676" "677" "678" "679" "680" "681" "682" "683" "684" "685" "686" [673] "687" "688" "689" "690" "691" "692" "693" "694" "695" "696" "697" "698" [685] "699" "700" "701" "702" "703" "704" "705" "706" "707" "708" "709" "710" [697] "711" "712" "713" "714" "715" "716" "717" "718" "719" "720" "721" "722" [709] "723" "724" "725" "726" "727" "728" "729" "730" "731" "732" "733" "734" [721] "735" "736" "737" "738" "739" "740" "741" "742" "743" "744" "745" "746" [733] "747" "748" "749" "750" "751" "752" "753" "754" "755" "756" "757" "758" [745] "759" "760" "761" "762" "763" "764" "765" "766" "767" "768" "769" "770" [757] "771" "772" "773" "774" "775" "776" "777" "778" "779" "780" "781" "782" [769] "783" "784" "785" "786" "787" "788" "789" "790" "791" "792" "793" "794" [781] "795" "796" "797" "798" "799" "800" "801" "802" "803" "804" "805" "806" [793] "807" "808" "809" "810" "811" "812" "813" "814" "815" "816" "817" "818" [805] "819" "820" "821" "822" "823" "824" "825" "826" "827" "828" "829" "830" [817] "831" "832" "833" "834" "835" "836" "837" "838" "839" "840" "841" "842" [829] "843" "844" "845" "846" "847" "848" "849" "850" "851" "852" "853" "854" [841] "855" "856" "857" "858" "859" "860" "861" "862" "863" "864" "865" "866" [853] "867" "868" "869" "870" "871" "872" "873" "874" "875" "876" "877" "878" [865] "879" "880" "881" "882" "883" "884" "885" "886" "887" "888" "889" "890" [877] "891" "892" "893" "894" "895" "896" "897" "898" "899" "900" "901" "902" [889] "903" "904" "905" "906" "907" "908" "909" "910" "911" "912" "913" "914" [901] "915" "916" "917" "918" "919" "920" "921" "922" "923" "924" "925" "926" [913] "927" "928" "929" "930" "931" "932" "933" "934" "935" "936" "937" "938" [925] "939" "940" "941" "942" "943" "944" "945" "946" "947" "948" "949" "950" [937] "951" "952" "953" "954" "955" "956" "957" "958" "959" "960" "961" "962" [949] "963" "964" "965" "966" "967" "968" "969" "970" "971" "972" "973" "974" [961] "975" "976" "977" "978" "979" "980" "981" "982" "983" "984" "985" "986" [973] "987" "988" "989" "990" "991" "992" "993" "994" "995" "996" "997" "998" [985] "999" "1000" "1001" "1002" "1003" "1004" "1005" "1006" "1007" "1008" "1009" "1010" [997] "1011" "1012" "1013" "1014" "1015" "1016" "1017" "1018" "1019" "1020" "1021" "1022" [1009] "1023" "1024" "1025" "1026" "1027" "1028" "1029" "1030" "1031" "1032" "1033" "1034" [1021] "1035" "1036" "1037" "1038" "1039" "1040" "1041" "1042" "1043" "1044" "1045" "1046" [1033] "1047" "1048" "1049" "1050" "1051" "1052" "1053" "1054" "1055" "1056" "1057" "1058" [1045] "1059" "1060" "1061" "1062" "1063" "1064" "1065" "1066" "1067" "1068" "1069" "1070" [1057] "1071" "1072" "1073" "1074" "1075" "1076" "1077" "1078" "1079" "1080" "1081" "1082" [1069] "1083" "1084" "1085" "1086" "1087" "1088" "1089" "1090" "1091" "1092" "1093" "1094" [1081] "1095" "1096" "1097" "1098" "1099" "1100" "1101" "1102" "1103" "1104" "1105" "1130" [1093] "1131" "1132" "1133" "1134" "1135" "1136" "1137" "1138" "1139" "1140" "1141" "1142" [1105] "1143" "1144" "1145" "1146" "1147" "1148" "1149" "1150" "1151" "1152" "1153" "1154" [1117] "1155" "1156" "1157" "1158" "1159" "1160" "1161" "1162" "1163" "1164" "1165" "1166" [1129] "1167" "1168" "1169" "1170" "1171" "1172" "1173" "1174" "1175" "1176" "1177" "1178" [1141] "1179" "1180" "1181" "1182" "1183" "1184" "1185" "1186" "1187" "1188" "1189" "1190" [1153] "1191" "1192" "1193" "1195" "1196" "1197" "1198" "1199" "1200" "1202" "1204" "1205" [1165] "1206" "1207" "1208" "1209" "1210" "1211" "1212" "1213" "1214" "1216" "1217" "1218" [1177] "1219" "1220" "1221" "1222" "1223" "1224" "1225" "1226" "1227" "1228" "1229" "1230" [1189] "1231" "1232" "1233" "1234" "1235" "1236" "1237" "1238" "1239" "1240" "1241" "1242" [1201] "1243" "1244" "1245" "1246" "1247" "1248" "1249" "1250" "1251" "1252" "1253" "1254" [1213] "1255" "1256" "1257" "1258" "1259" "1260" "1261" "1262" "1263" "1264" "1265" "1266" [1225] "1267" "1268" "1269" "1270" "1271" "1272" "1273" "1274" "1275" "1276" "1277" "1278" [1237] "1279" "1280" "1281" "1282" "1283" "1284" "1285" "1286" "1287" "1288" "1289" "1290" [1249] "1291" "1292" "1293" "1294" "1295" "1296" "1297" "1298" "1299" "1300" "1301" "1302" [1261] "1303" "1304" "1305" "1306" "1307" "1308" "1309" "1310" "1311" "1312" "1313" "1314" [1273] "1315" "1316" "1317" "1318" "1319" "1320" "1321" "1322" "1323" "1324" "1325" "1326" [1285] "1327" "1328" "1329" "1330" "1331" "1332" "1333" "1334" "1335" "1336" "1337" "1338" [1297] "1339" "1340" "1341" "1342" "1343" "1344" "1345" "1346" "1347" "1348" "1349" "1350" [1309] "1351" "1352" "1353" "1354" "1355" "1356" "1357" "1358" "1359" "1360" "1361" "1362" [1321] "1363" "1364" "1365" "1366" "1367" "1368" "1369" "1370" "1371" "1372" "1373" "1374" [1333] "1375" "1376" "1377" "1378" "1379" "1380" "1381" "1382" "1383" "1384" "1385" "1386" [1345] "1387" "1388" "1389" "1390" "1391" "1392" "1393" "1394" "1395" "1396" "1397" "1398" [1357] "1399" "1400" "1401" "1402" "1403" "1404" "1405" "1406" "1407" "1409" "1410" "1411" [1369] "1412" "1413" "1414" "1415" "1416" "1417" "1418" "1419" "1420" "1421" "1422" "1423" [1381] "1424" "1425" "1426" "1427" "1428" "1429" "1430" "1431" "1432" "1433" "1434" "1435" [1393] "1436" "1437" "1438" "1439" "1440" "1441" "1442" "1443" "1444" "1445" "1446" "1447" [1405] "1448" "1449" "1450" "1451" "1452" "1453" "1454" "1455" "1456" "1457" "1458" "1459" [1417] "1460" "1461" "1462" "1463" "1464" "1465" "1466" "1467" "1468" "1469" "1470" "1471" [1429] "1472" "1473" "1474" "1475" "1476" "1477" "1478" "1479" "1480" "1481" "1482" "1483" [1441] "1484" "1485" "1486" "1487" "1488" "1489" "1490" "1491" "1492" "1493" "1494" "1495" [1453] "1496" "1497" "1498" "1499" "1500" "1501" "1502" "1503" "1504" "1505" "1506" "1507" [1465] "1508" "1509" "1510" "1511" "1512" "1513" "1514" "1515" "1516" "1517" "1518" "1519" [1477] "1520" "1521" "1522" "1523" "1524" "1525" "1526" "1527" "1528" "1529" "1530" "1531" [1489] "1532" "1533" "1534" "1535" "1536" "1537" "1538" "1539" "1540" "1541" "1542" "1543" [1501] "1544" "1545" "1546" "1547" "1548" "1549" "1551" "1552" "1554" "1556" "1557" "1558" [1513] "1559" "1560" "1561" "1562" "1563" "1564" "1565" "1566" "1567" "1568" "1569" "1570" [1525] "1571" "1572" "1573" "1574" "1575" "1576" "1577" "1578" "1579" "1580" "1581" "1582" [1537] "1583" "1584" "1585" "1586" "1587" "1588" "1589" "1590" "1591" "1592" "1593" "1594" [1549] "1595" "1596" "1597" "1598" "1599" "1600" "1601" "1602" "1603" "1604" "1605" "1606" [1561] "1607" "1608" "1609" "1612" "1613" "1614" "1615" "1616" "1617" "1618" "1619" "1620" [1573] "1621" "1622" "1623" "1624" "1625" "1626" "1627" "1628" "1629" "1630" "1631" "1632" [1585] "1633" "1634" "1635" "1636" "1637" "1638" "1639" "1640" "1641" "1642" "1643" "1644" [1597] "1645" "1646" "1647" "1648" "1649" "1650" "1651" "1652" "1653" "1654" "1655" "1656" [1609] "1657" "1658" "1659" "1660" "1662" "1663" "1664" "1665" "1666" "1667" "1668" "1669" [1621] "1670" "1671" "1672" "1673" "1674" "1675" "1676" "1677" "1678" "1679" "1680" "1681" [1633] "1682" "1683" "1684" "1685" "1686" "1687" "1688" "1689" "1690" "1691" "1692" "1693" [1645] "1694" "1695" "1696" "1697" "1698" "1699" "1700" "1701" "1702" "1703" "1704" "1705" [1657] "1706" "1707" "1708" "1709" "1710" "1711" "1712" "1713" "1714" "1715" "1716" "1717" [1669] "1718" "1719" "1720" "1721" "1722" "1723" "1724" "1725" "1726" "1727" "1728" "1729" [1681] "1730" "1731" "1732" "1733" "1734" "1735" "1736" "1737" "1738" "1739" "1740" "1741" [1693] "1742" "1743" "1744" "1745" "1746" "1747" "1748" "1749" "1750" "1751" "1752" "1753" [1705] "1754" "1755" "1756" "1757" "1758" "1759" "1760" "1761" "1762" "1763" "1764" "1765" [1717] "1766" "1768" "1769" "1770" "1771" "1772" "1773" "1774" "1775" "1776" "1777" "1778" [1729] "1779" "1780" "1781" "1782" "1783" "1784" "1785" "1786" "1787" "1788" "1789" "1790" [1741] "1791" "1792" "1793" "1794" "1795" "1796" "1797" "1798" "1799" "1800" "1801" "1802" [1753] "1803" "1804" "1805" "1806" "1807" "1808" "1809" "1810" "1811" "1812" "1813" "1814" [1765] "1815" "1816" "1817" "1818" "1819" "1820" "1821" "1822" "1823" "1824" "1825" "1826" [1777] "1827" "1828" "1829" "1830" "1831" "1832" "1833" "1834" "1835" "1836" "1837" "1838" [1789] "1839" "1840" "1841" "1842" "1843" "1844" "1845" "1846" "1847" "1848" "1849" "1850" [1801] "1851" "1852" "1853" "1854" "1855" "1856" "1857" "1858" "1859" "1860" "1861" "1862" [1813] "1863" "1864" "1865" "1866" "1867" "1868" "1869" "1870" "1871" "1872" "1873" "1874" [1825] "1875" "1876" "1877" "1878" "1879" "1880" "1881" "1882" "1883" "1884" "1885" "1886" [1837] "1887" "1888" "1889" "1890" "1891" "1892" "1893" "1894" "1895" "1896" "1897" "1898" [1849] "1899" "1900" "1901" "1902" "1903" "1904" "1905" "1906" "1907" "1908" "1909" "1910" [1861] "1911" "1912" "1913" "1914" "1915" "1916" "1917" "1918" "1919" "1920" "1921" "1922" [1873] "1923" "1924" "1925" "1938" "1939" "1940" "1941" "1942" "1943" "1944" "1945" "1946" [1885] "1947" "1948" "1949" "1950" "1951" "1952" "1953" "1954" "1955" "1956" "1957" "1958" [1897] "1959" "1960" "1961" "1963" "1964" "1965" "1966" "1967" "1968" "1969" "1970" "1971" [1909] "1972" "1973" "1974" "1975" "1976" "1977" "1978" "1979" "1980" "1981" "1982" "1983" [1921] "1984" "1985" "1986" "1987" "1989" "1990" "1991" "1992" "1993" "1994" "1995" "1996" [1933] "1997" "1998" "1999" "2000" "2002" "2003" "2004" "2005" "2006" "2007" "2008" "2009" [1945] "2010" "2011" "2012" "2013" "2014" "2015" "2016" "2017" "2018" "2019" "2020" "2021" [1957] "2022" "2023" "2024" "2025" "2026" "2027" "2028" "2029" "2030" "2031" "2032" "2033" [1969] "2034" "2035" "2036" "2037" "2038" "2039" "2040" "2041" "2042" "2043" "2044" "2045" [1981] "2046" "2047" "2048" "2049" "2050" "2051" "2052" "2053" "2054" "2055" "2056" "2057" [1993] "2058" "2059" "2060" 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"3407" "3408" "3409" "3410" "3411" "3412" "3413" "3414" "3415" [3325] "3416" "3417" "3418" "3419" "3420" "3421" "3422" "3423" "3424" "3425" "3426" "3427" [3337] "3428" "3429" "3430" "3431" "3432" "3433" "3434" "3435" "3436" "3437" "3438" "3439" [3349] "3440" "3441" "3442" "3443" "3444" "3445" "3446" "3447" "3448" "3449" "3452" "3453" [3361] "3454" "3455" "3456" "3457" "3458" "3459" "3460" "3461" "3462" "3463" "3464" "3465" [3373] "3467" "3468" "3469" "3470" "3471" "3472" "3473" "3474" "3475" "3476" "3477" "3478" [3385] "3479" "3480" "3481" "3482" "3483" "3484" "3485" "3486" "3487" "3488" "3489" "3490" [3397] "3493" "3494" "3495" "3497" "3498" "3500" "3501" "3502" "3503" "3504" "3505" "3506" [3409] "3507" "3509" "3510" "3511" "3513" "3514" "3515" "3516" "3518" "3519" "3520" "3521" [3421] "3522" "3523" "3524" "3525" "3526" "3527" "3528" "3530" "3531" "3532" "3533" "3534" [3433] "3535" "3537" "3538" "3539" "3540" "3542" "3543" "3544" "3545" "3546" "3547" "3548" [3445] "3549" "3550" "3551" "3552" "3553" "3554" "3555" "3556" "3557" "3558" "3559" "3560" [3457] "3561" "3563" "3564" "3565" "3566" "3567" "3568" "3569" "3570" "3571" "3572" "3573" [3469] "3574" "3575" "3576" "3577" "3578" "3579" "3580" "3581" "3582" "3583" "3584" "3585" [3481] "3586" "3587" "3588" "3589" "3590" "3591" "3592" "3593" "3594" "3595" "3596" "3597" [3493] "3598" "3599" "3600" "3601" "3602" "3603" "3604" "3605" "3606" "3607" "3608" "3610" [3505] "3611" "3612" "3613" "3614" "3616" "3617" "3619" "3621" "3622" "3623" "3624" "3625" [3517] "3626" "3627" "3628" "3629" "3630" "3631" "3632" "3633" "3635" "3636" "3637" "3639" [3529] "3640" "3641" "3642" "3643" "3644" "3645" "3646" "3647" "3648" "3649" "3650" "3651" [3541] "3652" "3653" "3654" "3655" "3656" "3657" "3658" "3659" "3660" "3661" "3662" "3663" [3553] "3664" "3665" "3666" "3667" "3668" "3669" "3670" "3671" "3672" "3673" "3674" "3675" [3565] "3676" "3677" "3678" "3679" "3680" "3681" "3682" "3683" "3684" "3685" "3686" "3687" [3577] "3688" "3689" "3690" "3691" "3692" "3693" "3694" "3695" "3696" "3697" "3698" "3699" [3589] "3700" "3701" "3702" "3703" "3704" "3705" "3706" "3707" "3708" "3709" "3710" "3711" [3601] "3712" "3713" "3714" "3715" "3716" "3717" "3718" "3719" "3720" "3721" "3722" "3723" [3613] "3724" "3725" "3728" "3729" "3730" "3731" "3732" "3733" "3734" "3735" "3736" "3737" [3625] "3738" "3739" "3740" "3741" "3742" "3743" "3744" "3745" "3746" "3747" "3748" "3749" [3637] "3750" "3751" "3752" "3753" "3754" "3755" "3756" "3757" "3758" "3759" "3760" "3761" [3649] "3762" "3763" "3764" "3765" "3766" "3767" "3768" "3769" "3770" "3771" "3772" "3773" [3661] "3774" "3775" "3776" "3777" "3778" "3779" "3780" "3781" "3782" "3783" "3784" "3785" [3673] "3786" "3787" "3788" "3789" "3790" "3791" "3792" "3793" "3794" "3795" "3796" "3797" [3685] "3798" "3799" "3800" "3801" "3802" "3803" "3804" "3805" "3806" "3807" "3808" "3809" [3697] "3810" "3811" "3812" "3813" "3814" "3815" "3816" "3817" "3818" "3819" "3820" "3824" [3709] "3825" "3826" "3827" "3828" "3829" "3830" "3831" "3832" "3833" "3834" "3835" "3836" [3721] "3837" "3838" "3839" "3840" "3841" "3842" "3843" "3844" "3845" "3846" "3847" "3848" [3733] "3849" "3850" "3851" "3852" "3853" "3854" "3855" "3856" "3857" "3858" "3859" "3860" [3745] "3861" "3862" "3863" "3864" "3865" "3866" "3867" "3868" "3869" "3870" "3871" "3872" [3757] "3873" "3874" "3875" "3876" "3877" "3878" "3879" "3880" "3881" "3882" "3883" "3884" [3769] "3885" "3886" "3887" "3888" "3889" "3890" "3891" "3892" "3893" "3894" "3895" "3896" [3781] "3897" "3898" "3899" "3900" "3901" "3902" "3903" "3904" "3905" "3906" "3907" "3908" [3793] "3909" "3910" "3911" "3912" "3913" "3914" "3915" "3916" "3917" "3918" "3919" "3920" [3805] "3921" "3922" "3923" "3924" "3925" "3926" ``` --- * This is true for all vector objects, including lists ```r attr(fm2, "names") ``` ``` [1] "coefficients" "residuals" "effects" "rank" "fitted.values" [6] "assign" "qr" "df.residual" "na.action" "xlevels" [11] "call" "terms" "model" ``` --- * Attributes are arbitrary R objects that can be attached to any other object * Typically used for programming convenience, normally not seen by users * However, some attributes are "special" --- * The "names" attribute can be extracted using the function `names()` * `dimnames()` similarly gives row / column names for matrices and arrays ```r dimnames(Titanic) ``` ``` $Class [1] "1st" "2nd" "3rd" "Crew" $Sex [1] "Male" "Female" $Age [1] "Child" "Adult" $Survived [1] "No" "Yes" ``` --- * The "names" attribute can be extracted using the function `names()` * `dimnames()` similarly gives row / column names for matrices and arrays ```r attr(Titanic, "dimnames") ``` ``` $Class [1] "1st" "2nd" "3rd" "Crew" $Sex [1] "Male" "Female" $Age [1] "Child" "Adult" $Survived [1] "No" "Yes" ``` --- * For example, column names can be obtained as ```r dimnames(Titanic)[[2]] ``` ``` [1] "Male" "Female" ``` * There are convenient shortcuts called `rownames()` and `colnames()` ```r colnames(Titanic) ``` ``` [1] "Male" "Female" ``` --- * In fact, we can easily verify that this is what `colnames()` is doing by printing it ```r colnames ``` ``` function (x, do.NULL = TRUE, prefix = "col") { if (is.data.frame(x) && do.NULL) names(x) else dimnames(x)[[2L]] %||% if (do.NULL) NULL else { nc <- NCOL(x) if (nc > 0L) paste0(prefix, seq_len(nc)) else character() } } <bytecode: 0x7fb403a14358> <environment: namespace:base> ``` * All R functions can be easily inspected in this way --- * Another very important attribute is "class" * For example, the return value of `lm()` has class "lm" ```r attr(fm2, "class") ``` ``` [1] "lm" ``` * The class of an object can also be obtained using the function `class()` ```r class(fm2) ``` ``` [1] "lm" ``` --- * The class of an object can (usually) be "removed" by setting it to `NULL` * This is not something you should actually do! ```r class(fm2) <- NULL ``` * Such objects will no longer have a "class" attribute ```r attr(fm2, "class") ``` ``` NULL ``` * However, it will still have a class (implicitly) ```r class(fm2) ``` ``` [1] "list" ``` <!-- * We will discuss the class attribute in more details next day --> --- layout: true # Class, generic functions, and methods --- * The class of an object is fundamental to how R works * Every object in R must have a class * This is true even if the object does not have a class attribute ```r class(colnames) ``` ``` [1] "function" ``` ```r attr(colnames, "class") ``` ``` NULL ``` ```r class(Titanic) ``` ``` [1] "table" ``` ```r attr(Titanic, "class") ``` ``` [1] "table" ``` --- * The main use of the class of an object is in how _generic functions_ behave * Generic functions are intended to perform general tasks, like - `print()` - `plot()` - `summary()` * But details of what these functions should do depends on the input --- ```r print(Titanic[, , 1, 1]) ``` ``` Sex Class Male Female 1st 0 0 2nd 0 0 3rd 35 17 Crew 0 0 ``` ```r fm1 <- lm(pm10 ~ no2, aqi, subset = (no2 < 100)) print(fm1) ``` ``` Call: lm(formula = pm10 ~ no2, data = aqi, subset = (no2 < 100)) Coefficients: (Intercept) no2 52.011 4.229 ``` --- ```r summary(Titanic) ``` ``` Number of cases in table: 2201 Number of factors: 4 Test for independence of all factors: Chisq = 1637.4, df = 25, p-value = 0 Chi-squared approximation may be incorrect ``` --- ```r summary(fm1) ``` ``` Call: lm(formula = pm10 ~ no2, data = aqi, subset = (no2 < 100)) Residuals: Min 1Q Median 3Q Max -235.21 -51.35 -17.91 28.72 830.25 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 52.0114 2.7786 18.72 <2e-16 *** no2 4.2286 0.1014 41.71 <2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 83.24 on 3808 degrees of freedom (118 observations deleted due to missingness) Multiple R-squared: 0.3136, Adjusted R-squared: 0.3135 F-statistic: 1740 on 1 and 3808 DF, p-value: < 2.2e-16 ``` --- * But suppose we make a copy of `fm1` and remove the class attribute from it ```r fm2 <- fm1 class(fm2) <- NULL class(fm1) ``` ``` [1] "lm" ``` ```r class(fm2) ``` ``` [1] "list" ``` * `fm1` and `fm2` represent the same model fit * But the different class means that `print()` and `summary()` behave differently --- ```r summary(fm2) ``` ``` Length Class Mode coefficients 2 -none- numeric residuals 3810 -none- numeric effects 3810 -none- numeric rank 1 -none- numeric fitted.values 3810 -none- numeric assign 2 -none- numeric qr 5 qr list df.residual 1 -none- numeric na.action 118 omit numeric xlevels 0 -none- list call 4 -none- call terms 3 terms call model 2 data.frame list ``` --- .scrollable400[ ```r print(fm2) ``` ``` $coefficients (Intercept) no2 52.011396 4.228562 $residuals 1 2 3 4 5 6 127.64591729 129.33160471 157.18879234 143.24585460 111.56016718 131.33160471 7 8 9 10 11 12 98.01729213 107.93154203 109.16010450 173.93154203 98.61722945 414.61722945 13 15 16 17 18 19 255.07435439 337.41735482 653.33160471 55.58885503 113.64591729 36.27454245 20 21 22 23 24 25 95.90316761 104.04597998 -2.49689508 73.58885503 -16.72545755 -3.95402002 26 28 29 30 31 32 9.27454245 17.70297955 19.47441708 27.93154203 26.16010450 11.38866697 33 34 35 36 37 38 13.93154203 9.93154203 3.70297955 4.93154203 1.70297955 4.56016718 39 40 41 42 43 44 14.56016718 22.33160471 7.10304224 32.41735482 60.87447976 54.56016718 45 46 47 48 49 50 46.33160471 76.01729213 82.78872966 51.33160471 78.33160471 -16.06814455 51 52 53 54 55 56 12.44604266 54.76035524 31.13173008 19.27454245 11.73166740 60.58885503 57 58 59 60 61 62 134.64591729 171.47441708 -30.21127034 -51.66839529 -67.43983282 -63.12552024 63 64 65 66 67 68 -49.75414540 -57.29702045 -54.98270787 -61.12552024 -51.35408271 -31.43983282 69 70 71 72 73 74 -45.21127034 -66.58264518 -60.35408271 -49.66839529 -24.06845797 -22.06845797 75 76 77 78 79 80 -60.12552024 -19.21127034 -20.06845797 -22.06845797 -34.98270787 -35.98270787 81 82 83 84 85 86 -34.52558292 -44.43983282 -35.98270787 -41.29702045 -31.06845797 -32.52558292 87 88 89 90 91 92 -41.52558292 -53.83989550 -52.98270787 -37.83989550 -45.83989550 -31.15420808 93 94 95 96 97 98 -32.15420808 -42.38277055 -46.61133303 -42.83989550 -46.29702045 -47.61133303 99 100 101 102 103 104 -58.52558292 -44.61133303 -29.83989550 -29.15420808 -37.61133303 -34.83989550 105 106 107 108 109 110 -21.38277055 -22.38277055 -30.61133303 29.61722945 -11.15420808 -31.83989550 111 112 113 114 115 116 -5.83989550 -16.92564561 -13.69708313 -23.69708313 -27.92564561 -22.69708313 117 118 119 120 121 122 -7.92564561 -0.69708313 3.30291687 -3.46852066 -5.69708313 -20.69708313 123 124 125 126 127 128 -19.61133303 -23.15420808 -26.38277055 -10.38277055 -8.38277055 -6.38277055 129 130 131 132 133 134 -21.61133303 -29.92564561 -34.92564561 -9.92564561 20.07435439 13.38866697 135 136 137 138 139 140 -10.98270787 -58.66839529 -55.98270787 -34.75414540 -34.75414540 -10.75414540 141 142 143 144 145 146 -3.21127034 -4.98270787 -35.29702045 -42.06845797 -36.06845797 -34.06845797 147 148 149 150 151 152 -10.52558292 18.70297955 -92.78251981 -77.55395734 -65.09683239 -38.49689508 153 154 155 156 157 158 -38.03977013 -68.32539487 -60.09683239 -62.32539487 -49.46820723 -31.18258250 159 160 161 162 163 164 -86.35376928 -35.03977013 -72.18258250 -71.46820723 -20.95402002 -31.66839529 165 167 168 169 170 171 1.10304224 108.53147934 -9.75414540 2.24585460 0.78872966 -18.43983282 172 173 174 175 176 177 -19.21127034 13.93154203 50.84579192 14.93154203 71.16010450 45.38866697 178 179 180 181 182 183 78.61722945 68.38866697 180.38866697 187.07435439 97.30291687 164.84579192 184 185 186 187 188 189 140.93154203 87.76004181 52.76004181 47.07435439 16.56016718 9.87447976 190 191 192 193 194 195 6.64591729 30.10304224 46.73166740 9.67460513 -50.15389465 -17.01108229 196 197 198 199 200 201 -19.32539487 -28.92533218 -3.18258250 15.78872966 -0.75414540 -27.66839529 202 203 204 205 206 207 -51.95402002 -97.06814455 -65.61101960 -61.23964476 -4.18258250 -29.49689508 208 209 210 211 212 213 -35.41114497 -13.86826992 -73.63970745 -27.81120766 -2.95402002 44.18879234 214 215 216 217 218 219 -35.58264518 133.10304224 68.56016718 -42.26833260 -60.95402002 -56.46820723 220 221 222 223 224 225 -51.32539487 -23.26833260 73.93154203 -2.66839529 -39.41114497 -11.81120766 226 227 228 229 230 231 93.78872966 48.24585460 6.01729213 -11.12552024 -13.35408271 33.53147934 232 233 234 235 236 239 -14.83989550 17.84579192 15.07435439 -9.69708313 -4.69708313 51.73166740 240 241 242 243 244 245 51.96022987 150.47441708 71.33160471 78.93154203 94.53147934 199.47441708 246 247 248 249 250 251 82.93154203 45.56016718 79.41735482 35.18879234 315.44604266 378.27454245 252 253 254 255 256 257 123.73166740 119.87447976 72.64591729 15.41735482 22.10304224 50.87447976 258 259 260 261 262 263 51.10304224 67.87447976 90.96022987 205.50310492 113.73166740 300.04597998 264 265 266 267 268 269 120.41735482 114.50310492 106.50310492 115.50310492 148.33160471 -25.52558292 270 271 272 273 274 275 -27.35408271 -28.72545755 -14.26833260 -25.43983282 -37.58264518 -32.26833260 276 277 278 279 280 281 -28.12552024 -46.49689508 17.36029255 -9.63970745 -7.55395734 45.90316761 282 283 284 285 286 287 27.50310492 -1.81120766 25.64591729 37.10304224 75.70297955 59.38866697 288 289 290 291 292 293 41.93154203 45.16010450 0.70297955 -2.29702045 27.24585460 23.70297955 294 295 296 297 298 299 23.47441708 7.93154203 11.38866697 9.93154203 25.70297955 0.07435439 300 301 302 303 304 305 22.93154203 -12.52558292 -7.52558292 -16.98270787 -13.89695776 -15.43983282 306 307 308 309 310 311 -17.21127034 -2.66839529 25.47441708 3.01729213 -3.75414540 1.78872966 312 313 314 315 316 317 -14.52558292 -5.43983282 3.33160471 21.78872966 26.78872966 40.78872966 318 319 320 321 322 323 22.16010450 46.61722945 84.53147934 4.70297955 45.61722945 3.38866697 324 325 326 327 328 329 13.16010450 11.16010450 9.47441708 33.70297955 -4.03977013 -76.18258250 330 331 332 333 334 335 -31.81120766 -39.26833260 -36.72545755 -15.72545755 -23.18258250 -8.58264518 336 337 338 339 340 341 -2.43983282 -2.98270787 -71.78251981 -22.26833260 -17.72545755 1.81741750 342 343 344 345 346 347 16.96022987 5.41735482 27.24585460 -29.43983282 -19.75414540 -18.98270787 348 349 350 351 352 353 18.70297955 -7.58264518 11.18879234 19.10304224 7.33160471 6.41735482 354 355 356 357 358 359 -8.03977013 1.27454245 71.36029255 41.04597998 85.50310492 95.50310492 360 361 362 363 364 365 90.87447976 215.04597998 316.96022987 320.96022987 252.21748019 181.36029255 366 367 368 369 370 371 130.44604266 50.36060598 160.24616803 76.38866697 31.38866697 97.16010450 372 373 374 375 376 377 362.70297955 128.01729213 162.47441708 249.78872966 112.96022987 -39.01108229 378 379 380 381 382 383 -9.95402002 -10.63970745 13.53179277 74.90316761 82.67460513 124.36029255 384 385 386 387 388 389 66.84610535 70.13173008 55.58885503 16.73166740 109.64591729 74.18879234 390 391 392 393 394 395 55.50310492 21.96022987 -28.86826992 -21.41114497 -11.95402002 -33.09683239 396 397 398 399 400 401 -58.46820723 -32.55395734 -5.32539487 -16.63970745 32.13173008 136.50310492 402 403 404 405 407 408 109.98891771 -3.09683239 25.84610535 68.30323029 -63.15389465 -55.78251981 409 410 411 412 413 414 -68.92533218 53.30323029 199.33191814 204.67460513 98.76035524 -2.32539487 415 416 417 418 419 420 77.50310492 83.13173008 27.58885503 72.90316761 89.61754287 43.76035524 421 422 423 424 425 426 21.27454245 -41.52558292 -35.89695776 -29.21127034 -41.75414540 -50.52558292 427 428 429 430 431 432 -45.52558292 -76.43983282 -47.61133303 -83.66839529 -53.21127034 -53.98270787 433 435 436 437 438 439 -40.52558292 -22.83989550 -34.06845797 -27.83989550 -14.83989550 -63.98270787 440 441 442 443 444 445 -55.43983282 -61.21127034 -60.83989550 -51.52558292 -38.75414540 -28.06845797 446 447 448 449 450 451 -21.83989550 -20.83989550 -37.75414540 -56.21127034 -18.06845797 -7.83989550 452 453 454 458 460 461 -0.06845797 3.70297955 -2.29702045 -47.12552024 -5.52558292 -8.21127034 462 463 465 466 467 468 -6.43983282 -15.21127034 -25.06845797 6.30291687 -14.46852066 7.76004181 469 470 471 472 473 474 28.30291687 51.30291687 71.07435439 37.38866697 15.24585460 -11.66839529 475 476 477 478 479 480 -14.66839529 -0.98270787 -9.98270787 -2.98270787 -11.21127034 -72.72545755 481 482 483 484 485 486 -58.26833260 -63.12552024 -48.75414540 -52.61133303 -52.61133303 -71.12552024 487 488 489 490 491 492 -66.58264518 -39.43983282 -49.35408271 -113.09683239 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102.75415 111.21127 111.21127 115.43983 128.12552 140.81121 149.26833 699 700 701 702 703 704 705 706 707 174.63971 161.95402 170.41114 170.41114 170.41114 149.26833 178.86827 178.86827 191.55396 708 709 710 711 712 713 714 715 716 183.09683 183.09683 157.72546 174.63971 174.63971 238.06814 216.92533 195.78252 208.46821 717 718 719 720 721 722 723 724 725 200.01108 183.09683 225.38246 208.46821 174.63971 157.72546 178.86827 170.41114 178.86827 726 727 728 729 730 731 732 733 734 174.63971 183.09683 204.23964 195.78252 191.55396 191.55396 183.09683 187.32539 200.01108 735 736 737 738 739 740 741 742 743 216.92533 221.15389 216.92533 208.46821 204.23964 204.23964 216.92533 238.06814 225.38246 744 745 746 747 748 749 750 751 752 242.29671 276.12521 233.83958 229.61102 208.46821 178.86827 212.69677 221.15389 208.46821 753 754 755 756 757 758 759 760 761 195.78252 204.23964 200.01108 195.78252 157.72546 153.49690 149.26833 132.35408 161.95402 762 763 764 765 766 767 768 769 770 170.41114 178.86827 174.63971 174.63971 174.63971 170.41114 149.26833 191.55396 145.03977 771 772 773 774 775 776 777 778 779 119.66840 132.35408 132.35408 149.26833 195.78252 145.03977 106.98271 128.12552 136.58265 780 781 782 783 784 785 786 787 788 94.29702 94.29702 119.66840 102.75415 85.83990 102.75415 98.52558 94.29702 94.29702 789 790 791 792 793 794 795 796 797 132.35408 140.81121 119.66840 111.21127 98.52558 94.29702 102.75415 102.75415 128.12552 798 799 800 801 802 803 804 805 806 106.98271 128.12552 128.12552 90.06846 85.83990 81.61133 81.61133 81.61133 94.29702 807 808 809 810 811 812 813 814 815 85.83990 85.83990 94.29702 94.29702 94.29702 119.66840 94.29702 85.83990 106.98271 816 817 818 819 820 821 822 823 824 94.29702 94.29702 94.29702 90.06846 85.83990 85.83990 81.61133 85.83990 85.83990 825 826 827 828 829 830 831 832 833 73.15421 73.15421 81.61133 77.38277 94.29702 94.29702 94.29702 68.92565 77.38277 834 835 836 837 838 839 840 841 842 98.52558 81.61133 77.38277 77.38277 85.83990 94.29702 90.06846 90.06846 128.12552 843 844 845 846 847 848 849 850 851 123.89696 111.21127 123.89696 140.81121 153.49690 132.35408 77.38277 73.15421 73.15421 852 853 854 855 856 857 858 859 860 73.15421 77.38277 77.38277 77.38277 77.38277 81.61133 90.06846 98.52558 94.29702 861 862 863 864 865 866 867 868 869 94.29702 85.83990 90.06846 73.15421 68.92565 94.29702 115.43983 123.89696 132.35408 870 871 872 873 874 875 876 877 878 140.81121 145.03977 174.63971 145.03977 149.26833 149.26833 166.18258 183.09683 183.09683 879 880 881 882 883 884 885 886 887 178.86827 161.95402 174.63971 166.18258 157.72546 111.21127 149.26833 174.63971 195.78252 888 889 890 891 892 893 894 895 896 200.01108 161.95402 145.03977 145.03977 140.81121 157.72546 145.03977 119.66840 166.18258 897 898 899 900 901 902 903 904 905 178.86827 183.09683 157.72546 145.03977 123.89696 98.52558 115.43983 111.21127 174.63971 906 907 908 909 910 911 912 913 914 157.72546 178.86827 123.89696 102.75415 102.75415 111.21127 115.43983 149.26833 174.63971 915 916 917 918 919 920 921 922 923 166.18258 123.89696 132.35408 145.03977 161.95402 128.12552 102.75415 119.66840 85.83990 924 925 926 927 928 929 930 931 932 102.75415 132.35408 161.95402 119.66840 136.58265 98.52558 115.43983 161.95402 161.95402 933 934 935 936 937 938 939 940 941 166.18258 183.09683 183.09683 136.58265 166.18258 145.03977 132.35408 132.35408 136.58265 942 943 944 945 946 947 948 949 950 128.12552 128.12552 111.21127 94.29702 85.83990 94.29702 102.75415 111.21127 106.98271 951 952 953 954 955 956 957 958 959 119.66840 119.66840 111.21127 111.21127 128.12552 161.95402 98.52558 90.06846 85.83990 960 961 962 963 964 965 966 967 968 90.06846 111.21127 102.75415 174.63971 149.26833 115.43983 98.52558 102.75415 98.52558 969 970 971 972 973 974 975 976 977 119.66840 111.21127 119.66840 153.49690 132.35408 102.75415 111.21127 119.66840 115.43983 978 979 980 981 982 983 984 985 986 132.35408 119.66840 123.89696 106.98271 94.29702 119.66840 102.75415 111.21127 132.35408 987 988 989 990 991 992 993 994 995 140.81121 145.03977 132.35408 140.81121 153.49690 123.89696 106.98271 106.98271 123.89696 996 997 998 999 1000 1001 1002 1003 1004 128.12552 145.03977 136.58265 128.12552 119.66840 136.58265 136.58265 123.89696 115.43983 1005 1006 1007 1008 1009 1010 1011 1012 1013 128.12552 140.81121 145.03977 140.81121 136.58265 132.35408 132.35408 115.43983 73.15421 1014 1015 1016 1017 1018 1019 1020 1021 1022 85.83990 81.61133 77.38277 77.38277 77.38277 81.61133 85.83990 111.21127 115.43983 1023 1024 1025 1026 1027 1028 1029 1030 1031 123.89696 166.18258 212.69677 161.95402 149.26833 140.81121 136.58265 149.26833 153.49690 1032 1033 1034 1035 1036 1037 1038 1039 1040 161.95402 170.41114 149.26833 170.41114 111.21127 115.43983 115.43983 128.12552 98.52558 1041 1042 1043 1044 1045 1046 1047 1048 1049 136.58265 136.58265 128.12552 123.89696 149.26833 145.03977 145.03977 178.86827 170.41114 1050 1051 1052 1053 1054 1055 1056 1057 1058 132.35408 106.98271 94.29702 98.52558 106.98271 111.21127 123.89696 140.81121 136.58265 1059 1060 1061 1062 1063 1064 1065 1066 1067 119.66840 128.12552 136.58265 123.89696 166.18258 157.72546 191.55396 140.81121 102.75415 1068 1069 1070 1071 1072 1073 1074 1075 1076 119.66840 149.26833 149.26833 128.12552 123.89696 115.43983 132.35408 123.89696 166.18258 1077 1078 1079 1080 1081 1082 1083 1084 1085 153.49690 178.86827 174.63971 153.49690 161.95402 153.49690 161.95402 153.49690 149.26833 1086 1087 1088 1089 1090 1091 1092 1093 1094 187.32539 170.41114 204.23964 195.78252 208.46821 208.46821 200.01108 208.46821 191.55396 1095 1096 1097 1098 1099 1100 1101 1102 1103 216.92533 204.23964 200.01108 212.69677 208.46821 216.92533 187.32539 170.41114 161.95402 1104 1105 1130 1131 1132 1133 1134 1135 1136 191.55396 178.86827 191.55396 170.41114 145.03977 166.18258 157.72546 132.35408 149.26833 1137 1138 1139 1140 1141 1142 1143 1144 1145 132.35408 149.26833 157.72546 166.18258 157.72546 161.95402 94.29702 98.52558 119.66840 1146 1147 1148 1149 1150 1151 1152 1153 1154 102.75415 102.75415 111.21127 106.98271 94.29702 90.06846 73.15421 77.38277 140.81121 1155 1156 1157 1158 1159 1160 1161 1162 1163 128.12552 106.98271 106.98271 98.52558 90.06846 94.29702 115.43983 111.21127 94.29702 1164 1165 1166 1167 1168 1169 1170 1171 1172 94.29702 111.21127 98.52558 94.29702 90.06846 90.06846 106.98271 111.21127 90.06846 1173 1174 1175 1176 1177 1178 1179 1180 1181 115.43983 115.43983 60.46852 68.92565 64.69708 68.92565 64.69708 64.69708 60.46852 1182 1183 1184 1185 1186 1187 1188 1189 1190 60.46852 68.92565 68.92565 64.69708 64.69708 64.69708 60.46852 60.46852 60.46852 1191 1192 1193 1195 1196 1197 1198 1199 1200 64.69708 56.23996 56.23996 56.23996 56.23996 56.23996 64.69708 106.98271 64.69708 1202 1204 1205 1206 1207 1208 1209 1210 1211 56.23996 60.46852 56.23996 60.46852 60.46852 60.46852 56.23996 56.23996 56.23996 1212 1213 1214 1216 1217 1218 1219 1220 1221 56.23996 60.46852 60.46852 56.23996 64.69708 64.69708 56.23996 60.46852 68.92565 1222 1223 1224 1225 1226 1227 1228 1229 1230 56.23996 56.23996 56.23996 56.23996 56.23996 60.46852 60.46852 56.23996 60.46852 1231 1232 1233 1234 1235 1236 1237 1238 1239 64.69708 56.23996 81.61133 115.43983 123.89696 170.41114 161.95402 225.38246 195.78252 1240 1241 1242 1243 1244 1245 1246 1247 1248 128.12552 140.81121 161.95402 170.41114 161.95402 170.41114 157.72546 140.81121 161.95402 1249 1250 1251 1252 1253 1254 1255 1256 1257 140.81121 111.21127 123.89696 132.35408 90.06846 98.52558 98.52558 106.98271 136.58265 1258 1259 1260 1261 1262 1263 1264 1265 1266 157.72546 166.18258 161.95402 140.81121 153.49690 106.98271 106.98271 106.98271 106.98271 1267 1268 1269 1270 1271 1272 1273 1274 1275 98.52558 85.83990 128.12552 128.12552 85.83990 106.98271 81.61133 94.29702 90.06846 1276 1277 1278 1279 1280 1281 1282 1283 1284 85.83990 111.21127 145.03977 153.49690 85.83990 98.52558 73.15421 106.98271 98.52558 1285 1286 1287 1288 1289 1290 1291 1292 1293 94.29702 102.75415 115.43983 132.35408 132.35408 90.06846 77.38277 94.29702 140.81121 1294 1295 1296 1297 1298 1299 1300 1301 1302 132.35408 136.58265 166.18258 149.26833 149.26833 153.49690 153.49690 123.89696 106.98271 1303 1304 1305 1306 1307 1308 1309 1310 1311 85.83990 98.52558 85.83990 106.98271 106.98271 81.61133 106.98271 102.75415 85.83990 1312 1313 1314 1315 1316 1317 1318 1319 1320 85.83990 85.83990 106.98271 123.89696 132.35408 94.29702 102.75415 106.98271 128.12552 1321 1322 1323 1324 1325 1326 1327 1328 1329 115.43983 153.49690 106.98271 90.06846 145.03977 128.12552 102.75415 132.35408 119.66840 1330 1331 1332 1333 1334 1335 1336 1337 1338 98.52558 111.21127 128.12552 94.29702 85.83990 90.06846 161.95402 128.12552 132.35408 1339 1340 1341 1342 1343 1344 1345 1346 1347 111.21127 111.21127 136.58265 132.35408 123.89696 119.66840 115.43983 128.12552 145.03977 1348 1349 1350 1351 1352 1353 1354 1355 1356 183.09683 166.18258 187.32539 153.49690 157.72546 178.86827 161.95402 157.72546 145.03977 1357 1358 1359 1360 1361 1362 1363 1364 1365 140.81121 145.03977 157.72546 157.72546 161.95402 170.41114 161.95402 161.95402 153.49690 1366 1367 1368 1369 1370 1371 1372 1373 1374 153.49690 174.63971 174.63971 174.63971 178.86827 145.03977 153.49690 161.95402 170.41114 1375 1376 1377 1378 1379 1380 1381 1382 1383 161.95402 161.95402 149.26833 128.12552 153.49690 153.49690 123.89696 145.03977 178.86827 1384 1385 1386 1387 1388 1389 1390 1391 1392 140.81121 145.03977 161.95402 140.81121 153.49690 153.49690 153.49690 132.35408 153.49690 1393 1394 1395 1396 1397 1398 1399 1400 1401 145.03977 149.26833 136.58265 149.26833 161.95402 149.26833 153.49690 157.72546 161.95402 1402 1403 1404 1405 1406 1407 1409 1410 1411 140.81121 123.89696 106.98271 132.35408 157.72546 140.81121 94.29702 94.29702 174.63971 1412 1413 1414 1415 1416 1417 1418 1419 1420 174.63971 170.41114 166.18258 170.41114 191.55396 187.32539 204.23964 191.55396 191.55396 1421 1422 1423 1424 1425 1426 1427 1428 1429 178.86827 98.52558 115.43983 115.43983 132.35408 132.35408 119.66840 123.89696 128.12552 1430 1431 1432 1433 1434 1435 1436 1437 1438 98.52558 119.66840 157.72546 187.32539 200.01108 204.23964 140.81121 178.86827 276.12521 1439 1440 1441 1442 1443 1444 1445 1446 1447 187.32539 170.41114 157.72546 166.18258 166.18258 212.69677 225.38246 407.21064 301.49658 1448 1449 1450 1451 1452 1453 1454 1455 1456 322.63939 364.92502 288.81089 111.21127 212.69677 212.69677 208.46821 132.35408 115.43983 1457 1458 1459 1460 1461 1462 1463 1464 1465 145.03977 161.95402 183.09683 187.32539 153.49690 157.72546 225.38246 212.69677 216.92533 1466 1467 1468 1469 1470 1471 1472 1473 1474 153.49690 145.03977 157.72546 174.63971 174.63971 208.46821 208.46821 195.78252 187.32539 1475 1476 1477 1478 1479 1480 1481 1482 1483 170.41114 98.52558 68.92565 60.46852 128.12552 161.95402 187.32539 136.58265 111.21127 1484 1485 1486 1487 1488 1489 1490 1491 1492 106.98271 85.83990 60.46852 60.46852 68.92565 73.15421 68.92565 64.69708 60.46852 1493 1494 1495 1496 1497 1498 1499 1500 1501 64.69708 64.69708 60.46852 106.98271 145.03977 145.03977 149.26833 128.12552 149.26833 1502 1503 1504 1505 1506 1507 1508 1509 1510 174.63971 178.86827 140.81121 119.66840 106.98271 85.83990 60.46852 90.06846 90.06846 1511 1512 1513 1514 1515 1516 1517 1518 1519 81.61133 77.38277 73.15421 90.06846 106.98271 115.43983 136.58265 140.81121 106.98271 1520 1521 1522 1523 1524 1525 1526 1527 1528 106.98271 111.21127 102.75415 111.21127 128.12552 136.58265 106.98271 161.95402 166.18258 1529 1530 1531 1532 1533 1534 1535 1536 1537 123.89696 161.95402 132.35408 140.81121 115.43983 115.43983 123.89696 94.29702 106.98271 1538 1539 1540 1541 1542 1543 1544 1545 1546 115.43983 115.43983 123.89696 94.29702 102.75415 123.89696 115.43983 94.29702 111.21127 1547 1548 1549 1551 1552 1554 1556 1557 1558 98.52558 102.75415 90.06846 77.38277 60.46852 56.23996 98.52558 102.75415 115.43983 1559 1560 1561 1562 1563 1564 1565 1566 1567 111.21127 90.06846 85.83990 102.75415 90.06846 85.83990 106.98271 102.75415 111.21127 1568 1569 1570 1571 1572 1573 1574 1575 1576 94.29702 98.52558 77.38277 178.86827 204.23964 153.49690 149.26833 149.26833 136.58265 1577 1578 1579 1580 1581 1582 1583 1584 1585 128.12552 153.49690 145.03977 153.49690 145.03977 128.12552 136.58265 136.58265 157.72546 1586 1587 1588 1589 1590 1591 1592 1593 1594 136.58265 166.18258 166.18258 166.18258 174.63971 68.92565 77.38277 94.29702 102.75415 1595 1596 1597 1598 1599 1600 1601 1602 1603 90.06846 68.92565 106.98271 94.29702 77.38277 90.06846 85.83990 128.12552 106.98271 1604 1605 1606 1607 1608 1609 1612 1613 1614 77.38277 73.15421 115.43983 98.52558 85.83990 111.21127 85.83990 73.15421 64.69708 1615 1616 1617 1618 1619 1620 1621 1622 1623 85.83990 85.83990 77.38277 77.38277 68.92565 68.92565 85.83990 98.52558 106.98271 1624 1625 1626 1627 1628 1629 1630 1631 1632 90.06846 90.06846 77.38277 81.61133 90.06846 90.06846 94.29702 102.75415 94.29702 1633 1634 1635 1636 1637 1638 1639 1640 1641 102.75415 132.35408 166.18258 132.35408 132.35408 166.18258 136.58265 106.98271 111.21127 1642 1643 1644 1645 1646 1647 1648 1649 1650 149.26833 106.98271 77.38277 85.83990 73.15421 73.15421 119.66840 111.21127 123.89696 1651 1652 1653 1654 1655 1656 1657 1658 1659 115.43983 94.29702 77.38277 90.06846 153.49690 123.89696 119.66840 136.58265 123.89696 1660 1662 1663 1664 1665 1666 1667 1668 1669 94.29702 85.83990 132.35408 77.38277 68.92565 73.15421 102.75415 111.21127 111.21127 1670 1671 1672 1673 1674 1675 1676 1677 1678 81.61133 90.06846 115.43983 115.43983 102.75415 102.75415 102.75415 81.61133 90.06846 1679 1680 1681 1682 1683 1684 1685 1686 1687 81.61133 64.69708 81.61133 166.18258 178.86827 145.03977 128.12552 106.98271 140.81121 1688 1689 1690 1691 1692 1693 1694 1695 1696 157.72546 174.63971 149.26833 115.43983 115.43983 123.89696 106.98271 123.89696 102.75415 1697 1698 1699 1700 1701 1702 1703 1704 1705 115.43983 157.72546 153.49690 68.92565 111.21127 136.58265 183.09683 233.83958 98.52558 1706 1707 1708 1709 1710 1711 1712 1713 1714 60.46852 119.66840 123.89696 106.98271 119.66840 140.81121 204.23964 140.81121 123.89696 1715 1716 1717 1718 1719 1720 1721 1722 1723 111.21127 132.35408 178.86827 183.09683 216.92533 174.63971 111.21127 123.89696 140.81121 1724 1725 1726 1727 1728 1729 1730 1731 1732 208.46821 216.92533 212.69677 195.78252 136.58265 132.35408 132.35408 149.26833 145.03977 1733 1734 1735 1736 1737 1738 1739 1740 1741 149.26833 166.18258 132.35408 106.98271 102.75415 106.98271 123.89696 161.95402 119.66840 1742 1743 1744 1745 1746 1747 1748 1749 1750 102.75415 98.52558 102.75415 98.52558 94.29702 98.52558 94.29702 119.66840 128.12552 1751 1752 1753 1754 1755 1756 1757 1758 1759 145.03977 166.18258 161.95402 166.18258 187.32539 178.86827 221.15389 178.86827 128.12552 1760 1761 1762 1763 1764 1765 1766 1768 1769 123.89696 106.98271 85.83990 73.15421 73.15421 77.38277 68.92565 132.35408 178.86827 1770 1771 1772 1773 1774 1775 1776 1777 1778 187.32539 170.41114 149.26833 157.72546 106.98271 106.98271 111.21127 157.72546 233.83958 1779 1780 1781 1782 1783 1784 1785 1786 1787 145.03977 115.43983 115.43983 153.49690 140.81121 123.89696 132.35408 166.18258 123.89696 1788 1789 1790 1791 1792 1793 1794 1795 1796 119.66840 115.43983 115.43983 106.98271 68.92565 64.69708 85.83990 106.98271 119.66840 1797 1798 1799 1800 1801 1802 1803 1804 1805 128.12552 161.95402 242.29671 170.41114 106.98271 81.61133 68.92565 68.92565 85.83990 1806 1807 1808 1809 1810 1811 1812 1813 1814 178.86827 123.89696 102.75415 94.29702 140.81121 140.81121 140.81121 85.83990 123.89696 1815 1816 1817 1818 1819 1820 1821 1822 1823 123.89696 119.66840 145.03977 145.03977 153.49690 183.09683 161.95402 157.72546 153.49690 1824 1825 1826 1827 1828 1829 1830 1831 1832 136.58265 115.43983 145.03977 153.49690 111.21127 119.66840 102.75415 94.29702 94.29702 1833 1834 1835 1836 1837 1838 1839 1840 1841 111.21127 90.06846 128.12552 132.35408 81.61133 111.21127 157.72546 111.21127 115.43983 1842 1843 1844 1845 1846 1847 1848 1849 1850 128.12552 77.38277 81.61133 115.43983 132.35408 157.72546 119.66840 119.66840 136.58265 1851 1852 1853 1854 1855 1856 1857 1858 1859 149.26833 221.15389 170.41114 178.86827 166.18258 161.95402 216.92533 208.46821 161.95402 1860 1861 1862 1863 1864 1865 1866 1867 1868 225.38246 183.09683 157.72546 140.81121 161.95402 178.86827 161.95402 178.86827 178.86827 1869 1870 1871 1872 1873 1874 1875 1876 1877 128.12552 161.95402 183.09683 187.32539 195.78252 200.01108 161.95402 195.78252 128.12552 1878 1879 1880 1881 1882 1883 1884 1885 1886 111.21127 132.35408 140.81121 136.58265 157.72546 174.63971 229.61102 208.46821 183.09683 1887 1888 1889 1890 1891 1892 1893 1894 1895 128.12552 149.26833 153.49690 174.63971 98.52558 123.89696 153.49690 132.35408 166.18258 1896 1897 1898 1899 1900 1901 1902 1903 1904 183.09683 145.03977 161.95402 157.72546 140.81121 132.35408 140.81121 140.81121 166.18258 1905 1906 1907 1908 1909 1910 1911 1912 1913 166.18258 149.26833 161.95402 145.03977 145.03977 149.26833 161.95402 212.69677 157.72546 1914 1915 1916 1917 1918 1919 1920 1921 1922 178.86827 132.35408 140.81121 170.41114 174.63971 153.49690 229.61102 170.41114 145.03977 1923 1924 1925 1938 1939 1940 1941 1942 1943 136.58265 140.81121 140.81121 145.03977 119.66840 132.35408 136.58265 157.72546 170.41114 1944 1945 1946 1947 1948 1949 1950 1951 1952 204.23964 221.15389 208.46821 178.86827 166.18258 174.63971 195.78252 208.46821 212.69677 1953 1954 1955 1956 1957 1958 1959 1960 1961 195.78252 157.72546 187.32539 212.69677 225.38246 238.06814 208.46821 250.75383 254.98239 1963 1964 1965 1966 1967 1968 1969 1970 1971 200.01108 221.15389 233.83958 200.01108 216.92533 187.32539 191.55396 254.98239 208.46821 1972 1973 1974 1975 1976 1977 1978 1979 1980 183.09683 174.63971 191.55396 187.32539 204.23964 183.09683 178.86827 170.41114 170.41114 1981 1982 1983 1984 1985 1986 1987 1989 1990 178.86827 178.86827 195.78252 195.78252 221.15389 200.01108 204.23964 183.09683 221.15389 1991 1992 1993 1994 1995 1996 1997 1998 1999 225.38246 276.12521 208.46821 221.15389 250.75383 204.23964 204.23964 166.18258 208.46821 2000 2002 2003 2004 2005 2006 2007 2008 2009 183.09683 204.23964 195.78252 170.41114 166.18258 254.98239 297.26802 259.21096 254.98239 2010 2011 2012 2013 2014 2015 2016 2017 2018 276.12521 284.58233 293.03946 178.86827 153.49690 132.35408 149.26833 153.49690 140.81121 2019 2020 2021 2022 2023 2024 2025 2026 2027 153.49690 153.49690 136.58265 191.55396 204.23964 166.18258 149.26833 174.63971 183.09683 2028 2029 2030 2031 2032 2033 2034 2035 2036 229.61102 246.52527 331.09652 322.63939 221.15389 221.15389 216.92533 242.29671 238.06814 2037 2038 2039 2040 2041 2042 2043 2044 2045 259.21096 276.12521 267.66808 326.86796 280.35377 216.92533 259.21096 280.35377 225.38246 2046 2047 2048 2049 2050 2051 2052 2053 2054 233.83958 259.21096 271.89664 246.52527 233.83958 288.81089 280.35377 246.52527 271.89664 2055 2056 2057 2058 2059 2060 2061 2062 2063 229.61102 250.75383 263.43952 225.38246 259.21096 204.23964 187.32539 200.01108 200.01108 2064 2065 2066 2067 2068 2069 2070 2071 2072 191.55396 276.12521 250.75383 250.75383 242.29671 259.21096 305.72514 216.92533 233.83958 2073 2074 2075 2076 2077 2078 2080 2081 2082 191.55396 225.38246 212.69677 166.18258 166.18258 157.72546 119.66840 178.86827 170.41114 2083 2084 2085 2086 2087 2088 2089 2090 2091 221.15389 216.92533 191.55396 246.52527 166.18258 119.66840 123.89696 119.66840 123.89696 2092 2093 2094 2095 2096 2097 2098 2099 2100 132.35408 111.21127 128.12552 123.89696 132.35408 115.43983 140.81121 157.72546 128.12552 2101 2103 2104 2105 2106 2107 2108 2109 2110 183.09683 60.46852 85.83990 85.83990 56.23996 56.23996 60.46852 60.46852 77.38277 2111 2112 2113 2114 2115 2116 2117 2118 2119 81.61133 85.83990 132.35408 73.15421 73.15421 68.92565 68.92565 81.61133 81.61133 2120 2121 2122 2123 2124 2125 2126 2127 2128 64.69708 90.06846 161.95402 195.78252 128.12552 170.41114 94.29702 98.52558 90.06846 2129 2130 2131 2132 2133 2134 2135 2136 2137 102.75415 85.83990 81.61133 73.15421 81.61133 68.92565 68.92565 81.61133 102.75415 2138 2139 2140 2141 2142 2143 2144 2145 2146 73.15421 68.92565 73.15421 77.38277 77.38277 81.61133 77.38277 73.15421 73.15421 2147 2148 2149 2150 2151 2152 2153 2154 2155 85.83990 73.15421 64.69708 64.69708 68.92565 68.92565 64.69708 68.92565 77.38277 2156 2157 2158 2159 2160 2161 2162 2163 2164 90.06846 64.69708 73.15421 85.83990 136.58265 106.98271 106.98271 94.29702 200.01108 2165 2166 2167 2168 2169 2170 2171 2172 2173 128.12552 153.49690 161.95402 170.41114 195.78252 174.63971 200.01108 263.43952 263.43952 2174 2175 2176 2177 2178 2179 2180 2181 2183 242.29671 250.75383 216.92533 170.41114 149.26833 166.18258 178.86827 157.72546 335.32508 2184 2185 2186 2187 2188 2189 2190 2191 2192 250.75383 161.95402 166.18258 221.15389 170.41114 145.03977 170.41114 183.09683 216.92533 2193 2194 2195 2196 2197 2198 2199 2200 2201 221.15389 297.26802 208.46821 149.26833 166.18258 174.63971 136.58265 140.81121 145.03977 2202 2203 2204 2205 2206 2207 2208 2209 2210 170.41114 187.32539 216.92533 204.23964 119.66840 221.15389 191.55396 250.75383 284.58233 2211 2212 2213 2214 2215 2216 2217 2218 2219 111.21127 161.95402 191.55396 174.63971 200.01108 293.03946 191.55396 195.78252 216.92533 2220 2221 2222 2223 2224 2225 2226 2227 2228 238.06814 293.03946 263.43952 221.15389 221.15389 225.38246 254.98239 191.55396 123.89696 2229 2230 2231 2233 2234 2235 2236 2237 2238 94.29702 128.12552 145.03977 128.12552 161.95402 157.72546 115.43983 123.89696 157.72546 2239 2240 2241 2242 2243 2244 2245 2246 2247 161.95402 145.03977 102.75415 98.52558 94.29702 98.52558 98.52558 132.35408 140.81121 2248 2249 2250 2251 2252 2253 2254 2255 2256 183.09683 178.86827 212.69677 276.12521 242.29671 195.78252 187.32539 195.78252 178.86827 2257 2258 2259 2260 2261 2262 2263 2264 2265 166.18258 195.78252 136.58265 221.15389 187.32539 229.61102 195.78252 208.46821 200.01108 2266 2267 2268 2269 2270 2271 2272 2273 2274 208.46821 221.15389 204.23964 204.23964 246.52527 238.06814 225.38246 216.92533 191.55396 2275 2276 2277 2278 2279 2280 2281 2282 2283 208.46821 208.46821 221.15389 187.32539 233.83958 233.83958 242.29671 170.41114 149.26833 2284 2285 2286 2287 2288 2289 2290 2291 2292 157.72546 191.55396 183.09683 166.18258 157.72546 128.12552 145.03977 115.43983 115.43983 2293 2294 2295 2296 2297 2298 2299 2300 2301 106.98271 187.32539 246.52527 238.06814 280.35377 284.58233 301.49658 221.15389 136.58265 2302 2303 2304 2305 2306 2307 2308 2309 2310 174.63971 140.81121 157.72546 187.32539 204.23964 200.01108 225.38246 229.61102 246.52527 2311 2312 2313 2314 2315 2316 2317 2318 2319 225.38246 280.35377 233.83958 161.95402 136.58265 187.32539 216.92533 238.06814 229.61102 2320 2321 2322 2323 2324 2325 2326 2327 2328 204.23964 187.32539 191.55396 191.55396 229.61102 267.66808 204.23964 187.32539 187.32539 2329 2330 2331 2332 2333 2334 2335 2336 2337 204.23964 216.92533 221.15389 200.01108 200.01108 170.41114 166.18258 212.69677 229.61102 2338 2339 2340 2341 2342 2343 2344 2345 2346 195.78252 161.95402 145.03977 161.95402 153.49690 191.55396 216.92533 195.78252 174.63971 2347 2348 2349 2350 2351 2352 2353 2354 2355 212.69677 221.15389 212.69677 161.95402 200.01108 216.92533 221.15389 229.61102 233.83958 2356 2357 2358 2359 2360 2361 2362 2363 2364 216.92533 238.06814 216.92533 204.23964 233.83958 246.52527 242.29671 238.06814 293.03946 2365 2366 2367 2368 2369 2370 2371 2372 2373 254.98239 259.21096 259.21096 263.43952 259.21096 267.66808 301.49658 276.12521 309.95371 2374 2375 2376 2377 2378 2379 2380 2381 2382 284.58233 263.43952 276.12521 293.03946 297.26802 263.43952 280.35377 284.58233 267.66808 2383 2384 2385 2386 2387 2388 2389 2390 2391 216.92533 280.35377 259.21096 297.26802 381.83927 343.78221 386.06783 424.12489 432.58202 2392 2393 2394 2395 2396 2397 2398 2399 2400 335.32508 276.12521 246.52527 254.98239 225.38246 254.98239 204.23964 238.06814 221.15389 2401 2402 2403 2404 2405 2406 2407 2408 2409 238.06814 254.98239 267.66808 271.89664 284.58233 267.66808 271.89664 335.32508 297.26802 2410 2411 2412 2413 2414 2415 2416 2417 2418 301.49658 305.72514 352.23933 297.26802 301.49658 259.21096 242.29671 263.43952 301.49658 2419 2420 2421 2422 2423 2424 2425 2426 2427 348.01077 301.49658 195.78252 174.63971 166.18258 166.18258 149.26833 191.55396 200.01108 2428 2429 2430 2431 2432 2433 2434 2435 2436 225.38246 242.29671 229.61102 191.55396 183.09683 208.46821 195.78252 221.15389 204.23964 2437 2438 2439 2440 2441 2442 2443 2444 2445 229.61102 238.06814 221.15389 225.38246 242.29671 132.35408 145.03977 140.81121 115.43983 2446 2447 2448 2449 2450 2451 2452 2453 2454 153.49690 153.49690 149.26833 157.72546 136.58265 149.26833 145.03977 157.72546 145.03977 2455 2456 2457 2458 2459 2460 2461 2462 2463 140.81121 132.35408 145.03977 145.03977 170.41114 153.49690 149.26833 132.35408 128.12552 2464 2465 2466 2467 2468 2469 2470 2471 2472 123.89696 128.12552 123.89696 136.58265 128.12552 102.75415 94.29702 119.66840 111.21127 2473 2474 2475 2476 2477 2478 2479 2480 2481 115.43983 111.21127 111.21127 136.58265 140.81121 149.26833 140.81121 166.18258 132.35408 2482 2483 2484 2485 2486 2487 2488 2489 2490 102.75415 98.52558 98.52558 111.21127 98.52558 115.43983 123.89696 136.58265 132.35408 2491 2492 2493 2494 2495 2496 2497 2498 2499 115.43983 123.89696 149.26833 149.26833 119.66840 174.63971 145.03977 145.03977 123.89696 2500 2501 2502 2503 2505 2506 2507 2508 2510 119.66840 115.43983 123.89696 106.98271 111.21127 111.21127 111.21127 140.81121 140.81121 2511 2512 2513 2514 2515 2516 2517 2518 2519 161.95402 145.03977 157.72546 183.09683 161.95402 187.32539 136.58265 161.95402 208.46821 2520 2521 2522 2523 2524 2525 2526 2527 2528 195.78252 140.81121 119.66840 111.21127 119.66840 157.72546 166.18258 191.55396 204.23964 2529 2530 2531 2532 2533 2534 2535 2536 2537 200.01108 170.41114 170.41114 94.29702 111.21127 136.58265 132.35408 145.03977 128.12552 2538 2539 2540 2541 2543 2544 2545 2546 2547 128.12552 119.66840 136.58265 106.98271 98.52558 106.98271 102.75415 115.43983 106.98271 2548 2549 2550 2551 2552 2553 2554 2555 2556 128.12552 119.66840 119.66840 166.18258 157.72546 111.21127 140.81121 161.95402 153.49690 2557 2558 2559 2560 2561 2562 2563 2564 2565 166.18258 115.43983 90.06846 132.35408 132.35408 140.81121 128.12552 123.89696 115.43983 2566 2567 2568 2569 2570 2571 2572 2573 2574 111.21127 119.66840 115.43983 119.66840 119.66840 128.12552 140.81121 153.49690 136.58265 2575 2576 2577 2578 2579 2580 2581 2582 2583 140.81121 98.52558 90.06846 106.98271 136.58265 174.63971 208.46821 238.06814 229.61102 2584 2585 2586 2587 2588 2589 2590 2591 2592 174.63971 136.58265 128.12552 140.81121 157.72546 178.86827 187.32539 166.18258 225.38246 2593 2594 2595 2596 2597 2598 2599 2600 2601 183.09683 166.18258 161.95402 170.41114 136.58265 128.12552 153.49690 170.41114 174.63971 2602 2603 2604 2605 2606 2607 2608 2609 2610 153.49690 153.49690 183.09683 170.41114 136.58265 153.49690 149.26833 140.81121 153.49690 2611 2612 2613 2614 2615 2616 2617 2618 2619 149.26833 204.23964 161.95402 187.32539 166.18258 123.89696 161.95402 178.86827 174.63971 2620 2621 2622 2623 2624 2625 2626 2627 2628 191.55396 204.23964 233.83958 229.61102 200.01108 170.41114 157.72546 233.83958 187.32539 2629 2630 2631 2632 2633 2634 2635 2636 2637 187.32539 221.15389 200.01108 208.46821 242.29671 208.46821 161.95402 128.12552 132.35408 2638 2639 2640 2641 2642 2643 2644 2645 2646 174.63971 195.78252 183.09683 233.83958 233.83958 174.63971 149.26833 157.72546 170.41114 2647 2648 2649 2650 2651 2652 2653 2654 2655 191.55396 212.69677 208.46821 161.95402 221.15389 221.15389 242.29671 238.06814 229.61102 2656 2657 2658 2659 2660 2661 2662 2663 2664 200.01108 204.23964 183.09683 191.55396 166.18258 178.86827 170.41114 140.81121 161.95402 2665 2666 2667 2668 2669 2670 2671 2672 2673 149.26833 195.78252 267.66808 221.15389 200.01108 157.72546 153.49690 149.26833 149.26833 2674 2675 2676 2677 2678 2679 2680 2681 2682 174.63971 157.72546 140.81121 115.43983 111.21127 119.66840 102.75415 140.81121 136.58265 2683 2684 2685 2686 2687 2688 2689 2690 2691 128.12552 111.21127 145.03977 161.95402 149.26833 145.03977 153.49690 166.18258 187.32539 2692 2693 2694 2695 2696 2697 2698 2699 2700 212.69677 233.83958 254.98239 187.32539 242.29671 314.18227 229.61102 216.92533 200.01108 2701 2702 2703 2704 2705 2706 2707 2708 2709 204.23964 208.46821 250.75383 208.46821 229.61102 242.29671 221.15389 204.23964 187.32539 2710 2711 2712 2713 2714 2715 2716 2717 2718 183.09683 208.46821 208.46821 187.32539 242.29671 271.89664 246.52527 352.23933 250.75383 2719 2720 2721 2722 2723 2724 2725 2726 2727 284.58233 284.58233 348.01077 331.09652 301.49658 309.95371 318.41083 242.29671 238.06814 2728 2729 2730 2731 2732 2733 2734 2735 2736 259.21096 309.95371 297.26802 259.21096 254.98239 318.41083 318.41083 297.26802 276.12521 2737 2738 2739 2740 2741 2742 2743 2744 2745 288.81089 318.41083 288.81089 271.89664 267.66808 259.21096 293.03946 195.78252 200.01108 2746 2747 2748 2749 2750 2751 2752 2753 2754 200.01108 221.15389 326.86796 238.06814 178.86827 259.21096 267.66808 267.66808 250.75383 2755 2756 2757 2758 2759 2760 2761 2762 2763 318.41083 276.12521 221.15389 242.29671 212.69677 212.69677 212.69677 229.61102 195.78252 2764 2765 2766 2767 2768 2769 2770 2771 2772 208.46821 242.29671 238.06814 221.15389 229.61102 187.32539 174.63971 208.46821 280.35377 2773 2774 2775 2776 2777 2778 2779 2780 2781 246.52527 153.49690 166.18258 149.26833 161.95402 149.26833 115.43983 157.72546 149.26833 2782 2783 2784 2785 2786 2787 2788 2789 2790 136.58265 136.58265 132.35408 153.49690 145.03977 123.89696 136.58265 115.43983 136.58265 2791 2792 2793 2794 2795 2796 2797 2798 2799 123.89696 123.89696 128.12552 128.12552 111.21127 132.35408 187.32539 145.03977 128.12552 2800 2801 2802 2803 2804 2805 2806 2807 2808 140.81121 153.49690 149.26833 136.58265 136.58265 123.89696 132.35408 119.66840 128.12552 2809 2810 2811 2812 2813 2814 2815 2816 2817 132.35408 132.35408 102.75415 98.52558 132.35408 153.49690 149.26833 123.89696 98.52558 2818 2819 2820 2821 2822 2823 2824 2825 2826 111.21127 132.35408 123.89696 115.43983 145.03977 145.03977 136.58265 145.03977 136.58265 2827 2828 2829 2830 2831 2832 2833 2834 2835 140.81121 132.35408 128.12552 123.89696 111.21127 102.75415 111.21127 123.89696 119.66840 2836 2837 2838 2839 2840 2841 2842 2843 2844 123.89696 115.43983 119.66840 128.12552 145.03977 140.81121 174.63971 178.86827 200.01108 2845 2846 2847 2848 2849 2850 2851 2852 2853 174.63971 187.32539 204.23964 178.86827 128.12552 136.58265 170.41114 153.49690 195.78252 2854 2855 2856 2857 2858 2859 2860 2861 2862 157.72546 216.92533 183.09683 187.32539 187.32539 208.46821 318.41083 166.18258 187.32539 2863 2864 2865 2866 2867 2868 2869 2870 2871 225.38246 187.32539 204.23964 250.75383 187.32539 145.03977 123.89696 153.49690 191.55396 2872 2873 2874 2875 2876 2877 2878 2879 2880 212.69677 200.01108 157.72546 161.95402 238.06814 170.41114 145.03977 191.55396 191.55396 2881 2882 2883 2884 2885 2886 2887 2888 2889 238.06814 238.06814 318.41083 318.41083 267.66808 254.98239 276.12521 288.81089 233.83958 2890 2891 2892 2893 2894 2895 2896 2897 2898 238.06814 221.15389 174.63971 178.86827 221.15389 225.38246 170.41114 178.86827 183.09683 2899 2900 2901 2902 2903 2904 2905 2906 2907 212.69677 212.69677 233.83958 314.18227 216.92533 225.38246 288.81089 229.61102 195.78252 2908 2909 2910 2911 2912 2913 2914 2915 2916 140.81121 166.18258 174.63971 153.49690 204.23964 183.09683 195.78252 153.49690 111.21127 2917 2918 2919 2920 2921 2922 2923 2924 2925 161.95402 170.41114 161.95402 178.86827 263.43952 246.52527 140.81121 157.72546 174.63971 2926 2927 2928 2929 2930 2931 2932 2933 2934 200.01108 170.41114 174.63971 136.58265 140.81121 140.81121 136.58265 161.95402 132.35408 2935 2936 2937 2938 2939 2940 2941 2943 2944 157.72546 140.81121 115.43983 153.49690 128.12552 149.26833 140.81121 157.72546 149.26833 2945 2946 2947 2948 2949 2950 2951 2952 2953 128.12552 115.43983 280.35377 259.21096 250.75383 191.55396 200.01108 204.23964 178.86827 2954 2955 2956 2957 2958 2959 2960 2961 2962 191.55396 221.15389 200.01108 170.41114 161.95402 200.01108 233.83958 229.61102 208.46821 2963 2964 2965 2966 2967 2968 2969 2970 2971 208.46821 233.83958 238.06814 229.61102 225.38246 259.21096 238.06814 233.83958 280.35377 2972 2973 2974 2975 2976 2977 2978 2979 2980 301.49658 238.06814 221.15389 233.83958 233.83958 233.83958 259.21096 221.15389 250.75383 2981 2982 2983 2984 2985 2986 2987 2988 2989 195.78252 216.92533 246.52527 191.55396 271.89664 216.92533 183.09683 221.15389 229.61102 2990 2991 2992 2993 2994 2995 2996 2997 2998 263.43952 233.83958 216.92533 157.72546 115.43983 157.72546 212.69677 331.09652 276.12521 2999 3000 3001 3002 3003 3004 3005 3006 3007 212.69677 271.89664 238.06814 208.46821 280.35377 271.89664 271.89664 212.69677 174.63971 3008 3009 3010 3011 3012 3013 3014 3015 3016 170.41114 178.86827 166.18258 208.46821 216.92533 212.69677 153.49690 140.81121 216.92533 3017 3018 3019 3020 3021 3022 3023 3024 3025 174.63971 170.41114 200.01108 208.46821 242.29671 191.55396 149.26833 115.43983 161.95402 3026 3027 3028 3029 3030 3031 3032 3033 3034 221.15389 233.83958 170.41114 136.58265 187.32539 191.55396 246.52527 254.98239 174.63971 3035 3036 3037 3038 3039 3040 3041 3042 3043 216.92533 373.38214 390.29639 326.86796 301.49658 280.35377 271.89664 246.52527 242.29671 3044 3045 3046 3047 3048 3049 3051 3052 3054 204.23964 191.55396 204.23964 183.09683 267.66808 246.52527 280.35377 242.29671 297.26802 3055 3056 3057 3058 3059 3060 3061 3062 3063 246.52527 233.83958 242.29671 200.01108 242.29671 208.46821 288.81089 259.21096 221.15389 3064 3065 3066 3067 3068 3069 3070 3071 3072 276.12521 204.23964 229.61102 221.15389 343.78221 293.03946 212.69677 271.89664 221.15389 3073 3074 3075 3076 3077 3079 3080 3081 3082 212.69677 259.21096 233.83958 246.52527 225.38246 271.89664 271.89664 250.75383 246.52527 3083 3084 3085 3086 3087 3088 3089 3090 3091 267.66808 263.43952 229.61102 216.92533 216.92533 170.41114 200.01108 276.12521 195.78252 3092 3093 3094 3095 3096 3097 3098 3099 3100 216.92533 204.23964 174.63971 191.55396 166.18258 153.49690 166.18258 191.55396 212.69677 3101 3102 3103 3104 3105 3106 3107 3108 3109 208.46821 187.32539 212.69677 229.61102 225.38246 191.55396 221.15389 216.92533 221.15389 3110 3111 3112 3113 3114 3115 3116 3117 3118 233.83958 170.41114 254.98239 187.32539 204.23964 212.69677 233.83958 208.46821 216.92533 3119 3121 3122 3123 3124 3126 3127 3128 3129 233.83958 284.58233 145.03977 102.75415 102.75415 145.03977 153.49690 187.32539 161.95402 3130 3131 3132 3133 3134 3135 3136 3137 3138 153.49690 115.43983 123.89696 132.35408 119.66840 132.35408 136.58265 128.12552 170.41114 3139 3140 3141 3142 3143 3144 3145 3146 3147 183.09683 140.81121 128.12552 119.66840 145.03977 140.81121 111.21127 106.98271 106.98271 3148 3149 3150 3151 3152 3153 3154 3155 3156 123.89696 119.66840 123.89696 119.66840 119.66840 123.89696 140.81121 157.72546 149.26833 3157 3158 3159 3160 3161 3162 3163 3164 3165 136.58265 191.55396 132.35408 174.63971 161.95402 132.35408 149.26833 161.95402 145.03977 3166 3167 3168 3169 3170 3171 3172 3173 3174 136.58265 166.18258 140.81121 128.12552 111.21127 128.12552 140.81121 119.66840 140.81121 3175 3176 3177 3178 3180 3181 3182 3183 3184 153.49690 140.81121 153.49690 166.18258 140.81121 119.66840 149.26833 153.49690 149.26833 3186 3187 3188 3189 3191 3192 3193 3194 3195 94.29702 149.26833 115.43983 140.81121 123.89696 166.18258 200.01108 153.49690 191.55396 3196 3197 3198 3199 3200 3201 3202 3204 3205 166.18258 123.89696 115.43983 136.58265 157.72546 170.41114 170.41114 174.63971 191.55396 3206 3207 3208 3209 3210 3211 3212 3213 3214 242.29671 145.03977 140.81121 136.58265 161.95402 157.72546 153.49690 166.18258 140.81121 3215 3216 3217 3218 3219 3220 3221 3222 3223 187.32539 170.41114 140.81121 145.03977 145.03977 140.81121 145.03977 145.03977 174.63971 3224 3225 3226 3227 3228 3229 3230 3231 3232 233.83958 170.41114 153.49690 178.86827 208.46821 259.21096 170.41114 90.06846 102.75415 3233 3234 3235 3236 3237 3238 3239 3240 3241 119.66840 166.18258 174.63971 225.38246 263.43952 326.86796 284.58233 343.78221 301.49658 3242 3243 3244 3245 3246 3247 3250 3251 3252 195.78252 187.32539 174.63971 102.75415 208.46821 178.86827 191.55396 229.61102 178.86827 3253 3254 3255 3256 3257 3258 3259 3260 3261 119.66840 208.46821 183.09683 157.72546 191.55396 183.09683 183.09683 191.55396 157.72546 3262 3263 3264 3265 3266 3267 3268 3269 3270 233.83958 216.92533 204.23964 191.55396 132.35408 161.95402 191.55396 183.09683 191.55396 3271 3272 3273 3274 3276 3277 3278 3279 3280 229.61102 195.78252 229.61102 111.21127 119.66840 102.75415 149.26833 170.41114 166.18258 3281 3282 3283 3284 3285 3286 3287 3288 3289 178.86827 136.58265 276.12521 195.78252 132.35408 149.26833 225.38246 174.63971 149.26833 3290 3291 3292 3293 3294 3295 3296 3297 3298 204.23964 145.03977 128.12552 140.81121 106.98271 183.09683 145.03977 183.09683 153.49690 3299 3300 3301 3302 3303 3304 3305 3306 3307 115.43983 123.89696 178.86827 153.49690 170.41114 191.55396 161.95402 153.49690 183.09683 3308 3309 3310 3311 3312 3313 3314 3315 3316 178.86827 187.32539 170.41114 183.09683 174.63971 161.95402 191.55396 149.26833 153.49690 3317 3318 3319 3320 3321 3322 3323 3324 3325 157.72546 161.95402 119.66840 128.12552 136.58265 153.49690 136.58265 166.18258 170.41114 3326 3327 3328 3329 3330 3331 3332 3333 3334 174.63971 153.49690 166.18258 149.26833 153.49690 170.41114 208.46821 212.69677 170.41114 3335 3337 3338 3339 3340 3341 3342 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2535 262 19 2536 233 22 2537 228 18 2538 250 18 2539 241 16 2540 198 20 2541 291 13 2543 163 11 2544 181 13 2545 104 12 2546 165 15 2547 138 13 2548 166 18 2549 141 16 2550 129 16 2551 196 27 2552 257 25 2553 207 14 2554 346 21 2555 230 26 2556 230 24 2557 315 27 2558 198 15 2559 116 9 2560 150 19 2561 426 19 2562 173 21 2563 163 18 2564 187 17 2565 245 15 2566 119 14 2567 150 16 2568 157 15 2569 222 16 2570 355 16 2571 214 18 2572 157 21 2573 140 24 2574 139 20 2575 137 21 2576 159 11 2577 71 9 2578 69 13 2579 76 20 2580 75 29 2581 98 37 2582 159 44 2583 179 42 2584 228 29 2585 148 20 2586 169 18 2587 91 21 2588 70 25 2589 106 30 2590 139 32 2591 99 27 2592 102 41 2593 118 31 2594 160 27 2595 276 26 2596 104 28 2597 73 20 2598 197 18 2599 169 24 2600 63 28 2601 78 29 2602 78 24 2603 83 24 2604 101 31 2605 100 28 2606 80 20 2607 74 24 2608 58 23 2609 40 21 2610 38 24 2611 54 23 2612 263 36 2613 236 26 2614 239 32 2615 180 27 2616 118 17 2617 132 26 2618 147 30 2619 123 29 2620 175 33 2621 261 36 2622 320 43 2623 354 42 2624 212 35 2625 169 28 2626 129 25 2627 180 43 2628 131 32 2629 156 32 2630 266 40 2631 212 35 2632 202 37 2633 278 45 2634 252 37 2635 117 26 2636 64 18 2637 118 19 2638 199 29 2639 215 34 2640 188 31 2641 287 43 2642 316 43 2643 214 29 2644 166 23 2645 226 25 2646 125 28 2647 202 33 2648 194 38 2649 173 37 2650 162 26 2651 204 40 2652 179 40 2653 203 45 2654 195 44 2655 173 42 2656 163 35 2657 163 36 2658 143 31 2659 257 33 2660 236 27 2661 178 30 2662 215 28 2663 172 21 2664 171 26 2665 144 23 2666 173 34 2667 291 51 2668 226 40 2669 184 35 2670 148 25 2671 143 24 2672 152 23 2673 148 23 2674 185 29 2675 149 25 2676 113 21 2677 91 15 2678 92 14 2679 111 16 2680 113 12 2681 132 21 2682 168 20 2683 190 18 2684 159 14 2685 168 22 2686 178 26 2687 188 23 2688 202 22 2689 211 24 2690 119 27 2691 119 32 2692 159 38 2693 95 43 2694 130 48 2695 136 32 2696 141 45 2697 180 62 2698 146 42 2699 179 39 2700 156 35 2701 157 36 2702 136 37 2703 203 47 2704 169 37 2705 218 42 2706 234 45 2707 227 40 2708 225 36 2709 217 32 2710 237 31 2711 197 37 2712 235 37 2713 210 32 2714 269 45 2715 434 52 2716 446 46 2717 398 71 2718 386 47 2719 622 55 2720 508 55 2721 710 70 2722 535 66 2723 421 59 2724 813 61 2725 792 63 2726 578 45 2727 410 44 2728 343 49 2729 415 61 2730 442 58 2731 370 49 2732 260 48 2733 272 63 2734 279 63 2735 330 58 2736 294 53 2737 308 56 2738 370 63 2739 332 56 2740 348 52 2741 221 51 2742 254 49 2743 315 57 2744 279 34 2745 183 35 2746 150 35 2747 206 40 2748 355 65 2749 273 44 2750 287 30 2751 321 49 2752 242 51 2753 275 51 2754 276 47 2755 352 63 2756 306 53 2757 230 40 2758 360 45 2759 171 38 2760 204 38 2761 247 38 2762 274 42 2763 214 34 2764 256 37 2765 292 45 2766 300 44 2767 283 40 2768 246 42 2769 178 32 2770 213 29 2771 259 37 2772 504 54 2773 514 46 2774 47 24 2775 74 27 2776 64 23 2777 87 26 2778 106 23 2779 70 15 2780 88 25 2781 79 23 2782 59 20 2783 68 20 2784 56 19 2785 72 24 2786 72 22 2787 56 17 2788 52 20 2789 53 15 2790 56 20 2791 85 17 2792 107 17 2793 107 18 2794 110 18 2795 73 14 2796 82 19 2797 120 32 2798 85 22 2799 55 18 2800 52 21 2801 62 24 2802 40 23 2803 51 20 2804 52 20 2805 55 17 2806 55 19 2807 43 16 2808 88 18 2809 51 19 2810 59 19 2811 82 12 2812 84 11 2813 98 19 2814 72 24 2815 79 23 2816 68 17 2817 44 11 2818 51 14 2819 63 19 2820 61 17 2821 54 15 2822 75 22 2823 74 22 2824 71 20 2825 54 22 2826 57 20 2827 53 21 2828 43 19 2829 69 18 2830 78 17 2831 84 14 2832 82 12 2833 91 14 2834 99 17 2835 98 16 2836 98 17 2837 81 15 2838 71 16 2839 87 18 2840 87 22 2841 68 21 2842 99 29 2843 91 30 2844 110 35 2845 109 29 2846 105 32 2847 121 36 2848 112 30 2849 59 18 2850 65 20 2851 101 28 2852 86 24 2853 112 34 2854 84 25 2855 126 39 2856 129 31 2857 100 32 2858 148 32 2859 116 37 2860 157 63 2861 145 27 2862 134 32 2863 128 41 2864 190 32 2865 98 36 2866 201 47 2867 115 32 2868 177 22 2869 158 17 2870 227 24 2871 206 33 2872 221 38 2873 258 35 2874 210 25 2875 162 26 2876 278 44 2877 220 28 2878 246 22 2879 195 33 2880 148 33 2881 245 44 2882 195 44 2883 289 63 2884 305 63 2885 407 51 2886 305 48 2887 397 53 2888 331 56 2889 244 43 2890 413 44 2891 254 40 2892 113 29 2893 81 30 2894 114 40 2895 138 41 2896 115 28 2897 108 30 2898 79 31 2899 105 38 2900 148 38 2901 157 43 2902 210 62 2903 155 39 2904 165 41 2905 193 56 2906 177 42 2907 144 34 2908 200 21 2909 144 27 2910 142 29 2911 84 24 2912 210 36 2913 302 31 2914 499 34 2915 305 24 2916 140 14 2917 91 26 2918 119 28 2919 119 26 2920 144 30 2921 195 50 2922 195 46 2923 125 21 2924 192 25 2925 108 29 2926 105 35 2927 167 28 2928 205 29 2929 138 20 2930 122 21 2931 284 21 2932 109 20 2933 240 26 2934 97 19 2935 117 25 2936 117 21 2937 82 15 2938 122 24 2939 88 18 2940 262 23 2941 158 21 2943 71 25 2944 66 23 2945 74 18 2946 62 15 2947 404 54 2948 402 49 2949 508 47 2950 379 33 2951 197 35 2952 227 36 2953 354 30 2954 262 33 2955 280 40 2956 217 35 2957 164 28 2958 160 26 2959 188 35 2960 337 43 2961 302 42 2962 291 37 2963 242 37 2964 351 43 2965 461 44 2966 298 42 2967 251 41 2968 363 49 2969 281 44 2970 355 43 2971 413 54 2972 420 59 2973 257 44 2974 163 40 2975 232 43 2976 293 43 2977 243 43 2978 345 49 2979 260 40 2980 172 47 2981 132 34 2982 210 39 2983 312 46 2984 259 33 2985 246 52 2986 213 39 2987 129 31 2988 188 40 2989 170 42 2990 189 50 2991 168 43 2992 217 39 2993 161 25 2994 91 15 2995 108 25 2996 196 38 2997 283 66 2998 194 53 2999 211 38 3000 270 52 3001 144 44 3002 150 37 3003 178 54 3004 158 52 3005 204 52 3006 167 38 3007 127 29 3008 65 28 3009 143 30 3010 105 27 3011 159 37 3012 188 39 3013 157 38 3014 69 24 3015 52 21 3016 89 39 3017 122 29 3018 89 28 3019 103 35 3020 125 37 3021 178 45 3022 131 33 3023 91 23 3024 90 15 3025 117 26 3026 250 40 3027 179 43 3028 111 28 3029 83 20 3030 81 32 3031 113 33 3032 168 46 3033 200 48 3034 137 29 3035 114 39 3036 217 76 3037 250 80 3038 212 65 3039 175 59 3040 162 54 3041 181 52 3042 191 46 3043 294 45 3044 162 36 3045 132 33 3046 153 36 3047 108 31 3048 129 51 3049 140 46 3051 181 54 3052 254 45 3054 250 58 3055 181 46 3056 184 43 3057 140 45 3058 156 35 3059 144 45 3060 146 37 3061 354 56 3062 375 49 3063 274 40 3064 299 53 3065 338 36 3066 151 42 3067 210 40 3068 384 69 3069 233 57 3070 270 38 3071 320 52 3072 380 40 3073 356 38 3074 345 49 3075 253 43 3076 278 46 3077 267 41 3079 375 52 3080 389 52 3081 415 47 3082 365 46 3083 483 51 3084 460 50 3085 203 42 3086 274 39 3087 286 39 3088 239 28 3089 345 35 3090 393 53 3091 234 34 3092 309 39 3093 274 36 3094 315 29 3095 434 33 3096 400 27 3097 337 24 3098 374 27 3099 258 33 3100 285 38 3101 345 37 3102 220 32 3103 192 38 3104 191 42 3105 329 41 3106 449 33 3107 377 40 3108 399 39 3109 376 40 3110 306 43 3111 564 28 3112 526 48 3113 250 32 3114 178 36 3115 239 38 3116 262 43 3117 185 37 3118 230 39 3119 296 43 3121 450 55 3122 87 22 3123 188 12 3124 280 12 3126 63 22 3127 68 24 3128 68 32 3129 52 26 3130 36 24 3131 32 15 3132 53 17 3133 72 19 3134 117 16 3135 129 19 3136 91 20 3137 79 18 3138 89 28 3139 58 31 3140 54 21 3141 53 18 3142 59 16 3143 73 22 3144 64 21 3145 50 14 3146 36 13 3147 39 13 3148 42 17 3149 38 16 3150 58 17 3151 64 16 3152 65 16 3153 78 17 3154 72 21 3155 72 25 3156 65 23 3157 72 20 3158 88 33 3159 63 19 3160 89 29 3161 70 26 3162 67 19 3163 82 23 3164 76 26 3165 54 22 3166 55 20 3167 73 27 3168 70 21 3169 52 18 3170 79 14 3171 114 18 3172 123 21 3173 81 16 3174 66 21 3175 106 24 3176 94 21 3177 114 24 3178 113 27 3180 153 21 3181 229 16 3182 200 23 3183 140 24 3184 115 23 3186 240 10 3187 155 23 3188 116 15 3189 111 21 3191 106 17 3192 112 27 3193 192 35 3194 87 24 3195 77 33 3196 53 27 3197 64 17 3198 65 15 3199 55 20 3200 106 25 3201 117 28 3202 128 28 3204 95 29 3205 96 33 3206 105 45 3207 106 22 3208 73 21 3209 53 20 3210 90 26 3211 127 25 3212 82 24 3213 105 27 3214 98 21 3215 142 32 3216 155 28 3217 80 21 3218 74 22 3219 86 22 3220 104 21 3221 76 22 3222 108 22 3223 188 29 3224 313 43 3225 244 28 3226 245 24 3227 177 30 3228 194 37 3229 196 49 3230 259 28 3231 142 9 3232 128 12 3233 106 16 3234 93 27 3235 112 29 3236 108 41 3237 105 50 3238 159 65 3239 166 55 3240 215 69 3241 250 59 3242 196 34 3243 238 32 3244 177 29 3245 168 12 3246 267 37 3247 267 30 3250 156 33 3251 104 42 3252 123 30 3253 110 16 3254 157 37 3255 98 31 3256 190 25 3257 128 33 3258 182 31 3259 137 31 3260 289 33 3261 209 25 3262 160 43 3263 430 39 3264 223 36 3265 150 33 3266 372 19 3267 170 26 3268 56 33 3269 85 31 3270 105 33 3271 189 42 3272 116 34 3273 140 42 3274 85 14 3276 263 16 3277 122 12 3278 91 23 3279 80 28 3280 104 27 3281 121 30 3282 148 20 3283 141 53 3284 138 34 3285 121 19 3286 68 23 3287 84 41 3288 88 29 3289 219 23 3290 75 36 3291 98 22 3292 109 18 3293 121 21 3294 71 13 3295 76 31 3296 88 22 3297 188 31 3298 94 24 3299 112 15 3300 109 17 3301 149 30 3302 216 24 3303 176 28 3304 318 33 3305 229 26 3306 306 24 3307 295 31 3308 220 30 3309 172 32 3310 188 28 3311 186 31 3312 266 29 3313 212 26 3314 217 33 3315 178 23 3316 184 24 3317 139 25 3318 112 26 3319 95 16 3320 127 18 3321 119 20 3322 127 24 3323 133 20 3324 167 27 3325 150 28 3326 161 29 3327 288 24 3328 172 27 3329 138 23 3330 111 24 3331 162 28 3332 213 37 3333 291 38 3334 189 28 3335 130 27 3337 233 40 3338 276 45 3339 240 36 3340 300 53 3341 188 33 3342 215 33 3343 301 43 3344 187 33 3345 148 35 3346 135 26 3347 89 14 3348 257 54 3349 247 39 3350 159 48 3351 274 41 3352 179 43 3353 188 55 3355 87 34 3356 135 31 3357 226 41 3358 182 27 3359 66 27 3360 63 26 3361 115 20 3362 131 35 3363 128 34 3364 141 48 3365 162 67 3366 91 40 3367 38 19 3368 60 27 3369 82 25 3370 81 23 3371 125 38 3372 132 35 3373 133 29 3374 104 17 3375 159 31 3376 195 32 3377 209 38 3378 214 40 3379 138 24 3380 155 33 3382 78 21 3383 130 26 3384 98 5 3386 131 21 3387 115 17 3388 99 26 3389 123 27 3390 188 30 3392 56 14 3393 97 26 3394 123 25 3395 189 63 3396 193 41 3397 258 37 3398 226 44 3399 252 40 3400 271 60 3401 244 50 3402 807 42 3404 363 57 3405 389 35 3406 327 29 3407 285 35 3408 324 26 3409 315 21 3410 270 26 3411 249 23 3412 271 23 3413 241 14 3414 331 26 3415 282 28 3416 335 31 3417 343 27 3418 319 22 3419 415 31 3420 389 29 3421 237 24 3422 269 22 3423 167 30 3424 367 28 3425 352 25 3426 413 29 3427 241 19 3428 327 31 3429 279 23 3430 248 27 3431 309 35 3432 219 37 3433 269 34 3434 360 50 3435 318 43 3436 409 58 3437 271 38 3438 199 22 3439 286 24 3440 163 30 3441 213 28 3442 270 34 3443 197 29 3444 170 24 3445 262 27 3446 380 33 3447 259 32 3448 272 24 3449 414 25 3452 80 31 3453 129 14 3454 141 13 3455 170 12 3456 473 16 3457 159 12 3458 162 21 3459 353 14 3460 316 22 3461 448 21 3462 215 12 3463 188 22 3464 157 26 3465 473 30 3467 94 33 3468 94 25 3469 116 25 3470 146 28 3471 125 28 3472 138 26 3473 260 25 3474 225 18 3475 340 16 3476 220 21 3477 120 19 3478 100 16 3479 100 23 3480 92 18 3481 70 24 3482 56 18 3483 40 21 3484 63 20 3485 81 18 3486 97 23 3487 122 26 3488 85 18 3489 52 19 3490 81 21 3493 88 21 3494 65 19 3495 53 18 3497 152 31 3498 125 25 3500 91 18 3501 77 19 3502 69 23 3503 73 15 3504 95 11 3505 90 9 3506 46 9 3507 41 10 3509 185 30 3510 161 21 3511 134 22 3513 98 22 3514 88 21 3515 90 32 3516 92 30 3518 52 14 3519 51 13 3520 41 10 3521 41 15 3522 75 17 3523 120 11 3524 68 14 3525 83 16 3526 71 17 3527 81 20 3528 81 12 3530 57 12 3531 113 22 3532 86 13 3533 96 14 3534 93 18 3535 93 14 3537 118 11 3538 133 15 3539 85 11 3540 106 10 3542 94 15 3543 97 11 3544 170 23 3545 108 13 3546 150 12 3547 118 21 3548 205 17 3549 129 15 3550 87 8 3551 122 15 3552 114 15 3553 142 19 3554 177 28 3555 147 21 3556 115 19 3557 248 46 3558 199 29 3559 134 22 3560 77 14 3561 91 12 3563 147 14 3564 138 17 3565 269 23 3566 251 24 3567 204 19 3568 176 19 3569 252 21 3570 270 24 3571 253 27 3572 283 23 3573 257 16 3574 142 11 3575 219 16 3576 123 16 3577 107 21 3578 154 24 3579 232 20 3580 162 20 3581 245 20 3582 98 15 3583 164 17 3584 68 13 3585 71 12 3586 119 14 3587 125 13 3588 163 17 3589 123 12 3590 104 11 3591 117 14 3592 123 15 3593 169 19 3594 162 13 3595 93 13 3596 94 12 3597 65 13 3598 137 24 3599 169 22 3600 166 26 3601 362 21 3602 135 8 3603 97 7 3604 154 12 3605 165 16 3606 163 20 3607 161 17 3608 142 14 3610 170 14 3611 163 11 3612 173 10 3613 118 8 3614 102 8 3616 119 13 3617 109 7 3619 146 22 3621 146 30 3622 194 49 3623 253 63 3624 480 67 3625 661 74 3626 320 53 3627 299 42 3628 384 60 3629 551 64 3630 602 65 3631 202 39 3632 233 34 3633 319 3 3635 456 91 3636 484 37 3637 253 15 3639 101 13 3640 85 10 3641 102 12 3642 120 11 3643 127 11 3644 135 12 3645 163 20 3646 185 18 3647 241 21 3648 326 25 3649 301 25 3650 332 16 3651 239 21 3652 367 21 3653 190 19 3654 137 9 3655 202 28 3656 125 8 3657 113 5 3658 131 18 3659 270 20 3660 228 19 3661 234 17 3662 195 17 3663 83 13 3664 175 17 3665 170 11 3666 122 10 3667 192 9 3668 351 26 3669 395 30 3670 225 16 3671 376 23 3672 707 22 3673 129 27 3674 234 31 3675 253 39 3676 121 19 3677 69 22 3678 85 19 3679 95 16 3680 144 23 3681 145 28 3682 146 25 3683 111 23 3684 127 24 3685 118 23 3686 123 23 3687 61 24 3688 80 14 3689 69 23 3690 92 17 3691 126 18 3692 185 28 3693 141 22 3694 146 16 3695 91 11 3696 98 17 3697 114 25 3698 149 30 3699 257 30 3700 184 23 3701 82 20 3702 133 23 3703 129 46 3704 118 32 3705 131 33 3706 99 20 3707 86 13 3708 105 18 3709 172 25 3710 369 28 3711 73 13 3712 67 18 3713 98 12 3714 72 13 3715 139 23 3716 517 5 3717 22 22 3718 37 10 3719 34 9 3720 80 21 3721 49 4 3722 61 5 3723 60 8 3724 98 20 3725 675 7 3728 91 21 3729 99 26 3730 99 25 3731 93 22 3732 87 18 3733 113 25 3734 118 31 3735 121 34 3736 145 29 3737 167 27 3738 169 31 3739 75 14 3740 71 15 3741 72 23 3742 55 15 3743 65 17 3744 94 20 3745 206 15 3746 266 17 3747 281 15 3748 253 15 3749 189 15 3750 135 19 3751 68 25 3752 52 18 3753 56 19 3754 93 22 3755 85 23 3756 67 18 3757 83 22 3758 104 23 3759 46 17 3760 46 18 3761 42 12 3762 49 15 3763 60 18 3764 58 16 3765 57 15 3766 61 18 3767 67 18 3768 72 18 3769 49 19 3770 59 19 3771 47 17 3772 47 13 3773 36 11 3774 45 11 3775 41 14 3776 39 15 3777 52 19 3778 54 16 3779 30 14 3780 15 8 3781 10 6 3782 65 18 3783 75 16 3784 93 18 3785 81 20 3786 82 17 3787 89 20 3788 54 14 3789 45 15 3790 63 18 3791 62 16 3792 61 14 3793 67 16 3794 83 23 3795 62 20 3796 106 25 3797 107 25 3798 64 15 3799 65 15 3800 60 14 3801 72 17 3802 99 20 3803 130 26 3804 94 18 3805 60 14 3806 52 12 3807 56 12 3808 90 22 3809 90 25 3810 63 15 3811 47 13 3812 62 16 3813 76 14 3814 51 17 3815 56 16 3816 68 18 3817 55 18 3818 47 13 3819 37 3 3820 51 6 3824 110 30 3825 152 39 3826 157 40 3827 181 38 3828 187 33 3829 182 31 3830 124 33 3831 151 32 3832 108 28 3833 152 31 3834 103 24 3835 44 14 3836 66 25 3837 103 30 3838 136 36 3839 119 37 3840 172 41 3841 170 41 3842 164 35 3843 176 29 3844 160 37 3845 176 30 3846 159 33 3847 247 72 3848 192 45 3849 149 35 3850 93 29 3851 127 34 3852 142 32 3853 161 37 3854 206 36 3855 285 33 3856 167 23 3857 178 32 3858 238 29 3859 225 34 3860 85 22 3861 68 21 3862 78 10 3863 75 18 3864 399 22 3865 141 38 3866 93 29 3867 109 30 3868 172 84 3869 121 27 3870 90 20 3871 147 12 3872 156 17 3873 239 35 3874 139 20 3875 207 32 3876 147 31 3877 226 39 3878 323 43 3879 332 33 3880 184 25 3881 196 29 3882 221 27 3883 64 19 3884 47 17 3885 66 22 3886 66 19 3887 62 25 3888 58 19 3889 51 18 3890 74 24 3891 56 21 3892 98 28 3893 135 27 3894 245 49 3895 462 52 3896 243 39 3897 293 37 3898 42 31 3899 163 9 3900 121 12 3901 145 31 3902 111 34 3903 104 14 3904 425 66 3905 187 36 3906 37 29 3907 62 15 3908 163 48 3909 101 17 3910 450 50 3911 82 23 3912 71 17 3913 64 20 3914 216 25 3915 93 27 3916 61 16 3917 90 10 3918 122 13 3919 88 14 3920 130 23 3921 270 27 3922 124 15 3923 124 15 3924 367 29 3925 199 45 3926 145 22 ``` ] * This kind of customized output is achieved by _methods_ --- * Methods are specific implementations of a generic function customized to its input * The appropriate method is chosen by looking at the _class_ of the input argument -- * The methods available for a generic function can be obtained using the `methods()` function .scrollable400[ ```r methods("summary") ``` ``` [1] summary.aov summary.aovlist* [3] summary.aspell* summary.check_packages_in_dir* [5] summary.connection summary.data.frame [7] summary.Date summary.default [9] summary.ecdf* summary.factor [11] summary.glm summary.hcl_palettes* [13] summary.infl* summary.lm [15] summary.loess* summary.manova [17] summary.matrix summary.mlm* [19] summary.nls* summary.packageStatus* [21] summary.POSIXct summary.POSIXlt [23] summary.ppr* summary.prcomp* [25] summary.princomp* summary.proc_time [27] summary.rlang_error* summary.rlang_message* [29] summary.rlang_trace* summary.rlang_warning* [31] summary.rlang:::list_of_conditions* summary.srcfile [33] summary.srcref summary.stepfun [35] summary.stl* summary.table [37] summary.tukeysmooth* summary.warnings see '?methods' for accessing help and source code ``` ] --- .scrollable500[ ```r methods("print") # similar but much longer list ``` ``` [1] print.acf* [2] print.activeConcordance* [3] print.AES* [4] print.anova* [5] print.aov* [6] print.aovlist* [7] print.ar* [8] print.Arima* [9] print.arima0* [10] print.AsIs [11] print.aspell* [12] print.aspell_inspect_context* [13] print.bibentry* [14] print.Bibtex* [15] print.browseVignettes* [16] print.by [17] print.changedFiles* [18] print.check_bogus_return* [19] print.check_code_usage_in_package* [20] print.check_compiled_code* [21] print.check_demo_index* [22] print.check_depdef* [23] print.check_details* [24] print.check_details_changes* [25] print.check_doi_db* [26] print.check_dotInternal* [27] print.check_make_vars* [28] print.check_nonAPI_calls* [29] print.check_package_code_assign_to_globalenv* [30] print.check_package_code_attach* [31] print.check_package_code_data_into_globalenv* [32] print.check_package_code_startup_functions* [33] print.check_package_code_syntax* [34] print.check_package_code_unload_functions* [35] print.check_package_compact_datasets* [36] print.check_package_CRAN_incoming* [37] print.check_package_datalist* [38] print.check_package_datasets* [39] print.check_package_depends* [40] print.check_package_description* [41] print.check_package_description_encoding* [42] print.check_package_license* [43] print.check_packages_in_dir* [44] print.check_packages_used* [45] print.check_po_files* [46] print.check_pragmas* [47] print.check_Rd_line_widths* [48] print.check_Rd_metadata* [49] print.check_Rd_xrefs* [50] print.check_RegSym_calls* [51] print.check_S3_methods_needing_delayed_registration* [52] print.check_so_symbols* [53] print.check_T_and_F* [54] print.check_url_db* [55] print.check_vignette_index* [56] print.checkDocFiles* [57] print.checkDocStyle* [58] print.checkFF* [59] print.checkRd* [60] print.checkRdContents* [61] print.checkReplaceFuns* [62] print.checkS3methods* [63] print.checkTnF* [64] print.checkVignettes* [65] print.citation* [66] print.cli_ansi_html_style* [67] print.cli_ansi_string* [68] print.cli_ansi_style* [69] print.cli_boxx* [70] print.cli_diff_chr* [71] print.cli_doc* [72] print.cli_progress_demo* [73] print.cli_rule* [74] print.cli_sitrep* [75] print.cli_spark* [76] print.cli_spinner* [77] print.cli_tree* [78] print.codoc* [79] print.codocClasses* [80] print.codocData* [81] print.colorConverter* [82] print.compactPDF* [83] print.condition [84] print.connection [85] print.CRAN_package_reverse_dependencies_and_views* [86] print.data.frame [87] print.Date [88] print.default [89] print.dendrogram* [90] print.density* [91] print.difftime [92] print.dist* [93] print.Dlist [94] print.DLLInfo [95] print.DLLInfoList [96] print.DLLRegisteredRoutines [97] print.document_context* [98] print.document_position* [99] print.document_range* [100] print.document_selection* [101] print.dummy_coef* [102] print.dummy_coef_list* [103] print.ecdf* [104] print.eigen [105] print.factanal* [106] print.factor [107] print.family* [108] print.fileSnapshot* [109] print.findLineNumResult* [110] print.font_face* [111] print.font_feature* [112] print.formula* [113] print.fseq* [114] print.ftable* [115] print.function [116] print.getAnywhere* [117] print.glm* [118] print.glue* [119] print.hashtab* [120] print.hcl_palettes* [121] print.hclust* [122] print.help_files_with_topic* [123] print.hexmode [124] print.HoltWinters* [125] print.hsearch* [126] print.hsearch_db* [127] print.htest* [128] print.html* [129] print.html_dependency* [130] print.htmltools.selector* [131] print.htmltools.selector.list* [132] print.infl* [133] print.integrate* [134] print.isoreg* [135] print.kableExtra* [136] print.kableExtraInlinePlots* [137] print.key_missing* [138] print.kmeans* [139] print.knitr_kable* [140] print.Latex* [141] print.LaTeX* [142] print.libraryIQR [143] print.lifecycle_warnings* [144] print.listof [145] print.lm* [146] print.loadings* [147] print.loess* [148] print.logLik* [149] print.ls_str* [150] print.medpolish* [151] print.MethodsFunction* [152] print.mtable* [153] print.NativeRoutineList [154] print.news_db* [155] print.nls* [156] print.noquote [157] print.numeric_version [158] print.object_size* [159] print.octmode [160] print.packageDescription* [161] print.packageInfo [162] print.packageIQR* [163] print.packageStatus* [164] print.pairwise.htest* [165] print.person* [166] print.POSIXct [167] print.POSIXlt [168] print.power.htest* [169] print.ppr* [170] print.prcomp* [171] print.princomp* [172] print.proc_time [173] print.quosure* [174] print.quosures* [175] print.R6* [176] print.R6ClassGenerator* [177] print.raster* [178] print.Rconcordance* [179] print.Rd* [180] print.recordedplot* [181] print.restart [182] print.RGBcolorConverter* [183] print.RGlyphFont* [184] print.rlang_box_done* [185] print.rlang_box_splice* [186] print.rlang_data_pronoun* [187] print.rlang_dict* [188] print.rlang_dyn_array* [189] print.rlang_envs* [190] print.rlang_error* [191] print.rlang_fake_data_pronoun* [192] print.rlang_lambda_function* [193] print.rlang_message* [194] print.rlang_trace* [195] print.rlang_warning* [196] print.rlang_zap* [197] print.rlang:::list_of_conditions* [198] print.rle [199] print.rlib_bytes* [200] print.rlib_error_3_0* [201] print.rlib_trace_3_0* [202] print.roman* [203] print.sessionInfo* [204] print.shiny.tag* [205] print.shiny.tag.env* [206] print.shiny.tag.list* [207] print.shiny.tag.query* [208] print.simple.list [209] print.smooth.spline* [210] print.socket* [211] print.srcfile [212] print.srcref [213] print.stepfun* [214] print.stl* [215] print.stringr_view* [216] print.StructTS* [217] print.subdir_tests* [218] print.summarize_CRAN_check_status* [219] print.summary.aov* [220] print.summary.aovlist* [221] print.summary.ecdf* [222] print.summary.glm* [223] print.summary.lm* [224] print.summary.loess* [225] print.summary.manova* [226] print.summary.nls* [227] print.summary.packageStatus* [228] print.summary.ppr* [229] print.summary.prcomp* [230] print.summary.princomp* [231] print.summary.table [232] print.summary.warnings [233] print.summaryDefault [234] print.svg* [235] print.table [236] print.tables_aov* [237] print.terms* [238] print.trans* [239] print.ts* [240] print.tskernel* [241] print.TukeyHSD* [242] print.tukeyline* [243] print.tukeysmooth* [244] print.undoc* [245] print.vignette* [246] print.warnings [247] print.xfun_raw_string* [248] print.xfun_record_results* [249] print.xfun_rename_seq* [250] print.xfun_strict_list* [251] print.xgettext* [252] print.xml_document* [253] print.xml_missing* [254] print.xml_namespace* [255] print.xml_node* [256] print.xml_nodeset* [257] print.xngettext* [258] print.xtabs* see '?methods' for accessing help and source code ``` ] --- * All available methods for a given class can be similarly obtained ```r methods(class = "lm") ``` ``` [1] add1 alias anova case.names coerce [6] confint cooks.distance deviance dfbeta dfbetas [11] drop1 dummy.coef effects extractAIC family [16] formula hatvalues influence initialize kappa [21] labels logLik model.frame model.matrix nobs [26] plot predict print proj qr [31] residuals rstandard rstudent show simulate [36] slotsFromS3 summary variable.names vcov see '?methods' for accessing help and source code ``` --- * The name of a specific method appears to have the form `generic.class` * However, one should always call the generic function, not the method directly * This is not OK: ```r summary.lm(fm1) ``` * Instead, use ```r summary(fm1) ``` * In fact, many methods cannot be called directly because they are "hidden" --- * This is a form of _Object Oriented Programming_ (OOP) in R * Python also has OOP, but * Methods are usually tied to a class, not a _generic_ function * One notable exception is the `__str__()` method, which is used by `print()` --- layout: false # Getting help * R has an extensive collection of functions (even more if we include add-on packages) * It is impossible for anyone to know them all, or remember details * Fortunately, R also has an excellent help system -- * Every function and dataset in R (and add-on packages) must be documented * The documentation can be accessed by `help(name)` or `?name` * For example: `help(seq)`, `help(summary)`, etc. * A more general (but limited) search can be performed using `help.search("search-string")` --- # Getting help * How the help is shown depends on the _interface_ being used * RStudio has a separate help tab (which also allows searching) -- * However, before using the help system, you should know how methods are documented --- # Help for generic functions and methods * Generic functions and methods are distinct functions * They often have different help pages * In fact, many add-on packages define new methods for generics in another package * These are always documented in a separate help page --- # Help for generic functions and methods * To get help for the generic function `summary()`, type `help(summary)` * To get help for the `summary()` method for "matrix" objects, type `help(summary.matrix)` * To get help for the `summary()` method for "lm" objects, type `help(summary.lm)` * The first two happen to be the same help page, but the third is different -- * This is slightly confusing because you are __not__ supposed to call `summary.lm()` directly * More importantly, there may not actually be a `summary()` method for all classes * For example, "list" objects are handled by a _fallback_ method `summary.default()` -- * The list of available methods are obtained by `methods("summary")` as shown earlier * All these should have a corresponding help page --- # Help for generic functions and methods - The system we described is called "S3" (short for "S version 3") - The documentation refers to specific methods implemented using this system as "S3 methods" - To make things more complicated, there are other systems of defining classes and methods - We will skip the details of these for now --- # Replacement Functions * R functions are not allowed to modify its arguments -- * Consider a function that sets negative inputs to 0 $$ f(x) = \begin{cases} 0 & \text{ if } x < 0 \cr x & \text{ otherwise. } \end{cases} $$ -- * We will refer to this as the *ReLU* function --- # The Scalar ReLU Function in R ```r sReLU <- function(u) { if (u < 0) u = 0 u } ``` ??? Now suppose we want to implement this function for scalar input. This is a very natural implementation. -- ```r x <- -5 sReLU(x) ``` ``` [1] 0 ``` ```r x ``` ``` [1] -5 ``` ??? Let's see what happens when we call it. If we just call the function with a variable X as input, it returns the expected value, but the variable X does not change. --- # The Scalar ReLU Function in R ```r sReLU <- function(u) { if (u < 0) u = 0 u } ``` ```r x <- -5 y <- sReLU(x) ``` -- ```r x ``` ``` [1] -5 ``` ```r y ``` ``` [1] 0 ``` ??? This time, when we assign the value of the function to a new variable Y, X remains unchanged but Y now has the expected value. This is of course more or less what we would expect. --- # The Scalar ReLU Function in Python ```python def sReLU(u): if u < 0: u = 0 return u ``` -- ```python x = -5 y = sReLU(x) ``` -- ```python x ``` ``` -5 ``` ```python y ``` ``` 0 ``` --- # The Vectorized ReLU Function in Python ```python def vReLU(u): for i in range(len(u)): if u[i] < 0: u[i] = 0 return u ``` -- ```python from numpy import * x = random.normal(size = 10) y = x x ``` ``` array([-1.2993973 , -1.11465781, -0.31597779, -1.29471625, 0.38419193, 0.31589083, 1.17404954, 0.06097466, 0.33193044, 0.62444228]) ``` ```python y ``` ``` array([-1.2993973 , -1.11465781, -0.31597779, -1.29471625, 0.38419193, 0.31589083, 1.17404954, 0.06097466, 0.33193044, 0.62444228]) ``` --- # The Vectorized ReLU Function in Python ```python z = vReLU(y) ``` ??? When we now call the function with Y as argument and assign the result to a NEW variable called Z, several interesting things happen. -- ```python z ``` ``` array([0. , 0. , 0. , 0. , 0.38419193, 0.31589083, 1.17404954, 0.06097466, 0.33193044, 0.62444228]) ``` ??? First of all, Z has the negative values set to 0, as expected. -- ```python y ``` ``` array([0. , 0. , 0. , 0. , 0.38419193, 0.31589083, 1.17404954, 0.06097466, 0.33193044, 0.62444228]) ``` ??? But Y _also_ has the same values set to 0. Even though this behaviour is not consistent with the scalar case, it is not _too_ surprising, because the argument Y WAS actually modified in the function. -- ```python x ``` ``` array([0. , 0. , 0. , 0. , 0.38419193, 0.31589083, 1.17404954, 0.06097466, 0.33193044, 0.62444228]) ``` ??? Perhaps more surprisingly, the value of _X_ has also changed, even though it was not involved in the call at all. There is actually a very good reason for this behaviour, and depending on your PRIOR experience with programming, you may find this perfectly natural. But this behaviour is often surprising to beginners, and R avoids it by following what is known as a FUNCTIONAL programming paradigm, which says that functions should not modify its arguments. --- # The Vectorized ReLU Function in R ```r vReLU <- function(u) { for (i in seq_len(length(u))) { if (u[i] < 0) u[i] <- 0 } return(u) } ``` ??? Here is the equivalent function in R. -- ```r x <- rnorm(10) y <- x ``` -- ```r x ``` ``` [1] -0.5915339 -0.1111574 1.5757390 -1.5187443 -0.3405556 1.4324608 1.3858898 [8] 0.8068037 -0.6983363 -0.8147241 ``` ```r y ``` ``` [1] -0.5915339 -0.1111574 1.5757390 -1.5187443 -0.3405556 1.4324608 1.3858898 [8] 0.8068037 -0.6983363 -0.8147241 ``` ??? As before, we first generate a normal random vector X and make a copy named Y. --- # The Vectorized ReLU Function in R ```r z <- vReLU(y) z ``` ``` [1] 0.0000000 0.0000000 1.5757390 0.0000000 0.0000000 1.4324608 1.3858898 0.8068037 [9] 0.0000000 0.0000000 ``` -- ```r y ``` ``` [1] -0.5915339 -0.1111574 1.5757390 -1.5187443 -0.3405556 1.4324608 1.3858898 [8] 0.8068037 -0.6983363 -0.8147241 ``` ```r x ``` ``` [1] -0.5915339 -0.1111574 1.5757390 -1.5187443 -0.3405556 1.4324608 1.3858898 [8] 0.8068037 -0.6983363 -0.8147241 ``` ??? This time, when we run the function with Y as argument and assign the result to Z, only Z is modified. X and Y remain unchanged. --- # Alternative: Direct Assignment * We can instead use direct assignment to a 'subset' ```r y[y < 0] <- 0 ``` -- ```r x ``` ``` [1] -0.5915339 -0.1111574 1.5757390 -1.5187443 -0.3405556 1.4324608 1.3858898 [8] 0.8068037 -0.6983363 -0.8147241 ``` ```r y ``` ``` [1] 0.0000000 0.0000000 1.5757390 0.0000000 0.0000000 1.4324608 1.3858898 0.8068037 [9] 0.0000000 0.0000000 ``` --- # Functional programming * R generally follows a _functional programming_ paradigm * Among other things, this says that functions should not modify its arguments -- * This is a key difference between Python and R -- * One consequence: R needs an _unusual_ approach when it needs to modify objects --- layout: true # Example: Modifying Names of a Data Frame --- ```r d <- data.frame(1, rnorm(5), rexp(5)) names(d) ``` ``` [1] "X1" "rnorm.5." "rexp.5." ``` ```r d ``` ``` X1 rnorm.5. rexp.5. 1 1 0.5137902 2.3629231 2 1 -1.2638761 0.3281384 3 1 2.4951138 2.8728996 4 1 -1.7135097 0.7648846 5 1 0.2733856 0.5005642 ``` * Default names are not very nice --- * Want to change the names to `"Constant", "Normal", "Exponential"` * Possible solution using the `setNames()` function ```r setNames(d, c("Constant", "Normal", "Exponential")) ``` ``` Constant Normal Exponential 1 1 0.5137902 2.3629231 2 1 -1.2638761 0.3281384 3 1 2.4951138 2.8728996 4 1 -1.7135097 0.7648846 5 1 0.2733856 0.5005642 ``` --- * But names of `d` are not modified by this ```r d ``` ``` X1 rnorm.5. rexp.5. 1 1 0.5137902 2.3629231 2 1 -1.2638761 0.3281384 3 1 2.4951138 2.8728996 4 1 -1.7135097 0.7648846 5 1 0.2733856 0.5005642 ``` -- * Best we can hope for ```r d <- setNames(d, c("Constant", "Normal", "Exponential")) ``` --- * In fact, the __dplyr__ package has a more convenient version of this approach ```r d <- dplyr::rename(d, Constant = X1, Normal = rnorm.5., Exponential = rexp.5.) d ``` ``` Constant Normal Exponential 1 1 0.5137902 2.3629231 2 1 -1.2638761 0.3281384 3 1 2.4951138 2.8728996 4 1 -1.7135097 0.7648846 5 1 0.2733856 0.5005642 ``` --- * The classic R Alternative: Replacement Functions ```r names(d) <- c("Constant", "Normal", "Exponential") d ``` ``` Constant Normal Exponential 1 1 0.5137902 2.3629231 2 1 -1.2638761 0.3281384 3 1 2.4951138 2.8728996 4 1 -1.7135097 0.7648846 5 1 0.2733856 0.5005642 ``` --- layout: false # Replacement Functions * Other similar examples: ```r y[ y < 0 ] <- 0 class(x) <- NULL d$Normal[ d$Normal < 0 ] <- 0 attr(x, "name") <- value ``` * Common feature: "complex" expression on the LHS of the assignment