The crps() and scrps() functions and their loo_*() counterparts can be
used to compute the continuously ranked probability score (CRPS) and scaled
CRPS (SCRPS) (see Bolin and Wallin, 2022). CRPS is a proper scoring rule, and
strictly proper when the first moment of the predictive distribution is
finite. Both can be expressed in terms of samples form the predictive
distribution. See e.g. Gneiting and Raftery (2007) for a comprehensive
discussion on CRPS.
crps(x, ...)
scrps(x, ...)
loo_crps(x, ...)
loo_scrps(x, ...)
# S3 method for matrix
crps(x, x2, y, ..., permutations = 1)
# S3 method for numeric
crps(x, x2, y, ..., permutations = 1)
# S3 method for matrix
loo_crps(
x,
x2,
y,
log_lik,
...,
permutations = 1,
r_eff = 1,
cores = getOption("mc.cores", 1)
)
# S3 method for matrix
scrps(x, x2, y, ..., permutations = 1)
# S3 method for numeric
scrps(x, x2, y, ..., permutations = 1)
# S3 method for matrix
loo_scrps(
x,
x2,
y,
log_lik,
...,
permutations = 1,
r_eff = 1,
cores = getOption("mc.cores", 1)
)A S by N matrix (draws by observations), or a vector of length
S when only single observation is provided in y.
Passed on to E_loo() in the loo_*() version of these
functions.
Independent draws from the same distribution as draws in x.
Should be of the identical dimension.
A vector of observations or a single value.
An integer, with default value of 1, specifying how many
times the expected value of |X - X'| (|x - x2|) is computed. The row
order of x2 is shuffled as elements x and x2 are typically drawn
given the same values of parameters. This happens, e.g., when one calls
posterior_predict() twice for a fitted rstanarm or brms
model. Generating more permutations is expected to decrease the variance of
the computed expected value.
A log-likelihood matrix the same size as x.
An optional vector of relative effective sample size estimates
containing one element per observation. See psis() for details.
The number of cores to use for parallelization of [psis()].
See psis() for details.
A list containing two elements: estimates and pointwise.
The former reports estimator and standard error and latter the pointwise
values.
To compute (S)CRPS, the user needs to provide two sets of draws, x and
x2, from the predictive distribution. This is due to the fact that formulas
used to compute CRPS involve an expectation of the absolute difference of x
and x2, both having the same distribution. See the permutations argument,
as well as Gneiting and Raftery (2007) for details.
Bolin, D., & Wallin, J. (2022). Local scale invariance and robustness of proper scoring rules. arXiv. doi:10.48550/arXiv.1912.05642
Gneiting, T., & Raftery, A. E. (2007). Strictly Proper Scoring Rules, Prediction, and Estimation. Journal of the American Statistical Association, 102(477), 359–378.
# \dontrun{
# An example using rstanarm
library(rstanarm)
data("kidiq")
fit <- stan_glm(kid_score ~ mom_hs + mom_iq, data = kidiq)
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 0.000213 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 2.13 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1:
#> Chain 1:
#> Chain 1: Iteration: 1 / 2000 [ 0%] (Warmup)
#> Chain 1: Iteration: 200 / 2000 [ 10%] (Warmup)
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#> Chain 1: Iteration: 1800 / 2000 [ 90%] (Sampling)
#> Chain 1: Iteration: 2000 / 2000 [100%] (Sampling)
#> Chain 1:
#> Chain 1: Elapsed Time: 0.076 seconds (Warm-up)
#> Chain 1: 0.09 seconds (Sampling)
#> Chain 1: 0.166 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 1.8e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.18 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2:
#> Chain 2:
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#> Chain 2: Iteration: 2000 / 2000 [100%] (Sampling)
#> Chain 2:
#> Chain 2: Elapsed Time: 0.1 seconds (Warm-up)
#> Chain 2: 0.1 seconds (Sampling)
#> Chain 2: 0.2 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 1.5e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.15 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3:
#> Chain 3:
#> Chain 3: Iteration: 1 / 2000 [ 0%] (Warmup)
#> Chain 3: Iteration: 200 / 2000 [ 10%] (Warmup)
#> Chain 3: Iteration: 400 / 2000 [ 20%] (Warmup)
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#> Chain 3: Iteration: 2000 / 2000 [100%] (Sampling)
#> Chain 3:
#> Chain 3: Elapsed Time: 0.076 seconds (Warm-up)
#> Chain 3: 0.08 seconds (Sampling)
#> Chain 3: 0.156 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 1.5e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.15 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4:
#> Chain 4:
#> Chain 4: Iteration: 1 / 2000 [ 0%] (Warmup)
#> Chain 4: Iteration: 200 / 2000 [ 10%] (Warmup)
#> Chain 4: Iteration: 400 / 2000 [ 20%] (Warmup)
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#> Chain 4: Iteration: 2000 / 2000 [100%] (Sampling)
#> Chain 4:
#> Chain 4: Elapsed Time: 0.081 seconds (Warm-up)
#> Chain 4: 0.086 seconds (Sampling)
#> Chain 4: 0.167 seconds (Total)
#> Chain 4:
ypred1 <- posterior_predict(fit)
ypred2 <- posterior_predict(fit)
crps(ypred1, ypred2, y = fit$y)
#> $estimates
#> Estimate SE
#> -10.2289025 0.3452416
#>
#> $pointwise
#> 1 2 3 4 5 6 7
#> -24.990099 -9.657260 -7.446334 -4.754076 -21.781540 -6.939491 -31.090066
#> 8 9 10 11 12 13 14
#> -4.377036 -15.248244 -6.142388 -5.870952 -17.920907 -4.157854 -11.234899
#> 15 16 17 18 19 20 21
#> -11.631440 -6.526809 -5.191024 -4.660734 -4.408232 -12.844526 -5.705731
#> 22 23 24 25 26 27 28
#> -7.449201 -7.832754 -4.233057 -10.975285 -4.664320 -8.833488 -5.819207
#> 29 30 31 32 33 34 35
#> -4.244282 -4.281328 -15.787284 -30.400706 -6.543652 -11.570146 -5.365094
#> 36 37 38 39 40 41 42
#> -10.039881 -12.907251 -5.066769 -5.872869 -18.649738 -10.353712 -5.780542
#> 43 44 45 46 47 48 49
#> -7.948311 -4.292212 -19.103743 -5.837223 -24.585742 -4.507041 -17.647253
#> 50 51 52 53 54 55 56
#> -4.679263 -6.744241 -5.743212 -19.628792 -15.193175 -5.510478 -8.599100
#> 57 58 59 60 61 62 63
#> -5.879795 -5.264341 -5.302749 -5.459433 -9.509765 -9.016372 -4.417851
#> 64 65 66 67 68 69 70
#> -6.229068 -23.648908 -25.582084 -13.216741 -4.283352 -5.174674 -5.239265
#> 71 72 73 74 75 76 77
#> -5.740032 -4.890393 -4.694475 -15.123030 -11.614009 -16.269604 -4.425453
#> 78 79 80 81 82 83 84
#> -3.985703 -19.914424 -4.869179 -11.808568 -15.787271 -8.495717 -10.265243
#> 85 86 87 88 89 90 91
#> -6.284692 -4.447534 -35.372187 -5.701886 -4.676611 -7.591189 -6.973633
#> 92 93 94 95 96 97 98
#> -10.311266 -8.454235 -4.948828 -12.294160 -4.893688 -4.091693 -6.053500
#> 99 100 101 102 103 104 105
#> -13.841933 -15.967705 -5.257588 -15.755618 -10.709925 -5.230169 -8.366270
#> 106 107 108 109 110 111 112
#> -4.699766 -13.235634 -7.650373 -5.811290 -8.271055 -30.021811 -12.808877
#> 113 114 115 116 117 118 119
#> -25.690976 -7.579447 -5.295584 -4.823782 -25.034339 -29.766666 -4.703118
#> 120 121 122 123 124 125 126
#> -15.814769 -11.796324 -5.321439 -4.894910 -4.866492 -18.634125 -13.633340
#> 127 128 129 130 131 132 133
#> -10.435583 -5.558117 -4.323684 -8.764079 -24.145383 -8.272895 -6.855163
#> 134 135 136 137 138 139 140
#> -5.680793 -7.667393 -27.461299 -12.903087 -8.124463 -4.205648 -6.941802
#> 141 142 143 144 145 146 147
#> -4.304912 -4.712101 -4.145628 -5.963981 -4.414419 -4.172647 -6.444587
#> 148 149 150 151 152 153 154
#> -22.406717 -15.149492 -13.283477 -5.211036 -28.989773 -4.116259 -9.207646
#> 155 156 157 158 159 160 161
#> -8.349683 -4.497955 -7.351267 -7.698720 -10.160305 -5.044352 -5.914767
#> 162 163 164 165 166 167 168
#> -7.305688 -5.942986 -5.897738 -5.220435 -8.471887 -8.523607 -4.379158
#> 169 170 171 172 173 174 175
#> -15.717298 -6.463893 -5.090596 -5.174440 -5.968395 -7.052573 -12.503024
#> 176 177 178 179 180 181 182
#> -4.257968 -4.151539 -6.436262 -5.037180 -4.481005 -5.102446 -5.276279
#> 183 184 185 186 187 188 189
#> -4.303738 -14.279255 -20.152343 -13.826707 -4.959577 -20.592879 -4.310917
#> 190 191 192 193 194 195 196
#> -10.245202 -8.206200 -4.327003 -7.032611 -6.891264 -7.731745 -4.833334
#> 197 198 199 200 201 202 203
#> -6.818601 -11.206995 -6.681627 -4.293995 -5.121928 -5.164004 -7.386156
#> 204 205 206 207 208 209 210
#> -4.545963 -8.056687 -7.632516 -7.401501 -4.169851 -16.423923 -4.641051
#> 211 212 213 214 215 216 217
#> -8.037991 -13.010750 -39.523277 -19.259364 -11.365988 -6.901006 -5.784103
#> 218 219 220 221 222 223 224
#> -8.667910 -4.746662 -6.066699 -8.123340 -23.402953 -4.773369 -5.343618
#> 225 226 227 228 229 230 231
#> -5.511369 -4.480713 -4.879941 -6.744629 -15.529893 -9.500877 -4.837963
#> 232 233 234 235 236 237 238
#> -4.352123 -9.559010 -6.614750 -6.257536 -4.833381 -7.676039 -4.607315
#> 239 240 241 242 243 244 245
#> -4.086174 -5.994318 -4.352820 -21.581770 -4.497125 -13.824002 -6.868494
#> 246 247 248 249 250 251 252
#> -16.223607 -4.326620 -27.018989 -6.645658 -5.128369 -5.138602 -8.851399
#> 253 254 255 256 257 258 259
#> -11.606303 -5.176009 -4.551235 -5.608509 -10.513894 -10.912436 -8.979487
#> 260 261 262 263 264 265 266
#> -4.479191 -5.093842 -6.521964 -6.528088 -15.025644 -6.212760 -6.337282
#> 267 268 269 270 271 272 273
#> -6.278270 -21.348668 -19.386367 -8.453199 -4.610565 -27.265781 -42.444792
#> 274 275 276 277 278 279 280
#> -9.983181 -5.747517 -8.426348 -16.350867 -4.237857 -17.538770 -6.118426
#> 281 282 283 284 285 286 287
#> -13.649757 -19.936231 -21.592292 -4.719718 -4.338067 -42.461423 -18.667060
#> 288 289 290 291 292 293 294
#> -22.304015 -4.352973 -17.930965 -9.453872 -11.265408 -22.047764 -6.028313
#> 295 296 297 298 299 300 301
#> -7.565701 -22.713323 -4.669132 -17.469485 -7.750113 -15.119233 -5.552908
#> 302 303 304 305 306 307 308
#> -22.968198 -4.752557 -6.035101 -4.152780 -4.523250 -36.486349 -5.668813
#> 309 310 311 312 313 314 315
#> -4.459167 -29.144196 -12.710347 -29.830408 -11.802292 -4.519779 -4.568674
#> 316 317 318 319 320 321 322
#> -4.474499 -6.063925 -6.560643 -9.741876 -7.005294 -4.986478 -4.376984
#> 323 324 325 326 327 328 329
#> -12.441260 -17.577987 -8.799388 -9.197150 -7.202683 -26.762243 -5.024355
#> 330 331 332 333 334 335 336
#> -5.951065 -4.354393 -18.842222 -27.313189 -15.229946 -6.646184 -12.086292
#> 337 338 339 340 341 342 343
#> -15.402234 -19.601663 -8.238985 -5.861255 -21.171311 -4.523171 -12.616253
#> 344 345 346 347 348 349 350
#> -4.303528 -17.032356 -15.928884 -29.005767 -16.610456 -5.470516 -7.039136
#> 351 352 353 354 355 356 357
#> -9.268780 -10.102659 -11.594327 -4.354506 -20.154592 -26.965057 -15.562839
#> 358 359 360 361 362 363 364
#> -19.996679 -4.328289 -5.086946 -4.522773 -10.254820 -4.486712 -4.616746
#> 365 366 367 368 369 370 371
#> -11.284862 -4.404655 -12.710210 -30.091597 -11.704346 -4.222466 -9.221634
#> 372 373 374 375 376 377 378
#> -4.233227 -9.017408 -5.236285 -15.313609 -15.953206 -18.322510 -4.285541
#> 379 380 381 382 383 384 385
#> -4.501041 -6.166170 -14.049663 -9.343921 -8.076468 -10.511973 -11.436564
#> 386 387 388 389 390 391 392
#> -13.109214 -11.114473 -4.760043 -5.524175 -9.720257 -9.001863 -4.996974
#> 393 394 395 396 397 398 399
#> -19.563169 -20.812703 -4.575574 -5.277737 -20.523523 -23.311206 -4.393605
#> 400 401 402 403 404 405 406
#> -6.755891 -4.453456 -4.687144 -6.067947 -4.303689 -13.890801 -5.598087
#> 407 408 409 410 411 412 413
#> -16.181107 -5.265551 -27.431619 -4.230414 -4.475243 -12.686568 -10.125541
#> 414 415 416 417 418 419 420
#> -4.770692 -6.470942 -7.015164 -4.135998 -18.012834 -4.656753 -17.092118
#> 421 422 423 424 425 426 427
#> -19.233416 -4.555802 -11.404717 -10.396064 -23.909090 -4.384484 -4.651913
#> 428 429 430 431 432 433 434
#> -7.312231 -16.149626 -12.577736 -5.289945 -19.919982 -4.350266 -7.985258
#>
loo_crps(ypred1, ypred2, y = fit$y, log_lik = log_lik(fit))
#> $estimates
#> Estimate SE
#> -10.297895 0.349367
#>
#> $pointwise
#> [1] -25.323066 -9.581274 -7.333264 -4.771654 -21.817076 -7.111284
#> [7] -31.636504 -4.375844 -15.709887 -6.362892 -5.860400 -17.696360
#> [13] -4.384776 -11.046362 -11.795203 -6.450202 -4.881834 -4.836524
#> [19] -4.322119 -12.908182 -5.699659 -7.371158 -7.954181 -4.511666
#> [25] -11.162667 -4.570261 -8.822485 -5.951959 -4.210367 -4.379016
#> [31] -15.599730 -30.394103 -6.635047 -11.848290 -5.111847 -9.957581
#> [37] -13.002766 -5.031810 -5.733377 -18.717288 -10.409610 -5.930287
#> [43] -7.958221 -4.291444 -19.435025 -5.714599 -24.851734 -4.393779
#> [49] -17.920274 -4.795613 -6.772483 -5.697908 -19.861015 -15.271874
#> [55] -5.628905 -8.572231 -5.940575 -5.253002 -5.552601 -5.586480
#> [61] -9.538507 -9.043637 -4.300605 -6.265852 -23.571526 -25.703470
#> [67] -13.281449 -4.168291 -5.245350 -5.378360 -5.823852 -5.028659
#> [73] -4.848045 -15.191654 -11.648210 -16.356790 -4.394697 -4.033373
#> [79] -19.800091 -4.855264 -11.791753 -15.990180 -8.432755 -10.260557
#> [85] -6.348277 -4.516187 -35.704623 -5.701929 -4.756378 -7.801426
#> [91] -6.780100 -10.166189 -8.374773 -5.068930 -12.453335 -4.692069
#> [97] -4.230535 -6.093269 -13.986344 -16.346876 -5.437638 -15.709583
#> [103] -10.645165 -4.926769 -8.438678 -4.641027 -13.396489 -7.674940
#> [109] -5.846945 -8.401542 -30.740352 -12.737675 -25.590064 -7.730896
#> [115] -5.319714 -5.311793 -25.229401 -30.127934 -4.755415 -15.822724
#> [121] -12.011677 -5.285513 -4.766052 -4.857271 -18.738646 -13.700495
#> [127] -10.696280 -5.638844 -4.510100 -8.854396 -24.263804 -8.416096
#> [133] -6.730203 -5.442742 -7.560449 -27.979180 -13.005563 -8.388394
#> [139] -4.077047 -6.728151 -4.474575 -4.508987 -4.226735 -6.056109
#> [145] -4.429539 -4.206556 -6.562797 -22.472391 -15.290575 -13.226522
#> [151] -5.301596 -29.641532 -4.042417 -9.422937 -8.579525 -4.380486
#> [157] -7.487607 -7.593975 -10.062067 -5.191255 -6.016954 -7.405230
#> [163] -5.993372 -6.004881 -5.219774 -8.529474 -8.578177 -4.235529
#> [169] -16.297825 -6.464058 -4.919672 -5.483101 -6.073631 -7.222031
#> [175] -12.682422 -4.462732 -4.137003 -6.533833 -4.987452 -4.614787
#> [181] -5.177199 -5.258140 -4.119735 -14.013084 -20.498019 -13.987045
#> [187] -4.966601 -20.649895 -4.289259 -10.348290 -8.235060 -4.370131
#> [193] -7.210363 -6.740741 -7.782064 -4.641546 -6.949840 -11.276550
#> [199] -6.644119 -4.303550 -5.193623 -5.231955 -7.508080 -4.235240
#> [205] -8.196982 -7.585475 -7.317194 -4.085927 -16.372279 -4.670358
#> [211] -8.276114 -13.283535 -40.425339 -19.554877 -11.721488 -6.997021
#> [217] -5.901739 -8.572120 -4.622897 -6.107494 -8.068724 -23.427451
#> [223] -4.854310 -5.462829 -5.762697 -4.744221 -4.609074 -6.885982
#> [229] -15.358112 -9.633684 -4.796686 -4.387893 -9.710259 -6.590185
#> [235] -6.343350 -4.660701 -7.708385 -4.588995 -4.191914 -6.209133
#> [241] -4.302192 -21.733465 -4.434126 -13.904171 -6.949406 -16.101320
#> [247] -4.331816 -27.247407 -6.752032 -4.989421 -5.246057 -8.983518
#> [253] -11.636927 -5.202781 -4.433059 -5.692421 -10.623190 -10.946523
#> [259] -9.137151 -4.483126 -5.190455 -6.679914 -6.754904 -14.930892
#> [265] -6.441519 -6.382255 -6.531150 -21.732611 -19.791705 -8.495155
#> [271] -4.736656 -27.448068 -42.840195 -10.215245 -5.866869 -8.623630
#> [277] -16.458886 -4.185269 -17.733238 -6.275329 -13.713137 -20.125012
#> [283] -21.820561 -4.575891 -4.313169 -42.988532 -18.984070 -22.407340
#> [289] -4.079013 -18.322654 -9.491792 -11.370774 -21.918458 -5.766787
#> [295] -7.477094 -22.804475 -4.739492 -18.025290 -7.772326 -15.552381
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#> [307] -36.463451 -5.681294 -4.516625 -29.499711 -12.863909 -29.920053
#> [313] -11.625466 -4.585405 -4.547222 -4.569650 -5.937145 -6.517771
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#> [427] -4.254964 -7.503028 -16.388643 -12.866242 -5.560751 -20.528146
#> [433] -4.419933 -8.159600
#>
# }