GUMBEL: Two parameter Gumbel distribution and L-moments

Description Usage Arguments Details Value Note See Also Examples

Description

GUMBEL provides the link between L-moments of a sample and the two parameter Gumbel distribution.

Usage

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f.gumb (x, xi, alfa)
F.gumb (x, xi, alfa)
invF.gumb (F, xi, alfa)
Lmom.gumb (xi, alfa)
par.gumb (lambda1, lambda2)
rand.gumb (numerosita, xi, alfa)

Arguments

x

vector of quantiles

xi

vector of gumb location parameters

alfa

vector of gumb scale parameters

F

vector of probabilities

lambda1

vector of sample means

lambda2

vector of L-variances

numerosita

numeric value indicating the length of the vector to be generated

Details

See http://en.wikipedia.org/wiki/Fisher-Tippett_distribution for an introduction to the Gumbel distribution.

Definition

Parameters (2): ξ (location), α (scale).

Range of x: -∞ < x < ∞.

Probability density function:

f(x) = α^{-1} \exp[-(x-ξ)/α] \exp\{- \exp[-(x-ξ)/α]\}

Cumulative distribution function:

F(x) = \exp[-\exp(-(x-ξ)/α)]

Quantile function: x(F) = ξ - α \log(-\log F).

L-moments

λ_1 = ξ + α γ

λ_2 = α \log 2

τ_3 = 0.1699 = \log(9/8)/ \log 2

τ_4 = 0.1504 = (16 \log 2 - 10 \log 3)/ \log 2

Here γ is Euler's constant, 0.5772...

Parameters

α=λ_2 / \log 2

ξ = λ_1 - γ α

Lmom.gumb and par.gumb accept input as vectors of equal length. In f.gumb, F.gumb, invF.gumb and rand.gumb parameters (xi, alfa) must be atomic.

Value

f.gumb gives the density f, F.gumb gives the distribution function F, invF.gumb gives the quantile function x, Lmom.gumb gives the L-moments (λ_1, λ_2, τ_3, τ_4)), par.gumb gives the parameters (xi, alfa), and rand.gumb generates random deviates.

Note

For information on the package and the Author, and for all the references, see nsRFA.

See Also

rnorm, runif, EXP, GENLOGIS, GENPAR, GEV, KAPPA, LOGNORM, P3; DISTPLOTS, GOFmontecarlo, Lmoments.

Examples

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data(hydroSIMN)
annualflows[1:10,]
summary(annualflows)
x <- annualflows["dato"][,]
fac <- factor(annualflows["cod"][,])
split(x,fac)

camp <- split(x,fac)$"45"
ll <- Lmoments(camp)
parameters <- par.gumb(ll[1],ll[2])
f.gumb(1800,parameters$xi,parameters$alfa)
F.gumb(1800,parameters$xi,parameters$alfa)
invF.gumb(0.7686843,parameters$xi,parameters$alfa)
Lmom.gumb(parameters$xi,parameters$alfa)
rand.gumb(100,parameters$xi,parameters$alfa)

Rll <- regionalLmoments(x,fac); Rll
parameters <- par.gumb(Rll[1],Rll[2])
Lmom.gumb(parameters$xi,parameters$alfa)

Example output

   cod anno dato
1    1 1956 1494
2    1 1957 1309
3    1 1958 1699
4    1 1959 1467
5    1 1960 1918
6    1 1961 1469
7    1 1962 1267
8    1 1963 1523
9    1 1964 1338
10   1 1965 1438
      cod            anno           dato       
 Min.   : 1.0   Min.   :1921   Min.   : 172.0  
 1st Qu.:13.0   1st Qu.:1940   1st Qu.: 725.2  
 Median :22.0   Median :1951   Median : 981.0  
 Mean   :23.7   Mean   :1951   Mean   :1041.4  
 3rd Qu.:34.0   3rd Qu.:1960   3rd Qu.:1308.8  
 Max.   :49.0   Max.   :1985   Max.   :3045.0  
$`1`
 [1] 1494 1309 1699 1467 1918 1469 1267 1523 1338 1438 1788 1591 1697 1780 1769

$`2`
 [1] 1144 1652 1807 1881 1741 1124 2064 1434 1678 1239  921  983 1093 1744 1213
[16] 1590  956 1124 2181 1077 1345 1219  988 1325 1277 1479 1307 2053 1232  973
[31] 1407  912

$`3`
 [1] 2596  954 1115 1248  867 1280 1588 1055 1764 3045

$`4`
 [1]  871 1238 1505 1636 1553 1936 1739 1867 1184 1630 1311 1520 1201 1614 1971
[16] 1829 1781 1093 1996 1328 1662 1199  860  961  949 1536 1016 1386  820 1023
[31] 2329 1209 1305 1334 1024 1364 1310 1410 1247 2393 1317  909 1808 1020 1181
[46] 1365 1218 1644 1160 1002 1243 1332 1033 1170 1685 1478 2434 1600 1369 1215
[61] 1614 1449 1518 1490 1191

$`7`
 [1] 1481 1758 1774 1625 1607 2826 1488  928 2379 1173 1801 1824 1309 2220 1733

$`8`
 [1] 1086 1810 2244 2138 2028 1308 1947 1528 2244 1594  861 1378 1795 1344 1558
[16]  696  724 2497  660 1388 1484  952 1987 2646 1689 1443 2688 1249 1145 2392
[31] 1001 1380

$`9`
 [1] 2075 1607 1717 1261 1824 1330  963 1313 2276  682 1440 1304 1193

$`10`
 [1] 1096 1387 1289 1461 1054 1474 1137 1256  981 1696 1468 1850 1644 1248 1498
[16] 1317 1500 1109  859  931 1020 1493  954 1133 1144 1056

$`11`
 [1] 1320 1706  948 1643  944 1402 1202 1788 1665 1833 1679 1166 1833 1661 1938
[16] 1457  830 1221 1398 1674 1311 1611 1003 1021

$`12`
 [1]  890 1247 1040 1047  875 1060  913  968  749 1218 1104 1489 1300  833  994
[16] 1002 1134  854  826  695  939 1230  830 1096  876  704 1111  780  791  709
[31]  812  686  812  755  802 1098  868  735  829  750  635  887  711  753  935
[46]  862  830  924  735  766  930  783 1623 1359 1015  922  963  848  975  760
[61]  766

$`13`
 [1] 1288  854 1324  741 1043  756 1477 1160 1426 1360 1109 1211 1094 1666 1002
[16]  772 1124  997  649 1436  762 1293  930  721  838 1063  710 1002 1625 1002
[31]  848 1104  869  823  992  588  894 1073  675 1181 1568  817 1068  978

$`14`
 [1] 1505  928 1223  805 1449 1084 1588 1509 1137 1014 1181 1394  922  811 1428
[16] 1137 1240 1034  581 1501  700 1263  962  780  919 1068  855 1198 1569 1134
[31] 1007 1205  973  871 1188  581 1027 1192  578  875 1553  774  958 1187 2152
[46]  836  834  753 1110

$`15`
 [1]  969  811 1107  769  567  925  508  598  818  495

$`16`
 [1]  957  625  625  658 1022  555  496  625  593 1115  718  957  707  332  821
[16]  469  913  663  418  523  799  469 1000 1104  761  598 1033  707  469  614
[31]  270  609 1017  367

$`17`
 [1]  595  718  518  548  389  567  506  985  530 1097  934  675  614  587  722
[16]  499  459 1087  550  860  648  296  658

$`18`
 [1]  686  863  488  937  453  621  484  851  599 1161  894  598  645  606  772
[16]  449  486  510  559  829  545  898  529  392  856  625  773  651  674  432

$`19`
 [1]  589  715  479  696  394  533  430  845  519 1012  805  559  569  580  725
[16]  448  412  411  407  638  506  729  538  350  736  513  787

$`20`
 [1] 1237 1908 1263 1066 1401 1263 1134  799  919  971 1057 1710 1555 1667 1212
[16]  799 1366  962 1779 1504  808 1031 1186 1031 1796 1882 1487  945 1710 1194
[31]  919 1418  722 1160 1409  894 1279 1884 1307

$`21`
 [1] 489 704 310 665 259 501 428 820 551 994 658 425 423 409 736 440 401 398 342
[20] 658 449 665 535 247 584 338 580 569 311 412 565 403 846 917 525 411 717 526
[39] 248 451 185 356 564 256

$`22`
 [1] 1197  863 1382 1104  649  745  615 1116  618  739  761  720 1147  838 1057
[16]  739  529  962  470  881  495  417  553  819  711 1410 1472  727  671 1163
[31]  751  476  819  399  612  860  507  844 1245  953  976

$`23`
 [1]  835 1345 1085 1655 1291  838  974  862 1106  699  854  721  699 1033  892
[16] 1213  631  554  833  911  796  721  727

$`24`
 [1] 1795 1761 1962 1541 1007 1276 1144 1302  947 1210 1113 1532  764  849 1412
[16] 1105 1048  843 1048 1157

$`25`
 [1] 1498  880 1028 1046  589 1088 1179 1471  761 1106 2017  649 1129 1149 1355
[16] 1107

$`26`
 [1] 1634 1300 1715 1643 1295 1459 1020 1531  919 1095  876  857 1534 1183 1405
[16] 1051 1159 1478 1472 1364 1140 1126 1007

$`27`
 [1] 1157 1759 1245  842 1056  800 1244  806  925  839  782 1236 1601  886  768
[16] 1109  722  440

$`28`
 [1] 1121 1488 1158 1287 1210 1468 1445 1304 1967 1408

$`29`
 [1] 1121 1482 1163 1378 1201 1677 1360 2230 1117 1093 1647 1358

$`30`
 [1]  395  342  463  649  400  703  388  570  292  490  440  885  671 1035  729
[16]  360  467  351  765  418  455  339  311  493  432  686  353  337  449  513
[31]  374  475  628  496  844  974  375  419  651  441  226  438  218  461  504
[46]  309  543  870  433  724  604  712  865  395  324  436  607  399

$`31`
 [1]  754 1025  829 1428 1828 1472  771 1144  980 1728  720  850  995  901 1138
[16]  678  805 1509  616  629  716  848  767  720 1426 1370 1826 1046 1172  869
[31]  793 1008  571 1161

$`32`
 [1]  920 1674 1153 1512 1226  647  945  822 1665  632  746  705  759  932  617
[16]  632 1259  506  590  743  598  747  988  855 1229 1461  458  804  867  652
[31]  580

$`33`
 [1]  684  701  486  792  727 1086  564  624 1205  463  846  894  707  733  892
[16]  869 1283 1444  474  798  935  719  445  749  428  772  854  545 1002  939
[31]  643  603  785  775 1025  584

$`34`
 [1]  636  998 1014 1965 1333 1730 1330  825 1112  851 1423  960 1031  976  561
[16] 1055 1076 1224  658  707 1453  445  966  930  939  862 1115 1158 1573

$`35`
 [1]  845  803  746 1036 1160 1038 1285  369 1093  732  613  620  863  579  765
[16]  819  505  594  667  651  950 1583  688  622 1068

$`36`
 [1]  924 1676 1765  841  796  745 1363  663  714  382  771  796  956 1153  669
[16]  796 1879  643  796  994  733 1185

$`37`
 [1]  597  833  902 1207  793  598 1328  323  561  726  663  919 1139 1040 1264
[16] 1214

$`38`
 [1]  492  608  368  393 1123  172  281  539  424  585  632  528

$`39`
 [1]  339  929  560  727  490  684  979 1466  404  865  533  462  287  767  653
[16] 1176 1906  883

$`40`
 [1]  755.00  871.00  938.00 1175.00 1218.00  621.00  432.25  913.20  840.15
[10]  827.97  919.29  724.48  602.72

$`41`
[1] 1449 1449 1546 1516 1254 1382

$`42`
[1]  895 1006 1351 1215 1215 1279 1006 1156  821

$`43`
 [1]  948 1308 1185  801  848  926  932  755  764  891  677  835 1112  918  742
[16]  685  927

$`44`
 [1] 1607 1275 1613 1484 1487 1205 1367 1158 1583 1342 1848 1640 1225 1320 1202
[16] 1476 1190 1435  894 1326 1230 1042 1127

$`45`
 [1] 1953 1939 1677 1692 2051 2371 2022 1521 1448 1825 1363 1760 1672 1603 1244
[16] 1521 1783 1560 1357 1673 1625 1425 1688 1577 1736 1640 1584 1293 1277 1742
[31] 1491

$`46`
 [1] 1223 1077  671 1063  969  842 1037  903 1407 1153 1107 1293  813  834 1118
[16]  901  981

$`47`
 [1]  986  996 1335  964 1018  821  945  844 1133  975 1082 1252 1031  940 1078
[16]  933  709  923  899  747 1010  873  962  965  674  763  915 1029 1452 1486

$`48`
 [1]  872 1528 1062 1345 1158  998 1197 1234 1469 1343 2103 1745 1084 1717 1131
[16]  990 1186  884 1118 1383  877 1072 1906  830

$`49`
 [1]  808 1088 1435 1265 1065  911  992 1273 1031 1100  769  865  781 1019 1761

         l2 
0.001013982 
       l1 
0.7686843 
  l1 
1800 
$lambda1
      l1 
1648.806 

$lambda2
      l2 
138.2366 

$tau3
[1] 0.169925

$tau4
[1] 0.150375

  [1] 1903.708 1428.466 1285.814 1838.794 1665.145 2076.468 1566.537 1429.013
  [9] 1692.607 1680.064 1684.374 1640.741 1397.152 1574.904 1776.577 1376.031
 [17] 1974.668 1395.482 1960.947 1841.113 1603.785 1569.229 1648.876 1748.820
 [25] 1796.972 1611.675 1476.852 1438.388 1768.885 1576.190 1449.831 1454.273
 [33] 1595.974 1563.343 1388.448 1691.372 1761.212 1645.744 1466.284 1910.041
 [41] 1568.843 1409.652 1895.027 1540.044 1612.490 1636.033 2084.872 1227.158
 [49] 1338.268 1485.079 1279.252 1506.150 1678.622 1427.825 1566.434 2026.936
 [57] 1772.937 1485.905 1621.129 1396.367 1442.355 1464.600 1590.519 1717.804
 [65] 1573.774 1465.997 2320.100 1605.166 1530.286 1429.199 1547.450 1245.505
 [73] 1419.271 1691.497 1602.634 1494.855 1497.171 1582.278 1503.432 1385.969
 [81] 1547.634 1762.102 1745.194 1469.328 1957.102 1580.540 2322.846 1327.557
 [89] 2400.977 1717.974 1801.186 1524.829 1549.947 1766.201 1826.389 1644.157
 [97] 1601.854 1319.043 1339.373 1766.457
         l1R          l2R         lcvR         lcaR        lkurR 
1041.4321277  160.1783511    0.1597701    0.1489629    0.1356834 
$lambda1
     l1R 
1041.432 

$lambda2
     l2R 
160.1784 

$tau3
[1] 0.169925

$tau4
[1] 0.150375

nsRFA documentation built on Feb. 26, 2020, 5:19 p.m.