-
Notifications
You must be signed in to change notification settings - Fork 315
/
stats-Ex.Rout.save
17603 lines (16450 loc) · 502 KB
/
stats-Ex.Rout.save
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
R Under development (unstable) (2012-09-30 r60839) -- "Unsuffered Consequences"
Copyright (C) 2012 The R Foundation for Statistical Computing
ISBN 3-900051-07-0
Platform: x86_64-unknown-linux-gnu (64-bit)
R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.
Natural language support but running in an English locale
R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.
> pkgname <- "stats"
> source(file.path(R.home("share"), "R", "examples-header.R"))
> options(warn = 1)
> library('stats')
>
> assign(".oldSearch", search(), pos = 'CheckExEnv')
> cleanEx()
> nameEx("AIC")
> ### * AIC
>
> flush(stderr()); flush(stdout())
>
> ### Name: AIC
> ### Title: Akaike's An Information Criterion
> ### Aliases: AIC BIC
> ### Keywords: models
>
> ### ** Examples
>
> lm1 <- lm(Fertility ~ . , data = swiss)
> AIC(lm1)
[1] 326.0716
> stopifnot(all.equal(AIC(lm1),
+ AIC(logLik(lm1))))
> BIC(lm1)
[1] 339.0226
>
> lm2 <- update(lm1, . ~ . -Examination)
> AIC(lm1, lm2)
df AIC
lm1 7 326.0716
lm2 6 325.2408
> BIC(lm1, lm2)
df BIC
lm1 7 339.0226
lm2 6 336.3417
>
>
>
> cleanEx()
> nameEx("ARMAacf")
> ### * ARMAacf
>
> flush(stderr()); flush(stdout())
>
> ### Name: ARMAacf
> ### Title: Compute Theoretical ACF for an ARMA Process
> ### Aliases: ARMAacf
> ### Keywords: ts
>
> ### ** Examples
>
> ARMAacf(c(1.0, -0.25), 1.0, lag.max = 10)
0 1 2 3 4 5
1.000000000 0.875000000 0.625000000 0.406250000 0.250000000 0.148437500
6 7 8 9 10
0.085937500 0.048828125 0.027343750 0.015136719 0.008300781
>
> ## Example from Brockwell & Davis (1991, pp.92-4)
> ## answer 2^(-n) * (32/3 + 8 * n) /(32/3)
> n <- 1:10; 2^(-n) * (32/3 + 8 * n) /(32/3)
[1] 0.875000000 0.625000000 0.406250000 0.250000000 0.148437500 0.085937500
[7] 0.048828125 0.027343750 0.015136719 0.008300781
> ARMAacf(c(1.0, -0.25), 1.0, lag.max = 10, pacf = TRUE)
[1] 0.8750000 -0.6000000 0.3750000 -0.2727273 0.2142857 -0.1764706
[7] 0.1500000 -0.1304348 0.1153846 -0.1034483
> zapsmall(ARMAacf(c(1.0, -0.25), lag.max = 10, pacf = TRUE))
[1] 0.80 -0.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
>
> ## Cov-Matrix of length-7 sub-sample of AR(1) example:
> toeplitz(ARMAacf(0.8, lag.max = 7))
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
[1,] 1.0000000 0.800000 0.64000 0.5120 0.4096 0.32768 0.262144 0.2097152
[2,] 0.8000000 1.000000 0.80000 0.6400 0.5120 0.40960 0.327680 0.2621440
[3,] 0.6400000 0.800000 1.00000 0.8000 0.6400 0.51200 0.409600 0.3276800
[4,] 0.5120000 0.640000 0.80000 1.0000 0.8000 0.64000 0.512000 0.4096000
[5,] 0.4096000 0.512000 0.64000 0.8000 1.0000 0.80000 0.640000 0.5120000
[6,] 0.3276800 0.409600 0.51200 0.6400 0.8000 1.00000 0.800000 0.6400000
[7,] 0.2621440 0.327680 0.40960 0.5120 0.6400 0.80000 1.000000 0.8000000
[8,] 0.2097152 0.262144 0.32768 0.4096 0.5120 0.64000 0.800000 1.0000000
>
>
>
> cleanEx()
> nameEx("ARMAtoMA")
> ### * ARMAtoMA
>
> flush(stderr()); flush(stdout())
>
> ### Name: ARMAtoMA
> ### Title: Convert ARMA Process to Infinite MA Process
> ### Aliases: ARMAtoMA
> ### Keywords: ts
>
> ### ** Examples
>
> ARMAtoMA(c(1.0, -0.25), 1.0, 10)
[1] 2.00000000 1.75000000 1.25000000 0.81250000 0.50000000 0.29687500
[7] 0.17187500 0.09765625 0.05468750 0.03027344
> ## Example from Brockwell & Davis (1991, p.92)
> ## answer (1 + 3*n)*2^(-n)
> n <- 1:10; (1 + 3*n)*2^(-n)
[1] 2.00000000 1.75000000 1.25000000 0.81250000 0.50000000 0.29687500
[7] 0.17187500 0.09765625 0.05468750 0.03027344
>
>
>
> cleanEx()
> nameEx("Beta")
> ### * Beta
>
> flush(stderr()); flush(stdout())
>
> ### Name: Beta
> ### Title: The Beta Distribution
> ### Aliases: Beta dbeta pbeta qbeta rbeta
> ### Keywords: distribution
>
> ### ** Examples
>
> x <- seq(0, 1, length=21)
> dbeta(x, 1, 1)
[1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
> pbeta(x, 1, 1)
[1] 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70
[16] 0.75 0.80 0.85 0.90 0.95 1.00
>
>
>
> cleanEx()
> nameEx("Binomial")
> ### * Binomial
>
> flush(stderr()); flush(stdout())
>
> ### Name: Binomial
> ### Title: The Binomial Distribution
> ### Aliases: Binomial dbinom pbinom qbinom rbinom
> ### Keywords: distribution
>
> ### ** Examples
>
> require(graphics)
> # Compute P(45 < X < 55) for X Binomial(100,0.5)
> sum(dbinom(46:54, 100, 0.5))
[1] 0.6317984
>
> ## Using "log = TRUE" for an extended range :
> n <- 2000
> k <- seq(0, n, by = 20)
> plot (k, dbinom(k, n, pi/10, log=TRUE), type='l', ylab="log density",
+ main = "dbinom(*, log=TRUE) is better than log(dbinom(*))")
> lines(k, log(dbinom(k, n, pi/10)), col='red', lwd=2)
> ## extreme points are omitted since dbinom gives 0.
> mtext("dbinom(k, log=TRUE)", adj=0)
> mtext("extended range", adj=0, line = -1, font=4)
> mtext("log(dbinom(k))", col="red", adj=1)
>
>
>
> cleanEx()
> nameEx("Cauchy")
> ### * Cauchy
>
> flush(stderr()); flush(stdout())
>
> ### Name: Cauchy
> ### Title: The Cauchy Distribution
> ### Aliases: Cauchy dcauchy pcauchy qcauchy rcauchy
> ### Keywords: distribution
>
> ### ** Examples
>
> dcauchy(-1:4)
[1] 0.15915494 0.31830989 0.15915494 0.06366198 0.03183099 0.01872411
>
>
>
> cleanEx()
> nameEx("Chisquare")
> ### * Chisquare
>
> flush(stderr()); flush(stdout())
>
> ### Name: Chisquare
> ### Title: The (non-central) Chi-Squared Distribution
> ### Aliases: Chisquare dchisq pchisq qchisq rchisq
> ### Keywords: distribution
>
> ### ** Examples
>
> require(graphics)
>
> dchisq(1, df=1:3)
[1] 0.2419707 0.3032653 0.2419707
> pchisq(1, df= 3)
[1] 0.198748
> pchisq(1, df= 3, ncp = 0:4)# includes the above
[1] 0.19874804 0.13229855 0.08787311 0.05824691 0.03853592
>
> x <- 1:10
> ## Chi-squared(df = 2) is a special exponential distribution
> all.equal(dchisq(x, df=2), dexp(x, 1/2))
[1] TRUE
> all.equal(pchisq(x, df=2), pexp(x, 1/2))
[1] TRUE
>
> ## non-central RNG -- df=0 with ncp > 0: Z0 has point mass at 0!
> Z0 <- rchisq(100, df = 0, ncp = 2.)
> graphics::stem(Z0)
The decimal point is at the |
0 | 0000000000000000000000000000000000000013356778899
1 | 0001333456678888899
2 | 0011444467
3 | 00233345888
4 | 111246
5 |
6 |
7 | 178
8 | 23
>
> ## Not run:
> ##D ## visual testing
> ##D ## do P-P plots for 1000 points at various degrees of freedom
> ##D L <- 1.2; n <- 1000; pp <- ppoints(n)
> ##D op <- par(mfrow = c(3,3), mar= c(3,3,1,1)+.1, mgp= c(1.5,.6,0),
> ##D oma = c(0,0,3,0))
> ##D for(df in 2^(4*rnorm(9))) {
> ##D plot(pp, sort(pchisq(rr <- rchisq(n,df=df, ncp=L), df=df, ncp=L)),
> ##D ylab="pchisq(rchisq(.),.)", pch=".")
> ##D mtext(paste("df = ",formatC(df, digits = 4)), line= -2, adj=0.05)
> ##D abline(0,1,col=2)
> ##D }
> ##D mtext(expression("P-P plots : Noncentral "*
> ##D chi^2 *"(n=1000, df=X, ncp= 1.2)"),
> ##D cex = 1.5, font = 2, outer=TRUE)
> ##D par(op)
> ## End(Not run)
>
> ## "analytical" test
> lam <- seq(0,100, by=.25)
> p00 <- pchisq(0, df=0, ncp=lam)
> p.0 <- pchisq(1e-300, df=0, ncp=lam)
> stopifnot(all.equal(p00, exp(-lam/2)),
+ all.equal(p.0, exp(-lam/2)))
>
>
>
> cleanEx()
> nameEx("Exponential")
> ### * Exponential
>
> flush(stderr()); flush(stdout())
>
> ### Name: Exponential
> ### Title: The Exponential Distribution
> ### Aliases: Exponential dexp pexp qexp rexp
> ### Keywords: distribution
>
> ### ** Examples
>
> dexp(1) - exp(-1) #-> 0
[1] 0
>
>
>
> cleanEx()
> nameEx("Fdist")
> ### * Fdist
>
> flush(stderr()); flush(stdout())
>
> ### Name: FDist
> ### Title: The F Distribution
> ### Aliases: FDist df pf qf rf
> ### Keywords: distribution
>
> ### ** Examples
>
> ## the density of the square of a t_m is 2*dt(x, m)/(2*x)
> # check this is the same as the density of F_{1,m}
> x <- seq(0.001, 5, len=100)
> all.equal(df(x^2, 1, 5), dt(x, 5)/x)
[1] TRUE
>
> ## Identity: qf(2*p - 1, 1, df)) == qt(p, df)^2) for p >= 1/2
> p <- seq(1/2, .99, length=50); df <- 10
> rel.err <- function(x,y) ifelse(x==y,0, abs(x-y)/mean(abs(c(x,y))))
>
>
>
> cleanEx()
> nameEx("GammaDist")
> ### * GammaDist
>
> flush(stderr()); flush(stdout())
>
> ### Name: GammaDist
> ### Title: The Gamma Distribution
> ### Aliases: GammaDist dgamma pgamma qgamma rgamma
> ### Keywords: distribution
>
> ### ** Examples
>
> -log(dgamma(1:4, shape=1))
[1] 1 2 3 4
> p <- (1:9)/10
> pgamma(qgamma(p,shape=2), shape=2)
[1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
> 1 - 1/exp(qgamma(p, shape=1))
[1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
>
>
>
> cleanEx()
> nameEx("Geometric")
> ### * Geometric
>
> flush(stderr()); flush(stdout())
>
> ### Name: Geometric
> ### Title: The Geometric Distribution
> ### Aliases: Geometric dgeom pgeom qgeom rgeom
> ### Keywords: distribution
>
> ### ** Examples
>
> qgeom((1:9)/10, prob = .2)
[1] 0 0 1 2 3 4 5 7 10
> Ni <- rgeom(20, prob = 1/4); table(factor(Ni, 0:max(Ni)))
0 1 2 3 4 5 6 7 8 9 10 11
5 3 3 1 2 2 0 1 0 1 1 1
>
>
>
> cleanEx()
> nameEx("HoltWinters")
> ### * HoltWinters
>
> flush(stderr()); flush(stdout())
>
> ### Name: HoltWinters
> ### Title: Holt-Winters Filtering
> ### Aliases: HoltWinters print.HoltWinters residuals.HoltWinters
> ### Keywords: ts
>
> ### ** Examples
>
> ## Don't show:
> od <- options(digits=5)
> ## End Don't show
> require(graphics)
>
> ## Seasonal Holt-Winters
> (m <- HoltWinters(co2))
Holt-Winters exponential smoothing with trend and additive seasonal component.
Call:
HoltWinters(x = co2)
Smoothing parameters:
alpha: 0.51265
beta : 0.0094977
gamma: 0.47289
Coefficients:
[,1]
a 364.76162
b 0.12474
s1 0.22153
s2 0.95528
s3 1.59847
s4 2.87580
s5 3.28201
s6 2.44070
s7 0.89694
s8 -1.37964
s9 -3.41124
s10 -3.25702
s11 -1.91349
s12 -0.58442
> plot(m)
> plot(fitted(m))
>
> (m <- HoltWinters(AirPassengers, seasonal = "mult"))
Holt-Winters exponential smoothing with trend and multiplicative seasonal component.
Call:
HoltWinters(x = AirPassengers, seasonal = "mult")
Smoothing parameters:
alpha: 0.27559
beta : 0.032693
gamma: 0.87073
Coefficients:
[,1]
a 469.32322
b 3.02154
s1 0.94646
s2 0.88292
s3 0.97174
s4 1.03048
s5 1.04769
s6 1.18053
s7 1.35908
s8 1.33317
s9 1.10834
s10 0.98688
s11 0.83613
s12 0.92099
> plot(m)
>
> ## Non-Seasonal Holt-Winters
> x <- uspop + rnorm(uspop, sd = 5)
> m <- HoltWinters(x, gamma = FALSE)
> plot(m)
>
> ## Exponential Smoothing
> m2 <- HoltWinters(x, gamma = FALSE, beta = FALSE)
> lines(fitted(m2)[,1], col = 3)
> ## Don't show:
> options(od)
> ## End Don't show
>
>
>
> cleanEx()
> nameEx("Hypergeometric")
> ### * Hypergeometric
>
> flush(stderr()); flush(stdout())
>
> ### Name: Hypergeometric
> ### Title: The Hypergeometric Distribution
> ### Aliases: Hypergeometric dhyper phyper qhyper rhyper
> ### Keywords: distribution
>
> ### ** Examples
>
> m <- 10; n <- 7; k <- 8
> x <- 0:(k+1)
> rbind(phyper(x, m, n, k), dhyper(x, m, n, k))
[,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 0 0.0004113534 0.01336898 0.117030 0.4193747 0.7821884 0.9635952
[2,] 0 0.0004113534 0.01295763 0.103661 0.3023447 0.3628137 0.1814068
[,8] [,9] [,10]
[1,] 0.99814891 1.00000000 1
[2,] 0.03455368 0.00185109 0
> all(phyper(x, m, n, k) == cumsum(dhyper(x, m, n, k)))# FALSE
[1] FALSE
>
>
> cleanEx()
> nameEx("IQR")
> ### * IQR
>
> flush(stderr()); flush(stdout())
>
> ### Name: IQR
> ### Title: The Interquartile Range
> ### Aliases: IQR
> ### Keywords: univar robust distribution
>
> ### ** Examples
>
> IQR(rivers)
[1] 370
>
>
>
> cleanEx()
> nameEx("Logistic")
> ### * Logistic
>
> flush(stderr()); flush(stdout())
>
> ### Name: Logistic
> ### Title: The Logistic Distribution
> ### Aliases: Logistic dlogis plogis qlogis rlogis
> ### Keywords: distribution
>
> ### ** Examples
>
> var(rlogis(4000, 0, scale = 5))# approximately (+/- 3)
[1] 86.93007
> pi^2/3 * 5^2
[1] 82.2467
>
>
>
> cleanEx()
> nameEx("Lognormal")
> ### * Lognormal
>
> flush(stderr()); flush(stdout())
>
> ### Name: Lognormal
> ### Title: The Log Normal Distribution
> ### Aliases: Lognormal dlnorm plnorm qlnorm rlnorm
> ### Keywords: distribution
>
> ### ** Examples
>
> dlnorm(1) == dnorm(0)
[1] TRUE
>
>
>
> cleanEx()
> nameEx("Multinom")
> ### * Multinom
>
> flush(stderr()); flush(stdout())
>
> ### Name: Multinom
> ### Title: The Multinomial Distribution
> ### Aliases: Multinomial rmultinom dmultinom
> ### Keywords: distribution
>
> ### ** Examples
>
> rmultinom(10, size = 12, prob=c(0.1,0.2,0.8))
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 0 1 0 3 1 0 1 2 2 1
[2,] 2 4 4 2 0 1 2 2 5 3
[3,] 10 7 8 7 11 11 9 8 5 8
>
> pr <- c(1,3,6,10) # normalization not necessary for generation
> rmultinom(10, 20, prob = pr)
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
[1,] 3 0 0 0 1 1 1 1 1 1
[2,] 1 2 3 3 2 4 3 4 4 4
[3,] 7 6 9 7 8 3 8 6 2 7
[4,] 9 12 8 10 9 12 8 9 13 8
>
> ## all possible outcomes of Multinom(N = 3, K = 3)
> X <- t(as.matrix(expand.grid(0:3, 0:3))); X <- X[, colSums(X) <= 3]
> X <- rbind(X, 3:3 - colSums(X)); dimnames(X) <- list(letters[1:3], NULL)
> X
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
a 0 1 2 3 0 1 2 0 1 0
b 0 0 0 0 1 1 1 2 2 3
c 3 2 1 0 2 1 0 1 0 0
> round(apply(X, 2, function(x) dmultinom(x, prob = c(1,2,5))), 3)
[1] 0.244 0.146 0.029 0.002 0.293 0.117 0.012 0.117 0.023 0.016
>
>
>
> cleanEx()
> nameEx("NLSstAsymptotic")
> ### * NLSstAsymptotic
>
> flush(stderr()); flush(stdout())
>
> ### Name: NLSstAsymptotic
> ### Title: Fit the Asymptotic Regression Model
> ### Aliases: NLSstAsymptotic NLSstAsymptotic.sortedXyData
> ### Keywords: manip
>
> ### ** Examples
>
> Lob.329 <- Loblolly[ Loblolly$Seed == "329", ]
> print(NLSstAsymptotic(sortedXyData(expression(age),
+ expression(height),
+ Lob.329)), digits=3)
b0 b1 lrc
-8.25 102.38 -3.22
>
>
>
> cleanEx()
> nameEx("NLSstClosestX")
> ### * NLSstClosestX
>
> flush(stderr()); flush(stdout())
>
> ### Name: NLSstClosestX
> ### Title: Inverse Interpolation
> ### Aliases: NLSstClosestX NLSstClosestX.sortedXyData
> ### Keywords: manip
>
> ### ** Examples
>
> DNase.2 <- DNase[ DNase$Run == "2", ]
> DN.srt <- sortedXyData(expression(log(conc)), expression(density), DNase.2)
> NLSstClosestX(DN.srt, 1.0)
[1] 0.9795406
>
>
>
> cleanEx()
> nameEx("NLSstLfAsymptote")
> ### * NLSstLfAsymptote
>
> flush(stderr()); flush(stdout())
>
> ### Name: NLSstLfAsymptote
> ### Title: Horizontal Asymptote on the Left Side
> ### Aliases: NLSstLfAsymptote NLSstLfAsymptote.sortedXyData
> ### Keywords: manip
>
> ### ** Examples
>
> DNase.2 <- DNase[ DNase$Run == "2", ]
> DN.srt <- sortedXyData( expression(log(conc)), expression(density), DNase.2 )
> NLSstLfAsymptote( DN.srt )
[1] -0.1869375
>
>
>
> cleanEx()
> nameEx("NLSstRtAsymptote")
> ### * NLSstRtAsymptote
>
> flush(stderr()); flush(stdout())
>
> ### Name: NLSstRtAsymptote
> ### Title: Horizontal Asymptote on the Right Side
> ### Aliases: NLSstRtAsymptote NLSstRtAsymptote.sortedXyData
> ### Keywords: manip
>
> ### ** Examples
>
> DNase.2 <- DNase[ DNase$Run == "2", ]
> DN.srt <- sortedXyData( expression(log(conc)), expression(density), DNase.2 )
> NLSstRtAsymptote( DN.srt )
[1] 2.157437
>
>
>
> cleanEx()
> nameEx("NegBinomial")
> ### * NegBinomial
>
> flush(stderr()); flush(stdout())
>
> ### Name: NegBinomial
> ### Title: The Negative Binomial Distribution
> ### Aliases: NegBinomial dnbinom pnbinom qnbinom rnbinom
> ### Keywords: distribution
>
> ### ** Examples
>
> require(graphics)
> x <- 0:11
> dnbinom(x, size = 1, prob = 1/2) * 2^(1 + x) # == 1
[1] 1 1 1 1 1 1 1 1 1 1 1 1
> 126 / dnbinom(0:8, size = 2, prob = 1/2) #- theoretically integer
[1] 504.0 504.0 672.0 1008.0 1612.8 2688.0 4608.0 8064.0 14336.0
>
>
> x <- 0:15
> size <- (1:20)/4
> persp(x,size, dnb <- outer(x, size, function(x,s) dnbinom(x,s, prob= 0.4)),
+ xlab = "x", ylab = "s", zlab="density", theta = 150)
> title(tit <- "negative binomial density(x,s, pr = 0.4) vs. x & s")
>
> image (x,size, log10(dnb), main= paste("log [",tit,"]"))
> contour(x,size, log10(dnb),add=TRUE)
>
> ## Alternative parametrization
> x1 <- rnbinom(500, mu = 4, size = 1)
> x2 <- rnbinom(500, mu = 4, size = 10)
> x3 <- rnbinom(500, mu = 4, size = 100)
> h1 <- hist(x1, breaks = 20, plot = FALSE)
> h2 <- hist(x2, breaks = h1$breaks, plot = FALSE)
> h3 <- hist(x3, breaks = h1$breaks, plot = FALSE)
> barplot(rbind(h1$counts, h2$counts, h3$counts),
+ beside = TRUE, col = c("red","blue","cyan"),
+ names.arg = round(h1$breaks[-length(h1$breaks)]))
>
>
>
> cleanEx()
> nameEx("Normal")
> ### * Normal
>
> flush(stderr()); flush(stdout())
>
> ### Name: Normal
> ### Title: The Normal Distribution
> ### Aliases: Normal dnorm pnorm qnorm rnorm
> ### Keywords: distribution
>
> ### ** Examples
>
> require(graphics)
>
> dnorm(0) == 1/ sqrt(2*pi)
[1] TRUE
> dnorm(1) == exp(-1/2)/ sqrt(2*pi)
[1] TRUE
> dnorm(1) == 1/ sqrt(2*pi*exp(1))
[1] TRUE
>
> ## Using "log = TRUE" for an extended range :
> par(mfrow=c(2,1))
> plot(function(x) dnorm(x, log=TRUE), -60, 50,
+ main = "log { Normal density }")
> curve(log(dnorm(x)), add=TRUE, col="red",lwd=2)
> mtext("dnorm(x, log=TRUE)", adj=0)
> mtext("log(dnorm(x))", col="red", adj=1)
>
> plot(function(x) pnorm(x, log.p=TRUE), -50, 10,
+ main = "log { Normal Cumulative }")
> curve(log(pnorm(x)), add=TRUE, col="red",lwd=2)
> mtext("pnorm(x, log=TRUE)", adj=0)
> mtext("log(pnorm(x))", col="red", adj=1)
>
> ## if you want the so-called 'error function'
> erf <- function(x) 2 * pnorm(x * sqrt(2)) - 1
> ## (see Abramowitz and Stegun 29.2.29)
> ## and the so-called 'complementary error function'
> erfc <- function(x) 2 * pnorm(x * sqrt(2), lower = FALSE)
> ## and the inverses
> erfinv <- function (x) qnorm((1 + x)/2)/sqrt(2)
> erfcinv <- function (x) qnorm(x/2, lower = FALSE)/sqrt(2)
>
>
>
> graphics::par(get("par.postscript", pos = 'CheckExEnv'))
> cleanEx()
> nameEx("Poisson")
> ### * Poisson
>
> flush(stderr()); flush(stdout())
>
> ### Name: Poisson
> ### Title: The Poisson Distribution
> ### Aliases: Poisson dpois ppois qpois rpois
> ### Keywords: distribution
>
> ### ** Examples
>
> require(graphics)
>
> -log(dpois(0:7, lambda=1) * gamma(1+ 0:7)) # == 1
[1] 1 1 1 1 1 1 1 1
> Ni <- rpois(50, lambda = 4); table(factor(Ni, 0:max(Ni)))
0 1 2 3 4 5 6 7 8 9 10
1 2 7 9 8 13 5 4 0 0 1
>
> 1 - ppois(10*(15:25), lambda=100) # becomes 0 (cancellation)
[1] 1.233094e-06 1.261664e-08 7.085799e-11 2.252643e-13 4.440892e-16
[6] 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00
[11] 0.000000e+00
> ppois(10*(15:25), lambda=100, lower.tail=FALSE) # no cancellation
[1] 1.233094e-06 1.261664e-08 7.085800e-11 2.253110e-13 4.174239e-16
[6] 4.626179e-19 3.142097e-22 1.337219e-25 3.639328e-29 6.453883e-33
[11] 7.587807e-37
>
> par(mfrow = c(2, 1))
> x <- seq(-0.01, 5, 0.01)
> plot(x, ppois(x, 1), type="s", ylab="F(x)", main="Poisson(1) CDF")
> plot(x, pbinom(x, 100, 0.01),type="s", ylab="F(x)",
+ main="Binomial(100, 0.01) CDF")
>
>
>
> graphics::par(get("par.postscript", pos = 'CheckExEnv'))
> cleanEx()
> nameEx("SSD")
> ### * SSD
>
> flush(stderr()); flush(stdout())
>
> ### Name: SSD
> ### Title: SSD Matrix and Estimated Variance Matrix in Multivariate Models
> ### Aliases: SSD estVar
> ### Keywords: models multivariate
>
> ### ** Examples
>
> # Lifted from Baron+Li:
> # "Notes on the use of R for psychology experiments and questionnaires"
> # Maxwell and Delaney, p. 497
> reacttime <- matrix(c(
+ 420, 420, 480, 480, 600, 780,
+ 420, 480, 480, 360, 480, 600,
+ 480, 480, 540, 660, 780, 780,
+ 420, 540, 540, 480, 780, 900,
+ 540, 660, 540, 480, 660, 720,
+ 360, 420, 360, 360, 480, 540,
+ 480, 480, 600, 540, 720, 840,
+ 480, 600, 660, 540, 720, 900,
+ 540, 600, 540, 480, 720, 780,
+ 480, 420, 540, 540, 660, 780),
+ ncol = 6, byrow = TRUE,
+ dimnames=list(subj=1:10,
+ cond=c("deg0NA", "deg4NA", "deg8NA",
+ "deg0NP", "deg4NP", "deg8NP")))
>
> mlmfit <- lm(reacttime~1)
> SSD(mlmfit)
$SSD
cond
cond deg0NA deg4NA deg8NA deg0NP deg4NP deg8NP
deg0NA 29160 30600 26640 23760 32400 25560
deg4NA 30600 66600 32400 7200 36000 30600
deg8NA 26640 32400 56160 41040 57600 69840
deg0NP 23760 7200 41040 70560 72000 63360
deg4NP 32400 36000 57600 72000 108000 100800
deg8NP 25560 30600 69840 63360 100800 122760
$call
lm(formula = reacttime ~ 1)
$df
[1] 9
attr(,"class")
[1] "SSD"
> estVar(mlmfit)
cond
cond deg0NA deg4NA deg8NA deg0NP deg4NP deg8NP
deg0NA 3240 3400 2960 2640 3600 2840
deg4NA 3400 7400 3600 800 4000 3400
deg8NA 2960 3600 6240 4560 6400 7760
deg0NP 2640 800 4560 7840 8000 7040
deg4NP 3600 4000 6400 8000 12000 11200
deg8NP 2840 3400 7760 7040 11200 13640
>
>
>
> cleanEx()
> nameEx("SSasymp")
> ### * SSasymp
>
> flush(stderr()); flush(stdout())
>
> ### Name: SSasymp
> ### Title: Self-Starting Nls Asymptotic Regression Model
> ### Aliases: SSasymp
> ### Keywords: models
>
> ### ** Examples
>
> ## Don't show:
> options(show.nls.convergence=FALSE)
> ## End Don't show
> Lob.329 <- Loblolly[ Loblolly$Seed == "329", ]
> SSasymp( Lob.329$age, 100, -8.5, -3.2 ) # response only
[1] 3.988924 11.505611 27.822517 41.130854 51.985354 60.838463
> Asym <- 100
> resp0 <- -8.5
> lrc <- -3.2
> SSasymp( Lob.329$age, Asym, resp0, lrc ) # response and gradient
[1] 3.988924 11.505611 27.822517 41.130854 51.985354 60.838463
attr(,"gradient")
Asym resp0 lrc
[1,] 0.1151053 0.8848947 11.74087
[2,] 0.1843835 0.8156165 18.03613
[3,] 0.3347697 0.6652303 29.42113
[4,] 0.4574272 0.5425728 35.99454
[5,] 0.5574687 0.4425313 39.14366
[6,] 0.6390642 0.3609358 39.90776
> getInitial(height ~ SSasymp( age, Asym, resp0, lrc), data = Lob.329)
$Asym
[1] 94.1282
$resp0
[1] -8.250753
$lrc
[1] -3.217578
> ## Initial values are in fact the converged values
> fm1 <- nls(height ~ SSasymp( age, Asym, resp0, lrc), data = Lob.329)
> summary(fm1)
Formula: height ~ SSasymp(age, Asym, resp0, lrc)
Parameters:
Estimate Std. Error t value Pr(>|t|)
Asym 94.1282 8.4030 11.202 0.001525 **
resp0 -8.2508 1.2261 -6.729 0.006700 **
lrc -3.2176 0.1386 -23.218 0.000175 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.7493 on 3 degrees of freedom
> ## Don't show:
> require(graphics)
>
> xx <- seq(0, 5, len = 101)
> yy <- 5 - 4 * exp(-xx/(2*log(2)))
> par(mar = c(0, 0, 4.1, 0))
> plot(xx, yy, type = "l", axes = FALSE, ylim = c(0,6), xlim = c(-1, 5),
+ xlab = "", ylab = "", lwd = 2,
+ main = "Parameters in the SSasymp model")
> usr <- par("usr")
> arrows(usr[1], 0, usr[2], 0, length = 0.1, angle = 25)
> arrows(0, usr[3], 0, usr[4], length = 0.1, angle = 25)
> text(usr[2] - 0.2, 0.1, "x", adj = c(1, 0))
> text(-0.1, usr[4], "y", adj = c(1, 1))
> abline(h = 5, lty = 2, lwd = 0)
> arrows(-0.8, 2.1, -0.8, 0, length = 0.1, angle = 25)
> arrows(-0.8, 2.9, -0.8, 5, length = 0.1, angle = 25)
> text(-0.8, 2.5, expression(phi[1]), adj = c(0.5, 0.5))
> segments(-0.4, 1, 0, 1, lty = 2, lwd = 0.75)
> arrows(-0.3, 0.25, -0.3, 0, length = 0.07, angle = 25)
> arrows(-0.3, 0.75, -0.3, 1, length = 0.07, angle = 25)
> text(-0.3, 0.5, expression(phi[2]), adj = c(0.5, 0.5))
> segments(1, 3.025, 1, 4, lty = 2, lwd = 0.75)
> arrows(0.2, 3.5, 0, 3.5, length = 0.08, angle = 25)
> arrows(0.8, 3.5, 1, 3.5, length = 0.08, angle = 25)
> text(0.5, 3.5, expression(t[0.5]), adj = c(0.5, 0.5))
> ## End Don't show
>
>
>
> graphics::par(get("par.postscript", pos = 'CheckExEnv'))
> cleanEx()
> nameEx("SSasympOff")
> ### * SSasympOff
>
> flush(stderr()); flush(stdout())
>
> ### Name: SSasympOff
> ### Title: Self-Starting Nls Asymptotic Regression Model with an Offset
> ### Aliases: SSasympOff
> ### Keywords: models
>
> ### ** Examples
>
> CO2.Qn1 <- CO2[CO2$Plant == "Qn1", ]
> SSasympOff(CO2.Qn1$conc, 32, -4, 43) # response only
[1] 19.65412 29.14785 31.27791 31.88435 31.99259 31.99970 32.00000
> Asym <- 32; lrc <- -4; c0 <- 43
> SSasympOff(CO2.Qn1$conc, Asym, lrc, c0) # response and gradient
[1] 19.65412 29.14785 31.27791 31.88435 31.99259 31.99970 32.00000
attr(,"gradient")
Asym lrc c0
[1,] 0.6141911 1.175838e+01 -2.261227e-01
[2,] 0.9108704 6.895531e+00 -5.223887e-02
[3,] 0.9774346 2.737698e+00 -1.322559e-02
[4,] 0.9963859 6.503026e-01 -2.118250e-03
[5,] 0.9997683 6.204920e-02 -1.357751e-04
[6,] 0.9999906 3.479529e-03 -5.505583e-06
[7,] 1.0000000 1.369435e-05 -1.430967e-08
> getInitial(uptake ~ SSasympOff(conc, Asym, lrc, c0), data = CO2.Qn1)
$Asym
[1] 38.13978
$lrc
[1] -4.380647
$c0
[1] 51.22324
> ## Initial values are in fact the converged values
> fm1 <- nls(uptake ~ SSasympOff(conc, Asym, lrc, c0), data = CO2.Qn1)
> summary(fm1)
Formula: uptake ~ SSasympOff(conc, Asym, lrc, c0)
Parameters:
Estimate Std. Error t value Pr(>|t|)
Asym 38.1398 0.9164 41.620 1.99e-06 ***
lrc -4.3806 0.2042 -21.457 2.79e-05 ***
c0 51.2232 11.6698 4.389 0.0118 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.663 on 4 degrees of freedom
> ## Don't show:
> require(graphics)
>
> xx <- seq(0.5, 5, len = 101)
> yy <- 5 * (1 - exp(-(xx - 0.5)/(2*log(2))))
> par(mar = c(0, 0, 4.0, 0))
> plot(xx, yy, type = "l", axes = FALSE, ylim = c(0,6), xlim = c(-1, 5),