363_5
2025-06-19
df <- read.csv("male_players.csv")
df <- df %>%
filter(Position != "GK")
df <- df %>%
filter(League %in% c("Premier League", "LALIGA EA SPORTS",
"Serie A Enilive", "Bundesliga", "Ligue 1 McDonald's")) %>%
## pick the top 1000 players
arrange(desc(OVR)) %>%
slice(1:1000)- All 37 attributes are continuous player ratings, so after z scoring a simple metric based on straight line (Euclidean) or city block (Manhattan) distance preserves their geometry. No additional transforms are needed once scaling removes unit and variance differences; if we had categorical traits (e.g., preferred foot) we would switch to a mixed type metric such as Gower.
## recreate above process but drop all GK's
clen_noGK <- df %>%
filter(Position != "GK") %>%
mutate(Height_cm = as.numeric(str_extract(Height, "\\d+")),
Weight_kg = as.numeric(str_extract(Weight, "\\d+")),
Position = as.factor(Position), Preferred_foot = as.factor(Preferred.foot)) %>%
replace_na(list(GK.Diving = 0, GK.Handling = 0, GK.Kicking = 0,
GK.Positioning = 0, GK.Reflexes = 0))num_data_noGK <- clen_noGK %>%
dplyr::select(PAC, SHO, PAS, DRI, DEF, PHY, Acceleration,
Sprint.Speed, Positioning, Finishing, Shot.Power, Long.Shots,
Volleys, Penalties, Vision, Crossing, Free.Kick.Accuracy,
Short.Passing, Long.Passing, Curve, Dribbling, Agility,
Balance, Reactions, Ball.Control, Composure, Interceptions,
Heading.Accuracy, Def.Awareness, Standing.Tackle, Sliding.Tackle,
Jumping, Stamina, Strength, Aggression, Height_cm, Weight_kg)# 2. scale + PCA, without GK
X_sc_noGK <- scale(num_data_noGK)
# 1) find any zero-variance columns in the raw numeric data
zero.var_noGK <- which(apply(num_data_noGK, 2, var, na.rm = TRUE) ==
0)
if (length(zero.var_noGK) > 0) {
message("Dropping ", length(zero.var_noGK), " constant columns: ",
paste(colnames(num_data_noGK)[zero.var_noGK], collapse = ", "))
num_data2_noGK <- num_data_noGK[, -zero.var_noGK]
} else {
num_data2_noGK <- num_data_noGK
}
X_sc_noGK <- scale(num_data2_noGK)# 2) now run PCA with centering & scaling
pca_noGK <- prcomp(X_sc_noGK, center = TRUE, scale. = TRUE)
# 3) plot cumulative variance explained
plot(cumsum(pca_noGK$sdev^2/sum(pca_noGK$sdev^2)), xlab = "Principal components",
ylab = "Cumulative variance explained", type = "b", pch = 19)
# examine variance explained → choose ~10 PCs (~95%+)
fviz_eig(pca_noGK, addlabels = TRUE)
pc_df_noGK_noGK <- as_tibble(pca_noGK$x[, 1:10]) # retain the first 10 components
fviz_pca_var(pca_noGK, col.var = "contrib", gradient.cols = c("blue",
"white", "red"), title = "PCA variable contributions (without GKs)")
# 2.2. visualize PCA biplot
fviz_pca_biplot(pca_noGK, col.var = "contrib", gradient.cols = c("blue",
"white", "red"), title = "PCA biplot (without GKs)", repel = TRUE)
# 2.3. visualize PCA correlation circle
fviz_pca_var(pca_noGK, col.var = "coord", gradient.cols = c("blue",
"white", "red"), title = "PCA correlation circle (without GKs)")
## 1. Load helper that draws the 4 classic H-clust diagnostic
## plots (linkage distance, R², semi-partial R², pseudo-t²)
source("https://raw.githubusercontent.com/jreuning/sds363_code/refs/heads/main/HClusEval3.R.txt")
## pick linkage & distance you want to test ------------------
dist.m <- "euclidean"
clus.m <- "complete" # or "ward.D2", "average", etc.
## run and plot – shows the first 15 merges by default --------
hclus_eval(X_sc_noGK,
dist_m = dist.m,
clus_m = clus.m,
plot_op = TRUE, # four-panel diagnostic figure
print_num = 15) # how many merges to list## [1] "Creating Distance Matrix using euclidean"
## [1] "Clustering using complete"
## [1] "Clustering Complete. Access the Cluster object in first element of output"
## [1] "Calculating RMSSTD"
## [1] "RMSSTD Done. Access in Element 2"
## [1] "Calculating RSQ"
## [1] "RSQ Done. Access in Element 3"
## [1] "Calculating SPRSQ"
## [1] "SPRSQ Done. Access in Element 4"
## [1] "Calculating Cluster Dist. "
## [1] "CD Done. Access in Element 5"
## [[1]]
##
## Call:
## hclust(d = dist1, method = clus_m)
##
## Cluster method : complete
## Distance : euclidean
## Number of objects: 1000
##
##
## [[2]]
## [1] 1.0000000 0.8915704 0.8809982 0.8268769 0.7879687 0.7714563 0.7151679
## [8] 0.7200195 0.7206121 0.6838090 0.6795547 0.6349088 0.6117317 0.6805728
## [15] 0.6171045 0.6410949 0.6255472 0.6457260 0.6176260 0.5968483 0.6305155
## [22] 0.6167342 0.5989513 0.6410703 0.6238845 0.6163793 0.6370258 0.5903831
## [29] 0.6183178 0.5748971 0.5719099 0.5454820 0.5500797 0.5729849 0.6474852
## [36] 0.5825269 0.5569698 0.5890784 0.6103281 0.5646589 0.6101242 0.5657868
## [43] 0.5748255 0.5535857 0.6496824 0.5530960 0.5763284 0.6113915 0.5659521
## [50] 0.5737801 0.5306585 0.5362316 0.5464446 0.4962919 0.6335439 0.5353926
## [57] 0.5165637 0.5183046 0.5159883 0.5636047 0.5258733 0.5986118 0.5308772
## [64] 0.5174257 0.5348024 0.5607226 0.5337782 0.5516143 0.5421247 0.5110504
## [71] 0.5462217 0.5173514 0.5140475 0.5211661 0.6098378 0.5442220 0.6406117
## [78] 0.4657663 0.5986996 0.5681539 0.5529703 0.4923189 0.4363577 0.5000318
## [85] 0.6133387 0.5085657 0.5939041 0.5017216 0.5117050 0.5215549 0.4859679
## [92] 0.4471053 0.5514927 0.5651455 0.5736805 0.5456017 0.5079009 0.5303252
## [99] 0.5576185 0.4662371 0.4781246 0.5513754 0.5447947 0.4976472 0.5329400
## [106] 0.5566307 0.5025052 0.5570414 0.4889924 0.4917808 0.5171884 0.4998692
## [113] 0.4661423 0.4646436 0.5347979 0.4945524 0.5781460 0.5895046 0.4940146
## [120] 0.5720294 0.4869230 0.4770861 0.5572757 0.5608718 0.4749196 0.4632798
## [127] 0.5243447 0.4978471 0.4747371 0.6193411 0.5223043 0.4893462 0.4967341
## [134] 0.5064523 0.5066442 0.4873892 0.4376596 0.4941483 0.4874892 0.4669681
## [141] 0.5249745 0.4775041 0.5114924 0.4401129 0.4668732 0.4575279 0.4504602
## [148] 0.4851990 0.4879208 0.4509864 0.5152089 0.4684615 0.5017582 0.5019157
## [155] 0.4445636 0.6346059 0.5477331 0.4381542 0.4659014 0.4700393 0.4321612
## [162] 0.4451123 0.4810694 0.5322309 0.4482632 0.5152067 0.4715937 0.4667612
## [169] 0.4431364 0.4592051 0.4412076 0.6152918 0.5051213 0.4599653 0.4990356
## [176] 0.4753253 0.4556400 0.5318699 0.4683152 0.4318292 0.4456846 0.4693691
## [183] 0.4710532 0.4333109 0.4423605 0.4719402 0.4308003 0.5447034 0.4933543
## [190] 0.4423538 0.4818428 0.4514915 0.5266573 0.5377028 0.5424325 0.4635969
## [197] 0.4915564 0.4685071 0.4994908 0.5876851 0.4679141 0.5072229 0.4265284
## [204] 0.4787561 0.5816176 0.4317888 0.4585336 0.5394676 0.5596191 0.4130581
## [211] 0.4537877 0.4477231 0.4773314 0.4585514 0.4573835 0.4584653 0.5163469
## [218] 0.4281972 0.4053486 0.5711827 0.4342169 0.4259671 0.4305165 0.5172513
## [225] 0.4540036 0.4278931 0.4304711 0.4654464 0.4253339 0.4615631 0.4695344
## [232] 0.5222156 0.4269405 0.4064885 0.4477320 0.4760348 0.4165538 0.5526666
## [239] 0.4906840 0.4310045 0.4869929 0.4898567 0.4193421 0.4622049 0.4276974
## [246] 0.4559242 0.3935771 0.3977179 0.4244205 0.4757249 0.4465093 0.4478641
## [253] 0.5475471 0.3942994 0.5461346 0.4178043 0.4232623 0.4895771 0.5440332
## [260] 0.4250843 0.5049935 0.4598826 0.4047770 0.4331604 0.3903129 0.4258810
## [267] 0.4876189 0.4053118 0.4368188 0.4532410 0.4169776 0.4226487 0.4148958
## [274] 0.4381583 0.3973794 0.4736822 0.4183718 0.4297934 0.4331453 0.4282562
## [281] 0.4363798 0.5256308 0.4404345 0.4882649 0.4442318 0.4165881 0.4160885
## [288] 0.4199958 0.4751188 0.4146579 0.4083104 0.4362975 0.4042311 0.4381879
## [295] 0.3755263 0.4484574 0.3873204 0.3592901 0.4128501 0.4504567 0.3969116
## [302] 0.5088826 0.3826673 0.4156262 0.4107599 0.4265061 0.4032867 0.3766618
## [309] 0.4054718 0.4438064 0.4345895 0.4044113 0.4504621 0.3962285 0.4249966
## [316] 0.3948428 0.5003510 0.4535696 0.4484880 0.4223871 0.4989147 0.3803080
## [323] 0.4465205 0.4983245 0.4346735 0.4168507 0.4091532 0.4297770 0.4605367
## [330] 0.4387846 0.3855640 0.4231637 0.4722243 0.4577922 0.4894384 0.4124197
## [337] 0.4024030 0.4876114 0.4393573 0.3787603 0.3992811 0.3976213 0.4223945
## [344] 0.4297383 0.4390307 0.4132237 0.4339491 0.4209441 0.4003209 0.4271231
## [351] 0.4314026 0.3909935 0.3797582 0.3896204 0.4494499 0.4378573 0.4764489
## [358] 0.4009187 0.4463116 0.4124938 0.4061639 0.4172024 0.3844369 0.4727732
## [365] 0.4568526 0.4085743 0.3787238 0.3811349 0.4090151 0.3700244 0.4219949
## [372] 0.4057803 0.4686048 0.4032960 0.3804433 0.3855645 0.4050165 0.4407377
## [379] 0.3745414 0.3936147 0.4230467 0.4227531 0.4036024 0.4376908 0.3993851
## [386] 0.3937363 0.4133686 0.4236292 0.3480211 0.4620000 0.4619351 0.3894199
## [393] 0.4139150 0.3755560 0.3951915 0.4595161 0.4211320 0.3670760 0.3916315
## [400] 0.4218260 0.4571137 0.4036267 0.4179725 0.4284197 0.3809422 0.3974367
## [407] 0.3939595 0.4550071 0.4546041 0.3841233 0.3743977 0.3985195 0.3890400
## [414] 0.3608835 0.3901829 0.3572283 0.3851935 0.3920973 0.4502216 0.3864519
## [421] 0.4102619 0.4498244 0.3545058 0.4488545 0.3550048 0.3654496 0.4483763
## [428] 0.4476395 0.4061893 0.3979106 0.3751358 0.3831159 0.3505944 0.3715316
## [435] 0.4459758 0.3837274 0.4454065 0.3923858 0.3465201 0.3791071 0.3789797
## [442] 0.3576609 0.4026227 0.3981916 0.4007932 0.3835976 0.3764491 0.4403269
## [449] 0.4395425 0.3821564 0.3858133 0.3921713 0.3880188 0.4111197 0.3878249
## [456] 0.4369680 0.3904145 0.4361475 0.3389558 0.4023210 0.4356566 0.3578328
## [463] 0.4111644 0.3957296 0.3629207 0.3987181 0.3856223 0.4332718 0.4332202
## [470] 0.3772909 0.3813334 0.4327213 0.3985859 0.3446084 0.4318347 0.3830881
## [477] 0.3510233 0.3645578 0.4124283 0.3897173 0.4299114 0.3609164 0.4290662
## [484] 0.3580697 0.4115809 0.3684092 0.4277467 0.4122234 0.3595105 0.3786110
## [491] 0.3770180 0.3723145 0.4257503 0.3751705 0.3656883 0.3867711 0.3661488
## [498] 0.3913686 0.3818225 0.3594484 0.3583782 0.3412789 0.4210258 0.3344473
## [505] 0.4203372 0.4202987 0.3963343 0.4200457 0.3687340 0.3511803 0.3669741
## [512] 0.3823960 0.4192871 0.4192680 0.3404917 0.3804372 0.4076466 0.4181071
## [519] 0.3791263 0.3600784 0.3510859 0.3298724 0.4162580 0.3747782 0.3672290
## [526] 0.3493949 0.3722701 0.3839685 0.3581967 0.4014913 0.3625451 0.4127015
## [533] 0.4126357 0.3644326 0.4123273 0.3617684 0.3562987 0.3801678 0.3331123
## [540] 0.3552839 0.4110238 0.3724184 0.3678710 0.4100892 0.3603486 0.3610865
## [547] 0.3455567 0.4094845 0.3326895 0.3915278 0.3756905 0.3935083 0.3379026
## [554] 0.4077522 0.3399932 0.3808155 0.4066979 0.4062128 0.3460368 0.4054913
## [561] 0.3812437 0.3436812 0.3690858 0.3376669 0.3823821 0.3357204 0.3219170
## [568] 0.4003364 0.3666705 0.3996988 0.3996929 0.3992642 0.3421393 0.3548468
## [575] 0.3864279 0.3965212 0.3653189 0.3956484 0.3599096 0.3947817 0.3590892
## [582] 0.3646527 0.3550887 0.3418956 0.3464692 0.3933017 0.3929945 0.3177316
## [589] 0.3805610 0.3238969 0.3923179 0.3918534 0.3914981 0.3914300 0.3547303
## [596] 0.3678566 0.3906884 0.3894622 0.3888676 0.3596320 0.3567887 0.3876943
## [603] 0.3367345 0.3494338 0.3579647 0.3859506 0.3585032 0.3333839 0.3489492
## [610] 0.3447497 0.3852222 0.3316204 0.3845201 0.3843502 0.3840943 0.3840696
## [617] 0.3836512 0.3818295 0.3816422 0.3286044 0.3320942 0.3809888 0.3807812
## [624] 0.3807437 0.3804905 0.3802756 0.3799266 0.3322805 0.3798481 0.3275869
## [631] 0.3463626 0.3791571 0.3355865 0.3476874 0.3442153 0.3525303 0.3590337
## [638] 0.3778826 0.3540315 0.3437862 0.3334704 0.3773266 0.3254420 0.3770343
## [645] 0.3768454 0.3568723 0.3767255 0.3630637 0.3105337 0.3752122 0.3746378
## [652] 0.3495348 0.3741569 0.3238960 0.3319725 0.3313501 0.3639212 0.3734426
## [659] 0.3733027 0.3729362 0.3453168 0.3230475 0.3195252 0.3469846 0.3386619
## [666] 0.3719945 0.3476090 0.3719051 0.3712893 0.3709796 0.3329667 0.3552976
## [673] 0.3707365 0.3233005 0.3390697 0.3694991 0.3694435 0.3382784 0.3687374
## [680] 0.3134971 0.3356164 0.3678576 0.3678281 0.3117262 0.3671400 0.3670010
## [687] 0.3374845 0.3651553 0.3651051 0.3649726 0.3635715 0.3268929 0.3394579
## [694] 0.3167852 0.3601710 0.3600708 0.3600633 0.3379978 0.3564570 0.3414755
## [701] 0.3454445 0.3550526 0.3550368 0.3548594 0.3541420 0.3305380 0.3538903
## [708] 0.3538469 0.3170829 0.3231355 0.3531202 0.3527179 0.3525117 0.3045958
## [715] 0.3520886 0.2945021 0.3333720 0.3517099 0.3254792 0.3514293 0.3512777
## [722] 0.3510523 0.3502045 0.3493231 0.3492149 0.3091475 0.3032021 0.3484398
## [729] 0.3481279 0.3478556 0.3470729 0.3205951 0.3464866 0.3463680 0.3196944
## [736] 0.3021322 0.3130570 0.3458683 0.3203021 0.3184073 0.3454941 0.3452897
## [743] 0.3451992 0.3449454 0.3446894 0.3438946 0.3252767 0.3436328 0.3436068
## [750] 0.3434270 0.3431797 0.3430234 0.3142649 0.3206472 0.2961651 0.3422144
## [757] 0.3198688 0.2991007 0.3206260 0.3416904 0.2892602 0.3410976 0.3408113
## [764] 0.3403277 0.3402153 0.3393775 0.3391233 0.3023246 0.3176754 0.3383797
## [771] 0.3378073 0.3375571 0.3176257 0.3101513 0.3366982 0.3359272 0.3355613
## [778] 0.3350522 0.3349566 0.3344924 0.3343614 0.3341481 0.2950820 0.3119347
## [785] 0.3334782 0.3334246 0.3328253 0.3035097 0.3318715 0.3310181 0.3308880
## [792] 0.3306269 0.3304295 0.3073057 0.3300760 0.3300587 0.3293226 0.3288785
## [799] 0.3288322 0.3275982 0.3275458 0.2989898 0.2974754 0.2830339 0.3272685
## [806] 0.3270794 0.3269468 0.3266631 0.3261870 0.3261352 0.3258646 0.3257874
## [813] 0.3254715 0.3250441 0.3249260 0.2952872 0.3246169 0.2960504 0.3238138
## [820] 0.3235853 0.3235290 0.2879131 0.3224458 0.3140653 0.2848514 0.3211720
## [827] 0.3204247 0.3202915 0.2949944 0.3196105 0.3194023 0.3192092 0.3182969
## [834] 0.3181069 0.3175556 0.2883982 0.3172175 0.3002479 0.3169145 0.3168055
## [841] 0.3003980 0.2988091 0.2871159 0.3156044 0.3150404 0.3146603 0.3138161
## [848] 0.3136511 0.3132244 0.2988220 0.3127851 0.3127481 0.3122629 0.3109494
## [855] 0.3106543 0.3104508 0.3103626 0.3094719 0.3093845 0.3081330 0.3070934
## [862] 0.2844815 0.3065084 0.3064944 0.3006293 0.3041529 0.3034802 0.2801845
## [869] 0.3019904 0.3018691 0.3018580 0.3013706 0.3010284 0.3008171 0.2999281
## [876] 0.2998247 0.2995483 0.2658679 0.2985706 0.2983149 0.2978317 0.2977640
## [883] 0.2975798 0.2965920 0.2865897 0.2963213 0.2962009 0.2960881 0.2944390
## [890] 0.2943422 0.2937890 0.2551328 0.2932307 0.2620217 0.2913240 0.2910197
## [897] 0.2906226 0.2903787 0.2903441 0.2900203 0.2887931 0.2884617 0.2877630
## [904] 0.2603245 0.2874353 0.2873520 0.2870879 0.2869982 0.2869273 0.2867752
## [911] 0.2778405 0.2861865 0.2854966 0.2853636 0.2852294 0.2848990 0.2583263
## [918] 0.2840388 0.2838708 0.2838360 0.2838282 0.2829050 0.2822796 0.2822086
## [925] 0.2820458 0.2819135 0.2813726 0.2809944 0.2803353 0.2791535 0.2784816
## [932] 0.2780229 0.2780190 0.2772036 0.2771768 0.2770177 0.2768453 0.2767229
## [939] 0.2755336 0.2752222 0.2741886 0.2734069 0.2733788 0.2722726 0.2722202
## [946] 0.2716805 0.2708793 0.2704938 0.2692896 0.2691940 0.2690145 0.2683760
## [953] 0.2682512 0.2679751 0.2668364 0.2649655 0.2647533 0.2643738 0.2642738
## [960] 0.2622543 0.2621682 0.2609062 0.2599341 0.2598898 0.2590887 0.2586308
## [967] 0.2564886 0.2559664 0.2558490 0.2556358 0.2545122 0.2536568 0.2533513
## [974] 0.2528391 0.2527392 0.2527053 0.2513140 0.2510030 0.2496681 0.2493187
## [981] 0.2478091 0.2457410 0.2421385 0.2395493 0.2369358 0.2365501 0.2313769
## [988] 0.2303389 0.2255772 0.2254612 0.2252803 0.2241263 0.2225102 0.2196887
## [995] 0.2169635 0.2165784 0.2125222 0.2069702 0.1937258 0.0000000
##
## [[3]]
## [1] 0.0000000 0.2464694 0.3243645 0.3855254 0.4340403 0.5012862 0.5421796
## [8] 0.5494284 0.5688944 0.5846148 0.5940455 0.6035042 0.6097910 0.6127777
## [15] 0.6214563 0.6270984 0.6356415 0.6369099 0.6453687 0.6520751 0.6581177
## [22] 0.6611420 0.6645852 0.6771754 0.6789067 0.6878790 0.6904887 0.6928888
## [29] 0.6946430 0.6972222 0.6984626 0.7012825 0.7044670 0.7077194 0.7098430
## [36] 0.7109207 0.7125057 0.7132993 0.7147571 0.7156501 0.7170477 0.7190264
## [43] 0.7203668 0.7222744 0.7239809 0.7250945 0.7290052 0.7305708 0.7326949
## [50] 0.7343084 0.7348564 0.7372127 0.7393044 0.7412834 0.7433616 0.7440642
## [57] 0.7467633 0.7481981 0.7489029 0.7494858 0.7508098 0.7536373 0.7543124
## [64] 0.7552628 0.7570176 0.7577523 0.7586811 0.7608122 0.7616304 0.7622785
## [71] 0.7642053 0.7653250 0.7663426 0.7688879 0.7705151 0.7710717 0.7722475
## [78] 0.7729354 0.7739454 0.7748314 0.7755950 0.7764347 0.7774505 0.7781307
## [85] 0.7796511 0.7806359 0.7814693 0.7820335 0.7829856 0.7839486 0.7850751
## [92] 0.7865348 0.7871453 0.7876852 0.7881338 0.7888135 0.7897203 0.7905529
## [99] 0.7914894 0.7919865 0.7924654 0.7934669 0.7941125 0.7947442 0.7954871
## [106] 0.7961915 0.7968644 0.7978918 0.7984561 0.7994336 0.7998898 0.8006980
## [113] 0.8014358 0.8019504 0.8028527 0.8034994 0.8050201 0.8054801 0.8060477
## [120] 0.8066860 0.8071932 0.8084738 0.8091986 0.8099769 0.8104703 0.8112169
## [127] 0.8121463 0.8128468 0.8137681 0.8142579 0.8147666 0.8152660 0.8159030
## [134] 0.8166028 0.8173330 0.8179904 0.8188653 0.8198572 0.8203934 0.8208394
## [141] 0.8214698 0.8219338 0.8223837 0.8228834 0.8234180 0.8241542 0.8248635
## [148] 0.8259707 0.8269580 0.8273609 0.8280090 0.8284646 0.8292440 0.8300547
## [155] 0.8308730 0.8313369 0.8317400 0.8321638 0.8327213 0.8334103 0.8339182
## [162] 0.8347807 0.8354630 0.8358970 0.8363433 0.8370576 0.8374362 0.8381673
## [169] 0.8391551 0.8397370 0.8403640 0.8410055 0.8413845 0.8418744 0.8421872
## [176] 0.8427884 0.8431366 0.8436079 0.8440456 0.8445452 0.8452069 0.8457481
## [183] 0.8461284 0.8466064 0.8471017 0.8478677 0.8482905 0.8488507 0.8492334
## [190] 0.8496884 0.8503308 0.8507401 0.8513915 0.8518002 0.8521647 0.8524656
## [197] 0.8530813 0.8534629 0.8538397 0.8541536 0.8544993 0.8549811 0.8553050
## [204] 0.8556667 0.8560678 0.8564064 0.8568912 0.8572647 0.8576461 0.8579745
## [211] 0.8585222 0.8588796 0.8593935 0.8597357 0.8601875 0.8605377 0.8610295
## [218] 0.8613497 0.8618614 0.8622826 0.8626092 0.8632226 0.8637115 0.8643445
## [225] 0.8646772 0.8650618 0.8656831 0.8660165 0.8665579 0.8670957 0.8674038
## [232] 0.8677345 0.8680299 0.8683979 0.8687147 0.8690819 0.8694073 0.8697841
## [239] 0.8700899 0.8704067 0.8708466 0.8712446 0.8715684 0.8721134 0.8725324
## [246] 0.8728972 0.8731981 0.8736372 0.8739281 0.8743481 0.8746197 0.8749457
## [253] 0.8753777 0.8756778 0.8761516 0.8764501 0.8768746 0.8772666 0.8775760
## [260] 0.8778723 0.8782989 0.8786069 0.8789013 0.8792769 0.8796155 0.8799422
## [267] 0.8802648 0.8805730 0.8809053 0.8811616 0.8814943 0.8817779 0.8821315
## [274] 0.8825124 0.8829944 0.8832957 0.8835756 0.8838974 0.8842655 0.8845158
## [281] 0.8849382 0.8852453 0.8855218 0.8857881 0.8860780 0.8863561 0.8867059
## [288] 0.8869471 0.8874912 0.8877629 0.8882530 0.8888143 0.8890632 0.8893219
## [295] 0.8896199 0.8899112 0.8901733 0.8906033 0.8908645 0.8911250 0.8913608
## [302] 0.8916899 0.8919491 0.8923418 0.8928226 0.8931759 0.8934550 0.8938025
## [309] 0.8941189 0.8944125 0.8946756 0.8950029 0.8953841 0.8956706 0.8960036
## [316] 0.8963353 0.8966838 0.8969344 0.8971863 0.8974675 0.8977294 0.8979786
## [323] 0.8982769 0.8985275 0.8987760 0.8990487 0.8993192 0.8995865 0.8998111
## [330] 0.9000493 0.9003034 0.9005485 0.9008970 0.9011494 0.9014176 0.9016574
## [337] 0.9019157 0.9022158 0.9024538 0.9027495 0.9030958 0.9034342 0.9037624
## [344] 0.9041194 0.9044324 0.9047413 0.9049588 0.9053213 0.9056315 0.9058889
## [351] 0.9061284 0.9063857 0.9066518 0.9069682 0.9072190 0.9074630 0.9077024
## [358] 0.9079296 0.9081923 0.9084227 0.9086372 0.9088454 0.9091022 0.9093656
## [365] 0.9095893 0.9098222 0.9101238 0.9103978 0.9106648 0.9109178 0.9111502
## [372] 0.9113619 0.9116335 0.9118534 0.9121429 0.9123660 0.9126051 0.9128077
## [379] 0.9130420 0.9133201 0.9135110 0.9137345 0.9139698 0.9142024 0.9144319
## [386] 0.9146470 0.9148439 0.9150526 0.9152897 0.9155780 0.9157917 0.9160053
## [393] 0.9161880 0.9164162 0.9166607 0.9168986 0.9171100 0.9173206 0.9175664
## [400] 0.9178931 0.9181303 0.9183394 0.9185612 0.9188217 0.9190668 0.9193439
## [407] 0.9195588 0.9198071 0.9200144 0.9202213 0.9204342 0.9206589 0.9208819
## [414] 0.9211084 0.9212941 0.9214969 0.9217195 0.9219554 0.9221702 0.9223731
## [421] 0.9227251 0.9229379 0.9231404 0.9234521 0.9236538 0.9238145 0.9241178
## [428] 0.9243190 0.9245196 0.9247201 0.9249170 0.9251545 0.9253757 0.9256060
## [435] 0.9258492 0.9260483 0.9262604 0.9264590 0.9266768 0.9268733 0.9270501
## [442] 0.9273078 0.9275269 0.9277303 0.9279436 0.9281512 0.9283575 0.9285604
## [449] 0.9287545 0.9289479 0.9291597 0.9293570 0.9296016 0.9297962 0.9299916
## [456] 0.9302398 0.9304310 0.9306449 0.9308353 0.9310683 0.9312741 0.9314641
## [463] 0.9316950 0.9318926 0.9321051 0.9323322 0.9325594 0.9327586 0.9329466
## [470] 0.9331344 0.9333613 0.9335405 0.9337279 0.9339419 0.9340999 0.9342866
## [477] 0.9344764 0.9347071 0.9349052 0.9350981 0.9353012 0.9354862 0.9356615
## [484] 0.9358458 0.9360686 0.9362573 0.9364361 0.9366193 0.9368028 0.9370064
## [491] 0.9371871 0.9374000 0.9376435 0.9378249 0.9380314 0.9382492 0.9384479
## [498] 0.9386209 0.9388027 0.9389674 0.9391509 0.9393182 0.9395002 0.9396776
## [505] 0.9398525 0.9400294 0.9402062 0.9403953 0.9405719 0.9407649 0.9409906
## [512] 0.9411778 0.9413684 0.9415444 0.9417204 0.9419333 0.9421099 0.9422934
## [519] 0.9424684 0.9426662 0.9428798 0.9431200 0.9433021 0.9434755 0.9436679
## [526] 0.9438628 0.9440449 0.9442784 0.9444754 0.9446518 0.9448322 0.9450590
## [533] 0.9452295 0.9454000 0.9455589 0.9457291 0.9459010 0.9460529 0.9462124
## [540] 0.9464372 0.9466375 0.9468066 0.9469608 0.9471396 0.9473080 0.9474730
## [547] 0.9476476 0.9478338 0.9480016 0.9481875 0.9483589 0.9485204 0.9486899
## [554] 0.9488811 0.9490476 0.9492447 0.9494150 0.9495806 0.9497458 0.9499845
## [561] 0.9501490 0.9503164 0.9504969 0.9506636 0.9508248 0.9510006 0.9511759
## [568] 0.9514082 0.9515686 0.9517345 0.9518945 0.9520544 0.9522139 0.9523855
## [575] 0.9525708 0.9527302 0.9528876 0.9530580 0.9532147 0.9533848 0.9535408
## [582] 0.9536872 0.9538603 0.9540033 0.9542233 0.9544279 0.9545827 0.9547373
## [589] 0.9548914 0.9550464 0.9552350 0.9553891 0.9555428 0.9556962 0.9558496
## [596] 0.9560138 0.9561833 0.9563360 0.9564879 0.9566392 0.9568017 0.9569589
## [603] 0.9571094 0.9572551 0.9574115 0.9575626 0.9577117 0.9578563 0.9580062
## [610] 0.9581812 0.9583366 0.9584852 0.9586489 0.9587969 0.9589448 0.9590925
## [617] 0.9592401 0.9593875 0.9595334 0.9596792 0.9598277 0.9599693 0.9601146
## [624] 0.9602597 0.9604048 0.9605498 0.9606945 0.9608390 0.9609834 0.9611278
## [631] 0.9612876 0.9614502 0.9615941 0.9617348 0.9618723 0.9620223 0.9621761
## [638] 0.9623277 0.9624706 0.9626120 0.9627827 0.9629405 0.9630831 0.9632604
## [645] 0.9634027 0.9635448 0.9636852 0.9638273 0.9639656 0.9640928 0.9642337
## [652] 0.9643742 0.9645184 0.9646585 0.9648230 0.9649984 0.9651276 0.9652734
## [659] 0.9654130 0.9655525 0.9656917 0.9658297 0.9659851 0.9661129 0.9662443
## [666] 0.9663919 0.9665304 0.9666666 0.9668051 0.9669431 0.9670808 0.9672421
## [673] 0.9673835 0.9675211 0.9676570 0.9678048 0.9679414 0.9680781 0.9682302
## [680] 0.9683663 0.9685271 0.9686710 0.9688065 0.9689419 0.9690667 0.9692016
## [687] 0.9693365 0.9694865 0.9696200 0.9697534 0.9698867 0.9700190 0.9701708
## [694] 0.9703127 0.9704747 0.9706045 0.9707343 0.9708641 0.9709901 0.9711173
## [701] 0.9712542 0.9713873 0.9715135 0.9716397 0.9717657 0.9718913 0.9720362
## [708] 0.9721616 0.9722869 0.9724004 0.9725404 0.9726652 0.9727897 0.9729141
## [715] 0.9730366 0.9731607 0.9732921 0.9734255 0.9735493 0.9736888 0.9738124
## [722] 0.9739359 0.9740593 0.9741821 0.9743042 0.9744263 0.9745747 0.9746939
## [729] 0.9748154 0.9749367 0.9750578 0.9751784 0.9752930 0.9754131 0.9755332
## [736] 0.9756566 0.9757692 0.9759094 0.9760291 0.9761544 0.9762728 0.9763923
## [743] 0.9765117 0.9766309 0.9767500 0.9768690 0.9769874 0.9771024 0.9772206
## [750] 0.9773388 0.9774568 0.9775747 0.9776925 0.9778203 0.9779417 0.9780995
## [757] 0.9782168 0.9783418 0.9784555 0.9785865 0.9787034 0.9788320 0.9789485
## [764] 0.9790647 0.9791807 0.9792965 0.9794118 0.9795269 0.9796541 0.9797802
## [771] 0.9798948 0.9800090 0.9801231 0.9802454 0.9803574 0.9804708 0.9805838
## [778] 0.9806965 0.9808089 0.9809212 0.9810332 0.9811451 0.9812569 0.9813829
## [785] 0.9814962 0.9816076 0.9817188 0.9818297 0.9819472 0.9820574 0.9821671
## [792] 0.9822767 0.9823861 0.9824954 0.9826126 0.9827217 0.9828307 0.9829393
## [799] 0.9830476 0.9831558 0.9832632 0.9833706 0.9834903 0.9836151 0.9837200
## [806] 0.9838272 0.9839343 0.9840413 0.9841481 0.9842546 0.9843610 0.9844673
## [813] 0.9845736 0.9846796 0.9847854 0.9848911 0.9850032 0.9851087 0.9852129
## [820] 0.9853179 0.9854227 0.9855275 0.9856612 0.9857653 0.9858716 0.9859766
## [827] 0.9860798 0.9861826 0.9862853 0.9863991 0.9865013 0.9866034 0.9867054
## [834] 0.9868069 0.9869081 0.9870091 0.9871225 0.9872232 0.9873295 0.9874300
## [841] 0.9875305 0.9876283 0.9877389 0.9878424 0.9879421 0.9880415 0.9881406
## [848] 0.9882392 0.9883377 0.9884359 0.9885330 0.9886310 0.9887289 0.9888265
## [855] 0.9889233 0.9890199 0.9891164 0.9892128 0.9893086 0.9894045 0.9894995
## [862] 0.9895939 0.9896838 0.9897779 0.9898719 0.9899642 0.9900568 0.9901490
## [869] 0.9902362 0.9903275 0.9904188 0.9905100 0.9906009 0.9906916 0.9907822
## [876] 0.9908722 0.9909622 0.9910520 0.9911560 0.9912452 0.9913343 0.9914231
## [883] 0.9915118 0.9916005 0.9916885 0.9917771 0.9918650 0.9919528 0.9920406
## [890] 0.9921274 0.9922141 0.9923005 0.9923805 0.9924666 0.9925479 0.9926328
## [897] 0.9927176 0.9928021 0.9928865 0.9929709 0.9930551 0.9931386 0.9932219
## [904] 0.9933048 0.9933896 0.9934723 0.9935549 0.9936374 0.9937199 0.9938023
## [911] 0.9938846 0.9939689 0.9940509 0.9941325 0.9942140 0.9942954 0.9943767
## [918] 0.9944651 0.9945458 0.9946265 0.9947071 0.9947878 0.9948679 0.9949476
## [925] 0.9950274 0.9951070 0.9951865 0.9952658 0.9953448 0.9954235 0.9955015
## [932] 0.9955791 0.9956565 0.9957339 0.9958108 0.9958877 0.9959645 0.9960412
## [939] 0.9961179 0.9961939 0.9962697 0.9963450 0.9964198 0.9964946 0.9965688
## [946] 0.9966430 0.9967169 0.9967903 0.9968635 0.9969361 0.9970087 0.9970811
## [953] 0.9971532 0.9972252 0.9972971 0.9973684 0.9974387 0.9975088 0.9975788
## [960] 0.9976487 0.9977176 0.9977864 0.9978545 0.9979221 0.9979897 0.9980569
## [967] 0.9981239 0.9981898 0.9982553 0.9983209 0.9983863 0.9984511 0.9985155
## [974] 0.9985798 0.9986438 0.9987077 0.9987716 0.9988349 0.9988979 0.9989603
## [981] 0.9990225 0.9990840 0.9991445 0.9992031 0.9992606 0.9993168 0.9993728
## [988] 0.9994264 0.9994795 0.9995304 0.9995813 0.9996321 0.9996824 0.9997320
## [995] 0.9997803 0.9998274 0.9998743 0.9999196 0.9999624 1.0000000
##
## [[4]]
## [1] 2.464694e-01 7.789508e-02 6.116092e-02 4.851489e-02 6.724593e-02
## [6] 4.089334e-02 7.248887e-03 1.946592e-02 1.572046e-02 9.430710e-03
## [11] 9.458650e-03 6.286770e-03 2.986726e-03 8.678614e-03 5.642148e-03
## [16] 8.543059e-03 1.268435e-03 8.458780e-03 6.706412e-03 6.042572e-03
## [21] 3.024302e-03 3.443190e-03 1.259017e-02 1.731353e-03 8.972328e-03
## [26] 2.609642e-03 2.400092e-03 1.754203e-03 2.579202e-03 1.240446e-03
## [31] 2.819845e-03 3.184558e-03 3.252363e-03 2.123604e-03 1.077721e-03
## [36] 1.585018e-03 7.935488e-04 1.457803e-03 8.929760e-04 1.397596e-03
## [41] 1.978693e-03 1.340406e-03 1.907685e-03 1.706483e-03 1.113609e-03
## [46] 3.910670e-03 1.565596e-03 2.124140e-03 1.613428e-03 5.480717e-04
## [51] 2.356286e-03 2.091678e-03 1.978956e-03 2.078194e-03 7.026366e-04
## [56] 2.699116e-03 1.434773e-03 7.048067e-04 5.828759e-04 1.324045e-03
## [61] 2.827509e-03 6.750502e-04 9.504278e-04 1.754800e-03 7.346919e-04
## [66] 9.288564e-04 2.131095e-03 8.181593e-04 6.480524e-04 1.926819e-03
## [71] 1.119699e-03 1.017610e-03 2.545323e-03 1.627224e-03 5.565253e-04
## [76] 1.175797e-03 6.879784e-04 1.009974e-03 8.859530e-04 7.636652e-04
## [81] 8.397244e-04 1.015786e-03 6.801922e-04 1.520348e-03 9.848599e-04
## [86] 8.334112e-04 5.641156e-04 9.520935e-04 9.630206e-04 1.126541e-03
## [91] 1.459652e-03 6.105199e-04 5.399031e-04 4.485799e-04 6.796848e-04
## [96] 9.068784e-04 8.326130e-04 9.365038e-04 4.970316e-04 4.788962e-04
## [101] 1.001515e-03 6.456385e-04 6.316998e-04 7.428245e-04 7.044493e-04
## [106] 6.728735e-04 1.027436e-03 5.642822e-04 9.774864e-04 4.562299e-04
## [111] 8.081631e-04 7.377907e-04 5.146330e-04 9.023199e-04 6.466507e-04
## [116] 1.520735e-03 4.600126e-04 5.675701e-04 6.383317e-04 5.072174e-04
## [121] 1.280612e-03 7.247460e-04 7.783202e-04 4.933596e-04 7.465832e-04
## [126] 9.294812e-04 7.004944e-04 9.212622e-04 4.897746e-04 5.087141e-04
## [131] 4.994311e-04 6.370267e-04 6.997793e-04 7.301575e-04 6.574175e-04
## [136] 8.749244e-04 9.919120e-04 5.361461e-04 4.459950e-04 6.304585e-04
## [141] 4.640063e-04 4.498524e-04 4.996951e-04 5.346173e-04 7.362320e-04
## [146] 7.092998e-04 1.107139e-03 9.873360e-04 4.028803e-04 6.480807e-04
## [151] 4.556698e-04 7.793765e-04 8.106962e-04 8.182675e-04 4.639272e-04
## [156] 4.031278e-04 4.237960e-04 5.574907e-04 6.889959e-04 5.079277e-04
## [161] 8.624077e-04 6.823526e-04 4.340298e-04 4.462449e-04 7.143595e-04
## [166] 3.785946e-04 7.310609e-04 9.878145e-04 5.818519e-04 6.269909e-04
## [171] 6.415504e-04 3.789629e-04 4.899718e-04 3.127985e-04 6.011391e-04
## [176] 3.482123e-04 4.713088e-04 4.376612e-04 4.996584e-04 6.616602e-04
## [181] 5.411765e-04 3.803573e-04 4.779629e-04 4.953617e-04 7.659465e-04
## [186] 4.227859e-04 5.602561e-04 3.826311e-04 4.550936e-04 6.423091e-04
## [191] 4.093166e-04 6.514266e-04 4.087482e-04 3.644462e-04 3.009494e-04
## [196] 6.156094e-04 3.816500e-04 3.768421e-04 3.138064e-04 3.457195e-04
## [201] 4.818057e-04 3.239330e-04 3.617027e-04 4.010627e-04 3.386176e-04
## [206] 4.848401e-04 3.735094e-04 3.813907e-04 3.284125e-04 5.476614e-04
## [211] 3.573602e-04 5.139514e-04 3.421306e-04 4.518908e-04 3.501398e-04
## [216] 4.918305e-04 3.201645e-04 5.117714e-04 4.211794e-04 3.265763e-04
## [221] 6.134448e-04 4.888377e-04 6.329649e-04 3.327310e-04 3.845924e-04
## [226] 6.213194e-04 3.333643e-04 5.414708e-04 5.377968e-04 3.081171e-04
## [231] 3.306762e-04 2.953625e-04 3.680492e-04 3.167634e-04 3.671938e-04
## [236] 3.254485e-04 3.767978e-04 3.057461e-04 3.168497e-04 4.398689e-04
## [241] 3.979778e-04 3.237771e-04 5.450061e-04 4.190719e-04 3.647202e-04
## [246] 3.009355e-04 4.390821e-04 2.909521e-04 4.199613e-04 2.716368e-04
## [251] 3.259643e-04 4.320093e-04 3.001080e-04 4.737828e-04 2.985616e-04
## [256] 4.244423e-04 3.920608e-04 3.093582e-04 2.962684e-04 4.265759e-04
## [261] 3.080029e-04 2.944682e-04 3.755411e-04 3.386066e-04 3.267700e-04
## [266] 3.225659e-04 3.081751e-04 3.323003e-04 2.563303e-04 3.326501e-04
## [271] 2.836794e-04 3.535526e-04 3.808795e-04 4.820598e-04 3.012843e-04
## [276] 2.798761e-04 3.217821e-04 3.681464e-04 2.502863e-04 4.223981e-04
## [281] 3.070730e-04 2.765643e-04 2.663021e-04 2.898474e-04 2.781420e-04
## [286] 3.497612e-04 2.411841e-04 5.441154e-04 2.716735e-04 4.901212e-04
## [291] 5.613267e-04 2.489249e-04 2.586949e-04 2.979138e-04 2.913458e-04
## [296] 2.620966e-04 4.299629e-04 2.612541e-04 2.605295e-04 2.357902e-04
## [301] 3.290397e-04 2.592207e-04 3.926819e-04 4.808106e-04 3.532889e-04
## [306] 2.791334e-04 3.475391e-04 3.163239e-04 2.936760e-04 2.630509e-04
## [311] 3.273479e-04 3.811958e-04 2.864939e-04 3.329262e-04 3.316925e-04
## [316] 3.485211e-04 2.506018e-04 2.519484e-04 2.811539e-04 2.619206e-04
## [321] 2.491650e-04 2.982969e-04 2.506156e-04 2.485759e-04 2.726023e-04
## [326] 2.705813e-04 2.672446e-04 2.246469e-04 2.381320e-04 2.541254e-04
## [331] 2.451408e-04 3.485046e-04 2.523561e-04 2.681976e-04 2.397897e-04
## [336] 2.583597e-04 3.000420e-04 2.380029e-04 2.957001e-04 3.462809e-04
## [341] 3.384314e-04 3.282144e-04 3.569941e-04 3.129653e-04 3.089298e-04
## [346] 2.174606e-04 3.625752e-04 3.101251e-04 2.573991e-04 2.395850e-04
## [351] 2.572967e-04 2.660492e-04 3.164286e-04 2.508214e-04 2.439852e-04
## [356] 2.393929e-04 2.272308e-04 2.626254e-04 2.304454e-04 2.144655e-04
## [361] 2.081954e-04 2.568182e-04 2.634110e-04 2.237382e-04 2.328376e-04
## [366] 3.016666e-04 2.739171e-04 2.670864e-04 2.529653e-04 2.323627e-04
## [371] 2.117615e-04 2.716108e-04 2.198103e-04 2.895051e-04 2.231144e-04
## [376] 2.390776e-04 2.026644e-04 2.342891e-04 2.780546e-04 1.909825e-04
## [381] 2.234708e-04 2.352786e-04 2.326084e-04 2.294625e-04 2.151100e-04
## [386] 1.968878e-04 2.087510e-04 2.371339e-04 2.882754e-04 2.136577e-04
## [391] 2.135977e-04 1.826776e-04 2.282606e-04 2.444489e-04 2.379588e-04
## [396] 2.113664e-04 2.105707e-04 2.458166e-04 3.267670e-04 2.371241e-04
## [401] 2.091621e-04 2.218031e-04 2.604454e-04 2.450853e-04 2.771016e-04
## [406] 2.149348e-04 2.483426e-04 2.072387e-04 2.068718e-04 2.129455e-04
## [411] 2.247363e-04 2.229878e-04 2.264913e-04 1.857096e-04 2.027937e-04
## [416] 2.226222e-04 2.358783e-04 2.148024e-04 2.029024e-04 3.520116e-04
## [421] 2.127251e-04 2.025446e-04 3.117252e-04 2.016721e-04 1.606878e-04
## [426] 3.033025e-04 2.012425e-04 2.005817e-04 2.005291e-04 1.968917e-04
## [431] 2.374189e-04 2.212492e-04 2.302614e-04 2.432647e-04 1.990935e-04
## [436] 2.120864e-04 1.985855e-04 2.177686e-04 1.965472e-04 1.767469e-04
## [441] 2.577270e-04 2.191120e-04 2.034124e-04 2.133119e-04 2.076053e-04
## [446] 2.062734e-04 2.029396e-04 1.940819e-04 1.933910e-04 2.118036e-04
## [451] 1.972729e-04 2.446082e-04 1.946030e-04 1.954390e-04 2.482005e-04
## [456] 1.911321e-04 2.139532e-04 1.904151e-04 2.329562e-04 2.058467e-04
## [461] 1.899867e-04 2.308317e-04 1.976686e-04 2.124479e-04 2.271348e-04
## [466] 2.271991e-04 1.992313e-04 1.879124e-04 1.878676e-04 2.268391e-04
## [471] 1.792119e-04 1.874352e-04 2.139842e-04 1.580262e-04 1.866679e-04
## [476] 1.897825e-04 2.307328e-04 1.981209e-04 1.928586e-04 2.031372e-04
## [481] 1.850088e-04 1.752914e-04 1.842821e-04 2.227793e-04 1.886794e-04
## [486] 1.788700e-04 1.831504e-04 1.835022e-04 2.036251e-04 1.806856e-04
## [491] 2.129227e-04 2.434718e-04 1.814447e-04 2.065036e-04 2.177988e-04
## [496] 1.986392e-04 1.729840e-04 1.818266e-04 1.646799e-04 1.834906e-04
## [501] 1.673081e-04 1.820295e-04 1.774402e-04 1.748574e-04 1.768602e-04
## [506] 1.768278e-04 1.891125e-04 1.766150e-04 1.929685e-04 2.257172e-04
## [511] 1.871999e-04 1.906262e-04 1.759777e-04 1.759616e-04 2.129367e-04
## [516] 1.765637e-04 1.835772e-04 1.749885e-04 1.977764e-04 2.135889e-04
## [521] 2.402010e-04 1.820984e-04 1.734442e-04 1.924060e-04 1.948344e-04
## [526] 1.821371e-04 2.334387e-04 1.970539e-04 1.763716e-04 1.804160e-04
## [531] 2.268553e-04 1.704930e-04 1.704386e-04 1.588869e-04 1.701840e-04
## [536] 1.719678e-04 1.518986e-04 1.594915e-04 2.247987e-04 2.003065e-04
## [541] 1.691097e-04 1.541490e-04 1.788357e-04 1.683415e-04 1.650027e-04
## [546] 1.746296e-04 1.861725e-04 1.678454e-04 1.858714e-04 1.714619e-04
## [551] 1.614447e-04 1.695137e-04 1.912402e-04 1.664283e-04 1.971039e-04
## [556] 1.703544e-04 1.655687e-04 1.651740e-04 2.386940e-04 1.645878e-04
## [561] 1.673583e-04 1.804875e-04 1.666836e-04 1.612005e-04 1.758557e-04
## [566] 1.753129e-04 2.322529e-04 1.604297e-04 1.659090e-04 1.599191e-04
## [571] 1.599143e-04 1.595715e-04 1.715189e-04 1.853606e-04 1.593531e-04
## [576] 1.573865e-04 1.704811e-04 1.566943e-04 1.700956e-04 1.560086e-04
## [581] 1.463815e-04 1.730889e-04 1.430048e-04 2.199687e-04 2.045893e-04
## [586] 1.548410e-04 1.545992e-04 1.540660e-04 1.550166e-04 1.886371e-04
## [591] 1.540674e-04 1.537028e-04 1.534242e-04 1.533708e-04 1.641631e-04
## [596] 1.694935e-04 1.527902e-04 1.518327e-04 1.513694e-04 1.624532e-04
## [601] 1.572456e-04 1.504573e-04 1.457095e-04 1.563975e-04 1.510523e-04
## [606] 1.491070e-04 1.445923e-04 1.499742e-04 1.749287e-04 1.554815e-04
## [611] 1.485447e-04 1.637329e-04 1.480037e-04 1.478730e-04 1.476761e-04
## [616] 1.476571e-04 1.473355e-04 1.459397e-04 1.457966e-04 1.485445e-04
## [621] 1.415441e-04 1.452978e-04 1.451394e-04 1.451108e-04 1.449179e-04
## [626] 1.447543e-04 1.444887e-04 1.443895e-04 1.444290e-04 1.598146e-04
## [631] 1.625446e-04 1.439040e-04 1.406848e-04 1.375203e-04 1.500401e-04
## [636] 1.537630e-04 1.515977e-04 1.429382e-04 1.413311e-04 1.707886e-04
## [641] 1.577864e-04 1.425179e-04 1.773330e-04 1.422971e-04 1.421546e-04
## [646] 1.403551e-04 1.420641e-04 1.383523e-04 1.272032e-04 1.409251e-04
## [651] 1.404940e-04 1.441275e-04 1.401335e-04 1.645331e-04 1.753786e-04
## [656] 1.292240e-04 1.457981e-04 1.395989e-04 1.394944e-04 1.392207e-04
## [661] 1.379985e-04 1.553395e-04 1.278277e-04 1.313551e-04 1.476287e-04
## [666] 1.385185e-04 1.362233e-04 1.384518e-04 1.379937e-04 1.377636e-04
## [671] 1.612340e-04 1.414065e-04 1.375831e-04 1.359869e-04 1.477163e-04
## [676] 1.366662e-04 1.366251e-04 1.521749e-04 1.361034e-04 1.607155e-04
## [681] 1.439884e-04 1.354547e-04 1.354330e-04 1.247856e-04 1.349267e-04
## [686] 1.348246e-04 1.500150e-04 1.334718e-04 1.334352e-04 1.333383e-04
## [691] 1.323165e-04 1.517098e-04 1.419418e-04 1.619961e-04 1.298530e-04
## [696] 1.297808e-04 1.297753e-04 1.260244e-04 1.271888e-04 1.368419e-04
## [701] 1.331433e-04 1.261885e-04 1.261773e-04 1.260512e-04 1.255421e-04
## [706] 1.449613e-04 1.253637e-04 1.253330e-04 1.134616e-04 1.399755e-04
## [711] 1.248187e-04 1.245345e-04 1.243889e-04 1.225210e-04 1.240905e-04
## [716] 1.313523e-04 1.334154e-04 1.238237e-04 1.394960e-04 1.236262e-04
## [721] 1.235196e-04 1.233610e-04 1.227660e-04 1.221488e-04 1.220731e-04
## [726] 1.484194e-04 1.191633e-04 1.215318e-04 1.213143e-04 1.211246e-04
## [731] 1.205802e-04 1.145520e-04 1.201732e-04 1.200909e-04 1.233649e-04
## [736] 1.125860e-04 1.401936e-04 1.197446e-04 1.252769e-04 1.184234e-04
## [741] 1.194856e-04 1.193443e-04 1.192818e-04 1.191065e-04 1.189297e-04
## [746] 1.183819e-04 1.150353e-04 1.182017e-04 1.181838e-04 1.180602e-04
## [751] 1.178902e-04 1.177829e-04 1.277591e-04 1.214309e-04 1.578248e-04
## [756] 1.172279e-04 1.250754e-04 1.136865e-04 1.309817e-04 1.168692e-04
## [761] 1.286349e-04 1.164640e-04 1.162686e-04 1.159389e-04 1.158623e-04
## [766] 1.152924e-04 1.151198e-04 1.271899e-04 1.260426e-04 1.146155e-04
## [771] 1.142280e-04 1.140588e-04 1.223446e-04 1.119367e-04 1.134792e-04
## [776] 1.129601e-04 1.127141e-04 1.123724e-04 1.123082e-04 1.119972e-04
## [781] 1.119095e-04 1.117667e-04 1.260097e-04 1.133641e-04 1.113191e-04
## [786] 1.112832e-04 1.108836e-04 1.174638e-04 1.102489e-04 1.096826e-04
## [791] 1.095964e-04 1.094236e-04 1.092929e-04 1.171801e-04 1.090592e-04
## [796] 1.090478e-04 1.085619e-04 1.082694e-04 1.082388e-04 1.074280e-04
## [801] 1.073937e-04 1.197023e-04 1.247459e-04 1.048918e-04 1.072119e-04
## [806] 1.070880e-04 1.070012e-04 1.068156e-04 1.065045e-04 1.064707e-04
## [811] 1.062940e-04 1.062437e-04 1.060378e-04 1.057594e-04 1.056826e-04
## [816] 1.121670e-04 1.054816e-04 1.041942e-04 1.049603e-04 1.048122e-04
## [821] 1.047758e-04 1.337860e-04 1.040753e-04 1.062620e-04 1.050018e-04
## [826] 1.032547e-04 1.027748e-04 1.026893e-04 1.137685e-04 1.022531e-04
## [831] 1.021199e-04 1.019965e-04 1.014143e-04 1.012933e-04 1.009425e-04
## [836] 1.134045e-04 1.007277e-04 1.062715e-04 1.005354e-04 1.004662e-04
## [841] 9.776815e-05 1.106124e-04 1.035653e-04 9.970583e-05 9.934979e-05
## [846] 9.911019e-05 9.857915e-05 9.847547e-05 9.820773e-05 9.717808e-05
## [851] 9.793244e-05 9.790926e-05 9.760570e-05 9.678631e-05 9.660267e-05
## [856] 9.647618e-05 9.642136e-05 9.586874e-05 9.581459e-05 9.504099e-05
## [861] 9.440077e-05 8.992363e-05 9.404144e-05 9.403283e-05 9.229454e-05
## [866] 9.260160e-05 9.219244e-05 8.725336e-05 9.128950e-05 9.121618e-05
## [871] 9.120948e-05 9.091517e-05 9.070878e-05 9.058150e-05 9.004693e-05
## [876] 8.998481e-05 8.981899e-05 1.039458e-04 8.923363e-05 8.908088e-05
## [881] 8.879252e-05 8.875217e-05 8.864236e-05 8.805488e-05 8.860872e-05
## [886] 8.789421e-05 8.782281e-05 8.775590e-05 8.678108e-05 8.672404e-05
## [891] 8.639837e-05 8.003295e-05 8.607033e-05 8.125341e-05 8.495462e-05
## [896] 8.477722e-05 8.454603e-05 8.440419e-05 8.438407e-05 8.419594e-05
## [901] 8.348491e-05 8.329346e-05 8.289046e-05 8.478970e-05 8.270175e-05
## [906] 8.265384e-05 8.250196e-05 8.245043e-05 8.240971e-05 8.232236e-05
## [911] 8.426821e-05 8.198470e-05 8.158990e-05 8.151388e-05 8.143724e-05
## [916] 8.124868e-05 8.838768e-05 8.075879e-05 8.066330e-05 8.064351e-05
## [921] 8.063907e-05 8.011535e-05 7.976154e-05 7.972143e-05 7.962947e-05
## [926] 7.955477e-05 7.924981e-05 7.903691e-05 7.866652e-05 7.800470e-05
## [931] 7.762964e-05 7.737409e-05 7.737194e-05 7.691875e-05 7.690390e-05
## [936] 7.681560e-05 7.672006e-05 7.665223e-05 7.599474e-05 7.582308e-05
## [941] 7.525467e-05 7.482614e-05 7.481076e-05 7.420657e-05 7.417799e-05
## [946] 7.388419e-05 7.344903e-05 7.324013e-05 7.258946e-05 7.253795e-05
## [951] 7.244126e-05 7.209780e-05 7.203073e-05 7.188253e-05 7.127294e-05
## [956] 7.027700e-05 7.016446e-05 6.996347e-05 6.991054e-05 6.884619e-05
## [961] 6.880098e-05 6.814017e-05 6.763336e-05 6.761030e-05 6.719413e-05
## [966] 6.695685e-05 6.585227e-05 6.558438e-05 6.552426e-05 6.541506e-05
## [971] 6.484131e-05 6.440618e-05 6.425115e-05 6.399159e-05 6.394107e-05
## [976] 6.392388e-05 6.322197e-05 6.306560e-05 6.239654e-05 6.222206e-05
## [981] 6.147081e-05 6.044907e-05 5.868975e-05 5.744131e-05 5.619475e-05
## [986] 5.601195e-05 5.358884e-05 5.310911e-05 5.093601e-05 5.088362e-05
## [991] 5.080200e-05 5.028289e-05 4.956035e-05 4.831145e-05 4.712030e-05
## [996] 4.695315e-05 4.521091e-05 4.287953e-05 3.756724e-05 0.000000e+00
##
## [[5]]
## [1] 21.403395 16.390398 16.210405 15.050736 13.661975 13.322118 12.786402
## [8] 12.759505 11.662350 11.287004 11.163952 10.891453 10.743218 10.249856
## [15] 10.182215 10.175150 10.023846 10.017858 9.903857 9.865903 9.699700
## [22] 9.659129 9.470359 9.439507 9.318211 9.108384 9.009192 8.836800
## [29] 8.795705 8.284948 8.272759 8.251199 8.121289 8.113277 8.110448
## [36] 8.083847 8.040045 8.030247 8.024227 7.937467 7.858767 7.791056
## [43] 7.748554 7.743347 7.621897 7.618669 7.597199 7.514686 7.447793
## [50] 7.440815 7.324888 7.296124 7.270856 7.264259 7.263409 7.260572
## [57] 7.247053 7.232328 7.202297 7.159382 7.148179 7.071584 7.061409
## [64] 7.053746 7.043287 6.993600 6.992998 6.954421 6.907661 6.884762
## [71] 6.779510 6.770512 6.710106 6.707324 6.665210 6.637143 6.598426
## [78] 6.591238 6.553512 6.534618 6.429240 6.396448 6.382800 6.379238
## [85] 6.366920 6.363570 6.325625 6.313641 6.312395 6.304780 6.284451
## [92] 6.255665 6.236267 6.231489 6.226926 6.222032 6.217640 6.217268
## [99] 6.210509 6.182208 6.171098 6.168306 6.161418 6.148744 6.147989
## [106] 6.128080 6.108081 6.064337 6.049623 6.039912 6.029312 5.987151
## [113] 5.969664 5.947051 5.937940 5.934862 5.932198 5.922726 5.914222
## [120] 5.908591 5.895376 5.880568 5.845091 5.830307 5.826234 5.821516
## [127] 5.795388 5.773205 5.767061 5.758784 5.756113 5.756074 5.737430
## [134] 5.711981 5.690795 5.687744 5.675998 5.661280 5.653305 5.631399
## [141] 5.614396 5.612547 5.585253 5.582938 5.539100 5.517747 5.516535
## [148] 5.507279 5.500449 5.488202 5.476567 5.473056 5.472504 5.471416
## [155] 5.463893 5.459086 5.452645 5.451427 5.450629 5.426829 5.416871
## [162] 5.411678 5.405038 5.392292 5.388927 5.367334 5.363620 5.363598
## [169] 5.349042 5.324734 5.294343 5.292940 5.290539 5.255857 5.253515
## [176] 5.253114 5.236298 5.231248 5.226273 5.225869 5.200640 5.199227
## [183] 5.186464 5.185245 5.173291 5.138707 5.127339 5.120091 5.112646
## [190] 5.101014 5.100592 5.091117 5.084201 5.080333 5.080159 5.079453
## [197] 5.078715 5.074037 5.070331 5.055459 5.054636 5.048659 5.034669
## [204] 5.026194 5.003264 5.003044 4.989529 4.987982 4.978436 4.966466
## [211] 4.957963 4.955249 4.946654 4.946375 4.943269 4.938965 4.938888
## [218] 4.936403 4.916143 4.913500 4.903277 4.889708 4.889366 4.869063
## [225] 4.868491 4.860355 4.831115 4.825603 4.816979 4.805682 4.793627
## [232] 4.790715 4.785685 4.784162 4.781130 4.778851 4.766690 4.754218
## [239] 4.753204 4.750929 4.748560 4.740903 4.736376 4.730456 4.730160
## [246] 4.729279 4.726910 4.722531 4.722006 4.721325 4.720083 4.718911
## [253] 4.710179 4.699636 4.698027 4.696905 4.696629 4.690003 4.679950
## [260] 4.660112 4.656495 4.649386 4.641561 4.633910 4.619114 4.618427
## [267] 4.606657 4.603622 4.597236 4.593697 4.587418 4.583609 4.574967
## [274] 4.574469 4.546299 4.545338 4.544096 4.539268 4.539061 4.532778
## [281] 4.531772 4.521647 4.516263 4.504158 4.492650 4.474967 4.460749
## [288] 4.448651 4.440570 4.435992 4.424642 4.422142 4.417982 4.417152
## [295] 4.417033 4.414261 4.397456 4.392051 4.389211 4.380265 4.378893
## [302] 4.377574 4.376634 4.374694 4.373422 4.368339 4.360739 4.353507
## [309] 4.351188 4.351054 4.343640 4.341042 4.337136 4.335888 4.321588
## [316] 4.314082 4.304182 4.304068 4.302533 4.300821 4.291826 4.290661
## [323] 4.287228 4.286750 4.286713 4.275907 4.260271 4.258396 4.245420
## [330] 4.238500 4.234641 4.231878 4.216336 4.215658 4.210308 4.206386
## [337] 4.201761 4.194592 4.193078 4.192901 4.188998 4.186020 4.162062
## [344] 4.161852 4.156761 4.153063 4.144944 4.144035 4.143887 4.141175
## [351] 4.129304 4.127502 4.121334 4.112763 4.104726 4.103229 4.098568
## [358] 4.095267 4.089592 4.088075 4.084308 4.071424 4.067862 4.066949
## [365] 4.060692 4.060301 4.053070 4.051850 4.046545 4.038231 4.035694
## [372] 4.033305 4.031091 4.029904 4.022413 4.019886 4.016875 4.015800
## [379] 4.010734 4.008054 4.002386 4.002098 3.981351 3.981157 3.981069
## [386] 3.980974 3.980459 3.976657 3.975768 3.974274 3.973716 3.973384
## [393] 3.972334 3.970910 3.962082 3.952907 3.947659 3.943364 3.937817
## [400] 3.934923 3.932241 3.928869 3.928554 3.923287 3.921229 3.916941
## [407] 3.915406 3.914119 3.910653 3.910001 3.909127 3.907168 3.895822
## [414] 3.890034 3.888219 3.879047 3.874579 3.874319 3.872953 3.872309
## [421] 3.871642 3.869536 3.869498 3.861193 3.859872 3.859346 3.857079
## [428] 3.850740 3.849973 3.848778 3.845366 3.839395 3.838373 3.837071
## [435] 3.836429 3.833497 3.831531 3.824352 3.814332 3.811932 3.811251
## [442] 3.809967 3.808291 3.802930 3.798777 3.793880 3.791712 3.787836
## [449] 3.781088 3.774666 3.770240 3.768986 3.768266 3.766864 3.761140
## [456] 3.758941 3.755074 3.751883 3.749708 3.747660 3.747660 3.746853
## [463] 3.745898 3.739757 3.733949 3.732828 3.729218 3.727145 3.726701
## [470] 3.725270 3.725179 3.722409 3.720063 3.715134 3.714783 3.714296
## [477] 3.711373 3.704786 3.704094 3.700608 3.698238 3.693482 3.690967
## [484] 3.688500 3.686110 3.680496 3.679617 3.676276 3.672591 3.672458
## [491] 3.669221 3.667376 3.662442 3.658669 3.657085 3.653603 3.647181
## [498] 3.642362 3.640106 3.639816 3.633202 3.632644 3.621801 3.619695
## [505] 3.615877 3.615546 3.614844 3.613369 3.612637 3.611963 3.611793
## [512] 3.610032 3.606844 3.606680 3.604842 3.604601 3.600227 3.596693
## [519] 3.593198 3.591025 3.589482 3.587926 3.580787 3.572510 3.571737
## [526] 3.571216 3.570313 3.569316 3.568050 3.562783 3.551965 3.550192
## [533] 3.549626 3.548071 3.546974 3.546121 3.539306 3.538958 3.536939
## [540] 3.536671 3.535761 3.532696 3.529274 3.527721 3.526796 3.525947
## [547] 3.524153 3.522519 3.520152 3.519817 3.512909 3.511088 3.507796
## [554] 3.507617 3.507526 3.506695 3.498547 3.494375 3.488633 3.488168
## [561] 3.486043 3.485347 3.484788 3.480343 3.476446 3.474712 3.456446
## [568] 3.443824 3.440088 3.438339 3.438288 3.434601 3.430290 3.429000
## [575] 3.424143 3.411005 3.408701 3.403496 3.397258 3.396040 3.392658
## [582] 3.390350 3.390116 3.385810 3.385006 3.383309 3.380666 3.378595
## [589] 3.378455 3.377248 3.374846 3.370851 3.367794 3.367208 3.366908
## [596] 3.365580 3.360829 3.350281 3.345165 3.340975 3.336035 3.335073
## [603] 3.329504 3.329245 3.327347 3.320073 3.318686 3.316688 3.316654
## [610] 3.314902 3.313807 3.308719 3.307767 3.306306 3.304104 3.303892
## [617] 3.300292 3.284622 3.283011 3.278656 3.278273 3.277390 3.275603
## [624] 3.275281 3.273103 3.271255 3.268252 3.268252 3.267577 3.265597
## [631] 3.262149 3.261633 3.255054 3.254511 3.254052 3.254030 3.251223
## [638] 3.250669 3.249498 3.249046 3.248394 3.245886 3.243594 3.243371
## [645] 3.241747 3.240999 3.240715 3.236365 3.236283 3.227698 3.222756
## [652] 3.220259 3.218620 3.217137 3.216154 3.215551 3.213077 3.212474
## [659] 3.211271 3.208119 3.206960 3.205542 3.205540 3.200130 3.200112
## [666] 3.200018 3.199676 3.199248 3.193951 3.191287 3.190620 3.190358
## [673] 3.189196 3.187396 3.182070 3.178551 3.178073 3.176224 3.171999
## [680] 3.168692 3.168437 3.164431 3.164177 3.159808 3.158258 3.157062
## [687] 3.150673 3.141184 3.140753 3.139613 3.127560 3.109988 3.106827
## [694] 3.104794 3.098308 3.097446 3.097382 3.079886 3.066359 3.064289
## [701] 3.056774 3.054278 3.054142 3.052616 3.046444 3.045779 3.044279
## [708] 3.043906 3.039796 3.037919 3.037655 3.034194 3.032420 3.029758
## [715] 3.028781 3.026383 3.026268 3.025523 3.024457 3.023109 3.021805
## [722] 3.019866 3.012573 3.004991 3.004060 2.998166 2.997786 2.997393
## [729] 2.994709 2.992367 2.985634 2.985290 2.980591 2.979570 2.979131
## [736] 2.977856 2.977180 2.975272 2.974055 2.972482 2.972052 2.970294
## [743] 2.969516 2.967333 2.965131 2.958293 2.957922 2.956041 2.955817
## [750] 2.954271 2.952144 2.950799 2.950259 2.947588 2.945816 2.943840
## [757] 2.942425 2.941781 2.941169 2.939332 2.936338 2.934232 2.931770
## [764] 2.927610 2.926643 2.919435 2.917249 2.913157 2.911221 2.910853
## [771] 2.905928 2.903776 2.901675 2.897511 2.896387 2.889755 2.886608
## [778] 2.882228 2.881405 2.877412 2.876286 2.874451 2.871231 2.869704
## [785] 2.868688 2.868227 2.863071 2.859306 2.854866 2.847525 2.846406
## [792] 2.844160 2.842462 2.840288 2.839421 2.839273 2.832940 2.829120
## [799] 2.828721 2.818106 2.817656 2.817294 2.816105 2.815334 2.815270
## [806] 2.813643 2.812503 2.810062 2.805967 2.805521 2.803193 2.802529
## [813] 2.799812 2.796135 2.795119 2.793415 2.792460 2.787484 2.785552
## [820] 2.783586 2.783102 2.774602 2.773783 2.770087 2.764371 2.762826
## [827] 2.756397 2.755251 2.751853 2.749393 2.747602 2.745941 2.738093
## [834] 2.736459 2.731717 2.730654 2.728808 2.728379 2.726202 2.725264
## [841] 2.721911 2.720736 2.718672 2.714932 2.710080 2.706810 2.699549
## [848] 2.698129 2.694458 2.691115 2.690679 2.690361 2.686187 2.674888
## [855] 2.672349 2.670599 2.669840 2.662178 2.661426 2.650660 2.641717
## [862] 2.637328 2.636685 2.636564 2.627656 2.616422 2.610636 2.606139
## [869] 2.597820 2.596776 2.596681 2.592488 2.589544 2.587726 2.580079
## [876] 2.579189 2.576812 2.573601 2.568401 2.566202 2.562045 2.561463
## [883] 2.559878 2.551381 2.550665 2.549052 2.548017 2.547046 2.532860
## [890] 2.532027 2.527269 2.522783 2.522466 2.509237 2.506064 2.503446
## [897] 2.500030 2.497932 2.497634 2.494849 2.484292 2.481442 2.475431
## [904] 2.474431 2.472612 2.471896 2.469623 2.468852 2.468242 2.466934
## [911] 2.463569 2.461869 2.455935 2.454790 2.453636 2.450794 2.448543
## [918] 2.443394 2.441949 2.441649 2.441582 2.433641 2.428261 2.427650
## [925] 2.426250 2.425112 2.420459 2.417206 2.411535 2.401369 2.395589
## [932] 2.391643 2.391610 2.384595 2.384365 2.382996 2.381514 2.380461
## [939] 2.370229 2.367551 2.358660 2.351935 2.351693 2.342177 2.341726
## [946] 2.337084 2.330192 2.326876 2.316516 2.315694 2.314150 2.308658
## [953] 2.307584 2.305209 2.295414 2.279319 2.277494 2.274229 2.273369
## [960] 2.255997 2.255256 2.244400 2.236038 2.235656 2.228765 2.224826
## [967] 2.206399 2.201906 2.200897 2.199062 2.189397 2.182038 2.179411
## [974] 2.175004 2.174145 2.173853 2.161885 2.159210 2.147726 2.144721
## [981] 2.131734 2.113944 2.082954 2.060681 2.038198 2.034881 1.990379
## [988] 1.981450 1.940489 1.939490 1.937934 1.928007 1.914105 1.889834
## [995] 1.866391 1.863078 1.828185 1.780425 1.666492## 2. Run NbClust to compute 30+ “best-k” indices
library(NbClust)palette(rainbow(60))
set.seed(2024) # NbClust uses random starts for k-means-based indices
nb_out <- NbClust(X_sc_noGK, distance = dist.m, min.nc = 2, max.nc = 15,
method = clus.m, index = "all", alphaBeale = 0.1)
## *** : The Hubert index is a graphical method of determining the number of clusters.
## In the plot of Hubert index, we seek a significant knee that corresponds to a
## significant increase of the value of the measure i.e the significant peak in Hubert
## index second differences plot.
## 
## *** : The D index is a graphical method of determining the number of clusters.
## In the plot of D index, we seek a significant knee (the significant peak in Dindex
## second differences plot) that corresponds to a significant increase of the value of
## the measure.
##
## *******************************************************************
## * Among all indices:
## * 8 proposed 2 as the best number of clusters
## * 3 proposed 3 as the best number of clusters
## * 1 proposed 6 as the best number of clusters
## * 7 proposed 7 as the best number of clusters
## * 3 proposed 13 as the best number of clusters
## * 2 proposed 15 as the best number of clusters
##
## ***** Conclusion *****
##
## * According to the majority rule, the best number of clusters is 2
##
##
## *******************************************************************## Quick summary of how many indices vote for each k
table(nb_out$Best.nc[1, ]) # majority vote##
## 0 2 3 6 7 13 15
## 2 8 3 1 7 3 2# 3. determine optimal k (elbow & silhouette) elbow in PCA
# space
wss <- map_dbl(2:10, ~kmeans(pc_df_noGK_noGK, .x, nstart = 25)$tot.withinss)
ggplot(data.frame(k = 2:10, wss), aes(k, wss)) + geom_line() +
geom_point() + labs(title = "Elbow plot (PCA space)", x = "# clusters",
y = "Within‑cluster SS") + theme_minimal()
# average silhouette width
sil <- map_dbl(2:10, function(k) {
km <- kmeans(pc_df_noGK_noGK, k, nstart = 25)
mean(silhouette(km$cluster, dist(pc_df_noGK_noGK))[, 3])
})
ggplot(data.frame(k = 2:10, sil), aes(k, sil)) + geom_line() +
geom_point() + labs(title = "Avg silhouette width vs k",
x = "# clusters", y = "Silhouette") + theme_minimal()
# optional: NbClust suggestion nb <- NbClust(pc_df_noGK,
# distance = 'euclidean', min.nc = 2, max.nc = 10, method =
# 'kmeans') fviz_nbclust(nb)str(X_sc_noGK) # should list only numeric columns## num [1:1000, 1:37] 2.057 -0.583 1.29 0.609 1.886 ...
## - attr(*, "dimnames")=List of 2
## ..$ : NULL
## ..$ : chr [1:37] "PAC" "SHO" "PAS" "DRI" ...
## - attr(*, "scaled:center")= Named num [1:37] 72.8 64.7 71.2 75.3 63.8 ...
## ..- attr(*, "names")= chr [1:37] "PAC" "SHO" "PAS" "DRI" ...
## - attr(*, "scaled:scale")= Named num [1:37] 11.74 14.14 7.66 7.44 17.66 ...
## ..- attr(*, "names")= chr [1:37] "PAC" "SHO" "PAS" "DRI" ...anyNA(X_sc_noGK) # must be FALSE## [1] FALSEapply(X_sc_noGK, 2, sd)## PAC SHO PAS DRI
## 1 1 1 1
## DEF PHY Acceleration Sprint.Speed
## 1 1 1 1
## Positioning Finishing Shot.Power Long.Shots
## 1 1 1 1
## Volleys Penalties Vision Crossing
## 1 1 1 1
## Free.Kick.Accuracy Short.Passing Long.Passing Curve
## 1 1 1 1
## Dribbling Agility Balance Reactions
## 1 1 1 1
## Ball.Control Composure Interceptions Heading.Accuracy
## 1 1 1 1
## Def.Awareness Standing.Tackle Sliding.Tackle Jumping
## 1 1 1 1
## Stamina Strength Aggression Height_cm
## 1 1 1 1
## Weight_kg
## 1- The silhouette plot levels off after k = 4: the average silhouette width drops sharply from k = 2 to k = 3, slips only slightly at k = 4, and then declines steadily thereafter. k = 4 is the optimal choice, as though k = 2 solciits the largest width, in the context of player ratings, it is not very sensible to have only two clusters, as there are not only two types of players in soccer.
NbClust’s majority prefers k = 2 because it optimises a few variance indices, but both the pseudo T^2 jump and the sharp drop in SPRSQ after k = 4 show an additional structural split; hence I carried both 2 and 4 group solutions forward
# 4. finalize k‑means (k = 4)
set.seed(123)
km4 <- kmeans(pc_df_noGK_noGK, centers = 4, nstart = 50)
pc_df_noGK_noGK <- pc_df_noGK_noGK %>%
mutate(cluster = factor(km4$cluster))
fviz_cluster(list(data = pc_df_noGK_noGK[, 1:2], cluster = km4$cluster),
geom = "point", main = "k‑means clusters (k = 4) in PC1–PC2")
# summarize clusters on original ratings
clen_noGK %>%
mutate(cluster = km4$cluster) %>%
group_by(cluster) %>%
summarise(across(c(OVR, PAC, SHO, PAS, DRI, DEF, PHY), mean,
.names = "avg_{col}"), Count = n()) %>%
arrange(cluster) %>%
print(n = Inf)## # A tibble: 4 × 9
## cluster avg_OVR avg_PAC avg_SHO avg_PAS avg_DRI avg_DEF avg_PHY Count
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
## 1 1 79.1 71.3 66.2 75.2 77.0 74.2 75.1 418
## 2 2 79.2 80.8 74.0 75.4 81.8 46.3 62.0 210
## 3 3 78.8 76.9 78.2 67.9 77.0 38.3 75.4 153
## 4 4 78.1 65.5 43.6 61.9 64.4 78.5 77.9 219# 5. hierarchical examples (i) Ward + Euclidean
dist_euc <- dist(pc_df_noGK_noGK[, 1:10], method = "euclidean")
hc1 <- hclust(dist_euc, method = "ward.D2")
fviz_dend(hc1, k = 4, cex = 0.6, main = "Ward / Euclidean")## Registered S3 method overwritten by 'dendextend':
## method from
## rev.hclust vegan## Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
## of ggplot2 3.3.4.
## ℹ The deprecated feature was likely used in the factoextra package.
## Please report the issue at <https://github.com/kassambara/factoextra/issues>.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
# (ii) Complete + Manhattan
dist_man <- dist(pc_df_noGK_noGK[, 1:10], method = "manhattan")
hc2 <- hclust(dist_man, method = "complete")
fviz_dend(hc2, k = 4, cex = 0.6, main = "Complete / Manhattan")
# print the readouts of the clusters# k means = 2
km2 <- kmeans(pc_df_noGK_noGK, centers = 2, nstart = 50)
fviz_cluster(list(data = pc_df_noGK_noGK[, 1:2], cluster = km2$cluster),
geom = "point", main = "k‑means clusters (k = 2) in PC1–PC2")
# summarize clusters on original ratings
clen_noGK %>%
mutate(cluster = km2$cluster) %>%
group_by(cluster) %>%
summarise(across(c(OVR, PAC, SHO, PAS, DRI, DEF, PHY), mean,
.names = "avg_{col}"), Count = n()) %>%
arrange(cluster) %>%
print(n = Inf)## # A tibble: 2 × 9
## cluster avg_OVR avg_PAC avg_SHO avg_PAS avg_DRI avg_DEF avg_PHY Count
## <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <int>
## 1 1 78.3 65.4 44.8 62.7 65.1 78.5 78.1 242
## 2 2 79.0 75.2 71.1 73.9 78.5 59.1 71.4 758dist_euc2 <- dist(pc_df_noGK_noGK[, 1:10], method = "euclidean")
hc3 <- hclust(dist_euc2, method = "ward.D2")
fviz_dend(hc3, k = 2, cex = 0.6, main = "Ward / Euclidean (k = 2)")
dist_man2 <- dist(pc_df_noGK_noGK[, 1:10], method = "manhattan")
hc4 <- hclust(dist_man2, method = "complete")
fviz_dend(hc4, k = 2, cex = 0.6, main = "Complete / Manhattan (k = 2)")
- Ward/Euclidean cuts set at four clusters with large jumps after merge 996, while Complete/Manhattan shows longer chains and less separation, especially among attacking players; both suggest 3,4 natural groups.