Supongo que todos sabemos lo de las escalas de medida, ¿verdad? Nominal, ordinal, intervalo y de razón. Y que todos sabemos lo que es un PCA y que sólo sirve para variables numéricas, ¿seguro?. Un PCA y otras técnicas relacionadas se basan en diagonalizar una matriz, ya sea mediante SVD, autovalores o a lo Gifi en dónde se utiliza Alternative Least Squares.
La materia prima de la que parte un PCA es una matriz de covarianzas o de correlaciones, pero, ¿qué pasa si nuestras variables no son numéricas sino ordinales? Pues entonces podemos utilizar lo que se conoce como correlaciones policóricas, propuestas por Pearson en 1900, aunque os aconsejo leer el artículo que viene en la Encyclopedia Of Statistica Sciences, ojo, 9686 páginas y 7680$ que cuestan todos los volúmenes.
Básicamente, las correlaciones policóricas suponen la existencia de variables latentes continuas asociadas a las variables ordinales observadas. La estimación de la correlación policórica entre dos variables se basa en encontrar la distribución normal bivariante subyacente y el coeficiente de correlación de pearson que mejor aproxima las frecuencias observadas en la tabla de contingencia entre las dos varibles ordinales.
La librería polycor de R incorpora la función polychor que implementa el cálculo. Se puede ver el código simplemente poniendo polycor::polychor, si tienes la librería instalada, por supuesto.
Pues una vez tenemos estas correlaciones podemos aplicar las técnicas de PCA o de Análisis factorial sobre nuestra matriz. Veamos un ejemplo, de un curso que di hace unos años en la Pablo de Olavide.
Utilizamos el conjunto de datos Science: Consumer Protection and Perceptions of Science and Technology section of the 1992 Euro-Barometer Survey (Karlheinz and Melich, 1992) based on a sample from Great Britain. Se pregunta por diferentes aspectos de la ciencia, y las categorías de respuesta son “strongly disagree”,“disagree”, “agree” y “strongly agree” .
Variables
Comfort: Science and technology are making our lives healthier, easier and more comfortable.
Environment: Scientific and technological research cannot play an important role in protecting the environment and repairing it.
Work: The application of science and new technology will make work more interesting.
Future: Thanks to science and technology, there will be more opportunities for the future generations.
Technology: New technology does not depend on basic scientific research.
Industry: Scientific and technological research do not play an important role in industrial development.
Benefit: The benefits of science are greater than any harmful effect it may have.
Tengo los datos guardados en un rds y subidos al github
datos <- readRDS("../../data/science.rds")
head(datos)## Comfort Environment Work Future Technology
## 1 strongly agree strongly agree strongly agree agree strongly agree
## 2 agree strongly agree agree agree agree
## 3 agree disagree disagree disagree strongly agree
## 4 agree agree disagree disagree strongly agree
## 5 agree strongly disagree strongly agree strongly agree disagree
## 6 strongly agree agree strongly agree agree agree
## Industry Benefit
## 1 agree disagree
## 2 agree agree
## 3 strongly agree agree
## 4 strongly agree agree
## 5 agree strongly disagree
## 6 strongly agree agreesummary(datos)## Comfort Environment Work
## strongly disagree: 5 strongly disagree: 29 strongly disagree: 33
## disagree : 32 disagree : 90 disagree : 98
## agree :266 agree :145 agree :206
## strongly agree : 89 strongly agree :128 strongly agree : 55
## Future Technology Industry
## strongly disagree: 14 strongly disagree: 18 strongly disagree: 10
## disagree : 72 disagree : 91 disagree : 47
## agree :210 agree :157 agree :173
## strongly agree : 96 strongly agree :126 strongly agree :162
## Benefit
## strongly disagree: 21
## disagree :100
## agree :193
## strongly agree : 78Nos aseguramos de que los niveles de los factores están codificados correctamente y en el orden que queremos
levels(datos$Work)## [1] "strongly disagree" "disagree" "agree"
## [4] "strongly agree"Pues ya podemos utilizar las correlaciones policóricas y el análisis factorial o un PCA.
library(psych)
library(polycor)##
## Attaching package: 'polycor'## The following object is masked from 'package:psych':
##
## polyserialUtilizamos la función hetcor que nos permite calcular correlaciones entre variables continuas, entre continuas con dicotómicas, continuas con ordinales y entre ordinales.
cor_poly <- hetcor(datos)## Warning in log(P): Se han producido NaNscor_poly##
## Two-Step Estimates
##
## Correlations/Type of Correlation:
## Comfort Environment Work Future Technology Industry
## Comfort 1 Polychoric Polychoric Polychoric Polychoric Polychoric
## Environment 0.09934 1 Polychoric Polychoric Polychoric Polychoric
## Work 0.2012 -0.08277 1 Polychoric Polychoric Polychoric
## Future 0.3463 -0.02804 0.4786 1 Polychoric Polychoric
## Technology 0.08963 0.4638 -0.1039 -0.03596 1 Polychoric
## Industry 0.1857 0.4108 -0.007643 0.1027 0.4348 1
## Benefit 0.408 -0.03652 0.2086 0.3769 -0.01434 0.118
## Benefit
## Comfort Polychoric
## Environment Polychoric
## Work Polychoric
## Future Polychoric
## Technology Polychoric
## Industry Polychoric
## Benefit 1
##
## Standard Errors:
## Comfort Environment Work Future Technology Industry
## Comfort
## Environment 0.06309
## Work 0.06062 0.0594
## Future 0.05651 0.06057 0.04566
## Technology 0.06331 0.04529 0.0591 0.06049
## Industry 0.06306 0.04979 0.06213 0.06176 0.04825
## Benefit 0.05315 0.05985 0.0566 0.05062 0.05977 0.06078
##
## n = 392
##
## P-values for Tests of Bivariate Normality:
## Comfort Environment Work Future Technology Industry
## Comfort
## Environment 0.1763
## Work 0.00592 0.01337
## Future 0.3163 0.00311 0.3021
## Technology 0.01695 0.001547 0.07819 0.009519
## Industry 0.4694 1.891e-05 0.0001493 0.0005283 4.817e-07
## Benefit 0.08673 0.005932 0.01871 0.009679 0.06082 0.007907corrplot::corrplot(cor_poly$correlations)
Y sin más hacemos el análisis factorial. Veamos qué número de factores elegimos
VSS(cor_poly$correlations)## n.obs was not specified and was arbitrarily set to 1000. This only affects the chi square values.
##
## Very Simple Structure
## Call: vss(x = x, n = n, rotate = rotate, diagonal = diagonal, fm = fm,
## n.obs = n.obs, plot = plot, title = title, use = use, cor = cor)
## VSS complexity 1 achieves a maximimum of 0.7 with 2 factors
## VSS complexity 2 achieves a maximimum of 0.78 with 7 factors
##
## The Velicer MAP achieves a minimum of NA with 2 factors
## BIC achieves a minimum of NA with 3 factors
## Sample Size adjusted BIC achieves a minimum of NA with 3 factors
##
## Statistics by number of factors
## vss1 vss2 map dof chisq prob sqresid fit RMSEA BIC SABIC complex
## 1 0.39 0.00 0.083 14 6.6e+02 2.4e-132 6.0 0.39 0.215 566 610 1.0
## 2 0.70 0.73 0.074 8 7.7e+01 2.1e-13 2.6 0.73 0.093 22 47 1.1
## 3 0.67 0.78 0.131 3 7.6e-01 8.6e-01 2.0 0.80 0.000 -20 -10 1.3
## 4 0.67 0.78 0.244 -1 2.5e-07 NA 2.0 0.80 NA NA NA 1.2
## 5 0.67 0.78 0.429 -4 9.0e-08 NA 1.9 0.80 NA NA NA 1.2
## 6 0.67 0.78 1.000 -6 8.8e-13 NA 1.9 0.80 NA NA NA 1.2
## 7 0.67 0.78 NA -7 0.0e+00 NA 1.9 0.80 NA NA NA 1.2
## eChisq SRMR eCRMS eBIC
## 1 1.3e+03 1.8e-01 0.2174 1227
## 2 6.8e+01 4.0e-02 0.0651 13
## 3 4.8e-01 3.4e-03 0.0089 -20
## 4 1.6e-07 1.9e-06 NA NA
## 5 6.2e-08 1.2e-06 NA NA
## 6 3.5e-13 2.9e-09 NA NA
## 7 3.5e-13 2.9e-09 NA NAres_factorial <- fa(cor_poly$correlations, nfactors = 3, n.obs = nrow(datos))## Loading required namespace: GPArotationdiagram(res_factorial)
Y listo ya tenemos el análsis factorial hecho. Ahora habría que interpretar y demás, ver las cargas factoriales y las comunalidades (cómo de bien está representada la variable en la estructura factorial)
res_factorial$loadings##
## Loadings:
## MR2 MR1 MR3
## Comfort 0.168 0.222 0.357
## Environment 0.666
## Work 0.617
## Future 0.813
## Technology 0.695
## Industry 0.632
## Benefit 0.861
##
## MR2 MR1 MR3
## SS loadings 1.362 1.096 0.885
## Proportion Var 0.195 0.157 0.126
## Cumulative Var 0.195 0.351 0.478res_factorial$communalities## Comfort Environment Work Future Technology Industry
## 0.2940500 0.4429437 0.3519061 0.6846375 0.4854217 0.4208926
## Benefit
## 0.7309609Aunque el análisis lo hemos hecho utilizando las correlaciones policóricas, para obtener las puntuaciones para cada fila hay que convertir los datos a numéricos.
science.num <- data.frame(sapply(datos, as.numeric))
table(datos$Comfort, science.num$Comfort)##
## 1 2 3 4
## strongly disagree 5 0 0 0
## disagree 0 32 0 0
## agree 0 0 266 0
## strongly agree 0 0 0 89Science data: Puntuaciones factoriales
puntuaciones <- factor.scores(science.num, res_factorial)
puntuaciones$scores## MR2 MR1 MR3
## [1,] 0.983783784 0.586931403 -0.704801916
## [2,] 0.316308160 0.075403203 0.072119271
## [3,] 0.511214907 -1.267647217 0.111088544
## [4,] 0.971164636 -1.283190166 0.069652282
## [5,] -1.601261361 1.428869944 -1.845504751
## [6,] 0.483437248 0.817348346 0.546986739
## [7,] 0.511546440 -0.225286108 1.304894201
## [8,] -0.035027006 -0.161537530 1.334985660
## [9,] 0.980349283 1.002260263 -0.808566327
## [10,] 0.016011850 0.319878827 0.455157733
## [11,] -0.666247071 1.124247892 0.248753651
## [12,] -0.059594604 -1.574845059 0.129568131
## [13,] -0.057017852 0.042740522 0.124900336
## [14,] -1.656341385 0.019112260 -0.995177681
## [15,] 0.948905669 -0.480305368 -1.046938254
## [16,] 1.392694657 0.089972097 0.133589749
## [17,] -0.549010849 -1.868036524 -1.172346199
## [18,] -0.690215015 0.154694729 0.143646991
## [19,] -0.517567236 -0.385470894 -0.933974271
## [20,] -1.220028066 0.076377257 0.052085055
## [21,] -0.897804321 -2.013344274 -0.408168528
## [22,] 0.510946786 -0.669040473 0.204583332
## [23,] -0.143641569 0.090946151 0.113555533
## [24,] -1.555827177 -0.838028790 -0.602960738
## [25,] -1.039904617 0.049224102 -0.892270941
## [26,] -0.057617506 -0.401013842 -0.975410533
## [27,] 0.378695812 -0.343748845 0.071852202
## [28,] 0.016011850 0.319878827 0.455157733
## [29,] 1.541106372 -0.821139944 -0.955643386
## [30,] 0.378695812 -0.343748845 0.071852202
## [31,] 0.012845469 0.136600942 0.257898535
## [32,] -0.603922832 -0.935872010 -1.038813862
## [33,] 0.211250129 0.019189516 1.687932664
## [34,] 0.822484801 0.226528620 1.252380205
## [35,] 1.431445899 -0.256372005 1.222021677
## [36,] -0.841499957 -0.068187324 0.537763188
## [37,] -0.143641569 0.090946151 0.113555533
## [38,] -1.150164745 0.170237677 0.185083253
## [39,] -0.046375802 0.739030874 0.455424802
## [40,] -1.293389301 -1.243122156 -1.471977999
## [41,] -0.143641569 0.090946151 0.113555533
## [42,] -0.286802713 0.318554172 -0.256205275
## [43,] 0.015080662 -1.166236647 -1.838958794
## [44,] -1.174132688 -0.799315486 0.079976594
## [45,] 0.315976627 -0.966957907 -1.121686386
## [46,] 1.599952430 1.215649990 -0.508400388
## [47,] -0.665978950 0.525641147 0.155258864
## [48,] 0.288242649 -2.563543319 -2.524363114
## [49,] -0.296087439 0.973632688 -0.001286612
## [50,] -1.679977795 0.091920206 0.093521317
## [51,] -0.119673625 1.060499315 0.218662193
## [52,] 1.392694657 0.089972097 0.133589749
## [53,] -0.119673625 1.060499315 0.218662193
## [54,] 1.393026191 1.132333206 1.327395406
## [55,] 1.393026191 1.132333206 1.327395406
## [56,] -0.603591298 0.106489100 0.154991795
## [57,] -0.057617506 -0.401013842 -0.975410533
## [58,] -1.402003864 -0.990638475 -2.693408126
## [59,] -1.543960787 0.393660827 1.523822514
## [60,] 1.306339062 -0.460429019 0.028750158
## [61,] 1.440299012 0.986717314 -0.850002589
## [62,] 1.629519025 -1.416144540 1.458250149
## [63,] 0.475961579 0.304335878 0.413721471
## [64,] 1.469534074 -2.687438325 -0.077157708
## [65,] -0.603591298 0.106489100 0.154991795
## [66,] -1.112076570 -2.260828643 -1.114096132
## [67,] -0.168209167 -1.322361377 -1.091861997
## [68,] -1.033028602 0.118482204 -1.859316543
## [69,] -1.237119995 -0.823917804 -1.020067207
## [70,] -1.555202491 -0.746383889 0.092987179
## [71,] -0.493930825 -0.458278840 -2.022673269
## [72,] 1.431114366 -1.298733114 0.028216020
## [73,] 1.490291957 1.780417929 1.669264675
## [74,] 0.051328590 0.388863585 1.439825251
## [75,] 0.971496169 -0.240829056 1.263457939
## [76,] 0.622995850 -1.336853014 1.529777870
## [77,] -0.666578605 0.081886782 -0.945052006
## [78,] -0.822465989 0.479986357 0.010915862
## [79,] -0.081253917 -0.328205896 0.113288464
## [80,] 1.513928368 1.707609983 0.580565678
## [81,] 0.005369800 -0.376411525 0.124633267
## [82,] -1.586146530 1.155333789 0.331626175
## [83,] -0.373743025 1.471643304 1.823397851
## [84,] 1.552348076 0.318904772 0.475191949
## [85,] 1.662340082 0.796497942 -0.508667457
## [86,] 1.541106372 -0.821139944 -0.955643386
## [87,] -0.943368166 -0.095290420 0.757745590
## [88,] -1.483881767 -2.241683546 1.434728452
## [89,] -0.555986944 1.003234317 -0.828600543
## [90,] -1.111745036 -1.218467534 0.079709525
## [91,] -2.015152221 -0.730840941 0.134423441
## [92,] 0.463010898 -2.608146799 -0.005629988
## [93,] 1.304282004 0.684976693 -2.280303786
## [94,] 0.465319529 -0.391954473 0.083197005
## [95,] -3.438023559 1.325474742 0.925957626
## [96,] 0.909108518 0.178322991 1.263725008
## [97,] 0.621206913 -0.790054047 -0.872770862
## [98,] -0.081585450 -1.370567006 -1.080517193
## [99,] -0.839226384 0.622052405 0.132569257
## [100,] 0.932413395 -0.936846064 -1.018779646
## [101,] 1.502686664 0.567565267 -0.850269658
## [102,] 1.068162282 -0.036498698 0.505016339
## [103,] 0.300147419 0.661223616 1.294083536
## [104,] 0.476561233 0.748090243 1.514032341
## [105,] -0.167277980 0.163754097 1.202254530
## [106,] -0.468173945 -0.035524643 0.484982123
## [107,] 0.741647896 -1.605983261 -0.404948117
## [108,] 0.967354922 1.771358561 0.610657136
## [109,] 1.392694657 0.089972097 0.133589749
## [110,] -1.975979755 -1.481114201 -2.872187738
## [111,] 0.364180636 0.373541675 -1.004967854
## [112,] -0.151448772 -1.464427425 -1.213515392
## [113,] -1.047380287 -0.463788365 -1.025536208
## [114,] -2.139927525 0.107463154 0.134957579
## [115,] 1.416331068 0.017164151 -0.955109248
## [116,] -1.656341385 0.019112260 -0.995177681
## [117,] 0.409807892 0.096455676 -0.883581528
## [118,] -3.118716722 1.783340093 1.609162027
## [119,] -1.519992844 1.363213991 1.628929174
## [120,] -0.049810304 1.154359734 0.351660391
## [121,] 0.943718511 1.844166507 1.699356134
## [122,] -0.555986944 1.003234317 -0.828600543
## [123,] -1.150164745 0.170237677 0.185083253
## [124,] -1.126528334 0.097429731 -0.903615744
## [125,] 0.347751774 1.557968833 0.310491198
## [126,] 0.763638742 -1.810261314 0.805137207
## [127,] -0.270725504 -1.286942079 0.025262678
## [128,] 1.392363124 -0.952389013 -1.060215908
## [129,] -0.628158897 -1.306818428 -1.050425735
## [130,] 1.415999535 -1.025196959 -2.148914906
## [131,] 0.932744928 0.105515045 0.175026011
## [132,] -0.603591298 0.106489100 0.154991795
## [133,] -0.081253917 -0.328205896 0.113288464
## [134,] 1.455413843 0.713181159 1.327128337
## [135,] 1.022866558 1.282948411 1.577435670
## [136,] 1.439967479 -0.055643795 -2.043808246
## [137,] 1.106913523 -0.382842799 1.593448267
## [138,] 0.275579821 -0.752083912 0.088666006
## [139,] 1.393026191 1.132333206 1.327395406
## [140,] -0.081253917 -0.328205896 0.113288464
## [141,] 0.078399502 -0.099273221 0.454890664
## [142,] -1.006940188 1.583597510 1.842144506
## [143,] -0.541203646 -0.312662948 0.154724726
## [144,] -0.033381441 -0.030067424 -0.963798661
## [145,] -0.143641569 0.090946151 0.113555533
## [146,] 1.493502018 -1.717885162 0.027948951
## [147,] -1.815994803 -0.209820416 -1.336779881
## [148,] -0.143973102 -0.951414958 -1.080250124
## [149,] -1.108836751 1.441479157 1.268847388
## [150,] -1.286181752 -0.131502945 -1.245217944
## [151,] 0.910154428 -0.133961267 -2.135370183
## [152,] 1.552679609 1.361265882 1.668997606
## [153,] 1.454750776 -1.371541060 -1.060482977
## [154,] -0.603591298 0.106489100 0.154991795
## [155,] -0.517567236 -0.385470894 -0.933974271
## [156,] -1.196391655 0.003569311 -1.036613943
## [157,] 0.784065093 1.615233831 1.357753934
## [158,] 0.316308160 0.075403203 0.072119271
## [159,] 0.337109724 0.861678482 -0.020033268
## [160,] -0.081253917 -0.328205896 0.113288464
## [161,] 0.608212552 -0.020955749 0.546452601
## [162,] -0.081585450 -1.370567006 -1.080517193
## [163,] 0.838913663 -0.957898538 -0.063078847
## [164,] -0.019197798 -1.789719053 -1.080784262
## [165,] -0.082962894 -2.100643857 1.124772341
## [166,] -0.081253917 -0.328205896 0.113288464
## [167,] -0.189868480 -0.075722214 -1.108141663
## [168,] 0.378695812 -0.343748845 0.071852202
## [169,] -0.327262931 -1.107539688 -0.333153327
## [170,] -0.022407859 1.708584038 0.560531462
## [171,] -0.035090418 -1.802505385 0.047685216
## [172,] 1.552348076 0.318904772 0.475191949
## [173,] 1.392363124 -0.952389013 -1.060215908
## [174,] 1.006106163 1.425014459 1.699089065
## [175,] -0.024384956 0.534752821 1.665510126
## [176,] 1.576316020 1.288457936 0.580298609
## [177,] 1.552348076 0.318904772 0.475191949
## [178,] -0.627559242 -0.863064064 0.049885135
## [179,] -0.143641569 0.090946151 0.113555533
## [180,] -0.738150903 -1.784411598 -0.066566328
## [181,] 0.507405192 1.786901509 0.652093398
## [182,] -1.077135043 0.447375981 0.515340650
## [183,] -0.935331223 -1.131600907 0.299658330
## [184,] -1.150164745 0.170237677 0.185083253
## [185,] -0.143973102 -0.951414958 -1.080250124
## [186,] -0.136765554 0.160204254 -0.853490069
## [187,] -0.547321992 0.525693453 0.606902587
## [188,] -0.690215015 0.154694729 0.143646991
## [189,] 1.493833551 -0.675524052 1.221754609
## [190,] 1.599952430 1.215649990 -0.508400388
## [191,] -1.239954842 0.035165421 -0.023520748
## [192,] 1.615067261 0.942113834 1.668730537
## [193,] -0.610798847 -1.005130112 -0.071768260
## [194,] -1.558744085 1.709558092 0.540497246
## [195,] -0.134688376 -1.606493474 -1.335168787
## [196,] 1.416662601 1.059525260 0.238696409
## [197,] -0.771051920 -1.677817152 -1.513681330
## [198,] -0.787480782 -0.493389994 -0.198222278
## [199,] 0.956712872 1.075068209 0.280132671
## [200,] -0.127212707 -1.093481007 -1.201903520
## [201,] -1.529277569 2.018292506 1.883847837
## [202,] 0.569792844 1.367749461 0.651826330
## [203,] -0.603259765 1.148850209 1.348797452
## [204,] -0.073778247 0.184806571 0.246553732
## [205,] -0.604190953 -0.337265265 -0.945319075
## [206,] 0.943718511 1.844166507 1.699356134
## [207,] 0.260796524 0.563813353 -0.894659262
## [208,] -0.081585450 -1.370567006 -1.080517193
## [209,] 0.362535071 0.242071569 1.293816467
## [210,] -0.264875279 -1.526691735 -0.333420396
## [211,] -1.519992844 1.363213991 1.628929174
## [212,] 0.492721974 0.162269830 0.292068076
## [213,] -0.492141889 -1.005077807 0.379875463
## [214,] 1.614735728 -0.100247275 0.474924880
## [215,] -0.143641569 0.090946151 0.113555533
## [216,] -1.174400809 -0.200708742 0.173471381
## [217,] 0.796128266 -0.639979941 -2.355586056
## [218,] -0.143641569 0.090946151 0.113555533
## [219,] -0.112197956 1.573511782 0.351927460
## [220,] 1.392694657 0.089972097 0.133589749
## [221,] 0.355059402 -0.270940898 1.160551199
## [222,] -0.616585659 0.875587398 1.574215258
## [223,] 0.909108518 0.178322991 1.263725008
## [224,] -0.690546549 -0.887666381 -1.050158666
## [225,] -0.697422564 -0.956924483 -0.083113064
## [226,] 1.552348076 0.318904772 0.475191949
## [227,] -1.610714128 -0.257973739 -0.873791354
## [228,] 0.608812207 0.422798615 1.646763471
## [229,] -1.008629045 -0.810132467 0.062895720
## [230,] -1.109767939 -0.044636318 -1.025269139
## [231,] -0.627227709 0.179297046 1.243690792
## [232,] -0.589407655 -1.653162530 0.038006194
## [233,] -0.381550228 -0.083730272 0.496326926
## [234,] -0.119673625 1.060499315 0.218662193
## [235,] 0.464987996 -1.434315583 -1.110608652
## [236,] 0.386503015 1.211624732 1.398923127
## [237,] -0.530561597 0.383627404 0.485249192
## [238,] 0.956381339 0.032707099 -0.913672986
## [239,] 0.386171482 0.169263623 0.205117470
## [240,] -1.150164745 0.170237677 0.185083253
## [241,] -1.610114474 0.185780625 0.226519515
## [242,] 1.552348076 0.318904772 0.475191949
## [243,] -0.389918704 -0.089780895 -0.239391471
## [244,] -0.534059510 -0.842011590 -0.905815664
## [245,] 0.933076461 1.147876155 1.368831668
## [246,] 1.354274949 1.478677308 0.238963478
## [247,] 1.479050253 0.640373213 0.238429340
## [248,] -0.213504890 -0.002914268 -0.019442666
## [249,] -0.326931398 -0.065178578 0.860652330
## [250,] 0.085875171 0.413739246 0.588155932
## [251,] -1.071016697 -0.390980419 0.063162789
## [252,] -0.074377902 -0.258947794 -0.853757138
## [253,] 0.355327523 -0.869547643 1.067056412
## [254,] 0.496431609 0.048250048 -0.872236724
## [255,] -0.443937880 0.335421775 0.496593995
## [256,] -2.092323170 1.004208371 -0.848634759
## [257,] 0.886518017 -0.061153320 -1.046671185
## [258,] -0.057617506 -0.401013842 -0.975410533
## [259,] -0.990511326 0.399170352 0.526685454
## [260,] -1.656341385 0.019112260 -0.995177681
## [261,] 0.995132580 -0.313637002 0.174758942
## [262,] 1.552348076 0.318904772 0.475191949
## [263,] 1.537564778 1.634802037 -0.508133320
## [264,] -0.635134991 1.564452413 -0.706680079
## [265,] -2.867813702 1.188918218 -0.192832830
## [266,] 0.316308160 0.075403203 0.072119271
## [267,] 0.760097149 0.645680668 1.252647274
## [268,] 0.054763091 -0.026465275 1.543589662
## [269,] 0.210318942 -1.466925958 -0.606183863
## [270,] 1.295097358 -1.600473736 -1.402085177
## [271,] -1.087777093 -0.248914371 0.184816185
## [272,] 1.493833551 -0.675524052 1.221754609
## [273,] -1.634019005 0.857195316 1.408713300
## [274,] -0.649486676 0.982181844 0.127100256
## [275,] -1.512517174 1.876226458 1.762194442
## [276,] 0.846121211 0.153720674 0.163681208
## [277,] 0.464719875 -0.835708838 -1.017113864
## [278,] 0.862881607 0.011654626 0.042027812
## [279,] -0.530561597 0.383627404 0.485249192
## [280,] -0.745395119 1.685465988 0.370674115
## [281,] 1.046502969 1.210140465 0.488736672
## [282,] -1.087777093 -0.248914371 0.184816185
## [283,] 0.402931877 0.027197574 0.083464074
## [284,] 0.402931877 0.027197574 0.083464074
## [285,] -1.688930988 1.789359831 1.542245637
## [286,] 1.393026191 1.132333206 1.327395406
## [287,] -1.247762044 -1.520208156 -1.350591673
## [288,] -0.701188598 -1.583956733 -1.380683131
## [289,] -2.059090620 1.939975035 1.792285900
## [290,] -0.541203646 -0.312662948 0.154724726
## [291,] -0.603591298 0.106489100 0.154991795
## [292,] 0.472795199 0.121057994 0.216462273
## [293,] -0.850199967 -1.116599056 -1.391760866
## [294,] 0.869757622 0.080912728 -0.925017790
## [295,] 0.039648260 0.247070881 -0.633541264
## [296,] 0.378963933 -0.942355589 -0.021642585
## [297,] 0.005369800 -0.376411525 0.124633267
## [298,] -0.105221861 -1.297759060 0.008181804
## [299,] -1.656341385 0.019112260 -0.995177681
## [300,] 1.537564778 1.634802037 -0.508133320
## [301,] -0.049810304 1.154359734 0.351660391
## [302,] 0.448559134 -0.249888425 0.204850401
## [303,] -0.059594604 -1.574845059 0.129568131
## [304,] -1.220028066 0.076377257 0.052085055
## [305,] -1.126528334 0.097429731 -0.903615744
## [306,] -1.220028066 0.076377257 0.052085055
## [307,] -1.133672470 0.626778373 0.156924646
## [308,] 1.493833551 -0.675524052 1.221754609
## [309,] 0.410139426 1.138816786 0.310224129
## [310,] -1.019646308 1.132797048 0.377140519
## [311,] 0.862550073 -1.030706484 -1.151777845
## [312,] 1.354274949 1.478677308 0.238963478
## [313,] 1.478718720 -0.401987897 -0.955376317
## [314,] -1.133404349 0.028171629 0.063429858
## [315,] 0.346820586 0.071853359 -1.983625329
## [316,] -1.236188807 0.662197671 1.274049320
## [317,] -0.666247071 1.124247892 0.248753651
## [318,] 0.101474638 1.377241787 -0.042455805
## [319,] -1.033296723 0.717088949 -1.765821755
## [320,] -0.143641569 0.090946151 0.113555533
## [321,] -0.120005158 0.018138205 -0.975143465
## [322,] -0.081585450 -1.370567006 -1.080517193
## [323,] -0.096037214 0.987691369 -0.870036805
## [324,] -0.143641569 0.090946151 0.113555533
## [325,] 1.552348076 0.318904772 0.475191949
## [326,] -0.666578605 0.081886782 -0.945052006
## [327,] -2.037143067 -0.526562888 -1.075661883
## [328,] -0.081253917 -0.328205896 0.113288464
## [329,] -0.229665632 0.582906145 1.202521599
## [330,] -1.676479882 1.317559200 1.484586172
## [331,] 0.316639694 1.117764312 1.265924928
## [332,] 1.368726714 -0.879581066 0.028483089
## [333,] 0.869757622 0.080912728 -0.925017790
## [334,] 1.393026191 1.132333206 1.327395406
## [335,] -0.711874328 1.401333891 0.127367325
## [336,] 0.389937516 0.796295872 1.502687537
## [337,] 0.423660002 -2.705557063 -2.194372786
## [338,] 1.392363124 -0.952389013 -1.060215908
## [339,] -0.057617506 -0.401013842 -0.975410533
## [340,] 1.392694657 0.089972097 0.133589749
## [341,] -0.849868434 -0.074237947 -0.197955209
## [342,] -0.080985796 -0.926812641 0.019793677
## [343,] -1.507330016 -0.448245417 -0.984099946
## [344,] 1.416662601 1.059525260 0.238696409
## [345,] 0.555109626 -0.256882218 0.291801007
## [346,] -1.133072816 1.070532738 1.257235515
## [347,] -0.199421327 1.177963046 -0.759728213
## [348,] 1.092398347 0.334447721 0.516628211
## [349,] -0.081253917 -0.328205896 0.113288464
## [350,] 0.981806686 -0.586899814 0.400176748
## [351,] 0.884540919 -1.234984537 0.058307479
## [352,] -0.143641569 0.090946151 0.113555533
## [353,] -0.714182959 -0.814858435 0.038540332
## [354,] 1.057520232 -0.732789050 0.174491873
## [355,] -0.279658577 -0.210794470 -1.316745665
## [356,] 0.869757622 0.080912728 -0.925017790
## [357,] 0.598323260 -0.078814150 -2.685309315
## [358,] 1.590767784 -1.069800439 0.369818221
## [359,] -1.805309073 -3.195111040 -2.844830337
## [360,] -0.143909690 0.689552896 0.207050320
## [361,] -0.565171590 -1.282216111 0.049618066
## [362,] -0.184038375 0.305820146 1.323907926
## [363,] 0.023219398 1.431498039 0.681917788
## [364,] -0.081253917 -0.328205896 0.113288464
## [365,] 1.005774630 0.382653350 0.505283408
## [366,] 0.039648260 0.247070881 -0.633541264
## [367,] 0.386171482 0.169263623 0.205117470
## [368,] 0.402931877 0.027197574 0.083464074
## [369,] 1.392694657 0.089972097 0.133589749
## [370,] -1.512517174 1.876226458 1.762194442
## [371,] -0.230265286 0.139151780 0.102210729
## [372,] 1.369058247 0.162780043 1.222288746
## [373,] -0.119405504 0.461892570 0.125167405
## [374,] 0.594028909 1.738695880 0.663438202
## [375,] 1.490291957 1.780417929 1.669264675
## [376,] -0.143641569 0.090946151 0.113555533
## [377,] 1.197500058 -3.290919568 -2.937760104
## [378,] 0.110442769 1.827046775 1.793573461
## [379,] 0.005101679 0.222195220 0.218128055
## [380,] 0.032772245 0.177812778 0.333504338
## [381,] 0.029006211 -0.449219471 -0.964065730
## [382,] 1.490291957 1.780417929 1.669264675
## [383,] 1.018437457 -1.428806058 -2.107745712
## [384,] -0.677220654 -0.614403569 -1.275576472
## [385,] -0.689883482 1.197055838 1.337452649
## [386,] -1.512517174 1.876226458 1.762194442
## [387,] -0.689883482 1.197055838 1.337452649
## [388,] 1.541106372 -0.821139944 -0.955643386
## [389,] -1.117243608 -0.557648785 -1.158534407
## [390,] 0.297527375 -3.218621835 -2.779281777
## [391,] -1.710821754 -0.946891060 0.955460259
## [392,] -0.096368748 -0.054669741 -2.063842462Y ya con esas tres dimensiones ya podemos hacer lo que queramos.