Ejercicio 4

El conjunto de datos alzheimer.csv estĆ” relacionado a la enfermedad del Alzheimer. Aplique alguna tĆ©cnica de reducciĆ³n de dimensionalidad para visualizar en dos dimensiones la relaciĆ³n entre variables que obtuve luego de aplicar la tĆ©cnica de reducciĆ³n de dimensionalidad. Discuta los resultados.

Nota: Puede omitir la primer columna ā€œIdentifierā€.

Puntos extra: Genere un correlograma (o diagramas que considere Ćŗtiles) para analizar la relaciĆ³n de las variables del conjunto de datos.

AnƔlisis exploratorio

setwd("/Users/christianeugeniomartinez/Downloads")
D = read.csv("alzheimer.csv")
library(cowplot)
library(grid)
library(ggplot2)
summary(D)
##   IDENTIFIER           GSM21215          GSM21217          GSM21218      
##  Length:22215       Min.   :    0.5   Min.   :    0.9   Min.   :    0.4  
##  Class :character   1st Qu.:   54.5   1st Qu.:   57.4   1st Qu.:   61.4  
##  Mode  :character   Median :  182.5   Median :  190.4   Median :  197.6  
##                     Mean   :  666.6   Mean   :  694.7   Mean   :  664.0  
##                     3rd Qu.:  563.0   3rd Qu.:  564.7   3rd Qu.:  584.5  
##                     Max.   :23313.3   Max.   :37098.7   Max.   :25044.2  
##     GSM21219          GSM21220          GSM21221          GSM21226      
##  Min.   :    0.9   Min.   :    0.5   Min.   :    0.5   Min.   :    0.5  
##  1st Qu.:   68.7   1st Qu.:   60.1   1st Qu.:   60.1   1st Qu.:   75.9  
##  Median :  222.4   Median :  201.2   Median :  197.8   Median :  241.5  
##  Mean   :  665.9   Mean   :  691.1   Mean   :  668.9   Mean   :  663.9  
##  3rd Qu.:  618.2   3rd Qu.:  586.9   3rd Qu.:  576.2   3rd Qu.:  618.1  
##  Max.   :30867.1   Max.   :31601.2   Max.   :25266.9   Max.   :30183.3  
##     GSM21231          GSM21232          GSM21204          GSM21205      
##  Min.   :    0.6   Min.   :    0.5   Min.   :    0.9   Min.   :    1.7  
##  1st Qu.:   71.9   1st Qu.:   77.6   1st Qu.:   66.5   1st Qu.:   88.5  
##  Median :  227.7   Median :  248.7   Median :  217.8   Median :  274.4  
##  Mean   :  686.9   Mean   :  668.9   Mean   :  669.0   Mean   :  652.4  
##  3rd Qu.:  596.6   3rd Qu.:  618.9   3rd Qu.:  611.1   3rd Qu.:  637.9  
##  Max.   :36909.9   Max.   :35048.4   Max.   :36174.3   Max.   :46027.2  
##     GSM21216           GSM21222          GSM21225          GSM21228       
##  Min.   :    0.50   Min.   :    0.8   Min.   :    0.5   Min.   :    0.60  
##  1st Qu.:   58.15   1st Qu.:   68.5   1st Qu.:   63.0   1st Qu.:   75.05  
##  Median :  191.50   Median :  219.6   Median :  206.3   Median :  231.80  
##  Mean   :  680.36   Mean   :  661.9   Mean   :  675.0   Mean   :  677.70  
##  3rd Qu.:  574.95   3rd Qu.:  598.6   3rd Qu.:  607.9   3rd Qu.:  606.90  
##  Max.   :28901.30   Max.   :27498.3   Max.   :26635.7   Max.   :32476.40  
##     GSM21233          GSM21209           GSM21210          GSM21211      
##  Min.   :    0.9   Min.   :    1.20   Min.   :    0.8   Min.   :    1.2  
##  1st Qu.:   76.2   1st Qu.:   73.75   1st Qu.:   65.2   1st Qu.:   72.3  
##  Median :  235.8   Median :  235.40   Median :  213.0   Median :  224.4  
##  Mean   :  664.0   Mean   :  668.11   Mean   :  679.5   Mean   :  682.5  
##  3rd Qu.:  607.2   3rd Qu.:  618.45   3rd Qu.:  600.4   3rd Qu.:  602.5  
##  Max.   :30130.7   Max.   :39691.40   Max.   :30653.1   Max.   :42913.5  
##     GSM21214           GSM21223          GSM21224          GSM21227      
##  Min.   :    0.40   Min.   :    0.8   Min.   :    0.4   Min.   :    1.3  
##  1st Qu.:   66.75   1st Qu.:   69.5   1st Qu.:   66.0   1st Qu.:   72.4  
##  Median :  214.90   Median :  218.5   Median :  212.7   Median :  228.9  
##  Mean   :  667.45   Mean   :  675.4   Mean   :  694.2   Mean   :  694.2  
##  3rd Qu.:  601.35   3rd Qu.:  597.9   3rd Qu.:  599.2   3rd Qu.:  610.2  
##  Max.   :32213.70   Max.   :32358.7   Max.   :35002.8   Max.   :44696.8  
##     GSM21230          GSM21203           GSM21206          GSM21207      
##  Min.   :    1.1   Min.   :    1.70   Min.   :    0.5   Min.   :    1.6  
##  1st Qu.:   75.1   1st Qu.:   78.45   1st Qu.:   60.9   1st Qu.:   90.0  
##  Median :  241.4   Median :  248.50   Median :  203.7   Median :  272.5  
##  Mean   :  699.7   Mean   :  675.11   Mean   :  694.6   Mean   :  679.6  
##  3rd Qu.:  611.5   3rd Qu.:  614.00   3rd Qu.:  590.1   3rd Qu.:  628.9  
##  Max.   :52274.1   Max.   :43540.90   Max.   :35972.2   Max.   :78292.3  
##     GSM21208          GSM21212           GSM21213           GSM21229      
##  Min.   :    1.6   Min.   :    0.40   Min.   :    1.30   Min.   :    0.7  
##  1st Qu.:   71.6   1st Qu.:   80.95   1st Qu.:   75.55   1st Qu.:   67.6  
##  Median :  240.1   Median :  246.40   Median :  252.60   Median :  208.5  
##  Mean   :  664.1   Mean   :  660.91   Mean   :  665.41   Mean   :  673.0  
##  3rd Qu.:  612.9   3rd Qu.:  626.55   3rd Qu.:  626.60   3rd Qu.:  589.3  
##  Max.   :33719.0   Max.   :41252.90   Max.   :40707.20   Max.   :27179.6
cor(D[,2:32])
##           GSM21215  GSM21217  GSM21218  GSM21219  GSM21220  GSM21221  GSM21226
## GSM21215 1.0000000 0.9353832 0.9369323 0.8829778 0.9050716 0.9574731 0.8759577
## GSM21217 0.9353832 1.0000000 0.9097081 0.9000105 0.8926460 0.9191428 0.8769255
## GSM21218 0.9369323 0.9097081 1.0000000 0.9155014 0.9480002 0.9670626 0.9156078
## GSM21219 0.8829778 0.9000105 0.9155014 1.0000000 0.9536572 0.9234915 0.9672967
## GSM21220 0.9050716 0.8926460 0.9480002 0.9536572 1.0000000 0.9473730 0.9677615
## GSM21221 0.9574731 0.9191428 0.9670626 0.9234915 0.9473730 1.0000000 0.9170045
## GSM21226 0.8759577 0.8769255 0.9156078 0.9672967 0.9677615 0.9170045 1.0000000
## GSM21231 0.8907292 0.8754427 0.9295498 0.9554966 0.9695406 0.9436310 0.9621211
## GSM21232 0.8708923 0.8916677 0.9134709 0.9619263 0.9601073 0.9213772 0.9656313
## GSM21204 0.9102571 0.9226251 0.9061181 0.9424531 0.9235420 0.9004018 0.9390573
## GSM21205 0.7757866 0.8380126 0.8050809 0.9008846 0.8612062 0.8242918 0.8914688
## GSM21216 0.9485618 0.9365034 0.9635799 0.9311208 0.9457520 0.9812273 0.9164434
## GSM21222 0.9227848 0.8975757 0.9427718 0.9498366 0.9615706 0.9663345 0.9491900
## GSM21225 0.8895407 0.8605836 0.9205218 0.9527685 0.9543768 0.9333685 0.9604083
## GSM21228 0.8707796 0.8677730 0.9032156 0.9527764 0.9519923 0.9218126 0.9577838
## GSM21233 0.9202750 0.9189612 0.9571396 0.9496125 0.9562572 0.9530827 0.9455148
## GSM21209 0.8327451 0.8771304 0.8749440 0.9510477 0.9146536 0.8792508 0.9342113
## GSM21210 0.9155392 0.9221748 0.9420970 0.9638890 0.9589149 0.9602522 0.9556036
## GSM21211 0.8357918 0.8822178 0.8810223 0.9545664 0.9302910 0.8798769 0.9494094
## GSM21214 0.9137557 0.9304949 0.9622727 0.9438096 0.9442149 0.9533812 0.9375295
## GSM21223 0.8804141 0.8981663 0.9177721 0.9582545 0.9539457 0.9299030 0.9532457
## GSM21224 0.8668052 0.8533126 0.9100001 0.9467834 0.9503249 0.9221100 0.9541779
## GSM21227 0.8342587 0.8483455 0.8797820 0.9487702 0.9486456 0.8788347 0.9578691
## GSM21230 0.7980119 0.8058380 0.8441170 0.9348993 0.9160812 0.8555063 0.9361030
## GSM21203 0.8315814 0.8807255 0.8429234 0.9367617 0.9024829 0.8398039 0.9336650
## GSM21206 0.8410284 0.8312559 0.8942855 0.9104330 0.9503944 0.8843787 0.9287425
## GSM21207 0.6541584 0.7275266 0.6974657 0.8219366 0.7646096 0.6955096 0.8219120
## GSM21208 0.8977119 0.9317319 0.8813558 0.9192311 0.8994759 0.8852151 0.9096120
## GSM21212 0.7792472 0.8214978 0.8301819 0.9372894 0.8964881 0.8324948 0.9345452
## GSM21213 0.8311145 0.8606136 0.8358762 0.9326212 0.8979802 0.8399231 0.9341520
## GSM21229 0.9150516 0.8810147 0.9409905 0.9377914 0.9531923 0.9576165 0.9377179
##           GSM21231  GSM21232  GSM21204  GSM21205  GSM21216  GSM21222  GSM21225
## GSM21215 0.8907292 0.8708923 0.9102571 0.7757866 0.9485618 0.9227848 0.8895407
## GSM21217 0.8754427 0.8916677 0.9226251 0.8380126 0.9365034 0.8975757 0.8605836
## GSM21218 0.9295498 0.9134709 0.9061181 0.8050809 0.9635799 0.9427718 0.9205218
## GSM21219 0.9554966 0.9619263 0.9424531 0.9008846 0.9311208 0.9498366 0.9527685
## GSM21220 0.9695406 0.9601073 0.9235420 0.8612062 0.9457520 0.9615706 0.9543768
## GSM21221 0.9436310 0.9213772 0.9004018 0.8242918 0.9812273 0.9663345 0.9333685
## GSM21226 0.9621211 0.9656313 0.9390573 0.8914688 0.9164434 0.9491900 0.9604083
## GSM21231 1.0000000 0.9590432 0.9081820 0.8660865 0.9421595 0.9676899 0.9560211
## GSM21232 0.9590432 1.0000000 0.9298719 0.9207286 0.9246538 0.9554746 0.9418241
## GSM21204 0.9081820 0.9298719 1.0000000 0.8689672 0.9042754 0.9191051 0.9199948
## GSM21205 0.8660865 0.9207286 0.8689672 1.0000000 0.8306715 0.8540553 0.8533325
## GSM21216 0.9421595 0.9246538 0.9042754 0.8306715 1.0000000 0.9618449 0.9214736
## GSM21222 0.9676899 0.9554746 0.9191051 0.8540553 0.9618449 1.0000000 0.9511271
## GSM21225 0.9560211 0.9418241 0.9199948 0.8533325 0.9214736 0.9511271 1.0000000
## GSM21228 0.9635618 0.9712237 0.9107672 0.8950367 0.9212517 0.9616646 0.9579677
## GSM21233 0.9443647 0.9458992 0.9357922 0.8615858 0.9505144 0.9473387 0.9304035
## GSM21209 0.9146071 0.9457953 0.9123814 0.9364932 0.8886191 0.9036591 0.9118186
## GSM21210 0.9602454 0.9621961 0.9318886 0.8973078 0.9640286 0.9614947 0.9516589
## GSM21211 0.9250589 0.9618063 0.9247127 0.9274742 0.8937857 0.9152187 0.9118840
## GSM21214 0.9350317 0.9500159 0.9392231 0.8850182 0.9569880 0.9420284 0.9282461
## GSM21223 0.9563713 0.9735143 0.9260527 0.8962114 0.9410446 0.9625016 0.9347330
## GSM21224 0.9593379 0.9590888 0.9002408 0.8579019 0.9180618 0.9566258 0.9532290
## GSM21227 0.9382405 0.9604796 0.9168877 0.8949529 0.8761743 0.9216461 0.9470527
## GSM21230 0.9306288 0.9473402 0.8952608 0.8830376 0.8509939 0.9120474 0.9300339
## GSM21203 0.9034878 0.9254139 0.9355011 0.8757557 0.8573052 0.8832277 0.8895711
## GSM21206 0.9299546 0.9124623 0.8759237 0.8051345 0.8927748 0.9116110 0.9008833
## GSM21207 0.7701360 0.8328114 0.7958050 0.8938662 0.7031994 0.7375880 0.7917405
## GSM21208 0.8978396 0.9109487 0.9474235 0.8520005 0.9023861 0.9115182 0.8710626
## GSM21212 0.8997396 0.9410798 0.8968142 0.9256117 0.8456394 0.8879670 0.9059730
## GSM21213 0.9046633 0.9337760 0.9435984 0.9065818 0.8503508 0.8874933 0.9023270
## GSM21229 0.9538355 0.9327523 0.9000234 0.8259457 0.9552943 0.9672369 0.9574021
##           GSM21228  GSM21233  GSM21209  GSM21210  GSM21211  GSM21214  GSM21223
## GSM21215 0.8707796 0.9202750 0.8327451 0.9155392 0.8357918 0.9137557 0.8804141
## GSM21217 0.8677730 0.9189612 0.8771304 0.9221748 0.8822178 0.9304949 0.8981663
## GSM21218 0.9032156 0.9571396 0.8749440 0.9420970 0.8810223 0.9622727 0.9177721
## GSM21219 0.9527764 0.9496125 0.9510477 0.9638890 0.9545664 0.9438096 0.9582545
## GSM21220 0.9519923 0.9562572 0.9146536 0.9589149 0.9302910 0.9442149 0.9539457
## GSM21221 0.9218126 0.9530827 0.8792508 0.9602522 0.8798769 0.9533812 0.9299030
## GSM21226 0.9577838 0.9455148 0.9342113 0.9556036 0.9494094 0.9375295 0.9532457
## GSM21231 0.9635618 0.9443647 0.9146071 0.9602454 0.9250589 0.9350317 0.9563713
## GSM21232 0.9712237 0.9458992 0.9457953 0.9621961 0.9618063 0.9500159 0.9735143
## GSM21204 0.9107672 0.9357922 0.9123814 0.9318886 0.9247127 0.9392231 0.9260527
## GSM21205 0.8950367 0.8615858 0.9364932 0.8973078 0.9274742 0.8850182 0.8962114
## GSM21216 0.9212517 0.9505144 0.8886191 0.9640286 0.8937857 0.9569880 0.9410446
## GSM21222 0.9616646 0.9473387 0.9036591 0.9614947 0.9152187 0.9420284 0.9625016
## GSM21225 0.9579677 0.9304035 0.9118186 0.9516589 0.9118840 0.9282461 0.9347330
## GSM21228 1.0000000 0.9248756 0.9298796 0.9588593 0.9449169 0.9302105 0.9684000
## GSM21233 0.9248756 1.0000000 0.9079591 0.9559674 0.9206651 0.9705739 0.9380066
## GSM21209 0.9298796 0.9079591 1.0000000 0.9478485 0.9603123 0.9309922 0.9334896
## GSM21210 0.9588593 0.9559674 0.9478485 1.0000000 0.9455111 0.9686284 0.9648186
## GSM21211 0.9449169 0.9206651 0.9603123 0.9455111 1.0000000 0.9356373 0.9515491
## GSM21214 0.9302105 0.9705739 0.9309922 0.9686284 0.9356373 1.0000000 0.9484492
## GSM21223 0.9684000 0.9380066 0.9334896 0.9648186 0.9515491 0.9484492 1.0000000
## GSM21224 0.9759500 0.9231870 0.9200381 0.9564032 0.9297973 0.9241197 0.9612192
## GSM21227 0.9535323 0.9163270 0.9252712 0.9251677 0.9439120 0.9074515 0.9393593
## GSM21230 0.9500890 0.8872305 0.9110371 0.9086692 0.9283367 0.8829986 0.9359246
## GSM21203 0.9085978 0.8945546 0.9166614 0.9035943 0.9386201 0.8911623 0.9172872
## GSM21206 0.9061445 0.8960000 0.8716084 0.9022874 0.8914899 0.8841154 0.9217672
## GSM21207 0.8206373 0.7537538 0.8766557 0.7941095 0.8593297 0.7821910 0.7963008
## GSM21208 0.8874151 0.9139497 0.8794176 0.9121732 0.9065108 0.9115450 0.9221008
## GSM21212 0.9372529 0.8760736 0.9552284 0.9172282 0.9564975 0.8949556 0.9369720
## GSM21213 0.9229057 0.8865736 0.9284100 0.9090600 0.9394077 0.8932314 0.9245303
## GSM21229 0.9509479 0.9337889 0.8940929 0.9539362 0.8992849 0.9294250 0.9400248
##           GSM21224  GSM21227  GSM21230  GSM21203  GSM21206  GSM21207  GSM21208
## GSM21215 0.8668052 0.8342587 0.7980119 0.8315814 0.8410284 0.6541584 0.8977119
## GSM21217 0.8533126 0.8483455 0.8058380 0.8807255 0.8312559 0.7275266 0.9317319
## GSM21218 0.9100001 0.8797820 0.8441170 0.8429234 0.8942855 0.6974657 0.8813558
## GSM21219 0.9467834 0.9487702 0.9348993 0.9367617 0.9104330 0.8219366 0.9192311
## GSM21220 0.9503249 0.9486456 0.9160812 0.9024829 0.9503944 0.7646096 0.8994759
## GSM21221 0.9221100 0.8788347 0.8555063 0.8398039 0.8843787 0.6955096 0.8852151
## GSM21226 0.9541779 0.9578691 0.9361030 0.9336650 0.9287425 0.8219120 0.9096120
## GSM21231 0.9593379 0.9382405 0.9306288 0.9034878 0.9299546 0.7701360 0.8978396
## GSM21232 0.9590888 0.9604796 0.9473402 0.9254139 0.9124623 0.8328114 0.9109487
## GSM21204 0.9002408 0.9168877 0.8952608 0.9355011 0.8759237 0.7958050 0.9474235
## GSM21205 0.8579019 0.8949529 0.8830376 0.8757557 0.8051345 0.8938662 0.8520005
## GSM21216 0.9180618 0.8761743 0.8509939 0.8573052 0.8927748 0.7031994 0.9023861
## GSM21222 0.9566258 0.9216461 0.9120474 0.8832277 0.9116110 0.7375880 0.9115182
## GSM21225 0.9532290 0.9470527 0.9300339 0.8895711 0.9008833 0.7917405 0.8710626
## GSM21228 0.9759500 0.9535323 0.9500890 0.9085978 0.9061445 0.8206373 0.8874151
## GSM21233 0.9231870 0.9163270 0.8872305 0.8945546 0.8960000 0.7537538 0.9139497
## GSM21209 0.9200381 0.9252712 0.9110371 0.9166614 0.8716084 0.8766557 0.8794176
## GSM21210 0.9564032 0.9251677 0.9086692 0.9035943 0.9022874 0.7941095 0.9121732
## GSM21211 0.9297973 0.9439120 0.9283367 0.9386201 0.8914899 0.8593297 0.9065108
## GSM21214 0.9241197 0.9074515 0.8829986 0.8911623 0.8841154 0.7821910 0.9115450
## GSM21223 0.9612192 0.9393593 0.9359246 0.9172872 0.9217672 0.7963008 0.9221008
## GSM21224 1.0000000 0.9361580 0.9308156 0.8930135 0.9172574 0.7796873 0.8751130
## GSM21227 0.9361580 1.0000000 0.9651219 0.9254819 0.9060811 0.8583386 0.8752800
## GSM21230 0.9308156 0.9651219 1.0000000 0.9153163 0.8844677 0.8574508 0.8548898
## GSM21203 0.8930135 0.9254819 0.9153163 1.0000000 0.8758111 0.8384629 0.9354166
## GSM21206 0.9172574 0.9060811 0.8844677 0.8758111 1.0000000 0.7241101 0.8700015
## GSM21207 0.7796873 0.8583386 0.8574508 0.8384629 0.7241101 1.0000000 0.7444494
## GSM21208 0.8751130 0.8752800 0.8548898 0.9354166 0.8700015 0.7444494 1.0000000
## GSM21212 0.9211414 0.9326141 0.9313565 0.9275345 0.8708196 0.8955919 0.8744143
## GSM21213 0.9071476 0.9270945 0.9200618 0.9567559 0.8686445 0.8565382 0.9214023
## GSM21229 0.9548155 0.9165759 0.8946151 0.8671144 0.9086429 0.7400790 0.8774467
##           GSM21212  GSM21213  GSM21229
## GSM21215 0.7792472 0.8311145 0.9150516
## GSM21217 0.8214978 0.8606136 0.8810147
## GSM21218 0.8301819 0.8358762 0.9409905
## GSM21219 0.9372894 0.9326212 0.9377914
## GSM21220 0.8964881 0.8979802 0.9531923
## GSM21221 0.8324948 0.8399231 0.9576165
## GSM21226 0.9345452 0.9341520 0.9377179
## GSM21231 0.8997396 0.9046633 0.9538355
## GSM21232 0.9410798 0.9337760 0.9327523
## GSM21204 0.8968142 0.9435984 0.9000234
## GSM21205 0.9256117 0.9065818 0.8259457
## GSM21216 0.8456394 0.8503508 0.9552943
## GSM21222 0.8879670 0.8874933 0.9672369
## GSM21225 0.9059730 0.9023270 0.9574021
## GSM21228 0.9372529 0.9229057 0.9509479
## GSM21233 0.8760736 0.8865736 0.9337889
## GSM21209 0.9552284 0.9284100 0.8940929
## GSM21210 0.9172282 0.9090600 0.9539362
## GSM21211 0.9564975 0.9394077 0.8992849
## GSM21214 0.8949556 0.8932314 0.9294250
## GSM21223 0.9369720 0.9245303 0.9400248
## GSM21224 0.9211414 0.9071476 0.9548155
## GSM21227 0.9326141 0.9270945 0.9165759
## GSM21230 0.9313565 0.9200618 0.8946151
## GSM21203 0.9275345 0.9567559 0.8671144
## GSM21206 0.8708196 0.8686445 0.9086429
## GSM21207 0.8955919 0.8565382 0.7400790
## GSM21208 0.8744143 0.9214023 0.8774467
## GSM21212 1.0000000 0.9479129 0.8762190
## GSM21213 0.9479129 1.0000000 0.8710343
## GSM21229 0.8762190 0.8710343 1.0000000
colnames(D)
##  [1] "IDENTIFIER" "GSM21215"   "GSM21217"   "GSM21218"   "GSM21219"  
##  [6] "GSM21220"   "GSM21221"   "GSM21226"   "GSM21231"   "GSM21232"  
## [11] "GSM21204"   "GSM21205"   "GSM21216"   "GSM21222"   "GSM21225"  
## [16] "GSM21228"   "GSM21233"   "GSM21209"   "GSM21210"   "GSM21211"  
## [21] "GSM21214"   "GSM21223"   "GSM21224"   "GSM21227"   "GSM21230"  
## [26] "GSM21203"   "GSM21206"   "GSM21207"   "GSM21208"   "GSM21212"  
## [31] "GSM21213"   "GSM21229"
#pairs(D[,2:32], main = "Datos alzheimer")

CƔlculo de componentes principales

Por defecto, la funciĆ³n prcomp() centra las variables para que tengan media de 0. Con el argumento scale = TRUE indicamos que queremos escalar las variables para que tengan desviaciĆ³n estĆ”ndar igual a 1.

pca_D <- prcomp(D[,2:32], scale = TRUE)
names(pca_D)
## [1] "sdev"     "rotation" "center"   "scale"    "x"

Muestra de los primeros 6 elementos del vector de loadings de los 5 primeros componentes

head(pca_D$rotation)[, 1:5]
##                 PC1         PC2          PC3         PC4          PC5
## GSM21215 -0.1724615 -0.32121197  0.224466505 -0.01516863  0.323072359
## GSM21217 -0.1741319 -0.17046472  0.445171166 -0.04265692 -0.003400844
## GSM21218 -0.1778878 -0.26025106  0.004798824 -0.14010214  0.049145007
## GSM21219 -0.1846141  0.03427660 -0.007839905  0.03303118 -0.013908558
## GSM21220 -0.1836665 -0.09009127 -0.148673511  0.06627419 -0.029124958
## GSM21221 -0.1791785 -0.27150471 -0.016802856 -0.20373098  0.060042035

#DimenciĆ³n

dim(pca_D$rotation)
## [1] 31 31

Hay un total de 31 componentes principales distintas, ya que en general pueden haber min(nāˆ’1,p) componentes en un set de datos nƗp.Ā En este caso min (31, 31) = 64.

Podemos acceder a los vectores de los scores de la siguiente manera:

head(pca_D$x)[,1:5]
##              PC1         PC2        PC3         PC4         PC5
## [1,] -14.9630735  4.53946688  0.8718804 -0.47007428 -0.62903225
## [2,]   1.9975394 -0.03608750 -0.1353925  0.02104120 -0.01231705
## [3,]   1.0101081 -0.14802532 -0.8790234  1.01826654 -0.94460721
## [4,]  -2.5488601  0.68045326  0.4893180 -0.19788775 -0.10246445
## [5,]   1.6491719 -0.01297025  0.1338780 -0.01762962 -0.01101674
## [6,]   0.8183589  0.33244759  0.1019599 -0.08054798 -0.12982242

Otro de los outputs de la funciĆ³n prcomp() es la desviaciĆ³n estĆ”ndar de cada componente principal:

pca_D$sdev
##  [1] 5.3163854 0.9534608 0.6419103 0.5087700 0.3888509 0.3650124 0.3117415
##  [8] 0.3052550 0.2728228 0.2535404 0.2362288 0.2195945 0.2107304 0.1965574
## [15] 0.1855211 0.1805151 0.1761691 0.1705069 0.1651225 0.1583912 0.1554586
## [22] 0.1462308 0.1369696 0.1343752 0.1282888 0.1241790 0.1204769 0.1177992
## [29] 0.1095716 0.1053027 0.1013918

La varianza explicada por cada componente principal (correspondiente a los eigenvalores) la obtenemos elevando al cuadrado la desviaciĆ³n estĆ”ndar:

pca_D$sdev^2
##  [1] 28.26395337  0.90908746  0.41204887  0.25884690  0.15120504  0.13323407
##  [7]  0.09718276  0.09318061  0.07443230  0.06428275  0.05580404  0.04822176
## [13]  0.04440728  0.03863481  0.03441809  0.03258569  0.03103555  0.02907260
## [19]  0.02726544  0.02508777  0.02416738  0.02138345  0.01876067  0.01805670
## [25]  0.01645802  0.01542042  0.01451468  0.01387666  0.01200593  0.01108866
## [31]  0.01028029

Como es de esperar, la varianza explicada es mayor en la primera componente que en las subsiguientes.

RepresentaciĆ³n

Cabe destacar que la representaciĆ³n grĆ”fica de las observaciones y las variables es distinta: las observaciones se representan mediante sus proyecciones, mientras que las variables se representan mediante sus correlaciones. La correlaciĆ³n entre una componente y una variable estima la informaciĆ³n que comparten -> loadings, por lo que las variables se pueden representar como puntos en el espacio de los componentes utilizando sus loadings como coordenadas.

Se muestran dos ejemplos para representar las observaciones sobre las dos primeras componentes principales:

library(factoextra)
## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBa
fviz_pca_ind(pca_D, geom.ind = "point", 
             col.ind = "#FC4E07", 
             axes = c(1, 2), 
             pointsize = 1.5)

Para representar las variables sobre las dos primeras componentes principales podemos utilizar la funciĆ³n fviz_pca_var() del paquete factoextra. La correlaciĆ³n entre una variable y una componente principal se utiliza como la coordenada de dicha variable sobre la componente principal. De esta manera podemos obtener un grĆ”fico de correlaciĆ³n de variables:

fviz_pca_var(pca_D, col.var = "cos2", 
             geom.var = "arrow", 
             labelsize = 2, 
             repel = FALSE)

En este tipo de grĆ”fico, ademĆ”s de indicarse el % de varianza explicada por la primera (Dim1) y segunda componente (Dim2), las variables positivamente correlacionadas se agrupan juntas o prĆ³ximas, mientras que las negativamente correlacionadas se representan en lados opuestos del origen o cuadrantes opuestos. AdemĆ”s, la distancia entre las variables y el origen mide la calidad de la representaciĆ³n de las variables (mayor cuanto mĆ”s prĆ³xima a la circunferencia o cĆ­rculo de correlaciĆ³n, siendo Ć©stas las que mĆ”s contribuyen en los dos primeros componentes). La calidad de esta representaciĆ³n se mide por el valor al cuadrado del coseno (cos2 ) del Ć”ngulo del triĆ”ngulo formado por el punto del origen, la observaciĆ³n y su proyecciĆ³n sobre el componente. Para una variable dada, la suma del cos2 sobre todos los componentes principales serĆ” igual a 1, y si ademĆ”s la variable es perfectamente representable por solo los dos primeros componentes principales, la suma de cos2 sobre estos dos serĆ” igual a 1. Variables posicionadas cerca del origen puede ser un indicativo de que serĆ­an necesarios mĆ”s de dos componentes principales para su representaciĆ³n.

ElecciĆ³n del nĆŗmero de componentes principales

Podemos hacer uso de multitud de representaciones a la hora de escoger el nĆŗmero Ć³ptimo de componentes principales:

Scree plot

Una forma es generando un scree plot que represente los eigenvalores ordenador de mayor a menor. Con la funciĆ³n fviz_screeplot() del paquete factoextra podemos obtener esta representaciĆ³n, sin importar quĆ© funciĆ³n hemos utilizado para generar los componentes principales.

fviz_screeplot(pca_D, addlabels = TRUE)

ContribuciĆ³n de variables

Si contamos con un gran nĆŗmero de variables, podrĆ­amos decidir mostrar solo aquellas con mayor contribuciĆ³n.

fviz_contrib(pca_D, choice = "var", axes = 1)

La lĆ­nea roja discontinua indica el valor medio de contribuciĆ³n. Para una determinada componente, una variable con una contribuciĆ³n mayor a este lĆ­mite puede considerarse importante a la hora de contribuir a esta componente. En la representaciĆ³n anterior, la v ariable GSM21232 es la que mĆ”s contribuye a la PC1.

ProporciĆ³n de varianza explicada y acumulada

Para calcular la proporciĆ³n de varianza explicada (PVE) por cada componente principal, simplemente dividimos la varianza explicada de cada uno entre la varianza total de todos:

PVE <- 100*pca_D$sdev^2/sum(pca_D$sdev^2)
PVE
##  [1] 91.17404313  2.93254018  1.32918991  0.83499000  0.48775818  0.42978733
##  [7]  0.31349278  0.30058262  0.24010419  0.20736371  0.18001302  0.15555406
## [13]  0.14324930  0.12462842  0.11102610  0.10511512  0.10011469  0.09378258
## [19]  0.08795302  0.08092827  0.07795929  0.06897887  0.06051828  0.05824740
## [25]  0.05309039  0.04974330  0.04682154  0.04476341  0.03872879  0.03576987
## [31]  0.03316223

La primera componente principal explica el 91.2% de la varianza, mientras que la segunda solo un 2.9%.

par(mfrow = c(1,2))

plot(PVE, type = "o", 
     ylab = "PVE", 
     xlab = "Componente principal", 
     col = "blue")
plot(cumsum(PVE), type = "o", 
     ylab = "PVE acumulada", 
     xlab = "Componente principal", 
     col = "brown3")

De manera conjunta, los primeros 7 componentes principales explican en torno al 40% de la varianza de los datos, lo cual no es una cantidad muy alta.

prop_varianza <- pca_D$sdev^2 / sum(pca_D$sdev^2)
prop_varianza_acum <- cumsum(prop_varianza)
prop_varianza_acum
##  [1] 0.9117404 0.9410658 0.9543577 0.9627076 0.9675852 0.9718831 0.9750180
##  [8] 0.9780238 0.9804249 0.9824985 0.9842987 0.9858542 0.9872867 0.9885330
## [15] 0.9896432 0.9906944 0.9916955 0.9926334 0.9935129 0.9943222 0.9951018
## [22] 0.9957915 0.9963967 0.9969792 0.9975101 0.9980075 0.9984758 0.9989234
## [29] 0.9993107 0.9996684 1.0000000
ggplot(data = data.frame(prop_varianza_acum, pc = 1:31),
       aes(x = pc, y = prop_varianza_acum, group = 1)) +
  geom_point() +
  geom_line() +
  geom_label(aes(label = round(prop_varianza_acum,2))) +
  theme_bw() +
  labs(x = "Componente principal",
       y = "Prop. varianza explicada acumulada")

De manera conjunta, los primeros 7 componentes principales explican en torno al 98% de la varianza de los datos, lo cual es una cantidad muy alta.