Pajaros<-read_excel("Pajaros.xlsx")
Data<-Pajaros[-1]

Matriz de Correlaciones y Covarianzas

cov(Data)
##           X1        X2        X3        X4        X5
## X1 13.353741 13.610969 1.9220663 1.3306122 2.1922194
## X2 13.610969 25.682823 2.7136054 2.1977041 2.6578231
## X3  1.922066  2.713605 0.6316327 0.3422662 0.4146471
## X4  1.330612  2.197704 0.3422662 0.3184184 0.3393707
## X5  2.192219  2.657823 0.4146471 0.3393707 0.9828231
R_Pajaros<-cor(Data)
R_Pajaros
##           X1        X2        X3        X4        X5
## X1 1.0000000 0.7349642 0.6618119 0.6452841 0.6051247
## X2 0.7349642 1.0000000 0.6737411 0.7685087 0.5290138
## X3 0.6618119 0.6737411 1.0000000 0.7631899 0.5262701
## X4 0.6452841 0.7685087 0.7631899 1.0000000 0.6066493
## X5 0.6051247 0.5290138 0.5262701 0.6066493 1.0000000

Prueba de Esfericidad de Bartlett

cortest.bartlett(R_Pajaros,n=nrow(Data))
## $chisq
## [1] 150.193
## 
## $p.value
## [1] 3.401733e-27
## 
## $df
## [1] 10

Se concluye que no hay esfericidad en los datos.

Matriz de correlación con analisis gráfico

rcorr(as.matrix(Data),type = "pearson")
##      X1   X2   X3   X4   X5
## X1 1.00 0.73 0.66 0.65 0.61
## X2 0.73 1.00 0.67 0.77 0.53
## X3 0.66 0.67 1.00 0.76 0.53
## X4 0.65 0.77 0.76 1.00 0.61
## X5 0.61 0.53 0.53 0.61 1.00
## 
## n= 49 
## 
## 
## P
##    X1    X2    X3    X4    X5   
## X1       0e+00 0e+00 0e+00 0e+00
## X2 0e+00       0e+00 0e+00 0e+00
## X3 0e+00 0e+00       0e+00 1e-04
## X4 0e+00 0e+00 0e+00       0e+00
## X5 0e+00 0e+00 1e-04 0e+00
corrplot(R_Pajaros,method = "ellipse")

corrplot.mixed(R_Pajaros)

Analisis de componentes principales

PCA<-prcomp(Data,scale. = T)
PCA
## Standard deviations (1, .., p=5):
## [1] 1.9015726 0.7290433 0.6216306 0.5491498 0.4056199
## 
## Rotation (n x k) = (5 x 5):
##          PC1         PC2        PC3         PC4        PC5
## X1 0.4517989 -0.05072137  0.6904702 -0.42041399  0.3739091
## X2 0.4616809  0.29956355  0.3405484  0.54786307 -0.5300805
## X3 0.4505416  0.32457242 -0.4544927 -0.60629605 -0.3427923
## X4 0.4707389  0.18468403 -0.4109350  0.38827811  0.6516665
## X5 0.3976754 -0.87648935 -0.1784558  0.06887199 -0.1924341
summary(PCA)
## Importance of components:
##                           PC1    PC2     PC3     PC4     PC5
## Standard deviation     1.9016 0.7290 0.62163 0.54915 0.40562
## Proportion of Variance 0.7232 0.1063 0.07728 0.06031 0.03291
## Cumulative Proportion  0.7232 0.8295 0.90678 0.96709 1.00000

Dado que el summary devuelve las desviaciones estandar, se debe encontrar aun el valor del autovalor de las matrices.

Autovalores

eig.val<-get_eigenvalue(PCA)
eig.val
##       eigenvalue variance.percent cumulative.variance.percent
## Dim.1  3.6159783        72.319567                    72.31957
## Dim.2  0.5315041        10.630082                    82.94965
## Dim.3  0.3864245         7.728491                    90.67814
## Dim.4  0.3015655         6.031310                    96.70945
## Dim.5  0.1645275         3.290550                   100.00000

Autovectores

PCA$rotation
##          PC1         PC2        PC3         PC4        PC5
## X1 0.4517989 -0.05072137  0.6904702 -0.42041399  0.3739091
## X2 0.4616809  0.29956355  0.3405484  0.54786307 -0.5300805
## X3 0.4505416  0.32457242 -0.4544927 -0.60629605 -0.3427923
## X4 0.4707389  0.18468403 -0.4109350  0.38827811  0.6516665
## X5 0.3976754 -0.87648935 -0.1784558  0.06887199 -0.1924341

Valores del ACP para cada individuo

PCA$x[1:10,]
##               PC1         PC2         PC3          PC4        PC5
##  [1,]  0.06428901  0.60083713 -0.17123335  0.515825561 -0.5487904
##  [2,] -2.18031283  0.44230082  0.40006959  0.645459959 -0.2310766
##  [3,] -1.14556567 -0.01925412 -0.67612688  0.716298164 -0.2088714
##  [4,] -2.31106565 -0.17199267 -0.30596210 -0.149289289 -0.4781034
##  [5,] -0.29504203  0.66520783 -0.47421381  0.545862110 -0.2444780
##  [6,]  1.91626198  0.59525444  0.62093302 -0.006608669  0.2855166
##  [7,] -1.05036763  0.11981084  0.07446084  0.088396192  0.5303822
##  [8,]  0.43854156  0.16397253 -1.64844050 -0.815773999 -0.5615014
##  [9,]  2.69147373  0.78226687  0.36794761 -0.464857885  0.0579804
## [10,]  0.18568959 -1.31372223 -0.40908826  0.297345077  0.7021169

Grafico de Sedimentación

screeplot(PCA,main="Grafico de sedimentación")

fviz_eig(PCA)

Autovalor<-1:length(eig.val$eigenvalue)
ggplot(eig.val,aes(x=Autovalor))+
  geom_col(aes(y=variance.percent))+
  geom_line(aes(y=cumulative.variance.percent))+
    geom_point(aes(y=cumulative.variance.percent))+
  geom_hline(yintercept=80)+
  ggtitle("Grafico de Sedimentació")

Graficos de los individuos

fviz_pca_ind(PCA,
             col.ind = "cos2", # Color por la calidad de la representación
             gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
             repel = TRUE)     # Para evitar la superposición de texto

Grafica de las variables

fviz_pca_var(PCA,
             col.var = "contrib", # Color por contribuciones al CP
             gradient.cols = c("#00AFBB", "#E7B800", "#FC4E07"),
             repel = TRUE)     # Para evitar la superposición de texto

Biplot

fviz_pca_biplot(PCA, repel = TRUE,
                col.var = "red", # Color de las variables 
                col.ind = "blue", axes = c(1,2))  # Color de los indivíduos. Axes: es para seleccionar las componentes a graficar.