PCA: Utilisation de R pour générer et tracer des valeurs propres

Aug 17 2020

J'ai les données suivantes:

structure(list(V1 = c(0.94651, 0.5383, 0.96285, 0.91922, 0.46489, 
0.77343, 0.90285, 1.0069, 0.4285, -0.016644, 0.73531, -0.010766, 
0.72106, 0.71081, -0.086889, 0.37988, 0.32897, 0.52904, -0.0084557, 
0.10109), V2 = c(-0.56207, -0.31966, -0.57178, -0.54587, -0.27607, 
-0.45929, -0.53615, -0.59794, -0.25446, 0.0098841, -0.43665, 
0.006393, -0.42819, -0.4221, 0.051598, -0.22559, -0.19535, -0.31416, 
0.0050213, -0.060033), V3 = c(0.8266, 0.085098, 0.81386, 0.94114, 
-0.094576, 0.15722, 0.41302, 0.54406, -0.070067, -0.37905, 0.72315, 
0.037326, 0.72163, 0.84899, 0.024539, 0.073047, 0.12855, 0.19081, 
0.037335, -0.025786), V4 = c(0.085098, 0.77217, 0.087477, 0.095301, 
-0.11338, 0.029254, 0.055516, 0.013586, -0.07974, -0.058024, 
0.042291, 0.23548, 0.067229, 0.057507, 0.035961, 0.11495, 0.069248, 
0.03216, -0.064853, 0.044449), V5 = c(0.81386, 0.087477, 0.80893, 
0.92834, -0.042251, 0.23575, 0.46973, 0.58175, -0.012993, -0.30251, 
0.73196, 0.053843, 0.74423, 0.85489, 0.044913, 0.13302, 0.16572, 
0.22198, 0.076549, 0.017349), V6 = c(0.94114, 0.095301, 0.92834, 
1.0931, -0.082746, 0.18189, 0.4852, 0.62751, -0.085199, -0.4134, 
0.8921, 0.047127, 0.89089, 1.0642, 0.042064, 0.080492, 0.14708, 
0.21112, 0.042478, -0.028438), V7 = c(-0.094576, -0.11338, -0.042251, 
-0.082746, 0.80368, 0.6199, 0.53577, 0.43571, 0.52259, 0.72283, 
0.14265, 0.12527, 0.15794, 0.10972, 0.27052, 0.35122, 0.24159, 
0.1599, 0.28239, 0.32291), V8 = c(0.15722, 0.029254, 0.23575, 
0.18189, 0.6199, 1.045, 0.81547, 0.65866, 0.71941, 0.82931, 0.32166, 
0.1449, 0.43432, 0.30115, 0.19978, 0.78488, 0.46836, 0.40675, 
0.49992, 0.5234), V9 = c(0.41302, 0.055516, 0.46973, 0.4852, 
0.53577, 0.81547, 0.88555, 0.7562, 0.51057, 0.47113, 0.54481, 
0.15711, 0.61531, 0.59215, 0.26507, 0.59573, 0.48186, 0.39617, 
0.38826, 0.37683), V10 = c(0.54406, 0.013586, 0.58175, 0.62751, 
0.43571, 0.65866, 0.7562, 0.78475, 0.39652, 0.27494, 0.61592, 
0.1222, 0.66145, 0.66101, 0.16319, 0.40111, 0.31974, 0.34904, 
0.28275, 0.25133), V11 = c(-0.070067, -0.07974, -0.012993, -0.085199, 
0.52259, 0.71941, 0.51057, 0.39652, 1.0164, 0.69192, 0.12521, 
0.15234, 0.18136, 0.066519, 0.074701, 0.50289, 0.29894, 0.25127, 
0.67174, 0.3459), V12 = c(-0.37905, -0.058024, -0.30251, -0.4134, 
0.72283, 0.82931, 0.47113, 0.27494, 0.69192, 1.1506, -0.1139, 
0.14507, -0.027262, -0.2119, 0.16865, 0.60408, 0.30354, 0.21503, 
0.40543, 0.55259), V13 = c(0.72315, 0.042291, 0.73196, 0.8921, 
0.14265, 0.32166, 0.54481, 0.61592, 0.12521, -0.1139, 1.5481, 
0.2868, 1.5331, 1.604, 0.32047, 0.29921, 0.30592, 0.49005, 0.391, 
0.17689), V14 = c(0.037326, 0.23548, 0.053843, 0.047127, 0.12527, 
0.1449, 0.15711, 0.1222, 0.15234, 0.14507, 0.2868, 0.76717, 0.36333, 
0.21414, 0.33952, 0.35815, 0.24875, 0.35865, 0.4172, 0.28988), 
    V15 = c(0.72163, 0.067229, 0.74423, 0.89089, 0.15794, 0.43432, 
    0.61531, 0.66145, 0.18136, -0.027262, 1.5331, 0.36333, 1.6139, 
    1.5989, 0.3219, 0.41679, 0.37344, 0.59587, 0.493, 0.24522
    ), V16 = c(0.84899, 0.057507, 0.85489, 1.0642, 0.10972, 0.30115, 
    0.59215, 0.66101, 0.066519, -0.2119, 1.604, 0.21414, 1.5989, 
    1.8191, 0.2722, 0.25619, 0.28694, 0.40081, 0.29757, 0.12688
    ), V17 = c(0.024539, 0.035961, 0.044913, 0.042064, 0.27052, 
    0.19978, 0.26507, 0.16319, 0.074701, 0.16865, 0.32047, 0.33952, 
    0.3219, 0.2722, 0.92673, 0.42773, 0.41275, 0.47912, 0.37034, 
    0.35505), V18 = c(0.073047, 0.11495, 0.13302, 0.080492, 0.35122, 
    0.78488, 0.59573, 0.40111, 0.50289, 0.60408, 0.29921, 0.35815, 
    0.41679, 0.25619, 0.42773, 1.0542, 0.61743, 0.57759, 0.67009, 
    0.67155), V19 = c(0.12855, 0.069248, 0.16572, 0.14708, 0.24159, 
    0.46836, 0.48186, 0.31974, 0.29894, 0.30354, 0.30592, 0.24875, 
    0.37344, 0.28694, 0.41275, 0.61743, 0.51219, 0.46494, 0.47372, 
    0.39662), V20 = c(0.19081, 0.03216, 0.22198, 0.21112, 0.1599, 
    0.40675, 0.39617, 0.34904, 0.25127, 0.21503, 0.49005, 0.35865, 
    0.59587, 0.40081, 0.47912, 0.57759, 0.46494, 0.79956, 0.50647, 
    0.40812), V21 = c(0.037335, -0.064853, 0.076549, 0.042478, 
    0.28239, 0.49992, 0.38826, 0.28275, 0.67174, 0.40543, 0.391, 
    0.4172, 0.493, 0.29757, 0.37034, 0.67009, 0.47372, 0.50647, 
    1.4825, 0.40309), V22 = c(-0.025786, 0.044449, 0.017349, 
    -0.028438, 0.32291, 0.5234, 0.37683, 0.25133, 0.3459, 0.55259, 
    0.17689, 0.28988, 0.24522, 0.12688, 0.35505, 0.67155, 0.39662, 
    0.40812, 0.40309, 0.59524), V23 = c(0.25527, -0.017435, 0.24472, 
    0.19172, -0.08942, 0.019515, 0.023886, 0.085312, -0.1077, 
    -0.25759, -0.0056936, -0.16772, 0.00016361, 0.032313, -0.29165, 
    -0.055895, -0.061876, 0.011813, -0.18098, -0.14357), V24 = c(-0.017435, 
    0.86281, -0.01802, -0.013843, -0.20764, -0.034841, -0.011684, 
    -0.054282, -0.05775, -0.079061, 0.039397, 0.48237, 0.04116, 
    0.017332, 0.033632, 0.044072, 0.019504, -0.027936, 0.057321, 
    0.0088914), V25 = c(0.24472, -0.01802, 0.23612, 0.18369, 
    -0.069095, 0.042049, 0.039087, 0.092742, -0.091879, -0.22268, 
    -0.0020998, -0.1605, 0.0058407, 0.032342, -0.27061, -0.025945, 
    -0.034951, 0.03467, -0.16788, -0.11148), V26 = c(0.19172, 
    -0.013843, 0.18369, 0.14561, -0.074105, 0.012767, 0.019197, 
    0.063111, -0.08197, -0.19216, -0.0014697, -0.12429, 0.0022122, 
    0.025654, -0.21522, -0.039968, -0.042864, 0.0073339, -0.13347, 
    -0.10308), V27 = c(-0.08942, -0.20764, -0.069095, -0.074105, 
    0.91404, 0.34296, 0.1169, 0.18089, 0.39112, 0.51196, 0.071998, 
    0.039649, 0.060645, 0.014709, 0.40831, 0.13309, 0.10797, 
    0.27916, 0.1567, 0.2165), V28 = c(0.019515, -0.034841, 0.042049, 
    0.012767, 0.34296, 0.40838, 0.26292, 0.19653, 0.22056, 0.42274, 
    0.063293, -0.0042594, 0.089042, 0.026711, 0.16124, 0.43457, 
    0.39117, 0.39529, 0.076837, 0.425), V29 = c(0.023886, -0.011684, 
    0.039087, 0.019197, 0.1169, 0.26292, 0.29695, 0.15903, 0.129, 
    0.27237, 0.055221, 0.037713, 0.071066, 0.02503, 0.074612, 
    0.39081, 0.53542, 0.40199, 0.061141, 0.40689), V30 = c(0.085312, 
    -0.054282, 0.092742, 0.063111, 0.18089, 0.19653, 0.15903, 
    0.16809, 0.093384, 0.11927, 0.04596, -0.027867, 0.052697, 
    0.030282, 0.030441, 0.18169, 0.19679, 0.26977, -0.0073689, 
    0.1654), V31 = c(-0.1077, -0.05775, -0.091879, -0.08197, 
    0.39112, 0.22056, 0.129, 0.093384, 0.81882, 0.36239, 0.11077, 
    0.11839, 0.10007, 0.028829, 0.28994, 0.2202, 0.23569, 0.26342, 
    0.3219, 0.28324), V32 = c(-0.25759, -0.079061, -0.22268, 
    -0.19216, 0.51196, 0.42274, 0.27237, 0.11927, 0.36239, 0.9133, 
    0.079302, 0.17579, 0.091098, -0.0070463, 0.54, 0.54119, 0.53789, 
    0.4253, 0.31003, 0.772), V33 = c(-0.0056936, 0.039397, -0.0020998, 
    -0.0014697, 0.071998, 0.063293, 0.055221, 0.04596, 0.11077, 
    0.079302, 0.16478, 0.17777, 0.14017, 0.071905, 0.13318, 0.12564, 
    0.11937, 0.15937, 0.15798, 0.18798), V34 = c(-0.16772, 0.48237, 
    -0.1605, -0.12429, 0.039649, -0.0042594, 0.037713, -0.027867, 
    0.11839, 0.17579, 0.17777, 1.1407, 0.1481, 0.058347, 0.43169, 
    0.16129, 0.16705, 0.16025, 0.40938, 0.27552), V35 = c(0.00016361, 
    0.04116, 0.0058407, 0.0022122, 0.060645, 0.089042, 0.071066, 
    0.052697, 0.10007, 0.091098, 0.14017, 0.1481, 0.14548, 0.061918, 
    0.10971, 0.15723, 0.13187, 0.17229, 0.13877, 0.175), V36 = c(0.032313, 
    0.017332, 0.032342, 0.025654, 0.014709, 0.026711, 0.02503, 
    0.030282, 0.028829, -0.0070463, 0.071905, 0.058347, 0.061918, 
    0.037792, 0.017188, 0.046331, 0.040203, 0.06882, 0.04331, 
    0.05934), V37 = c(-0.29165, 0.033632, -0.27061, -0.21522, 
    0.40831, 0.16124, 0.074612, 0.030441, 0.28994, 0.54, 0.13318, 
    0.43169, 0.10971, 0.017188, 1.0391, 0.31502, 0.21118, 0.28367, 
    0.43956, 0.47978), V38 = c(-0.055895, 0.044072, -0.025945, 
    -0.039968, 0.13309, 0.43457, 0.39081, 0.18169, 0.2202, 0.54119, 
    0.12564, 0.16129, 0.15723, 0.046331, 0.31502, 0.83371, 0.79387, 
    0.6541, 0.23566, 0.84715), V39 = c(-0.061876, 0.019504, -0.034951, 
    -0.042864, 0.10797, 0.39117, 0.53542, 0.19679, 0.23569, 0.53789, 
    0.11937, 0.16705, 0.13187, 0.040203, 0.21118, 0.79387, 1.1892, 
    0.79066, 0.21556, 0.89389), V40 = c(0.011813, -0.027936, 
    0.03467, 0.0073339, 0.27916, 0.39529, 0.40199, 0.26977, 0.26342, 
    0.4253, 0.15937, 0.16025, 0.17229, 0.06882, 0.28367, 0.6541, 
    0.79066, 0.85596, 0.20311, 0.68227), V41 = c(-0.18098, 0.057321, 
    -0.16788, -0.13347, 0.1567, 0.076837, 0.061141, -0.0073689, 
    0.3219, 0.31003, 0.15798, 0.40938, 0.13877, 0.04331, 0.43956, 
    0.23566, 0.21556, 0.20311, 0.71678, 0.35077), V42 = c(-0.14357, 
    0.0088914, -0.11148, -0.10308, 0.2165, 0.425, 0.40689, 0.1654, 
    0.28324, 0.772, 0.18798, 0.27552, 0.175, 0.05934, 0.47978, 
    0.84715, 0.89389, 0.68227, 0.35077, 1.233)), class = "data.frame", row.names = c(NA, 
-20L))

Mon code est le suivant:

data <- read.csv("data.csv", header = FALSE)
eigen_fun <- function() {
  sigma1 <- as.matrix((data[,3:22]))
  sigma2 <- as.matrix((data[,23:42]))
  sample1 <- mvrnorm(n = 250, mu = as_vector(data[,1]), Sigma = sigma1)
  sample2 <- mvrnorm(n = 250, mu = as_vector(data[,2]), Sigma = sigma2)
  sampCombined <- rbind(sample1, sample2);
  covCombined <- cov(sampCombined);
  covCombinedPCA <- prcomp(sampCombined);
  eigenvalues <- covCombinedPCA$sdev^2;
}
mat <- replicate(50, eigen_fun())
colMeans(mat)
scree.plot(mat, title = "Scree Plot", type = "E", use = "complete.obs", simu = "F");

Je commence par générer 250 échantillons aléatoires à partir de la distribution normale $ N (\ mu_1, \ Sigma_1) $ et 250 échantillons à partir de la distribution normale $ N (\ mu_2, \ Sigma_2) $ . Je calcule ensuite la matrice de covariance de l'échantillon de ces 500 échantillons aléatoires, puis je trouve les valeurs propres de la matrice de covariance de l'échantillon.

Je répète ensuite cette partie pour avoir un total de 50 vecteurs de valeurs propres mat.

Je calcule ensuite le vecteur moyen des valeurs propres sur les 50 répétitions en utilisant colMeans(mat).

Enfin, je veux afficher les 50 vecteurs de valeurs propres et leur vecteur moyen dans un tracé de valeurs propres ou de scree. En dépit d'être un novice, en regardant cette intrigue, il me semble que quelque chose ne va pas? Ai-je fait quelque chose de mal?

ÉDITER:

Il y a aussi screeplots() https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/screeplot , mais je ne sais pas comment faire fonctionner celui-ci avec les 50 vecteurs de valeurs propres.

Réponses

1 StupidWolf Aug 17 2020 at 21:37

J'ai légèrement modifié votre fonction:

library(MASS)
eigen_fun <- function() {
  sigma1 <- as.matrix((data[,3:22]))
  sigma2 <- as.matrix((data[,23:42]))
  sample1 <- mvrnorm(n = 250, mu = data[,1], Sigma = sigma1)
  sample2 <- mvrnorm(n = 250, mu = data[,2], Sigma = sigma2)
  sampCombined <- rbind(sample1, sample2);
  covCombined <- cov(sampCombined);
  covCombinedPCA <- prcomp(sampCombined);
  eigenvalues <- covCombinedPCA$sdev^2;
}
set.seed(111)
mat <- replicate(50, eigen_fun())

Si vous regardez les dimensions de mat, chaque ligne est une valeur propre et chaque colonne est une réplique. Donc, la première ligne est votre 1ère valeur propre: