ANÁLISIS MULTIVARIABLE

CARGA DE DATOS Y LIBRERÍAS

CARGA DE LIBRERÍAS

library(readxl)
library(gt)
library(dplyr)
library(e1071)
library(htmltools)

CARGA DE DATOS

datos <- read_excel(
  "C:/Users/klaus/Downloads/SiO2_Al2O3_modelo_polinomico_cubico_2500.xlsx",
  sheet = 1
)

names(datos)
## [1] "SiO2"  "Al2O3"

DEFINICIÓN DE LA VARIABLE

Causa y efecto:

El contenido de SiO2 influye en la variación del contenido de Al2O3 dentro de las muestras geoquímicas. Esta relación no es completamente lineal, por lo que se propone un modelo polinómico cúbico para representar mejor el comportamiento entre ambas variables.

# Variable independiente (X): Dióxido de silicio

X <- datos$SiO2

# Variable dependiente (Y): Óxido de aluminio

Y <- datos$Al2O3

# Conversión a numérico

X <- as.numeric(X)

Y <- as.numeric(Y)

# Eliminar datos faltantes

TVP_original <- data.frame(X, Y)

TVP_original <- na.omit(TVP_original)

X <- TVP_original$X

Y <- TVP_original$Y

TABLA DE PARES DE VALORES ORIGINAL

TVP_original <- data.frame(X, Y)

n_original <- nrow(TVP_original)

cat("Tamaño muestral del modelo =", n_original)
## Tamaño muestral del modelo = 2500
# Tabla pares de valores original

tabla <- TVP_original %>%
  gt() %>%
  cols_align(
    align = "center",
    columns = everything()
  ) %>%
  fmt_number(
    columns = everything(),
    decimals = 2
  ) %>%
  tab_header(
    title = md("*Tabla N°1*"),
    subtitle = md("*Pares de valores original de SiO2 y Al2O3 en muestras geoquímicas de depósitos minerales*")
  ) %>%
  tab_source_note(
    source_note = md(paste0(
      "**Nota:** Tamaño muestral n = ", nrow(TVP_original),
      " observaciones utilizadas en el análisis."
    ))
  )

div(
  style = "height:400px; overflow-y:auto;",
  tabla
)
Tabla N°1
Pares de valores original de SiO2 y Al2O3 en muestras geoquímicas de depósitos minerales
X Y
45.51 15.41
63.14 16.36
75.25 13.69
66.48 15.67
54.93 15.27
66.88 16.24
46.32 14.72
49.37 14.66
66.06 16.48
53.99 14.40
69.29 16.33
46.85 13.59
45.00 14.77
66.58 17.58
65.22 16.07
53.87 15.01
72.12 15.18
52.24 15.44
66.44 17.47
50.60 14.98
62.21 16.60
69.66 15.65
60.98 15.85
54.03 15.00
63.30 16.02
75.38 14.02
57.19 15.13
74.76 14.13
50.49 15.10
46.33 14.67
62.17 16.58
48.08 13.52
69.68 15.95
60.85 15.31
46.90 14.83
57.78 15.81
62.39 16.39
51.47 14.40
48.87 14.36
58.35 16.79
69.09 16.43
75.04 14.67
68.70 15.37
64.65 16.39
59.49 15.97
67.25 15.64
59.35 16.46
76.47 13.46
59.82 15.86
67.94 16.30
65.79 15.99
60.24 15.24
58.40 15.81
63.47 17.01
74.23 15.08
50.15 14.14
59.00 15.41
61.09 15.50
66.28 15.55
62.09 16.60
45.45 14.98
62.03 16.29
71.48 15.79
59.73 15.37
57.81 15.67
70.98 15.07
68.63 17.13
60.06 15.84
60.81 15.52
74.10 14.13
63.99 16.19
71.97 16.43
58.08 15.63
72.21 15.75
55.86 14.79
49.29 15.16
69.00 15.39
51.69 14.95
50.52 14.35
66.09 16.39
70.44 15.89
53.79 14.89
64.67 16.22
50.41 14.90
61.92 16.30
72.19 15.64
52.68 14.84
70.31 15.64
55.31 15.06
54.99 15.29
70.21 15.44
60.11 16.85
55.33 15.78
64.89 15.92
55.19 15.38
60.88 15.57
62.06 15.90
62.39 16.71
57.77 16.23
49.44 14.46
66.79 17.11
72.06 15.32
61.44 16.15
50.61 15.22
48.52 15.11
62.33 16.06
69.22 16.16
71.87 15.25
57.40 15.44
62.49 16.38
64.76 15.99
62.63 16.54
52.20 14.48
65.42 15.02
60.72 15.80
53.11 15.34
66.54 15.57
67.32 15.50
55.52 16.02
66.34 16.30
71.77 14.92
54.15 15.33
71.83 14.96
68.71 15.46
66.15 16.19
45.78 15.22
75.34 14.15
49.21 13.97
63.19 16.02
48.45 14.58
57.27 14.41
64.80 16.21
69.37 15.60
52.90 14.98
65.91 16.48
66.11 16.32
57.97 16.07
71.96 15.14
63.08 16.45
48.31 14.84
57.69 14.69
56.26 15.55
45.58 14.97
69.61 15.46
58.26 15.70
70.59 15.16
58.00 15.96
68.23 15.90
48.38 14.89
65.05 16.34
52.63 14.84
50.30 14.50
66.53 15.87
60.64 16.59
57.27 16.44
65.08 16.16
51.72 15.13
45.94 14.94
51.46 14.77
50.26 14.63
45.00 14.86
68.05 15.50
60.06 15.77
57.29 15.71
52.76 14.53
48.65 14.97
50.67 14.18
63.04 16.42
65.87 16.02
69.16 15.83
71.96 15.59
64.02 16.18
61.66 15.84
53.26 14.44
50.67 14.40
62.00 15.84
64.25 16.29
53.20 15.17
73.46 15.22
62.56 15.54
65.09 16.81
73.49 14.69
58.02 15.39
61.87 15.73
62.71 16.93
47.99 14.71
50.48 14.77
49.03 15.54
58.05 14.90
51.70 14.84
68.05 16.24
69.65 15.23
72.43 14.72
45.00 14.94
63.17 16.54
56.32 14.71
55.24 15.67
59.39 15.77
69.73 15.20
66.16 16.84
78.00 11.91
56.79 15.90
63.82 16.39
53.19 14.90
60.30 15.79
47.65 14.63
61.28 15.66
60.78 14.82
69.23 15.60
75.49 13.12
51.36 15.44
75.79 13.81
72.97 15.12
63.64 16.55
58.82 15.95
72.61 14.64
61.30 15.84
60.17 15.64
55.61 16.32
58.92 16.24
60.34 16.24
54.24 14.97
71.99 15.76
54.50 15.10
50.01 14.23
72.51 15.15
58.69 16.39
52.70 14.87
55.97 15.19
50.44 13.01
63.47 16.40
68.53 15.68
61.97 16.39
65.57 16.26
69.54 15.71
72.76 14.80
48.37 14.24
67.69 16.22
61.47 15.38
62.65 16.11
67.39 15.98
72.83 15.24
53.03 14.21
57.28 15.90
53.55 14.19
55.38 15.71
68.31 15.52
51.53 14.77
62.79 16.49
52.38 14.86
72.84 14.11
55.60 15.57
65.81 16.76
63.48 16.48
74.48 14.71
64.76 16.10
58.17 15.54
75.59 14.59
48.67 14.89
66.12 16.42
61.32 15.59
68.81 15.34
61.81 16.48
53.84 16.21
71.49 15.23
62.31 15.76
50.33 15.69
57.30 15.91
66.93 16.19
54.21 15.43
66.24 16.64
60.78 15.92
70.19 14.97
50.35 14.57
71.22 16.23
64.93 16.64
65.83 15.76
65.58 16.33
52.30 14.37
46.72 14.42
72.69 15.42
67.39 16.31
64.77 15.83
71.84 15.07
55.45 15.53
64.24 16.28
45.00 14.49
64.42 15.77
48.95 14.85
59.35 15.39
61.80 16.36
61.46 16.06
63.42 16.16
59.58 15.61
58.15 15.79
62.01 15.92
58.89 16.29
78.00 13.22
62.90 16.11
73.19 14.71
68.84 16.03
46.51 14.88
69.14 16.29
53.89 15.88
62.39 15.19
66.67 15.69
64.11 15.41
72.25 15.43
69.65 15.31
70.70 15.24
57.02 16.15
52.33 15.01
63.10 16.76
61.13 15.15
45.21 14.78
65.98 16.65
62.02 16.42
57.39 15.12
48.03 14.31
64.45 17.04
45.00 14.00
50.36 14.62
56.58 15.47
68.22 16.32
66.22 16.55
76.72 13.29
65.55 16.49
45.59 14.68
68.37 15.56
70.90 15.64
73.82 14.70
60.20 16.18
50.28 14.73
51.95 15.01
52.20 14.41
69.73 16.60
54.34 15.16
56.46 16.02
76.28 13.38
45.98 15.25
58.18 16.13
50.70 14.21
54.74 14.89
50.78 14.85
69.17 15.85
61.92 15.78
62.72 16.13
71.54 15.07
61.52 15.70
73.74 15.26
58.77 15.63
56.58 15.29
53.10 14.49
71.73 15.06
57.52 15.25
74.23 14.36
50.44 14.69
60.84 16.34
54.03 14.27
61.03 15.90
63.19 16.35
61.85 15.52
69.43 15.72
56.23 14.63
61.36 15.39
46.27 14.26
45.00 14.52
70.91 15.87
58.86 15.31
58.29 15.78
54.26 14.73
77.51 12.88
70.32 15.23
50.55 14.52
68.17 15.94
70.26 16.32
60.77 16.08
61.90 15.83
71.99 15.00
51.90 14.25
66.63 15.66
47.50 14.49
65.26 15.45
48.94 14.65
61.63 15.81
75.36 13.37
70.53 16.12
63.38 16.36
59.07 15.37
45.00 14.57
66.08 16.47
70.08 14.86
57.98 15.50
63.98 16.03
54.36 14.85
68.69 15.81
72.05 14.96
56.59 15.43
46.88 14.88
56.71 15.23
67.72 15.84
57.63 15.65
62.59 16.94
67.33 16.22
68.74 16.15
59.22 15.17
72.71 14.51
51.78 15.07
64.75 16.19
47.69 14.46
63.48 15.89
46.94 14.43
48.25 14.66
45.00 14.61
59.42 16.65
71.02 14.96
69.38 15.66
60.92 15.80
70.56 15.27
72.10 15.00
60.46 15.51
58.15 16.11
62.72 16.67
69.73 15.55
60.10 15.89
73.01 14.34
68.00 15.97
66.27 16.27
65.36 16.90
57.25 15.86
65.21 17.12
72.69 15.35
52.96 14.83
67.34 15.75
70.64 15.59
51.63 15.23
71.02 15.32
47.67 13.98
63.84 16.21
58.59 15.56
69.92 16.08
67.66 16.00
56.43 15.55
69.99 16.22
67.19 15.48
53.89 15.46
51.57 14.92
58.61 15.39
60.98 16.25
60.01 16.41
48.64 14.75
65.34 16.23
58.92 14.63
61.59 16.24
59.85 15.78
48.50 14.46
76.63 13.07
63.96 15.47
63.16 15.77
59.39 15.57
65.88 15.88
51.24 14.06
68.55 16.25
76.35 13.82
48.63 15.67
54.67 15.19
57.51 15.21
62.54 15.95
75.13 14.18
47.23 14.28
47.76 15.36
56.82 15.26
75.88 13.19
62.38 16.68
73.27 15.15
51.25 15.37
63.25 16.66
55.46 14.96
67.03 15.24
63.34 16.72
55.50 15.23
50.08 14.68
63.00 15.66
50.96 14.31
65.76 16.17
55.32 15.41
49.59 14.90
59.11 15.86
60.61 15.40
52.78 15.46
53.68 14.44
65.01 16.02
69.21 17.09
46.70 14.44
52.29 14.71
59.76 15.83
64.37 15.85
61.03 15.32
64.03 16.07
49.87 15.05
65.05 16.37
57.64 16.18
57.40 14.55
66.27 16.08
70.22 15.20
46.71 14.44
47.52 14.38
68.40 16.40
57.28 15.16
58.88 16.16
57.41 15.21
53.31 15.17
63.10 16.09
70.83 15.68
68.82 16.55
67.66 15.31
63.36 16.30
60.60 15.91
62.28 15.91
71.49 15.66
61.56 16.73
64.49 15.91
64.80 15.92
50.12 15.14
48.73 14.99
49.63 14.39
67.65 15.80
75.69 14.43
59.13 15.87
47.53 13.91
65.73 16.30
55.06 15.63
68.30 15.84
66.55 15.79
60.78 15.86
48.46 14.35
70.46 15.44
58.37 15.58
56.71 15.67
51.31 14.83
50.23 14.87
60.81 15.56
47.08 14.37
51.60 14.95
58.34 15.35
52.75 15.38
45.00 14.71
64.73 16.04
72.21 14.73
62.43 15.59
52.90 14.31
74.71 14.17
70.21 16.43
59.60 16.41
70.12 15.38
62.49 15.78
65.25 16.58
64.36 16.08
53.08 15.32
54.25 14.37
52.22 15.02
73.41 15.13
65.03 16.93
58.77 15.27
73.89 14.17
73.50 14.91
77.13 12.78
64.39 16.37
69.86 15.31
63.32 15.36
52.92 14.92
48.55 14.03
49.98 14.12
59.81 15.31
60.06 15.91
49.78 14.70
61.68 15.51
51.71 13.88
73.00 14.38
57.26 15.75
53.41 15.59
61.36 16.20
68.28 15.86
65.47 15.17
54.61 14.31
75.88 13.13
50.04 14.34
63.23 15.84
51.24 14.70
72.80 14.60
66.74 16.65
63.72 15.70
56.79 15.21
51.26 15.00
49.98 14.51
52.01 15.53
66.90 15.84
52.07 14.57
65.22 15.18
73.59 14.99
53.17 14.20
59.67 14.90
62.31 16.72
63.01 15.80
51.64 14.14
49.94 14.83
62.21 15.82
74.28 14.66
52.01 15.07
59.05 16.12
56.38 15.87
59.95 16.49
47.57 14.83
78.00 12.36
62.19 16.29
70.37 16.00
55.98 14.72
71.09 15.79
63.14 16.19
57.12 15.89
65.91 15.63
65.02 15.63
74.95 14.16
62.77 15.97
70.05 15.27
57.34 14.92
60.70 16.17
74.17 14.48
60.76 15.73
67.11 16.39
62.17 16.26
68.06 16.38
62.24 16.39
60.06 16.26
62.94 15.52
60.38 14.93
46.06 13.86
60.64 15.57
47.04 14.26
63.64 15.39
52.15 15.23
50.54 15.41
69.84 15.15
68.94 16.26
67.79 15.19
46.30 14.70
75.11 15.09
59.82 15.83
53.21 14.61
64.66 16.28
53.16 14.61
49.66 14.03
52.19 14.90
63.66 16.90
49.89 15.33
50.68 14.42
68.72 16.69
62.64 16.35
71.04 15.66
47.94 14.76
46.60 14.56
66.86 16.32
70.95 14.47
65.27 15.39
62.17 16.60
74.54 14.80
45.19 14.56
72.65 15.47
72.94 14.52
61.23 15.59
60.24 15.55
69.15 16.35
53.12 14.13
46.94 15.26
67.84 16.11
49.91 14.29
61.14 16.17
73.35 15.18
53.59 14.94
53.83 15.25
60.40 15.89
77.10 12.09
53.60 15.45
75.07 13.97
78.00 12.97
67.15 16.64
59.96 15.63
48.55 14.39
45.56 15.11
71.28 16.26
48.96 14.75
65.36 15.41
63.57 15.74
70.32 15.66
50.64 15.33
69.50 15.92
64.53 16.20
73.67 14.92
66.11 15.12
55.71 16.12
59.03 14.92
61.14 16.13
53.51 15.04
56.60 15.70
52.61 15.39
56.06 15.82
49.84 14.63
71.41 15.60
53.26 15.42
50.18 14.53
62.57 15.41
65.35 16.31
51.76 14.41
66.54 17.09
49.28 14.28
76.82 13.41
71.21 15.31
51.80 14.36
58.67 15.41
70.74 15.85
72.00 15.84
58.66 15.77
73.66 15.12
58.55 15.78
61.57 15.97
56.14 15.17
59.75 15.61
53.51 15.79
65.81 16.51
65.27 16.08
60.87 16.01
47.99 13.97
61.06 15.93
54.79 15.12
56.05 14.78
64.99 15.41
68.04 15.61
56.92 15.87
77.74 11.33
47.81 14.54
72.59 15.43
71.91 15.92
67.92 16.22
68.54 15.69
61.83 16.57
63.34 16.08
52.57 14.79
66.10 16.32
58.51 15.45
47.48 14.87
70.38 16.23
63.93 16.52
64.92 16.41
68.12 15.84
47.58 14.44
65.19 15.68
55.62 15.04
60.14 16.20
60.11 15.87
51.02 14.29
57.92 16.18
54.10 15.45
64.31 16.00
69.78 16.38
62.50 15.63
64.66 15.38
49.28 14.39
58.53 15.56
48.56 14.14
59.25 15.95
51.16 14.36
55.68 15.26
50.73 14.75
57.21 14.86
69.15 16.54
64.85 16.39
57.01 15.33
48.14 14.40
58.84 15.42
49.67 14.94
56.86 15.29
57.90 14.53
71.48 15.53
51.39 14.75
58.62 15.35
60.88 15.97
78.00 12.38
53.94 14.91
71.79 15.42
62.31 16.19
66.61 16.48
71.81 15.93
54.73 14.99
57.63 15.62
58.06 15.33
63.77 16.87
74.07 14.93
51.95 14.74
62.62 15.53
65.74 15.99
71.55 15.21
57.79 15.72
71.69 16.07
69.42 15.71
73.41 14.47
47.75 14.65
73.46 14.52
72.00 15.70
54.65 14.64
64.02 16.11
63.14 15.31
66.53 15.94
69.32 16.46
63.31 16.18
59.27 15.69
68.64 15.72
60.78 15.75
71.29 15.43
61.82 15.30
58.98 14.54
51.37 14.03
52.64 15.20
56.18 16.14
70.65 15.65
59.04 15.55
59.78 15.12
62.67 16.28
61.75 15.44
64.46 16.27
72.13 15.07
57.47 15.05
64.43 15.52
52.97 15.06
68.42 15.69
65.87 16.07
71.32 15.15
63.48 16.62
73.23 14.74
62.84 15.69
74.38 14.72
65.02 15.88
59.27 16.27
49.83 14.57
62.83 15.94
62.78 16.49
56.61 15.22
47.09 14.15
62.96 16.62
77.18 12.52
72.78 15.06
71.98 15.96
66.32 16.22
48.05 13.86
67.54 16.56
48.14 14.40
76.24 13.31
62.94 15.60
70.92 15.24
57.42 16.16
52.05 15.10
54.79 14.39
77.12 12.59
66.48 16.62
63.68 15.55
70.28 15.50
52.80 15.31
71.38 13.91
66.21 16.44
58.94 14.92
66.14 15.85
56.74 14.99
53.27 15.54
60.79 15.83
46.66 14.60
50.59 14.61
72.91 14.99
55.61 14.72
60.16 15.83
64.41 15.88
73.63 15.04
51.62 15.55
68.49 16.22
58.27 15.84
51.10 15.18
68.13 15.99
59.61 15.70
58.42 15.79
50.48 15.05
63.46 15.98
66.14 16.93
60.86 15.81
45.00 15.87
51.44 14.06
54.63 15.42
45.11 14.12
56.32 15.74
72.59 15.43
61.85 16.15
62.05 16.45
74.16 14.41
60.12 16.03
70.68 15.87
54.82 14.62
73.38 14.38
58.94 15.66
73.00 15.25
70.06 15.56
63.63 16.26
63.98 16.23
51.43 14.79
76.83 14.04
56.02 14.89
49.05 14.66
62.04 15.15
61.21 16.09
51.86 14.45
61.89 16.46
59.94 16.02
59.86 15.72
62.91 14.85
46.65 14.36
73.16 14.95
71.19 15.33
63.85 16.80
48.36 14.89
63.41 15.84
61.85 16.10
51.90 15.16
59.08 15.69
69.91 16.29
63.30 16.45
63.56 16.04
73.77 15.05
61.26 15.92
71.79 15.63
50.50 13.58
63.33 16.03
47.21 14.10
72.68 14.47
65.56 15.99
52.97 15.30
46.70 13.97
55.37 14.44
51.95 14.08
54.18 14.92
59.30 15.55
70.46 15.25
72.94 14.94
71.36 15.47
74.32 14.80
60.82 15.08
71.29 15.13
58.07 16.00
60.60 16.28
61.58 16.28
66.49 16.50
48.14 14.83
65.22 15.66
67.85 16.13
56.51 15.64
57.80 15.53
71.93 15.25
48.51 14.54
62.45 16.02
49.58 13.95
50.90 14.37
70.65 15.88
70.49 15.29
61.92 17.02
62.13 16.56
62.11 16.11
71.47 15.73
73.36 15.03
74.03 14.28
73.66 14.05
59.97 15.77
74.97 13.50
71.92 15.33
57.44 15.24
47.09 14.11
49.81 13.98
62.17 16.13
65.14 16.51
61.02 15.61
47.85 14.33
66.40 15.60
54.34 14.99
64.94 16.50
49.75 15.36
52.24 14.34
73.91 14.43
65.88 15.43
69.78 16.01
47.49 14.87
67.68 16.32
51.51 14.53
61.66 16.35
69.20 15.28
55.98 15.83
48.91 14.38
49.91 14.43
64.63 16.33
66.37 16.04
54.96 14.81
69.90 15.09
62.26 15.84
68.43 15.96
58.95 15.21
49.66 14.89
73.14 14.41
53.58 14.95
55.52 15.70
65.52 15.73
59.64 15.75
55.94 14.48
63.73 15.72
60.82 15.49
53.43 14.07
71.96 15.75
67.70 16.01
48.85 14.78
52.84 15.00
72.00 15.48
59.97 15.47
58.45 15.70
75.35 13.63
57.90 15.72
65.23 16.33
59.86 15.57
55.14 15.18
60.57 15.83
63.00 17.03
68.94 16.84
52.58 15.31
56.03 15.54
56.58 15.11
60.81 15.83
48.47 14.27
71.59 15.01
47.54 14.37
45.52 15.03
67.84 15.89
63.24 15.81
73.75 14.18
49.94 14.94
49.49 14.33
66.57 15.98
63.52 16.16
69.78 16.01
66.61 15.62
53.90 15.53
65.88 15.11
69.01 15.69
54.70 14.69
66.99 16.24
63.75 16.02
67.12 16.20
57.54 15.68
73.31 14.93
70.27 15.80
52.13 15.24
53.99 15.04
63.18 15.80
45.72 15.33
49.97 14.17
49.55 13.65
66.23 16.64
62.80 16.23
49.00 14.62
55.50 15.14
54.39 14.83
62.46 16.38
62.70 16.06
57.58 15.54
73.04 15.17
50.94 15.31
50.92 13.96
52.73 15.17
68.29 15.74
63.78 16.07
70.41 15.11
72.41 14.62
51.98 15.93
63.09 15.93
47.06 15.51
70.28 15.91
63.80 16.30
62.56 16.00
73.86 14.15
45.00 14.40
72.03 15.77
47.59 13.83
50.46 14.53
48.96 14.37
52.45 14.67
48.64 14.41
71.65 15.70
47.21 14.37
68.73 15.59
45.00 14.66
49.28 13.93
58.90 16.39
70.33 15.49
74.63 14.02
76.55 12.77
70.80 15.31
67.35 15.92
50.36 13.70
58.58 15.46
66.45 15.86
59.20 15.96
57.51 15.10
54.38 14.08
61.10 16.22
73.75 15.26
75.42 13.59
58.29 15.18
73.97 15.13
51.68 15.25
61.02 15.95
50.69 14.89
48.04 15.40
75.65 14.06
59.41 15.46
61.39 15.86
75.79 13.88
52.79 14.43
68.65 17.08
66.50 15.90
60.80 16.53
67.39 15.46
51.10 14.14
71.96 13.83
66.57 16.21
49.50 15.19
55.80 15.31
77.43 13.82
65.04 16.52
73.14 15.45
70.09 15.97
56.58 15.09
73.83 13.96
60.13 16.35
69.51 15.61
65.80 15.51
60.85 16.33
51.03 14.65
61.12 15.90
69.12 16.01
71.79 15.25
59.66 14.81
50.93 15.17
71.26 16.23
59.18 16.39
54.15 14.67
72.87 15.49
55.24 15.33
49.24 14.90
71.58 15.85
49.92 13.94
58.64 16.51
67.74 16.64
46.70 14.27
64.36 16.35
75.59 13.36
60.90 15.54
61.15 15.41
54.09 15.25
46.53 13.39
47.59 14.82
62.52 16.26
50.89 15.37
74.51 13.97
51.59 14.28
65.68 16.04
71.68 15.19
49.03 14.78
67.95 15.81
69.68 15.47
57.25 15.64
56.23 15.75
74.55 13.88
53.75 14.90
62.18 15.33
61.11 16.08
72.02 15.85
54.20 14.84
63.97 15.86
70.40 16.55
59.79 15.98
61.97 15.33
48.02 14.14
59.39 14.84
50.46 14.45
65.42 15.80
60.82 16.10
56.19 15.74
61.63 16.22
59.62 16.08
56.01 15.64
74.53 14.18
64.67 15.79
70.84 16.05
49.44 14.26
51.53 15.41
74.26 13.60
65.71 15.54
51.44 14.35
57.42 16.41
45.70 14.05
74.95 14.10
55.36 15.25
64.65 16.02
63.70 15.79
69.18 16.03
65.10 15.61
62.38 15.43
58.35 15.76
67.32 15.89
63.28 16.74
45.00 15.09
57.40 14.74
61.48 15.45
60.37 14.90
55.93 15.06
64.95 15.39
69.97 15.51
61.54 15.94
61.88 15.74
74.23 13.82
66.86 15.81
53.31 15.35
49.35 14.39
71.34 15.67
53.89 14.59
55.92 15.34
52.78 14.82
72.60 15.00
66.63 16.13
71.30 14.49
55.32 15.31
45.00 13.85
57.00 14.99
60.72 15.63
58.33 15.83
46.10 14.95
45.94 14.55
70.84 16.21
74.47 14.38
50.74 14.10
77.56 12.00
69.07 15.59
69.52 16.06
60.05 16.51
70.25 15.28
57.84 14.70
46.47 14.72
66.77 16.45
49.97 14.44
60.33 15.98
50.95 15.06
56.52 15.20
53.60 14.73
70.39 15.18
58.26 14.97
60.02 15.23
63.21 15.82
60.54 15.38
45.81 15.28
56.10 15.59
54.28 14.67
60.15 15.99
46.39 14.12
50.69 15.16
49.69 14.53
52.03 14.94
78.00 12.85
71.16 16.47
55.30 15.40
53.66 15.60
60.49 15.96
61.78 15.73
61.15 15.84
66.10 16.20
64.62 15.71
64.02 16.77
62.73 15.52
69.43 16.12
66.18 15.95
47.86 14.10
66.30 16.38
58.94 16.13
58.05 14.69
61.75 16.36
63.85 16.32
71.00 15.89
72.30 15.47
59.86 15.76
51.66 14.23
61.22 16.07
70.15 15.88
61.03 16.71
62.44 16.35
68.92 15.71
70.08 15.45
64.84 17.00
63.72 16.68
58.77 15.86
61.29 15.78
53.15 15.32
67.23 16.60
52.85 14.56
62.88 16.39
70.92 15.83
54.63 14.98
64.32 15.96
58.37 15.56
46.71 13.92
52.97 15.93
50.94 14.52
71.63 15.54
49.95 14.07
62.29 16.85
58.95 15.26
63.41 16.11
60.17 16.75
68.76 15.40
57.23 15.39
70.73 16.03
51.46 15.12
52.72 14.32
50.16 15.55
65.38 16.67
57.66 15.87
56.50 16.28
52.30 15.27
52.43 14.50
66.92 15.68
59.35 15.32
74.41 14.50
51.43 14.82
45.00 14.37
57.84 14.18
63.88 16.16
58.87 15.56
51.06 14.69
73.32 14.51
53.18 14.87
62.48 16.00
71.60 14.23
50.14 14.68
59.17 15.60
51.92 14.32
54.75 15.31
59.80 15.26
51.52 15.19
68.21 15.38
70.00 15.58
59.83 14.89
71.66 14.11
67.04 16.39
51.17 14.39
62.06 15.64
60.07 15.83
52.93 15.07
61.65 15.57
48.47 13.79
53.43 15.09
56.10 15.88
45.00 14.19
52.70 15.33
60.47 15.32
69.62 15.00
74.63 14.70
64.82 16.39
61.59 15.94
52.94 14.24
63.43 16.35
61.51 16.17
56.99 15.13
73.53 15.58
71.74 15.16
46.01 14.90
53.40 15.30
64.86 16.03
63.25 16.87
51.23 14.82
52.86 14.49
62.42 15.45
59.80 15.58
53.22 15.56
51.80 14.72
77.58 12.21
67.61 15.87
53.03 14.73
64.09 15.74
60.63 16.20
61.93 15.95
65.33 16.22
64.35 16.29
72.65 15.08
62.89 16.32
68.54 15.87
49.80 15.13
63.23 16.23
57.33 16.01
62.96 16.21
63.34 16.16
58.59 16.18
74.04 13.93
61.45 16.96
72.78 14.62
63.67 15.98
72.10 15.05
71.24 15.65
75.42 14.11
59.83 16.29
48.93 14.27
68.85 15.77
64.50 16.16
45.51 15.25
45.00 14.65
49.25 14.19
73.91 14.53
70.66 15.79
63.74 15.68
53.43 15.26
57.37 15.53
56.43 14.65
69.02 16.80
46.90 14.95
57.22 15.22
63.71 16.96
59.80 16.20
63.74 16.10
58.58 15.08
54.79 14.35
63.73 16.28
53.11 15.00
58.00 15.87
68.99 15.65
70.55 14.44
71.52 15.54
66.58 15.62
55.97 15.12
69.07 15.37
70.33 16.06
60.87 16.43
49.41 15.05
64.26 16.11
46.91 14.53
59.28 16.64
69.38 15.48
53.09 15.07
55.09 15.21
69.45 15.91
52.68 14.69
63.65 15.80
62.77 15.39
69.39 16.63
71.02 15.68
51.75 14.98
49.39 14.69
60.69 15.39
65.78 16.94
69.68 16.60
64.16 15.75
59.48 16.33
67.77 16.74
50.95 14.88
70.58 15.61
62.07 16.13
51.33 14.88
69.60 16.59
66.73 15.93
63.77 15.22
59.79 15.27
53.01 14.87
72.11 15.35
50.94 14.44
47.87 14.45
49.77 14.96
57.77 15.97
45.90 14.58
65.62 15.98
59.96 15.46
48.87 14.68
61.23 16.44
58.65 16.51
63.93 15.62
71.13 14.49
64.23 16.62
73.39 14.76
64.92 16.28
61.59 16.66
74.68 14.57
48.06 14.94
57.09 15.69
71.75 15.86
55.90 14.55
47.85 14.45
56.15 15.58
60.57 16.53
46.05 14.43
65.45 16.07
45.54 14.77
68.68 15.78
74.76 14.73
57.32 15.35
61.05 16.19
71.11 15.37
50.87 14.07
63.25 15.81
54.64 16.00
55.27 14.54
49.16 14.85
69.00 15.86
51.71 14.76
54.53 15.32
66.98 15.89
59.52 15.48
72.01 15.25
69.44 15.39
51.34 14.73
64.58 16.72
63.92 16.05
60.34 15.72
59.17 15.58
52.96 15.35
67.07 16.36
72.58 15.19
60.29 15.83
68.57 15.60
65.98 16.18
59.04 15.67
54.85 15.11
46.60 13.63
69.02 15.77
63.69 15.79
68.92 15.46
70.59 15.91
65.64 16.29
77.79 12.69
50.37 13.95
73.21 14.23
60.60 16.22
67.14 16.10
70.43 15.29
62.36 16.57
71.52 15.05
67.88 16.66
62.32 15.88
74.75 14.13
69.74 15.20
63.15 16.67
61.90 15.29
71.56 15.81
74.29 14.17
47.63 14.40
51.32 14.80
64.09 16.56
46.86 14.95
72.34 15.45
54.67 15.13
51.69 15.29
57.54 14.95
63.49 15.68
58.33 15.35
74.15 14.54
52.56 14.52
71.12 15.51
57.91 15.78
66.43 16.15
49.54 14.84
48.86 14.18
57.24 15.44
70.48 15.79
48.68 13.60
59.40 16.08
55.86 15.32
55.71 14.97
72.12 15.22
68.84 16.48
63.71 15.98
48.09 14.13
75.63 13.94
64.24 16.43
76.23 13.10
66.15 16.41
50.51 15.64
67.51 15.99
70.39 15.47
66.95 16.08
64.12 16.37
51.30 14.80
58.95 15.56
64.55 15.50
47.97 14.76
62.36 16.23
50.09 15.04
71.21 14.94
47.72 13.93
64.94 15.41
52.33 14.49
67.31 16.29
51.16 13.73
67.11 16.19
72.16 14.56
69.40 16.36
71.69 15.30
74.93 14.23
58.90 15.25
53.01 15.34
69.50 16.02
55.33 14.72
67.64 16.30
62.65 16.58
49.56 14.74
58.90 16.03
72.97 14.72
66.59 16.50
47.91 14.34
64.29 16.54
54.50 14.94
53.76 15.01
73.92 13.81
54.83 15.07
69.89 15.84
71.07 15.52
74.17 13.79
62.96 16.72
58.92 15.86
51.30 14.16
67.52 16.76
67.99 16.67
50.04 13.90
66.47 16.39
66.81 16.06
47.17 15.08
63.58 16.24
56.46 15.27
58.75 14.93
57.15 15.93
76.44 13.16
52.68 14.87
56.94 16.04
75.32 14.17
49.09 15.09
73.41 15.16
56.50 14.06
67.65 16.31
50.77 15.38
62.02 15.60
59.13 15.12
49.89 15.24
73.82 14.88
71.58 15.50
65.51 15.85
47.10 14.49
68.42 16.14
58.94 16.25
61.63 16.18
74.31 15.16
52.51 13.87
59.98 16.08
47.55 14.98
59.68 16.07
56.46 15.65
58.69 15.27
66.28 16.22
59.71 15.69
57.70 15.48
56.19 14.64
45.05 13.77
69.88 15.55
53.84 14.72
70.19 16.01
52.57 14.74
70.13 15.46
73.15 15.16
50.22 14.16
69.93 15.82
55.72 15.45
71.10 15.41
58.44 15.18
59.45 14.28
73.43 14.72
45.00 14.09
71.69 15.90
52.74 14.81
64.53 16.40
47.31 14.29
54.69 14.74
62.16 16.14
51.32 14.24
57.47 15.63
61.32 15.61
59.32 14.89
57.62 16.14
51.11 14.09
65.14 15.99
58.12 15.96
62.99 16.03
54.68 15.25
70.71 15.12
60.91 15.68
60.58 15.27
70.32 15.48
70.35 15.19
56.64 14.93
59.26 15.90
70.57 15.70
50.59 13.94
70.05 15.55
51.57 15.02
46.85 14.53
62.82 16.24
62.73 16.31
58.55 16.04
69.76 15.58
76.56 13.51
65.48 16.14
75.18 14.10
74.19 14.40
63.12 16.66
50.67 16.20
60.76 14.74
58.67 15.83
64.91 15.73
71.06 15.17
54.06 14.80
74.41 15.32
69.67 16.21
60.95 15.90
66.56 15.41
76.12 12.43
61.57 15.78
69.21 16.29
45.00 15.02
61.99 16.26
62.56 15.86
56.50 15.45
67.13 16.33
55.12 14.84
46.39 14.87
77.48 12.60
69.65 15.33
51.18 14.20
58.25 15.38
53.66 14.55
52.01 15.45
65.46 16.28
66.29 16.23
67.24 16.11
68.06 16.24
48.74 14.39
66.96 15.78
69.70 16.25
53.03 15.48
74.69 14.84
61.23 15.21
51.97 15.40
67.00 16.48
52.57 15.53
58.02 14.66
73.35 15.80
51.07 14.52
68.83 16.13
68.70 16.30
71.56 15.50
46.44 14.98
48.85 15.36
49.89 14.69
73.50 14.84
49.20 15.09
58.34 15.81
52.92 14.60
53.19 15.30
54.18 15.27
55.26 15.10
71.54 15.45
59.85 15.65
66.30 16.28
73.10 15.04
65.59 16.17
61.18 16.08
63.02 15.64
68.47 16.20
51.63 15.38
51.64 15.56
61.52 15.92
68.81 15.85
64.96 16.28
72.65 15.07
66.69 17.16
62.42 16.01
72.85 14.86
60.16 15.80
64.19 15.94
78.00 12.92
53.90 14.89
61.30 16.34
52.22 14.57
71.96 15.41
45.00 14.68
45.27 14.22
61.09 16.19
49.59 13.80
49.19 14.30
61.00 16.77
55.84 15.15
65.44 15.86
47.95 14.72
61.63 15.91
66.71 15.92
66.25 16.07
46.55 14.57
60.18 15.42
64.84 16.66
68.73 16.64
59.36 15.52
73.69 15.08
73.48 15.00
72.91 14.89
49.25 14.53
51.17 14.00
59.64 15.89
71.86 15.08
58.30 15.02
69.24 15.62
56.25 14.87
53.87 15.30
54.28 14.50
60.78 17.06
65.50 16.20
67.91 16.15
67.62 16.60
47.69 14.69
46.47 14.32
60.98 16.17
77.17 12.51
51.75 14.64
76.00 13.90
49.13 14.40
65.34 16.05
63.42 16.39
60.22 15.70
60.88 16.38
53.10 14.91
53.61 15.30
61.26 16.32
56.09 14.47
62.54 15.90
72.46 15.33
59.71 15.97
50.75 14.92
75.83 13.95
71.92 15.51
62.77 15.81
68.66 16.84
63.45 16.04
66.35 15.70
59.02 15.74
73.33 15.15
64.12 16.44
61.75 16.17
69.93 16.02
66.21 15.67
58.00 14.39
56.28 15.82
45.47 14.63
49.73 14.57
50.95 14.20
49.37 14.72
72.29 14.84
77.51 12.61
64.27 15.57
60.47 16.11
70.38 15.54
58.05 15.90
50.89 13.82
74.25 15.15
65.75 15.78
65.08 16.34
53.17 14.78
64.41 15.98
46.04 14.47
52.77 15.04
72.39 15.00
62.51 15.92
66.77 17.12
52.92 14.27
73.99 14.01
59.02 16.05
67.84 16.57
48.69 14.52
64.53 16.78
57.68 15.70
45.60 14.51
68.04 15.59
47.66 14.48
49.76 14.91
56.05 15.55
61.34 16.07
57.09 15.82
78.00 13.30
61.53 16.16
55.01 15.29
45.00 14.26
63.08 15.66
48.85 13.76
48.74 15.17
72.90 15.20
54.74 14.83
61.80 15.81
47.63 14.24
62.15 16.17
45.00 13.86
49.49 14.07
48.68 14.88
58.29 16.12
64.53 15.70
63.83 15.79
62.30 15.79
49.90 14.65
66.48 15.87
67.85 15.67
68.98 15.32
63.93 16.01
49.76 14.09
74.09 14.99
51.95 14.28
74.15 15.15
68.73 16.45
65.97 16.78
70.88 15.27
72.36 15.19
62.07 16.71
61.35 15.79
55.44 15.09
65.41 16.15
68.74 15.26
68.94 16.48
65.58 16.36
63.32 15.46
62.65 16.07
72.64 15.73
64.46 16.34
52.32 14.86
54.01 15.23
75.23 14.63
45.17 14.42
53.02 15.91
65.73 16.14
60.76 16.32
64.59 16.08
50.74 13.92
63.55 16.45
55.37 14.82
62.93 15.33
60.06 16.70
61.20 15.66
69.89 15.69
56.52 15.29
50.46 14.91
66.57 16.13
53.22 15.05
71.74 15.86
62.02 17.23
70.65 15.81
61.07 15.59
70.16 15.90
67.82 16.21
73.77 14.51
60.74 16.67
65.09 16.34
60.09 15.11
60.92 15.53
45.21 14.88
47.95 14.27
56.40 15.16
47.94 14.76
58.95 16.10
69.51 16.45
52.13 14.49
50.65 14.45
69.94 15.75
69.77 16.48
71.51 15.54
48.57 14.90
48.41 14.60
63.98 16.49
49.08 14.84
61.38 16.86
55.86 15.27
60.90 16.44
52.04 15.26
61.49 15.52
73.06 14.85
70.85 15.07
49.41 13.84
67.30 16.50
67.22 16.31
55.55 14.78
60.02 16.01
71.97 15.78
54.03 15.66
71.32 15.25
59.04 15.68
65.54 16.28
52.18 15.45
55.30 15.71
73.29 15.11
69.05 15.31
55.30 15.25
52.24 15.87
74.52 14.29
62.50 16.38
59.37 15.06
54.33 14.93
64.05 16.24
67.30 15.80
73.73 14.22
58.62 15.79
61.68 15.60
67.46 16.40
70.78 15.56
47.66 15.00
70.58 15.66
50.57 14.98
57.28 14.74
45.00 14.34
73.42 14.57
69.37 16.04
66.09 16.50
54.06 15.09
63.68 15.82
65.55 16.13
67.06 16.50
67.78 15.44
78.00 12.29
70.62 16.30
50.59 14.77
51.48 15.08
68.03 16.35
68.25 16.26
47.75 15.78
49.44 14.47
57.47 15.66
47.34 14.43
57.35 15.09
61.05 16.14
78.00 12.00
71.21 14.72
62.57 16.50
50.44 14.14
70.38 16.56
61.91 15.16
60.17 15.35
56.75 14.80
49.21 14.76
71.19 15.97
63.75 16.81
61.12 15.91
65.27 17.58
63.94 16.53
67.27 15.31
48.48 14.14
47.35 15.19
61.76 15.61
56.33 15.37
45.54 14.49
47.67 14.97
60.04 15.65
54.47 14.31
67.10 15.59
45.91 14.24
68.75 16.14
54.98 15.32
64.33 15.56
57.02 14.78
64.29 16.11
67.55 15.80
60.72 16.04
59.54 15.75
65.26 15.99
65.25 16.03
61.21 16.10
76.12 13.63
72.86 15.29
63.19 15.33
62.57 17.64
59.49 15.54
63.78 16.29
70.70 15.95
71.55 15.54
57.60 15.09
45.00 15.25
64.19 15.98
55.82 15.54
55.43 14.48
58.76 15.86
70.66 15.44
54.09 15.16
47.46 14.34
72.40 14.59
64.58 16.99
71.98 15.21
63.32 16.10
62.66 15.80
66.02 15.83
49.29 15.57
61.72 15.87
64.94 15.13
70.15 15.95
57.13 14.72
74.44 14.30
62.26 15.78
47.16 14.63
51.54 14.26
51.51 15.20
55.02 14.92
59.93 16.47
56.31 14.85
57.45 15.26
52.63 14.90
68.73 15.51
51.01 14.60
71.70 15.32
74.72 14.08
47.28 14.53
61.32 15.95
66.14 15.46
45.00 14.22
69.12 14.52
49.42 13.85
53.55 14.69
64.82 16.05
58.47 14.89
66.95 16.21
61.72 15.99
61.21 15.87
62.86 16.23
68.82 15.91
59.21 15.82
56.24 14.96
73.29 14.47
65.50 16.18
64.93 15.77
59.86 15.86
66.46 15.70
65.92 16.32
66.16 15.71
63.91 16.69
52.38 14.61
61.94 16.47
65.09 15.77
62.92 16.07
66.07 16.95
61.54 15.92
58.72 15.88
62.66 16.15
51.80 14.73
53.71 15.34
72.01 15.50
71.40 14.72
63.68 16.36
56.33 16.05
71.50 15.53
59.51 16.29
45.56 14.80
53.91 14.75
63.17 15.82
58.46 15.69
70.65 15.28
59.73 15.73
60.94 15.06
66.35 16.43
45.86 13.75
52.19 15.45
71.81 15.15
61.78 16.56
62.22 15.70
68.41 15.50
61.18 16.18
48.17 14.39
70.79 16.18
70.72 15.08
74.30 14.47
69.28 15.98
49.39 14.96
58.03 15.54
62.44 15.94
50.31 14.24
72.20 15.52
70.00 16.56
62.64 15.90
61.81 16.56
49.85 14.47
49.16 14.24
51.03 15.41
54.93 14.71
60.51 16.96
52.40 15.07
47.54 14.55
45.00 14.02
55.85 15.54
63.99 15.96
57.14 14.68
71.10 15.70
54.40 15.04
75.06 13.68
69.19 16.26
53.37 14.39
53.51 14.98
71.90 15.10
66.94 16.69
64.51 16.18
56.68 14.50
58.78 15.30
70.39 16.05
57.49 15.85
45.00 14.15
45.39 14.46
58.41 14.93
63.15 16.48
70.37 15.89
57.85 15.96
64.29 16.08
59.29 15.74
53.35 15.73
61.34 16.36
54.74 14.79
63.67 15.68
57.95 15.39
63.18 15.52
71.99 15.02
64.39 15.91
45.00 14.78
56.47 15.79
72.49 15.28
66.86 15.44
49.16 14.05
47.58 14.61
46.19 14.95
68.70 15.82
75.79 14.10
68.64 15.33
53.68 15.82
57.83 15.66
69.72 15.63
46.69 14.26
54.14 14.57
50.90 14.48
58.69 16.00
51.78 14.98
49.09 14.78
57.32 14.96
63.01 15.76
51.36 14.51
56.50 15.51
59.51 15.56
74.45 14.93
45.86 15.10
60.63 16.78
59.15 15.94
65.45 15.92
66.62 16.12
59.70 16.23
51.78 15.00
64.09 16.44
50.86 15.84
71.22 15.09
48.31 15.60
73.79 15.08
58.41 15.65
65.92 16.41
67.08 15.69
62.12 16.19
67.55 15.89
67.62 16.40
67.02 16.08
59.67 16.35
48.88 15.48
59.07 16.08
52.54 14.30
72.67 15.24
48.40 14.75
65.65 16.43
68.39 16.18
71.22 15.65
45.00 15.05
50.28 14.13
58.80 15.45
72.25 15.30
67.55 16.15
57.21 15.77
73.28 14.91
45.00 14.27
71.30 15.13
67.17 16.79
62.20 15.20
58.30 15.74
68.29 16.54
61.88 16.88
60.67 16.24
65.55 16.09
58.37 15.02
57.36 16.11
68.20 15.84
57.29 15.20
57.20 14.71
45.00 14.33
68.86 15.40
46.73 13.83
53.44 13.99
56.67 15.26
63.47 16.09
62.64 16.35
59.73 15.96
55.88 15.37
64.87 15.78
49.22 14.90
68.83 15.97
50.39 14.90
70.41 15.22
55.94 14.37
49.85 15.13
64.49 16.01
75.23 13.71
58.71 15.83
68.29 15.68
46.19 14.64
64.71 15.90
59.03 15.02
59.88 15.34
61.60 16.19
64.46 15.53
57.87 15.45
60.71 15.63
64.45 16.06
72.04 15.02
54.60 13.65
49.44 14.37
60.52 16.00
50.92 14.44
53.36 14.45
55.88 15.10
60.76 15.85
59.72 15.86
78.00 12.32
62.74 15.79
72.32 16.34
72.51 15.49
48.19 14.22
59.51 15.48
65.24 16.64
63.37 16.31
77.52 13.00
76.32 13.53
74.65 14.82
58.33 16.00
60.54 15.71
62.18 15.12
65.10 16.43
66.20 16.01
58.98 16.36
59.22 15.84
46.17 15.16
61.45 15.56
54.43 14.56
52.99 14.91
60.84 15.72
45.53 15.24
62.93 16.74
73.57 14.23
59.94 16.44
49.92 14.40
64.59 15.87
57.75 16.66
64.99 15.29
52.78 13.99
69.73 16.17
69.23 16.24
65.09 16.22
52.36 13.89
66.99 15.64
57.69 15.31
46.72 14.76
52.90 15.73
61.07 16.09
77.04 14.09
56.39 14.83
50.80 14.51
66.52 16.09
57.95 15.18
47.27 14.35
47.86 15.00
53.92 15.07
52.64 14.70
59.45 14.27
53.04 14.56
60.06 15.94
57.83 15.61
75.27 13.96
59.17 15.84
60.36 16.08
66.23 16.75
49.49 14.58
69.11 16.46
65.87 16.10
62.79 16.83
67.19 16.97
70.22 16.25
65.94 16.22
69.55 16.14
66.75 16.11
72.10 15.31
54.49 15.33
50.20 14.39
48.18 14.36
73.53 15.50
63.41 16.36
52.19 14.40
51.50 14.09
51.35 14.42
74.31 14.15
62.20 15.97
66.73 15.92
55.79 14.81
70.04 15.53
57.05 15.94
50.27 14.55
57.14 14.94
57.98 15.43
70.35 16.22
62.43 16.07
52.57 14.88
55.22 14.81
71.23 15.46
58.08 15.96
49.95 14.67
71.67 14.51
50.98 14.39
53.02 14.53
58.89 14.95
60.56 15.73
67.32 16.72
51.79 15.04
61.79 16.82
72.77 14.79
51.30 14.98
71.61 15.36
64.95 16.59
45.00 15.03
62.86 16.46
59.60 15.86
65.55 16.79
72.31 16.43
54.02 15.00
70.28 15.66
64.38 15.85
64.88 16.28
55.93 16.73
55.98 15.24
51.01 14.87
51.55 15.80
65.95 15.56
49.57 14.33
55.13 15.42
75.24 14.00
49.86 13.94
68.49 15.66
56.40 14.37
52.10 14.86
49.21 14.91
67.90 15.59
52.75 15.23
59.02 16.15
74.38 13.69
64.39 16.38
49.78 14.85
74.28 13.99
52.91 14.65
53.82 15.08
48.86 15.29
61.32 15.77
61.69 15.94
49.58 14.29
69.35 16.09
60.03 15.96
71.96 14.83
62.47 15.68
49.72 14.50
78.00 12.64
48.92 13.98
69.97 15.65
49.19 14.73
60.91 16.00
48.94 15.25
Nota: Tamaño muestral n = 2500 observaciones utilizadas en el análisis.

GRÁFICA DE DISPERSIÓN ORIGINAL

plot(
  X, Y,
  type = "n",
  main = "Gráfica N°1:\nDiagrama original de dispersión entre SiO2 y Al2O3\nen muestras geoquímicas de depósitos minerales",
  xlab = "SiO2 (%)",
  ylab = "Al2O3 (%)",
  xlim = c(min(X, na.rm = TRUE), max(X, na.rm = TRUE)),
  ylim = c(min(Y, na.rm = TRUE), max(Y, na.rm = TRUE)),
  cex.main = 1.1,
  cex.lab = 1.1,
  cex.axis = 0.9
)

grid(nx = NULL, ny = NULL, col = "gray85", lty = 1)

points(
  X, Y,
  col = "blue",
  pch = 16,
  cex = 1.2
)

box(lwd = 1.5)

TRATAMIENTO DE DATOS

Promediación de los datos

Inicialmente se realizó una depuración de datos faltantes. Posteriormente, se aplicó una agregación de datos agrupando los valores de SiO2 y calculando el promedio de Al2O3. Esto permite suavizar la dispersión de los datos y observar mejor la tendencia polinómica cúbica.

# Eliminar datos faltantes

datos <- na.omit(datos)

# Conversión a numérico

datos$SiO2 <- as.numeric(datos$SiO2)

datos$Al2O3 <- as.numeric(datos$Al2O3)

# Ordenar los datos

datos <- datos[order(datos$SiO2, datos$Al2O3), ]

# Agrupar SiO2 para simplificar los pares de valores
# Se redondea SiO2 a 1 decimal para obtener grupos similares

datos$SiO2_grupo <- round(datos$SiO2, 1)

# Calcular el promedio de Al2O3 para cada grupo de SiO2

datos_prom <- aggregate(
  Al2O3 ~ SiO2_grupo,
  data = datos,
  mean
)

# Variable independiente simplificada

X <- datos_prom$SiO2_grupo

# Variable dependiente simplificada

Y <- datos_prom$Al2O3

TABLA DE PARES DE VALORES SIMPLIFICADA

TVP_SIO2_AL2O3 <- data.frame(X, Y)

n_modelo <- nrow(TVP_SIO2_AL2O3)

cat("Tamaño muestral del modelo =", n_modelo)
## Tamaño muestral del modelo = 326
tabla <- TVP_SIO2_AL2O3 %>%
  gt() %>%
  cols_align(
    align = "center",
    columns = everything()
  ) %>%
  fmt_number(
    columns = everything(),
    decimals = 2
  ) %>%
  tab_header(
    title = md("*Tabla N°2*"),
    subtitle = md("*Pares de valores simplificada de SiO2 y Al2O3 en muestras geoquímicas de depósitos minerales*")
  ) %>%
  tab_source_note(
    source_note = md(paste0(
      "**Nota:** Tamaño muestral n = ", nrow(TVP_SIO2_AL2O3),
      " observaciones utilizadas en el análisis."
    ))
  )

div(
  style = "height:400px; overflow-y:auto;",
  tabla
)
Tabla N°2
Pares de valores simplificada de SiO2 y Al2O3 en muestras geoquímicas de depósitos minerales
X Y
45.00 14.54
45.10 14.12
45.20 14.66
45.30 14.22
45.40 14.46
45.50 14.97
45.60 14.81
45.70 14.69
45.80 15.25
45.90 14.53
46.00 14.76
46.10 14.40
46.20 14.92
46.30 14.59
46.40 14.66
46.50 14.38
46.60 14.18
46.70 14.29
46.90 14.66
47.00 14.26
47.10 14.53
47.20 14.49
47.30 14.40
47.40 15.19
47.50 14.53
47.60 14.47
47.70 14.50
47.80 15.08
47.90 14.52
48.00 14.46
48.10 14.37
48.20 14.41
48.30 15.22
48.40 14.67
48.50 14.37
48.60 14.81
48.70 14.63
48.90 14.65
49.00 14.80
49.10 14.78
49.20 14.57
49.30 14.67
49.40 14.48
49.50 14.44
49.60 14.34
49.70 14.58
49.80 14.72
49.90 14.67
50.00 14.27
50.10 14.74
50.20 14.70
50.30 14.64
50.40 14.28
50.50 14.76
50.60 14.79
50.70 14.62
50.80 14.91
50.90 14.65
51.00 14.67
51.10 14.52
51.20 14.40
51.30 14.71
51.40 14.57
51.50 14.88
51.60 15.00
51.70 14.79
51.80 14.79
51.90 14.54
52.00 15.07
52.10 14.88
52.20 14.94
52.30 14.79
52.40 14.59
52.50 14.28
52.60 14.98
52.70 14.86
52.80 14.92
52.90 14.71
53.00 15.07
53.10 14.95
53.20 14.90
53.30 15.18
53.40 14.87
53.50 14.94
53.60 15.07
53.70 15.15
53.80 15.15
53.90 15.14
54.00 14.94
54.10 15.04
54.20 14.97
54.30 14.83
54.40 14.67
54.50 15.00
54.60 14.83
54.70 14.94
54.80 14.81
54.90 15.03
55.00 15.13
55.10 15.26
55.20 15.30
55.30 15.23
55.40 14.96
55.50 15.34
55.60 15.41
55.70 15.45
55.80 15.20
55.90 15.16
56.00 15.25
56.10 15.42
56.20 15.25
56.30 15.44
56.40 15.07
56.50 15.47
56.60 15.28
56.70 15.13
56.80 15.29
56.90 15.73
57.00 15.39
57.10 15.38
57.20 15.34
57.30 15.42
57.40 15.42
57.50 15.36
57.60 15.70
57.70 15.41
57.80 15.61
57.90 15.60
58.00 15.35
58.10 15.83
58.20 15.68
58.30 15.59
58.40 15.61
58.50 15.59
58.60 15.76
58.70 15.80
58.80 15.52
58.90 15.71
59.00 15.58
59.10 15.70
59.20 15.79
59.30 15.81
59.40 15.65
59.50 15.49
59.60 15.90
59.70 15.72
59.80 15.62
59.90 15.86
60.00 15.89
60.10 16.04
60.20 15.75
60.30 15.91
60.40 15.45
60.50 15.87
60.60 16.03
60.70 15.95
60.80 15.81
60.90 15.86
61.00 16.06
61.10 15.87
61.20 15.93
61.30 15.93
61.40 16.04
61.50 15.89
61.60 16.11
61.70 15.87
61.80 16.14
61.90 16.03
62.00 16.08
62.10 16.22
62.20 16.01
62.30 16.08
62.40 16.05
62.50 16.02
62.60 16.22
62.70 16.16
62.80 16.11
62.90 15.97
63.00 16.19
63.10 16.23
63.20 16.01
63.30 16.12
63.40 16.24
63.50 16.25
63.60 16.00
63.70 16.06
63.80 16.20
63.90 16.30
64.00 16.13
64.10 16.16
64.20 16.18
64.30 16.02
64.40 16.08
64.50 16.13
64.60 16.17
64.70 16.00
64.80 16.25
64.90 15.99
65.00 16.05
65.10 16.22
65.20 16.14
65.30 16.14
65.40 15.98
65.50 15.98
65.60 16.26
65.70 16.07
65.80 16.18
65.90 15.97
66.00 16.20
66.10 16.26
66.20 16.30
66.30 16.15
66.40 16.22
66.50 16.10
66.60 16.09
66.70 16.21
66.80 16.57
66.90 16.03
67.00 16.00
67.10 16.15
67.20 16.32
67.30 15.99
67.40 15.92
67.50 16.43
67.60 16.14
67.70 16.05
67.80 15.99
67.90 16.18
68.00 15.97
68.10 16.11
68.20 15.94
68.30 15.86
68.40 15.92
68.50 15.89
68.60 16.01
68.70 16.05
68.80 15.96
68.90 16.03
69.00 15.72
69.10 15.81
69.20 16.09
69.30 16.21
69.40 15.87
69.50 15.96
69.60 15.80
69.70 15.76
69.80 15.94
69.90 15.74
70.00 15.73
70.10 15.45
70.20 15.73
70.30 15.70
70.40 15.77
70.50 15.58
70.60 15.52
70.70 15.61
70.80 15.72
70.90 15.57
71.00 15.29
71.10 15.37
71.20 15.53
71.30 15.42
71.40 14.93
71.50 15.47
71.60 15.36
71.70 15.29
71.80 15.35
71.90 15.35
72.00 15.41
72.10 15.19
72.20 15.28
72.30 15.71
72.40 14.82
72.50 15.31
72.60 15.24
72.70 15.08
72.80 14.76
72.90 15.05
73.00 14.83
73.10 14.94
73.20 14.76
73.30 15.02
73.40 14.78
73.50 15.03
73.60 14.75
73.70 14.78
73.80 14.70
73.90 14.22
74.00 14.34
74.10 14.68
74.20 14.52
74.30 14.38
74.40 14.58
74.50 14.39
74.60 14.20
74.70 14.50
74.80 14.33
74.90 14.23
75.00 14.11
75.10 14.23
75.20 14.03
75.30 13.98
75.40 13.77
75.50 13.12
75.60 13.96
75.70 14.25
75.80 13.94
75.90 13.16
76.00 13.90
76.10 13.03
76.20 13.21
76.30 13.58
76.40 13.16
76.50 13.46
76.60 13.12
76.70 13.29
76.80 13.72
77.00 14.09
77.10 12.49
77.20 12.52
77.40 13.82
77.50 12.77
77.60 12.11
77.70 11.33
77.80 12.69
78.00 12.60
Nota: Tamaño muestral n = 326 observaciones utilizadas en el análisis.

GRÁFICA DE DISPERSIÓN SIMPLIFICADA

# Asegurar formato numérico

X <- as.numeric(as.character(X))

Y <- as.numeric(as.character(Y))

# Definir límites

x_max <- max(X, na.rm = TRUE) * 1.05

y_max <- max(Y, na.rm = TRUE) * 1.05

x_min <- min(X, na.rm = TRUE) * 0.95

y_min <- min(Y, na.rm = TRUE) * 0.95

# Crear gráfico

plot(
  X, Y,
  type = "n",
  main = "Gráfica N°2\nDiagrama simplificado de dispersión entre SiO2 y Al2O3\nen muestras geoquímicas de depósitos minerales",
  xlab = "SiO2 (%)",
  ylab = "Al2O3 (%)",
  xlim = c(x_min, x_max),
  ylim = c(y_min, y_max),
  cex.main = 1.1,
  cex.lab = 1.1,
  cex.axis = 0.9
)

grid(nx = NULL, ny = NULL, col = "gray85", lty = 1)

points(
  X, Y,
  col = "blue",
  pch = 16,
  cex = 1.2
)

box(lwd = 1.5)

CONJETURA

La distribución de los puntos en el gráfico muestra una relación curvada entre SiO2 y Al2O3, lo que sugiere el uso de un modelo polinómico. A medida que cambia el contenido de SiO2, el porcentaje de Al2O3 no varía de forma totalmente lineal, sino que presenta una tendencia no lineal. Por ello, se propone un modelo polinómico cúbico, incluyendo el término X³, para representar mejor el comportamiento geoquímico entre ambas variables.

PARÁMETROS

Modelo polinómico general:

# Y = a + bX + cX^2 + dX^3

Modelo polinómico aplicado al estudio:

# Al2O3 = a + b(SiO2) + c(SiO2)^2 + d(SiO2)^3

Ajuste del modelo:

modelo_pol <- lm(
  Y ~ X + I(X^2) + I(X^3),
  data = TVP_SIO2_AL2O3
)

summary(modelo_pol)
## 
## Call:
## lm(formula = Y ~ X + I(X^2) + I(X^3), data = TVP_SIO2_AL2O3)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.2353 -0.1010  0.0039  0.1155  1.0708 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  1.175e+02  4.052e+00   28.99   <2e-16 ***
## X           -5.694e+00  2.027e-01  -28.09   <2e-16 ***
## I(X^2)       1.031e-01  3.339e-03   30.89   <2e-16 ***
## I(X^3)      -6.078e-04  1.811e-05  -33.57   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2206 on 322 degrees of freedom
## Multiple R-squared:  0.9332, Adjusted R-squared:  0.9326 
## F-statistic:  1500 on 3 and 322 DF,  p-value: < 2.2e-16

Parámetros:

param <- coef(modelo_pol)

a_est <- param[1]

b_est <- param[2]

c_est <- param[3]

d_est <- param[4]

a_est
## (Intercept) 
##    117.4675
b_est
##         X 
## -5.694486
c_est
##    I(X^2) 
## 0.1031378
d_est
##        I(X^3) 
## -0.0006077919

Ecuación polinómica cúbica:

cat(
  "Y = ",
  round(a_est, 4),
  " + ",
  round(b_est, 4),
  "X + ",
  round(c_est, 6),
  "X^2 + ",
  round(d_est, 8),
  "X^3"
)
## Y =  117.4675  +  -5.6945 X +  0.103138 X^2 +  -0.00060779 X^3

Justificación del uso de la regresión lineal lm()

Aunque el modelo planteado presenta una relación polinómica, se utiliza la función lm() porque el modelo puede expresarse como una combinación lineal de sus parámetros al incluir términos como X² y X³. Esto permite ajustar relaciones no lineales entre las variables mediante una regresión lineal respecto a sus coeficientes.

Modelo polinómico de grado 3:

# Y = a + bX + cX^2 + dX^3

Para aplicar regresión lineal se define cada término del polinomio como una variable explicativa:

# X1 = X
# X2 = X^2
# X3 = X^3

Por lo tanto, el modelo queda expresado como:

# Y = β0 + β1X1 + β2X2 + β3X3

Dado que la ecuación es lineal en términos de los parámetros, puede ser estimada en R utilizando:

# lm(Al2O3 ~ SiO2 + I(SiO2^2) + I(SiO2^3))

Finalmente, tras obtener los estimadores, reconstruimos el modelo polinómico cúbico aplicado al estudio:

# Al2O3 = a + b(SiO2) + c(SiO2)^2 + d(SiO2)^3

COMPARACIÓN DE LA REALIDAD CON EL MODELO

# Límites

x_max <- max(X, na.rm = TRUE) * 1.05

y_max <- max(Y, na.rm = TRUE) * 1.05

x_min <- min(X, na.rm = TRUE) * 0.95

y_min <- min(Y, na.rm = TRUE) * 0.95

# Gráfico vacío

plot(
  X, Y,
  type = "n",
  main = "Gráfica N°3:\nModelo polinómico cúbico entre SiO2 y Al2O3\nen muestras geoquímicas de depósitos minerales",
  xlab = "SiO2 (%)",
  ylab = "Al2O3 (%)",
  xlim = c(x_min, x_max),
  ylim = c(y_min, y_max),
  cex.main = 1.1,
  cex.lab = 1.1,
  cex.axis = 0.9
)

# Puntos reales

points(
  X, Y,
  col = "blue",
  pch = 16,
  cex = 1.2
)

# Curva del modelo polinómico cúbico

x_seq <- seq(
  min(X, na.rm = TRUE),
  max(X, na.rm = TRUE),
  length.out = 1000
)

y_pred <- a_est + b_est * x_seq + c_est * x_seq^2 + d_est * x_seq^3

lines(
  x_seq,
  y_pred,
  col = "red",
  lwd = 3
)

box(lwd = 1.5)

legend(
  "topleft",
  legend = c("Datos reales", "Modelo polinómico"),
  col = c("blue", "red"),
  pch = c(16, NA),
  lwd = c(NA, 3),
  bty = "o",
  bg = "white",
  cex = 0.8,
  x.intersp = 0.6,
  y.intersp = 0.8
)

TEST DE APROBACIÓN

# Coeficiente de correlación de Pearson

Y_estimado <- predict(modelo_pol, newdata = TVP_SIO2_AL2O3)

r <- cor(TVP_SIO2_AL2O3$Y, Y_estimado) * 100

r
## [1] 96.60299
# Coeficiente de determinación R^2

R2 <- summary(modelo_pol)$r.squared * 100

R2
## [1] 93.32137
# Tabla resumen del modelo

Variable <- c("SiO2 - Al2O3")

tabla_resumen_polinomico <- data.frame(
  Variable,
  round(r, 2),
  round(R2, 2)
)

colnames(tabla_resumen_polinomico) <- c(
  "Variable",
  "Test Pearson (%)",
  "R cuadrado (%)"
)

knitr::kable(
  tabla_resumen_polinomico,
  format = "markdown",
  caption = "Tabla resumen del modelo polinómico cúbico"
)
Tabla resumen del modelo polinómico cúbico
Variable Test Pearson (%) R cuadrado (%)
SiO2 - Al2O3 96.6 93.32

RESTRICCIONES

Dominios

Dominio de X:

# D(X) = {R+}

Dominio de Y:

# D(Y) = {R+}

¿Existe algún valor de X que reemplazado en el modelo matemático genere un valor fuera del dominio de Y?

El modelo polinómico cúbico no presenta restricciones dentro del rango de datos analizado, ya que tanto SiO2 como Al2O3 son variables geoquímicas positivas expresadas en porcentaje. Sin embargo, el modelo debe utilizarse únicamente dentro del rango observado de SiO2, debido a que fuera de ese rango podría generar estimaciones poco realistas para Al2O3.

ESTIMACIONES

PREGUNTA DE CANTIDAD

SiO2_objetivo <- median(X, na.rm = TRUE)

if(SiO2_objetivo < 0){
  stop("Error: El contenido de SiO2 debe ser válido.")
}

Al2O3_est <- a_est +
  b_est * SiO2_objetivo +
  c_est * (SiO2_objetivo^2) +
  d_est * (SiO2_objetivo^3)

plot(1, type = "n", axes = FALSE, xlab = "", ylab = "")

text(
  1, 1,
  labels = paste(
    "¿Cuál es el Al2O3 esperado\n",
    "cuando el SiO2 es",
    round(SiO2_objetivo, 2), "%?\n\n",
    "Resultado estimado (Al2O3):",
    round(Al2O3_est, 2), "%"
  ),
  cex = 1.2,
  col = "blue",
  font = 2
)

PREGUNTA DE PORCENTAJE

incremento_pct <- 0.10

nuevo_SiO2 <- SiO2_objetivo * (1 + incremento_pct)

# Cálculo del Al2O3 actual

Al2O3_actual <- a_est +
  b_est * SiO2_objetivo +
  c_est * (SiO2_objetivo^2) +
  d_est * (SiO2_objetivo^3)

# Cálculo del nuevo Al2O3

nuevo_Al2O3 <- a_est +
  b_est * nuevo_SiO2 +
  c_est * (nuevo_SiO2^2) +
  d_est * (nuevo_SiO2^3)

porcentaje_cambio_Al2O3 <- ((nuevo_Al2O3 - Al2O3_actual) / Al2O3_actual) * 100

par(mar = c(0, 0, 0, 0))

plot(
  1,
  type = "n",
  axes = FALSE,
  xlab = "",
  ylab = "",
  xlim = c(0.9, 1.1),
  ylim = c(0.9, 1.1)
)

text(
  1, 1,
  labels = paste0(
    "¿En qué porcentaje cambia el Al2O3 esperado\n",
    "al incrementar el SiO2 un ", (incremento_pct * 100), "%?\n\n",
    "Cambio estimado de Al2O3: ",
    round(porcentaje_cambio_Al2O3, 2), "%"
  ),
  cex = 1.2,
  col = "firebrick3",
  font = 2
)

CONCLUSIÓN

Entre SiO2 y Al2O3 existe una relación de tipo polinómica cúbica, representada por el modelo Y = 117.47 + -5.69X + 0.1031X² + -0.000608X³. El modelo incluye el término cúbico X³, por lo que permite representar una relación no lineal entre ambas variables geoquímicas. El coeficiente de Pearson obtenido fue de 96.6% y el coeficiente de determinación fue de 93.32%. Esto indica que el modelo polinómico cúbico permite explicar la variación del Al2O3 a partir del contenido de SiO2 dentro del rango de datos analizado.