ANÁLISIS MULTIVARIABLE

CARGA DE DATOS Y LIBRERÍAS

# =============================================================================
# CARGA DE DATOS
# =============================================================================

library(readr)
library(gt)
library(dplyr)
library(e1071)
library(htmltools)
library(knitr)

datos <- read_csv("D:/tabla_regresion_exponencial_origen_2500.csv")
names(datos)
## [1] "SIO2_WT_PERCENT" "MGO_WT_PERCENT"

DEFINICIÓN DE LA VARIABLE

El contenido de SiO2 influye en la variación del porcentaje de MgO dentro de las muestras geoquímicas. A medida que se incrementa la concentración de sílice, el magnesio muestra una disminución progresiva con una tendencia no lineal que tiende a estabilizarse en contenidos altos de sílice. Esta relación se modela mediante una función logarítmica para capturar dicho comportamiento geoquímico.

## DEFINICIÓN DE LA VARIABLE

# Variable independiente (X): Dióxido de silicio
X <- datos$SIO2_WT_PERCENT

# Variable dependiente (Y): Óxido de magnesio
Y <- datos$MGO_WT_PERCENT

# 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)

# FILTRADO CLAVE: El modelo logarítmico requiere estrictamente X > 0 e Y > 0
TVP_original <- TVP_original[TVP_original$X > 0 & TVP_original$Y > 0, ]

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 = 2499
# 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 MgO 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 MgO en muestras geoquímicas de depósitos minerales
X Y
0.29 0.17
0.22 0.13
1.20 0.70
0.90 0.50
1.92 1.11
0.17 0.11
1.91 1.16
1.03 0.59
0.23 0.13
1.44 0.87
0.24 0.15
1.79 1.08
1.00 0.53
0.50 0.31
0.26 0.16
0.41 0.24
1.15 0.66
1.87 1.05
1.58 0.85
76.79 16.65
31.26 11.40
64.60 14.86
3.27 1.86
16.64 7.58
43.03 12.24
22.78 9.24
96.63 13.84
90.49 13.61
86.57 13.88
3.22 1.86
79.95 12.89
90.62 17.73
70.39 14.10
22.97 9.61
41.14 11.01
97.86 14.38
32.90 9.81
59.55 14.53
53.58 12.14
46.11 12.14
22.58 9.21
92.06 10.62
61.08 14.18
22.01 8.34
3.93 2.12
54.79 12.44
9.02 4.73
81.81 15.07
68.36 14.76
93.58 14.59
8.99 4.54
1.34 0.81
21.61 9.45
38.37 11.69
73.75 16.43
25.51 9.27
25.54 9.44
23.89 8.18
10.65 5.42
18.23 7.84
89.29 15.00
5.97 3.11
16.60 7.20
3.70 2.08
70.09 14.98
99.66 12.40
11.10 5.08
72.46 11.50
33.74 11.69
17.68 6.94
94.19 15.69
97.59 15.62
76.48 15.87
40.25 14.21
41.80 11.59
64.50 13.11
3.57 2.04
13.47 6.62
63.48 13.32
93.06 15.54
95.04 12.07
56.33 15.67
21.20 8.85
54.09 14.75
17.33 7.65
14.75 6.70
72.65 15.75
20.98 8.89
2.21 1.25
62.60 12.29
95.09 17.26
89.36 12.59
36.93 13.26
92.87 16.64
51.49 13.93
94.37 13.92
56.95 13.14
1.33 0.78
33.83 11.48
11.86 5.84
34.06 12.15
36.40 13.34
65.47 13.97
93.60 14.36
19.95 8.49
62.38 13.26
15.72 7.22
8.26 4.26
69.31 11.26
27.37 10.84
40.49 11.78
99.72 16.15
50.41 14.20
93.80 16.35
35.68 11.03
29.80 10.25
11.55 5.43
45.20 13.05
97.79 14.41
46.70 13.39
3.24 1.92
65.58 11.91
2.86 1.59
30.17 11.05
97.81 10.65
7.74 3.99
34.41 10.73
0.22 0.13
68.74 14.75
3.40 1.90
27.66 9.58
1.80 1.10
9.12 4.37
95.77 15.64
88.81 15.26
65.70 13.26
97.98 16.79
46.07 12.78
7.53 3.87
80.47 12.74
63.67 14.97
70.42 12.42
97.42 16.12
65.52 12.66
67.98 14.65
33.52 11.43
79.64 13.61
30.85 11.30
0.24 0.15
97.07 16.70
27.02 8.94
86.78 16.75
48.58 14.44
91.88 17.64
60.72 16.30
93.67 15.22
39.07 11.21
97.84 15.42
83.35 13.80
94.29 16.59
94.87 14.22
67.76 15.15
36.58 11.95
32.43 10.79
16.57 6.68
14.46 6.27
0.11 0.07
10.08 5.00
51.46 14.09
96.93 15.92
43.79 11.26
40.33 11.93
55.51 12.22
44.95 12.84
60.45 13.83
48.07 13.83
16.95 7.77
21.82 9.26
72.51 13.85
2.96 1.68
62.61 15.87
36.91 12.88
62.18 14.79
38.18 11.68
49.93 12.20
66.70 12.37
76.66 14.34
26.60 10.10
76.78 17.39
40.62 11.77
7.23 3.86
18.73 7.71
46.49 12.20
74.62 14.58
53.01 14.37
21.32 8.20
33.84 10.92
49.55 13.04
63.47 13.88
25.58 10.07
28.43 11.06
69.50 12.35
74.82 14.64
4.59 2.40
49.33 13.56
21.58 8.20
43.02 11.84
11.46 5.48
42.11 13.18
65.40 12.54
24.75 9.84
71.93 13.44
34.23 10.56
48.49 13.29
40.61 12.61
46.98 13.64
94.99 14.27
56.68 13.47
8.51 4.07
82.14 15.10
8.44 4.32
92.79 13.46
3.88 2.02
16.41 7.39
88.58 16.79
16.54 7.27
79.14 13.62
46.62 11.64
52.58 14.21
53.85 12.43
36.68 11.19
10.48 5.17
18.07 8.03
73.73 13.42
69.59 12.97
72.54 14.39
86.50 16.87
76.74 14.32
91.24 16.50
15.74 6.81
98.32 14.96
56.84 13.31
89.78 17.02
5.32 2.84
81.45 13.61
59.83 14.11
35.34 11.28
24.40 8.68
2.47 1.46
92.21 16.57
24.32 10.11
94.52 13.55
87.00 13.87
68.28 14.33
91.63 17.82
14.12 6.76
7.02 3.74
3.14 1.79
27.42 9.53
2.95 1.65
85.11 13.16
54.67 14.73
8.03 4.53
33.90 11.35
83.69 15.59
95.56 15.12
84.52 14.72
92.03 11.86
71.36 16.87
19.48 8.25
49.72 12.84
9.25 4.75
68.29 12.96
4.09 2.22
40.30 12.25
8.62 4.46
82.72 15.68
50.46 14.55
87.38 17.10
64.74 13.04
3.69 2.01
10.63 4.61
44.12 12.78
45.12 11.52
53.21 15.81
98.65 14.94
84.80 13.46
82.67 18.85
76.53 14.16
8.00 3.75
7.43 3.69
99.75 13.38
88.02 16.21
52.65 11.19
2.21 1.18
91.02 13.78
23.44 8.76
24.66 8.80
91.26 13.17
91.12 13.05
3.61 2.05
64.31 13.65
57.41 13.85
49.90 13.40
82.18 16.43
48.89 13.99
43.87 12.58
98.83 11.85
10.11 4.92
98.64 16.73
15.69 6.94
66.24 13.36
14.99 6.72
40.89 12.73
60.88 13.56
36.42 10.73
21.26 8.17
96.58 15.50
81.66 15.22
27.78 10.58
1.89 1.12
20.46 9.71
6.84 3.48
79.48 16.42
97.67 16.09
47.19 12.59
44.65 11.90
59.60 13.55
64.18 14.37
82.71 16.94
26.93 10.09
11.05 5.78
68.18 12.59
46.16 13.43
13.84 6.58
84.33 12.85
83.86 12.66
77.82 12.07
48.18 13.02
7.73 3.99
6.23 3.15
95.24 16.33
88.87 14.44
63.07 14.54
44.70 12.32
10.29 4.93
38.87 9.82
98.92 15.29
64.77 14.71
97.03 13.01
64.14 13.63
5.89 3.24
39.34 11.18
46.54 12.57
95.19 10.16
72.73 12.64
20.17 7.59
0.10 0.06
76.26 16.07
16.72 7.12
72.78 14.95
78.65 16.55
96.32 13.74
97.53 17.69
57.45 11.56
6.89 3.60
74.50 12.27
63.39 14.67
15.12 6.79
94.40 11.69
10.97 5.36
60.76 13.72
47.62 11.71
1.04 0.61
56.83 17.00
2.89 1.65
9.36 4.85
55.33 14.63
53.23 14.41
69.04 14.95
83.57 12.41
83.20 13.23
36.66 11.18
35.02 10.77
80.96 18.70
76.39 14.38
40.13 14.61
73.63 12.63
83.33 15.20
62.08 13.17
59.76 14.47
26.79 9.26
81.44 17.16
73.79 12.39
46.50 12.26
60.01 14.42
1.40 0.82
88.42 19.08
10.94 5.11
94.08 13.91
2.92 1.49
71.78 14.09
77.18 14.18
38.87 11.84
45.71 10.82
52.77 13.36
46.70 12.97
15.22 6.74
83.14 12.71
88.29 19.25
93.24 14.29
82.42 13.05
93.52 15.93
55.03 11.88
69.35 15.44
53.15 12.92
8.91 4.57
24.95 9.30
33.70 11.85
66.26 14.83
79.60 13.47
91.04 18.73
61.35 14.40
51.21 10.98
87.22 12.00
96.75 14.15
87.04 14.21
37.01 12.19
51.67 14.47
19.19 8.82
38.94 10.94
70.51 15.16
77.70 12.84
28.77 8.93
2.58 1.37
33.05 12.18
28.38 9.24
7.70 3.86
15.28 6.81
38.74 13.37
38.25 11.40
79.25 14.50
35.11 11.33
33.38 11.43
24.27 10.09
35.64 10.46
65.50 13.66
42.60 13.38
33.04 11.02
7.19 3.99
3.38 2.01
46.50 13.80
31.51 10.37
1.18 0.69
82.50 12.86
59.88 13.53
66.38 12.96
37.87 11.82
59.90 13.37
44.39 11.96
41.16 12.01
36.55 11.34
22.67 8.51
43.49 13.62
84.03 12.80
21.29 8.80
92.95 13.36
93.81 15.38
32.99 10.96
89.68 15.30
80.40 13.73
34.93 10.76
68.18 15.07
53.18 14.77
17.75 7.34
17.45 7.72
50.73 10.35
28.17 10.83
92.60 12.39
17.17 7.74
18.73 8.31
80.84 17.22
96.63 15.64
16.06 6.74
7.09 3.55
70.08 14.31
63.72 12.86
57.22 12.87
80.81 13.11
40.56 12.30
66.35 12.68
81.74 11.15
49.69 14.47
8.49 4.19
5.86 3.28
6.20 3.32
91.09 20.45
71.21 13.38
63.24 14.19
51.73 15.11
21.23 8.95
62.77 13.36
26.03 9.56
52.63 12.90
68.87 14.21
95.57 14.45
41.92 11.21
79.03 14.27
38.21 11.53
27.27 9.88
26.77 10.15
62.51 13.60
54.81 13.24
9.28 4.97
48.69 11.47
88.18 12.49
83.32 15.77
46.97 12.97
14.12 6.01
52.83 11.45
71.94 13.18
75.06 12.13
41.15 13.04
88.87 14.37
89.02 15.89
43.65 10.88
0.23 0.14
21.89 8.55
10.95 5.26
95.62 13.60
65.98 15.36
67.04 13.73
87.33 19.36
34.93 10.90
27.28 10.13
13.77 6.29
77.67 12.06
41.75 12.32
81.46 14.73
89.83 15.44
96.82 14.27
13.38 6.47
86.12 12.94
91.92 13.96
99.46 13.65
65.50 14.00
64.70 10.17
13.46 5.70
41.66 11.34
52.53 14.65
17.37 6.89
76.22 14.88
3.61 1.96
21.83 8.60
72.35 15.16
17.49 8.05
75.68 16.01
96.85 16.40
60.02 15.04
67.39 11.53
18.96 7.72
81.70 13.04
78.93 15.78
31.08 10.62
82.76 14.91
69.70 19.84
45.79 12.77
21.61 8.49
66.81 15.97
41.24 11.41
76.24 10.26
0.08 0.05
83.76 16.33
87.21 12.38
56.72 16.14
77.26 17.26
35.37 10.36
40.86 11.41
64.53 17.43
9.98 4.70
59.42 14.11
53.86 13.38
54.83 11.85
17.48 7.55
48.30 14.61
10.59 5.46
75.81 13.13
80.09 11.73
14.22 6.48
77.92 15.76
66.38 13.94
46.13 13.30
61.73 13.64
74.09 16.65
44.76 12.75
46.69 12.78
0.13 0.08
23.11 8.18
58.63 11.81
79.68 15.92
99.52 15.05
82.30 13.59
42.71 13.51
75.62 12.97
38.72 10.12
83.86 12.62
2.35 1.37
44.50 11.86
5.17 2.70
23.08 9.43
36.45 11.15
14.78 7.16
77.14 13.64
70.49 14.65
52.54 12.30
2.01 1.13
95.80 15.30
15.91 6.80
24.00 8.18
27.86 9.22
87.91 15.20
15.80 6.60
68.89 12.93
66.85 14.02
14.92 6.73
58.50 12.64
54.78 12.78
24.45 8.70
85.71 14.82
39.43 11.11
75.83 12.17
78.33 13.19
30.60 11.64
11.62 5.69
98.29 16.42
37.94 11.06
59.74 14.37
9.94 4.89
70.39 16.93
23.52 8.92
0.35 0.21
20.81 8.05
71.50 13.28
60.29 13.85
68.93 11.60
76.32 11.52
74.66 11.81
37.20 11.02
29.06 10.99
17.28 7.50
5.59 3.02
92.78 17.53
12.75 6.21
77.89 14.96
15.55 6.75
21.56 8.63
50.23 13.15
64.79 12.65
39.05 12.68
81.13 16.26
78.53 12.22
52.73 11.84
18.15 8.31
15.43 7.00
29.91 10.22
77.81 13.20
81.71 15.23
35.54 10.91
4.85 2.51
84.79 15.81
2.82 1.61
86.91 12.11
29.03 11.01
42.12 12.18
56.92 14.47
97.89 15.93
79.66 13.52
9.30 4.28
61.08 11.22
0.59 0.33
19.17 7.70
39.06 11.72
73.41 12.86
39.62 12.46
46.32 11.56
64.89 17.34
26.68 10.10
13.95 5.83
80.35 13.13
67.10 13.52
78.24 10.97
80.09 18.18
46.38 13.76
87.17 17.13
72.09 15.17
23.94 9.14
95.03 12.58
69.57 13.11
64.90 12.67
77.18 12.78
60.92 13.86
37.27 12.52
82.39 15.23
39.18 11.43
4.63 2.42
15.53 7.08
48.17 12.26
86.92 13.99
14.21 6.60
57.70 11.88
63.16 13.28
35.93 11.23
85.21 13.15
78.08 16.51
48.63 11.98
41.77 12.04
72.11 11.59
68.75 12.73
90.97 18.29
29.14 10.22
54.35 12.55
30.10 9.99
63.65 13.48
95.97 14.48
67.07 12.83
60.67 13.63
91.71 10.65
76.33 14.01
97.37 19.38
20.68 8.54
50.02 12.86
27.20 11.06
87.76 12.40
0.77 0.44
80.25 13.21
60.51 10.38
87.61 11.66
77.55 16.50
63.63 13.93
61.85 16.94
82.85 16.16
32.64 10.17
10.29 4.94
24.40 9.18
0.38 0.23
91.92 13.65
84.74 13.63
94.45 16.22
57.86 12.41
56.94 14.40
50.22 11.52
43.12 12.09
83.75 16.15
8.54 4.61
90.61 16.35
1.93 1.10
55.86 14.55
66.94 14.00
76.17 13.95
5.93 3.19
18.49 7.42
46.88 14.13
12.44 5.36
33.36 10.12
40.03 12.50
38.80 11.50
49.05 11.91
62.25 12.86
81.09 14.66
42.67 11.66
21.83 8.68
28.25 9.86
19.06 7.98
3.71 1.96
21.58 8.18
46.15 10.65
27.76 10.81
39.38 13.07
1.07 0.64
18.90 8.12
95.50 14.71
42.96 11.31
70.71 14.33
89.11 13.55
11.95 5.47
4.45 2.26
37.92 12.76
44.02 13.12
81.88 12.76
81.59 12.69
86.30 16.28
30.08 10.60
71.24 13.82
36.22 11.34
46.69 12.37
55.54 13.71
90.04 15.49
24.79 8.98
53.61 11.65
29.41 9.92
33.86 10.78
85.03 12.47
60.82 13.54
35.78 11.79
93.30 10.40
70.87 14.84
70.82 15.01
57.40 15.16
40.88 11.30
32.46 10.74
22.60 9.63
0.39 0.23
14.58 6.32
65.37 14.68
30.16 10.83
86.24 11.77
35.73 10.43
15.23 6.78
5.56 2.97
67.79 16.53
74.79 12.75
33.51 11.88
0.02 0.01
30.86 11.76
51.39 12.16
54.25 14.12
14.82 6.29
38.04 10.44
91.32 15.89
45.74 11.82
91.57 14.89
45.00 12.99
82.60 11.39
16.70 7.39
24.81 9.25
57.34 14.21
17.73 7.36
56.02 13.41
67.03 14.38
74.88 14.27
32.35 10.57
38.22 11.90
58.07 12.06
81.71 18.52
6.88 3.23
82.04 17.51
15.96 6.57
61.50 14.44
21.45 8.63
80.29 13.30
75.94 12.58
47.13 14.79
78.42 15.72
48.27 14.27
19.65 8.01
21.80 8.70
6.39 3.07
68.92 14.82
90.59 12.47
44.58 13.83
93.59 14.46
25.25 10.00
90.25 11.80
61.90 11.32
62.78 13.78
92.95 16.09
43.06 12.19
59.42 12.24
68.25 11.43
58.22 10.33
52.18 11.74
47.47 12.74
7.59 4.25
23.42 9.64
96.17 10.36
54.01 14.29
51.59 14.59
50.67 13.11
30.55 9.19
17.75 7.46
99.89 13.91
81.68 17.76
77.08 12.54
58.40 14.17
7.28 3.67
22.03 8.53
0.80 0.48
91.28 13.92
83.76 17.95
32.34 9.54
10.65 5.52
56.58 14.46
71.70 13.44
57.88 13.14
33.17 10.63
96.45 13.89
34.30 10.92
95.03 17.16
12.15 5.53
65.12 17.69
37.50 10.48
28.15 8.94
69.76 14.36
23.64 9.07
27.70 11.08
9.36 4.89
48.83 13.23
65.89 13.59
69.16 15.12
73.92 14.17
72.45 16.24
32.37 10.80
0.66 0.39
51.27 14.14
66.64 15.88
45.09 12.44
54.21 14.85
41.82 13.15
46.97 12.58
87.72 14.81
42.46 12.49
5.22 2.91
1.97 1.10
9.84 5.20
72.59 15.08
64.61 10.25
66.06 13.87
9.46 4.69
60.14 13.73
30.53 11.59
1.80 1.02
4.97 2.71
59.48 12.52
64.31 13.16
58.60 13.48
71.05 15.85
68.01 11.84
59.12 12.55
2.07 1.19
88.05 14.31
71.42 13.62
58.80 11.04
76.36 16.69
6.30 3.54
77.76 12.28
16.71 7.10
82.10 16.43
7.68 4.17
2.31 1.33
88.28 14.23
38.87 11.46
95.98 14.38
64.79 15.12
89.79 13.09
83.80 12.75
94.01 16.94
66.59 15.51
49.21 12.59
68.77 14.12
58.80 14.34
73.26 16.57
83.80 15.00
40.92 10.88
53.05 14.08
48.20 12.06
3.23 1.85
35.76 10.79
48.12 11.76
83.20 12.34
44.08 13.20
74.00 14.12
95.76 12.21
86.05 15.94
88.36 12.40
28.61 9.39
21.99 8.58
31.72 12.27
17.06 7.89
85.68 14.60
78.82 14.28
67.39 14.86
41.06 12.88
45.76 14.25
3.99 2.18
40.44 12.86
74.15 16.89
79.47 14.04
34.70 11.83
89.93 13.94
2.29 1.26
85.22 16.02
50.50 13.26
69.04 14.01
60.43 12.68
93.60 16.63
6.28 3.17
98.37 16.59
10.69 5.35
7.13 3.87
87.56 13.97
92.97 15.33
87.06 14.91
60.45 13.84
0.10 0.06
67.92 13.17
86.68 15.93
88.84 16.60
79.25 10.79
31.90 11.04
7.66 3.88
85.40 15.99
86.13 16.44
35.46 10.09
21.99 8.82
22.12 8.07
19.86 8.90
24.69 10.27
3.70 2.08
3.13 1.66
86.06 16.95
93.71 16.94
62.67 13.79
69.93 15.79
77.86 12.57
65.18 13.46
64.03 13.49
17.53 7.80
14.97 6.91
70.52 11.55
98.45 16.45
1.69 0.96
86.13 16.55
68.85 13.86
75.47 13.35
78.28 11.36
24.70 9.44
19.04 7.89
53.92 12.36
51.84 12.01
31.44 11.92
95.95 13.77
93.01 12.85
49.12 11.55
48.92 11.86
7.72 3.81
37.00 10.90
8.78 4.90
96.79 13.15
87.41 17.32
48.83 12.68
75.58 14.33
30.17 10.83
73.11 12.50
10.42 5.01
21.85 8.93
20.19 8.44
71.94 15.82
1.05 0.59
28.76 9.23
87.33 15.17
19.27 8.21
79.96 14.09
90.13 11.66
0.30 0.19
27.83 10.08
42.77 12.22
66.01 16.11
57.15 14.35
22.07 9.05
8.59 4.45
71.81 12.72
14.84 6.84
7.38 3.83
14.51 6.45
89.50 16.11
71.15 13.43
58.62 13.42
92.76 14.99
27.82 10.17
88.37 11.02
59.93 13.83
7.93 4.32
60.32 14.11
30.36 10.67
13.81 6.66
18.43 7.55
68.52 16.21
36.61 13.40
73.74 14.24
98.50 15.21
35.18 11.51
4.79 2.58
11.64 5.58
7.64 4.34
8.37 4.25
53.96 10.21
68.92 14.73
74.74 13.23
70.38 12.37
45.88 13.96
1.38 0.83
58.32 16.03
80.75 14.32
91.64 17.19
9.44 4.68
7.96 3.86
71.58 11.44
64.89 14.82
16.92 7.25
55.84 15.18
46.12 14.80
61.49 12.82
62.06 12.40
57.64 14.51
69.67 14.27
68.70 15.85
27.01 9.79
3.05 1.83
11.83 5.50
95.44 13.60
59.12 13.17
16.16 6.61
77.64 12.81
6.70 3.82
77.48 16.79
62.15 14.41
76.70 14.03
79.79 15.85
9.81 5.09
19.78 8.19
75.63 17.00
60.56 12.72
96.67 16.69
23.79 9.31
94.83 15.14
10.30 5.37
49.83 12.06
21.97 8.91
3.94 2.24
50.63 12.64
31.48 11.01
55.34 13.97
77.96 14.71
19.55 8.14
46.24 13.26
55.86 12.69
65.73 10.76
95.85 15.45
65.82 11.75
3.51 1.94
52.93 13.49
36.85 11.15
90.56 14.48
68.02 14.12
23.31 9.67
60.14 11.24
74.96 13.71
92.15 12.91
98.69 16.23
76.51 13.27
2.46 1.36
24.16 9.01
8.28 4.37
70.11 15.04
80.03 16.52
94.94 16.03
52.48 12.32
42.23 11.22
80.58 14.48
17.61 7.91
70.73 14.29
24.10 9.33
19.95 8.33
14.09 6.39
47.89 13.41
70.37 12.88
63.76 12.91
88.36 17.51
32.83 10.59
92.98 15.85
51.47 13.92
91.24 13.06
53.02 12.20
58.77 13.16
28.90 9.95
56.89 15.36
61.30 10.37
15.63 6.68
46.41 11.94
14.33 6.34
77.63 13.51
0.61 0.37
27.54 10.53
23.85 8.31
77.35 14.73
97.40 14.74
23.00 8.62
87.13 16.77
11.55 5.51
79.71 16.14
92.02 14.27
87.46 13.25
49.78 13.07
21.42 9.18
92.59 13.44
72.40 12.06
14.41 6.24
19.70 9.14
24.07 9.32
68.09 14.38
90.90 13.09
24.42 8.96
54.28 15.26
17.89 8.76
24.53 9.00
9.16 4.71
96.11 12.49
0.26 0.15
67.73 15.12
16.45 7.07
84.64 14.33
96.97 17.49
24.72 9.37
82.75 13.51
11.96 6.05
69.51 14.74
99.65 14.02
43.57 11.74
62.89 13.44
79.46 10.18
68.99 13.89
97.25 12.28
80.50 13.81
14.89 6.15
62.85 13.30
37.95 10.37
55.71 14.73
57.74 14.30
74.15 14.79
89.23 16.58
32.64 11.57
41.00 12.54
7.55 4.09
51.02 11.92
92.87 13.18
66.06 11.91
35.77 12.32
43.01 11.78
59.41 12.59
31.51 11.62
57.74 11.97
89.11 16.80
96.76 13.86
34.76 10.18
2.00 1.16
58.22 14.78
38.37 11.94
20.27 7.92
21.90 8.58
85.68 15.88
74.19 15.25
78.55 16.26
78.49 11.30
92.21 13.76
38.84 11.18
36.81 11.58
77.18 13.37
34.69 12.06
53.64 12.84
79.36 13.40
43.16 12.43
4.19 2.27
91.75 15.98
88.80 14.08
14.55 7.06
47.25 11.92
58.47 13.63
97.75 10.60
98.75 14.57
49.17 11.89
98.76 15.41
76.62 14.12
1.35 0.79
79.37 14.60
90.08 13.53
15.85 6.63
22.65 8.19
64.01 10.91
46.54 13.92
93.84 18.04
60.33 14.21
86.34 16.87
25.55 9.94
28.40 9.28
87.14 15.27
50.62 11.64
43.29 13.13
92.43 13.49
73.97 13.22
88.62 13.60
8.79 4.44
16.98 6.72
22.40 8.49
41.40 13.43
17.73 7.56
23.58 8.51
93.48 12.67
1.82 1.04
73.11 15.92
9.87 5.03
47.29 13.17
49.03 14.28
68.01 15.79
3.49 1.95
87.61 15.55
3.98 2.23
80.83 12.99
100.00 14.36
88.39 13.26
23.61 8.75
36.64 12.82
42.78 13.37
14.06 6.66
66.73 10.75
27.84 10.40
53.90 12.12
17.60 7.18
51.72 11.89
13.62 6.97
24.81 10.34
43.20 10.03
2.36 1.25
43.12 11.90
86.77 15.11
64.96 13.54
49.44 11.62
11.95 5.78
86.47 15.45
36.62 12.23
53.81 10.40
8.55 4.34
89.66 12.13
91.76 17.12
10.39 5.11
17.39 7.37
36.27 10.45
57.29 14.38
61.56 14.15
84.97 15.78
72.87 11.07
66.45 10.90
56.84 13.85
98.46 16.35
2.92 1.69
37.36 13.29
65.79 13.32
83.85 12.30
62.82 13.32
76.46 16.35
59.08 13.73
9.92 5.06
86.19 16.36
96.01 12.19
40.96 12.64
51.83 11.69
51.58 11.14
90.35 11.56
66.27 11.40
22.41 7.54
69.67 14.01
19.72 8.45
48.01 13.72
10.13 5.21
58.82 11.91
90.19 15.58
81.16 16.40
97.64 10.57
58.26 14.74
82.54 11.21
78.18 14.89
43.35 10.86
58.88 16.09
66.16 13.49
74.84 11.37
19.11 8.33
62.79 12.45
75.13 16.22
66.80 15.79
0.57 0.35
48.09 11.01
52.72 12.10
61.13 13.09
49.57 11.62
53.99 14.94
92.14 14.91
54.69 12.81
41.38 12.21
62.91 12.56
36.69 11.59
82.11 16.22
12.34 5.86
35.24 12.25
9.92 5.10
68.40 14.50
88.59 15.82
92.44 9.94
95.77 19.77
81.05 11.64
79.98 12.92
51.52 11.92
18.59 8.62
50.69 12.82
97.34 18.25
50.35 12.76
12.32 5.96
79.52 15.39
71.25 13.37
32.34 10.92
88.83 15.10
25.81 10.57
98.66 13.35
81.25 17.22
3.90 2.07
79.20 14.50
66.80 15.24
69.25 11.88
15.23 6.58
23.69 8.14
32.73 11.11
35.10 10.97
36.34 10.24
15.80 6.52
44.30 13.37
80.53 13.82
19.20 7.88
12.20 6.15
30.83 11.22
92.12 16.69
33.87 11.06
89.32 12.26
36.99 11.79
80.64 14.69
44.21 10.76
46.99 12.02
59.96 16.31
5.68 3.21
24.16 8.43
23.35 8.95
40.29 10.53
54.91 13.11
44.09 11.60
74.14 12.65
32.82 10.48
54.03 12.11
55.56 12.26
80.20 16.94
31.96 10.45
26.73 10.16
28.09 9.66
98.11 18.33
98.72 13.98
71.04 16.55
53.87 10.85
61.37 12.73
97.36 15.90
78.41 16.09
52.59 13.93
64.41 12.50
16.89 7.51
62.35 14.45
89.84 12.98
99.39 13.84
19.43 8.58
38.30 10.73
52.04 13.13
22.30 9.25
81.64 10.68
33.77 12.17
64.90 14.48
28.38 10.70
73.89 12.24
51.97 9.43
93.84 13.15
44.46 12.97
73.48 15.85
96.85 15.95
10.20 5.35
72.35 14.73
5.62 3.16
58.21 15.30
9.88 4.92
83.47 16.95
92.75 15.03
3.86 2.18
76.28 15.75
45.04 12.68
67.43 16.18
62.64 13.33
29.47 10.32
83.09 13.69
30.91 9.32
90.78 16.23
0.82 0.43
75.73 14.64
45.13 12.64
25.20 9.24
54.21 12.58
9.75 5.00
99.71 13.70
53.07 14.71
71.01 14.12
88.42 16.21
74.01 16.18
74.78 15.52
48.50 13.10
28.87 10.81
93.77 14.92
57.47 12.94
9.84 5.30
37.82 11.94
97.42 15.93
79.96 13.84
22.35 8.58
2.96 1.61
14.79 6.44
50.24 14.31
82.98 12.98
51.43 12.95
47.69 13.59
28.81 10.65
34.16 12.08
69.30 14.51
60.81 13.62
29.56 10.52
14.76 6.57
19.02 8.38
34.68 11.24
58.89 12.76
10.11 4.67
14.09 6.05
90.29 15.65
45.40 14.19
7.90 3.86
46.40 11.94
74.07 14.31
6.10 3.40
92.41 13.58
38.84 12.27
91.65 10.38
84.75 15.75
46.36 13.19
32.77 10.32
43.54 11.37
27.45 10.15
4.23 2.21
55.62 13.89
14.81 6.47
72.20 13.16
70.58 15.94
34.65 11.16
45.71 12.98
49.99 11.73
41.81 12.11
58.49 15.09
60.89 14.61
2.55 1.39
26.05 10.29
3.30 1.84
99.42 15.28
2.19 1.24
27.93 10.95
58.24 10.85
45.46 12.71
36.14 12.16
0.73 0.44
11.57 5.07
88.19 16.02
61.01 14.24
80.48 17.81
92.38 14.31
50.17 11.26
66.49 17.24
41.66 12.78
59.78 12.94
79.89 15.01
38.84 12.25
3.25 1.89
41.64 11.31
73.36 13.98
37.75 12.47
16.76 7.36
56.21 13.08
32.41 11.08
18.93 7.49
31.64 10.64
67.23 12.68
35.20 9.43
57.26 12.68
19.40 8.65
34.50 10.97
37.91 10.61
40.40 13.48
12.31 5.37
5.63 3.15
6.22 3.33
13.91 6.47
13.72 6.38
59.98 11.73
1.18 0.74
17.57 7.06
92.27 14.50
95.83 14.08
85.94 15.95
61.19 11.53
19.28 7.77
50.21 15.17
78.31 16.12
19.29 8.08
80.16 15.75
50.00 14.15
45.71 12.56
20.95 8.56
14.40 6.46
10.51 5.49
78.08 15.82
86.10 14.26
95.45 11.58
38.96 11.72
85.44 14.15
22.16 8.05
97.55 12.81
79.52 12.94
77.86 12.80
38.84 10.79
44.75 11.69
63.44 13.32
48.07 12.71
95.20 13.79
13.60 6.19
26.94 9.81
83.89 12.76
64.82 15.50
5.54 3.03
99.90 11.16
82.90 17.17
73.99 16.07
39.19 12.74
40.92 11.49
8.00 4.24
18.12 8.21
94.58 13.88
21.80 9.07
67.85 12.73
28.96 10.27
91.67 16.49
90.72 14.85
35.09 10.68
30.00 10.07
62.17 14.57
29.01 9.60
23.90 8.83
43.95 10.78
71.19 13.87
47.84 12.41
24.86 9.17
96.26 20.07
99.47 16.91
67.81 11.57
40.41 13.07
46.84 12.20
83.65 17.87
43.04 12.41
1.61 0.93
67.29 12.93
38.77 11.06
69.67 11.29
6.57 3.41
43.37 12.27
90.90 17.93
79.36 15.16
73.30 12.62
92.22 12.41
41.02 11.76
9.44 4.80
20.71 8.98
21.93 8.70
62.31 15.64
58.11 15.99
64.23 11.76
5.56 2.94
93.53 11.27
10.91 5.04
81.18 15.70
22.86 9.50
87.17 14.69
72.31 18.56
52.15 13.01
98.91 15.14
11.44 5.69
49.94 12.77
76.49 15.30
7.51 3.58
36.24 11.18
46.17 13.70
7.11 3.94
92.08 13.03
34.27 11.02
58.96 13.65
36.00 10.34
73.81 13.35
0.50 0.30
64.40 11.88
93.08 18.32
55.61 11.96
37.82 10.74
31.43 9.95
26.24 9.52
54.36 13.90
95.58 13.82
21.32 8.30
6.75 3.50
21.72 8.47
67.74 12.02
9.71 4.57
14.87 6.23
16.89 7.23
11.01 5.15
83.93 13.00
91.76 15.56
6.09 3.23
95.86 17.62
47.61 13.10
74.24 11.42
66.05 13.72
73.03 11.65
21.71 8.72
82.07 13.19
88.17 13.02
25.25 9.64
44.85 14.01
60.84 15.03
81.39 14.43
92.42 15.24
59.64 12.89
88.32 13.76
99.69 13.84
69.97 13.45
26.70 9.67
52.95 12.11
14.42 6.44
52.23 12.81
80.68 12.07
8.40 4.32
74.24 14.51
89.32 16.15
99.67 12.87
6.30 3.52
85.40 13.40
10.72 5.50
43.91 12.31
34.04 10.65
17.97 8.00
73.84 13.89
46.92 11.84
87.43 16.74
26.92 9.17
95.40 17.67
62.04 13.88
97.42 11.41
38.93 11.09
52.96 11.97
58.28 12.07
64.47 12.39
26.28 9.43
87.84 14.27
21.97 9.23
85.41 17.46
58.64 11.41
24.88 9.30
87.01 13.20
88.81 14.57
44.72 12.70
17.01 6.84
73.37 15.55
2.15 1.22
63.90 13.96
69.82 13.68
20.62 8.33
72.03 11.53
39.10 11.88
34.99 11.50
78.56 15.75
17.89 7.25
63.14 14.08
26.59 9.56
70.63 13.71
23.63 9.18
22.13 8.98
89.86 14.71
77.89 13.26
30.64 9.40
48.41 13.54
36.00 11.17
67.67 14.32
41.67 12.68
70.21 11.64
1.78 1.01
20.82 8.97
74.98 16.32
60.80 12.85
59.03 14.33
81.80 16.81
51.31 12.91
71.26 14.51
4.64 2.48
25.73 9.04
57.22 12.17
82.80 10.49
25.89 9.30
88.23 17.20
20.26 8.29
12.50 5.65
42.19 12.05
50.39 13.68
90.92 11.72
38.91 13.54
66.58 15.12
7.47 3.73
40.47 11.29
76.09 16.24
4.23 2.51
83.47 14.98
12.50 6.13
73.56 12.50
42.18 11.42
81.79 14.47
72.07 12.51
30.63 10.69
54.22 12.15
3.41 2.02
94.97 13.11
45.14 12.52
45.09 14.87
86.65 14.74
54.17 14.69
35.02 12.17
80.61 13.13
95.48 16.89
27.86 9.95
49.45 14.58
98.38 15.29
45.16 11.31
12.59 5.40
68.31 12.33
4.28 2.25
23.73 8.36
15.93 7.63
11.59 5.68
92.40 14.35
96.99 13.91
33.95 10.59
66.32 14.52
96.98 16.18
52.58 14.16
43.49 11.79
85.72 17.19
27.08 10.47
74.72 13.77
75.02 12.51
70.68 15.94
87.04 15.52
71.28 13.69
21.07 8.82
41.90 10.74
52.25 13.18
91.69 15.71
21.48 8.71
89.75 18.13
96.79 12.83
2.70 1.40
18.87 8.00
50.94 13.11
87.41 17.39
19.56 8.58
39.15 11.08
88.03 14.38
60.78 13.94
97.98 13.30
82.28 14.47
43.58 12.74
9.04 4.92
9.38 4.91
69.30 13.60
49.74 12.77
32.97 10.87
7.24 3.76
18.07 8.33
25.91 9.19
18.36 8.02
53.85 11.62
91.57 16.03
66.64 12.97
73.22 14.53
5.93 3.12
33.44 10.09
66.50 13.23
25.23 9.87
76.83 16.27
5.47 3.15
27.01 9.86
74.60 14.90
39.01 11.94
77.60 16.66
69.57 13.96
95.22 16.33
97.01 13.71
17.95 7.08
10.97 5.39
89.83 10.61
58.87 11.90
70.72 13.89
24.75 10.02
44.27 12.15
33.29 11.06
9.21 4.88
61.26 12.21
81.34 12.59
33.64 11.32
20.91 8.73
86.66 14.03
58.87 14.30
10.48 5.19
91.54 15.45
43.95 12.25
34.55 11.91
82.23 18.02
38.93 12.18
85.50 14.82
76.50 14.80
3.43 2.01
44.14 13.58
22.18 9.56
21.05 8.84
95.22 16.12
0.55 0.34
27.02 10.02
17.62 7.16
24.84 9.70
55.58 13.55
27.10 10.16
50.45 12.58
22.70 8.64
86.77 16.11
68.12 16.27
51.02 12.75
72.40 12.75
49.59 13.59
42.15 11.47
54.72 11.71
93.28 13.45
60.88 12.77
25.06 10.20
1.85 1.03
15.03 7.09
2.32 1.30
8.44 4.54
47.05 12.84
52.00 11.23
52.77 14.96
19.06 7.93
14.88 7.12
58.97 13.17
53.53 13.27
9.98 4.64
64.79 11.18
81.78 13.54
40.54 10.79
97.13 11.62
22.05 9.08
5.37 2.90
22.36 8.88
38.78 11.78
5.05 2.75
99.02 15.07
7.08 3.68
89.14 12.13
36.73 12.72
8.85 4.32
88.67 14.83
40.70 10.83
86.17 15.26
25.81 10.28
47.87 12.39
36.28 11.68
57.63 15.44
49.14 13.62
59.24 14.69
94.10 15.62
47.42 12.42
51.69 12.12
96.26 15.86
12.76 5.96
62.93 15.39
42.12 12.16
0.66 0.39
2.39 1.51
76.14 12.90
92.52 11.97
48.88 12.15
12.93 6.01
17.19 7.62
12.12 5.79
58.81 14.04
29.31 11.10
18.59 8.18
37.28 10.59
56.01 16.59
38.07 12.03
87.57 15.70
35.44 12.42
58.38 12.73
52.67 12.55
7.38 3.69
61.99 12.37
40.17 13.57
78.55 14.81
55.72 10.61
25.58 9.75
7.14 3.69
16.15 7.07
72.85 12.79
47.09 15.05
73.15 16.62
16.27 7.04
8.65 4.54
58.88 14.15
57.21 14.46
67.42 15.75
71.06 13.43
85.14 13.04
78.66 14.86
28.29 10.56
27.38 9.38
24.61 9.53
66.31 11.66
33.47 11.57
72.39 15.05
57.04 13.10
44.81 12.14
64.14 13.52
33.02 11.36
57.36 16.88
15.33 6.70
97.93 17.31
7.62 3.84
83.66 14.65
51.12 11.47
29.79 9.85
55.42 14.14
7.86 4.08
89.39 16.82
55.59 13.21
12.17 6.17
73.58 15.40
55.58 13.60
36.02 11.22
9.57 4.96
38.23 11.21
86.80 16.30
71.16 13.19
71.04 16.17
55.65 12.73
89.59 17.42
20.69 8.37
78.06 12.96
29.25 10.28
99.84 12.02
7.08 3.88
83.09 14.09
11.73 5.55
0.75 0.43
9.29 4.64
94.71 15.64
89.23 9.30
57.47 13.01
40.61 10.49
66.86 14.50
88.47 12.57
49.48 13.82
36.69 11.66
71.21 14.17
67.88 14.55
19.72 8.08
32.98 11.99
57.57 13.03
26.46 9.74
77.81 14.50
81.80 12.58
34.23 10.26
34.22 12.00
38.76 11.87
16.38 7.01
3.80 2.16
4.72 2.51
25.76 9.45
86.01 13.77
92.93 15.86
88.94 14.29
8.92 4.39
8.10 4.31
80.99 15.12
46.17 12.42
84.59 14.19
95.28 15.82
51.57 15.61
14.20 6.42
97.94 15.80
69.80 16.88
29.62 10.27
75.34 14.46
85.16 14.18
6.15 3.02
33.56 10.54
16.23 7.21
51.53 11.67
31.21 10.89
24.50 8.78
27.44 10.72
18.80 7.37
38.93 11.90
98.00 17.29
82.67 11.63
66.69 14.91
87.23 13.95
17.69 7.47
8.08 4.49
63.36 13.93
88.08 14.29
90.56 14.57
50.04 12.45
75.29 15.41
53.17 13.11
34.58 10.53
51.40 13.00
39.31 12.54
88.83 12.74
13.22 6.30
39.93 13.47
54.28 12.55
77.08 12.95
76.65 16.88
64.96 12.34
78.15 13.93
4.84 2.59
86.24 13.01
23.57 8.73
94.70 13.27
64.95 12.28
82.36 15.83
24.39 9.98
31.31 9.97
93.61 11.29
58.83 14.59
27.09 9.99
20.42 8.14
84.31 12.09
70.43 11.47
28.84 9.54
0.58 0.35
85.08 16.21
38.76 12.57
60.84 12.61
77.69 11.93
82.45 14.22
76.28 13.52
59.19 10.62
3.94 2.14
42.04 10.80
32.88 12.00
27.22 10.66
47.93 13.64
98.18 14.93
48.18 13.12
11.33 5.43
84.15 14.62
30.67 10.31
12.73 6.11
69.35 15.14
13.62 6.45
27.67 9.36
90.26 13.83
57.01 12.72
56.77 13.04
98.96 14.45
24.04 9.03
50.40 11.89
20.13 7.16
72.92 15.54
73.70 14.76
29.38 10.45
35.03 10.81
23.50 8.15
18.81 8.55
89.28 15.61
6.45 3.55
9.63 4.95
9.94 5.33
60.58 13.57
83.92 13.03
70.53 13.71
83.00 13.26
55.63 13.78
94.44 17.22
46.91 13.10
96.83 14.62
56.51 14.63
20.43 9.20
81.15 14.35
59.05 12.60
39.72 11.88
15.33 6.33
9.01 4.23
6.12 3.25
25.68 9.42
58.72 14.00
4.32 2.21
80.87 17.65
45.62 11.61
4.95 2.80
96.75 9.99
34.34 11.83
72.09 13.76
51.53 11.63
5.67 3.05
1.99 1.18
96.36 16.26
14.64 6.41
83.07 16.16
73.11 15.34
90.85 16.13
62.20 15.48
26.37 10.06
43.07 15.05
7.34 3.75
45.92 10.68
88.99 15.67
12.49 5.59
91.70 15.23
77.96 15.42
67.11 13.93
55.44 15.23
36.00 11.08
87.01 18.38
95.86 17.70
31.42 11.15
75.11 11.59
28.86 10.30
38.99 10.81
2.55 1.42
0.93 0.56
68.77 11.99
64.21 14.19
70.84 12.17
11.79 5.64
90.72 12.05
40.44 10.27
50.61 14.41
70.37 17.79
44.49 12.19
66.15 15.22
53.43 10.38
3.23 1.76
14.14 6.49
44.76 12.98
6.33 3.25
46.91 10.97
12.83 5.46
43.33 13.25
54.77 14.90
37.58 11.49
5.89 3.20
7.77 3.93
27.40 9.84
95.65 16.61
57.53 14.81
51.79 12.47
0.13 0.08
93.64 14.51
40.32 11.89
27.99 11.21
78.94 14.36
9.18 4.53
54.55 11.21
82.60 17.47
60.40 13.84
58.35 14.53
13.42 5.92
21.84 7.93
32.66 11.67
68.33 11.04
72.88 15.73
87.25 14.10
6.98 3.82
86.98 15.16
76.98 9.83
79.54 11.77
58.43 17.21
47.81 12.35
66.82 17.19
31.17 10.73
73.53 15.61
14.84 6.72
84.18 15.78
4.97 2.73
34.43 12.39
68.89 15.01
76.38 15.48
40.99 12.98
98.27 14.98
26.93 10.02
22.03 9.45
44.09 13.46
5.15 2.78
0.72 0.43
7.79 4.03
75.64 12.03
5.37 2.84
71.37 11.50
47.16 12.26
9.25 4.83
26.08 10.00
59.80 14.35
12.76 5.65
18.28 8.38
18.11 8.22
2.79 1.54
57.56 12.45
77.34 13.68
80.78 11.28
37.82 11.19
80.71 16.30
63.22 15.95
72.89 11.05
18.86 7.95
35.80 12.44
22.47 9.62
79.16 17.78
46.16 13.48
67.75 17.13
79.33 13.15
58.83 14.45
57.01 14.78
39.60 10.25
64.05 12.66
62.58 12.46
69.19 14.33
5.54 2.96
78.36 14.77
23.31 8.44
50.02 13.36
64.44 12.17
42.29 12.01
21.32 10.16
69.46 12.93
91.85 15.10
72.99 16.61
82.16 15.00
64.36 14.06
8.14 4.47
37.91 11.45
60.41 10.85
56.61 13.65
80.72 14.89
27.77 8.56
40.61 12.38
75.93 14.24
95.71 16.50
65.36 12.60
40.98 12.30
23.91 9.24
95.73 14.31
21.23 8.91
50.77 14.28
82.39 16.33
5.93 3.22
8.89 4.84
32.17 10.22
50.70 12.07
14.44 6.54
84.34 14.29
73.11 12.98
79.54 12.70
60.82 12.69
95.46 14.53
64.96 16.03
1.97 1.15
95.20 19.52
32.58 10.32
40.53 11.30
91.05 12.62
49.48 11.60
44.28 12.39
19.30 8.06
47.14 12.28
82.60 16.03
72.08 15.57
26.98 8.13
36.84 11.30
98.54 15.65
31.69 10.67
75.69 15.81
80.18 15.27
15.86 6.84
17.24 7.69
90.00 15.06
76.74 12.95
59.46 15.47
87.93 14.56
62.76 16.05
12.66 5.88
9.98 5.02
66.98 14.21
85.88 12.18
88.14 13.00
62.50 11.99
46.42 11.76
69.22 12.35
26.33 9.39
95.57 15.14
0.20 0.11
46.04 12.21
8.83 4.22
90.61 14.17
20.78 8.10
82.14 14.50
82.15 12.07
91.25 14.39
28.38 9.13
71.07 14.94
24.98 9.50
58.51 13.53
32.63 11.26
3.02 1.78
37.28 10.00
46.09 13.12
42.84 12.12
61.93 12.05
72.66 12.24
44.83 12.66
69.78 13.46
40.98 11.83
97.44 16.02
72.32 13.90
96.93 13.70
83.13 16.13
72.50 14.07
60.33 15.33
4.75 2.64
90.22 11.27
16.43 6.89
18.68 8.26
52.58 10.59
79.73 14.90
87.40 15.39
70.64 13.75
76.72 11.95
93.55 16.03
36.22 10.32
22.88 8.97
92.35 15.94
14.29 6.97
68.92 14.52
90.19 16.78
3.67 2.07
81.96 12.49
16.05 7.55
30.80 10.49
96.99 17.87
59.81 16.62
95.92 15.73
40.59 12.38
2.55 1.40
71.17 14.05
20.14 9.29
60.55 14.78
44.07 11.87
84.13 15.73
15.28 6.85
29.75 11.13
59.27 14.10
3.26 1.97
79.55 15.17
94.48 11.55
49.39 13.66
17.60 7.26
92.36 14.78
14.39 5.94
31.78 12.01
24.98 8.64
Nota: Tamaño muestral n = 2499 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 MgO\nen muestras geoquímicas de depósitos minerales",
  xlab = "SiO2 (%)",
  ylab = "MgO (%)",
  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

Inicialmente se realizó una depuración de datos faltantes y valores no válidos para el cálculo logarítmico. Posteriormente, se aplicó una agregación de datos agrupando los valores de SiO2 y calculando el promedio de MgO. Esto permite reducir la dispersión de los datos y observar claramente la tendencia logarítmica.

# Asignar datos filtrados previamente
datos_limpios <- TVP_original

# Ordenar los datos
datos_limpios <- datos_limpios[order(datos_limpios$X, datos_limpios$Y), ]

# Agrupar SiO2 para simplificar los pares de valores (redondeo a 1 decimal)
datos_limpios$SiO2_grupo <- round(datos_limpios$X, 1)

# Calcular el promedio de MgO para cada grupo de SiO2
datos_prom <- aggregate(
  Y ~ SiO2_grupo,
  data = datos_limpios,
  mean
)

# Variables simplificadas
X <- datos_prom$SiO2_grupo
Y <- datos_prom$Y

# Crear data.frame simplificado
TVP_SIO2_MGO <- data.frame(X, Y)

# FILTRADO DE SEGURIDAD: Garantizar que no existan valores <= 0 tras la agregación
TVP_SIO2_MGO <- TVP_SIO2_MGO[TVP_SIO2_MGO$X > 0 & TVP_SIO2_MGO$Y > 0, ]

# Actualizar variables X e Y
X <- TVP_SIO2_MGO$X
Y <- TVP_SIO2_MGO$Y

n_modelo <- nrow(TVP_SIO2_MGO)

cat("Tamaño muestral del modelo =", n_modelo)
## Tamaño muestral del modelo = 913
tabla <- TVP_SIO2_MGO %>%
  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 MgO en muestras geoquímicas de depósitos minerales*")
  ) %>%
  tab_source_note(
    source_note = md(paste0(
      "**Nota:** Tamaño muestral n = ", nrow(TVP_SIO2_MGO),
      " observaciones utilizadas en el análisis."
    ))
  )

div(
  style = "height:400px; overflow-y:auto;",
  tabla
)
Tabla N°2
Pares de valores simplificada de SiO2 y MgO en muestras geoquímicas de depósitos minerales
X Y
0.10 0.07
0.20 0.13
0.30 0.18
0.40 0.23
0.50 0.30
0.60 0.35
0.70 0.41
0.80 0.44
0.90 0.53
1.00 0.58
1.10 0.65
1.20 0.71
1.30 0.80
1.40 0.83
1.60 0.89
1.70 0.96
1.80 1.05
1.90 1.09
2.00 1.14
2.10 1.21
2.20 1.22
2.30 1.30
2.40 1.38
2.50 1.41
2.60 1.37
2.70 1.40
2.80 1.58
2.90 1.60
3.00 1.71
3.10 1.73
3.20 1.86
3.30 1.89
3.40 1.98
3.50 1.94
3.60 2.02
3.70 2.04
3.80 2.16
3.90 2.13
4.00 2.21
4.10 2.22
4.20 2.33
4.30 2.23
4.40 2.26
4.60 2.43
4.70 2.51
4.80 2.58
5.00 2.75
5.20 2.80
5.30 2.84
5.40 2.87
5.50 3.05
5.60 3.05
5.70 3.13
5.90 3.21
6.00 3.11
6.10 3.29
6.20 3.21
6.30 3.37
6.40 3.31
6.60 3.41
6.70 3.82
6.80 3.49
6.90 3.42
7.00 3.78
7.10 3.77
7.20 3.87
7.30 3.71
7.40 3.74
7.50 3.73
7.60 4.13
7.70 3.95
7.80 3.98
7.90 4.09
8.00 4.09
8.10 4.42
8.30 4.31
8.40 4.36
8.50 4.29
8.60 4.42
8.70 4.54
8.80 4.47
8.90 4.60
9.00 4.61
9.10 4.37
9.20 4.74
9.30 4.63
9.40 4.83
9.50 4.69
9.60 4.96
9.70 4.57
9.80 5.15
9.90 5.05
10.00 4.79
10.10 4.95
10.20 5.35
10.30 5.08
10.40 5.06
10.50 5.28
10.60 5.04
10.70 5.45
10.90 5.14
11.00 5.30
11.10 5.43
11.30 5.43
11.40 5.69
11.50 5.48
11.60 5.49
11.70 5.55
11.80 5.57
11.90 5.70
12.00 6.05
12.10 5.79
12.20 5.95
12.30 5.73
12.40 5.36
12.50 5.79
12.60 5.40
12.70 6.00
12.80 5.82
12.90 6.01
13.20 6.30
13.40 6.20
13.50 6.16
13.60 6.54
13.70 6.38
13.80 6.51
13.90 6.15
14.10 6.39
14.20 6.50
14.30 6.65
14.40 6.32
14.50 6.36
14.60 6.60
14.80 6.65
14.90 6.56
15.00 6.91
15.10 6.79
15.20 6.70
15.30 6.67
15.40 7.00
15.50 7.08
15.60 6.71
15.70 6.99
15.80 6.58
15.90 7.09
16.00 7.06
16.10 6.91
16.20 6.91
16.30 7.04
16.40 7.09
16.50 7.27
16.60 7.15
16.70 7.20
16.80 7.36
16.90 7.33
17.00 7.11
17.10 7.89
17.20 7.68
17.30 7.58
17.40 7.33
17.50 7.80
17.60 7.31
17.70 7.33
17.80 7.40
17.90 8.00
18.00 7.54
18.10 8.22
18.20 7.84
18.30 8.38
18.40 7.79
18.50 7.42
18.60 8.40
18.70 8.09
18.80 7.96
18.90 7.89
19.00 8.00
19.10 8.08
19.20 8.13
19.30 8.03
19.40 8.62
19.50 8.25
19.60 8.24
19.70 8.56
19.80 8.19
19.90 8.90
20.00 8.41
20.10 8.22
20.20 8.02
20.30 8.11
20.40 8.67
20.50 9.71
20.60 8.33
20.70 8.63
20.80 8.37
20.90 8.73
21.00 8.76
21.10 8.82
21.20 8.90
21.30 8.73
21.40 8.91
21.50 8.71
21.60 8.59
21.70 8.60
21.80 8.71
21.90 8.69
22.00 8.87
22.10 8.70
22.20 8.80
22.30 9.25
22.40 8.37
22.50 9.62
22.60 9.01
22.70 8.57
22.80 9.24
22.90 9.23
23.00 9.11
23.10 8.80
23.30 9.05
23.40 9.12
23.50 8.54
23.60 8.85
23.70 8.25
23.80 9.31
23.90 8.74
24.00 8.61
24.10 9.32
24.20 8.72
24.30 10.10
24.40 9.10
24.50 8.89
24.60 9.53
24.70 9.47
24.80 9.69
24.90 9.23
25.00 9.15
25.10 10.20
25.20 9.69
25.50 9.36
25.60 9.92
25.70 9.23
25.80 10.10
25.90 9.25
26.00 9.93
26.10 10.00
26.20 9.52
26.30 9.41
26.40 10.06
26.50 9.74
26.60 9.83
26.70 9.98
26.80 9.71
26.90 9.77
27.00 9.35
27.10 10.21
27.20 10.86
27.30 10.01
27.40 10.08
27.50 10.53
27.70 10.01
27.80 10.10
27.90 10.04
28.00 11.21
28.10 9.30
28.20 10.34
28.30 10.56
28.40 9.88
28.60 9.39
28.80 9.59
28.90 10.35
29.00 10.29
29.10 10.61
29.20 10.28
29.30 11.10
29.40 10.18
29.50 10.32
29.60 10.39
29.80 10.41
29.90 10.22
30.00 10.07
30.10 10.29
30.20 10.90
30.40 10.67
30.50 11.59
30.60 10.23
30.70 10.31
30.80 10.86
30.90 10.79
31.10 10.62
31.20 10.81
31.30 10.69
31.40 11.01
31.50 11.00
31.60 10.64
31.70 11.47
31.80 12.01
31.90 11.04
32.00 10.45
32.20 10.22
32.30 10.23
32.40 10.81
32.50 10.74
32.60 10.83
32.70 11.39
32.80 10.46
32.90 10.91
33.00 11.40
33.20 10.63
33.30 11.06
33.40 10.55
33.50 11.63
33.60 10.93
33.70 11.77
33.80 11.52
33.90 11.06
34.00 10.62
34.10 12.15
34.20 11.22
34.30 11.26
34.40 11.56
34.50 11.44
34.60 10.84
34.70 11.71
34.80 10.18
34.90 10.83
35.00 11.31
35.10 10.99
35.20 11.06
35.30 11.28
35.40 11.39
35.50 10.50
35.60 10.46
35.70 10.73
35.80 11.83
35.90 11.23
36.00 10.95
36.10 12.16
36.20 10.95
36.30 10.79
36.40 12.04
36.50 11.25
36.60 12.60
36.70 11.67
36.80 11.44
36.90 12.43
37.00 11.63
37.20 11.02
37.30 11.04
37.40 13.29
37.50 10.48
37.60 11.49
37.80 11.59
37.90 11.54
38.00 10.40
38.10 12.03
38.20 11.54
38.30 10.73
38.40 11.81
38.70 11.74
38.80 11.70
38.90 11.60
39.00 11.79
39.10 11.47
39.20 12.09
39.30 11.86
39.40 12.09
39.60 11.36
39.70 11.88
39.90 13.47
40.00 12.50
40.10 14.61
40.20 13.89
40.30 11.65
40.40 12.42
40.50 11.29
40.60 11.99
40.70 10.83
40.90 11.56
41.00 12.34
41.10 12.31
41.20 11.71
41.40 12.82
41.60 11.31
41.70 12.27
41.80 12.24
41.90 10.98
42.00 10.80
42.10 12.25
42.20 11.56
42.30 12.01
42.50 12.49
42.60 13.38
42.70 12.59
42.80 12.57
43.00 11.92
43.10 12.81
43.20 11.23
43.30 13.19
43.40 11.56
43.50 12.26
43.60 11.79
43.80 11.26
43.90 12.45
44.00 12.05
44.10 12.75
44.20 10.76
44.30 12.64
44.40 11.96
44.50 12.34
44.60 12.87
44.70 12.51
44.80 12.44
44.90 14.01
45.00 12.84
45.10 12.80
45.20 12.18
45.40 14.19
45.50 12.71
45.60 11.61
45.70 12.04
45.80 13.51
45.90 12.32
46.00 12.21
46.10 12.80
46.20 13.26
46.30 11.56
46.40 12.52
46.50 12.95
46.60 11.64
46.70 12.88
46.80 12.20
46.90 12.51
47.00 12.81
47.10 14.04
47.20 12.26
47.30 13.17
47.40 12.42
47.50 12.74
47.60 12.41
47.70 13.59
47.80 12.38
47.90 13.15
48.00 13.72
48.10 12.33
48.20 12.62
48.30 14.44
48.40 13.54
48.50 13.20
48.60 13.21
48.70 11.47
48.80 12.96
48.90 12.67
49.00 13.09
49.10 12.59
49.20 12.24
49.30 13.56
49.40 12.64
49.50 13.26
49.60 12.61
49.70 13.36
49.80 12.57
49.90 12.79
50.00 12.91
50.20 13.08
50.40 13.13
50.50 13.46
50.60 12.90
50.70 12.09
50.80 14.28
50.90 13.11
51.00 12.34
51.10 11.47
51.20 10.98
51.30 13.53
51.40 12.70
51.50 12.86
51.60 13.78
51.70 13.40
51.80 12.06
52.00 11.26
52.10 13.01
52.20 12.58
52.50 13.09
52.60 12.83
52.70 12.16
52.80 13.26
52.90 13.49
53.00 12.95
53.10 13.82
53.20 14.53
53.40 10.38
53.50 13.27
53.60 12.21
53.80 10.40
53.90 12.13
54.00 12.89
54.10 14.75
54.20 13.68
54.30 13.91
54.40 13.23
54.50 11.21
54.70 13.08
54.80 13.04
54.90 13.11
55.00 11.88
55.30 14.30
55.40 14.69
55.50 12.96
55.60 13.12
55.70 12.67
55.80 15.18
55.90 13.62
56.00 15.00
56.20 13.08
56.30 15.67
56.50 14.63
56.60 14.05
56.70 14.80
56.80 14.30
56.90 14.74
57.00 13.44
57.10 14.35
57.20 13.17
57.30 13.76
57.40 15.30
57.50 13.08
57.60 13.86
57.70 12.72
57.90 12.78
58.10 14.03
58.20 12.81
58.30 14.28
58.40 14.66
58.50 13.72
58.60 12.53
58.70 14.00
58.80 13.36
58.90 13.84
59.00 13.44
59.10 13.15
59.20 12.65
59.30 14.10
59.40 12.98
59.50 14.17
59.60 13.22
59.70 14.37
59.80 14.50
59.90 13.58
60.00 14.38
60.10 12.48
60.30 14.38
60.40 12.46
60.50 13.21
60.60 13.14
60.70 14.96
60.80 13.50
60.90 13.70
61.00 14.24
61.10 12.83
61.20 11.53
61.30 11.29
61.40 13.57
61.50 13.63
61.60 14.15
61.70 13.64
61.90 13.44
62.00 13.12
62.10 13.33
62.20 14.43
62.30 15.64
62.40 13.86
62.50 12.79
62.60 13.49
62.70 13.79
62.80 13.79
62.90 13.67
63.10 14.31
63.20 14.47
63.40 13.97
63.50 13.60
63.60 13.71
63.70 13.91
63.80 12.91
63.90 13.96
64.00 12.35
64.10 13.57
64.20 13.44
64.30 13.41
64.40 12.65
64.50 14.31
64.60 12.55
64.70 11.61
64.80 13.83
64.90 14.83
65.00 13.55
65.10 17.69
65.20 13.46
65.40 13.27
65.50 13.57
65.60 11.91
65.70 12.01
65.80 12.54
65.90 13.59
66.00 15.06
66.10 12.89
66.20 14.02
66.30 13.02
66.40 12.60
66.50 15.23
66.60 14.87
66.70 12.68
66.80 15.64
66.90 14.25
67.00 14.11
67.10 13.43
67.20 12.68
67.30 12.93
67.40 14.58
67.70 13.82
67.80 14.62
67.90 13.86
68.00 14.10
68.10 15.32
68.20 13.03
68.30 12.66
68.40 14.63
68.50 16.21
68.70 15.30
68.80 13.18
68.90 13.97
69.00 14.28
69.20 13.42
69.30 13.99
69.50 13.34
69.60 13.35
69.70 14.85
69.80 14.59
69.90 15.79
70.00 13.45
70.10 14.78
70.20 11.64
70.40 13.99
70.50 13.77
70.60 14.47
70.70 14.61
70.80 13.59
70.90 14.84
71.00 15.67
71.10 14.18
71.20 13.66
71.30 14.10
71.40 14.00
71.50 13.28
71.60 11.44
71.70 13.44
71.80 13.41
71.90 14.15
72.00 11.53
72.10 13.72
72.20 13.16
72.30 15.59
72.40 14.03
72.50 13.45
72.60 15.08
72.70 13.54
72.80 13.87
72.90 13.35
73.00 14.13
73.10 14.19
73.20 15.57
73.30 14.59
73.40 14.13
73.50 15.73
73.60 13.51
73.70 14.14
73.80 14.02
73.90 13.21
74.00 14.90
74.10 14.54
74.20 14.57
74.50 12.27
74.60 14.74
74.70 12.94
74.80 13.57
74.90 14.27
75.00 14.18
75.10 13.31
75.30 14.94
75.50 13.35
75.60 14.08
75.70 15.49
75.80 12.65
75.90 13.41
76.10 14.57
76.20 13.03
76.30 14.17
76.40 15.52
76.50 14.96
76.60 14.12
76.70 14.08
76.80 16.77
77.00 9.83
77.10 13.04
77.20 13.44
77.30 15.22
77.50 16.79
77.60 14.87
77.70 12.28
77.80 13.01
77.90 13.87
78.00 15.07
78.10 15.10
78.20 13.26
78.30 13.56
78.40 15.53
78.50 11.76
78.60 15.61
78.70 15.71
78.80 14.28
78.90 15.07
79.00 14.27
79.10 13.62
79.20 14.39
79.30 13.15
79.40 14.39
79.50 13.35
79.60 14.08
79.70 15.12
79.80 15.85
79.90 15.01
80.00 14.05
80.10 14.96
80.20 15.29
80.30 13.21
80.40 13.73
80.50 14.54
80.60 14.10
80.70 14.42
80.80 13.78
80.90 17.65
81.00 15.15
81.10 15.46
81.20 15.92
81.30 12.59
81.40 15.07
81.50 14.73
81.60 11.68
81.70 15.15
81.80 14.49
81.90 12.76
82.00 15.00
82.10 15.09
82.20 15.38
82.30 14.03
82.40 14.93
82.50 12.04
82.60 14.96
82.70 15.78
82.80 13.77
82.90 17.17
83.00 13.12
83.10 14.56
83.20 12.79
83.30 14.92
83.50 15.96
83.60 12.41
83.70 16.04
83.80 15.08
83.90 12.81
84.00 12.80
84.10 15.73
84.20 15.20
84.30 13.08
84.50 14.72
84.60 14.26
84.70 13.63
84.80 15.01
85.00 14.12
85.10 14.14
85.20 14.45
85.40 15.25
85.50 14.82
85.70 15.62
85.90 14.06
86.00 14.86
86.10 15.43
86.20 14.10
86.30 16.58
86.50 16.16
86.60 13.88
86.70 14.90
86.80 16.07
86.90 13.05
87.00 15.06
87.10 15.65
87.20 14.04
87.30 17.27
87.40 16.79
87.50 13.25
87.60 14.22
87.70 14.81
87.80 13.34
87.90 14.88
88.00 14.97
88.10 13.64
88.20 14.68
88.30 15.75
88.40 14.91
88.50 12.57
88.60 15.40
88.70 14.83
88.80 14.72
88.90 14.37
89.00 15.78
89.10 14.16
89.20 12.94
89.30 14.75
89.40 14.71
89.50 16.11
89.60 17.42
89.70 13.71
89.80 14.54
89.90 14.32
90.00 15.28
90.10 12.59
90.20 13.86
90.30 13.68
90.50 13.61
90.60 14.96
90.70 13.45
90.80 16.18
90.90 14.25
91.00 15.86
91.10 16.75
91.20 14.65
91.30 14.33
91.50 15.45
91.60 16.48
91.70 13.69
91.80 15.94
91.90 15.08
92.00 13.06
92.10 13.81
92.20 13.91
92.30 15.22
92.40 13.67
92.50 11.97
92.60 12.91
92.80 15.25
92.90 15.23
93.00 14.70
93.10 16.93
93.20 14.29
93.30 11.93
93.50 13.29
93.60 14.55
93.70 16.08
93.80 15.57
94.00 16.94
94.10 14.77
94.20 15.69
94.30 16.59
94.40 14.76
94.50 12.55
94.60 13.88
94.70 14.46
94.80 15.14
94.90 15.12
95.00 13.84
95.10 17.26
95.20 15.38
95.30 15.82
95.40 14.28
95.50 15.38
95.60 14.43
95.70 15.81
95.80 15.41
95.90 17.02
96.00 13.71
96.10 12.49
96.20 10.36
96.30 16.56
96.40 15.08
96.60 14.99
96.70 16.69
96.80 13.91
96.90 14.81
97.00 15.36
97.10 14.16
97.20 12.28
97.30 18.25
97.40 15.64
97.50 17.69
97.60 13.00
97.70 16.09
97.80 12.77
97.90 15.86
98.00 15.79
98.10 18.33
98.20 14.93
98.30 15.45
98.40 16.11
98.50 15.74
98.60 16.73
98.70 14.62
98.80 13.94
98.90 15.21
99.00 14.76
99.40 14.56
99.50 15.20
99.70 13.83
99.80 12.70
99.90 12.54
100.00 14.36
Nota: Tamaño muestral n = 913 observaciones utilizadas en el análisis. 
# 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 MgO\nen muestras geoquímicas de depósitos minerales",
  xlab = "SiO2 (%)",
  ylab = "MgO (%)",
  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 curvilínea con concavidad marcada entre SiO2 y MgO. A medida que aumenta el porcentaje de SiO2, el valor de MgO disminuye de forma acelerada en niveles bajos de sílice y luego atenúa su ritmo de cambio en niveles elevados. Por ello, se propone un modelo logarítmico de la forma \(Y = a + b \ln(X)\) para representar con precisión el comportamiento geoquímico entre ambas variables.

PARÁMETROS

# Modelo logarítmico general:
# Y = a + b * ln(X)

#Modelo logarítmico aplicado al estudio:
# MgO = a + b * ln(SiO2)

#Ajuste del modelo:

modelo_log <- lm(
  Y ~ log(X),
  data = TVP_SIO2_MGO
)

summary(modelo_log)
## 
## Call:
## lm(formula = Y ~ log(X), data = TVP_SIO2_MGO)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -4.5419 -0.8135 -0.0438  0.6278 10.8957 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2.20367    0.15350  -14.36   <2e-16 ***
## log(X)       3.74595    0.04109   91.16   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.237 on 911 degrees of freedom
## Multiple R-squared:  0.9012, Adjusted R-squared:  0.9011 
## F-statistic:  8310 on 1 and 911 DF,  p-value: < 2.2e-16
param <- coef(modelo_log)

a_est <- param[1]
b_est <- param[2]

a_est
## (Intercept) 
##   -2.203672
b_est
##   log(X) 
## 3.745951
cat(
  "Y = ",
  round(a_est, 4),
  " + ",
  round(b_est, 4),
  " * ln(X)"
)
## Y =  -2.2037  +  3.746  * ln(X)

Justificación del uso de la regresión lineal lm()

Aunque la relación subyacente es no lineal respecto a la variable original \(X\), se utiliza la función lm() mediante la transformación logarítmica de la variable explicativa \(X' = \ln(X)\). Esto convierte la estructura matemática en un modelo lineal respecto a sus parámetros (\(Y = a + b X'\)), permitiendo aplicar el método de mínimos cuadrados ordinarios en R.

# Modelo logarítmico:
# Y = a + b * ln(X)

# Definiendo la variable transformada:
# X1 = ln(X)

# El modelo queda expresado de forma lineal como:
# Y = β0 + β1 * X1

# Por lo tanto, la estimación en R se ejecuta mediante:
# lm(MgO ~ log(SiO2))

# Reconstruyendo la ecuación para el estudio geoquímico:
# MgO = a + b * ln(SiO2)

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 logarítmico entre SiO2 y MgO\nen muestras geoquímicas de depósitos minerales",
  xlab = "SiO2 (%)",
  ylab = "MgO (%)",
  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)

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

# Curva del modelo logarítmico
x_seq <- seq(
  min(X, na.rm = TRUE),
  max(X, na.rm = TRUE),
  length.out = 1000
)

y_pred <- a_est + b_est * log(x_seq)

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

box(lwd = 1.5)

legend(
  "topright",
  legend = c("Datos reales", "Modelo logarítmico"),
  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_log, newdata = TVP_SIO2_MGO)

r <- cor(TVP_SIO2_MGO$Y, Y_estimado) * 100
r
## [1] 94.93157
# Coeficiente de determinación R^2
R2 <- summary(modelo_log)$r.squared * 100
R2
## [1] 90.12003
# Tabla resumen del modelo
Variable <- c("SiO2 - MgO")

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

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

knitr::kable(
  tabla_resumen_logaritmico,
  format = "markdown",
  caption = "Tabla resumen del modelo logarítmico"
)
Tabla resumen del modelo logarítmico
Variable Test Pearson (%) R cuadrado (%)
SiO2 - MgO 94.93 90.12

RESTRICCIONES

# Dominio de X:
# D(X) = {R+ | X > 0}

# 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?

Debido a la naturaleza de la función logarítmica, la variable independiente \(X\) (SiO2) requiere estrictamente valores mayores a cero (\(X > 0\)). Geoquímicamente, tanto el SiO2 como el MgO representan porcentajes en peso y deben tomar valores positivos. El modelo debe aplicarse dentro del rango observado de SiO2, dado que extrapolaciones fuera de este dominio pueden derivar en estimaciones de MgO físicamente imposibles o negativas.

ESTIMACIONES

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

if(SiO2_objetivo <= 0){
  stop("Error: El contenido de SiO2 debe ser estrictamente mayor a 0.")
}

MgO_est <- a_est + b_est * log(SiO2_objetivo)

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

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

incremento_pct <- 0.10

nuevo_SiO2 <- SiO2_objetivo * (1 + incremento_pct)

# Cálculo del MgO actual
MgO_actual <- a_est + b_est * log(SiO2_objetivo)

# Cálculo del nuevo MgO
nuevo_MgO <- a_est + b_est * log(nuevo_SiO2)

porcentaje_cambio_MgO <- ((nuevo_MgO - MgO_actual) / MgO_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 MgO esperado\n",
    "al incrementar el SiO2 un ", (incremento_pct * 100), "%?\n\n",
    "Cambio estimado de MgO: ",
    round(porcentaje_cambio_MgO, 2), "%"
  ),
  cex = 1.2,
  col = "firebrick3",
  font = 2
)

CONCLUSIÓN

Entre el contenido de SiO2 y el porcentaje de MgO existe una relación de tipo logarítmica representada por el modelo \(Y = a + b \ln(X)\). El ajuste obtenido permite modelar el comportamiento geoquímico caracterizado por una variación acentuada en niveles bajos de sílice y una posterior estabilización. Con un coeficiente de correlación de Pearson de r round(r, 2)% y un coeficiente de determinación de r round(R2, 2)%, el modelo resulta adecuado para predecir el contenido de MgO dentro del rango de datos analizado.