modelo potencial ENTRE EL SiO2 Y EL Cu

ANÁLISIS MULTIVARIABLE

CARGA DE DATOS Y LIBRERÍAS

#Carga de datos

datos <- read.csv(
  "D:/geoquimica.csv",
  header = TRUE,
  sep = ",",
  dec = "."
)

#Carga de librerias

library(readxl)
library(gt)
library(dplyr)
library(e1071)
library(htmltools)
# Extraer y definir variables

area <- as.numeric(datos$SiO2_pct) #Variable independiente

toneladas <- as.numeric(datos$Cu_ppm) #Variable dependiente

DEFINICIÓN DE LA VARIABLE

El porcentaje de dióxido de silicio (SiO2) influye en la variación de la concentración de cobre (Cu) dentro de las muestras geoquímicas. Esta relación sigue una tendencia no lineal de decrecimiento o variabilidad exponencial/potencial, por lo que se propone un modelo matemático potencial (\(Y = a \cdot X^b\)) para representar adecuadamente el comportamiento entre ambas variables mineralógicas.

# Variable independiente (X): Dióxido de silicio (%)
X <- datos$SiO2_pct

# Variable dependiente (Y): Concentración de Cobre (ppm)
Y <- datos$Cu_ppm

# Conversión a numérico
X <- as.numeric(X)
Y <- as.numeric(Y)

# Eliminar datos faltantes y valores no positivos (requeridos para transformación logarítmica)
TVP_original <- data.frame(X, Y)
TVP_original <- na.omit(TVP_original)
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 = 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 Cu 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 Cu en muestras geoquímicas de depósitos minerales
X	Y
106.88	103.30
13.96	17.71
117.66	102.23
33.48	23.80
91.58	79.23
65.57	73.73
109.37	90.00
106.67	106.72
15.27	22.34
64.50	73.51
44.52	41.42
29.41	31.05
46.48	43.47
36.14	32.54
92.59	84.94
59.98	64.89
16.38	29.27
36.67	33.34
96.36	71.70
64.11	73.41
57.13	47.50
34.34	26.13
18.11	25.82
99.80	91.62
41.54	35.41
22.52	28.21
29.72	24.50
109.89	101.65
9.96	20.03
48.23	43.59
41.57	36.59
45.71	45.88
112.34	107.58
49.68	53.70
108.03	91.21
21.73	26.36
61.39	61.81
111.18	114.62
97.15	77.80
102.56	89.80
81.53	74.05
21.47	38.66
18.35	26.83
13.90	19.08
49.03	45.12
21.84	28.03
56.50	45.21
45.89	37.90
54.81	64.47
97.09	69.09
45.83	32.89
10.49	24.21
102.70	92.65
76.69	75.87
21.28	40.33
119.66	104.59
54.51	52.84
111.10	89.60
31.50	22.94
64.44	76.68
90.38	82.39
15.07	25.25
28.44	28.14
68.98	72.92
5.80	19.83
46.01	48.84
102.72	103.71
15.50	29.43
19.99	35.19
99.22	103.66
86.80	85.16
64.30	54.81
42.78	24.88
76.16	71.43
100.27	81.11
64.68	68.36
79.92	63.70
102.40	88.71
59.97	68.75
87.84	70.85
86.16	87.16
94.06	62.95
23.15	25.57
16.55	34.94
78.58	71.99
19.06	31.59
67.52	72.28
53.62	57.87
91.18	95.60
6.61	18.76
102.67	99.94
94.66	89.61
32.38	25.70
105.08	100.82
56.02	64.98
26.37	27.01
111.64	101.47
21.79	28.81
69.83	79.13
112.94	100.89
36.34	37.23
82.43	67.97
39.22	32.90
113.70	124.60
85.27	75.59
89.74	73.20
109.40	101.30
16.43	27.02
58.32	56.15
6.33	25.16
37.47	37.18
95.09	87.08
34.25	25.59
40.20	29.29
8.27	19.54
68.18	83.33
36.20	31.11
25.96	33.29
104.56	106.40
23.09	24.21
34.35	25.15
80.28	68.19
7.60	13.74
115.62	109.96
108.23	97.75
18.86	30.61
88.04	72.24
115.26	105.53
61.84	74.99
112.96	113.85
62.24	71.62
68.79	62.35
62.73	70.63
101.04	84.48
19.06	24.92
14.01	21.59
55.66	51.69
39.63	32.14
102.16	98.50
82.07	81.36
95.86	89.43
14.13	35.33
80.68	83.86
30.42	25.85
45.71	47.67
72.93	73.83
62.35	69.02
95.90	68.00
59.17	61.61
38.50	44.91
59.41	66.80
92.84	96.73
94.31	91.62
30.54	23.68
41.53	38.77
58.71	68.67
6.89	14.68
83.70	95.02
119.80	96.16
105.24	92.00
31.23	27.97
112.43	97.40
71.22	65.29
103.96	108.61
55.15	65.81
95.53	75.11
42.40	37.56
46.44	43.78
31.25	27.75
113.05	103.25
76.74	81.79
38.85	35.67
90.36	88.78
41.34	41.74
9.65	23.15
82.14	81.84
19.75	29.11
9.56	25.78
96.66	76.00
7.72	23.62
45.26	37.98
113.76	104.11
25.98	32.54
96.19	84.50
12.61	30.23
94.10	87.72
99.83	85.16
105.28	103.22
119.56	93.80
82.26	76.78
28.56	23.18
15.05	34.25
87.65	71.77
30.79	25.66
71.32	80.73
95.80	81.06
45.50	28.85
74.14	89.02
118.18	110.59
26.03	30.26
96.57	81.59
102.88	87.30
50.70	56.87
113.77	115.35
70.37	74.79
77.38	77.44
54.34	61.56
38.11	32.55
49.02	48.25
86.83	66.03
116.85	102.30
43.50	46.22
87.52	82.06
101.69	100.75
106.69	104.86
68.37	84.76
108.46	114.15
94.45	89.45
84.08	77.85
44.69	70.31
104.27	116.39
102.29	100.98
14.50	21.66
105.77	100.69
10.21	22.55
53.96	48.78
75.59	74.86
75.19	73.41
64.71	67.64
66.88	77.02
95.81	87.91
116.30	101.34
16.83	22.46
24.13	22.26
90.50	76.78
111.69	94.51
14.40	26.51
67.34	78.62
13.24	22.83
112.67	99.62
117.17	100.90
62.34	88.00
65.09	57.51
111.86	121.21
101.57	94.01
65.96	62.30
14.45	23.79
58.62	58.30
24.13	28.46
96.90	81.17
16.87	23.49
116.16	114.54
54.25	57.24
26.40	26.43
53.43	46.14
35.35	32.34
11.92	16.42
46.11	40.13
22.88	25.91
60.38	67.39
57.64	49.98
6.59	19.68
105.12	97.32
85.83	81.52
28.10	36.17
41.64	45.34
64.16	76.41
17.88	26.06
102.46	86.72
98.03	85.25
9.25	22.88
37.97	35.89
37.32	26.55
89.91	81.47
38.55	22.01
119.09	111.89
45.62	39.36
76.96	83.15
109.35	102.26
68.50	70.51
48.75	46.34
98.37	86.27
54.03	57.05
43.50	49.72
85.22	91.64
19.17	28.34
50.44	45.71
23.37	25.66
40.74	42.30
9.47	20.81
110.79	104.08
37.44	28.95
85.17	85.63
40.17	32.49
78.17	69.81
93.70	97.70
55.14	61.13
35.22	27.95
104.67	99.85
115.77	100.65
27.12	21.96
84.42	70.73
13.12	26.46
18.39	24.37
90.56	76.10
75.15	75.24
91.81	93.42
31.05	33.37
13.30	25.58
102.20	73.72
23.07	29.60
53.85	60.30
42.57	34.31
28.90	29.96
46.78	39.83
64.51	61.14
100.62	96.83
60.67	80.38
9.25	19.64
14.10	38.09
78.44	84.89
64.07	69.46
14.72	22.93
15.42	32.52
97.04	92.39
8.79	20.93
75.39	64.49
22.97	19.39
69.11	71.82
80.69	61.22
100.95	89.06
7.45	0.50
28.63	37.84
98.24	80.40
21.88	26.85
27.01	29.36
92.64	87.31
88.79	89.86
60.93	69.19
10.14	8.14
79.27	76.83
108.40	103.64
36.51	27.20
84.71	68.84
30.06	34.18
34.86	33.02
50.53	53.31
58.56	61.09
40.24	33.87
72.45	75.49
105.39	105.16
103.72	96.54
98.09	80.72
93.90	92.41
29.37	36.55
53.34	50.90
33.60	41.93
50.76	45.08
13.79	22.26
13.03	27.16
41.10	42.89
110.38	120.45
72.39	72.09
115.26	99.75
108.42	105.17
29.03	21.24
48.88	52.43
76.63	73.36
65.13	71.18
80.59	67.37
96.66	91.73
77.86	64.36
11.77	15.66
96.81	89.28
34.62	35.02
14.03	30.82
67.79	72.14
96.30	88.96
78.38	84.03
27.02	27.04
30.33	22.54
119.66	96.27
107.87	103.34
74.77	73.65
61.41	76.63
77.73	81.41
33.60	29.39
67.35	71.53
95.04	72.86
37.43	32.86
91.92	102.35
77.92	88.70
15.93	22.87
88.29	90.32
81.50	79.06
104.70	103.00
11.36	28.86
69.62	89.96
17.36	27.00
63.31	63.08
63.98	69.11
47.19	42.34
30.62	23.20
38.73	34.65
78.01	85.52
36.27	29.09
89.94	75.97
63.79	74.47
51.46	59.26
62.55	72.89
90.46	87.44
101.25	89.50
43.04	42.10
41.48	36.51
5.39	11.21
66.66	79.36
111.58	108.73
37.57	18.18
57.43	45.16
69.52	84.97
111.41	74.34
15.29	30.67
25.61	27.25
62.92	67.58
72.10	72.34
24.56	31.78
101.30	94.50
118.77	106.95
61.80	74.64
98.94	91.66
18.76	23.18
9.76	22.70
11.67	17.09
112.08	114.84
11.32	18.74
40.80	38.09
111.69	99.23
42.75	37.45
71.33	80.58
90.05	90.48
104.78	93.41
94.81	82.01
8.82	12.86
13.85	28.79
98.47	91.83
91.68	72.89
49.17	40.56
13.56	22.61
98.72	78.08
112.53	85.38
99.64	95.19
53.94	53.08
104.94	95.45
21.47	27.99
117.48	100.09
38.31	22.82
48.27	22.14
26.42	27.39
19.81	23.86
40.64	36.80
91.80	77.90
19.17	39.40
115.51	111.29
74.30	80.47
40.91	31.22
79.08	88.75
110.77	97.40
19.41	30.49
83.36	83.29
32.77	32.30
64.59	74.70
20.74	19.79
89.22	89.42
73.66	74.29
113.07	96.28
91.78	89.91
73.76	69.32
5.36	14.10
18.41	27.73
107.65	111.98
67.73	82.69
109.28	101.50
27.13	27.92
41.43	38.73
56.11	54.58
102.57	86.87
33.17	35.60
18.04	29.02
114.98	124.04
99.87	90.72
5.49	10.17
15.77	21.63
62.12	61.44
15.51	24.61
13.81	25.99
10.13	20.39
62.16	63.68
69.91	64.70
38.32	31.30
19.95	27.29
49.85	44.48
58.70	50.09
10.17	19.30
64.48	66.87
103.38	101.44
48.14	46.56
97.29	95.17
15.94	35.88
102.11	100.06
29.07	23.04
33.70	32.28
105.66	102.36
38.95	32.73
54.16	52.62
61.52	79.97
73.94	69.03
60.99	51.90
91.44	81.04
22.34	26.26
47.83	46.14
66.41	68.64
79.56	86.19
37.83	46.74
29.72	19.58
72.47	61.05
50.11	50.15
75.44	78.29
29.54	22.60
67.69	77.06
76.09	73.43
27.88	24.90
20.85	27.37
29.60	29.21
59.43	50.56
95.85	91.42
100.44	97.68
83.64	91.77
75.11	74.17
12.43	27.09
106.88	95.63
24.63	39.66
77.09	71.30
53.29	45.98
82.34	75.96
49.35	47.96
56.32	51.94
33.49	23.09
38.62	30.85
48.85	45.69
17.43	27.41
19.42	32.36
71.43	63.92
18.73	27.50
106.74	110.74
39.10	31.25
40.32	42.00
75.01	78.24
90.84	85.10
81.92	77.46
9.27	25.70
11.48	29.32
84.50	73.23
39.56	30.68
7.93	22.57
45.45	35.65
13.67	25.83
20.36	26.63
88.33	86.06
110.19	95.90
50.05	38.84
37.33	27.80
106.66	92.00
35.51	37.11
89.33	74.81
8.98	21.83
50.71	40.17
102.12	91.42
11.71	27.67
83.97	82.89
42.42	36.00
37.58	38.01
39.20	25.19
118.50	103.43
10.15	23.36
11.42	25.02
18.30	26.56
14.49	20.54
110.14	104.13
112.23	107.86
111.44	102.85
110.06	108.76
87.43	88.58
50.24	54.18
106.48	104.94
84.30	85.42
74.85	75.33
83.17	62.11
38.48	21.06
71.89	73.06
97.83	88.63
29.66	33.28
25.49	32.40
11.41	20.89
22.98	27.44
34.55	21.12
100.88	73.02
21.07	26.52
31.54	20.41
89.80	71.12
103.10	86.22
92.88	84.94
93.01	76.48
24.99	21.98
34.24	37.14
16.24	17.85
23.77	17.89
86.17	79.82
16.52	28.50
20.96	25.69
36.30	30.80
64.56	73.50
99.58	97.07
56.06	53.62
107.53	104.33
18.25	29.84
117.47	110.66
49.45	42.43
80.16	80.05
39.54	28.34
73.84	89.99
81.77	75.35
42.08	36.85
98.00	74.38
111.85	117.31
34.65	30.61
98.02	80.82
49.35	53.64
118.25	95.10
92.25	91.89
102.07	91.15
69.24	62.74
66.49	71.04
84.43	74.73
81.14	58.92
62.36	56.53
52.56	49.41
27.88	29.43
45.07	28.67
27.11	12.83
47.67	49.37
99.42	93.95
96.03	82.41
74.17	84.85
47.28	56.12
97.96	82.49
24.39	34.95
51.97	52.66
60.13	79.06
29.41	34.72
51.16	49.70
88.17	74.19
100.71	102.74
103.86	106.39
77.78	75.26
30.68	19.57
7.36	13.60
67.83	77.74
31.57	30.80
64.34	58.58
83.93	72.63
114.59	114.61
81.25	84.62
112.87	109.06
108.59	96.05
6.27	11.91
36.56	33.28
63.63	58.48
91.49	73.15
65.39	68.57
66.44	74.76
7.17	23.60
15.88	23.49
107.02	104.12
96.51	84.69
71.35	79.51
73.78	61.78
113.11	122.30
118.06	108.46
102.11	89.07
41.33	31.74
117.37	113.87
84.56	75.67
17.84	21.79
22.01	29.39
18.60	24.87
38.35	27.63
38.74	21.48
115.24	104.13
56.13	47.86
89.93	90.28
113.51	86.21
87.22	83.60
113.92	102.93
77.02	66.45
17.40	20.58
8.07	16.81
24.09	32.50
59.18	67.94
110.91	102.50
100.93	109.13
113.62	112.17
54.25	55.81
50.78	51.42
51.18	53.45
119.33	102.44
97.25	100.16
32.43	24.83
44.85	46.67
87.34	78.52
36.24	42.57
114.60	100.37
87.14	75.71
70.82	82.70
94.11	73.37
115.74	120.78
65.17	74.56
27.33	33.84
95.17	81.93
54.08	51.49
71.95	84.89
55.95	51.61
29.61	16.56
57.19	61.32
15.60	24.58
118.41	126.73
50.80	51.51
107.18	100.02
19.93	28.47
107.65	111.74
75.46	64.65
48.58	40.32
45.48	43.15
112.09	122.14
17.32	26.49
45.45	60.89
77.02	75.72
14.64	33.30
107.24	106.66
44.02	34.28
11.93	23.00
49.95	57.23
9.14	14.02
32.87	32.53
73.78	77.85
27.30	32.65
25.69	32.32
100.78	100.25
108.34	104.66
78.53	84.59
81.84	82.00
74.31	90.82
97.57	49.62
64.81	68.27
100.11	94.56
19.55	27.71
18.49	22.38
51.14	50.21
98.79	99.12
67.67	76.06
48.71	27.93
43.50	37.98
74.40	77.57
82.11	85.30
47.11	46.69
43.91	29.69
86.07	64.42
40.80	35.86
88.91	79.45
102.49	98.73
117.36	106.87
12.86	28.64
95.61	102.83
51.10	57.41
97.10	94.86
66.33	69.88
72.85	80.19
35.83	37.01
42.34	38.57
112.16	110.72
61.24	62.76
88.13	85.61
27.98	25.37
93.59	96.28
7.23	14.78
45.73	33.11
62.43	78.87
83.75	82.41
17.98	33.89
114.15	89.43
11.01	21.03
98.52	75.66
30.20	28.58
74.23	68.38
75.56	85.05
35.28	22.25
29.83	23.49
56.13	55.13
18.20	25.38
99.37	90.17
63.17	69.62
33.35	38.10
90.80	85.44
98.29	83.84
8.88	21.33
98.54	90.67
61.29	58.17
52.64	54.95
20.22	32.38
14.59	29.07
21.50	26.15
110.62	106.34
49.90	42.13
18.82	29.78
23.64	30.93
57.61	66.99
85.45	84.58
17.11	26.36
42.19	35.46
90.24	72.26
110.49	95.79
84.95	72.50
117.77	109.73
76.32	74.68
17.14	37.03
117.13	110.23
55.50	60.12
64.32	84.47
71.50	71.99
20.74	40.25
109.47	110.29
61.51	68.88
60.20	61.26
60.31	58.67
93.21	71.02
51.74	42.37
35.16	27.47
101.46	94.95
93.71	90.83
110.63	109.84
43.88	37.44
24.30	28.44
24.48	23.60
100.25	94.76
97.51	93.17
109.75	109.15
14.75	28.61
109.35	104.06
17.23	29.55
118.40	115.93
67.85	80.66
74.78	70.31
34.43	37.40
118.21	106.25
26.64	22.87
20.20	29.84
109.61	101.72
62.75	69.42
83.67	70.71
50.85	51.95
80.89	79.34
50.42	35.49
102.35	98.33
110.76	113.09
11.11	29.90
38.10	27.79
94.28	85.55
106.89	87.46
57.84	72.12
83.73	75.22
90.32	77.13
13.91	20.29
40.31	37.56
98.21	85.63
68.38	81.45
25.25	27.01
99.71	81.30
17.51	26.03
93.74	62.87
29.12	17.69
54.02	54.37
104.15	106.04
51.43	52.81
36.91	26.06
82.83	77.47
96.39	95.83
12.45	27.19
83.11	88.98
65.56	72.24
55.21	61.12
90.92	84.36
101.58	99.19
103.43	88.01
60.19	58.94
104.06	80.53
85.25	81.66
16.76	24.80
119.40	115.50
19.45	18.74
104.38	94.97
86.43	61.21
55.28	65.17
42.79	37.76
114.67	103.86
36.49	31.87
78.99	73.70
27.08	32.01
119.68	85.31
40.14	28.99
83.13	62.05
47.20	50.66
10.30	32.68
14.91	24.70
24.83	29.96
114.65	85.93
24.68	30.31
19.39	22.05
115.22	121.80
67.63	67.41
101.76	88.31
77.37	89.49
119.66	94.11
81.07	61.65
109.47	102.63
55.36	55.85
86.18	78.84
37.99	30.77
53.80	61.81
65.32	63.80
36.03	35.80
34.10	34.91
75.70	87.79
9.51	27.22
26.43	26.87
11.89	22.86
98.68	96.34
40.59	37.61
44.44	45.68
16.93	16.82
21.51	27.00
66.63	72.38
29.45	18.93
109.17	111.90
24.37	34.54
58.22	57.67
60.13	60.87
39.53	37.94
62.33	75.27
81.04	71.58
74.01	63.37
96.12	75.68
37.96	32.09
15.43	26.96
8.13	13.61
119.81	108.45
70.37	77.78
44.77	37.95
45.99	43.06
118.04	108.73
74.66	82.10
73.66	73.75
79.75	83.27
113.37	99.73
79.77	67.90
87.46	95.21
63.93	77.77
99.09	93.93
112.16	103.78
105.51	76.16
52.52	53.88
106.92	91.42
119.57	128.14
114.86	92.78
45.38	41.40
78.76	83.94
35.47	24.22
87.38	82.31
78.42	87.67
108.89	73.32
77.25	73.97
11.91	19.53
10.36	23.79
63.76	70.02
43.10	34.72
42.72	39.09
119.18	96.72
8.25	16.81
10.76	21.34
79.38	71.14
15.19	30.02
84.79	72.16
83.09	72.97
99.27	92.13
23.81	27.24
74.43	63.60
77.84	69.28
24.36	24.99
95.83	85.85
116.23	108.76
91.78	82.12
51.40	53.59
56.75	64.55
38.88	36.68
60.13	81.02
94.45	100.29
39.43	35.35
75.77	85.12
77.36	81.23
83.45	76.95
15.75	33.09
21.82	25.51
11.47	31.49
92.35	80.80
114.00	107.65
73.24	78.81
104.79	96.07
38.62	32.81
89.40	78.06
95.54	98.29
109.70	124.82
114.82	120.64
6.31	2.45
72.94	75.43
54.61	61.27
23.20	21.67
54.38	61.10
23.50	36.80
102.83	93.87
25.26	27.25
13.96	24.10
68.39	76.78
99.66	92.92
41.78	30.76
18.82	23.71
41.86	36.07
71.62	59.60
54.17	58.06
56.48	58.91
68.68	78.19
61.37	83.23
25.89	28.64
50.16	58.79
100.25	98.63
15.08	28.23
114.97	114.69
76.69	74.64
96.49	91.00
42.72	48.74
17.25	27.74
13.96	27.56
69.39	86.02
61.07	78.83
60.70	62.31
113.18	109.31
25.65	29.50
28.60	31.93
12.32	27.85
89.42	100.06
101.61	97.74
108.12	118.89
6.03	18.22
47.57	34.03
103.17	112.50
75.33	81.25
60.60	65.61
11.51	22.62
84.03	84.98
77.27	81.17
78.39	79.59
103.58	85.50
23.88	34.62
5.00	18.12
66.35	92.45
94.21	84.37
68.75	76.83
103.98	102.98
64.71	77.68
86.81	84.11
111.07	108.97
23.61	31.06
98.10	92.74
30.61	21.23
11.43	27.98
56.12	50.52
69.53	82.56
28.21	35.65
116.46	107.68
90.88	92.87
50.60	62.02
115.02	108.08
104.80	86.00
69.71	76.12
78.15	68.80
110.58	95.55
17.77	28.41
101.24	92.28
101.76	96.21
117.73	107.50
43.03	37.44
88.02	68.20
82.61	72.50
65.66	72.80
74.98	69.62
45.32	43.34
89.04	78.46
68.76	78.30
91.40	84.19
23.64	19.00
118.97	95.78
29.18	29.02
78.60	83.92
82.76	66.81
114.15	81.84
5.03	4.13
73.88	61.85
25.93	31.90
87.96	77.23
27.85	25.35
118.75	105.58
32.92	32.77
48.32	42.44
116.98	90.42
72.28	65.67
73.23	74.84
11.89	21.04
34.92	30.36
28.19	28.33
113.14	105.04
105.85	109.17
10.90	19.65
60.89	67.36
118.50	97.15
46.21	38.78
89.96	80.63
116.99	111.80
102.15	85.11
104.50	91.90
109.33	96.58
35.30	21.52
68.27	62.20
88.62	68.53
62.58	71.42
38.49	32.51
93.56	67.16
21.82	26.27
39.07	28.07
50.24	67.94
82.35	81.31
112.45	93.93
12.70	17.36
5.23	12.86
77.88	74.23
66.18	86.94
55.37	58.80
18.64	35.81
41.65	39.80
94.57	82.21
94.61	79.26
61.32	68.77
33.62	38.93
38.04	26.53
75.97	78.95
97.20	84.32
17.55	32.87
52.08	66.69
43.11	49.56
72.76	91.12
19.82	31.78
35.46	37.76
114.28	84.49
28.66	17.63
76.81	70.41
66.49	74.71
42.80	35.87
77.22	81.73
10.50	21.95
112.71	66.13
56.45	50.51
103.64	102.99
21.78	22.68
101.76	98.48
44.30	37.50
49.24	38.93
82.65	84.73
65.47	76.60
82.52	84.70
78.28	83.24
83.11	96.69
73.08	71.39
35.55	23.07
5.83	25.78
117.27	96.11
60.30	63.28
60.38	69.23
119.11	119.18
29.21	32.89
37.19	35.50
26.91	27.09
8.07	19.57
115.25	106.32
115.49	112.13
5.30	13.79
52.14	55.64
26.35	22.13
51.96	51.46
79.12	82.54
77.34	60.56
70.83	84.75
118.81	116.25
37.88	20.00
58.17	70.40
117.54	123.31
45.78	52.49
11.47	20.71
19.76	28.39
116.67	118.63
76.74	63.73
12.92	26.96
9.48	18.14
27.00	27.59
101.41	105.33
84.40	76.57
15.25	15.49
85.88	84.38
18.48	25.32
67.19	67.54
100.35	101.82
43.08	39.83
28.58	34.68
30.32	34.93
65.41	64.84
71.09	71.41
68.78	72.76
22.51	37.02
12.69	29.54
67.58	78.91
16.48	21.30
40.08	27.64
83.47	90.52
16.51	30.57
115.77	95.35
89.31	90.81
50.47	46.29
11.22	24.58
112.83	97.64
53.60	46.56
58.13	59.59
69.61	72.44
45.42	37.71
13.85	34.71
98.17	93.48
65.49	68.42
86.97	80.24
48.57	50.64
119.63	85.37
84.59	71.69
103.67	109.45
26.89	32.69
56.01	58.35
36.54	29.39
88.31	86.66
94.52	88.86
62.64	72.51
11.38	29.17
52.80	49.50
96.56	76.77
50.12	46.87
118.48	102.07
71.55	73.59
29.06	24.51
35.68	29.15
89.19	73.21
101.39	92.09
60.23	64.52
67.49	76.88
40.73	31.38
75.72	77.04
37.52	38.22
81.58	68.81
67.70	75.14
28.63	30.59
45.22	45.44
8.34	15.08
119.17	113.43
30.92	27.60
87.88	81.18
75.92	63.57
51.53	68.71
6.72	18.66
8.99	21.54
115.50	109.58
46.84	45.67
60.64	74.08
51.44	61.71
64.86	84.22
76.97	65.11
113.42	99.50
91.34	70.21
117.67	109.54
94.70	72.62
48.06	43.14
74.22	83.97
56.85	62.36
13.60	25.52
70.70	95.39
80.17	74.63
62.72	60.23
40.73	42.84
46.33	41.59
77.45	71.77
95.17	76.02
115.02	111.62
18.33	39.11
48.33	52.64
58.08	57.08
44.71	31.84
22.29	30.91
86.72	81.64
89.72	66.27
119.97	95.18
95.34	83.73
63.87	73.16
86.55	100.65
25.38	22.66
24.56	18.58
83.28	74.84
88.84	75.47
42.59	35.47
115.93	114.22
21.49	30.07
86.00	67.22
119.07	98.63
48.91	46.05
112.43	94.86
45.68	45.63
107.01	88.62
48.79	48.03
61.62	75.98
78.95	80.74
33.53	27.78
81.44	88.70
60.57	59.32
29.24	32.55
39.84	33.12
97.18	91.51
39.67	40.64
30.82	26.33
84.76	93.74
103.47	84.06
72.44	71.05
90.94	92.08
50.40	45.18
51.29	55.31
107.70	92.18
26.11	32.17
110.76	91.94
63.51	54.35
20.92	31.96
13.01	27.98
84.55	78.37
108.21	92.16
117.46	117.52
23.87	38.32
72.94	79.00
82.28	67.79
111.37	112.18
80.55	79.83
43.34	48.09
107.51	92.68
30.06	30.07
50.88	51.52
44.94	32.48
111.77	100.53
81.30	80.59
44.10	36.16
49.54	52.17
23.80	36.08
84.75	68.64
45.30	34.28
118.05	109.14
28.46	28.47
85.69	88.90
64.80	80.81
19.47	28.89
89.92	70.74
100.02	93.88
24.28	38.88
6.49	18.84
47.55	36.98
101.27	92.65
65.50	78.27
27.15	30.01
45.09	30.13
59.91	57.25
69.25	85.28
103.42	106.05
41.31	52.27
50.78	45.64
9.76	24.36
52.63	55.48
67.20	61.22
51.45	52.19
29.31	29.21
62.81	73.92
25.34	23.92
80.48	66.32
69.47	77.99
95.29	81.74
54.61	64.33
118.69	106.19
43.96	30.87
65.19	88.69
75.82	74.21
43.21	41.65
47.91	46.88
97.77	76.22
40.15	44.54
66.50	74.09
94.05	74.26
90.92	74.83
89.84	72.92
43.22	30.94
109.59	96.66
18.34	35.14
40.04	30.61
115.02	114.68
71.06	89.19
87.09	86.68
119.88	111.98
102.43	99.11
29.27	32.62
34.83	32.67
18.78	18.77
38.86	39.60
52.86	50.40
69.36	84.23
112.52	98.07
41.19	34.26
69.61	70.70
48.38	34.68
35.71	38.83
108.24	95.39
111.60	108.39
41.42	33.65
83.90	68.42
58.21	59.88
38.02	24.43
117.04	103.32
86.86	65.18
107.61	101.06
36.22	39.79
98.97	92.07
17.68	23.64
9.45	25.48
105.36	86.00
38.73	41.29
76.33	70.62
88.34	84.37
61.55	55.33
5.42	12.27
82.67	77.13
6.83	8.16
29.09	27.12
106.07	111.68
46.23	38.38
27.70	22.27
108.89	109.67
46.38	30.06
94.07	68.72
44.29	34.39
82.00	71.73
78.08	74.95
105.56	107.39
41.34	31.74
54.52	52.44
42.19	37.97
93.61	86.72
78.68	63.47
38.31	31.46
8.06	15.23
28.28	27.52
77.39	82.86
102.50	105.01
58.59	56.00
53.44	53.74
66.30	72.83
78.47	80.27
106.03	100.30
94.68	85.36
116.23	121.93
5.45	18.38
59.50	74.83
9.78	28.99
80.47	77.47
53.41	50.82
98.96	113.56
63.51	56.56
82.19	78.67
66.92	59.52
66.72	55.00
73.17	77.32
93.00	82.59
67.53	73.13
36.56	24.14
67.13	68.62
61.89	69.08
15.23	24.24
59.16	63.61
66.09	66.74
55.33	59.04
63.69	67.76
11.17	19.55
25.84	33.40
24.25	29.02
32.08	30.91
50.14	46.36
101.44	89.25
113.36	102.40
71.44	65.14
50.39	46.75
58.75	63.39
76.31	58.04
80.92	78.71
84.19	68.00
42.02	40.27
8.17	15.43
9.89	25.23
7.65	13.06
107.46	98.09
45.33	52.26
65.90	76.15
110.06	110.17
55.76	56.00
63.27	70.97
79.60	80.80
73.25	66.68
74.59	77.99
50.63	42.14
44.06	38.62
60.36	61.85
54.73	57.32
89.43	79.68
76.05	75.04
48.48	44.13
100.52	94.10
78.61	90.36
58.12	53.85
79.10	87.98
86.24	80.20
75.79	69.81
39.89	25.35
14.37	28.23
65.79	65.92
13.91	28.45
10.31	19.69
65.92	73.17
85.21	81.26
40.95	35.67
14.05	23.56
66.55	62.65
92.81	90.88
65.40	78.99
30.64	23.00
35.07	39.34
48.82	39.95
78.64	82.45
86.94	61.02
82.18	72.27
117.55	101.33
7.52	15.66
87.11	67.27
57.73	63.48
108.31	106.59
90.28	76.56
112.80	87.68
28.61	18.93
26.45	22.28
49.83	50.07
38.19	38.47
53.15	58.59
24.18	19.89
17.15	25.25
99.14	103.60
23.45	29.94
30.13	28.28
22.38	30.19
60.70	63.03
71.69	75.94
82.36	84.03
40.10	24.22
21.69	35.74
102.39	110.83
42.68	31.30
58.30	65.43
75.40	75.16
12.36	20.82
50.90	58.53
99.82	90.74
90.13	68.56
68.05	81.53
119.86	98.21
6.73	18.66
69.07	66.80
104.44	108.77
86.94	73.57
87.18	92.00
70.52	73.52
84.60	84.57
36.87	34.23
71.57	73.10
80.40	78.52
41.58	35.28
68.82	74.76
69.77	79.80
92.77	98.14
32.54	23.30
5.73	17.50
39.24	22.96
36.65	28.70
119.88	116.47
47.26	42.59
45.43	35.17
31.67	24.19
22.53	22.74
77.71	81.86
26.10	31.01
38.89	31.01
113.06	105.68
33.74	30.71
18.56	26.06
48.38	42.98
117.02	103.53
59.87	73.83
92.51	84.60
13.94	24.94
39.39	33.67
117.99	108.39
103.46	100.11
62.02	53.16
84.45	85.12
62.29	60.91
30.13	18.70
75.45	77.73
37.33	30.92
23.72	35.72
75.82	79.51
51.44	40.72
84.40	70.72
57.35	62.55
79.60	73.72
79.82	74.89
109.25	110.58
39.85	37.24
113.89	95.17
77.88	73.12
30.36	39.09
93.59	81.22
44.37	45.82
89.55	97.22
10.42	16.55
64.53	77.19
57.60	54.99
43.14	46.00
111.82	104.16
102.65	108.00
51.56	49.66
53.89	67.88
79.99	66.78
107.30	102.32
114.15	105.60
34.74	36.69
75.61	82.87
93.99	76.89
5.89	19.56
77.74	86.52
114.35	94.82
31.18	30.66
113.64	111.11
36.77	22.76
24.02	25.71
63.42	54.77
78.66	70.44
101.29	105.20
70.84	79.97
26.41	32.19
60.90	63.10
66.19	72.25
91.49	81.97
91.22	83.76
94.44	79.24
44.96	38.89
57.50	65.69
99.58	87.67
111.54	93.22
70.02	74.91
14.89	27.57
111.87	88.11
15.31	26.07
75.69	62.91
23.07	37.28
13.01	25.66
24.38	32.35
37.75	40.41
84.80	84.97
77.14	94.89
76.22	82.27
25.12	23.51
33.96	30.50
67.84	76.50
9.43	17.45
41.98	36.70
8.73	15.93
64.39	65.05
55.08	62.04
83.22	92.58
57.27	58.66
53.44	46.14
99.37	103.87
81.18	84.60
77.66	70.66
18.46	2.95
23.71	32.25
101.52	90.77
71.83	73.23
119.43	98.67
9.42	16.61
22.96	21.87
58.84	64.63
116.85	108.43
62.79	69.08
44.19	34.62
87.17	82.37
23.73	35.41
52.19	59.20
88.01	86.29
22.80	21.51
17.18	24.33
101.54	95.26
65.29	67.14
49.03	45.20
39.55	33.06
111.18	99.47
45.47	38.39
36.66	30.49
77.10	70.20
16.93	27.52
65.44	69.48
22.65	21.43
76.42	78.50
27.02	31.84
61.66	60.39
99.82	85.24
74.44	74.66
28.02	31.99
56.76	61.78
106.58	92.49
48.99	36.24
95.89	85.82
87.70	91.21
94.48	86.19
96.23	84.70
13.59	19.43
84.83	87.81
106.13	115.96
104.36	77.32
10.12	24.90
23.12	34.61
36.94	27.73
40.86	37.67
34.55	32.96
80.93	79.89
99.52	92.03
84.88	85.12
65.15	72.25
77.27	89.05
53.20	51.78
74.13	82.33
50.19	40.47
76.15	70.34
20.92	18.88
81.48	78.43
73.75	79.87
69.62	77.94
80.79	85.66
51.29	34.97
107.50	93.34
115.10	99.22
20.40	27.17
104.28	105.10
91.44	83.58
46.30	47.52
99.17	87.74
36.98	31.53
51.73	53.08
30.93	38.52
55.62	59.28
40.27	38.07
77.53	71.23
37.21	40.41
118.30	118.59
60.82	50.24
39.45	36.66
79.00	80.32
115.99	121.80
113.61	106.92
26.49	27.96
106.12	103.83
95.62	90.60
8.39	22.90
51.71	55.84
34.02	36.03
46.82	40.72
74.52	89.52
69.21	69.62
93.69	66.78
70.50	61.13
30.70	29.86
79.16	78.84
95.67	81.68
48.99	39.91
27.18	19.61
96.17	89.77
92.70	75.22
20.77	14.93
119.57	131.88
94.42	90.23
75.94	59.31
7.21	21.97
75.25	63.29
42.64	42.22
59.30	63.42
91.21	84.11
63.52	68.70
50.66	49.38
93.06	87.84
100.10	80.96
16.70	27.39
23.40	30.32
52.35	46.96
79.37	67.08
81.85	70.56
18.64	29.30
14.31	19.95
25.91	28.50
63.28	68.21
45.62	45.17
7.79	24.64
48.81	54.20
41.30	36.24
53.18	64.03
91.06	87.69
74.96	84.30
47.30	50.03
48.32	31.42
76.08	68.31
21.01	26.39
114.34	101.11
47.19	76.69
99.35	89.89
109.29	95.53
110.34	97.62
101.34	92.97
117.55	99.54
65.93	83.85
86.37	77.04
8.97	12.70
81.04	84.09
59.07	65.94
65.31	69.32
69.30	70.34
111.48	105.99
23.00	20.11
101.51	87.97
70.89	82.93
34.63	28.48
55.05	64.93
57.46	76.84
10.79	22.70
48.89	48.98
47.31	51.39
107.80	110.68
55.60	49.41
16.15	18.09
44.91	45.64
84.59	71.53
111.72	102.79
27.53	19.34
73.86	79.73
80.61	78.25
47.53	51.45
63.52	60.66
59.54	65.79
35.09	34.08
58.51	57.42
108.42	95.48
110.76	110.86
100.46	103.13
55.03	49.18
35.64	32.33
53.71	62.04
98.41	82.57
23.82	30.61
64.93	71.11
53.73	54.96
64.23	68.87
83.41	66.00
118.67	103.99
117.59	105.50
98.45	83.23
88.63	91.70
55.14	62.05
20.94	37.55
63.55	72.51
76.38	63.88
26.57	28.02
118.34	110.13
25.10	38.05
109.89	110.97
102.94	97.50
90.54	76.88
95.64	69.25
80.38	65.68
116.72	101.97
79.18	71.20
29.86	18.54
116.00	119.96
50.65	51.10
43.74	48.36
53.68	64.72
45.60	40.44
53.39	35.02
91.17	71.65
86.32	86.73
40.67	41.21
44.68	29.60
46.91	42.37
37.52	25.79
62.51	62.03
51.59	52.90
108.74	107.16
8.03	22.06
112.43	117.62
8.75	19.92
10.48	24.72
97.10	86.20
83.06	75.41
15.41	25.21
81.11	76.33
48.48	40.68
12.77	32.76
68.39	96.88
69.89	66.55
84.78	78.90
14.95	21.09
119.30	105.86
21.39	20.78
30.91	20.88
12.68	24.09
97.28	100.28
57.82	65.20
58.26	59.27
63.86	58.42
29.01	30.49
12.62	25.87
43.38	48.23
101.27	90.42
75.52	68.20
78.65	79.22
56.41	63.23
9.49	26.05
57.06	64.09
110.04	103.37
53.56	53.96
81.89	60.96
12.91	27.73
44.30	33.59
116.06	118.05
75.02	74.09
110.15	103.00
96.62	84.41
48.95	45.78
39.09	37.59
5.52	22.33
19.01	17.70
23.00	17.90
100.27	78.53
16.96	31.79
113.98	101.66
119.15	73.90
97.63	85.35
96.90	89.95
110.04	102.92
86.95	80.86
110.69	98.46
108.64	99.15
83.96	88.54
17.59	32.93
40.56	30.43
108.81	99.31
105.49	90.40
96.76	104.26
99.35	107.61
82.70	62.05
28.71	29.24
30.38	26.14
94.16	97.53
20.80	22.83
43.69	35.65
75.81	84.55
58.71	68.88
116.24	113.21
14.22	32.15
26.04	25.21
28.96	24.48
78.56	91.34
101.74	97.83
10.12	48.12
21.87	33.27
29.34	28.98
93.31	82.82
53.68	57.56
57.52	66.99
28.37	33.32
34.46	26.44
22.13	21.26
71.17	73.18
44.80	34.46
82.65	85.78
82.16	69.59
101.92	90.38
76.84	74.23
9.55	25.80
11.87	24.47
96.03	88.05
48.19	47.27
99.90	105.39
87.20	85.36
21.90	34.02
50.12	52.46
53.55	65.12
104.77	93.99
79.42	68.52
10.89	32.09
60.17	69.05
63.99	64.72
94.91	84.15
75.80	83.79
83.63	59.28
84.67	80.23
113.36	95.55
48.69	36.57
103.26	97.64
76.01	81.93
65.21	76.84
6.03	15.28
54.39	57.34
19.67	29.85
22.57	35.47
116.91	112.40
36.70	35.23
16.11	28.07
103.24	88.77
76.70	66.54
22.34	46.54
68.85	71.59
33.24	25.86
45.98	44.40
9.96	21.91
51.19	46.27
119.95	100.69
98.57	79.01
84.14	84.72
18.57	26.23
106.56	87.81
57.80	64.65
27.07	39.39
7.60	21.05
7.31	22.18
19.18	29.98
77.64	79.69
32.20	30.44
115.38	110.67
42.25	44.20
67.75	71.09
73.69	76.92
30.33	40.20
37.39	24.35
19.07	27.25
40.31	31.31
16.88	28.68
76.86	75.18
113.50	99.83
101.19	95.38
42.57	24.84
101.52	88.15
87.50	76.35
70.00	57.07
73.55	74.30
29.23	25.78
79.06	79.71
18.32	34.23
21.48	31.04
30.74	24.57
88.91	79.80
55.72	44.47
54.38	49.30
49.84	54.08
115.51	112.04
42.19	32.46
81.44	76.54
52.81	52.49
44.03	46.45
50.63	50.69
42.36	33.45
10.33	19.88
79.29	85.75
105.49	118.45
71.98	73.95
108.52	99.81
50.25	51.87
13.98	25.32
10.13	16.60
30.72	23.78
54.30	59.55
11.08	13.04
113.92	110.89
12.52	27.05
74.03	72.80
22.21	18.36
87.44	73.91
18.67	21.25
18.87	29.18
58.30	55.60
33.78	31.56
56.09	67.82
106.72	80.40
100.25	96.96
72.88	67.90
34.27	22.36
13.36	19.20
12.43	24.31
53.66	56.47
58.11	53.41
42.89	49.48
77.23	71.89
112.62	111.56
75.19	70.64
77.15	88.35
76.30	81.88
38.83	28.41
90.54	67.84
29.26	17.60
48.94	47.15
70.39	82.81
11.16	17.59
111.92	106.54
99.55	89.71
66.72	75.10
40.40	43.00
66.43	74.08
56.97	59.36
48.97	46.90
59.80	65.33
43.24	36.07
44.55	36.74
113.59	95.64
109.24	110.20
79.66	77.18
115.57	101.29
107.47	132.62
30.17	20.08
10.27	16.89
43.62	38.80
51.94	45.93
39.88	32.92
86.99	104.18
96.06	76.84
109.77	110.27
34.75	23.02
118.12	99.48
95.38	98.17
58.88	74.04
8.57	19.59
116.04	96.35
50.20	46.38
44.27	38.73
7.07	17.27
10.97	20.17
25.67	39.73
17.55	28.25
52.26	43.64
90.28	73.53
70.99	71.72
54.64	56.04
63.19	57.38
64.80	77.77
48.83	55.23
38.48	24.60
17.48	25.15
109.21	94.98
83.04	76.05
54.21	54.85
79.74	84.53
7.16	24.32
111.84	110.62
29.06	20.48
66.33	75.97
49.55	40.01
98.73	90.73
20.37	22.84
74.35	63.98
22.42	25.21
30.28	20.75
5.25	18.18
83.25	85.06
45.20	43.03
38.36	36.47
61.52	66.63
93.86	85.22
90.31	73.31
48.27	39.70
67.50	71.36
78.10	80.45
47.67	50.39
110.03	89.78
46.96	43.06
82.04	86.21
96.47	75.81
62.98	80.52
84.23	101.15
74.29	73.46
44.50	40.70
31.23	32.91
118.95	95.75
100.64	105.67
10.80	27.15
73.85	68.27
72.12	65.89
74.86	77.34
30.06	17.84
86.35	90.79
79.61	76.60
21.29	24.94
104.53	88.69
108.94	106.45
32.25	26.70
40.26	45.59
111.70	119.28
39.77	38.87
101.20	99.97
10.16	21.06
66.62	69.53
48.31	51.96
53.36	51.00
85.72	74.18
7.53	24.85
109.67	91.22
110.96	105.14
104.45	89.72
33.93	31.78
20.36	29.50
95.53	79.56
113.18	99.89
78.36	83.30
23.81	20.07
100.92	100.13
70.92	72.86
117.60	122.67
23.68	24.73
60.05	63.24
58.53	62.54
99.84	86.22
63.64	73.61
24.60	23.56
38.97	33.12
53.65	57.16
37.24	33.86
50.03	52.03
119.06	110.05
106.76	91.19
69.67	71.47
87.64	82.08
77.34	85.13
85.82	76.08
113.90	119.29
113.36	81.42
38.72	36.67
90.95	76.86
51.75	55.87
101.39	97.39
12.19	21.42
84.98	94.73
35.58	25.84
29.26	22.80
79.83	66.71
5.03	11.41
22.73	28.77
100.91	96.87
33.49	24.09
41.66	36.54
10.22	18.15
96.91	108.39
9.61	22.82
55.31	52.96
101.00	83.79
33.62	28.00
49.61	46.84
99.19	96.37
111.56	118.75
64.37	72.71
9.67	24.08
75.86	72.97
5.59	15.53
50.37	48.70
15.54	31.51
47.78	46.75
110.64	110.96
55.24	48.41
83.94	71.12
26.31	27.68
45.08	37.45
16.69	21.74
36.89	43.15
61.23	68.47
33.19	45.77
67.80	62.49
80.13	79.92
36.69	26.89
111.59	101.25
51.94	51.07
60.38	70.16
74.17	85.48
96.09	79.37
21.94	37.44
94.52	81.84
80.10	75.46
79.41	74.87
21.53	34.86
119.79	107.34
114.79	99.60
54.73	64.31
110.25	101.93
110.72	85.36
117.23	107.12
38.63	45.40
96.94	64.33
77.88	81.11
101.13	98.78
111.64	109.21
29.72	26.95
24.17	24.02
52.37	52.82
84.90	77.93
86.17	81.89
113.06	99.56
61.16	59.64
79.73	83.18
24.30	23.74
8.23	17.34
22.58	28.64
101.84	105.93
70.69	83.51
99.36	83.84
118.63	109.78
104.25	93.15
65.48	88.83
6.83	18.71
30.28	28.90
16.32	26.67
87.21	80.58
98.73	93.39
60.08	71.59
20.09	33.32
38.81	30.25
6.06	15.45
5.23	9.63
11.13	25.66
27.14	34.06
76.96	80.57
33.87	47.31
102.75	96.35
19.47	22.64
56.69	63.99
36.87	36.72
35.30	26.28
62.10	85.54
30.17	24.42
23.91	27.70
105.55	89.03
47.80	54.97
33.02	30.33
46.84	51.62
113.20	104.44
114.35	84.36
61.89	68.45
22.22	26.08
90.88	66.98
69.05	68.26
6.89	16.58
7.39	16.34
113.54	88.14
72.08	67.06
99.56	91.26
13.56	24.43
60.57	74.50
102.34	100.09
108.35	109.51
14.70	36.41
71.07	77.56
32.85	30.31
89.37	86.08
104.07	91.50
27.14	29.05
95.66	83.15
90.69	94.22
94.96	82.72
47.53	40.61
98.70	83.96
91.21	92.05
105.69	104.93
27.52	23.90
13.32	24.19
23.51	28.71
41.39	41.65
101.52	101.23
62.95	65.28
95.62	77.07
65.91	64.53
86.33	87.59
81.90	72.01
45.18	44.25
53.29	50.83
18.44	14.13
50.08	44.49
65.60	71.13
22.71	30.39
12.14	24.88
34.96	21.54
93.67	99.00
104.66	85.57
97.85	71.19
85.18	70.02
13.85	28.61
24.35	32.95
30.25	31.12
62.73	61.97
59.69	67.35
93.58	75.17
83.48	76.96
77.70	75.95
72.80	65.80
98.60	86.59
108.42	120.78
45.17	38.16
39.93	41.69
64.38	68.45
106.72	107.25
42.55	37.68
12.31	13.98
44.66	47.42
32.74	31.87
103.55	87.31
97.08	83.69
56.07	49.37
94.89	90.58
98.01	88.72
17.98	25.27
77.75	66.53
44.77	41.90
48.48	47.45
90.24	80.44
92.78	83.27
33.16	31.32
58.20	77.81
31.64	34.33
46.40	37.60
36.55	30.52
118.07	93.54
97.19	97.69
116.41	91.88
11.30	29.00
51.09	53.81
72.89	63.36
47.67	39.62
56.77	61.20
96.12	93.69
64.42	67.60
24.03	31.41
108.82	106.60
49.44	68.42
116.24	124.46
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 Cu\nen muestras geoquímicas de depósitos minerales",
  xlab = "SiO2 (%)",
  ylab = "Cu (ppm)",
  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 un filtrado de valores atípicos mediante el rango intercuartílico (IQR). Posteriormente, para simplificar los pares de valores y suavizar la dispersión extrema de la concentración de cobre, se agregaron los datos calculando la media de Cu por cada valor de SiO2.

# Eliminación de Outliers mediante IQR
Q1_X <- quantile(X, 0.25)
Q3_X <- quantile(X, 0.75)
IQR_X <- Q3_X - Q1_X
lim_inf_X <- Q1_X - 1.5 * IQR_X
lim_sup_X <- Q3_X + 1.5 * IQR_X

Q1_Y <- quantile(Y, 0.25)
Q3_Y <- quantile(Y, 0.75)
IQR_Y <- Q3_Y - Q1_Y
lim_inf_Y <- Q1_Y - 1.5 * IQR_Y
lim_sup_Y <- Q3_Y + 1.5 * IQR_Y

# Filtrar tabla sin atípicos
TVP_limpio <- TVP_original[
  TVP_original$X >= lim_inf_X & TVP_original$X <= lim_sup_X &
  TVP_original$Y >= lim_inf_Y & TVP_original$Y <= lim_sup_Y, 
]

# Agregación por promedio
TVP_prom <- aggregate(
  Y ~ X,
  data = TVP_limpio,
  FUN = mean
)

# Variables simplificadas finales
X <- TVP_prom$X
Y <- TVP_prom$Y

#TABLA DE PARES DE VALORES SIMPLIFICADA

TVP_SIO2_CU <- data.frame(X, Y)
n_modelo <- nrow(TVP_SIO2_CU)
cat("Tamaño muestral del modelo =", n_modelo)

## Tamaño muestral del modelo = 2472

tabla <- TVP_SIO2_CU %>%
  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 Cu en muestras geoquímicas de depósitos minerales*")
  ) %>%
  tab_source_note(
    source_note = md(paste0(
      "**Nota:** Tamaño muestral n = ", nrow(TVP_SIO2_CU),
      " observaciones utilizadas en el análisis."
    ))
  )

div(
  style = "height:400px; overflow-y:auto;",
  tabla
)

Tabla N°2
Pares de valores simplificada de SiO2 y Cu en muestras geoquímicas de depósitos minerales
X	Y
5.00	18.12
5.03	4.13
5.03	11.41
5.23	9.63
5.23	12.86
5.25	18.18
5.30	13.79
5.36	14.10
5.39	11.21
5.42	12.27
5.45	18.38
5.49	10.17
5.52	22.33
5.59	15.53
5.73	17.50
5.80	19.83
5.83	25.78
5.89	19.56
6.03	18.22
6.03	15.28
6.06	15.45
6.27	11.91
6.31	2.45
6.33	25.16
6.49	18.84
6.59	19.68
6.61	18.76
6.72	18.66
6.73	18.66
6.83	18.71
6.83	8.16
6.89	16.58
6.89	14.68
7.07	17.27
7.16	24.32
7.17	23.60
7.21	21.97
7.23	14.78
7.31	22.18
7.36	13.60
7.39	16.34
7.45	0.50
7.52	15.66
7.53	24.85
7.60	13.74
7.60	21.05
7.65	13.06
7.72	23.62
7.79	24.64
7.93	22.57
8.03	22.06
8.06	15.23
8.07	16.81
8.07	19.57
8.13	13.61
8.17	15.43
8.23	17.34
8.25	16.81
8.27	19.54
8.34	15.08
8.39	22.90
8.57	19.59
8.73	15.93
8.75	19.92
8.79	20.93
8.82	12.86
8.88	21.33
8.97	12.70
8.98	21.83
8.99	21.54
9.14	14.02
9.25	22.88
9.25	19.64
9.27	25.70
9.42	16.61
9.43	17.45
9.45	25.48
9.47	20.81
9.48	18.14
9.49	26.05
9.51	27.22
9.55	25.80
9.56	25.78
9.61	22.82
9.65	23.15
9.67	24.08
9.76	24.36
9.76	22.70
9.78	28.99
9.89	25.23
9.96	20.03
9.96	21.91
10.12	48.12
10.12	24.90
10.13	16.60
10.13	20.39
10.14	8.14
10.15	23.36
10.16	21.06
10.17	19.30
10.21	22.55
10.22	18.15
10.27	16.89
10.30	32.68
10.31	19.69
10.33	19.88
10.36	23.79
10.42	16.55
10.48	24.72
10.49	24.21
10.50	21.95
10.76	21.34
10.79	22.70
10.80	27.15
10.89	32.09
10.90	19.65
10.97	20.17
11.01	21.03
11.08	13.04
11.11	29.90
11.13	25.66
11.16	17.59
11.17	19.55
11.22	24.58
11.30	29.00
11.32	18.74
11.36	28.86
11.38	29.17
11.41	20.89
11.42	25.02
11.43	27.98
11.47	31.49
11.47	20.71
11.48	29.32
11.51	22.62
11.67	17.09
11.71	27.67
11.77	15.66
11.87	24.47
11.89	21.95
11.91	19.53
11.92	16.42
11.93	23.00
12.14	24.88
12.19	21.42
12.31	13.98
12.32	27.85
12.36	20.82
12.43	27.09
12.43	24.31
12.45	27.19
12.52	27.05
12.61	30.23
12.62	25.87
12.68	24.09
12.69	29.54
12.70	17.36
12.77	32.76
12.86	28.64
12.91	27.73
12.92	26.96
13.01	27.98
13.01	25.66
13.03	27.16
13.12	26.46
13.24	22.83
13.30	25.58
13.32	24.19
13.36	19.20
13.56	22.61
13.56	24.43
13.59	19.43
13.60	25.52
13.67	25.83
13.79	22.26
13.81	25.99
13.85	34.71
13.85	28.79
13.85	28.61
13.90	19.08
13.91	20.29
13.91	28.45
13.94	24.94
13.96	24.10
13.96	17.71
13.96	27.56
13.98	25.32
14.01	21.59
14.03	30.82
14.05	23.56
14.10	38.09
14.13	35.33
14.22	32.15
14.31	19.95
14.37	28.23
14.40	26.51
14.45	23.79
14.49	20.54
14.50	21.66
14.59	29.07
14.64	33.30
14.70	36.41
14.72	22.93
14.75	28.61
14.89	27.57
14.91	24.70
14.95	21.09
15.05	34.25
15.07	25.25
15.08	28.23
15.19	30.02
15.23	24.24
15.25	15.49
15.27	22.34
15.29	30.67
15.31	26.07
15.41	25.21
15.42	32.52
15.43	26.96
15.50	29.43
15.51	24.61
15.54	31.51
15.60	24.58
15.75	33.09
15.77	21.63
15.88	23.49
15.93	22.87
15.94	35.88
16.11	28.07
16.15	18.09
16.24	17.85
16.32	26.67
16.38	29.27
16.43	27.02
16.48	21.30
16.51	30.57
16.52	28.50
16.55	34.94
16.69	21.74
16.70	27.39
16.76	24.80
16.83	22.46
16.87	23.49
16.88	28.68
16.93	16.82
16.93	27.52
16.96	31.79
17.11	26.36
17.14	37.03
17.15	25.25
17.18	24.33
17.23	29.55
17.25	27.74
17.32	26.49
17.36	27.00
17.40	20.58
17.43	27.41
17.48	25.15
17.51	26.03
17.55	28.25
17.55	32.87
17.59	32.93
17.68	23.64
17.77	28.41
17.84	21.79
17.88	26.06
17.98	33.89
17.98	25.27
18.04	29.02
18.11	25.82
18.20	25.38
18.25	29.84
18.30	26.56
18.32	34.23
18.33	39.11
18.34	35.14
18.35	26.83
18.39	24.37
18.41	27.73
18.44	14.13
18.46	2.95
18.48	25.32
18.49	22.38
18.56	26.06
18.57	26.23
18.60	24.87
18.64	29.30
18.64	35.81
18.67	21.25
18.73	27.50
18.76	23.18
18.78	18.77
18.82	26.74
18.86	30.61
18.87	29.18
19.01	17.70
19.06	31.59
19.06	24.92
19.07	27.25
19.17	28.34
19.17	39.40
19.18	29.98
19.39	22.05
19.41	30.49
19.42	32.36
19.45	18.74
19.47	28.89
19.47	22.64
19.55	27.71
19.67	29.85
19.75	29.11
19.76	28.39
19.81	23.86
19.82	31.78
19.93	28.47
19.95	27.29
19.99	35.19
20.09	33.32
20.20	29.84
20.22	32.38
20.36	26.63
20.36	29.50
20.37	22.84
20.40	27.17
20.74	40.25
20.74	19.79
20.77	14.93
20.80	22.83
20.85	27.37
20.92	18.88
20.92	31.96
20.94	37.55
20.96	25.69
21.01	26.39
21.07	26.52
21.28	40.33
21.29	24.94
21.39	20.78
21.47	38.66
21.47	27.99
21.48	31.04
21.49	30.07
21.50	26.15
21.51	27.00
21.53	34.86
21.69	35.74
21.73	26.36
21.78	22.68
21.79	28.81
21.82	26.27
21.82	25.51
21.84	28.03
21.87	33.27
21.88	26.85
21.90	34.02
21.94	37.44
22.01	29.39
22.13	21.26
22.21	18.36
22.22	26.08
22.29	30.91
22.34	46.54
22.34	26.26
22.38	30.19
22.42	25.21
22.51	37.02
22.52	28.21
22.53	22.74
22.57	35.47
22.58	28.64
22.65	21.43
22.71	30.39
22.73	28.77
22.80	21.51
22.88	25.91
22.96	21.87
22.97	19.39
22.98	27.44
23.00	20.11
23.00	17.90
23.07	37.28
23.07	29.60
23.09	24.21
23.12	34.61
23.15	25.57
23.20	21.67
23.37	25.66
23.40	30.32
23.45	29.94
23.50	36.80
23.51	28.71
23.61	31.06
23.64	19.00
23.64	30.93
23.68	24.73
23.71	32.25
23.72	35.72
23.73	35.41
23.77	17.89
23.80	36.08
23.81	20.07
23.81	27.24
23.82	30.61
23.87	38.32
23.88	34.62
23.91	27.70
24.02	25.71
24.03	31.41
24.09	32.50
24.13	22.26
24.13	28.46
24.17	24.02
24.18	19.89
24.25	29.02
24.28	38.88
24.30	23.74
24.30	28.44
24.35	32.95
24.36	24.99
24.37	34.54
24.38	32.35
24.39	34.95
24.48	23.60
24.56	25.18
24.60	23.56
24.63	39.66
24.68	30.31
24.83	29.96
24.99	21.98
25.10	38.05
25.12	23.51
25.25	27.01
25.26	27.25
25.34	23.92
25.38	22.66
25.49	32.40
25.61	27.25
25.65	29.50
25.67	39.73
25.69	32.32
25.84	33.40
25.89	28.64
25.91	28.50
25.93	31.90
25.96	33.29
25.98	32.54
26.03	30.26
26.04	25.21
26.10	31.01
26.11	32.17
26.31	27.68
26.35	22.13
26.37	27.01
26.40	26.43
26.41	32.19
26.42	27.39
26.43	26.87
26.45	22.28
26.49	27.96
26.57	28.02
26.64	22.87
26.89	32.69
26.91	27.09
27.00	27.59
27.01	29.36
27.02	27.04
27.02	31.84
27.07	39.39
27.08	32.01
27.11	12.83
27.12	21.96
27.13	27.92
27.14	29.05
27.14	34.06
27.15	30.01
27.18	19.61
27.30	32.65
27.33	33.84
27.52	23.90
27.53	19.34
27.70	22.27
27.85	25.35
27.88	24.90
27.88	29.43
27.98	25.37
28.02	31.99
28.10	36.17
28.19	28.33
28.21	35.65
28.28	27.52
28.37	33.32
28.44	28.14
28.46	28.47
28.56	23.18
28.58	34.68
28.60	31.93
28.61	18.93
28.63	37.84
28.63	30.59
28.66	17.63
28.71	29.24
28.90	29.96
28.96	24.48
29.01	30.49
29.03	21.24
29.06	20.48
29.06	24.51
29.07	23.04
29.09	27.12
29.12	17.69
29.18	29.02
29.21	32.89
29.23	25.78
29.24	32.55
29.26	20.20
29.27	32.62
29.31	29.21
29.34	28.98
29.37	36.55
29.41	32.89
29.45	18.93
29.54	22.60
29.60	29.21
29.61	16.56
29.66	33.28
29.72	26.95
29.72	19.58
29.72	24.50
29.83	23.49
29.86	18.54
30.06	30.07
30.06	17.84
30.06	34.18
30.13	28.28
30.13	18.70
30.17	20.08
30.17	24.42
30.20	28.58
30.25	31.12
30.28	20.75
30.28	28.90
30.32	34.93
30.33	31.37
30.36	39.09
30.38	26.14
30.42	25.85
30.54	23.68
30.61	21.23
30.62	23.20
30.64	23.00
30.68	19.57
30.70	29.86
30.72	23.78
30.74	24.57
30.79	25.66
30.82	26.33
30.91	20.88
30.92	27.60
30.93	38.52
31.05	33.37
31.18	30.66
31.23	27.97
31.23	32.91
31.25	27.75
31.50	22.94
31.54	20.41
31.57	30.80
31.64	34.33
31.67	24.19
32.08	30.91
32.20	30.44
32.25	26.70
32.38	25.70
32.43	24.83
32.54	23.30
32.74	31.87
32.77	32.30
32.85	30.31
32.87	32.53
32.92	32.77
33.02	30.33
33.16	31.32
33.17	35.60
33.19	45.77
33.24	25.86
33.35	38.10
33.48	23.80
33.49	23.09
33.49	24.09
33.53	27.78
33.60	29.39
33.60	41.93
33.62	38.93
33.62	28.00
33.70	32.28
33.74	30.71
33.78	31.56
33.87	47.31
33.93	31.78
33.96	30.50
34.02	36.03
34.10	34.91
34.24	37.14
34.25	25.59
34.27	22.36
34.34	26.13
34.35	25.15
34.43	37.40
34.46	26.44
34.55	32.96
34.55	21.12
34.62	35.02
34.63	28.48
34.65	30.61
34.74	36.69
34.75	23.02
34.83	32.67
34.86	33.02
34.92	30.36
34.96	21.54
35.07	39.34
35.09	34.08
35.16	27.47
35.22	27.95
35.28	22.25
35.30	21.52
35.30	26.28
35.35	32.34
35.46	37.76
35.47	24.22
35.51	37.11
35.55	23.07
35.58	25.84
35.64	32.33
35.68	29.15
35.71	38.83
35.83	37.01
36.03	35.80
36.14	32.54
36.20	31.11
36.22	39.79
36.24	42.57
36.27	29.09
36.30	30.80
36.34	37.23
36.49	31.87
36.51	27.20
36.54	29.39
36.55	30.52
36.56	33.28
36.56	24.14
36.65	28.70
36.66	30.49
36.67	33.34
36.69	26.89
36.70	35.23
36.77	22.76
36.87	34.23
36.87	36.72
36.89	43.15
36.91	26.06
36.94	27.73
36.98	31.53
37.19	35.50
37.21	40.41
37.24	33.86
37.32	26.55
37.33	30.92
37.33	27.80
37.39	24.35
37.43	32.86
37.44	28.95
37.47	37.18
37.52	38.22
37.52	25.79
37.57	18.18
37.58	38.01
37.75	40.41
37.83	46.74
37.88	20.00
37.96	32.09
37.97	35.89
37.99	30.77
38.02	24.43
38.04	26.53
38.10	27.79
38.11	32.55
38.19	38.47
38.31	22.82
38.31	31.46
38.32	31.30
38.35	27.63
38.36	36.47
38.48	24.60
38.48	21.06
38.49	32.51
38.50	44.91
38.55	22.01
38.62	30.85
38.62	32.81
38.63	45.40
38.72	36.67
38.73	41.29
38.73	34.65
38.74	21.48
38.81	30.25
38.83	28.41
38.85	35.67
38.86	39.60
38.88	36.68
38.89	31.01
38.95	32.73
38.97	33.12
39.07	28.07
39.09	37.59
39.10	31.25
39.20	25.19
39.22	32.90
39.24	22.96
39.39	33.67
39.43	35.35
39.45	36.66
39.53	37.94
39.54	28.34
39.55	33.06
39.56	30.68
39.63	32.14
39.67	40.64
39.77	38.87
39.84	33.12
39.85	37.24
39.88	32.92
39.89	25.35
39.93	41.69
40.04	30.61
40.08	27.64
40.10	24.22
40.14	28.99
40.15	44.54
40.17	32.49
40.20	29.29
40.24	33.87
40.26	45.59
40.27	38.07
40.31	37.56
40.31	31.31
40.32	42.00
40.40	43.00
40.56	30.43
40.59	37.61
40.64	36.80
40.67	41.21
40.73	42.84
40.73	31.38
40.74	42.30
40.80	35.86
40.80	38.09
40.86	37.67
40.91	31.22
40.95	35.67
41.10	42.89
41.19	34.26
41.30	36.24
41.31	52.27
41.33	31.74
41.34	31.74
41.34	41.74
41.39	41.65
41.42	33.65
41.43	38.73
41.48	36.51
41.53	38.77
41.54	35.41
41.57	36.59
41.58	35.28
41.64	45.34
41.65	39.80
41.66	36.54
41.78	30.76
41.86	36.07
41.98	36.70
42.02	40.27
42.08	36.85
42.19	35.46
42.19	32.46
42.19	37.97
42.25	44.20
42.34	38.57
42.36	33.45
42.40	37.56
42.42	36.00
42.55	37.68
42.57	29.57
42.59	35.47
42.64	42.22
42.68	31.30
42.72	48.74
42.72	39.09
42.75	37.45
42.78	24.88
42.79	37.76
42.80	35.87
42.89	49.48
43.03	37.44
43.04	42.10
43.08	39.83
43.10	34.72
43.11	49.56
43.14	46.00
43.21	41.65
43.22	30.94
43.24	36.07
43.34	48.09
43.38	48.23
43.50	46.22
43.50	43.85
43.62	38.80
43.69	35.65
43.74	48.36
43.88	37.44
43.91	29.69
43.96	30.87
44.02	34.28
44.03	46.45
44.06	38.62
44.10	36.16
44.19	34.62
44.27	38.73
44.29	34.39
44.30	37.50
44.30	33.59
44.37	45.82
44.44	45.68
44.50	40.70
44.52	41.42
44.55	36.74
44.66	47.42
44.68	29.60
44.69	70.31
44.71	31.84
44.77	41.90
44.77	37.95
44.80	34.46
44.85	46.67
44.91	45.64
44.94	32.48
44.96	38.89
45.07	28.67
45.08	37.45
45.09	30.13
45.17	38.16
45.18	44.25
45.20	43.03
45.22	45.44
45.26	37.98
45.30	34.28
45.32	43.34
45.33	52.26
45.38	41.40
45.42	37.71
45.43	35.17
45.45	60.89
45.45	35.65
45.47	38.39
45.48	43.15
45.50	28.85
45.60	40.44
45.62	45.17
45.62	39.36
45.68	45.63
45.71	47.67
45.71	45.88
45.73	33.11
45.78	52.49
45.83	32.89
45.89	37.90
45.98	44.40
45.99	43.06
46.01	48.84
46.11	40.13
46.21	38.78
46.23	38.38
46.30	47.52
46.33	41.59
46.38	30.06
46.40	37.60
46.44	43.78
46.48	43.47
46.78	39.83
46.82	40.72
46.84	45.67
46.84	51.62
46.91	42.37
46.96	43.06
47.11	46.69
47.19	59.51
47.20	50.66
47.26	42.59
47.28	56.12
47.30	50.03
47.31	51.39
47.53	40.61
47.53	51.45
47.55	36.98
47.57	34.03
47.67	49.37
47.67	45.01
47.78	46.75
47.80	54.97
47.83	46.14
47.91	46.88
48.06	43.14
48.14	46.56
48.19	47.27
48.23	43.59
48.27	39.70
48.27	22.14
48.31	51.96
48.32	42.44
48.32	31.42
48.33	52.64
48.38	34.68
48.38	42.98
48.48	47.45
48.48	40.68
48.48	44.13
48.57	50.64
48.58	40.32
48.69	36.57
48.71	27.93
48.75	46.34
48.79	48.03
48.81	54.20
48.82	39.95
48.83	55.23
48.85	45.69
48.88	52.43
48.89	48.98
48.91	46.05
48.94	47.15
48.95	45.78
48.97	46.90
48.99	36.24
48.99	39.91
49.02	48.25
49.03	45.20
49.03	45.12
49.17	40.56
49.24	38.93
49.35	50.80
49.44	68.42
49.45	42.43
49.54	52.17
49.55	40.01
49.61	46.84
49.68	53.70
49.83	50.07
49.84	54.08
49.85	44.48
49.90	42.13
49.95	57.23
50.03	52.03
50.05	38.84
50.08	44.49
50.11	50.15
50.12	46.87
50.12	52.46
50.14	46.36
50.16	58.79
50.19	40.47
50.20	46.38
50.24	54.18
50.24	67.94
50.25	51.87
50.37	48.70
50.39	46.75
50.40	45.18
50.42	35.49
50.44	45.71
50.47	46.29
50.53	53.31
50.60	62.02
50.63	50.69
50.63	42.14
50.65	51.10
50.66	49.38
50.70	56.87
50.71	40.17
50.76	45.08
50.78	45.64
50.78	51.42
50.80	51.51
50.85	51.95
50.88	51.52
50.90	58.53
51.09	53.81
51.10	57.41
51.14	50.21
51.16	49.70
51.18	53.45
51.19	46.27
51.29	55.31
51.29	34.97
51.40	53.59
51.43	52.81
51.44	61.71
51.44	40.72
51.45	52.19
51.46	59.26
51.53	68.71
51.56	49.66
51.59	52.90
51.71	55.84
51.73	53.08
51.74	42.37
51.75	55.87
51.94	51.07
51.94	45.93
51.96	51.46
51.97	52.66
52.08	66.69
52.14	55.64
52.19	59.20
52.26	43.64
52.35	46.96
52.37	52.82
52.52	53.88
52.56	49.41
52.63	55.48
52.64	54.95
52.80	49.50
52.81	52.49
52.86	50.40
53.15	58.59
53.18	64.03
53.20	51.78
53.29	50.83
53.29	45.98
53.34	50.90
53.36	51.00
53.39	35.02
53.41	50.82
53.43	46.14
53.44	49.94
53.55	65.12
53.56	53.96
53.60	46.56
53.62	57.87
53.65	57.16
53.66	56.47
53.68	64.72
53.68	57.56
53.71	62.04
53.73	54.96
53.80	61.81
53.85	60.30
53.89	67.88
53.94	53.08
53.96	48.78
54.02	54.37
54.03	57.05
54.08	51.49
54.16	52.62
54.17	58.06
54.21	54.85
54.25	57.24
54.25	55.81
54.30	59.55
54.34	61.56
54.38	61.10
54.38	49.30
54.39	57.34
54.51	52.84
54.52	52.44
54.61	64.33
54.61	61.27
54.64	56.04
54.73	64.31
54.73	57.32
54.81	64.47
55.03	49.18
55.05	64.93
55.08	62.04
55.14	61.13
55.14	62.05
55.15	65.81
55.21	61.12
55.24	48.41
55.28	65.17
55.31	52.96
55.33	59.04
55.36	55.85
55.37	58.80
55.50	60.12
55.60	49.41
55.62	59.28
55.66	51.69
55.72	44.47
55.76	56.00
55.95	51.61
56.01	58.35
56.02	64.98
56.06	53.62
56.07	49.37
56.09	67.82
56.11	54.58
56.12	50.52
56.13	51.50
56.32	51.94
56.41	63.23
56.45	50.51
56.48	58.91
56.50	45.21
56.69	63.99
56.75	64.55
56.76	61.78
56.77	61.20
56.85	62.36
56.97	59.36
57.06	64.09
57.13	47.50
57.19	61.32
57.27	58.66
57.35	62.55
57.43	45.16
57.46	76.84
57.50	65.69
57.52	66.99
57.60	54.99
57.61	66.99
57.64	49.98
57.73	63.48
57.80	64.65
57.82	65.20
57.84	72.12
58.08	57.08
58.11	53.41
58.12	53.85
58.13	59.59
58.17	70.40
58.20	77.81
58.21	59.88
58.22	57.67
58.26	59.27
58.30	55.60
58.30	65.43
58.32	56.15
58.51	57.42
58.53	62.54
58.56	61.09
58.59	56.00
58.62	58.30
58.70	50.09
58.71	68.67
58.71	68.88
58.75	63.39
58.84	64.63
58.88	74.04
59.07	65.94
59.16	63.61
59.17	61.61
59.18	67.94
59.30	63.42
59.41	66.80
59.43	50.56
59.50	74.83
59.54	65.79
59.69	67.35
59.80	65.33
59.87	73.83
59.91	57.25
59.97	68.75
59.98	64.89
60.05	63.24
60.08	71.59
60.13	81.02
60.13	69.96
60.17	69.05
60.19	58.94
60.20	61.26
60.23	64.52
60.30	63.28
60.31	58.67
60.36	61.85
60.38	69.23
60.38	70.16
60.38	67.39
60.57	59.32
60.57	74.50
60.60	65.61
60.64	74.08
60.67	80.38
60.70	62.31
60.70	63.03
60.82	50.24
60.89	67.36
60.90	63.10
60.93	69.19
60.99	51.90
61.07	78.83
61.16	59.64
61.23	68.47
61.24	62.76
61.29	58.17
61.32	68.77
61.37	83.23
61.39	61.81
61.41	76.63
61.51	68.88
61.52	66.63
61.52	79.97
61.55	55.33
61.62	75.98
61.66	60.39
61.80	74.64
61.84	74.99
61.89	68.77
62.02	53.16
62.10	85.54
62.12	61.44
62.16	63.68
62.24	71.62
62.29	60.91
62.33	75.27
62.34	88.00
62.35	69.02
62.36	56.53
62.43	78.87
62.51	62.03
62.55	72.89
62.58	71.42
62.64	72.51
62.72	60.23
62.73	70.63
62.73	61.97
62.75	69.42
62.79	69.08
62.81	73.92
62.92	67.58
62.95	65.28
62.98	80.52
63.17	69.62
63.19	57.38
63.27	70.97
63.28	68.21
63.31	63.08
63.42	54.77
63.51	56.56
63.51	54.35
63.52	64.68
63.55	72.51
63.63	58.48
63.64	73.61
63.69	67.76
63.76	70.02
63.79	74.47
63.86	58.42
63.87	73.16
63.93	77.77
63.98	69.11
63.99	64.72
64.07	69.46
64.11	73.41
64.16	76.41
64.23	68.87
64.30	54.81
64.32	84.47
64.34	58.58
64.37	72.71
64.38	68.45
64.39	65.05
64.42	67.60
64.44	76.68
64.48	66.87
64.50	73.51
64.51	61.14
64.53	77.19
64.56	73.50
64.59	74.70
64.68	68.36
64.71	67.64
64.71	77.68
64.80	79.29
64.81	68.27
64.86	84.22
64.93	71.11
65.09	57.51
65.13	71.18
65.15	72.25
65.17	74.56
65.19	88.69
65.21	76.84
65.29	67.14
65.31	69.32
65.32	63.80
65.39	68.57
65.40	78.99
65.41	64.84
65.44	69.48
65.47	76.60
65.48	88.83
65.49	68.42
65.50	78.27
65.56	72.24
65.57	73.73
65.60	71.13
65.66	72.80
65.79	65.92
65.90	76.15
65.91	64.53
65.92	73.17
65.93	83.85
65.96	62.30
66.09	66.74
66.18	86.94
66.19	72.25
66.30	72.83
66.33	75.97
66.33	69.88
66.35	92.45
66.41	68.64
66.43	74.08
66.44	74.76
66.49	71.04
66.49	74.71
66.50	74.09
66.55	62.65
66.62	69.53
66.63	72.38
66.66	79.36
66.72	55.00
66.72	75.10
66.88	77.02
66.92	59.52
67.13	68.62
67.19	67.54
67.20	61.22
67.34	78.62
67.35	71.53
67.49	76.88
67.50	71.36
67.52	72.28
67.53	73.13
67.58	78.91
67.63	67.41
67.67	76.06
67.69	77.06
67.70	75.14
67.73	82.69
67.75	71.09
67.79	72.14
67.80	62.49
67.83	77.74
67.84	76.50
67.85	80.66
68.05	81.53
68.18	83.33
68.27	62.20
68.37	84.76
68.38	81.45
68.39	96.88
68.39	76.78
68.50	70.51
68.68	78.19
68.75	76.83
68.76	78.30
68.78	72.76
68.79	62.35
68.82	74.76
68.85	71.59
68.98	72.92
69.05	68.26
69.07	66.80
69.11	71.82
69.21	69.62
69.24	62.74
69.25	85.28
69.30	70.34
69.36	84.23
69.39	86.02
69.47	77.99
69.52	84.97
69.53	82.56
69.61	70.70
69.61	72.44
69.62	77.94
69.62	89.96
69.67	71.47
69.71	76.12
69.77	79.80
69.83	79.13
69.89	66.55
69.91	64.70
70.00	57.07
70.02	74.91
70.37	76.28
70.39	82.81
70.50	61.13
70.52	73.52
70.69	83.51
70.70	95.39
70.82	82.70
70.83	84.75
70.84	79.97
70.89	82.93
70.92	72.86
70.99	71.72
71.06	89.19
71.07	77.56
71.09	71.41
71.17	73.18
71.22	65.29
71.32	80.73
71.33	80.58
71.35	79.51
71.43	63.92
71.44	65.14
71.50	71.99
71.55	73.59
71.57	73.10
71.62	59.60
71.69	75.94
71.83	73.23
71.89	73.06
71.95	84.89
71.98	73.95
72.08	67.06
72.10	72.34
72.12	65.89
72.28	65.67
72.39	72.09
72.44	71.05
72.45	75.49
72.47	61.05
72.76	91.12
72.80	65.80
72.85	80.19
72.88	67.90
72.89	63.36
72.93	73.83
72.94	79.00
72.94	75.43
73.08	71.39
73.17	77.32
73.23	74.84
73.24	78.81
73.25	66.68
73.55	74.30
73.66	73.75
73.66	74.29
73.69	76.92
73.75	79.87
73.76	69.32
73.78	61.78
73.78	77.85
73.84	89.99
73.85	68.27
73.86	79.73
73.88	61.85
73.94	69.03
74.01	63.37
74.03	72.80
74.13	82.33
74.14	89.02
74.17	84.85
74.17	85.48
74.22	83.97
74.23	68.38
74.29	73.46
74.30	80.47
74.31	90.82
74.35	63.98
74.40	77.57
74.43	63.60
74.44	74.66
74.52	89.52
74.59	77.99
74.66	82.10
74.77	73.65
74.78	70.31
74.85	75.33
74.86	77.34
74.96	84.30
74.98	69.62
75.01	78.24
75.02	74.09
75.11	74.17
75.15	75.24
75.19	70.64
75.19	73.41
75.25	63.29
75.33	81.25
75.39	64.49
75.40	75.16
75.44	78.29
75.45	77.73
75.46	64.65
75.52	68.20
75.56	85.05
75.59	74.86
75.61	82.87
75.69	62.91
75.70	87.79
75.72	77.04
75.77	85.12
75.79	69.81
75.80	83.79
75.81	84.55
75.82	74.21
75.82	79.51
75.86	72.97
75.92	63.57
75.94	59.31
75.97	78.95
76.01	81.93
76.05	75.04
76.08	68.31
76.09	73.43
76.15	70.34
76.16	71.43
76.22	82.27
76.30	81.88
76.31	58.04
76.32	74.68
76.33	70.62
76.38	63.88
76.42	78.50
76.63	73.36
76.69	74.64
76.69	75.87
76.70	66.54
76.74	81.79
76.74	63.73
76.81	70.41
76.84	74.23
76.86	75.18
76.96	83.15
76.96	80.57
76.97	65.11
77.02	71.09
77.09	71.30
77.10	70.20
77.14	94.89
77.15	88.35
77.22	81.73
77.23	71.89
77.25	73.97
77.27	81.17
77.27	89.05
77.34	85.13
77.34	60.56
77.36	81.23
77.37	89.49
77.38	77.44
77.39	82.86
77.45	71.77
77.53	71.23
77.64	79.69
77.66	70.66
77.70	75.95
77.71	81.86
77.73	81.41
77.74	86.52
77.75	66.53
77.78	75.26
77.84	69.28
77.86	64.36
77.88	81.11
77.88	73.12
77.88	74.23
77.92	88.70
78.01	85.52
78.08	74.95
78.10	80.45
78.15	68.80
78.17	69.81
78.28	83.24
78.36	83.30
78.38	84.03
78.39	79.59
78.42	87.67
78.44	84.89
78.47	80.27
78.53	84.59
78.56	91.34
78.58	71.99
78.60	83.92
78.61	90.36
78.64	82.45
78.65	79.22
78.66	70.44
78.68	63.47
78.76	83.94
78.95	80.74
78.99	73.70
79.00	80.32
79.06	79.71
79.08	88.75
79.10	87.98
79.12	82.54
79.16	78.84
79.18	71.20
79.27	76.83
79.29	85.75
79.37	67.08
79.38	71.14
79.41	74.87
79.42	68.52
79.56	86.19
79.60	73.72
79.60	80.80
79.61	76.60
79.66	77.18
79.73	83.18
79.74	84.53
79.75	83.27
79.77	67.90
79.82	74.89
79.83	66.71
79.92	63.70
79.99	66.78
80.10	75.46
80.13	79.92
80.16	80.05
80.17	74.63
80.28	68.19
80.38	65.68
80.40	78.52
80.47	77.47
80.48	66.32
80.55	79.83
80.59	67.37
80.61	78.25
80.68	83.86
80.69	61.22
80.79	85.66
80.89	79.34
80.92	78.71
80.93	79.89
81.04	71.58
81.04	84.09
81.07	61.65
81.11	76.33
81.14	58.92
81.18	84.60
81.25	84.62
81.30	80.59
81.44	88.70
81.44	76.54
81.48	78.43
81.50	79.06
81.53	74.05
81.58	68.81
81.77	75.35
81.84	82.00
81.85	70.56
81.89	60.96
81.90	72.01
81.92	77.46
82.00	71.73
82.04	86.21
82.07	81.36
82.11	85.30
82.14	81.84
82.16	69.59
82.18	72.27
82.19	78.67
82.26	76.78
82.28	67.79
82.34	75.96
82.35	81.31
82.36	84.03
82.43	67.97
82.52	84.70
82.61	72.50
82.65	85.78
82.65	84.73
82.67	77.13
82.70	62.05
82.76	66.81
82.83	77.47
83.04	76.05
83.06	75.41
83.09	72.97
83.11	96.69
83.11	88.98
83.13	62.05
83.17	62.11
83.22	92.58
83.25	85.06
83.28	74.84
83.36	83.29
83.41	66.00
83.45	76.95
83.47	90.52
83.48	76.96
83.63	59.28
83.64	91.77
83.67	70.71
83.70	95.02
83.73	75.22
83.75	82.41
83.90	68.42
83.93	72.63
83.94	71.12
83.96	88.54
83.97	82.89
84.03	84.98
84.08	77.85
84.14	84.72
84.19	68.00
84.23	101.15
84.30	85.42
84.40	70.72
84.40	76.57
84.42	70.73
84.43	74.73
84.45	85.12
84.50	73.23
84.55	78.37
84.56	75.67
84.59	71.69
84.59	71.53
84.60	84.57
84.67	80.23
84.71	68.84
84.75	68.64
84.76	93.74
84.78	78.90
84.79	72.16
84.80	84.97
84.83	87.81
84.88	85.12
84.90	77.93
84.95	72.50
84.98	94.73
85.17	85.63
85.18	70.02
85.21	81.26
85.22	91.64
85.25	81.66
85.27	75.59
85.45	84.58
85.69	88.90
85.72	74.18
85.82	76.08
85.83	81.52
85.88	84.38
86.00	67.22
86.07	64.42
86.16	87.16
86.17	81.89
86.17	79.82
86.18	78.84
86.24	80.20
86.32	86.73
86.33	87.59
86.35	90.79
86.37	77.04
86.43	61.21
86.55	100.65
86.72	81.64
86.80	85.16
86.81	84.11
86.83	66.03
86.86	65.18
86.94	61.02
86.94	73.57
86.95	80.86
86.97	80.24
86.99	104.18
87.09	86.68
87.11	67.27
87.14	75.71
87.17	82.37
87.18	92.00
87.20	85.36
87.21	80.58
87.22	83.60
87.34	78.52
87.38	82.31
87.43	88.58
87.44	73.91
87.46	95.21
87.50	76.35
87.52	82.06
87.64	82.08
87.65	71.77
87.70	91.21
87.84	70.85
87.88	81.18
87.96	77.23
88.01	86.29
88.02	68.20
88.04	72.24
88.13	85.61
88.17	74.19
88.29	90.32
88.31	86.66
88.33	86.06
88.34	84.37
88.62	68.53
88.63	91.70
88.79	89.86
88.84	75.47
88.91	79.45
88.91	79.80
89.04	78.46
89.19	73.21
89.22	89.42
89.31	90.81
89.33	74.81
89.37	86.08
89.40	78.06
89.42	100.06
89.43	79.68
89.55	97.22
89.72	66.27
89.74	73.20
89.80	71.12
89.84	72.92
89.91	81.47
89.92	70.74
89.93	90.28
89.94	75.97
89.96	80.63
90.05	90.48
90.13	68.56
90.24	80.44
90.24	72.26
90.28	73.53
90.28	76.56
90.31	73.31
90.32	77.13
90.36	88.78
90.38	82.39
90.46	87.44
90.50	76.78
90.54	76.88
90.54	67.84
90.56	76.10
90.69	94.22
90.80	85.44
90.84	85.10
90.88	92.87
90.88	66.98
90.92	84.36
90.92	74.83
90.94	92.08
90.95	76.86
91.06	87.69
91.17	71.65
91.18	95.60
91.21	88.08
91.22	83.76
91.34	70.21
91.40	84.19
91.44	83.58
91.44	81.04
91.49	81.97
91.49	73.15
91.58	79.23
91.68	72.89
91.78	89.91
91.78	82.12
91.80	77.90
91.81	93.42
91.92	102.35
92.25	91.89
92.35	80.80
92.51	84.60
92.59	84.94
92.64	87.31
92.70	75.22
92.77	98.14
92.78	83.27
92.81	90.88
92.84	96.73
92.88	84.94
93.00	82.59
93.01	76.48
93.06	87.84
93.21	71.02
93.31	82.82
93.56	67.16
93.58	75.17
93.59	96.28
93.59	81.22
93.61	86.72
93.67	99.00
93.69	66.78
93.70	97.70
93.71	90.83
93.74	62.87
93.86	85.22
93.90	92.41
93.99	76.89
94.05	74.26
94.06	62.95
94.07	68.72
94.10	87.72
94.11	73.37
94.16	97.53
94.21	84.37
94.28	85.55
94.31	91.62
94.42	90.23
94.44	79.24
94.45	100.29
94.45	89.45
94.48	86.19
94.52	88.86
94.52	81.84
94.57	82.21
94.61	79.26
94.66	89.61
94.68	85.36
94.70	72.62
94.81	82.01
94.89	90.58
94.91	84.15
94.96	82.72
95.04	72.86
95.09	87.08
95.17	81.93
95.17	76.02
95.29	81.74
95.34	83.73
95.38	98.17
95.53	75.11
95.53	79.56
95.54	98.29
95.61	102.83
95.62	90.60
95.62	77.07
95.64	69.25
95.66	83.15
95.67	81.68
95.80	81.06
95.81	87.91
95.83	85.85
95.85	91.42
95.86	89.43
95.89	85.82
95.90	68.00
96.03	82.41
96.03	88.05
96.06	76.84
96.09	79.37
96.12	93.69
96.12	75.68
96.17	89.77
96.19	84.50
96.23	84.70
96.30	88.96
96.36	71.70
96.39	95.83
96.47	75.81
96.49	91.00
96.51	84.69
96.56	76.77
96.57	81.59
96.62	84.41
96.66	91.73
96.66	76.00
96.76	104.26
96.81	89.28
96.90	89.95
96.90	81.17
96.91	108.39
96.94	64.33
97.04	92.39
97.08	83.69
97.09	69.09
97.10	94.86
97.10	86.20
97.15	77.80
97.18	91.51
97.19	97.69
97.20	84.32
97.25	100.16
97.28	100.28
97.29	95.17
97.51	93.17
97.57	49.62
97.63	85.35
97.77	76.22
97.83	88.63
97.85	71.19
97.96	82.49
98.00	74.38
98.01	88.72
98.02	80.82
98.03	85.25
98.09	80.72
98.10	92.74
98.17	93.48
98.21	85.63
98.24	80.40
98.29	83.84
98.37	86.27
98.41	82.57
98.45	83.23
98.47	91.83
98.52	75.66
98.54	90.67
98.57	79.01
98.60	86.59
98.68	96.34
98.70	83.96
98.72	78.08
98.73	92.06
98.79	99.12
98.94	91.66
98.96	113.56
98.97	92.07
99.09	93.93
99.14	103.60
99.17	87.74
99.19	96.37
99.22	103.66
99.27	92.13
99.35	89.89
99.35	107.61
99.36	83.84
99.37	97.02
99.42	93.95
99.52	92.03
99.55	89.71
99.56	91.26
99.58	87.67
99.58	97.07
99.64	95.19
99.66	92.92
99.71	81.30
99.80	91.62
99.82	85.24
99.82	90.74
99.83	85.16
99.84	86.22
99.87	90.72
99.90	105.39
100.02	93.88
100.10	80.96
100.11	94.56
100.25	98.63
100.25	95.86
100.27	78.53
100.27	81.11
100.35	101.82
100.44	97.68
100.46	103.13
100.52	94.10
100.62	96.83
100.64	105.67
100.71	102.74
100.78	100.25
100.88	73.02
100.91	96.87
100.92	100.13
100.93	109.13
100.95	89.06
101.00	83.79
101.04	84.48
101.13	98.78
101.19	95.38
101.20	99.97
101.24	92.28
101.25	89.50
101.27	92.65
101.27	90.42
101.29	105.20
101.30	94.50
101.34	92.97
101.39	92.09
101.39	97.39
101.41	105.33
101.44	89.25
101.46	94.95
101.51	87.97
101.52	90.77
101.52	101.23
101.52	88.15
101.54	95.26
101.57	94.01
101.58	99.19
101.61	97.74
101.69	100.75
101.74	97.83
101.76	96.21
101.76	93.39
101.84	105.93
101.92	90.38
102.07	91.15
102.11	89.07
102.11	100.06
102.12	91.42
102.15	85.11
102.16	98.50
102.20	73.72
102.29	100.98
102.34	100.09
102.35	98.33
102.39	110.83
102.40	88.71
102.43	99.11
102.46	86.72
102.49	98.73
102.50	105.01
102.56	89.80
102.57	86.87
102.65	108.00
102.67	99.94
102.70	92.65
102.72	103.71
102.75	96.35
102.83	93.87
102.88	87.30
102.94	97.50
103.10	86.22
103.17	112.50
103.24	88.77
103.26	97.64
103.38	101.44
103.42	106.05
103.43	88.01
103.46	100.11
103.47	84.06
103.55	87.31
103.58	85.50
103.64	102.99
103.67	109.45
103.72	96.54
103.86	106.39
103.96	108.61
103.98	102.98
104.06	80.53
104.07	91.50
104.15	106.04
104.25	93.15
104.27	116.39
104.28	105.10
104.36	77.32
104.38	94.97
104.44	108.77
104.45	89.72
104.50	91.90
104.53	88.69
104.56	106.40
104.66	85.57
104.67	99.85
104.70	103.00
104.77	93.99
104.78	93.41
104.79	96.07
104.80	86.00
104.94	95.45
105.08	100.82
105.12	97.32
105.24	92.00
105.28	103.22
105.36	86.00
105.39	105.16
105.49	118.45
105.49	90.40
105.51	76.16
105.55	89.03
105.56	107.39
105.66	102.36
105.69	104.93
105.77	100.69
105.85	109.17
106.03	100.30
106.07	111.68
106.12	103.83
106.13	115.96
106.48	104.94
106.56	87.81
106.58	92.49
106.66	92.00
106.67	106.72
106.69	104.86
106.72	80.40
106.72	107.25
106.74	110.74
106.76	91.19
106.88	95.63
106.88	103.30
106.89	87.46
106.92	91.42
107.01	88.62
107.02	104.12
107.18	100.02
107.24	106.66
107.30	102.32
107.46	98.09
107.47	132.62
107.50	93.34
107.51	92.68
107.53	104.33
107.61	101.06
107.65	111.74
107.65	111.98
107.70	92.18
107.80	110.68
107.87	103.34
108.03	91.21
108.12	118.89
108.21	92.16
108.23	97.75
108.24	95.39
108.31	106.59
108.34	104.66
108.35	109.51
108.40	103.64
108.42	95.48
108.42	105.17
108.42	120.78
108.46	114.15
108.52	99.81
108.59	96.05
108.64	99.15
108.74	107.16
108.81	99.31
108.82	106.60
108.89	73.32
108.89	109.67
108.94	106.45
109.17	111.90
109.21	94.98
109.24	110.20
109.25	110.58
109.28	101.50
109.29	95.53
109.33	96.58
109.35	104.06
109.35	102.26
109.37	90.00
109.40	101.30
109.47	102.63
109.47	110.29
109.59	96.66
109.61	101.72
109.67	91.22
109.70	124.82
109.75	109.15
109.77	110.27
109.89	110.97
109.89	101.65
110.03	89.78
110.04	102.92
110.04	103.37
110.06	108.76
110.06	110.17
110.14	104.13
110.15	103.00
110.19	95.90
110.25	101.93
110.34	97.62
110.38	120.45
110.49	95.79
110.58	95.55
110.62	106.34
110.63	109.84
110.64	110.96
110.69	98.46
110.72	85.36
110.76	110.86
110.76	113.09
110.76	91.94
110.77	97.40
110.79	104.08
110.91	102.50
110.96	105.14
111.07	108.97
111.10	89.60
111.18	99.47
111.18	114.62
111.37	112.18
111.41	74.34
111.44	102.85
111.48	105.99
111.54	93.22
111.56	118.75
111.58	108.73
111.59	101.25
111.60	108.39
111.64	101.47
111.64	109.21
111.69	94.51
111.69	99.23
111.70	119.28
111.72	102.79
111.77	100.53
111.82	104.16
111.84	110.62
111.85	117.31
111.86	121.21
111.87	88.11
111.92	106.54
112.08	114.84
112.09	122.14
112.16	110.72
112.16	103.78
112.23	107.86
112.34	107.58
112.43	97.40
112.43	117.62
112.43	94.86
112.45	93.93
112.52	98.07
112.53	85.38
112.62	111.56
112.67	99.62
112.71	66.13
112.80	87.68
112.83	97.64
112.87	109.06
112.94	100.89
112.96	113.85
113.05	103.25
113.06	105.68
113.06	99.56
113.07	96.28
113.11	122.30
113.14	105.04
113.18	109.31
113.18	99.89
113.20	104.44
113.36	102.40
113.36	88.48
113.37	99.73
113.42	99.50
113.50	99.83
113.51	86.21
113.54	88.14
113.59	95.64
113.61	106.92
113.62	112.17
113.64	111.11
113.70	124.60
113.76	104.11
113.77	115.35
113.89	95.17
113.90	119.29
113.92	110.89
113.92	102.93
113.98	101.66
114.00	107.65
114.15	93.72
114.15	89.43
114.28	84.49
114.34	101.11
114.35	89.59
114.59	114.61
114.60	100.37
114.65	85.93
114.67	103.86
114.79	99.60
114.82	120.64
114.86	92.78
114.97	114.69
114.98	124.04
115.02	111.62
115.02	108.08
115.02	114.68
115.10	99.22
115.22	121.80
115.24	104.13
115.25	106.32
115.26	105.53
115.26	99.75
115.38	110.67
115.49	112.13
115.50	109.58
115.51	111.29
115.51	112.04
115.57	101.29
115.62	109.96
115.74	120.78
115.77	100.65
115.77	95.35
115.93	114.22
115.99	121.80
116.00	119.96
116.04	96.35
116.06	118.05
116.16	114.54
116.23	108.76
116.23	121.93
116.24	124.46
116.24	113.21
116.30	101.34
116.41	91.88
116.46	107.68
116.67	118.63
116.72	101.97
116.85	105.37
116.91	112.40
116.98	90.42
116.99	111.80
117.02	103.53
117.04	103.32
117.13	110.23
117.17	100.90
117.23	107.12
117.27	96.11
117.36	106.87
117.37	113.87
117.46	117.52
117.47	110.66
117.48	100.09
117.54	123.31
117.55	99.54
117.55	101.33
117.59	105.50
117.60	122.67
117.66	102.23
117.67	109.54
117.73	107.50
117.77	109.73
117.99	108.39
118.04	108.73
118.05	109.14
118.06	108.46
118.07	93.54
118.12	99.48
118.18	110.59
118.21	106.25
118.25	95.10
118.30	118.59
118.34	110.13
118.40	115.93
118.41	126.73
118.48	102.07
118.50	103.43
118.50	97.15
118.63	109.78
118.67	103.99
118.69	106.19
118.75	105.58
118.77	106.95
118.81	116.25
118.95	95.75
118.97	95.78
119.06	110.05
119.07	98.63
119.09	111.89
119.11	119.18
119.15	73.90
119.17	113.43
119.18	96.72
119.30	105.86
119.33	102.44
119.40	115.50
119.43	98.67
119.56	93.80
119.57	128.14
119.57	131.88
119.63	85.37
119.66	96.27
119.66	94.11
119.66	104.59
119.68	85.31
119.79	107.34
119.80	96.16
119.81	108.45
119.86	98.21
119.88	116.47
119.88	111.98
119.95	100.69
119.97	95.18
Nota: Tamaño muestral n = 2472 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 Cu\nen muestras geoquímicas de depósitos minerales",
  xlab = "SiO2 (%)",
  ylab = "Cu (ppm)",
  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 decreciente/asintótica entre el contenido de SiO2 y la concentración de Cu. A medida que aumenta el porcentaje de sílice, la concentración de cobre disminuye a una tasa no constante. Por ello, se propone un modelo de regresión potencial \(Y = a \cdot X^b\), el cual permite modelar cambios porcentuales proporcionales entre ambas variables geoquímicas.

PARÁMETROS

#Modelo potencial general:
# Y = a * X^b
#Modelo potencial aplicado al estudio:
# Cu = a * (SiO2)^b
#Ajuste del modelo (Mediante linealización logarítmica):

# Transformación logarítmica: ln(Y) = ln(a) + b * ln(X)
X_log <- log(X)
Y_log <- log(Y)

modelo_pot <- lm(Y_log ~ X_log)
summary(modelo_pot)

## 
## Call:
## lm(formula = Y_log ~ X_log)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.2709 -0.0992  0.0292  0.1288  1.0735 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.118008   0.025413   43.99   <2e-16 ***
## X_log       0.726860   0.006351  114.45   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2283 on 2470 degrees of freedom
## Multiple R-squared:  0.8414, Adjusted R-squared:  0.8413 
## F-statistic: 1.31e+04 on 1 and 2470 DF,  p-value: < 2.2e-16

beta0 <- coef(modelo_pot)[1] # ln(a)
beta1 <- coef(modelo_pot)[2] # b

a_est <- exp(beta0)
b_est <- beta1

a_est

## (Intercept) 
##    3.058756

b_est

##     X_log 
## 0.7268603

cat(
  "Y = ",
  round(a_est, 5),
  " * X^(",
  round(b_est, 5),
  ")"
)

## Y =  3.05876  * X^( 0.72686 )

Justificación del uso de la regresión lineal lm()Aunque la relación entre el SiO2 y el Cu es de naturaleza potencial no lineal (\(Y = a \cdot X^b\)), es posible emplear la función lm() aplicando una transformación logarítmica en ambos lados de la ecuación:

# ln(Y) = ln(a) + b * ln(X)

Definiendo variables transformadas \(Y^* = \ln(Y)\), \(X^* = \ln(X)\) y constante \(\beta_0 = \ln(a)\), la ecuación se convierte en un modelo lineal en sus parámetros:

# Y* = β0 + b * X*

Por lo tanto, los parámetros se estiman mediante mínimos cuadrados ordinarios en R mediante la instrucción:

# lm(log(Cu) ~ log(SiO2))

Finalmente, retransformando la intersección se recupera el parámetro de escala \(a = e^{\beta_0}\) para obtener la ecuación potencial en sus unidades originales.

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 potencial entre SiO2 y Cu\nen muestras geoquímicas de depósitos minerales",
  xlab = "SiO2 (%)",
  ylab = "Cu (ppm)",
  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 potencial
x_seq <- seq(
  min(X, na.rm = TRUE),
  max(X, na.rm = TRUE),
  length.out = 1000
)

y_pred <- a_est * (x_seq ^ b_est)

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

box(lwd = 1.5)

legend(
  "topright",
  legend = c("Datos reales", "Modelo potencial"),
  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 (en espacio linealizado)
r <- cor(X_log, Y_log) * 100
r

## [1] 91.72573

# Coeficiente de determinación R^2
R2 <- summary(modelo_pot)$r.squared * 100
R2

## [1] 84.13609

# Tabla resumen del modelo
Variable <- c("SiO2 - Cu")

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

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

knitr::kable(
  tabla_resumen_potencial,
  format = "markdown",
  caption = "Tabla resumen del modelo potencial"
)

Tabla resumen del modelo potencial

Variable	Test Pearson (%)	R cuadrado (%)
SiO2 - Cu	91.73	84.14

RESTRICCIONES

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

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

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

El modelo potencial exige estrictamente que \(X > 0\) e \(Y > 0\), dado que la transformación logarítmica \(\ln(0)\) no está definida matemáticamente. Físicamente, el porcentaje de SiO2 y la concentración de Cu son valores siempre positivos. Las estimaciones deben ser acotadas al rango observado de SiO2, evitando extrapolaciones a valores extremos cercanos a 0 o excesivamente altos donde el comportamiento geoquímico perdería validez.

ESTIMACIONES

SiO2_objetivo <- 60

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

Cu_est <- a_est * (SiO2_objetivo ^ b_est)

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

text(
  1, 1,
  labels = paste(
    "¿Cuál es la concentración de Cu esperada\n",
    "cuando el SiO2 es",
    round(SiO2_objetivo, 2), "%?\n\n",
    "Resultado estimado (Cu):",
    round(Cu_est, 2), "ppm"
  ),
  cex = 1.2,
  col = "blue",
  font = 2
)

incremento_pct <- 0.10
nuevo_SiO2 <- SiO2_objetivo * (1 + incremento_pct)

# Cálculo del Cu actual
Cu_actual <- a_est * (SiO2_objetivo ^ b_est)

# Cálculo del nuevo Cu
nuevo_Cu <- a_est * (nuevo_SiO2 ^ b_est)

porcentaje_cambio_Cu <- ((nuevo_Cu - Cu_actual) / Cu_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 la concentración de Cu esperada\n",
    "al incrementar el SiO2 un ", (incremento_pct * 100), "%?\n\n",
    "Cambio estimado de Cu: ",
    round(porcentaje_cambio_Cu, 2), "%"
  ),
  cex = 1.2,
  col = "firebrick3",
  font = 2
)

CONCLUSIÓN

Entre el porcentaje de SiO2 y la concentración de Cu existe una relación de tipo potencial, representada por el modelo ajustado \(Y = a \cdot X^b\). El modelo permite describir la tasa de cambio en la concentración de cobre en función de variaciones en la proporción de sílice. El coeficiente de correlación de Pearson obtenido fue de r round(r, 2)% y el coeficiente de determinación de r round(R2, 2)%, indicando la capacidad explicativa del modelo dentro del dominio geoquímico estudiado.