Estadística multivariable

# Relación entre variables
# Modelo Exponencial
#Autor: Grupo  1
#Fecha:11/06/2026

0.- Carga de Librerias

library(gt)

## Warning: package 'gt' was built under R version 4.5.3

library(dplyr)

## Warning: package 'dplyr' was built under R version 4.5.3

## 
## Adjuntando el paquete: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(e1071)

## Warning: package 'e1071' was built under R version 4.5.3

library(htmltools)

1.- Carga de Datos

#Cargar los datos 
datos <- read.csv("~/semestre 3 y 4/Estadistica/Datos Cambiados.csv",
                  header = TRUE, dec = ".", sep = ",", na.strings = "-")

datos <- na.omit(datos)

datos <- datos[order(datos$AQI, datos$NO2), ]

2.- Definición de variables

La concentración de dióxido de nitrogeno (NO2) es la CAUSA porque corresponde a un contaminante gaseoso primario de alta toxicidad que se acumula en la atmósfera, especialmente en zonas urbanas con alta actividad vehicular e industrial, y el Índice de Calidad del Aire (AQI) es el EFECTO, ya que varía en respuesta a los incrementos en la concentración de NO2 para sintetizar el nivel de riesgo para la salud humana, mostrando un comportamiento de tipo exponencial.

# Selección de variables

# Y es Variable dependiente
Y_original <- datos$AQI            #EFECTO

# X es Variable Independiente
X_original <- datos$NO2            #CAUSA

3.- Tabla pares de valores

# Variables originales 

Y_original <- datos$AQI      # Variable dependiente
X_original <- datos$NO2      # Variable independiente

n_original <- nrow(datos)

cat("Tamaño muestral =", n_original)

## Tamaño muestral = 4954

TO_NO2_AQI <- data.frame(X_original, Y_original)

# Tabla de pares de valores originales
tabla_original <- TO_NO2_AQI %>%
  gt() %>%
  cols_align(
    align = "center",
    columns = everything()
  ) %>%
  fmt_number(
    columns = everything(),
    decimals = 2
  ) %>%
  tab_header(
    title = md("*Tabla Nro. 1*"),
    subtitle = md("*Pares de valores originales de NO2 y AQI, Calidad del aire en India 2015-2020*")
  ) %>%
  tab_source_note(
    source_note = md(paste0(
      "**Nota:** Tamaño muestral n = ", nrow(TO_NO2_AQI),
      " observaciones utilizadas en el análisis."
    ))
  )

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

Tabla Nro. 1
Pares de valores originales de NO2 y AQI, Calidad del aire en India 2015-2020
X_original	Y_original
12.21	23.00
9.40	26.00
18.69	26.00
12.29	29.00
18.86	29.00
9.31	30.00
9.66	30.00
18.92	30.00
12.54	31.00
17.70	31.00
21.59	31.00
5.94	32.00
8.63	32.00
9.52	32.00
10.24	32.00
10.86	32.00
12.65	32.00
15.09	32.00
19.97	32.00
11.26	33.00
11.43	33.00
11.90	33.00
17.97	33.00
18.89	33.00
19.95	33.00
22.64	33.00
9.80	34.00
10.35	34.00
10.85	34.00
11.89	34.00
11.97	34.00
12.37	34.00
12.40	34.00
13.00	34.00
13.90	34.00
16.73	34.00
19.61	34.00
43.87	34.00
8.88	35.00
9.71	35.00
12.12	35.00
12.20	35.00
12.34	35.00
12.34	35.00
12.76	35.00
13.07	35.00
13.54	35.00
20.71	35.00
9.91	36.00
9.99	36.00
10.33	36.00
11.89	36.00
12.57	36.00
13.38	36.00
14.34	36.00
14.78	36.00
15.62	36.00
18.31	36.00
24.60	36.00
6.66	37.00
6.87	37.00
7.36	37.00
9.25	37.00
12.20	37.00
12.93	37.00
13.12	37.00
13.33	37.00
13.35	37.00
20.22	37.00
20.73	37.00
22.18	37.00
23.46	37.00
25.18	37.00
5.99	38.00
8.58	38.00
9.69	38.00
10.05	38.00
11.15	38.00
11.38	38.00
11.79	38.00
11.88	38.00
11.89	38.00
12.69	38.00
12.91	38.00
14.23	38.00
19.35	38.00
20.54	38.00
20.72	38.00
20.85	38.00
22.78	38.00
6.05	39.00
8.79	39.00
10.02	39.00
10.47	39.00
10.63	39.00
11.92	39.00
12.72	39.00
13.26	39.00
15.32	39.00
16.94	39.00
17.24	39.00
22.71	39.00
9.17	40.00
9.41	40.00
10.16	40.00
12.01	40.00
12.76	40.00
13.20	40.00
13.24	40.00
14.46	40.00
14.93	40.00
19.87	40.00
23.70	40.00
24.88	40.00
9.00	41.00
10.91	41.00
12.47	41.00
14.94	41.00
15.96	41.00
19.43	41.00
20.56	41.00
20.70	41.00
23.18	41.00
23.21	41.00
25.06	41.00
27.16	41.00
7.91	42.00
8.82	42.00
9.13	42.00
11.66	42.00
11.77	42.00
17.79	42.00
18.22	42.00
19.59	42.00
21.01	42.00
23.97	42.00
26.02	42.00
6.09	43.00
9.06	43.00
10.70	43.00
12.63	43.00
13.68	43.00
17.33	43.00
19.49	43.00
20.52	43.00
24.46	43.00
29.41	43.00
5.27	44.00
5.35	44.00
8.20	44.00
8.56	44.00
11.71	44.00
12.75	44.00
13.46	44.00
14.10	44.00
14.30	44.00
14.66	44.00
14.72	44.00
14.96	44.00
15.08	44.00
15.50	44.00
17.01	44.00
17.47	44.00
17.73	44.00
18.02	44.00
18.29	44.00
20.52	44.00
22.41	44.00
23.09	44.00
23.21	44.00
28.07	44.00
7.56	45.00
8.06	45.00
9.15	45.00
9.47	45.00
9.63	45.00
9.78	45.00
10.01	45.00
10.72	45.00
12.62	45.00
12.63	45.00
13.65	45.00
15.07	45.00
15.10	45.00
16.21	45.00
21.09	45.00
21.27	45.00
21.97	45.00
5.43	46.00
5.53	46.00
7.44	46.00
10.16	46.00
12.20	46.00
12.40	46.00
12.48	46.00
13.26	46.00
13.50	46.00
13.58	46.00
13.58	46.00
16.76	46.00
16.82	46.00
18.79	46.00
19.79	46.00
19.92	46.00
24.04	46.00
24.39	46.00
24.58	46.00
29.20	46.00
6.80	47.00
7.09	47.00
10.62	47.00
10.69	47.00
10.72	47.00
11.32	47.00
11.57	47.00
11.87	47.00
11.92	47.00
12.29	47.00
13.26	47.00
14.13	47.00
14.20	47.00
14.93	47.00
15.19	47.00
16.93	47.00
18.71	47.00
19.40	47.00
21.08	47.00
21.95	47.00
22.38	47.00
23.27	47.00
27.59	47.00
30.16	47.00
42.12	47.00
3.31	48.00
5.14	48.00
5.34	48.00
8.91	48.00
9.50	48.00
9.94	48.00
10.58	48.00
14.77	48.00
14.85	48.00
15.41	48.00
15.41	48.00
15.47	48.00
15.98	48.00
17.88	48.00
18.01	48.00
18.43	48.00
19.29	48.00
21.57	48.00
21.70	48.00
22.59	48.00
23.73	48.00
23.99	48.00
24.14	48.00
24.67	48.00
24.80	48.00
24.89	48.00
27.10	48.00
27.18	48.00
27.60	48.00
5.08	49.00
10.69	49.00
11.19	49.00
12.27	49.00
12.51	49.00
13.08	49.00
13.84	49.00
14.05	49.00
14.05	49.00
15.50	49.00
17.24	49.00
17.43	49.00
17.67	49.00
18.12	49.00
18.34	49.00
18.79	49.00
19.48	49.00
19.73	49.00
20.60	49.00
21.69	49.00
21.84	49.00
22.14	49.00
22.73	49.00
26.00	49.00
27.70	49.00
33.31	49.00
46.21	49.00
5.42	50.00
5.53	50.00
8.37	50.00
8.47	50.00
8.59	50.00
9.69	50.00
9.71	50.00
11.68	50.00
11.69	50.00
11.76	50.00
12.30	50.00
12.68	50.00
14.33	50.00
14.61	50.00
16.27	50.00
16.45	50.00
16.94	50.00
17.02	50.00
17.50	50.00
18.04	50.00
18.20	50.00
19.91	50.00
20.14	50.00
20.78	50.00
24.09	50.00
24.79	50.00
25.60	50.00
25.86	50.00
26.18	50.00
35.90	50.00
8.20	51.00
10.34	51.00
10.52	51.00
10.59	51.00
12.06	51.00
12.90	51.00
15.09	51.00
15.18	51.00
15.36	51.00
15.50	51.00
16.19	51.00
17.27	51.00
17.52	51.00
18.09	51.00
19.07	51.00
19.27	51.00
19.56	51.00
21.28	51.00
22.14	51.00
22.21	51.00
22.45	51.00
24.72	51.00
26.65	51.00
29.33	51.00
5.56	52.00
6.76	52.00
6.83	52.00
7.42	52.00
8.23	52.00
9.37	52.00
10.50	52.00
13.03	52.00
14.68	52.00
18.42	52.00
19.36	52.00
19.96	52.00
20.40	52.00
20.62	52.00
21.78	52.00
22.02	52.00
25.42	52.00
25.68	52.00
28.32	52.00
35.05	52.00
37.98	52.00
38.08	52.00
40.79	52.00
40.82	52.00
7.69	53.00
10.48	53.00
11.63	53.00
12.77	53.00
13.56	53.00
14.16	53.00
14.64	53.00
15.05	53.00
15.21	53.00
18.96	53.00
19.00	53.00
19.76	53.00
20.60	53.00
21.12	53.00
21.37	53.00
22.38	53.00
23.24	53.00
24.21	53.00
24.22	53.00
24.70	53.00
24.82	53.00
26.87	53.00
32.18	53.00
37.94	53.00
37.94	53.00
38.05	53.00
9.11	54.00
9.52	54.00
9.72	54.00
10.33	54.00
10.50	54.00
11.05	54.00
11.86	54.00
13.27	54.00
13.32	54.00
16.99	54.00
17.97	54.00
19.50	54.00
19.60	54.00
20.04	54.00
20.06	54.00
20.64	54.00
20.95	54.00
21.03	54.00
21.67	54.00
23.04	54.00
23.41	54.00
23.75	54.00
24.24	54.00
24.38	54.00
25.22	54.00
25.37	54.00
26.30	54.00
29.26	54.00
29.99	54.00
31.35	54.00
32.21	54.00
33.97	54.00
7.76	55.00
8.17	55.00
8.62	55.00
9.53	55.00
10.08	55.00
10.35	55.00
10.90	55.00
12.56	55.00
12.81	55.00
13.52	55.00
14.93	55.00
15.27	55.00
15.64	55.00
17.06	55.00
17.48	55.00
20.41	55.00
22.42	55.00
27.24	55.00
28.43	55.00
30.48	55.00
30.95	55.00
31.19	55.00
33.08	55.00
36.97	55.00
39.15	55.00
41.65	55.00
42.44	55.00
5.11	56.00
6.37	56.00
9.12	56.00
9.14	56.00
9.26	56.00
10.14	56.00
10.21	56.00
11.69	56.00
11.69	56.00
12.30	56.00
12.50	56.00
13.77	56.00
14.04	56.00
15.22	56.00
15.61	56.00
15.86	56.00
16.17	56.00
16.38	56.00
16.76	56.00
17.72	56.00
17.92	56.00
19.73	56.00
20.46	56.00
21.02	56.00
21.59	56.00
21.80	56.00
21.98	56.00
23.78	56.00
24.88	56.00
26.50	56.00
28.03	56.00
30.18	56.00
39.29	56.00
47.47	56.00
5.02	57.00
9.23	57.00
9.26	57.00
10.86	57.00
12.57	57.00
13.16	57.00
13.20	57.00
13.28	57.00
14.12	57.00
14.72	57.00
19.43	57.00
19.43	57.00
29.19	57.00
5.38	58.00
7.86	58.00
9.83	58.00
10.62	58.00
11.61	58.00
12.53	58.00
12.88	58.00
13.37	58.00
15.36	58.00
17.47	58.00
19.12	58.00
20.88	58.00
21.39	58.00
21.68	58.00
22.06	58.00
22.64	58.00
25.39	58.00
25.58	58.00
32.84	58.00
4.55	59.00
8.17	59.00
8.54	59.00
8.68	59.00
8.85	59.00
9.68	59.00
11.41	59.00
12.23	59.00
14.19	59.00
14.22	59.00
14.77	59.00
16.95	59.00
18.00	59.00
18.52	59.00
20.17	59.00
21.39	59.00
21.67	59.00
23.43	59.00
24.27	59.00
25.41	59.00
25.74	59.00
27.25	59.00
28.51	59.00
30.52	59.00
35.00	59.00
4.12	60.00
6.91	60.00
8.48	60.00
10.98	60.00
11.54	60.00
11.59	60.00
12.02	60.00
12.38	60.00
12.43	60.00
12.70	60.00
12.97	60.00
13.10	60.00
13.14	60.00
13.79	60.00
13.97	60.00
14.13	60.00
14.29	60.00
15.01	60.00
15.13	60.00
15.13	60.00
15.32	60.00
15.62	60.00
15.66	60.00
16.31	60.00
17.00	60.00
17.03	60.00
17.66	60.00
19.18	60.00
19.62	60.00
19.64	60.00
20.44	60.00
21.64	60.00
21.71	60.00
21.85	60.00
22.14	60.00
22.83	60.00
23.33	60.00
25.06	60.00
25.87	60.00
26.62	60.00
27.03	60.00
31.77	60.00
9.25	61.00
10.21	61.00
11.15	61.00
13.39	61.00
13.74	61.00
14.28	61.00
14.65	61.00
14.82	61.00
14.96	61.00
15.56	61.00
16.27	61.00
16.32	61.00
16.52	61.00
18.26	61.00
18.32	61.00
18.80	61.00
19.01	61.00
21.68	61.00
22.14	61.00
24.28	61.00
30.56	61.00
32.93	61.00
36.94	61.00
7.80	62.00
8.65	62.00
8.68	62.00
9.70	62.00
11.31	62.00
12.42	62.00
12.98	62.00
13.43	62.00
14.65	62.00
14.69	62.00
15.51	62.00
15.60	62.00
16.07	62.00
18.35	62.00
19.34	62.00
19.86	62.00
20.68	62.00
21.49	62.00
21.58	62.00
23.11	62.00
23.58	62.00
23.73	62.00
24.40	62.00
24.78	62.00
25.11	62.00
26.40	62.00
26.85	62.00
31.44	62.00
31.96	62.00
36.10	62.00
36.88	62.00
38.11	62.00
6.60	63.00
9.11	63.00
10.80	63.00
11.05	63.00
11.31	63.00
11.65	63.00
11.93	63.00
12.28	63.00
12.35	63.00
12.73	63.00
12.87	63.00
14.27	63.00
16.09	63.00
16.64	63.00
16.98	63.00
17.19	63.00
19.45	63.00
20.86	63.00
21.56	63.00
22.09	63.00
22.74	63.00
23.54	63.00
25.07	63.00
25.67	63.00
25.77	63.00
29.55	63.00
30.60	63.00
39.37	63.00
40.17	63.00
40.95	63.00
8.03	64.00
9.84	64.00
11.61	64.00
12.00	64.00
13.77	64.00
13.94	64.00
14.01	64.00
15.90	64.00
16.60	64.00
16.68	64.00
17.39	64.00
18.00	64.00
18.52	64.00
20.85	64.00
21.13	64.00
21.29	64.00
22.16	64.00
22.73	64.00
23.62	64.00
24.07	64.00
24.23	64.00
24.48	64.00
24.57	64.00
25.24	64.00
25.84	64.00
27.43	64.00
30.68	64.00
32.10	64.00
33.10	64.00
36.41	64.00
7.64	65.00
11.78	65.00
12.16	65.00
12.84	65.00
13.04	65.00
14.76	65.00
15.61	65.00
15.64	65.00
15.90	65.00
15.93	65.00
15.96	65.00
16.82	65.00
16.83	65.00
17.62	65.00
17.85	65.00
21.21	65.00
22.53	65.00
22.68	65.00
22.97	65.00
23.01	65.00
23.90	65.00
24.01	65.00
24.85	65.00
25.39	65.00
25.85	65.00
26.08	65.00
27.46	65.00
29.04	65.00
36.89	65.00
45.99	65.00
59.25	65.00
8.65	66.00
9.87	66.00
11.87	66.00
12.33	66.00
13.33	66.00
15.30	66.00
15.37	66.00
15.70	66.00
16.16	66.00
16.71	66.00
19.91	66.00
20.11	66.00
20.93	66.00
23.50	66.00
25.34	66.00
27.06	66.00
27.41	66.00
27.46	66.00
27.48	66.00
27.84	66.00
28.50	66.00
32.55	66.00
33.67	66.00
42.07	66.00
43.64	66.00
70.69	66.00
8.77	67.00
9.17	67.00
10.90	67.00
11.68	67.00
14.19	67.00
15.24	67.00
15.64	67.00
17.39	67.00
18.55	67.00
19.10	67.00
19.22	67.00
19.28	67.00
20.12	67.00
20.24	67.00
20.32	67.00
20.48	67.00
21.26	67.00
21.94	67.00
22.34	67.00
22.43	67.00
22.46	67.00
24.42	67.00
26.37	67.00
26.57	67.00
27.60	67.00
27.75	67.00
29.09	67.00
29.21	67.00
29.24	67.00
29.65	67.00
31.53	67.00
39.62	67.00
5.30	68.00
5.64	68.00
7.40	68.00
11.56	68.00
13.75	68.00
13.96	68.00
15.62	68.00
16.68	68.00
18.03	68.00
18.55	68.00
19.90	68.00
22.16	68.00
22.74	68.00
22.84	68.00
23.41	68.00
24.85	68.00
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39.00	120.00
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31.92	160.00
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34.85	160.00
35.58	160.00
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32.96	292.00
34.73	292.00
39.23	292.00
51.93	292.00
59.91	292.00
65.41	292.00
35.15	293.00
45.30	293.00
51.74	293.00
52.20	293.00
63.88	293.00
71.04	293.00
26.97	294.00
28.60	295.00
44.06	295.00
50.86	295.00
56.67	295.00
66.27	295.00
33.10	296.00
46.57	296.00
51.64	296.00
51.83	296.00
52.13	296.00
55.49	296.00
60.49	296.00
76.17	296.00
130.26	296.00
11.76	297.00
45.60	297.00
50.03	297.00
51.47	297.00
53.71	297.00
29.16	298.00
32.19	298.00
41.09	298.00
44.13	298.00
64.93	298.00
92.32	298.00
40.15	299.00
57.23	299.00
29.87	300.00
46.51	300.00
47.59	300.00
50.64	300.00
54.31	300.00
57.24	300.00
59.90	300.00
49.13	301.00
52.85	301.00
31.63	302.00
42.01	302.00
42.41	302.00
45.33	302.00
60.85	302.00
76.17	302.00
50.67	303.00
57.51	303.00
60.34	303.00
67.00	303.00
40.53	304.00
73.49	304.00
47.70	305.00
51.88	305.00
69.88	305.00
74.49	305.00
35.42	306.00
65.30	306.00
77.28	306.00
23.28	307.00
30.75	307.00
58.46	307.00
61.03	307.00
78.53	307.00
87.59	307.00
34.72	308.00
32.71	309.00
43.91	309.00
45.89	309.00
53.28	309.00
64.18	309.00
66.65	309.00
67.15	309.00
20.65	310.00
44.09	310.00
46.49	310.00
50.75	310.00
58.53	310.00
63.15	310.00
69.77	310.00
36.23	311.00
23.13	312.00
52.92	312.00
70.14	312.00
29.69	313.00
57.08	313.00
62.47	313.00
63.42	313.00
71.65	313.00
38.08	314.00
48.86	314.00
51.60	314.00
24.75	315.00
32.29	315.00
35.31	315.00
44.64	315.00
47.55	315.00
49.86	315.00
56.43	315.00
59.43	315.00
77.62	315.00
92.53	315.00
7.93	316.00
47.41	316.00
65.86	316.00
30.00	317.00
34.71	317.00
42.88	317.00
45.14	317.00
58.22	317.00
74.93	317.00
30.39	318.00
37.84	318.00
54.28	318.00
63.57	318.00
30.14	319.00
36.91	319.00
42.20	319.00
51.84	319.00
103.90	319.00
42.16	320.00
45.11	320.00
47.37	320.00
53.03	320.00
61.29	320.00
87.03	320.00
39.83	321.00
44.01	321.00
49.49	321.00
71.76	321.00
6.33	322.00
26.59	322.00
41.21	322.00
20.95	323.00
29.41	323.00
34.53	323.00
50.39	323.00
61.17	323.00
15.77	324.00
21.42	324.00
56.51	324.00
34.92	325.00
55.77	325.00
79.29	326.00
25.40	327.00
38.34	327.00
64.25	327.00
44.11	328.00
56.48	328.00
90.56	328.00
50.75	329.00
52.73	329.00
57.69	329.00
60.24	329.00
67.18	329.00
78.50	329.00
47.38	330.00
51.02	330.00
57.64	330.00
61.82	330.00
74.13	330.00
29.77	331.00
52.45	331.00
52.55	331.00
70.77	331.00
21.82	332.00
52.81	332.00
54.51	332.00
58.56	332.00
20.83	333.00
34.50	333.00
38.05	333.00
43.78	333.00
66.40	333.00
31.05	334.00
31.23	334.00
41.36	334.00
47.60	334.00
49.25	334.00
52.70	334.00
53.63	334.00
62.91	334.00
74.83	334.00
13.60	335.00
51.19	335.00
56.43	335.00
67.36	335.00
68.29	335.00
31.33	336.00
53.51	336.00
64.80	336.00
22.56	338.00
35.89	338.00
55.88	338.00
58.08	338.00
94.33	338.00
50.35	339.00
21.65	340.00
29.28	340.00
33.82	340.00
34.23	340.00
35.11	340.00
45.08	340.00
48.95	340.00
55.04	340.00
56.70	340.00
56.73	340.00
69.51	340.00
44.61	341.00
54.75	341.00
42.73	342.00
63.70	342.00
25.34	343.00
32.83	343.00
47.26	343.00
55.96	343.00
67.38	343.00
71.35	343.00
80.39	343.00
46.83	344.00
53.35	344.00
54.04	344.00
38.43	345.00
47.28	345.00
65.44	345.00
36.59	346.00
52.79	346.00
57.11	347.00
60.12	347.00
62.66	347.00
69.29	347.00
76.48	347.00
77.25	347.00
92.54	347.00
54.64	348.00
64.15	348.00
32.84	349.00
37.25	349.00
50.94	349.00
68.30	349.00
47.86	350.00
53.15	350.00
62.40	350.00
38.87	351.00
38.96	351.00
49.86	351.00
65.15	351.00
40.51	352.00
40.68	352.00
52.51	352.00
59.91	352.00
40.15	353.00
41.17	353.00
47.39	353.00
48.25	354.00
48.37	354.00
48.40	354.00
55.64	354.00
77.72	354.00
31.59	356.00
49.41	356.00
61.06	356.00
65.66	356.00
66.91	356.00
72.23	356.00
53.86	357.00
66.37	357.00
80.62	357.00
33.02	358.00
44.22	358.00
58.17	358.00
48.18	359.00
75.37	359.00
33.86	360.00
53.04	360.00
62.53	360.00
59.36	361.00
39.88	362.00
47.21	362.00
56.45	362.00
60.61	362.00
69.60	363.00
36.69	364.00
36.74	364.00
54.37	365.00
56.74	365.00
60.29	365.00
67.09	365.00
76.46	365.00
45.88	366.00
53.08	366.00
57.67	366.00
59.06	366.00
60.21	366.00
81.33	366.00
58.87	367.00
63.96	367.00
83.62	367.00
41.70	368.00
55.27	368.00
67.60	368.00
62.84	369.00
74.39	369.00
50.73	370.00
53.77	370.00
60.56	370.00
61.28	370.00
61.34	370.00
63.77	370.00
51.94	371.00
68.28	371.00
35.29	372.00
54.60	372.00
57.60	372.00
70.04	372.00
66.19	373.00
56.46	374.00
65.87	374.00
38.56	375.00
44.71	375.00
59.52	375.00
74.50	375.00
75.42	375.00
91.65	375.00
46.39	376.00
50.88	376.00
54.63	376.00
54.83	376.00
56.72	376.00
61.36	376.00
53.81	377.00
56.83	377.00
57.27	377.00
72.63	377.00
75.08	377.00
44.30	378.00
51.11	378.00
57.85	378.00
66.41	378.00
79.20	378.00
99.06	378.00
47.29	379.00
65.61	379.00
84.45	379.00
50.55	380.00
55.04	381.00
66.41	382.00
68.93	382.00
43.16	383.00
57.05	383.00
71.38	383.00
83.75	383.00
85.03	383.00
88.61	383.00
22.78	384.00
39.77	384.00
46.90	384.00
47.02	384.00
63.68	384.00
65.83	384.00
44.32	385.00
71.73	385.00
17.35	387.00
41.63	387.00
20.29	388.00
57.73	388.00
58.32	388.00
63.78	388.00
60.85	389.00
65.30	389.00
83.94	389.00
91.53	389.00
51.13	390.00
50.70	391.00
58.15	391.00
65.98	391.00
60.41	392.00
61.67	392.00
96.62	392.00
61.57	394.00
74.04	394.00
74.17	394.00
63.79	396.00
77.30	396.00
78.94	396.00
62.22	397.00
65.70	397.00
54.80	398.00
58.92	398.00
70.25	398.00
75.89	398.00
59.70	399.00
83.03	400.00
76.71	401.00
57.66	403.00
59.12	403.00
50.13	404.00
64.61	404.00
68.59	404.00
52.24	405.00
65.20	405.00
64.73	406.00
71.25	406.00
82.74	407.00
43.78	411.00
55.08	411.00
65.07	411.00
77.90	411.00
60.59	413.00
63.04	413.00
60.17	415.00
65.49	415.00
47.50	417.00
48.45	417.00
60.03	419.00
64.97	420.00
68.40	420.00
54.69	421.00
60.40	421.00
66.03	422.00
68.02	422.00
68.97	423.00
69.73	423.00
48.25	424.00
55.98	425.00
78.48	425.00
55.65	426.00
74.62	428.00
79.50	430.00
85.56	430.00
55.72	431.00
66.58	431.00
105.29	431.00
57.53	432.00
71.63	433.00
72.93	433.00
68.58	437.00
71.88	437.00
42.35	438.00
53.73	439.00
84.89	439.00
60.35	440.00
88.74	440.00
50.24	441.00
19.14	442.00
55.10	443.00
30.99	444.00
56.46	444.00
42.92	449.00
60.05	450.00
97.76	450.00
49.45	451.00
66.97	451.00
75.98	451.00
60.58	452.00
63.87	452.00
64.57	452.00
25.75	453.00
32.87	454.00
61.66	454.00
70.55	454.00
74.72	454.00
87.47	454.00
57.76	455.00
101.44	455.00
70.46	456.00
78.43	458.00
71.37	460.00
63.92	462.00
94.04	463.00
66.11	464.00
68.87	464.00
61.63	467.00
78.71	468.00
64.36	471.00
78.61	471.00
36.39	472.00
54.10	473.00
61.20	475.00
66.10	475.00
69.46	475.00
92.32	478.00
106.04	480.00
85.14	482.00
86.59	483.00
76.89	484.00
64.80	485.00
0.62	487.00
68.25	487.00
64.18	492.00
54.76	497.00
75.06	501.00
0.65	502.00
55.61	506.00
70.54	509.00
16.83	510.00
84.74	515.00
25.37	522.00
25.00	531.00
44.53	532.00
65.17	532.00
73.60	532.00
66.20	537.00
82.63	537.00
77.63	557.00
65.77	561.00
75.26	577.00
73.61	591.00
24.61	593.00
59.35	595.00
80.73	597.00
70.10	613.00
49.67	659.00
77.51	675.00
81.67	677.00
Nota: Tamaño muestral n = 4954 observaciones utilizadas en el análisis.

4. - Gráfica de Dispersión

# Gráfica de dispersión (Datos originales)

# Límites del gráfico
x_max_original <- max(X_original) * 1.05
y_max_original <- max(Y_original) * 1.05

# Crear gráfico vacío
plot(X_original, Y_original,
     type = "n",
     main = "Gráfica Nro. 1\nDiagrama de dispersión entre NO2 y AQI\nen el estudio de la calidad del aire en India 2015-2020",
     xlab = expression(NO[2]*" ("*mu*"g/m³)"),
     ylab = "AQI (Índice)",
     xlim = c(0, x_max_original),
     ylim = c(0, y_max_original),
     cex.main = 1.1,
     cex.lab = 1.1,
     cex.axis = 0.9)

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

# Puntos
points(X_original, Y_original,
       col = "deepskyblue3",
       pch = 16,
       cex = 1.2)

# Marco
box(lwd = 1.5)

5.- Tratamiento de datos

Promediación de los datos

Con el fin de reducir la dispersión de los datos, se promediaron las observaciones de las variables independiente y dependiente. Este procedimiento disminuye la variabilidad y facilita la identificación de la tendencia exponencial, favoreciendo un mejor ajuste del modelo.

#Posteriormente a definir promediamos cada variable
datos_prom <- aggregate(NO2 ~ AQI, data = datos, mean)

# variables promediadas

# Y es Variable dependiente
Y <- datos_prom$AQI              # Efecto

# X es Variable Independiente
X <- datos_prom$NO2              # Causa   


# Tabla de valores promediados
n_modelo <- nrow(datos_prom)

cat("Tamaño muestral del modelo =", n_modelo)

## Tamaño muestral del modelo = 448

TVP_NO2_AQI <- data.frame(X, Y)

5.1 .- Tabla pares de valores simplificada

# Tabla pares de valores
tabla <- TVP_NO2_AQI %>%
  gt() %>%
  cols_align(
    align = "center",
    columns = everything()
  ) %>%
  fmt_number(
    columns = everything(),
    decimals = 2
  ) %>%
  tab_header(
    title = md("*Tabla Nro. 2*"),
    subtitle = md("*Pares de valores de NO2 y AQI, Calidad del aire en India 2015-2020*")
  ) %>%
  tab_source_note(
    source_note = md(paste0(
      "**Nota:** Tamaño muestral n = ", nrow(TVP_NO2_AQI),
      " observaciones utilizadas en el análisis."
    ))
  )

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

Tabla Nro. 2
Pares de valores de NO2 y AQI, Calidad del aire en India 2015-2020
X	Y
12.21	23.00
14.05	26.00
15.57	29.00
12.63	30.00
17.28	31.00
11.61	32.00
16.29	33.00
15.56	34.00
12.77	35.00
14.16	36.00
14.77	37.00
13.91	38.00
13.01	39.00
14.82	40.00
18.55	41.00
15.99	42.00
16.34	43.00
15.60	44.00
13.18	45.00
15.69	46.00
16.81	47.00
17.66	48.00
18.94	49.00
16.28	50.00
17.56	51.00
20.71	52.00
21.25	53.00
20.30	54.00
21.08	55.00
18.05	56.00
14.11	57.00
17.29	58.00
18.08	59.00
16.84	60.00
18.44	61.00
20.48	62.00
19.71	63.00
20.87	64.00
21.98	65.00
24.36	66.00
21.62	67.00
22.90	68.00
22.80	69.00
23.96	70.00
22.93	71.00
22.42	72.00
22.88	73.00
23.81	74.00
24.58	75.00
25.74	76.00
22.33	77.00
29.57	78.00
26.62	79.00
27.04	80.00
32.59	81.00
26.76	82.00
28.76	83.00
28.84	84.00
26.66	85.00
25.98	86.00
28.91	87.00
27.30	88.00
29.43	89.00
33.25	90.00
30.69	91.00
30.10	92.00
30.69	93.00
24.98	94.00
27.15	95.00
31.89	96.00
34.26	97.00
29.97	98.00
28.26	99.00
31.74	100.00
30.38	101.00
28.46	102.00
35.02	103.00
36.42	104.00
33.60	105.00
33.50	106.00
32.57	107.00
36.29	108.00
31.30	109.00
33.74	110.00
31.69	111.00
33.50	112.00
29.75	113.00
39.15	114.00
37.00	115.00
37.45	116.00
34.70	117.00
34.27	118.00
34.44	119.00
34.84	120.00
27.57	121.00
34.65	122.00
33.76	123.00
37.28	124.00
40.20	125.00
38.42	126.00
34.88	127.00
32.78	128.00
35.66	129.00
34.39	130.00
35.58	131.00
37.77	132.00
43.83	133.00
30.80	134.00
33.41	135.00
40.24	136.00
32.83	137.00
32.14	138.00
34.25	139.00
33.30	140.00
34.32	141.00
36.76	142.00
36.04	143.00
37.61	144.00
32.40	145.00
44.40	146.00
29.67	147.00
31.97	148.00
29.64	149.00
41.31	150.00
40.71	151.00
29.33	152.00
37.87	153.00
35.43	154.00
37.16	155.00
42.96	156.00
31.87	157.00
40.25	158.00
40.46	159.00
39.69	160.00
34.79	161.00
42.16	162.00
46.05	163.00
43.81	164.00
37.25	165.00
32.63	166.00
41.96	167.00
44.24	168.00
39.12	169.00
32.61	170.00
33.44	171.00
37.00	172.00
39.16	173.00
34.08	174.00
39.37	175.00
33.36	176.00
39.89	177.00
45.18	178.00
38.55	179.00
42.42	180.00
39.33	181.00
45.71	182.00
34.73	183.00
50.48	184.00
43.02	185.00
37.00	186.00
34.85	187.00
27.97	188.00
38.87	189.00
38.00	190.00
35.86	191.00
50.45	192.00
38.54	193.00
47.53	194.00
46.70	195.00
44.59	196.00
50.95	197.00
42.98	198.00
41.16	199.00
41.08	200.00
46.33	201.00
37.48	202.00
47.06	203.00
25.82	204.00
43.28	205.00
34.38	206.00
49.01	207.00
44.25	208.00
49.29	209.00
22.50	210.00
34.34	211.00
19.65	212.00
23.55	213.00
41.62	214.00
44.66	215.00
31.22	216.00
41.25	217.00
40.56	218.00
47.72	219.00
43.23	220.00
44.93	221.00
47.54	222.00
40.96	223.00
46.93	224.00
58.95	225.00
45.87	226.00
38.89	227.00
48.34	228.00
48.65	229.00
49.43	230.00
43.08	231.00
50.44	232.00
43.17	233.00
44.67	234.00
41.21	235.00
48.51	236.00
55.18	237.00
34.69	238.00
47.27	239.00
53.38	240.00
55.91	241.00
56.01	242.00
44.49	243.00
43.64	244.00
36.99	245.00
44.84	246.00
48.53	247.00
47.39	248.00
40.19	249.00
50.26	251.00
40.70	252.00
41.94	253.00
53.43	254.00
49.45	255.00
28.89	256.00
47.07	257.00
46.74	258.00
43.23	259.00
57.92	260.00
39.22	261.00
42.01	262.00
46.40	263.00
45.50	264.00
44.30	265.00
41.88	266.00
55.63	267.00
36.91	268.00
47.42	269.00
34.14	270.00
44.17	271.00
40.94	272.00
43.46	273.00
50.00	274.00
38.79	275.00
56.83	276.00
55.62	277.00
66.51	278.00
46.24	279.00
42.91	280.00
47.55	281.00
42.08	282.00
37.85	283.00
48.00	284.00
47.13	285.00
51.97	286.00
52.55	287.00
42.52	288.00
50.64	289.00
43.60	290.00
65.89	291.00
44.26	292.00
53.22	293.00
26.97	294.00
49.29	295.00
61.96	296.00
42.51	297.00
50.64	298.00
48.69	299.00
49.44	300.00
50.99	301.00
49.73	302.00
58.88	303.00
57.01	304.00
60.99	305.00
59.33	306.00
56.61	307.00
34.72	308.00
53.40	309.00
50.49	310.00
36.23	311.00
48.73	312.00
56.86	313.00
46.18	314.00
52.04	315.00
40.40	316.00
47.65	317.00
46.52	318.00
53.00	319.00
56.00	320.00
51.27	321.00
24.71	322.00
39.29	323.00
31.23	324.00
45.34	325.00
79.29	326.00
42.66	327.00
63.72	328.00
61.18	329.00
58.40	330.00
51.38	331.00
46.92	332.00
40.71	333.00
49.40	334.00
51.37	335.00
49.88	336.00
53.35	338.00
50.35	339.00
44.19	340.00
49.68	341.00
53.22	342.00
54.36	343.00
51.41	344.00
50.38	345.00
44.69	346.00
70.78	347.00
59.40	348.00
47.33	349.00
54.47	350.00
48.21	351.00
48.40	352.00
42.90	353.00
55.68	354.00
57.81	356.00
66.95	357.00
45.14	358.00
61.78	359.00
49.81	360.00
59.36	361.00
51.04	362.00
69.60	363.00
36.72	364.00
62.99	365.00
59.54	366.00
68.82	367.00
54.86	368.00
68.62	369.00
58.58	370.00
60.11	371.00
54.38	372.00
66.19	373.00
61.17	374.00
64.06	375.00
54.13	376.00
63.12	377.00
66.32	378.00
65.78	379.00
50.55	380.00
55.04	381.00
67.67	382.00
71.50	383.00
47.66	384.00
58.03	385.00
29.49	387.00
50.03	388.00
75.41	389.00
51.13	390.00
58.28	391.00
72.90	392.00
69.93	394.00
73.34	396.00
63.96	397.00
64.97	398.00
59.70	399.00
83.03	400.00
76.71	401.00
58.39	403.00
61.11	404.00
58.72	405.00
67.99	406.00
82.74	407.00
60.46	411.00
61.81	413.00
62.83	415.00
47.98	417.00
60.03	419.00
66.69	420.00
57.55	421.00
67.03	422.00
69.35	423.00
48.25	424.00
67.23	425.00
55.65	426.00
74.62	428.00
82.53	430.00
75.86	431.00
57.53	432.00
72.28	433.00
70.23	437.00
42.35	438.00
69.31	439.00
74.55	440.00
50.24	441.00
19.14	442.00
55.10	443.00
43.73	444.00
42.92	449.00
78.91	450.00
64.13	451.00
63.01	452.00
25.75	453.00
65.45	454.00
79.60	455.00
70.46	456.00
78.43	458.00
71.37	460.00
63.92	462.00
94.04	463.00
67.49	464.00
61.63	467.00
78.71	468.00
71.48	471.00
36.39	472.00
54.10	473.00
65.59	475.00
92.32	478.00
106.04	480.00
85.14	482.00
86.59	483.00
76.89	484.00
64.80	485.00
34.44	487.00
64.18	492.00
54.76	497.00
75.06	501.00
0.65	502.00
55.61	506.00
70.54	509.00
16.83	510.00
84.74	515.00
25.37	522.00
25.00	531.00
61.10	532.00
74.41	537.00
77.63	557.00
65.77	561.00
75.26	577.00
73.61	591.00
24.61	593.00
59.35	595.00
80.73	597.00
70.10	613.00
49.67	659.00
77.51	675.00
81.67	677.00
Nota: Tamaño muestral n = 448 observaciones utilizadas en el análisis.

5.2 .- Gráfico de dispersión simplificada

#--------------------------------------------------------------------------------
# CREAR GRÁFICO DE DISPERSIÓN (Con datos promediados)
#--------------------------------------------------------------------------------
# Límites
x_max <- max(X) * 1.05
y_max <- max(Y) * 1.05

# Crear gráfico vacío
plot(X, Y,
     type = "n",
     main = "Gráfica Nro. 2\nDiagrama de dispersión entre NO2 y AQI\nen el estudio de la calidad del aire en India 2015-2020",
     xlab = expression(NO[2]*" ("*mu*"g/m³)"),
     ylab = "AQI (Índice)",
     xlim = c(0, x_max),
     ylim = c(0, y_max),
     cex.main = 1.1,
     cex.lab = 1.1,
     cex.axis = 0.9)

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

# Puntos
points(X, Y,
       col = "deepskyblue3",
       pch = 16,
       cex = 1.2)
box(lwd = 1.5)

6.- Conjetura

Conjetura.-
La distribución de los puntos en el gráfico muestra una curva ascendente, lo que sugiere un modelo exponencial. El AQI aumenta de forma acelerada a medida que se incrementa la concentración de NO2, indicando una relación no lineal y un impacto creciente del contaminante sobre la calidad del aire.

Modelo exponencial general:

\[ Y = ae^{bX} \]

Modelo exponencial aplicado al estudio:

\[ AQI = ae^{b(NO_2)} \]

7.- Parámetros

#------------------------------------------------------------
#Parámetros

Y1 <- log(Y)
regresionexponencial <- lm(Y1 ~ X)

# Ver los coeficientes (a,b)
beta0 <- regresionexponencial$coefficients[1]
beta1 <- regresionexponencial$coefficients[2]

b <- beta1
b

##          X 
## 0.03310434

a <- exp(beta0)
a

## (Intercept) 
##    47.92617

Modelo exponencial obtenido:

\[ AQI = 47.92617\,e^{(0.0331043)(NO_2)} \]

#La expresión anterior corresponde al modelo exponencial final del estudio, obtenido al reemplazar los parámetros (a) y (b) por los valores estimados. Esta ecuación describe la relación entre la concentración de dióxido de nitrógeno (NO₂) y el Índice de Calidad del Aire (AQI).

Justificación del uso de la regresión lineal (lm)

Aunque el modelo conjeturado es exponencial, se puede usar lm() porque el modelo se linealiza mediante logaritmos.

Modelo:
\[ Y = ae^{bX} \]

Al aplicar logaritmo natural:

\[ \ln(Y) = \ln(a) + bX \]

Definimos:
\[ Y_1 = \ln(Y), \; \beta_0 = \ln(a), \; \beta_1 = b \]

Obtenemos un modelo lineal:
\[ Y_1 = \beta_0 + \beta_1 X \]

Esta ecuación ya corresponde a una regresión lineal, por lo que se puede estimar usando lm(Y1 ~ X).
Luego, para recuperar el modelo exponencial original, se aplica la transformación inversa

8.- Comparación de la realidad con el modelo

# Límites
x_max <- max(X) * 1.05
y_max <- max(Y) * 1.05

# Gráfico vacío
plot(X, Y,
     type = "n",
     main = "Gráfica Nro. 3\nModelo exponencial entre NO2 y AQI\nen el estudio de la calidad del aire en India 2015-2020",
     xlab = expression(NO[2]*" ("*mu*"g/m³)"),
     ylab = "AQI (Índice)",
     xlim = c(0, x_max),
     ylim = c(0, y_max),
     cex.main = 1.1,
     cex.lab = 1.1,
     cex.axis = 0.9)

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

# Puntos reales
points(X, Y,
       col = "deepskyblue3",
       pch = 16,
       cex = 1.2)

# Curva del modelo exponencial
curve(a * exp(b * x),
      from = 0,
      to = x_max,
      col = "red",
      lwd = 3,
      add = TRUE)
box(lwd = 1.5)

legend("topleft",
       legend = c("Datos reales", "Modelo exponencial"),
       col = c("deepskyblue3", "red"),
       pch = c(16, NA),
       lwd = c(NA, 3),
       bty = "o",
       bg = "white",
       cex = 0.8,
       x.intersp = 0.6,
       y.intersp = 0.8)

9- Test de Bondad

# 7.- Test de bondad

# Coeficiente de correlación
r <- cor(X, Y1)*100
r

## [1] 79.74544

#El coeficiente de correlación fue r=0.7975, lo que indica una relación positiva fuerte entre la concentración de NO₂ y el logaritmo del AQI. Por tanto, el modelo exponencial presenta un buen ajuste a los datos observados.

10.- Restricciones

# Dominio [x]: D = {R+^0}
# Dominio [y]: D = {Z+^0}

¿Existe algún valor en el dominio de x que sustituido en el modelo matemático genere un valor en y fuera de su dominio?

No existen restricciones para el uso del modelo exponencial, ya que la concentración de NO₂ toma valores no negativos y el modelo genera valores positivos para el AQI. Aunque los resultados pueden ser decimales, estos pueden redondearse al entero más cercano sin salir del dominio práctico de la variable.

11 .- Estimación

#--------------------------------------
# Cálculos

AQI_60 <- round(47.92617 * exp(0.03310434 * 60))

AQI_40 <- 47.92617 * exp(0.03310434 * 40)
AQI_60_real <- 47.92617 * exp(0.03310434 * 60)

incremento_porcentaje <- ((AQI_60_real - AQI_40) / AQI_40) * 100

#--------------------------------------
# Visualización

par(mar = c(1,1,1,1))

plot(0, 0,
     type = "n",
     xlim = c(0,1),
     ylim = c(0,1),
     axes = FALSE,
     xlab = "",
     ylab = "")

text(
  x = 0.5,
  y = 0.48,   # Baja el texto para que no se corte arriba
  labels = paste(
    "Pregunta de cantidad\n\n",
    "¿Cuál es el AQI esperado cuando\n",
    "la concentración de NO2 es de 60 µg/m³?\n\n",
    "Resultado:", AQI_60, "\n\n",
    "-----------------------------------------\n\n",
    "Pregunta de porcentaje\n\n",
    "¿En qué porcentaje aumenta el AQI esperado\n",
    "al incrementar la concentración de NO2\n",
    "de 40 a 60 µg/m³?\n\n",
    "Resultado:", round(incremento_porcentaje,2), "%"
  ),
  cex = 1.15,
  col = "deepskyblue3",
  font = 2
)

12.- Conclusión

En conclusión:

Entre la concentración de dióxido de nitrógeno (NO2) y el Índice AQI existe una relación de tipo exponencial, cuyo modelo es y = 47.92617 * e^(0.0331043x), no se identifican restricciones dentro del dominio de estudio. El modelo indica que el AQI aumenta de forma acelerada a medida que se incrementa la concentración de Dioxido de Nitrogeno, evidenciando que este contaminante influye de manera significativa en la calidad del aire.