Modelo Logarítmico
Camila Zambrano
22/06/2026
# Tema: Estadística Multivariable
# Autor: Camila Zambrano
# Fecha: 22/06/20260.Carga de librerías
library(gt)
library(dplyr)##
## 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, unionlibrary(e1071)
library(htmltools)1. Carga de datos
setwd("~/CAMILA")
datos <- read.csv("Datos Cambiados..csv",
header = TRUE,
sep = ",",
dec = ".",
na.strings = "-")2. Definición de la variable
datos <- na.omit(datos)
datos <- datos[order(datos$AQI, datos$PM2.5), ]
datos_prom <- aggregate(PM2.5 ~ AQI, data = datos, mean)
# Variable independiente (X): Material Paticulado 2.5
X <- datos_prom$PM2.5
# Variable dependiente (Y): Índice de Calidad del Aire
Y <- datos_prom$AQI3. Tabla pares de valores
n_modelo <- nrow(datos_prom)
cat("Tamaño muestral del modelo =", n_modelo)## Tamaño muestral del modelo = 448TVP_PM2.5_AQI <- data.frame(X, Y)
# Tabla pares de valores
tabla <- TVP_PM2.5_AQI %>%
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 de PM2.5 y AQI, Calidad del aire en India 2015-2020*")
) %>%
tab_source_note(
source_note = md(paste0(
"**Nota:** Tamaño muestral n = ", nrow(TVP_PM2.5_AQI),
" observaciones utilizadas en el análisis."
))
)
div(
style = "height:400px; overflow-y:auto;",
tabla
)| Tabla N°1 | |
| Pares de valores de PM2.5 y AQI, Calidad del aire en India 2015-2020 | |
| X | Y |
|---|---|
| 6.30 | 23.00 |
| 9.46 | 26.00 |
| 9.97 | 29.00 |
| 14.48 | 30.00 |
| 9.55 | 31.00 |
| 9.26 | 32.00 |
| 11.97 | 33.00 |
| 11.09 | 34.00 |
| 12.64 | 35.00 |
| 12.97 | 36.00 |
| 11.00 | 37.00 |
| 12.30 | 38.00 |
| 12.25 | 39.00 |
| 14.34 | 40.00 |
| 15.01 | 41.00 |
| 13.65 | 42.00 |
| 14.97 | 43.00 |
| 15.36 | 44.00 |
| 14.38 | 45.00 |
| 15.60 | 46.00 |
| 13.92 | 47.00 |
| 15.70 | 48.00 |
| 16.57 | 49.00 |
| 17.76 | 50.00 |
| 16.41 | 51.00 |
| 17.23 | 52.00 |
| 18.41 | 53.00 |
| 20.50 | 54.00 |
| 17.43 | 55.00 |
| 19.32 | 56.00 |
| 17.60 | 57.00 |
| 20.98 | 58.00 |
| 19.58 | 59.00 |
| 19.34 | 60.00 |
| 21.04 | 61.00 |
| 21.36 | 62.00 |
| 18.75 | 63.00 |
| 22.79 | 64.00 |
| 23.59 | 65.00 |
| 22.55 | 66.00 |
| 24.27 | 67.00 |
| 26.38 | 68.00 |
| 26.01 | 69.00 |
| 28.40 | 70.00 |
| 25.72 | 71.00 |
| 27.57 | 72.00 |
| 28.03 | 73.00 |
| 27.01 | 74.00 |
| 30.28 | 75.00 |
| 28.96 | 76.00 |
| 27.05 | 77.00 |
| 29.27 | 78.00 |
| 26.94 | 79.00 |
| 31.22 | 80.00 |
| 29.49 | 81.00 |
| 32.09 | 82.00 |
| 32.00 | 83.00 |
| 30.96 | 84.00 |
| 30.32 | 85.00 |
| 32.49 | 86.00 |
| 34.56 | 87.00 |
| 32.35 | 88.00 |
| 34.07 | 89.00 |
| 35.04 | 90.00 |
| 36.45 | 91.00 |
| 35.69 | 92.00 |
| 34.87 | 93.00 |
| 37.22 | 94.00 |
| 39.05 | 95.00 |
| 40.78 | 96.00 |
| 40.51 | 97.00 |
| 39.87 | 98.00 |
| 40.09 | 99.00 |
| 40.38 | 100.00 |
| 40.52 | 101.00 |
| 40.96 | 102.00 |
| 43.77 | 103.00 |
| 44.81 | 104.00 |
| 42.54 | 105.00 |
| 44.96 | 106.00 |
| 47.63 | 107.00 |
| 47.35 | 108.00 |
| 43.19 | 109.00 |
| 47.10 | 110.00 |
| 44.35 | 111.00 |
| 48.74 | 112.00 |
| 48.85 | 113.00 |
| 50.28 | 114.00 |
| 50.34 | 115.00 |
| 50.27 | 116.00 |
| 49.89 | 117.00 |
| 48.10 | 118.00 |
| 51.55 | 119.00 |
| 53.24 | 120.00 |
| 52.90 | 121.00 |
| 54.51 | 122.00 |
| 53.48 | 123.00 |
| 55.31 | 124.00 |
| 56.30 | 125.00 |
| 54.89 | 126.00 |
| 54.32 | 127.00 |
| 54.47 | 128.00 |
| 54.89 | 129.00 |
| 52.51 | 130.00 |
| 55.60 | 131.00 |
| 62.52 | 132.00 |
| 60.26 | 133.00 |
| 65.95 | 134.00 |
| 56.13 | 135.00 |
| 60.66 | 136.00 |
| 62.69 | 137.00 |
| 55.83 | 138.00 |
| 61.89 | 139.00 |
| 58.18 | 140.00 |
| 60.36 | 141.00 |
| 58.11 | 142.00 |
| 63.89 | 143.00 |
| 64.11 | 144.00 |
| 66.35 | 145.00 |
| 65.22 | 146.00 |
| 59.73 | 147.00 |
| 64.35 | 148.00 |
| 62.86 | 149.00 |
| 67.02 | 150.00 |
| 66.26 | 151.00 |
| 59.04 | 152.00 |
| 65.44 | 153.00 |
| 64.06 | 154.00 |
| 63.50 | 155.00 |
| 67.36 | 156.00 |
| 70.14 | 157.00 |
| 66.94 | 158.00 |
| 67.52 | 159.00 |
| 67.32 | 160.00 |
| 68.04 | 161.00 |
| 64.65 | 162.00 |
| 69.87 | 163.00 |
| 70.04 | 164.00 |
| 68.10 | 165.00 |
| 66.98 | 166.00 |
| 61.03 | 167.00 |
| 72.45 | 168.00 |
| 73.21 | 169.00 |
| 64.66 | 170.00 |
| 63.71 | 171.00 |
| 75.49 | 172.00 |
| 70.00 | 173.00 |
| 66.49 | 174.00 |
| 67.95 | 175.00 |
| 70.81 | 176.00 |
| 87.40 | 177.00 |
| 87.60 | 178.00 |
| 77.36 | 179.00 |
| 74.55 | 180.00 |
| 88.92 | 181.00 |
| 77.27 | 182.00 |
| 71.55 | 183.00 |
| 86.52 | 184.00 |
| 91.66 | 185.00 |
| 84.86 | 186.00 |
| 79.56 | 187.00 |
| 53.58 | 188.00 |
| 87.53 | 189.00 |
| 73.82 | 190.00 |
| 71.45 | 191.00 |
| 81.65 | 192.00 |
| 85.22 | 193.00 |
| 77.52 | 194.00 |
| 85.81 | 195.00 |
| 76.63 | 196.00 |
| 80.45 | 197.00 |
| 84.96 | 198.00 |
| 92.31 | 199.00 |
| 84.65 | 200.00 |
| 69.83 | 201.00 |
| 82.01 | 202.00 |
| 90.29 | 203.00 |
| 84.66 | 204.00 |
| 85.60 | 205.00 |
| 82.25 | 206.00 |
| 83.98 | 207.00 |
| 88.14 | 208.00 |
| 80.99 | 209.00 |
| 84.07 | 210.00 |
| 94.77 | 211.00 |
| 69.56 | 212.00 |
| 136.85 | 213.00 |
| 90.86 | 214.00 |
| 87.83 | 215.00 |
| 108.04 | 216.00 |
| 81.83 | 217.00 |
| 81.84 | 218.00 |
| 93.00 | 219.00 |
| 92.40 | 220.00 |
| 91.30 | 221.00 |
| 93.40 | 222.00 |
| 95.33 | 223.00 |
| 101.84 | 224.00 |
| 95.24 | 225.00 |
| 90.94 | 226.00 |
| 96.40 | 227.00 |
| 93.45 | 228.00 |
| 93.18 | 229.00 |
| 95.00 | 230.00 |
| 101.24 | 231.00 |
| 90.38 | 232.00 |
| 84.29 | 233.00 |
| 86.52 | 234.00 |
| 96.96 | 235.00 |
| 100.96 | 236.00 |
| 89.56 | 237.00 |
| 89.01 | 238.00 |
| 85.69 | 239.00 |
| 90.45 | 240.00 |
| 104.71 | 241.00 |
| 105.95 | 242.00 |
| 91.67 | 243.00 |
| 112.96 | 244.00 |
| 98.60 | 245.00 |
| 108.40 | 246.00 |
| 91.70 | 247.00 |
| 106.70 | 248.00 |
| 88.03 | 249.00 |
| 110.71 | 251.00 |
| 81.63 | 252.00 |
| 113.33 | 253.00 |
| 115.57 | 254.00 |
| 108.68 | 255.00 |
| 82.70 | 256.00 |
| 100.67 | 257.00 |
| 92.68 | 258.00 |
| 74.83 | 259.00 |
| 104.84 | 260.00 |
| 135.89 | 261.00 |
| 111.32 | 262.00 |
| 111.54 | 263.00 |
| 110.73 | 264.00 |
| 112.05 | 265.00 |
| 103.95 | 266.00 |
| 115.13 | 267.00 |
| 99.59 | 268.00 |
| 102.04 | 269.00 |
| 108.02 | 270.00 |
| 103.37 | 271.00 |
| 120.98 | 272.00 |
| 106.87 | 273.00 |
| 118.64 | 274.00 |
| 105.02 | 275.00 |
| 119.99 | 276.00 |
| 119.11 | 277.00 |
| 121.38 | 278.00 |
| 110.89 | 279.00 |
| 101.28 | 280.00 |
| 116.54 | 281.00 |
| 112.27 | 282.00 |
| 103.33 | 283.00 |
| 112.64 | 284.00 |
| 112.50 | 285.00 |
| 127.54 | 286.00 |
| 107.12 | 287.00 |
| 107.80 | 288.00 |
| 122.35 | 289.00 |
| 99.56 | 290.00 |
| 133.94 | 291.00 |
| 126.08 | 292.00 |
| 108.66 | 293.00 |
| 76.35 | 294.00 |
| 124.61 | 295.00 |
| 132.99 | 296.00 |
| 127.31 | 297.00 |
| 128.31 | 298.00 |
| 128.90 | 299.00 |
| 119.47 | 300.00 |
| 135.55 | 301.00 |
| 116.18 | 302.00 |
| 130.03 | 303.00 |
| 140.84 | 304.00 |
| 129.19 | 305.00 |
| 155.91 | 306.00 |
| 120.20 | 307.00 |
| 95.34 | 308.00 |
| 148.25 | 309.00 |
| 144.66 | 310.00 |
| 177.59 | 311.00 |
| 109.88 | 312.00 |
| 145.90 | 313.00 |
| 125.86 | 314.00 |
| 136.00 | 315.00 |
| 123.61 | 316.00 |
| 146.44 | 317.00 |
| 149.69 | 318.00 |
| 158.10 | 319.00 |
| 132.54 | 320.00 |
| 157.12 | 321.00 |
| 103.64 | 322.00 |
| 135.63 | 323.00 |
| 126.70 | 324.00 |
| 143.88 | 325.00 |
| 177.20 | 326.00 |
| 165.92 | 327.00 |
| 161.30 | 328.00 |
| 161.80 | 329.00 |
| 145.13 | 330.00 |
| 159.33 | 331.00 |
| 154.26 | 332.00 |
| 140.87 | 333.00 |
| 142.14 | 334.00 |
| 160.54 | 335.00 |
| 171.97 | 336.00 |
| 168.43 | 338.00 |
| 192.35 | 339.00 |
| 168.99 | 340.00 |
| 172.55 | 341.00 |
| 192.06 | 342.00 |
| 163.94 | 343.00 |
| 166.04 | 344.00 |
| 178.64 | 345.00 |
| 147.25 | 346.00 |
| 166.24 | 347.00 |
| 182.69 | 348.00 |
| 184.40 | 349.00 |
| 178.97 | 350.00 |
| 134.21 | 351.00 |
| 180.78 | 352.00 |
| 180.61 | 353.00 |
| 194.88 | 354.00 |
| 178.31 | 356.00 |
| 201.08 | 357.00 |
| 158.00 | 358.00 |
| 218.99 | 359.00 |
| 156.19 | 360.00 |
| 136.20 | 361.00 |
| 186.54 | 362.00 |
| 193.10 | 363.00 |
| 205.02 | 364.00 |
| 179.71 | 365.00 |
| 201.20 | 366.00 |
| 181.70 | 367.00 |
| 141.17 | 368.00 |
| 167.03 | 369.00 |
| 201.44 | 370.00 |
| 187.30 | 371.00 |
| 217.49 | 372.00 |
| 233.10 | 373.00 |
| 164.63 | 374.00 |
| 191.17 | 375.00 |
| 208.32 | 376.00 |
| 207.70 | 377.00 |
| 200.85 | 378.00 |
| 209.35 | 379.00 |
| 177.25 | 380.00 |
| 218.98 | 381.00 |
| 216.06 | 382.00 |
| 190.39 | 383.00 |
| 174.49 | 384.00 |
| 204.24 | 385.00 |
| 215.61 | 387.00 |
| 155.18 | 388.00 |
| 218.00 | 389.00 |
| 137.20 | 390.00 |
| 168.00 | 391.00 |
| 176.83 | 392.00 |
| 221.08 | 394.00 |
| 207.18 | 396.00 |
| 218.04 | 397.00 |
| 251.23 | 398.00 |
| 227.28 | 399.00 |
| 183.28 | 400.00 |
| 278.95 | 401.00 |
| 228.78 | 403.00 |
| 227.89 | 404.00 |
| 229.00 | 405.00 |
| 172.20 | 406.00 |
| 244.78 | 407.00 |
| 217.09 | 411.00 |
| 212.36 | 413.00 |
| 269.67 | 415.00 |
| 273.46 | 417.00 |
| 287.56 | 419.00 |
| 240.26 | 420.00 |
| 266.02 | 421.00 |
| 238.69 | 422.00 |
| 249.90 | 423.00 |
| 361.98 | 424.00 |
| 279.18 | 425.00 |
| 228.90 | 426.00 |
| 198.10 | 428.00 |
| 294.36 | 430.00 |
| 250.42 | 431.00 |
| 246.46 | 432.00 |
| 238.82 | 433.00 |
| 254.46 | 437.00 |
| 174.03 | 438.00 |
| 263.72 | 439.00 |
| 252.07 | 440.00 |
| 249.87 | 441.00 |
| 525.09 | 442.00 |
| 209.63 | 443.00 |
| 186.80 | 444.00 |
| 242.58 | 449.00 |
| 257.71 | 450.00 |
| 259.14 | 451.00 |
| 338.57 | 452.00 |
| 261.77 | 453.00 |
| 258.61 | 454.00 |
| 299.11 | 455.00 |
| 263.39 | 456.00 |
| 290.10 | 458.00 |
| 314.32 | 460.00 |
| 303.41 | 462.00 |
| 295.71 | 463.00 |
| 273.71 | 464.00 |
| 293.14 | 467.00 |
| 326.79 | 468.00 |
| 323.98 | 471.00 |
| 313.22 | 472.00 |
| 320.26 | 473.00 |
| 313.29 | 475.00 |
| 247.27 | 478.00 |
| 314.59 | 480.00 |
| 311.03 | 482.00 |
| 353.86 | 483.00 |
| 336.42 | 484.00 |
| 327.04 | 485.00 |
| 408.56 | 487.00 |
| 372.14 | 492.00 |
| 426.52 | 497.00 |
| 331.20 | 501.00 |
| 270.21 | 502.00 |
| 333.43 | 506.00 |
| 319.69 | 509.00 |
| 353.58 | 510.00 |
| 358.91 | 515.00 |
| 225.49 | 522.00 |
| 127.32 | 531.00 |
| 391.20 | 532.00 |
| 363.41 | 537.00 |
| 401.58 | 557.00 |
| 354.56 | 561.00 |
| 378.68 | 577.00 |
| 388.45 | 591.00 |
| 153.28 | 593.00 |
| 423.52 | 595.00 |
| 370.73 | 597.00 |
| 465.08 | 613.00 |
| 582.28 | 659.00 |
| 639.19 | 675.00 |
| 495.90 | 677.00 |
| Nota: Tamaño muestral n = 448 observaciones utilizadas en el análisis. | |
4. Gráfica de dispersión
# Asegurar formato numérico
X <- as.numeric(as.character(X))
Y <- as.numeric(as.character(Y))
# Definir límites con manejo de valores faltantes
x_max <- max(X, na.rm = TRUE) * 1.05
y_max <- max(Y, na.rm = TRUE) * 1.05
# Crear gráfico
plot(X, Y,
type = "n",
main = "Gráfica N°1\nDiagrama de dispersión entre PM2.5 y AQI\nen el estudio de la calidad del aire en India 2015-2020",
xlab = expression("Material Particulado 2.5 ("*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 (usando na.omit para evitar errores con valores faltantes en puntos)
points(na.omit(data.frame(X, Y)),
col = "deepskyblue3",
pch = 16,
cex = 1.2)
box(lwd = 1.5)
5. Conjetura y parámetros
Conjetura
Conjetura.-
La distribución de los puntos en el gráfico muestra una curva
ascendente, lo que sugiere un modelo logarítmico. El AQI aumenta de
forma acelerada a medida que se incrementa el Material Particulado 2.5,
indicando una relación no lineal y un impacto creciente del contaminante
sobre la calidad del aire.
Modelo exponencial general:
\[ Y = a+b*ln(X) \]
Modelo logarítmico aplicado al estudio:
\[ AQI = a+b*ln(PM2.5) \]
Parámetros
modelo_log <- lm(Y ~ log(X), data = TVP_PM2.5_AQI)
a_est <- coef(modelo_log)[1]
a_est## (Intercept)
## -443.7451b_est <- coef(modelo_log)[2]
b_est## log(X)
## 154.4636. Gráfica de dispersión modelo - realidad
# Límites
x_max <- max(X) * 1.05
y_max <- max(Y) * 1.05
# Gráfico vacío
plot(X, Y,
type = "n",
main = "Gráfica N°2\nModelo logarítmico entre PM2.5 y AQI\nen el estudio de la calidad del aire en India 2015-2020",
xlab = expression("Material Particulado 2.5 ("*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 logarítmico
curve(a_est + b_est * log(x),
from = 0,
to = x_max,
col = "red",
lwd = 3,
add = TRUE)
box(lwd = 1.5)
legend("topleft",
legend = c("Datos reales", "Modelo logarítmico"),
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)
7. Test de Bondad
Coeficiente de correlación de Pearson
r <- cor(TVP_PM2.5_AQI$X, TVP_PM2.5_AQI$Y) * 100r## [1] 93.01206Coeficiente de determinación (r^2)
r2 <- (r^2) / 100r2## [1] 86.512448. Restricciones
# Dominio [x]: D = {x|x E R+^0}
# Dominio [y]: D = {x|x E N ^ 0 ≤ x ≤ 2934}El modelo logarítmico presenta una restricción importante en su dominio: no puede incluir valores de PM2.5 iguales a cero, ya que el logaritmo natural (ln) no está definido para dicho valor. Debido a esto, el modelo solo es aplicable cuando la concentración de material particulado es estrictamente mayor a cero. El modelo es creciente, lo que significa que el AQI aumenta a medida que se incrementa el material particulado, manteniendo la relación lógica del estudio.
9. Estimación
carga_objetivo <- median(X, na.rm = TRUE)
if(carga_objetivo <= 0){
stop("Error: La carga (PM2.5) debe ser mayor a 0 para utilizar el modelo logarítmico.")
}
# Cálculo del AQI estimado
aqi_est <- a_est + b_est * log(carga_objetivo)
# Visualización del resultado
plot(1, type = "n", axes = FALSE, xlab = "", ylab = "")
text(
1, 1,
labels = paste(
"¿Cuál es el AQI esperado\n",
"cuando el PM2.5 es", round(carga_objetivo, 2), "µg/m³?\n\n",
"Resultado estimado (AQI):", round(aqi_est, 0)
),
cex = 1.2,
col = "deepskyblue3",
font = 2
)
10. Conclusión
Entre la concentración de material particulado (PM2.5) y el Índice de Calidad del Aire (AQI) existe una relación de tipo logarítmica, representada por el modelo y = -443.7451 + 154.463*ln(x), siendo “y = Índice de Calidad de Agua (AQI)” y “x = Material Particulado 2.5 (PM2.5)”. El modelo presenta restricciones de que la concentración de PM2.5 debe ser extrictamente mayor a cero (PM2.5 > 0), dado que el logaritmo no está definido para valores nulos o negativos. El modelo indica que el AQI aumenta de forma constante con el incremento de PM2.5