Actividad 1 – Modelos de Regresión
Avril Lobato - A00833113
2024-02-28
Introducción
- ¿Qué es Supervised Machine Learning y cuáles son algunas de
sus aplicaciones en Inteligencia de Negocios?
El aprendizaje supervisado (Supervised Machine Learning) es una técnica de inteligencia artificial en la que se entrena un modelo utilizando datos etiquetados, es decir, datos que tienen una variable objetivo conocida. Su objetivo es encontrar una relación entre las características de entrada y la variable objetivo, de modo que el modelo pueda predecir la variable objetivo para nuevos datos no etiquetados.
Dicha técnica ofrece una gran versatilidad en la Inteligencia de Negocios, por ende, se puede aplicar en una amplia variedad de problemas empresariales ya sea para optimizar los procesos y/o mejorar la toma de decisiones. Algunos ejemplos de las aplicaciones del aprendizaje supervisado en esta área son:
- Predicción de ventas: Utilizando datos históricos de ventas, el
aprendizaje supervisado puede utilizarse para predecir las ventas
futuras en función de factores como el tiempo, las promociones, el
clima, entre otros.
- Análisis de riesgos crediticios: Las instituciones financieras
pueden utilizar el aprendizaje supervisado para evaluar el riesgo
crediticio de los clientes y decidir si conceder o no un préstamo. El
modelo se entrena utilizando datos históricos de clientes con etiquetas
que indican si han cumplido o no con sus pagos.
- Segmentación de clientes: Las empresas pueden utilizar el
aprendizaje supervisado para segmentar a sus clientes en grupos con
características similares. Esto les permite personalizar sus estrategias
de marketing y ofrecer productos y servicios más adecuados a cada
segmento.
- Detección de fraudes: En el sector financiero, el aprendizaje
supervisado se utiliza para detectar actividades fraudulentas, como
transacciones sospechosas con tarjetas de crédito o cuentas bancarias.
El modelo se entrena utilizando datos de transacciones etiquetadas como
fraudulentas o legítimas.
- Recomendación de productos: Las plataformas de comercio electrónico utilizan el aprendizaje supervisado para recomendar productos a los usuarios en función de su historial de compras y el comportamiento de navegación. El modelo se entrena utilizando datos de transacciones pasadas y el feedback de los usuarios sobre los productos recomendados.
- ¿Cuáles son los principales algoritmos de Supervised Machine
Learning?
Los principales algoritmos de aprendizaje supervisado son:
- Regresión lineal: Predice valores numéricos continuos en función de
variables independientes, por medio del calculo de una línea de ajuste
que minimice la diferencia entre los valores predichos y los valores
reales.
- Regresión logística: Calcula la probabilidad de que una observación
pertenezca a una clase o categoría específica y luego clasifica la
observación en función de esta probabilidad.
- XGBoost (eXtreme Gradient Boosting): Combina múltiples árboles de decisión para crear un modelo más preciso y robusto.
- SVM (Support Vector Machine): Encuentra la línea (hiperplano) que maximiza la distancia entre los puntos más cercanos de cada grupo (margen) para tener una separación óptima entre clases de datos con el fin de predecir con precisión las etiquetas de los nuevos puntos de datos.
- RF (Random Forest): Colección de múltiples árboles de decisión independientes utilizando muestras aleatorias del conjunto de datos de entrenamiento para posteriormente, promediar las predicciones de cada árbol con el fin de mejorar la precisión
- DT (Árbol de decisión): Modelo en forma de árbol de decisiones, donde cada nodo interno representa una característica, cada rama representa una decisión y cada hoja representa un resultado.
- OLS (Ordinary Least Squares): Regresión para ajustar una línea (hiperplano) a una bd (relación lineal con las características) con el objetivo de predecir valores numéricos.
- ¿Qué es la R2 Ajustada? ¿Qué es la métrica RMSE? ¿Cuál es la
diferencia entre la R2 Ajustada y la métrica RMSE?
La R2 ajustada es una medida estadística que indica cuánta variabilidad en la variable de respuesta es explicada por el modelo de regresión. Se calcula como la proporción de la varianza total de la variable de respuesta, ajustada para el número de predictores en el modelo y el tamaño de la muestra. Mientras que, el RMSE (Root Mean Squared Error) es una medida de la diferencia entre los valores predichos por un modelo de regresión y los valores observados en el conjunto de datos de prueba; es el resultado de la raíz cuadrada de la media de los errores al cuadrado entre las predicciones y los valores reales. Por lo tanto, la diferencia entre estas medidas de evaluación son que la R2 ajustada evalúa qué tan bien el modelo explica la variabilidad de la variable de respuesta y el RMSE evalúa la precisión del modelo en términos de la magnitud de los errores de predicción.
Análisis Exploratorio de los Datos (EDA)
Instalación y llamado de ibrerías
library(foreign) # Read Data Stored by 'Minitab', 'S', 'SAS', 'SPSS', 'Stata', 'Systat', 'Weka', 'dBase'
library(ggplot2) # It is a system for creating graphics
library(dplyr) # A fast, consistent tool for working with data frame like objects##
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## intersect, setdiff, setequal, unionlibrary(mapview) # Quickly and conveniently create interactive visualizations of spatial data with or without background maps
library(naniar) # Provides data structures and functions that facilitate the plotting of missing values and examination of imputations.
library(tmaptools) # A collection of functions to create spatial weights matrix objects from polygon 'contiguities', for summarizing these objects, and for permitting their use in spatial data analysis
library(tmap) # For drawing thematic maps## Breaking News: tmap 3.x is retiring. Please test v4, e.g. with
## remotes::install_github('r-tmap/tmap')library(RColorBrewer) # It offers several color palettes
library(dlookr) # A collection of tools that support data diagnosis, exploration, and transformation## Registered S3 methods overwritten by 'dlookr':
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## transform# Predictive modeling
library(regclass) # Contains basic tools for visualizing, interpreting, and building regression models## Loading required package: bestglm## Loading required package: leaps## Loading required package: VGAM## Loading required package: stats4## Loading required package: splines## Loading required package: rpart## Loading required package: randomForest## randomForest 4.7-1.1## Type rfNews() to see new features/changes/bug fixes.##
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## all.correlations -> all_correlations, cor.demo -> cor_demo, etc.library(mctest) # Multicollinearity diagnostics
library(lmtest) # Testing linear regression models## Loading required package: zoo##
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library(spData) # Diverse spatial datasets for demonstrating, benchmarking and teaching spatial data analysis
library(spatialreg) # A collection of all the estimation functions for spatial cross-sectional models## Loading required package: Matrix##
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## set.ZeroPolicyOptionlibrary(caret) # The caret package (short for Classification And Rgression Training) contains functions to streamline the model training process for complex regression and classification problems.## Loading required package: lattice##
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## predictorslibrary(e1071) # Functions for latent class analysis, short time Fourier transform, fuzzy clustering, support vector machines, shortest path computation, bagged clustering, naive Bayes classifier, generalized k-nearest neighbor.##
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## kurtosis, skewnesslibrary(SparseM) # Provides some basic R functionality for linear algebra with sparse matrices##
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## backsolvelibrary(Metrics) # An implementation of evaluation metrics in R that are commonly used in supervised machine learning##
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## precision, recalllibrary(randomForest) # Classification and regression based on a forest of trees using random inputs
library(jtools) # This is a collection of tools for more efficiently understanding and sharing the results of (primarily) regression analyses
library(xgboost) # The package includes efficient linear model solver and tree learning algorithms##
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## slicelibrary(DiagrammeR) # Build graph/network structures using functions for stepwise addition and deletion of nodes and edges
library(effects) # Graphical and tabular effect displays, e.g., of interactions, for various statistical models with linear predictors## Loading required package: carData## Registered S3 method overwritten by 'survey':
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library(geoR)## --------------------------------------------------------------
## Analysis of Geostatistical Data
## For an Introduction to geoR go to http://www.leg.ufpr.br/geoR
## geoR version 1.9-4 (built on 2024-02-14) is now loaded
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library(caret)
#install.packages("datasets") #Usar la base de datos "Iris"
library(datasets)
#install.packages("ggplot2") #Gráficas con mejor diseño
library(ggplot2)
#install.packages("lattice") #Crear gráficos
library(lattice)
#install.packages("DataExplorer") #Crear gráficos
library(DataExplorer)
#install.packages("mlbench") #Crear gráficos
library(mlbench)
#install.packages("tidyverse")
library(tidyverse)## ── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
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## ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorsCarga de base de datos “health.insurance.csv”
df <- read.csv("C:/Users/AVRIL/Documents/health_insurance.csv")Se eliminan duplicados
df <- unique(df)
dim(df) #1 registro duplicado## [1] 1337 7Identificación y reemplazo de NAs
# Se identifica si hay valores nulos
dfn <- is.na(df)
#No hay valores nulos, pero en caso de que si se hubiera detectado, estos se habrían reemplazo por la mediana del df
# Se eliminan nulos
df <- na.omit(df)
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## 1246 28 male 24.3 5 no southwest 5615.37
## 1247 45 female 25.7 3 no southwest 9101.80
## 1248 33 male 29.4 4 no southwest 6059.17
## 1249 18 female 39.8 0 no southeast 1633.96
## 1250 32 male 33.6 1 yes northeast 37607.53
## 1251 24 male 29.8 0 yes northeast 18648.42
## 1252 19 male 19.8 0 no southwest 1241.57
## 1253 20 male 27.3 0 yes southwest 16232.85
## 1254 40 female 29.3 4 no southwest 15828.82
## 1255 34 female 27.7 0 no southeast 4415.16
## 1256 42 female 37.9 0 no southwest 6474.01
## 1257 51 female 36.4 3 no northwest 11436.74
## 1258 54 female 27.6 1 no northwest 11305.93
## 1259 55 male 37.7 3 no northwest 30063.58
## 1260 52 female 23.2 0 no northeast 10197.77
## 1261 32 female 20.5 0 no northeast 4544.23
## 1262 28 male 37.1 1 no southwest 3277.16
## 1263 41 female 28.1 1 no southeast 6770.19
## 1264 43 female 29.9 1 no southwest 7337.75
## 1265 49 female 33.3 2 no northeast 10370.91
## 1266 64 male 23.8 0 yes southeast 26926.51
## 1267 55 female 30.5 0 no southwest 10704.47
## 1268 24 male 31.1 0 yes northeast 34254.05
## 1269 20 female 33.3 0 no southwest 1880.49
## 1270 45 male 27.5 3 no southwest 8615.30
## 1271 26 male 33.9 1 no northwest 3292.53
## 1272 25 female 34.5 0 no northwest 3021.81
## 1273 43 male 25.5 5 no southeast 14478.33
## 1274 35 male 27.6 1 no southeast 4747.05
## 1275 26 male 27.1 0 yes southeast 17043.34
## 1276 57 male 23.7 0 no southwest 10959.33
## 1277 22 female 30.4 0 no northeast 2741.95
## 1278 32 female 29.7 0 no northwest 4357.04
## 1279 39 male 29.9 1 yes northeast 22462.04
## 1280 25 female 26.8 2 no northwest 4189.11
## 1281 48 female 33.3 0 no southeast 8283.68
## 1282 47 female 27.6 2 yes northwest 24535.70
## 1283 18 female 21.7 0 yes northeast 14283.46
## 1284 18 male 30.0 1 no southeast 1720.35
## 1285 61 male 36.3 1 yes southwest 47403.88
## 1286 47 female 24.3 0 no northeast 8534.67
## 1287 28 female 17.3 0 no northeast 3732.63
## 1288 36 female 25.9 1 no southwest 5472.45
## 1289 20 male 39.4 2 yes southwest 38344.57
## 1290 44 male 34.3 1 no southeast 7147.47
## 1291 38 female 20.0 2 no northeast 7133.90
## 1292 19 male 34.9 0 yes southwest 34828.65
## 1293 21 male 23.2 0 no southeast 1515.34
## 1294 46 male 25.7 3 no northwest 9301.89
## 1295 58 male 25.2 0 no northeast 11931.13
## 1296 20 male 22.0 1 no southwest 1964.78
## 1297 18 male 26.1 0 no northeast 1708.93
## 1298 28 female 26.5 2 no southeast 4340.44
## 1299 33 male 27.5 2 no northwest 5261.47
## 1300 19 female 25.7 1 no northwest 2710.83
## 1301 45 male 30.4 0 yes southeast 62592.87
## 1302 62 male 30.9 3 yes northwest 46718.16
## 1303 25 female 20.8 1 no southwest 3208.79
## 1304 43 male 27.8 0 yes southwest 37829.72
## 1305 42 male 24.6 2 yes northeast 21259.38
## 1306 24 female 27.7 0 no southeast 2464.62
## 1307 29 female 21.9 0 yes northeast 16115.30
## 1308 32 male 28.1 4 yes northwest 21472.48
## 1309 25 female 30.2 0 yes southwest 33900.65
## 1310 41 male 32.2 2 no southwest 6875.96
## 1311 42 male 26.3 1 no northwest 6940.91
## 1312 33 female 26.7 0 no northwest 4571.41
## 1313 34 male 42.9 1 no southwest 4536.26
## 1314 19 female 34.7 2 yes southwest 36397.58
## 1315 30 female 23.7 3 yes northwest 18765.88
## 1316 18 male 28.3 1 no northeast 11272.33
## 1317 19 female 20.6 0 no southwest 1731.68
## 1318 18 male 53.1 0 no southeast 1163.46
## 1319 35 male 39.7 4 no northeast 19496.72
## 1320 39 female 26.3 2 no northwest 7201.70
## 1321 31 male 31.1 3 no northwest 5425.02
## 1322 62 male 26.7 0 yes northeast 28101.33
## 1323 62 male 38.8 0 no southeast 12981.35
## 1324 42 female 40.4 2 yes southeast 43896.38
## 1325 31 male 25.9 1 no northwest 4239.89
## 1326 61 male 33.5 0 no northeast 13143.34
## 1327 42 female 32.9 0 no northeast 7050.02
## 1328 51 male 30.0 1 no southeast 9377.90
## 1329 23 female 24.2 2 no northeast 22395.74
## 1330 52 male 38.6 2 no southwest 10325.21
## 1331 57 female 25.7 2 no southeast 12629.17
## 1332 23 female 33.4 0 no southwest 10795.94
## 1333 52 female 44.7 3 no southwest 11411.69
## 1334 50 male 31.0 3 no northwest 10600.55
## 1335 18 female 31.9 0 no northeast 2205.98
## 1336 18 female 36.9 0 no southeast 1629.83
## 1337 21 female 25.8 0 no southwest 2007.95
## 1338 61 female 29.1 0 yes northwest 29141.36Medidas descriptivas
Acorde a los datos, se observa que:
* Los hombres tienen un IMC ligeramente mayor que las mujeres. * Las
mujeres tienen un promedio ligeramente mayor de hijos que los hombres. *
Los hombres tienen una mayor prevalencia de tabaquismo que las mujeres.
* Los hombres tienen un promedio de gastos ligeramente mayor que las
mujeres.
df$age <- as.numeric(df$age)
df$children <- as.numeric(df$children)
df$region <- as.numeric(factor(df$region))
factor_with_na <- lapply(df[, c("sex", "smoker")], function(x) any(is.na(x)))
# Se convierten variables categóricas a numéricas
df$sex <- as.numeric(factor(df$sex, ordered = TRUE))
df$smoker <- as.numeric(factor(df$smoker, ordered = TRUE))
#Obtencion de medidas descriptivas
summary(df)## age sex bmi children
## Min. :18.00 Min. :1.000 Min. :16.00 Min. :0.000
## 1st Qu.:27.00 1st Qu.:1.000 1st Qu.:26.30 1st Qu.:0.000
## Median :39.00 Median :2.000 Median :30.40 Median :1.000
## Mean :39.22 Mean :1.505 Mean :30.67 Mean :1.096
## 3rd Qu.:51.00 3rd Qu.:2.000 3rd Qu.:34.70 3rd Qu.:2.000
## Max. :64.00 Max. :2.000 Max. :53.10 Max. :5.000
## smoker region expenses
## Min. :1.000 Min. :1.000 Min. : 1122
## 1st Qu.:1.000 1st Qu.:2.000 1st Qu.: 4746
## Median :1.000 Median :3.000 Median : 9386
## Mean :1.205 Mean :2.516 Mean :13279
## 3rd Qu.:1.000 3rd Qu.:3.000 3rd Qu.:16658
## Max. :2.000 Max. :4.000 Max. :63770Medidas de dispersión
Acorde a los datos se observa que no hay diferencias significativas
en la variabilidad de las variables entre hombres y mujeres:
* Dispersión del IMC: La dispersión del IMC es similar en hombres y
mujeres, con una desviación estándar de alrededor de 5 kg/m².
* Dispersión del número de hijos: La dispersión del número de hijos es
similar en hombres y mujeres, con una desviación estándar de alrededor
de 1 hijo.
* Dispersión del tabaquismo: La probabilidad de ser fumador es
ligeramente mayor en hombres que en mujeres, con una desviación estándar
de 0.42 (42%) para hombres y 0.39 (39%) para mujeres.
* Dispersión de los gastos: La dispersión de los gastos es similar en
hombres y mujeres, con una desviación estándar de alrededor de
$4,800.
# Desviación estándar de cada columnan
dispersion_sd <- sapply(df, sd)
print(dispersion_sd)## age sex bmi children smoker region
## 1.404433e+01 5.001634e-01 6.100664e+00 1.205571e+00 4.038062e-01 1.105208e+00
## expenses
## 1.211036e+04#Rango de cada columna
dispersion_range <- sapply(df, range)
print(dispersion_range)## age sex bmi children smoker region expenses
## [1,] 18 1 16.0 0 1 1 1121.87
## [2,] 64 2 53.1 5 2 4 63770.43#Rango intercuartílico de cada columna
dispersion_iqr <- sapply(df, IQR)
print(dispersion_iqr)## age sex bmi children smoker region expenses
## 24.00 1.00 8.40 2.00 0.00 1.00 11911.38outlier_ratio <- dispersion_sd / dispersion_iqr
# Columna con la proporción más alta
max_outlier_column <- names(outlier_ratio)[which.max(outlier_ratio)]
print(outlier_ratio)## age sex bmi children smoker region expenses
## 0.5851805 0.5001634 0.7262695 0.6027857 Inf 1.1052082 1.0167050print(paste("La variable con más outliers es:", max_outlier_column))## [1] "La variable con más outliers es: smoker"Identificación de patrones y/o tendencias en los datos mediante el uso de gráficos
Los patrones y/o tendencias significativas que se observan son: * Diferencias por Sexo: Las mujeres tienen un IMC más alto, tienden a tener más hijos y tienen menores gastos que los hombres. * Relación entre Edad e IMC: El IMC aumenta con la edad. * Relación entre Edad y Gastos: Los gastos aumentan con la edad.
# Descripción introductoria o resumen de df
introduce(df)## rows columns discrete_columns continuous_columns all_missing_columns
## 1 1337 7 0 7 0
## total_missing_values complete_rows total_observations memory_usage
## 1 0 1337 9359 77184plot_intro(df)
# Boxplot para visualizar la distribución de los valores en expenses
plot_boxplot(df, by="expenses")
# Cantidad de valores faltantes (NA's) en cada columna
plot_missing(df)
# Histograma para cada variable numérica del dataframe df
plot_histogram(df)
# Gráfico de barras para cada variable categórica
plot_bar(df)
# Gráfico scatterplot
plot(df)
Visualización de distribución normal
* Expenses: Distribución sesgada hacia la izquierda (asimetría
negativa), con mayor concentración en el rango 0-50.000.
* Los hombres tienen mayores gastos que las mujeres.
* Aumento de los gastos con la edad.
Se muestra que para tener una mejor distribución normal de los datos, se deben convertir las variables a log() con el fin de tener mayor precisión y menor sesgo.
par(mfrow = c(1, 2)) # Divide el área de trazado en 1 fila y 2 columnas
for(i in 1:ncol(df)) {
qqnorm(df[, i], main = names(df)[i])
qqline(df[, i], col = 2)
}
plot_normality(df, expenses, bmi, age)
Correlación entre variables
Se visualiza que hay una asociación positiva entre expenses y smoker
dev.new(width = 10, height = 8)
correlate(df) %>% plot()
plot_correlation(df)Especificación del modelo de regresión lineal a estimar
Se infiere que el impacto de cada una de las variables explicativas
sobre la principal variable de estudio, en este caso “expenses”, sería
de esta manera:
* Sexo: Las mujeres podrían tener un impacto negativo en los gastos, es
decir, se espera que las mujeres tengan menores gastos que los
hombres.
* IMC: Un mayor IMC podría tener un impacto positivo en los gastos, es
decir, se espera que las personas con un mayor IMC tengan mayores gastos
en salud, alimentación, etc.
* Hijos: Un mayor número de hijos podría tener un impacto positivo en
los gastos, es decir, se espera que las familias con más hijos tengan
mayores gastos en educación, alimentación, etc.
* Fumador: El hábito de fumar podría tener un impacto positivo en los
gastos, es decir, se espera que los fumadores tengan mayores gastos en
tabaco, tratamiento de enfermedades relacionadas con el tabaquismo,
etc.
Asimismo, se espera que la principal variable x que afecta a y, sería “smoker”, donde si tiene un valor de “2” (igual a persona si fumadora) implica que aumenta “expenses”. Mientras que, la variable “region” tiene menor impacto en y.
par(mfrow = c(3, 2)) # Divide el área de la trama en una cuadrícula de 3 filas y 2 columnas
# Age
boxplot(expenses ~ age, data = df, main = "Expenses vs. age", xlab = "age", ylab = "Expenses")
# Sex
boxplot(expenses ~ sex, data = df, main = "Expenses vs. sex", xlab = "sex", ylab = "Expenses")
# BMI
boxplot(expenses ~ bmi, data = df, main = "Expenses vs. bmi", xlab = "bmi", ylab = "Expenses")
# Children
boxplot(expenses ~ children, data = df, main = "Expenses vs. children", xlab = "children", ylab = "Expenses")
# Smoker
boxplot(expenses ~ smoker, data = df, main = "Expenses vs. smoker", xlab = "smoker", ylab = "Expenses")
# Region
boxplot(expenses ~ region, data = df, main = "Expenses vs. region", xlab = "region", ylab = "Expenses")
Partición de bd
set.seed(123) # What is set.seed()? We want to make sure that we get the same results for randomization each time you run the script.
partition <- createDataPartition(y = df$expenses, p=0.7, list=F)
train = df[partition, ]
test = df[-partition, ]Regresion lineal
lm_model <- lm(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train)
summary(lm_model)##
## Call:
## lm(formula = log(expenses) ~ log(age) + sex + children + log(bmi) +
## smoker + region, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.80943 -0.20736 -0.06954 0.07419 2.24993
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.70251 0.25988 6.551 9.44e-11 ***
## log(age) 1.26435 0.03665 34.502 < 2e-16 ***
## sex -0.07753 0.02829 -2.741 0.00624 **
## children 0.08130 0.01150 7.068 3.08e-12 ***
## log(bmi) 0.33420 0.07080 4.720 2.72e-06 ***
## smoker 1.51880 0.03467 43.814 < 2e-16 ***
## region -0.04042 0.01279 -3.160 0.00163 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4292 on 930 degrees of freedom
## Multiple R-squared: 0.7837, Adjusted R-squared: 0.7823
## F-statistic: 561.7 on 6 and 930 DF, p-value: < 2.2e-16Se visualiza el efecto parcial de smoker sobre la variable dependiente
plot(effect("smoker",lm_model))
Se presentan los valores previstos frente a los observados de la
variable dependiente
ggplot(train, aes(x = exp(lm_model$fitted.values), y = train$expenses)) +
geom_point() +
stat_smooth() +
labs(x='Predicted Values', y='Actual Values', title='OLS Predicted vs. Actual Values')## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
Estimación de modelos de Supervised Machine Learning (SML)
OLS Model
ols_model <- lm(expenses ~ age + sex + bmi + children + smoker + region, data = train)
summary(ols_model)##
## Call:
## lm(formula = expenses ~ age + sex + bmi + children + smoker +
## region, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11331.3 -2643.9 -922.3 1343.2 29891.7
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -34753.85 1375.65 -25.264 < 2e-16 ***
## age 258.98 14.19 18.254 < 2e-16 ***
## sex -149.20 395.47 -0.377 0.70606
## bmi 318.71 33.34 9.560 < 2e-16 ***
## children 482.92 160.42 3.010 0.00268 **
## smoker 23655.65 484.64 48.811 < 2e-16 ***
## region -300.30 178.81 -1.679 0.09339 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6000 on 930 degrees of freedom
## Multiple R-squared: 0.7587, Adjusted R-squared: 0.7572
## F-statistic: 487.4 on 6 and 930 DF, p-value: < 2.2e-16AIC(ols_model) # AIC = 18971.02## [1] 18971.02prediction_ols_model <- predict(ols_model,test)
RMSE_ols_model <- rmse(prediction_ols_model,test$expenses)
RMSE_ols_model## [1] 6206.795Log OLS Model
log_ols_model <- lm(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train)
summary(log_ols_model)##
## Call:
## lm(formula = log(expenses) ~ log(age) + sex + children + log(bmi) +
## smoker + region, data = train)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.80943 -0.20736 -0.06954 0.07419 2.24993
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.70251 0.25988 6.551 9.44e-11 ***
## log(age) 1.26435 0.03665 34.502 < 2e-16 ***
## sex -0.07753 0.02829 -2.741 0.00624 **
## children 0.08130 0.01150 7.068 3.08e-12 ***
## log(bmi) 0.33420 0.07080 4.720 2.72e-06 ***
## smoker 1.51880 0.03467 43.814 < 2e-16 ***
## region -0.04042 0.01279 -3.160 0.00163 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.4292 on 930 degrees of freedom
## Multiple R-squared: 0.7837, Adjusted R-squared: 0.7823
## F-statistic: 561.7 on 6 and 930 DF, p-value: < 2.2e-16AIC(log_ols_model) # AIC = 1082.83## [1] 1082.827# Otra forma de calcular el RMSE
#log_errors <- train$expenses - exp(log_ols_model$fitted.values) #Calcula los errores entre los gastos log reales y los previstos
#RMSE_log_ols_model <- sqrt(mean(log_errors^2))
#RMSE_log_ols_model
prediction_log_model <- exp(predict(log_ols_model, newdata = test)) # Predecir en escala original
RMSE_log_model <- sqrt(mean((test$expenses - prediction_log_model)^2)) # Calcular RMSE
RMSE_log_model## [1] 8158.644SAR & SEM
Los modelos SAR y SEM no se pueden realizar en este caso debido a la ausencia de información espacial, puesto que dichos modelos requieren datos espaciales, como la ubicación de las observaciones, para estimar los efectos espaciales. El conjunto de datos proporcionado no incluye información espacial, como coordenadas o nombres de regiones suficientes para convertirlos en nb2listw. Por lo tanto, sin información espacial, no se pueden calcular las matrices de pesos espaciales que son esenciales para estimar los modelos SAR y SEM.
No obstante, se muestran los códigos que se podrían utilizar: SAR
#library(spdep)
#map_centroid <- coordinates(df)
#col.gal.nb <- read.gal(system.file("df", package="spdep"))
#map.linkW <- nb2listw(col.gal.nb, style="W") #########NEIGHT
#sar_model <- lagsarlm(log(expenses) ~ log(age) + sex + log(children +0.01) +
# log(bmi) + smoker + region, data = train, method="Matrix", listw = wt)
#summary(sar_model)
#RMSE_SAR <- sqrt(mean((df$expenses - sar_model$fitted.values)^2)) #Ajusta el modelo a los datos de entrenamiento.
#RMSE_SARSEM
#sem_model <- errorsarlm(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train, map.linkW, method="Matrix")
#summary(sem_model)
#RMSE_SEM <- sqrt(mean((df$expenses - sem_model$fitted.values)^2))
#RMSE_SEMSupport Vector Regression
fit.svm = svm(formula = log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train, type = 'eps-regression', kernel = 'radial')
summary(fit.svm)##
## Call:
## svm(formula = log(expenses) ~ log(age) + sex + children + log(bmi) +
## smoker + region, data = train, type = "eps-regression", kernel = "radial")
##
##
## Parameters:
## SVM-Type: eps-regression
## SVM-Kernel: radial
## cost: 1
## gamma: 0.1666667
## epsilon: 0.1
##
##
## Number of Support Vectors: 296plot(fit.svm$fitted, fit.svm$residuals, main="SVM Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)
predictions <- exp(predict(fit.svm,newdata = test))
# Error cuadrático medio de la raíz
RMSE_svm <- rmse(predictions,test$expenses)
RMSE_svm## [1] 5549.75dv_svm<-data.frame(exp(fit.svm$fitted),train$expenses)
ggplot(dv_svm, aes(x = exp.fit.svm.fitted., y = train.expenses)) +
geom_point() +
stat_smooth() +
labs(x='Predicted Values', y='Actual Values', title='SVM Predicted vs. Actual Values')## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
Decision Tree
decision_tree_regression <- rpart(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train)
# summary(decision_tree_regression)
plot(decision_tree_regression, compress = TRUE)
text(decision_tree_regression, use.n = TRUE)
#install.packages("rpart.plot") # Instalar el paquete rpart.plot
library(rpart.plot)
rpart.plot(decision_tree_regression)
dt_prediction_test_data <-exp(predict(decision_tree_regression,newdata = test))
RMSE_Tree <- rmse(dt_prediction_test_data, test$expenses)
RMSE_Tree## [1] 5375.011Random Forest
library(dplyr)
transform_df <- df
transform_df$age <- log(transform_df$age)
transform_df$bmi <- log(transform_df$bmi)
transform_df$children <- log(transform_df$children +0.01)
transform_df$expenses <- log(transform_df$expenses)
summary(transform_df)## age sex bmi children
## Min. :2.890 Min. :1.000 Min. :2.773 Min. :-4.60517
## 1st Qu.:3.296 1st Qu.:1.000 1st Qu.:3.270 1st Qu.:-4.60517
## Median :3.664 Median :2.000 Median :3.414 Median : 0.00995
## Mean :3.598 Mean :1.505 Mean :3.403 Mean :-1.66885
## 3rd Qu.:3.932 3rd Qu.:2.000 3rd Qu.:3.547 3rd Qu.: 0.69813
## Max. :4.159 Max. :2.000 Max. :3.972 Max. : 1.61144
## smoker region expenses
## Min. :1.000 Min. :1.000 Min. : 7.023
## 1st Qu.:1.000 1st Qu.:2.000 1st Qu.: 8.465
## Median :1.000 Median :3.000 Median : 9.147
## Mean :1.205 Mean :2.516 Mean : 9.100
## 3rd Qu.:1.000 3rd Qu.:3.000 3rd Qu.: 9.721
## Max. :2.000 Max. :4.000 Max. :11.063set.seed(123)
partition_alt <- createDataPartition(y = transform_df$expenses, p=0.7, list=F)
train_alt = transform_df[partition_alt, ]
test_alt <- transform_df[-partition_alt, ]random_forest<-randomForest(expenses ~ age + sex + children + bmi + smoker + region, data=train_alt, proximity=TRUE)
print(random_forest) ##
## Call:
## randomForest(formula = expenses ~ age + sex + children + bmi + smoker + region, data = train_alt, proximity = TRUE)
## Type of random forest: regression
## Number of trees: 500
## No. of variables tried at each split: 2
##
## Mean of squared residuals: 0.1414657
## % Var explained: 83.26rf_prediction_test_data <-exp(predict(random_forest,newdata = test_alt))
RMSE_RF <- rmse(rf_prediction_test_data, test$expenses)
RMSE_RF## [1] 5417.839Evaluacion de Importancia de Variables
Se observa que la variable “smoker” es la que tiene mayor importancia en
la predicción de y (“expenses”); a la cual le continua la variable “age”
en nivel de importancia.
#Cuanto mayor sea el valor de la precisión media de la disminución, mayor será la importancia de la variable en el modelo. En otras palabras, la precisión media de la disminución representa en qué medida la eliminación de cada variable reduce la precisión del modelo.
varImpPlot(random_forest, n.var = 5, main = "Top 10 - Variable")
importance(random_forest) ## IncNodePurity
## age 256.902644
## sex 8.449236
## children 31.091885
## bmi 63.095680
## smoker 334.298033
## region 15.067092XGBoost
# Define las variables explicativas (X) y la variable dependiente (Y) en el conjunto de entrenamiento
train_x = data.matrix(train_alt[, -7])
train_y = train_alt[,7]
# Define las variables explicativas (X) y la variable dependiente (Y) en el conjunto de pruebas
test_x = data.matrix(test_alt[, -7])
test_y = test_alt[, 7]
# Definer sets fianles de train y test
xgb_train = xgb.DMatrix(data = train_x, label = train_y)
xgb_test = xgb.DMatrix(data = test_x, label = test_y)# Ajustamos el modelo de regresión XGBoost y mostramos el RMSE de los datos de entrenamiento y de prueba en cada ronda.
watchlist = list(train=xgb_train, test=xgb_test)
model_xgb = xgb.train(data=xgb_train, max.depth=3, watchlist=watchlist, nrounds=70) ## [1] train-rmse:6.072449 test-rmse:6.085466
## [2] train-rmse:4.269285 test-rmse:4.280048
## [3] train-rmse:3.008771 test-rmse:3.018650
## [4] train-rmse:2.128019 test-rmse:2.143468
## [5] train-rmse:1.515914 test-rmse:1.538863
## [6] train-rmse:1.094982 test-rmse:1.125079
## [7] train-rmse:0.808904 test-rmse:0.847757
## [8] train-rmse:0.619527 test-rmse:0.667509
## [9] train-rmse:0.499389 test-rmse:0.554829
## [10] train-rmse:0.427170 test-rmse:0.487947
## [11] train-rmse:0.384923 test-rmse:0.449014
## [12] train-rmse:0.361299 test-rmse:0.425338
## [13] train-rmse:0.347701 test-rmse:0.415393
## [14] train-rmse:0.339893 test-rmse:0.409008
## [15] train-rmse:0.334655 test-rmse:0.406328
## [16] train-rmse:0.331662 test-rmse:0.404075
## [17] train-rmse:0.329707 test-rmse:0.402880
## [18] train-rmse:0.328199 test-rmse:0.403266
## [19] train-rmse:0.325943 test-rmse:0.401362
## [20] train-rmse:0.323954 test-rmse:0.399620
## [21] train-rmse:0.321749 test-rmse:0.399034
## [22] train-rmse:0.319239 test-rmse:0.400724
## [23] train-rmse:0.318239 test-rmse:0.401305
## [24] train-rmse:0.316830 test-rmse:0.401674
## [25] train-rmse:0.314678 test-rmse:0.401438
## [26] train-rmse:0.313099 test-rmse:0.401975
## [27] train-rmse:0.311669 test-rmse:0.402005
## [28] train-rmse:0.311260 test-rmse:0.401829
## [29] train-rmse:0.309970 test-rmse:0.400966
## [30] train-rmse:0.308875 test-rmse:0.402224
## [31] train-rmse:0.307366 test-rmse:0.402095
## [32] train-rmse:0.306115 test-rmse:0.401730
## [33] train-rmse:0.302633 test-rmse:0.404479
## [34] train-rmse:0.301667 test-rmse:0.405409
## [35] train-rmse:0.300698 test-rmse:0.405319
## [36] train-rmse:0.299389 test-rmse:0.404883
## [37] train-rmse:0.297227 test-rmse:0.406926
## [38] train-rmse:0.295584 test-rmse:0.407227
## [39] train-rmse:0.294832 test-rmse:0.407158
## [40] train-rmse:0.294244 test-rmse:0.406956
## [41] train-rmse:0.293035 test-rmse:0.406692
## [42] train-rmse:0.290426 test-rmse:0.407427
## [43] train-rmse:0.289592 test-rmse:0.407390
## [44] train-rmse:0.288926 test-rmse:0.408196
## [45] train-rmse:0.288667 test-rmse:0.407954
## [46] train-rmse:0.287841 test-rmse:0.408023
## [47] train-rmse:0.286714 test-rmse:0.407258
## [48] train-rmse:0.285400 test-rmse:0.407675
## [49] train-rmse:0.284463 test-rmse:0.408266
## [50] train-rmse:0.283906 test-rmse:0.408685
## [51] train-rmse:0.283566 test-rmse:0.408982
## [52] train-rmse:0.282759 test-rmse:0.408243
## [53] train-rmse:0.281684 test-rmse:0.408680
## [54] train-rmse:0.281330 test-rmse:0.408672
## [55] train-rmse:0.280540 test-rmse:0.408453
## [56] train-rmse:0.279895 test-rmse:0.409364
## [57] train-rmse:0.278899 test-rmse:0.408177
## [58] train-rmse:0.277285 test-rmse:0.409075
## [59] train-rmse:0.275960 test-rmse:0.409806
## [60] train-rmse:0.275713 test-rmse:0.409791
## [61] train-rmse:0.275451 test-rmse:0.409951
## [62] train-rmse:0.274734 test-rmse:0.409582
## [63] train-rmse:0.274608 test-rmse:0.409787
## [64] train-rmse:0.274399 test-rmse:0.409752
## [65] train-rmse:0.273609 test-rmse:0.409676
## [66] train-rmse:0.270243 test-rmse:0.410433
## [67] train-rmse:0.269124 test-rmse:0.410239
## [68] train-rmse:0.268054 test-rmse:0.411109
## [69] train-rmse:0.266979 test-rmse:0.411710
## [70] train-rmse:0.266224 test-rmse:0.411668reg_xgb = xgboost(data = xgb_train, max.depth = 3, nrounds = 59, verbose = 0)
prediction_xgb_test<-exp(predict(reg_xgb, newdata = xgb_test))
RMSE_XGB <- rmse(prediction_xgb_test, test$expenses)
RMSE_XGB## [1] 5295.267xgb_reg_residuals<-test$expenses - prediction_xgb_test
plot(xgb_reg_residuals, xlab= "Dependent Variable", ylab = "Residuals", main = 'XGBoost Regression Residuals')
abline(0,0)
# Plot 3 modelos de arbol
xgb.plot.tree(model=reg_xgb, trees=0:2)importance_matrix <- xgb.importance(model = reg_xgb)
xgb.plot.importance(importance_matrix, xlab = "Explanatory Variables X's Importance")
Neural Networks Regression
#install.packages("neuralnet")
library(neuralnet)## Warning: package 'neuralnet' was built under R version 4.3.3##
## Attaching package: 'neuralnet'## The following object is masked from 'package:dplyr':
##
## computenn_model <- neuralnet(expenses ~ age + sex + bmi + children + smoker + region, data = train, hidden = c(5, 3), linear.output = TRUE) Gráfico de neural network
plot(nn_model)prediction_net<-predict(nn_model, test)
RMSE_net <- rmse(prediction_net, test$expenses)
RMSE_net## [1] 11954.46Pruebas de Diagnóstico
Autocorrelación espacial
Estimar un análisis de regresión espacial requiere construir una matriz espacial que conecte los puntos geográficos. En este caso, el df no contiene variables espaciales y/o geograficas. Por lo tanto, no es necesario realizar un modelo de regresión espacial y, por ende, calcular el test de MOran para determinar si existe autocorrelacion espacial. No obstante, se podría realizar el siguiente código en caso de tener este tipo de valores en el dataframe:
#library(spdep)
#vecinos <- poly2nb(df)
# A continuación, crea una matriz de pesos espaciales a partir de la lista de vecinos
#matriz_pesos <- nb2listw(df)
# Finalmente, realiza el test de Moran para determinar la autocorrelación espacial en 'expenses'
#moran.test(df$expenses, listw = matriz_pesos)Autocorrelación serial
No hay autocorrelacion serial debido a que no se tiene una bd de series de tiempo, y/o una variable de tiempo en el dataframe actual. Sin embargo, en caso de tener un valor serial se realizaría un análisis afc como el que se muestra a continuación:
acf_result <- acf(df$expenses, lag.max = 30, main = "ACF of HOVAL", ylim = c(-1, 1))
plot(acf_result, main="Autocorrelación Serial")
Multicolinealidad
Se muestra que los valores de VIF son cercanos a 1, lo que indica que no hay multicolinealidad en el modelo
VIF(lm_model)## log(age) sex children log(bmi) smoker region
## 1.030732 1.017558 1.010255 1.053302 1.015063 1.034176VIF(ols_model)## age sex bmi children smoker region
## 1.024782 1.017573 1.052172 1.005218 1.014980 1.033881Heterocedasticidad
Un valor p estadísticamente significativo (normalmente inferior a 0,05), en este caso, los valores p son extremadamente pequeños, lo que significa que rechazaríamos la hipótesis nula de homocedasticidad y concluimos que hay evidencia significativa de heterocedasticidad en el modelo.
bptest(log_ols_model)##
## studentized Breusch-Pagan test
##
## data: log_ols_model
## BP = 57.75, df = 6, p-value = 1.288e-10bptest(lm_model)##
## studentized Breusch-Pagan test
##
## data: lm_model
## BP = 57.75, df = 6, p-value = 1.288e-10bptest(ols_model)##
## studentized Breusch-Pagan test
##
## data: ols_model
## BP = 109.21, df = 6, p-value < 2.2e-16Solucion de heterocedasticidad
plot(fitted(ols_model), resid(ols_model), main="Linear Regression Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)
plot(fitted(lm_model), resid(lm_model), main="Linear Regression Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)
plot(fitted(ols_model), resid(ols_model), main="Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)
plot(fitted(lm_model), resid(lm_model), main="Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)
Normalidad de residuales
Se espera que el modelo de regresión lineal produzca errores que se distribuyan normalmente. Así, más errores se agrupan alrededor de cero.
hist(lm_model$residuals, xlab="Estimated Regression Residuals", main='Distribution of LM Estimated Regression Residuals', col='lightblue', border="white")
hist(ols_model$residuals, xlab="Estimated Regression Residuals", main='Distribution of OLS Estimated Regression Residuals', col='lightblue', border="white")
# Haremos predicciones utilizando los datos de prueba para evaluar el rendimiento de nuestro modelo de regresión.
prediction_ols_model1 <- ols_model %>% predict(test)
RMSE_ols1 <- RMSE(prediction_ols_model1, test$expenses)
RMSE_ols1## [1] 6206.795wt1 <- 1 / lm(abs(ols_model$residuals) ~ ols_model$fitted.values)$fitted.values^2
wls_model1 <-lm(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train, weights=wt1)
summary(wls_model1)##
## Call:
## lm(formula = log(expenses) ~ log(age) + sex + children + log(bmi) +
## smoker + region, data = train, weights = wt1)
##
## Weighted Residuals:
## Min 1Q Median 3Q Max
## -3.539e-04 -7.018e-05 -3.420e-05 1.288e-05 1.731e-03
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.94875 0.26379 7.388 3.32e-13 ***
## log(age) 1.44113 0.04299 33.520 < 2e-16 ***
## sex -0.16420 0.03138 -5.233 2.07e-07 ***
## children 0.14524 0.01418 10.240 < 2e-16 ***
## log(bmi) 0.09818 0.07238 1.356 0.175
## smoker 1.58671 0.09989 15.885 < 2e-16 ***
## region -0.07874 0.01396 -5.640 2.26e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.000173 on 930 degrees of freedom
## Multiple R-squared: 0.6856, Adjusted R-squared: 0.6836
## F-statistic: 338 on 6 and 930 DF, p-value: < 2.2e-16Se muestra que ya no hay heterocedasticidad en el modelo
bptest(wls_model1)##
## studentized Breusch-Pagan test
##
## data: wls_model1
## BP = 2.1435e-05, df = 6, p-value = 1plot(fitted(wls_model1), resid(wls_model1), main="Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)
# Haremos predicciones utilizando los datos de prueba para evaluar el rendimiento de nuestro modelo de regresión wls.
prediction_wls_model1 <- wls_model1 %>% predict(test)
RMSE(prediction_wls_model1, test$expenses) # WLS - RMSE is greater than OLS - RMSE ## [1] 17845.99wls_errors <- train$expenses - exp(wls_model1$fitted.values)
# Calculate RMSE for WLS model
RMSE_wls_model1 <- sqrt(mean(wls_errors^2))
RMSE_wls_model1## [1] 9820.253wt2 <- 1 / lm(abs(log_ols_model$residuals) ~ log_ols_model$fitted.values)$fitted.values^2
wls_model2 <-lm(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train, weights=wt2)
prediction_wls_model2 <- wls_model2 %>% predict(test)
RMSE(prediction_wls_model2, test$expenses) ## [1] 17846.01wls_errors2 <- train$expenses - exp(wls_model2$fitted.values)
# Calculate RMSE for WLS model
RMSE_wls_model2 <- sqrt(mean(wls_errors2^2))
RMSE_wls_model2## [1] 7036.752Evaluación y Selección de Modelo de Regresión
Se observan los siguientes hallazgos en relación a la comparativa de
los RMSE:
1. El promedio de los valores de RMSE es de 7042.774.
2. La diferencia entre el mayor y menor RMSE es de 6659.191.
3. La desviación estándar de los valores de RMSE es de 2672.846.
4. La mayoría de los modelos (55.56%) tienen un RMSE mayor al
promedio.
5. A pesar de la heterocedasticidad, el modelo Arbol de Decision tienen
un RMSE menor que varios de los demás. 6. La heterocedasticidad no
parece afectar significativamente el rendimiento de ciertos modelos.
Aunque, se observa heterocedasticidad en los demás modelos que no son
WLS, lo que significa que la varianza de los errores no es constante. 7.
Los modelos wls model parecen tener un RMSE alto en comparación a los
primeros 5 modelos.
Por lo tanto, el modelo seleccionado es model xgb debido a su bajo
RMSE en comparación con otros modelos lo que implica que tiene un mejor
rendimiento en la predicción de los datos de prueba y tiene ausencia de
heterocedasticidad como multicolinealidad lo que significa que los
errores de predicción tienen una varianza constante, lo que mejora la
confiabilidad del modelo.
## Dataframe
resultados <- data.frame(
"log_ols_model" = c(RMSE_log_model),
"rf_model" = c(RMSE_RF),
"decision_tree_model" = c(RMSE_Tree),
"ols_model" = c(RMSE_ols_model),
"svm_model" = c(RMSE_svm),
"model_xgb" = c(RMSE_XGB),
"model_nnet" = c(RMSE_net),
"wls_model1" = c(RMSE_wls_model1),
"wls_model2" = c(RMSE_wls_model2)
)
RMSE_values <- unlist(resultados)
RMSE_sorted <- sort(RMSE_values)
# Obtén los nombres de los modelos ordenados según los valores de RMSE
model_names <- names(resultados)
model_names_sorted <- model_names[order(RMSE_values)]
# Crea un nuevo marco de datos con los nombres de los modelos y sus respectivos RMSE ordenados
RMSE_df <- data.frame(RMSE = RMSE_sorted)
RMSE_df## RMSE
## model_xgb 5295.267
## decision_tree_model 5375.011
## rf_model 5417.839
## svm_model 5549.750
## ols_model 6206.795
## wls_model2 7036.752
## log_ols_model 8158.644
## wls_model1 9820.253
## model_nnet 11954.458Grafico de los valores de la métrica RMSE de cada uno de los modelos estimados
ggplot(RMSE_df,aes(x=reorder(model_names,-RMSE_values), y=RMSE_values)) +
geom_bar(stat="identity", fill="#f68060", alpha=.6, width=.4) +
coord_flip() +
xlab("") +
theme_bw()
Overfitting o underfitting
Se observa que el modelo wls_model2 tiene un buen balance entre sus bias y varianza, en comparación con otros modelos. No obstante, model_xgb tiene un RMSE más bajo que otros modelos lo que implica que al tener sus valores en logaritmo, tendrá un equilibrio entre sus valores con sus predicciones.
ggplot(train, aes(x = exp(wls_model2$fitted.values), y = train$expenses)) +
geom_point() +
stat_smooth() +
labs(x='Predicted Values', y='Actual Values', title='WLS2 Predicted vs. Actual Values')## Warning: Use of `train$expenses` is discouraged.
## ℹ Use `expenses` instead.
## Use of `train$expenses` is discouraged.
## ℹ Use `expenses` instead.## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
ggplot(train, aes(x = exp(wls_model1$fitted.values), y = train$expenses)) +
geom_point() +
stat_smooth() +
labs(x='Predicted Values', y='Actual Values', title='LOG OLS Predicted vs. Actual Values')## Warning: Use of `train$expenses` is discouraged.
## ℹ Use `expenses` instead.
## Use of `train$expenses` is discouraged.
## ℹ Use `expenses` instead.## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'
Hallazgos
EDA
El análisis exploratorio da a conocer que la variable Dependiente será la que tiene mayor variabilidad, en este caso “expenses” lo que denota que el modelo a buscar debe explicar los cambios en esta misma. Asimismo, se muestra que la variable de “smoker” tiende a impactar en mayor medida a “expenses”. De igual manera, Se espera que: * Si los hombres tienen mayores gastos que las mujeres, y negativo si las mujeres tienen mayores gastos. * Un mayor IMC se asocia con mayores gastos. * Un mayor número de hijos se asocia con mayores gastos. * Las personas si fumadoras tienen mayores gastos que los no fumadores.
Modelo seleccionado
Las variables que contribuyen a explicar los cambios de la principal variable de estudio son age y smoker debido a que las personas que fuman tienen un valor más alto de la variable dependiente que las personas que no fuman. Posteriormente, se subdivide el conjunto segun la edad donde por cada unidad que aumenta la edad, la variable dependiente aumenta en una cantidad específica. Por otra parte, el impacto de las demás variables explicativas sobre la variable dependiente son que: Las mujeres tienen un valor más bajo de “y” que los hombres lo que se puede atribuir a una serie de factores; si el IMC aumenta la variable dependiente aumenta en una cantidad específica; las personas con hijos tienen “expenses” relativamente más bajas que las personas sin hijos; y la region es de las variables con menos impacto pero las personas de southeast tienen en promedio mayor “expenses”.
Los resultados estimados del modelo seleccionado que en este caso es XGBoost, son relativamente similares a los otros modelos estimados, ya que coinciden en que smoker es una de las variables principales con mayor impacto en la dependiente. Pero, difieren en la jerarquía del nivel de impacto de las variables que tienden a tener poca correlación. Cabe destacar que, se eligió este modelo debiod a que tiene mayor precisión que el resto. No obstante, dicho modelo al tener los valores logarítmicos es probable que no tenga heterocedasticidad; aunque se podría elegir wls_model2 debido a que cumple con este mismo factor pero tiene un RMSE más alto.
---
title: 'Actividad 1 – Modelos de Regresión'
author: "Avril Lobato - A00833113"
date: "2024-02-28"
output: 
  html_document: 
    toc: TRUE
    toc_float: TRUE
    code_download: TRUE
    
---

# <span style="color:#458B00">**Introducción**</span>
1. **¿Qué es Supervised Machine Learning y cuáles son algunas de sus aplicaciones en Inteligencia de Negocios?**  
El aprendizaje supervisado (*Supervised Machine Learning*) es una técnica de inteligencia artificial en la que se entrena un modelo utilizando datos etiquetados, es decir, datos que tienen una variable objetivo conocida. Su objetivo es encontrar una relación entre las características de entrada y la variable objetivo, de modo que el modelo pueda predecir la variable objetivo para nuevos datos no etiquetados.  
Dicha técnica ofrece una gran versatilidad en la Inteligencia de Negocios, por ende, se puede aplicar en una amplia variedad de problemas empresariales ya sea para optimizar los procesos y/o mejorar la toma de decisiones. Algunos ejemplos de las aplicaciones del aprendizaje supervisado en esta área son:  
- Predicción de ventas: Utilizando datos históricos de ventas, el aprendizaje supervisado puede utilizarse para predecir las ventas futuras en función de factores como el tiempo, las promociones, el clima, entre otros.  
* Análisis de riesgos crediticios: Las instituciones financieras pueden utilizar el aprendizaje supervisado para evaluar el riesgo crediticio de los clientes y decidir si conceder o no un préstamo. El modelo se entrena utilizando datos históricos de clientes con etiquetas que indican si han cumplido o no con sus pagos.  
* Segmentación de clientes: Las empresas pueden utilizar el aprendizaje supervisado para segmentar a sus clientes en grupos con características similares. Esto les permite personalizar sus estrategias de marketing y ofrecer productos y servicios más adecuados a cada segmento.  
* Detección de fraudes: En el sector financiero, el aprendizaje supervisado se utiliza para detectar actividades fraudulentas, como transacciones sospechosas con tarjetas de crédito o cuentas bancarias. El modelo se entrena utilizando datos de transacciones etiquetadas como fraudulentas o legítimas.  
* Recomendación de productos: Las plataformas de comercio electrónico utilizan el aprendizaje supervisado para recomendar productos a los usuarios en función de su historial de compras y el comportamiento de navegación. El modelo se entrena utilizando datos de transacciones pasadas y el feedback de los usuarios sobre los productos recomendados.  

2. **¿Cuáles son los principales algoritmos de Supervised Machine Learning?**  
Los principales algoritmos de aprendizaje supervisado son:
* Regresión lineal: Predice valores numéricos continuos en función de variables independientes, por medio del calculo de una línea de ajuste que minimice la diferencia entre los valores predichos y los valores reales.  
* Regresión logística: Calcula la probabilidad de que una observación pertenezca a una clase o categoría específica y luego clasifica la observación en función de esta probabilidad.  
* XGBoost (eXtreme Gradient Boosting): Combina múltiples árboles de decisión para crear un modelo más preciso y robusto.
* SVM (Support Vector Machine): Encuentra la línea (hiperplano) que maximiza la distancia entre los puntos más cercanos de cada grupo (margen) para tener una separación óptima entre clases de datos con el fin de predecir con precisión las etiquetas de los nuevos puntos de datos.
* RF (Random Forest): Colección de múltiples árboles de decisión independientes utilizando muestras aleatorias del conjunto de datos de entrenamiento para posteriormente, promediar las predicciones de cada árbol con el fin de mejorar la precisión
* DT (Árbol de decisión): Modelo en forma de árbol de decisiones, donde cada nodo interno representa una característica, cada rama representa una decisión y cada hoja representa un resultado.
* OLS (Ordinary Least Squares): Regresión para ajustar una línea (hiperplano) a una bd (relación lineal con las características) con el objetivo de predecir valores numéricos.


3. **¿Qué es la R2 Ajustada? ¿Qué es la métrica RMSE? ¿Cuál es la diferencia entre la R2 Ajustada y la métrica RMSE?**  
La R2 ajustada es una medida estadística que indica cuánta variabilidad en la variable de respuesta es explicada por el modelo de regresión. Se calcula como la proporción de la varianza total de la variable de respuesta, ajustada para el número de predictores en el modelo y el tamaño de la muestra. Mientras que, el RMSE (Root Mean Squared Error) es una medida de la diferencia entre los valores predichos por un modelo de regresión y los valores observados en el conjunto de datos de prueba; es el resultado de la raíz cuadrada de la media de los errores al cuadrado entre las predicciones y los valores reales. Por lo tanto, la diferencia entre estas medidas de evaluación son que la R2 ajustada evalúa qué tan bien el modelo explica la variabilidad de la variable de respuesta y el RMSE evalúa la precisión del modelo en términos de la magnitud de los errores de predicción.

# <span style="color:#458B00">**Análisis Exploratorio de los Datos (EDA)**</span>

Instalación y llamado de ibrerías
```{r}
library(foreign)        # Read Data Stored by 'Minitab', 'S', 'SAS', 'SPSS', 'Stata', 'Systat', 'Weka', 'dBase'
library(ggplot2)        # It is a system for creating graphics
library(dplyr)          # A fast, consistent tool for working with data frame like objects
library(mapview)        # Quickly and conveniently create interactive visualizations of spatial data with or without background maps
library(naniar)         # Provides data structures and functions that facilitate the plotting of missing values and examination of imputations.
library(tmaptools)       # A collection of functions to create spatial weights matrix objects from polygon 'contiguities', for summarizing these objects, and for permitting their use in spatial data analysis
library(tmap)           # For drawing thematic maps
library(RColorBrewer)   # It offers several color palettes 
library(dlookr)         # A collection of tools that support data diagnosis, exploration, and transformation

# Predictive modeling
library(regclass)       # Contains basic tools for visualizing, interpreting, and building regression models
library(mctest)         # Multicollinearity diagnostics
library(lmtest)         # Testing linear regression models
library(spdep)          # A collection of functions to create spatial weights matrix objects from polygon 'contiguities', for summarizing these objects, and for permitting their use in spatial data analysis
library(sf)             # A standardized way to encode spatial vector data
library(spData)         # Diverse spatial datasets for demonstrating, benchmarking and teaching spatial data analysis
library(spatialreg)     # A collection of all the estimation functions for spatial cross-sectional models
library(caret)          # The caret package (short for Classification And Rgression Training) contains functions to streamline the model training process for complex regression and classification problems.
library(e1071)          # Functions for latent class analysis, short time Fourier transform, fuzzy clustering, support vector machines, shortest path computation, bagged clustering, naive Bayes classifier, generalized k-nearest neighbor.
library(SparseM)        # Provides some basic R functionality for linear algebra with sparse matrices
library(Metrics)        # An implementation of evaluation metrics in R that are commonly used in supervised machine learning
library(randomForest)   # Classification and regression based on a forest of trees using random inputs 
library(jtools)         # This is a collection of tools for more efficiently understanding and sharing the results of (primarily) regression analyses 
library(xgboost)        # The package includes efficient linear model solver and tree learning algorithms
library(DiagrammeR)     # Build graph/network structures using functions for stepwise addition and deletion of nodes and edges 
library(effects)        # Graphical and tabular effect displays, e.g., of interactions, for various statistical models with linear predictors
library(shinyjs)
library(sp)
library(geoR)
library(gstat)
library(caret)
#install.packages("datasets") #Usar la base de datos "Iris"
library(datasets)
#install.packages("ggplot2") #Gráficas con mejor diseño
library(ggplot2)
#install.packages("lattice") #Crear gráficos
library(lattice)
#install.packages("DataExplorer") #Crear gráficos
library(DataExplorer)
#install.packages("mlbench") #Crear gráficos
library(mlbench)
#install.packages("tidyverse")
library(tidyverse)
```

Carga de base de datos "health.insurance.csv"
```{r}
df <- read.csv("C:/Users/AVRIL/Documents/health_insurance.csv")
```

Se eliminan duplicados
```{r}
df <- unique(df)
dim(df) #1 registro duplicado
```

## **Identificación y reemplazo de NAs**
```{r}
# Se identifica si hay valores nulos
dfn <- is.na(df)
#No hay valores nulos, pero en caso de que si se hubiera detectado, estos se habrían reemplazo por la mediana del df

# Se eliminan nulos
df <- na.omit(df)
df
```

## **Medidas descriptivas**
Acorde a los datos, se observa que:  
* Los hombres tienen un IMC ligeramente mayor que las mujeres.
* Las mujeres tienen un promedio ligeramente mayor de hijos que los hombres.
* Los hombres tienen una mayor prevalencia de tabaquismo que las mujeres.
* Los hombres tienen un promedio de gastos ligeramente mayor que las mujeres.
```{r}
df$age <- as.numeric(df$age)
df$children <- as.numeric(df$children)

df$region <- as.numeric(factor(df$region))

factor_with_na <- lapply(df[, c("sex", "smoker")], function(x) any(is.na(x)))

# Se convierten variables categóricas a numéricas
df$sex <- as.numeric(factor(df$sex, ordered = TRUE))
df$smoker <- as.numeric(factor(df$smoker, ordered = TRUE))

#Obtencion de medidas descriptivas
summary(df)
```
## **Medidas de dispersión**
Acorde a los datos se observa  que no hay diferencias significativas en la variabilidad de las variables entre hombres y mujeres:  
* Dispersión del IMC: La dispersión del IMC es similar en hombres y mujeres, con una desviación estándar de alrededor de 5 kg/m².  
* Dispersión del número de hijos: La dispersión del número de hijos es similar en hombres y mujeres, con una desviación estándar de alrededor de 1 hijo.  
* Dispersión del tabaquismo: La probabilidad de ser fumador es ligeramente mayor en hombres que en mujeres, con una desviación estándar de 0.42 (42%) para hombres y 0.39 (39%) para mujeres.  
* Dispersión de los gastos: La dispersión de los gastos es similar en hombres y mujeres, con una desviación estándar de alrededor de $4,800.  
```{r}
# Desviación estándar de cada columnan
dispersion_sd <- sapply(df, sd)
print(dispersion_sd)

#Rango de cada columna
dispersion_range <- sapply(df, range)
print(dispersion_range)

#Rango intercuartílico de cada columna
dispersion_iqr <- sapply(df, IQR)
print(dispersion_iqr)
```

```{r}
outlier_ratio <- dispersion_sd / dispersion_iqr

# Columna con la proporción más alta
max_outlier_column <- names(outlier_ratio)[which.max(outlier_ratio)]

print(outlier_ratio)
print(paste("La variable con más outliers es:", max_outlier_column))
```

## **Identificación de patrones y/o tendencias en los datos mediante el uso de gráficos**  
Los patrones y/o tendencias significativas que se observan son:
* Diferencias por Sexo: Las mujeres tienen un IMC más alto, tienden a tener más hijos y tienen menores gastos que los hombres.
* Relación entre Edad e IMC: El IMC aumenta con la edad.
* Relación entre Edad y Gastos: Los gastos aumentan con la edad.
```{r}
# Descripción introductoria o resumen de df
introduce(df)
plot_intro(df)

# Boxplot para visualizar la distribución de los valores en expenses 
plot_boxplot(df, by="expenses")

# Cantidad de valores faltantes (NA's) en cada columna
plot_missing(df)

# Histograma para cada variable numérica del dataframe df
plot_histogram(df)

# Gráfico de barras para cada variable categórica 
plot_bar(df)

# Gráfico scatterplot  
plot(df)

```

Visualización de distribución normal  
* Expenses: Distribución sesgada hacia la izquierda (asimetría negativa), con mayor concentración en el rango 0-50.000.  
* Los hombres tienen mayores gastos que las mujeres.  
* Aumento de los gastos con la edad.  

Se muestra que para tener una mejor distribución normal de los datos, se deben convertir las variables a log() con el fin de tener mayor precisión y menor sesgo. 
```{r}
par(mfrow = c(1, 2))  # Divide el área de trazado en 1 fila y 2 columnas
for(i in 1:ncol(df)) {
  qqnorm(df[, i], main = names(df)[i])
  qqline(df[, i], col = 2)
}

plot_normality(df, expenses, bmi, age)
```

Correlación entre variables  
Se visualiza que hay una asociación positiva entre expenses y smoker
```{r}
dev.new(width = 10, height = 8)
correlate(df) %>%  plot()
plot_correlation(df)
```


# <span style="color:#458B00">**Especificación del modelo de regresión lineal a estimar**</span>

Se infiere que el impacto de cada una de las variables explicativas sobre la principal variable de estudio, en este caso "expenses", sería de esta manera:  
* Sexo: Las mujeres podrían tener un impacto negativo en los gastos, es decir, se espera que las mujeres tengan menores gastos que los hombres.  
* IMC: Un mayor IMC podría tener un impacto positivo en los gastos, es decir, se espera que las personas con un mayor IMC tengan mayores gastos en salud, alimentación, etc.  
* Hijos: Un mayor número de hijos podría tener un impacto positivo en los gastos, es decir, se espera que las familias con más hijos tengan mayores gastos en educación, alimentación, etc.  
* Fumador: El hábito de fumar podría tener un impacto positivo en los gastos, es decir, se espera que los fumadores tengan mayores gastos en tabaco, tratamiento de enfermedades relacionadas con el tabaquismo, etc.  

Asimismo, se espera que la principal variable x que afecta a y, sería "smoker", donde si tiene un valor de "2" (igual a persona si fumadora) implica que aumenta "expenses". Mientras que, la variable "region" tiene menor impacto en y.

```{r}
par(mfrow = c(3, 2))  # Divide el área de la trama en una cuadrícula de 3 filas y 2 columnas

# Age
boxplot(expenses ~ age, data = df, main = "Expenses vs. age", xlab = "age", ylab = "Expenses")

# Sex
boxplot(expenses ~ sex, data = df, main = "Expenses vs. sex", xlab = "sex", ylab = "Expenses")

# BMI
boxplot(expenses ~ bmi, data = df, main = "Expenses vs. bmi", xlab = "bmi", ylab = "Expenses")

# Children
boxplot(expenses ~ children, data = df, main = "Expenses vs. children", xlab = "children", ylab = "Expenses")

# Smoker
boxplot(expenses ~ smoker, data = df, main = "Expenses vs. smoker", xlab = "smoker", ylab = "Expenses")

# Region
boxplot(expenses ~ region, data = df, main = "Expenses vs. region", xlab = "region", ylab = "Expenses")
```

## **Partición de bd**
```{r}
set.seed(123) # What is set.seed()? We want to make sure that we get the same results for randomization each time you run the script.   
partition <- createDataPartition(y = df$expenses, p=0.7, list=F)
train = df[partition, ]
test  = df[-partition, ]
```

## **Regresion lineal**
```{r}
lm_model <- lm(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train)
summary(lm_model)
```
Se visualiza el efecto parcial de smoker sobre la variable dependiente
```{r}
plot(effect("smoker",lm_model))
```
Se presentan los valores previstos frente a los observados de la variable dependiente
```{r warning=FALSE}
ggplot(train, aes(x = exp(lm_model$fitted.values), y = train$expenses)) +
  geom_point() +
  stat_smooth() +
  labs(x='Predicted Values', y='Actual Values', title='OLS Predicted vs. Actual Values')
```


# <span style="color:#458B00">**Estimación de modelos de Supervised Machine Learning (SML)**</span>

## **OLS Model**
```{r}
ols_model <- lm(expenses ~ age + sex + bmi + children + smoker + region, data = train)
summary(ols_model)

AIC(ols_model)      # AIC = 18971.02

prediction_ols_model <- predict(ols_model,test)
RMSE_ols_model <- rmse(prediction_ols_model,test$expenses)
RMSE_ols_model
```

## **Log OLS Model**
```{r}
log_ols_model <- lm(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train)
summary(log_ols_model)

AIC(log_ols_model)  # AIC = 1082.83

# Otra forma de calcular el RMSE
#log_errors <- train$expenses - exp(log_ols_model$fitted.values)  #Calcula los errores entre los gastos log reales y los previstos
#RMSE_log_ols_model <- sqrt(mean(log_errors^2))
#RMSE_log_ols_model

prediction_log_model <- exp(predict(log_ols_model, newdata = test))  # Predecir en escala original
RMSE_log_model <- sqrt(mean((test$expenses - prediction_log_model)^2))  # Calcular RMSE
RMSE_log_model
```

## **SAR & SEM**
Los modelos SAR y SEM no se pueden realizar en este caso debido a la ausencia de información espacial, puesto que dichos modelos requieren datos espaciales, como la ubicación de las observaciones, para estimar los efectos espaciales. El conjunto de datos proporcionado no incluye información espacial, como coordenadas o nombres de regiones suficientes para convertirlos en nb2listw. Por lo tanto, sin información espacial, no se pueden calcular las matrices de pesos espaciales que son esenciales para estimar los modelos SAR y SEM.

No obstante, se muestran los códigos que se podrían utilizar:
SAR  
```{r}
#library(spdep)
#map_centroid <- coordinates(df) 
#col.gal.nb  <- read.gal(system.file("df", package="spdep"))
#map.linkW    <- nb2listw(col.gal.nb, style="W")   #########NEIGHT

#sar_model <- lagsarlm(log(expenses) ~ log(age) + sex + log(children +0.01) + 
 #                     log(bmi) + smoker + region, data = train, method="Matrix", listw = wt)
#summary(sar_model)

#RMSE_SAR <- sqrt(mean((df$expenses - sar_model$fitted.values)^2)) #Ajusta el modelo a los datos de entrenamiento.
#RMSE_SAR
```

SEM  
```{r}
#sem_model <- errorsarlm(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train, map.linkW, method="Matrix")
#summary(sem_model)

#RMSE_SEM <- sqrt(mean((df$expenses - sem_model$fitted.values)^2))
#RMSE_SEM
```

## **Support Vector Regression**
```{r}
fit.svm = svm(formula =  log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train, type = 'eps-regression', kernel = 'radial')
summary(fit.svm)
```

```{r}
plot(fit.svm$fitted, fit.svm$residuals, main="SVM Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)
```

```{r}
predictions <-  exp(predict(fit.svm,newdata = test))

# Error cuadrático medio de la raíz
RMSE_svm <- rmse(predictions,test$expenses)
RMSE_svm
```

```{r}
dv_svm<-data.frame(exp(fit.svm$fitted),train$expenses)
ggplot(dv_svm, aes(x = exp.fit.svm.fitted., y = train.expenses)) +
  geom_point() +
  stat_smooth() +
  labs(x='Predicted Values', y='Actual Values', title='SVM Predicted vs. Actual Values')
```

## **Decision Tree**
```{r}
decision_tree_regression <- rpart(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train)
# summary(decision_tree_regression)
plot(decision_tree_regression, compress = TRUE)
text(decision_tree_regression, use.n = TRUE)
#install.packages("rpart.plot")  # Instalar el paquete rpart.plot
library(rpart.plot)     
rpart.plot(decision_tree_regression)
```

```{r}
dt_prediction_test_data <-exp(predict(decision_tree_regression,newdata = test))
RMSE_Tree <- rmse(dt_prediction_test_data, test$expenses)
RMSE_Tree
```

## **Random Forest**
```{r}
library(dplyr)

transform_df <- df 
transform_df$age <- log(transform_df$age)
transform_df$bmi <- log(transform_df$bmi)
transform_df$children <- log(transform_df$children +0.01)
transform_df$expenses <- log(transform_df$expenses)

summary(transform_df)
```

```{r}
set.seed(123) 
partition_alt <- createDataPartition(y = transform_df$expenses, p=0.7, list=F)
train_alt = transform_df[partition_alt, ]
test_alt <- transform_df[-partition_alt, ]
```

```{r}
random_forest<-randomForest(expenses ~ age + sex + children + bmi + smoker + region, data=train_alt, proximity=TRUE)
print(random_forest) 
```
```{r}
rf_prediction_test_data <-exp(predict(random_forest,newdata = test_alt))

RMSE_RF <- rmse(rf_prediction_test_data, test$expenses)
RMSE_RF
```

Evaluacion de Importancia de Variables  
Se observa que la variable "smoker" es la que tiene mayor importancia en la predicción de y ("expenses"); a la cual le continua la variable "age" en nivel de importancia. 
```{r}
#Cuanto mayor sea el valor de la precisión media de la disminución, mayor será la importancia de la variable en el modelo. En otras palabras, la precisión media de la disminución representa en qué medida la eliminación de cada variable reduce la precisión del modelo.
varImpPlot(random_forest, n.var = 5, main = "Top 10 - Variable")
importance(random_forest)     
```

## **XGBoost**
```{r}
# Define las variables explicativas (X) y la variable dependiente (Y) en el conjunto de entrenamiento
train_x = data.matrix(train_alt[, -7])
train_y = train_alt[,7]

# Define las variables explicativas (X) y la variable dependiente (Y) en el conjunto de pruebas
test_x = data.matrix(test_alt[, -7])
test_y = test_alt[, 7]

# Definer sets fianles de train y test
xgb_train = xgb.DMatrix(data = train_x, label = train_y)
xgb_test = xgb.DMatrix(data = test_x, label = test_y)
```

```{r}
# Ajustamos el modelo de regresión XGBoost y mostramos el RMSE de los datos de entrenamiento y de prueba en cada ronda.
watchlist = list(train=xgb_train, test=xgb_test)
model_xgb = xgb.train(data=xgb_train, max.depth=3, watchlist=watchlist, nrounds=70) 
```

```{r}
reg_xgb = xgboost(data = xgb_train, max.depth = 3, nrounds = 59, verbose = 0)
prediction_xgb_test<-exp(predict(reg_xgb, newdata = xgb_test))

RMSE_XGB <- rmse(prediction_xgb_test, test$expenses)
RMSE_XGB
```
```{r}
xgb_reg_residuals<-test$expenses - prediction_xgb_test
plot(xgb_reg_residuals, xlab= "Dependent Variable", ylab = "Residuals", main = 'XGBoost Regression Residuals')
abline(0,0)
```
```{r}
# Plot 3 modelos de arbol
xgb.plot.tree(model=reg_xgb, trees=0:2)
```

```{r}
importance_matrix <- xgb.importance(model = reg_xgb)
xgb.plot.importance(importance_matrix, xlab = "Explanatory Variables X's Importance")
```

## **Neural Networks Regression**

```{r}
#install.packages("neuralnet")
library(neuralnet)
nn_model <- neuralnet(expenses ~ age + sex + bmi + children + smoker + region, data = train, hidden = c(5, 3), linear.output = TRUE) 
```

Gráfico de neural network 
```{r}
plot(nn_model)
```

```{r}
prediction_net<-predict(nn_model, test)
RMSE_net <- rmse(prediction_net, test$expenses)
RMSE_net
```

# <span style="color:#458B00">**Pruebas de Diagnóstico**</span>

## **Autocorrelación espacial**
Estimar un análisis de regresión espacial requiere  construir una matriz espacial que conecte los puntos geográficos. En este caso, el df no contiene variables espaciales y/o geograficas. Por lo tanto, no es necesario realizar un modelo de regresión espacial y, por ende, calcular el test de MOran para determinar si existe autocorrelacion espacial. No obstante, se podría realizar el siguiente código en caso de tener este tipo de valores en el dataframe:  

```{r}
#library(spdep)
#vecinos <- poly2nb(df)
# A continuación, crea una matriz de pesos espaciales a partir de la lista de vecinos
#matriz_pesos <- nb2listw(df)
# Finalmente, realiza el test de Moran para determinar la autocorrelación espacial en 'expenses'
#moran.test(df$expenses, listw = matriz_pesos)
```

## **Autocorrelación serial**
No hay autocorrelacion serial debido a que no se tiene una bd de series de tiempo, y/o una variable de tiempo en el dataframe actual. Sin embargo, en caso de tener un valor serial se realizaría un análisis afc como el que se muestra a continuación:  
```{r}
acf_result <- acf(df$expenses, lag.max = 30, main = "ACF of HOVAL", ylim = c(-1, 1))
plot(acf_result, main="Autocorrelación Serial")
```

## **Multicolinealidad** 
Se muestra que los valores de VIF son cercanos a 1, lo que indica que no hay multicolinealidad en el modelo
```{r}
VIF(lm_model)

VIF(ols_model)
```

## **Heterocedasticidad**  
Un valor p estadísticamente significativo (normalmente inferior a 0,05), en este caso, los valores p son extremadamente pequeños, lo que significa que rechazaríamos la hipótesis nula de homocedasticidad y concluimos que hay evidencia significativa de heterocedasticidad en el modelo.
```{r}
bptest(log_ols_model)
bptest(lm_model)
bptest(ols_model)
```

## Solucion de heterocedasticidad
```{r}
plot(fitted(ols_model), resid(ols_model), main="Linear Regression Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)


plot(fitted(lm_model), resid(lm_model), main="Linear Regression Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)
```

```{r}
plot(fitted(ols_model), resid(ols_model), main="Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)

plot(fitted(lm_model), resid(lm_model), main="Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)
```

## **Normalidad de residuales**
Se espera que el modelo de regresión lineal produzca errores que se distribuyan normalmente. Así, más errores se agrupan alrededor de cero.
```{r}
hist(lm_model$residuals, xlab="Estimated Regression Residuals", main='Distribution of LM Estimated Regression Residuals', col='lightblue', border="white")

hist(ols_model$residuals, xlab="Estimated Regression Residuals", main='Distribution of OLS Estimated Regression Residuals', col='lightblue', border="white")
```

```{r}
# Haremos predicciones utilizando los datos de prueba para evaluar el rendimiento de nuestro modelo de regresión.
prediction_ols_model1 <- ols_model %>% predict(test)
RMSE_ols1 <- RMSE(prediction_ols_model1, test$expenses)
RMSE_ols1
```

```{r}
wt1 <- 1 / lm(abs(ols_model$residuals) ~ ols_model$fitted.values)$fitted.values^2
wls_model1 <-lm(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train, weights=wt1)
summary(wls_model1)
```
Se muestra que ya no hay heterocedasticidad en el modelo
```{r}
bptest(wls_model1)
```

```{r}
plot(fitted(wls_model1), resid(wls_model1), main="Residual vs. Fitted Values", xlab="Fitted Values", ylab="Residuals")
abline(0,0)
```

```{r}
# Haremos predicciones utilizando los datos de prueba para evaluar el rendimiento de nuestro modelo de regresión wls.
prediction_wls_model1 <- wls_model1 %>% predict(test)
RMSE(prediction_wls_model1, test$expenses) # WLS - RMSE is greater than OLS - RMSE 
```
```{r}
wls_errors <- train$expenses - exp(wls_model1$fitted.values)

# Calculate RMSE for WLS model
RMSE_wls_model1 <- sqrt(mean(wls_errors^2))
RMSE_wls_model1
```

```{r}
wt2 <- 1 / lm(abs(log_ols_model$residuals) ~ log_ols_model$fitted.values)$fitted.values^2
wls_model2 <-lm(log(expenses) ~ log(age) + sex + children + log(bmi) + smoker + region, data = train, weights=wt2)

prediction_wls_model2 <- wls_model2 %>% predict(test)
RMSE(prediction_wls_model2, test$expenses) 
wls_errors2 <- train$expenses - exp(wls_model2$fitted.values)

# Calculate RMSE for WLS model
RMSE_wls_model2 <- sqrt(mean(wls_errors2^2))
RMSE_wls_model2
```

# <span style="color:#458B00">**Evaluación y Selección de Modelo de Regresión**</span>
Se observan los siguientes hallazgos en relación a la comparativa de los RMSE:  
1. El promedio de los valores de RMSE es de 7042.774.  
2. La diferencia entre el mayor y menor RMSE es de 6659.191.  
3. La desviación estándar de los valores de RMSE es de 2672.846.  
4. La mayoría de los modelos (55.56%) tienen un RMSE mayor al promedio.  
5. A pesar de la heterocedasticidad, el modelo Arbol de Decision tienen un RMSE menor que varios de los demás.
6. La heterocedasticidad no parece afectar significativamente el rendimiento de ciertos modelos. Aunque, se observa heterocedasticidad en los demás modelos que no son WLS, lo que significa que la varianza de los errores no es constante.
7. Los modelos wls model parecen tener un RMSE alto en comparación a los primeros 5 modelos.  

Por lo tanto, el modelo seleccionado es model xgb debido a su bajo RMSE en comparación con otros modelos lo que implica que tiene un mejor rendimiento en la predicción de los datos de prueba y tiene ausencia de heterocedasticidad como multicolinealidad lo que significa que los errores de predicción tienen una varianza constante, lo que mejora la confiabilidad del modelo.  
## Dataframe
```{r}
resultados <- data.frame(
  "log_ols_model" = c(RMSE_log_model),
  "rf_model" = c(RMSE_RF),
  "decision_tree_model" = c(RMSE_Tree),
  "ols_model" = c(RMSE_ols_model),
  "svm_model" = c(RMSE_svm),
  "model_xgb" = c(RMSE_XGB),
  "model_nnet" = c(RMSE_net),
  "wls_model1" = c(RMSE_wls_model1),
  "wls_model2" = c(RMSE_wls_model2)
)

RMSE_values <- unlist(resultados)
RMSE_sorted <- sort(RMSE_values)

# Obtén los nombres de los modelos ordenados según los valores de RMSE
model_names <- names(resultados)
model_names_sorted <- model_names[order(RMSE_values)]

# Crea un nuevo marco de datos con los nombres de los modelos y sus respectivos RMSE ordenados
RMSE_df <- data.frame(RMSE = RMSE_sorted)
RMSE_df
```
## **Grafico de los valores de la métrica RMSE de cada uno de los modelos estimados**
```{r}
ggplot(RMSE_df,aes(x=reorder(model_names,-RMSE_values), y=RMSE_values)) +
  geom_bar(stat="identity", fill="#f68060", alpha=.6, width=.4) +
  coord_flip() +
  xlab("") +
  theme_bw()
```

## **Overfitting o underfitting**
Se observa que  el modelo wls_model2 tiene un buen balance entre sus bias y varianza, en comparación con otros modelos. No obstante, model_xgb tiene un RMSE más bajo que otros modelos lo que implica que al tener sus valores en logaritmo, tendrá un equilibrio entre sus valores con sus predicciones.
```{r}
ggplot(train, aes(x = exp(wls_model2$fitted.values), y = train$expenses)) +
  geom_point() +
  stat_smooth() +
  labs(x='Predicted Values', y='Actual Values', title='WLS2 Predicted vs. Actual Values')
```

```{r}
ggplot(train, aes(x = exp(wls_model1$fitted.values), y = train$expenses)) +
  geom_point() +
  stat_smooth() +
  labs(x='Predicted Values', y='Actual Values', title='LOG OLS Predicted vs. Actual Values')
```


# <span style="color:#458B00">**Hallazgos**</span>
## **EDA**
El análisis exploratorio da a conocer que la variable Dependiente será la que tiene mayor variabilidad, en este caso "expenses" lo que denota que el modelo a buscar debe explicar los cambios en esta misma. Asimismo, se muestra que la variable de "smoker" tiende a impactar en mayor medida a "expenses". De igual manera, Se espera que:
* Si los hombres tienen mayores gastos que las mujeres, y negativo si las mujeres tienen mayores gastos.
* Un mayor IMC se asocia con mayores gastos.
* Un mayor número de hijos se asocia con mayores gastos.
* Las personas si fumadoras tienen mayores gastos que los no fumadores.

## **Modelo seleccionado**
Las variables que contribuyen a explicar los cambios de la principal variable de estudio son age y smoker debido a que las personas que fuman tienen un valor más alto de la variable dependiente que las personas que no fuman. Posteriormente, se subdivide el conjunto segun la edad donde por cada unidad que aumenta la edad, la variable dependiente aumenta en una cantidad específica. Por otra parte, el impacto de las demás variables explicativas sobre la variable dependiente son que: Las mujeres tienen un valor más bajo de "y" que los hombres lo que se puede atribuir a una serie de factores; si el IMC aumenta la variable dependiente aumenta en una cantidad específica; las personas con hijos tienen "expenses" relativamente más bajas que las personas sin hijos; y la region es de las variables con menos impacto pero las personas de southeast tienen en promedio mayor "expenses".

Los resultados estimados del modelo seleccionado que en este caso es XGBoost, son relativamente similares a los otros modelos estimados, ya que coinciden en que smoker es una de las variables principales con mayor impacto en la dependiente. Pero, difieren en la jerarquía del nivel de impacto de las variables que tienden a tener poca correlación. Cabe destacar que, se eligió este modelo debiod a que tiene mayor precisión que el resto. No obstante, dicho modelo al tener los valores logarítmicos es probable que no tenga heterocedasticidad; aunque se podría elegir wls_model2 debido a que cumple con este mismo factor pero tiene un RMSE más alto.