Homework 4

Predicting Housing Price in California

Direction

• Please submit one Quarto Document of Homework 4 to Brightspace using the following file naming convention:

• Example:

  • danl-320-hw4-choe-byeonghak.qmd

• Due: April 20, 2026, 11:59 P.M. (ET)

• Please send Byeong-Hak an email (bchoe@geneseo.edu) if you have any questions.

library(tidyverse)
library(broom)
library(stargazer)

library(glmnet)
library(gamlr)
library(Matrix)

library(janitor)
library(rpart)
library(rpart.plot)
library(ranger)
# library(xgboost)
library(vip)
library(pdp)

library(DT)
library(rmarkdown)
library(hrbrthemes)
library(ggthemes)

theme_set(
  theme_ipsum()
)

scale_colour_discrete <- function(...) scale_color_colorblind(...)
scale_fill_discrete <- function(...) scale_fill_colorblind(...)

Data

df <- read_csv('https://bcdanl.github.io/data/california_housing_cleaned.csv')

• The housing data at census tract-level in California include:
  • Latitude/Longitude of tract centers
  • Median Home age.
  • Median Income
  • Average room/bedroom numbers
  • Average occupancy
  • Median home values
• The goal is to predict the log of median housing value for census tracts.

Question 1

• Divide the df DataFrame into training and test DataFrames.
  • Use dtrain and dtest for training and test DataFrames, respectively.
  • 70% of observations in the df are assigned to dtrain; the rest is assigned to dtest.

Question 2

• Consider the following model:

\[ \begin{align} \log(\text{medianHouseValue})_{i} = &\beta_0 + \beta_1 \text{housingMedianAge}_{i} + \beta_2 \text{medianIncome}_{i}\\ &+ \beta_3 \text{AveBedrms}_{i} + \beta_4 \text{AveRooms}_{i} + \beta_5 \text{AveOccupancy}_{i}\\ &+ \beta_{6}\text{longitude}_{i} + \beta_{7}\text{latitude}_{i} + \epsilon_{i} \end{align} \]

• Provide a rationale behind the model why it includes \(\text{longitude}\) and \(\text{longitude}\) as predictors for \(\log(\text{medianHouseValue})\)

Question 3

• Fit the linear regression model in Question 2.

Question 4

• Interpret \(\hat{\beta_{2}}\), \(\hat{\beta_{4}}\), and \(\hat{\beta_{5}}\), obtained from the linear regression model in Question 3.

Questions 5-6

• Suppose the model allow for the relationship between the outcome and each of these predictors—\(\text{housingMedianAge}_{i}\), \(\text{medianIncome}_{i}\), \(\text{AveBedrms}_{i}\), \(\text{AveRooms}_{i}\), and \(\text{AveOccupancy}_{i}\)—varies by location of Census tract.

\[ \begin{align} \log(\text{medianHouseValue})_{i} = &\beta_0 + \beta_1 \text{housingMedianAge}_{i} + \beta_2 \text{medianIncome}_{i}\\ &+ \beta_3 \text{AveBedrms}_{i} + \beta_4 \text{AveRooms}_{i} + \beta_5 \text{AveOccupancy}_{i}\\ &+ \beta_6 \text{longitude}_{i} + \beta_7 \text{latitude}_{i}\\ &+ (\beta_8 \text{housingMedianAge}_{i} + \beta_9 \text{medianIncome}_{i}\\ &\qquad+ \beta_{10} \text{AveBedrms}_{i} + \beta_{11} \text{AveRooms}_{i} + \beta_{12} \text{AveOccupancy}_{i} ) \times \text{longitude}_{i}\\ &+ (\beta_{13} \text{housingMedianAge}_{i} + \beta_{14} \text{medianIncome}_{i}\\ &\qquad+ \beta_{15} \text{AveBedrms}_{i} + \beta_{16} \text{AveRooms}_{i} + \beta_{17} \text{AveOccupancy}_{i} ) \times \text{latitude}_{i}\\ &+ (\beta_{18} \text{housingMedianAge}_{i} + \beta_{19} \text{medianIncome}_{i} \\ &\qquad+ \beta_{20} \text{AveBedrms}_{i} + \beta_{21} \text{AveRooms}_{i} + \beta_{22} \text{AveOccupancy}_{i} )\times \text{longitude}_{i} \times \text{latitude}_{i}\\ &+ \epsilon_{i} \end{align} \]

Question 5

Fit a Lasso linear regression model.

Question 6

Fit a Ridge linear regression model.

Questions 7-10

Consider the tree-based model described below:

\[ \begin{align} \log(\text{medianHouseValue})_{i} = f(&\text{housingMedianAge}_{i}, \text{medianIncome}_{i},\\ &\text{AveBedrms}_{i}, \text{AveRooms}_{i}, \text{AveOccupancy}_{i},\\ &\text{latitude}_{i}, \text{longitude}_{i}) \end{align} \]

Question 7

Fit a regression tree model.

Question 8

Fit a random forest model with num.trees = 500.

Question 9

Fit a gradient-boosted tree model using the following hyperparameter grid:

xgb_grid <- tidyr::crossing(
  nrounds = seq(20, 200, by = 20),
  eta = c(0.025, 0.05, 0.1, 0.3),
  max_depth = c(2, 3),
  gamma = c(0),
  colsample_bytree = c(1),
  min_child_weight = c(1),
  subsample = c(1)
)

xgb_grid |> 
  paged_table()

Question 10

Compare the prediction performance across the models:

• Linear regression
• Lasso regression
• Ridge regression
• Regression tree
• Random forest
• Gradient-boosted tree