NikhilBharadwaj_DA_Lab8
2024-03-28
6) In this exercise, you will further analyze the Wage data set considered throughout this chapter.
(a) Perform polynomial regression to predict wage using age. Use cross-validation to select the optimal degree d for the polyno- mial. What degree was chosen, and how does this compare to the results of hypothesis testing using ANOVA? Make a plot of the resulting polynomial fit to the data.
# Load required libraries
library(ISLR2) # For the Wage dataset
library(boot) # For cross-validation
#library(bootStepAIC) # For stepAIC function
# Load the Wage dataset
data(Wage)
# Polynomial regression with cross-validation to select optimal degree
set.seed(123) # For reproducibility
cv.error <- rep(NA, 10) # Vector to store cross-validation errors
for (d in 1:10) {
fit <- glm(wage ~ poly(age, d), data = Wage)
cv.error[d] <- cv.glm(Wage, fit, K = 10)$delta[1]
}
# Optimal degree selected using cross-validation
optimal_degree <- which.min(cv.error)
cat("Optimal Degree selected using cross-validation:", optimal_degree, "\n")## Optimal Degree selected using cross-validation: 10# ANOVA hypothesis testing to select optimal degree
fit_ano <- lm(wage ~ poly(age, 10), data = Wage)
summary_aov <- anova(fit_ano, test = "F")
# Degree selected using ANOVA
degree_ANOVA <- which.max(summary_aov$'Pr(>F)'[1:10])
cat("Degree selected using ANOVA:", degree_ANOVA, "\n")## Degree selected using ANOVA: 1# Plotting the resulting polynomial fit to the data
plot(Wage$age, Wage$wage, col = "blue", xlab = "Age", ylab = "Wage", main = "Polynomial Regression Fit")
points(Wage$age, fitted(fit), col = "red", pch = 20)
legend("topright", legend = c("Actual", "Polynomial Fit"), col = c("blue", "red"), pch = c(1, 20))
# Plotting the resulting polynomial fit to the data based on ANOVA-selected degree
plot(Wage$age, Wage$wage, col = "blue", xlab = "Age", ylab = "Wage", main = "Polynomial Regression Fit (Degree 1)")
lines(sort(Wage$age), predict(fit_ano, newdata = data.frame(age = sort(Wage$age))), col = "red")
legend("topright", legend = c("Actual", "Polynomial Fit"), col = c("blue", "red"), lty = c(NA, 1), pch = c(1, NA))
(b) Fit a step function to predict wage using age, and perform cross- validation to choose the optimal number of cuts. Make a plot of the fit obtained.
# Load necessary libraries
library(boot) # For cross-validation
# Load the Wage dataset
data(Wage)
# Define the number of folds for cross-validation
k <- 10
# Define a function to compute the mean squared error for a step function
step_mse <- function(data, cuts) {
# Create age groups based on the specified number of cuts
data$age_group <- cut(data$age, cuts)
# Fit a linear regression model
lm_fit <- lm(wage ~ age_group, data = data)
# Compute mean squared error
mse <- mean(lm_fit$residuals^2)
return(mse)
}
# Perform cross-validation to choose the optimal number of cuts
cv_step <- function(data, max_cuts) {
cv_error <- rep(0, max_cuts - 1)
for (i in 2:max_cuts) {
cv_error[i - 1] <- mean(sapply(split(data, cut(data$age, i)), step_mse, cuts = i))
}
return(cv_error)
}
# Perform cross-validation
cv_errors <- cv_step(Wage, max_cuts = 10)
# Find the optimal number of cuts with minimum CV error
optimal_cuts <- which.min(cv_errors) + 1 # Adding 1 because indexing starts from 1
# Plot the cross-validation error
plot(2:10, cv_errors, type = 'b', xlab = 'Number of Cuts', ylab = 'Cross-Validation Error',
main = 'Cross-Validation Error vs. Number of Cuts')
points(optimal_cuts, cv_errors[optimal_cuts - 1], col = 'red', pch = 19)
legend("topright", legend = paste("Optimal Cuts =", optimal_cuts), col = 'red', pch = 19)
# Fit the final model with the optimal number of cuts
Wage$age_group <- cut(Wage$age, optimal_cuts)
lm_final <- lm(wage ~ age_group, data = Wage)
# Plot the fit obtained
plot(Wage$age, Wage$wage, xlab = 'Age', ylab = 'Wage', main = 'Step Function Fit', col = 'blue')
lines(Wage$age, predict(lm_final), col = 'red', lwd = 2)
7. The Wage data set contains a number of other features not explored in this chapter, such as marital status (maritl), job class (jobclass), and others. Explore the relationships between some of these other predictors and wage, and use non-linear fitting techniques in order to fit flexible models to the data. Create plots of the results obtained, and write a summary of your findings.
# Load necessary libraries
library(mgcv)## Loading required package: nlme## This is mgcv 1.8-42. For overview type 'help("mgcv-package")'.library(ggplot2)
# Fit a GAM to explore the relationship between wage and age, marital status, and job class
gam_model <- gam(wage ~ s(age) + maritl + jobclass, data = Wage)
# Summary of the GAM
summary(gam_model)##
## Family: gaussian
## Link function: identity
##
## Formula:
## wage ~ s(age) + maritl + jobclass
##
## Parametric coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 94.512 1.898 49.803 < 2e-16 ***
## maritl2. Married 14.501 2.061 7.036 2.44e-12 ***
## maritl3. Widowed -1.887 9.113 -0.207 0.836
## maritl4. Divorced -2.048 3.345 -0.612 0.540
## maritl5. Separated -3.324 5.533 -0.601 0.548
## jobclass2. Information 15.204 1.423 10.686 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Approximate significance of smooth terms:
## edf Ref.df F p-value
## s(age) 5.027 6.116 19.26 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## R-sq.(adj) = 0.141 Deviance explained = 14.4%
## GCV = 1501.8 Scale est. = 1496.3 n = 3000# Plot the results
plot(gam_model, pages = 1)
# Predicted vs. observed plot
plot(Wage$wage, predict(gam_model), xlab = "Observed Wage", ylab = "Predicted Wage",
main = "Predicted vs. Observed Wage")
# Explore the relationship between wage and marital status using boxplots
ggplot(Wage, aes(x = maritl, y = wage)) +
geom_boxplot() +
labs(x = "Marital Status", y = "Wage", title = "Relationship between Marital Status and Wage")
# Explore the relationship between wage and job class using boxplots
ggplot(Wage, aes(x = jobclass, y = wage)) +
geom_boxplot() +
labs(x = "Job Class", y = "Wage", title = "Relationship between Job Class and Wage")
Parametric Coefficients: The intercept represents the estimated average wage for individuals in the reference category of the categorical variables maritl (Never Married) and jobclass (Industrial).
Coefficients for other categories of maritl and jobclass represent the difference in average wage compared to the reference category. For example, individuals who are Married (maritl2. Married) have an estimated average wage that is 14.501 units higher than individuals who are Never Married, holding other variables constant.
Some categories, such as Widowed, Divorced, and Separated, do not show significant differences in average wage compared to the reference category (Never Married), as indicated by their high p-values.
Smooth Terms (Age): The smooth term for age (s(age)) suggests a non-linear relationship between age and wage. The estimated degrees of freedom (edf) indicate that the relationship is modeled using approximately 5.027 degrees of freedom, implying a moderately complex relationship.
The F-test and associated p-value assess the overall significance of the smooth term for age. In this case, the p-value is highly significant (p < 2e-16), indicating that age is a significant predictor of wage after accounting for other variables in the model.
Adjusted R-squared and Deviance Explained: The adjusted R-squared value (R-sq.(adj)) measures the proportion of variance in the response variable (wage) explained by the model, adjusted for the number of predictors. In this case, the adjusted R-squared is 0.141, indicating that approximately 14.1% of the variability in wage is explained by the model.
The Deviance explained represents the percentage reduction in deviance achieved by the model compared to a null model with no predictors. In this case, the model explains 14.4% of the deviance, indicating a modest but significant improvement over the null model.
Generalized Cross Validation (GCV) and Scale Estimation: GCV provides an estimate of the predictive performance of the model. A lower GCV value suggests better model fit, although the absolute value itself may not be interpretable. Here, the GCV is 1501.8.
Scale estimation (Scale est.) provides an estimate of the error variance in the model. In this case, the scale estimate is 1496.3.
Overall, the GAM suggests that age and job class are significant predictors of wage, with a non-linear relationship observed for age. Marital status, however, shows less significance in explaining wage variability in this model.
9. This question uses the variables dis (the weighted mean of distances to five Boston employment centers) and nox (nitrogen oxides concen- tration in parts per 10 million) from the Boston data. We will treat dis as the predictor and nox as the response.
(a) Use the poly() function to fit a cubic polynomial regression to predict nox using dis. Report the regression output, and plot the resulting data and polynomial fits.
# Load the Boston dataset
data(Boston)
# Fit cubic polynomial regression
model <- lm(nox ~ poly(dis, 3), data = Boston)
# Display regression output
summary(model)##
## Call:
## lm(formula = nox ~ poly(dis, 3), data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.121130 -0.040619 -0.009738 0.023385 0.194904
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.554695 0.002759 201.021 < 2e-16 ***
## poly(dis, 3)1 -2.003096 0.062071 -32.271 < 2e-16 ***
## poly(dis, 3)2 0.856330 0.062071 13.796 < 2e-16 ***
## poly(dis, 3)3 -0.318049 0.062071 -5.124 4.27e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06207 on 502 degrees of freedom
## Multiple R-squared: 0.7148, Adjusted R-squared: 0.7131
## F-statistic: 419.3 on 3 and 502 DF, p-value: < 2.2e-16# Plot the data and polynomial fits
plot(Boston$dis, Boston$nox, xlab = "dis", ylab = "nox", main = "Cubic Polynomial Regression")
lines(sort(Boston$dis), predict(model, data.frame(dis = sort(Boston$dis))), col = "red", lwd = 2)
(b) Plot the polynomial fits for a range of different polynomial degrees (say, from 1 to 10), and report the associated residual sum of squares.
# Define a function to fit polynomial regression and calculate RSS
fit_polynomial <- function(degree) {
model <- lm(nox ~ poly(dis, degree), data = Boston)
rss <- sum(model$residuals^2)
return(list(model = model, rss = rss))
}
# Initialize lists to store RSS and models
rss_values <- numeric(10)
models <- list()
# Fit polynomial regressions for degrees 1 to 10
for (degree in 1:10) {
fit <- fit_polynomial(degree)
rss_values[degree] <- fit$rss
models[[degree]] <- fit$model
}
# Plot the polynomial fits and report RSS
par(mfrow = c(2, 5), mar = c(4, 4, 2, 2))
for (degree in 1:10) {
plot(Boston$dis, Boston$nox, xlab = "dis", ylab = "nox", main = paste("Degree", degree))
lines(sort(Boston$dis), predict(models[[degree]], data.frame(dis = sort(Boston$dis))), col = "red", lwd = 2)
text(4, max(Boston$nox), paste("RSS:", round(rss_values[degree], digits = 2)), pos = 4, col = "blue")
}
(c) Perform cross-validation or another approach to select the opti- mal degree for the polynomial, and explain your results.
library(boot)
# Function to calculate mean squared error (MSE)
calculate_mse <- function(actual, predicted) {
return(mean((actual - predicted)^2))
}
# Define k-fold cross-validation function
k_fold_cv <- function(data, k, max_degree) {
# Initialize vector to store mean squared errors
mse_values <- numeric(max_degree)
# Perform k-fold cross-validation for each degree
for (degree in 1:max_degree) {
mse <- numeric(k)
for (i in 1:k) {
# Split data into training and validation sets
validation_indices <- ((i - 1) * floor(nrow(data) / k) + 1):(i * floor(nrow(data) / k))
validation_set <- data[validation_indices, ]
training_set <- data[-validation_indices, ]
# Fit polynomial regression model on training set
model <- lm(nox ~ poly(dis, degree), data = training_set)
# Predict on validation set
predicted <- predict(model, newdata = validation_set)
# Calculate mean squared error
mse[i] <- calculate_mse(validation_set$nox, predicted)
}
# Calculate average MSE across folds for this degree
mse_values[degree] <- mean(mse)
}
return(mse_values)
}
# Set number of folds for cross-validation
k <- 10
# Perform k-fold cross-validation for degrees 1 to 10
cv_errors <- k_fold_cv(Boston, k, max_degree = 10)
# Find the optimal degree that minimizes cross-validation error
optimal_degree <- which.min(cv_errors)
# Plot cross-validation errors
plot(1:10, cv_errors, type = "b", xlab = "Degree", ylab = "Cross-validation Error", main = "Cross-validation Error vs. Degree")
# Highlight the optimal degree
points(optimal_degree, cv_errors[optimal_degree], col = "red", pch = 19)
text(optimal_degree, cv_errors[optimal_degree], paste("Degree:", optimal_degree), pos = 4, col = "red")
uadratic Relationship: The quadratic model suggests that as the distance to employment centers (dis) increases, the nitrogen oxides concentration (nox) initially decreases, reaches a minimum point, and then starts increasing again. This indicates a non-linear relationship between the distance to employment centers and nitrogen oxides concentration.
Optimal Complexity: The selection of degree 2 implies that adding further complexity to the model (e.g., using higher degrees) does not significantly improve predictive performance. A quadratic model strikes a balance between simplicity and flexibility, capturing the curvature in the relationship without overfitting to noise in the data.
Interpretability: A quadratic polynomial is relatively easy to interpret compared to higher-order polynomials. It allows for clear visualization and understanding of how changes in distance to employment centers relate to changes in nitrogen oxides concentration.
Practical Implications: The finding that the nitrogen oxides concentration initially decreases with distance to employment centers but then increases suggests potential policy implications. For example, it could indicate that there might be a distance threshold beyond which efforts to reduce emissions from employment centers could have diminishing returns or even unintended consequences.
Overall, the selection of a quadratic polynomial suggests a nuanced relationship between distance to employment centers and nitrogen oxides concentration, providing insights that can inform decision-making in urban planning, environmental management, and public health.
(d) Use the bs() function to fit a regression spline to predict nox using dis. Report the output for the fit using four degrees of freedom. How did you choose the knots? Plot the resulting fit.
library(splines)
# Fit regression spline with four degrees of freedom
model_spline <- lm(nox ~ bs(dis, df = 4), data = Boston)
# Display regression output
summary(model_spline)##
## Call:
## lm(formula = nox ~ bs(dis, df = 4), data = Boston)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.124622 -0.039259 -0.008514 0.020850 0.193891
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.73447 0.01460 50.306 < 2e-16 ***
## bs(dis, df = 4)1 -0.05810 0.02186 -2.658 0.00812 **
## bs(dis, df = 4)2 -0.46356 0.02366 -19.596 < 2e-16 ***
## bs(dis, df = 4)3 -0.19979 0.04311 -4.634 4.58e-06 ***
## bs(dis, df = 4)4 -0.38881 0.04551 -8.544 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06195 on 501 degrees of freedom
## Multiple R-squared: 0.7164, Adjusted R-squared: 0.7142
## F-statistic: 316.5 on 4 and 501 DF, p-value: < 2.2e-16# Plot the resulting fit
plot(Boston$dis, Boston$nox, xlab = "dis", ylab = "nox", main = "Regression Spline Fit")
lines(sort(Boston$dis), predict(model_spline, data.frame(dis = sort(Boston$dis))), col = "red", lwd = 2)
The knots for the spline are chosen automatically by the bs() function based on the distribution of the predictor variable dis. This method ensures that the spline captures the non-linear relationship between dis and nox effectively while avoiding overfitting.
(e) Now fit a regression spline for a range of degrees of freedom, and plot the resulting fits and report the resulting RSS. Describe the results obtained.
library(splines)
# Initialize vectors to store RSS and models
rss_values <- numeric(10)
models <- list()
# Fit regression spline for degrees of freedom from 3 to 12
for (df in 3:12) {
model <- lm(nox ~ bs(dis, df = df), data = Boston)
rss <- sum(residuals(model)^2)
rss_values[df-2] <- rss
models[[df-2]] <- model
}
# Plot the resulting fits and report RSS
par(mfrow = c(2, 5), mar = c(4, 4, 2, 2))
for (df in 3:12) {
plot(Boston$dis, Boston$nox, xlab = "dis", ylab = "nox", main = paste("DF =", df))
lines(sort(Boston$dis), predict(models[[df-2]], data.frame(dis = sort(Boston$dis))), col = "red", lwd = 2)
text(4, max(Boston$nox), paste("RSS:", round(rss_values[df-2], digits = 2)), pos = 4, col = "blue")
}
Decreasing RSS: As the degrees of freedom increase (i.e., the flexibility of the spline increases), the models tend to have lower RSS values. This indicates that higher degrees of freedom allow the spline to better capture the non-linear relationship between the predictor variable dis and the response variable nox, resulting in a better fit to the data.
Overfitting: At very high degrees of freedom, there might be a tendency for the spline to overfit the data, capturing noise and variability that are not representative of the underlying relationship between dis and nox. This can be inferred if the RSS starts to increase again after reaching a minimum, indicating that the model is becoming overly complex.
Balancing Complexity and Fit: There is typically a trade-off between model complexity (determined by the degrees of freedom) and goodness of fit (indicated by the RSS). The goal is to select a model that adequately captures the underlying non-linear relationship while avoiding excessive complexity. This balance can often be achieved by choosing a moderate number of degrees of freedom that provides a good fit without overfitting.
Visual Inspection of Fits: By examining the plots of the spline fits for different degrees of freedom, it’s possible to visually assess how well the spline captures the curvature and variability in the data. Models with appropriate degrees of freedom should closely follow the data pattern without excessive oscillations or deviations.
Overall, the results obtained from fitting regression splines with varying degrees of freedom help in understanding the relationship between dis and nox and in selecting an appropriate level of flexibility for the spline model that strikes a balance between capturing the underlying non-linear relationship and avoiding overfitting.
(f) Perform cross-validation or another approach in order to select the best degrees of freedom for a regression spline on this data. Describe your results.
library(splines)
# Remove missing values from the data
Boston_clean <- na.omit(Boston)
# Initialize vectors to store cross-validated errors
cv_errors <- numeric(10)
# Perform k-fold cross-validation for degrees of freedom from 3 to 12
for (df in 3:12) {
cv <- cv.glm(Boston_clean, glm(nox ~ bs(dis, df = df), data = Boston_clean), K = 10)
cv_errors[df-2] <- mean(cv$delta^2)
}
# Find the optimal degrees of freedom with the lowest cross-validated error
optimal_df <- which.min(cv_errors) + 2 # Adding 2 to account for starting from degree 3
# Plot cross-validated errors
plot(3:12, cv_errors, type = "b", xlab = "Degrees of Freedom", ylab = "Cross-validated Error",
main = "Cross-validated Error vs. Degrees of Freedom")
points(optimal_df, cv_errors[optimal_df - 2], col = "red", pch = 19)
Based on the cross-validation results, the optimal degrees of freedom for the regression spline model on the Boston dataset were determined to be 5. This indicates that a spline with 5 degrees of freedom provides the best balance between model complexity and predictive performance.
Here’s a description of the results:
Optimal Degrees of Freedom: The selected degrees of freedom represent the flexibility of the spline model in capturing the non-linear relationship between the predictor variable (dis) and the response variable (nox). In this case, a spline with 5 degrees of freedom is deemed to be the most suitable for accurately representing the underlying relationship in the data.
Cross-validated Error: The plot of cross-validated errors against degrees of freedom shows how the model’s performance varies with different degrees of freedom. The cross-validated error tends to decrease initially as the degrees of freedom increase, indicating improved model flexibility. However, after reaching the optimal point (in this case, at 5 degrees of freedom), further increasing the degrees of freedom may lead to overfitting and higher cross-validated errors.
Model Interpretability and Complexity: While increasing the degrees of freedom beyond 5 may result in better fit to the training data, it could lead to a more complex model that may not generalize well to unseen data. Therefore, selecting the optimal degrees of freedom is crucial for achieving a balance between model interpretability and predictive accuracy.
In summary, the selection of 5 degrees of freedom for the regression spline model indicates a model that captures the non-linear relationship between dis and nox effectively while avoiding excessive complexity. This model can be used for prediction and interpretation with confidence in its predictive performance.
11) In Section 7.7, it was mentioned that GAMs are generally fit using a backfitting approach. The idea behind backfitting is actually quite simple. We will now explore backfitting in the context of multiple linear regression.Suppose that we would like to perform multiple linear regression, but we do not have software to do so. Instead, we only have software to perform simple linear regression. Therefore, we take the following iterative approach: we repeatedly hold all but one coefficient esti- mate fixed at its current value, and update only that coefficient estimate using a simple linear regression. The process is continued un- til convergence—that is, until the coefficient estimates stop changing.We now try this out on a toy example.
(a) Generate a response Y and two predictors X1 and X2, with n = 100.
# Set seed for reproducibility
set.seed(123)
# Number of observations
n <- 100
# Generate predictor variables X1 and X2
X1 <- rnorm(n)
X2 <- rnorm(n)
# Generate response variable Y
Y <- 2*X1 + 3*X2 + rnorm(n, mean = 0, sd = 0.5)(b) (b) Initialize βˆ1 to take on a value of your choice. It does not matter what value you choose.
# Initialize beta hat for coefficient beta1
beta_hat_1 <- 0.5(c) Keeping βˆ1 fixed, fit the model Y − βˆ 1 X 1 = β 0 + β 2 X 2 + ε .
# Calculate 'a' as Y - beta1*X1
a <- Y - beta_hat_1 * X1
# Fit simple linear regression with 'a' as response and X2 as predictor
lm_result <- lm(a ~ X2)
# Extract the coefficient beta2
beta2 <- coef(lm_result)[2](d) Keeping βˆ2 fixed, fit the model Y − βˆ 2 X 2 = β 0 + β 1 X 1 + ε .
# Calculate 'a' as Y - beta2*X2
a <- Y - beta2 * X2
# Fit simple linear regression with 'a' as response and X1 as predictor
lm_result <- lm(a ~ X1)
# Extract the coefficient beta1
beta1 <- coef(lm_result)[2](e) Write a for loop to repeat (c) and (d) 1,000 times. Report the estimates of βˆ0, βˆ1, and βˆ2 at each iteration of the for loop. Create a plot in which each of these values is displayed, with βˆ0, βˆ1, and βˆ2 each shown in a different color.
# Initialize vectors to store coefficient estimates
beta0_estimates <- numeric(1000)
beta1_estimates <- numeric(1000)
beta2_estimates <- numeric(1000)
# Initial value for beta_hat_1
beta_hat_1 <- 0.5
# Loop for 1000 iterations
for (i in 1:1000) {
# Fit model with beta_hat_1 fixed
a <- Y - beta_hat_1 * X1
lm_result <- lm(a ~ X2)
beta2 <- coef(lm_result)[2]
# Fit model with beta2 fixed
a <- Y - beta2 * X2
lm_result <- lm(a ~ X1)
beta1 <- coef(lm_result)[2]
# Store coefficient estimates
beta0_estimates[i] <- coef(lm_result)[1]
beta1_estimates[i] <- beta1
beta2_estimates[i] <- beta2
# Print estimates at each iteration
cat("Iteration:", i, "\tBeta0:", coef(lm_result)[1], "\tBeta1:", beta1, "\tBeta2:", beta2, "\n")
}## Iteration: 1 Beta0: 0.06064258 Beta1: 1.929897 Beta2: 2.944883
## Iteration: 2 Beta0: 0.06064258 Beta1: 1.929897 Beta2: 2.944883
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## Iteration: 1000 Beta0: 0.06064258 Beta1: 1.929897 Beta2: 2.944883# Plotting beta0
plot(1:1000, beta0_estimates, type = "l", col = "red", xlab = "Iteration", ylab = "Coefficient Estimate", main = "Coefficient Estimates Over Iterations - Beta0")
# Plotting beta1
plot(1:1000, beta1_estimates, type = "l", col = "blue", xlab = "Iteration", ylab = "Coefficient Estimate", main = "Coefficient Estimates Over Iterations - Beta1")
# Plotting beta2
plot(1:1000, beta2_estimates, type = "l", col = "green", xlab = "Iteration", ylab = "Coefficient Estimate", main = "Coefficient Estimates Over Iterations - Beta2")
(f) Compare your answer in (e) to the results of simply performing multiple linear regression to predict Y using X1 and X2. Use the abline() function to overlay those multiple linear regression coefficient estimates on the plot obtained in (e).
# Fit multiple linear regression model
lm_result_multiple <- lm(Y ~ X1 + X2)
# Extract coefficient estimates from the multiple linear regression model
beta0_multiple <- coef(lm_result_multiple)[1]
beta1_multiple <- coef(lm_result_multiple)[2]
beta2_multiple <- coef(lm_result_multiple)[3]
# Plotting beta0 with overlay of multiple linear regression coefficient estimate
plot(1:1000, beta0_estimates, type = "l", col = "red", xlab = "Iteration", ylab = "Coefficient Estimate", main = "Coefficient Estimates Over Iterations - Beta0")
abline(h = beta0_multiple, col = "black")
# Plotting beta1 with overlay of multiple linear regression coefficient estimate
plot(1:1000, beta1_estimates, type = "l", col = "blue", xlab = "Iteration", ylab = "Coefficient Estimate", main = "Coefficient Estimates Over Iterations - Beta1")
abline(h = beta1_multiple, col = "black")
# Plotting beta2 with overlay of multiple linear regression coefficient estimate
plot(1:1000, beta2_estimates, type = "l", col = "green", xlab = "Iteration", ylab = "Coefficient Estimate", main = "Coefficient Estimates Over Iterations - Beta2")
abline(h = beta2_multiple, col = "black")
(g) On this data set, how many backfitting iterations were required in order to obtain a “good” approximation to the multiple re- gression coefficient estimates?
# Define a threshold for the change in coefficient estimates
threshold <- 0.001
# Initialize vectors to store coefficient estimates
beta0_estimates <- numeric(1000)
beta1_estimates <- numeric(1000)
beta2_estimates <- numeric(1000)
# Initial value for beta_hat_1
beta_hat_1 <- 0.5
# Loop for up to 1000 iterations
for (i in 1:1000) {
# Fit model with beta_hat_1 fixed
a <- Y - beta_hat_1 * X1
lm_result <- lm(a ~ X2)
beta2 <- coef(lm_result)[2]
# Fit model with beta2 fixed
a <- Y - beta2 * X2
lm_result <- lm(a ~ X1)
beta1 <- coef(lm_result)[2]
# Store coefficient estimates
beta0_estimates[i] <- coef(lm_result)[1]
beta1_estimates[i] <- beta1
beta2_estimates[i] <- beta2
# Check convergence
if (i > 1) {
# Calculate change in coefficient estimates
delta_beta0 <- abs(beta0_estimates[i] - beta0_estimates[i - 1])
delta_beta1 <- abs(beta1_estimates[i] - beta1_estimates[i - 1])
delta_beta2 <- abs(beta2_estimates[i] - beta2_estimates[i - 1])
# Check if changes are below threshold
if (delta_beta0 < threshold && delta_beta1 < threshold && delta_beta2 < threshold) {
cat("Convergence achieved after", i, "iterations.\n")
break
}
}
}## Convergence achieved after 2 iterations.# If convergence is not achieved after 1000 iterations
if (i == 1000) {
cat("Convergence not achieved after 1000 iterations.\n")
}