# I am analyzing the RSS Contour Plot for my auto insurance dataset to better understand
# how the intercept (β0) and slope (β1) values influence the residual sum of squares (RSS).
# My goal is to find the optimal values that minimize RSS and explain their significance.
# I interpret the RSS Contour Plot by focusing on the key elements. Each contour line in the plot
# represents constant RSS values, and I see the colors transitioning from green to yellow and pink.
# This gradient tells me that areas closer to green have the lowest RSS, which means a better model fit.
# As the colors transition outward, RSS increases, indicating a worse fit between the model and data.
# I see a red dot at the center of the plot, and this represents the optimal combination of β0 and β1.
# In my case, the optimal intercept (β0) is around 270, and the optimal slope (β1) is approximately -0.15.
# These are the parameter values that minimize RSS and make my regression model the best fit.
# I notice that the intercept (β0 = 270) suggests that if advertising spend is zero,
# the expected number of claims would be around 270. This becomes the baseline for claims
# when no advertising occurs. The slope (β1 = -0.15) tells me that for every additional $1,000
# spent on advertising, the number of claims decreases by 0.15 on average. This indicates
# a negative correlation between advertising spend and claims, meaning advertising might
# reduce claims slightly.
# As I study the gradient, I see how sensitive RSS is to changes in β0 and β1. The green area shows me
# that small deviations in parameter values don’t drastically increase RSS, making my model
# robust around this region. However, moving further into the yellow and pink areas increases RSS,
# showing that larger deviations significantly worsen the model's performance.
# I conclude that the optimal values from the red dot are crucial. The intercept helps me predict
# baseline claims when advertising is zero, and the slope gives me insight into the effect
# of advertising on reducing claims. Although the impact of advertising is small,
# these findings can help me guide decisions about resource allocation for advertising strategies.
# I also consider the limitations of my model. While I assume a linear relationship
# between advertising spend and claims, the real-world relationship might be more complex
# and non-linear. I know this slope (-0.15) doesn’t necessarily mean causation—
# other factors could be influencing the observed relationship.
# In conclusion, I see the RSS Contour Plot as a powerful tool that helps me visualize
# the optimization process of my regression model. It confirms that the best-fit parameters
# are β0 = 270 and β1 = -0.15, which minimize RSS. These values tell me how advertising spend
# relates to claims and provide actionable insights, even with modest effect sizes.
# I am confident that this analysis helps me make informed, data-driven decisions
# to improve advertising strategies and understand my data better.
# I loaded the necessary library and my dataset from the specified path
library(readxl)
auto_insurance_data <- read_excel("C:/Users/jacob/Downloads/auto_insurance_data.xlsx")
View(auto_insurance_data)
# I extract the relevant columns for analysis
advertising_spend_i <- auto_insurance_data$Advertising_Spend_1000s
actual_claims_i <- auto_insurance_data$Number_of_Claims
# I calculate the means of advertising spend and claims
avg_advertising_spend <- mean(advertising_spend_i)
avg_claims <- mean(actual_claims_i)
# I derive slope_hat using the covariance formula and the sample mean
slope_hat <- sum((advertising_spend_i - avg_advertising_spend) *
(actual_claims_i - avg_claims)) /
sum((advertising_spend_i - avg_advertising_spend)^2)
# Let yˆi = βˆ0 + βˆ1xi be the prediction for Y based on the ith value of X,
# where x_i represents the advertising spend in $1,000s. This prediction is
# based on my linear regression model parameters.
# I calculate intercept_hat using the relationship between the means of advertising spend and claims
intercept_hat <- avg_claims - slope_hat * avg_advertising_spend
# The least squares approach chooses βˆ0 and βˆ1 to minimize the RSS.
# The slope is defined as:
# βˆ1 = Σ(xi − x¯)(yi − y¯) / Σ(xi − x¯)^2,
# where x¯ and y¯ are the sample means.
# I define the prediction for the number of claims (Y) based on the ith value of advertising expenditure (X)
predicted_claims_i <- intercept_hat + slope_hat * advertising_spend_i
# Then ei = yi − yˆi represents the ith residual—this is the difference between
# the ith observed response value (y_i) and the ith predicted response value
# (yˆi). Residuals quantify the error of my linear model for each observation.
# I calculate the ith residual, which is the difference between the actual claims (actual_claims_i)
# and the predicted claims (predicted_claims_i) using my linear model
residual_i <- actual_claims_i - predicted_claims_i
# I express the residual sum of squares (RSS) as the sum of squared residuals
RSS <- sum(residual_i^2)
# I define the residual sum of squares (RSS) as:
# RSS = e1^2 + e2^2 + ... + en^2,
# or equivalently:
# RSS = (y1 − βˆ0 − βˆ1x1)^2 + (y2 − βˆ0 − βˆ1x2)^2 + ... + (yn − βˆ0 − βˆ1xn)^2.
# This equation calculates the total squared error of my model.
# I visualize the results of my simple linear regression fit for auto insurance data
plot(x = advertising_spend_i, y = actual_claims_i,
main = "Linear Regression Fit: Auto Insurance Advertising",
xlab = "Advertising Spend ($1,000s)",
ylab = "Number of Claims",
col = "darkorange", pch = 16) # Points are orange for visibility
# Figure 1 displays the simple linear regression fit, where βˆ0 is the intercept,
# and βˆ1 is the slope. It shows the relationship between advertising spend and
# the number of claims.
# I add the regression line I derived from my model
abline(a = intercept_hat, b = slope_hat, col = "blue", lwd = 2) # Line is blue for emphasis
# The regression line illustrates the linear approximation between the predictor
# (advertising spend) and the response (number of claims).
# I create a contour plot for the RSS to visualize optimal parameter values
beta_0_grid <- seq(intercept_hat - 20, intercept_hat + 20, length.out = 50)
beta_1_grid <- seq(slope_hat - 0.1, slope_hat + 0.1, length.out = 50)
# I fix the function for RSS calculation to handle grids properly
RSS_function <- function(b0, b1) {
sapply(b0, function(b0_val) {
sapply(b1, function(b1_val) {
sum((actual_claims_i - (b0_val + b1_val * advertising_spend_i))^2)
})
})
}
# I compute RSS values for each combination of βˆ0 and βˆ1
RSS_values <- outer(beta_0_grid, beta_1_grid, Vectorize(function(b0, b1) {
sum((actual_claims_i - (b0 + b1 * advertising_spend_i))^2)
}))
# I plot the contour for RSS
filled.contour(beta_0_grid, beta_1_grid, RSS_values,
xlab = "Intercept (β0)", ylab = "Slope (β1)",
main = "RSS Contour Plot: Auto Insurance Data",
color.palette = terrain.colors) # Adds gradient color to contours
# Figure 2 computes the RSS for a grid of values for β0 and β1. The red dot
# represents the optimal pair of least squares estimates (βˆ0, βˆ1).
# I highlight the optimal parameter values that minimize the RSS with a red dot
points(intercept_hat, slope_hat, col = "red", pch = 19)
# These values of βˆ0 and βˆ1 minimize the residual sum of squares (RSS), indicating
# the best-fit line for the dataset.
# I conclude that for every $1,000 increase in advertising spend,
# approximately 3.5 additional claims are generated,
# based on my fitted model where slope_hat = 3.5 and intercept_hat = 200.