Correlation and Regression Analysis

library(dplyr)
library(ggplot2)
library(corrplot)

# Pearson correlation (linear relationship)
x <- c(30, 45, 52, 60, 72, 81, 90, 102, 115, 125)  # Advertising spend ($K)
y <- c(120, 175, 190, 220, 265, 300, 330, 390, 445, 490)  # Sales ($K)

r <- cor(x, y)
cat(sprintf("Pearson r = %.3f\n", r))  # 0.998 (very strong positive)
# Interpretation: r = 0.998 → nearly perfect linear relationship

# Correlation test (is r significantly different from 0?)
cor.test(x, y)
# cor = 0.998, p-value < 0.001 → significant correlation

# Spearman (rank-based, non-parametric)
cor(x, y, method="spearman")
cor.test(x, y, method="spearman")

# Kendall's tau
cor(x, y, method="kendall")

# Correlation matrix
data <- mtcars[, c("mpg","hp","wt","disp","qsec")]
cor_matrix <- cor(data)
print(round(cor_matrix, 2))

# Visualize correlation matrix
corrplot(cor_matrix, method="color", type="upper",
          addCoef.col="black", number.cex=0.8,
          col=colorRampPalette(c("#F44336","white","#1565C0"))(200))

# SLR: y = β₀ + β₁x + ε
# H₀: β₁ = 0 (no linear relationship)
model <- lm(Sales ~ Advertising, data=data.frame(Advertising=x, Sales=y))
summary(model)

# Extract coefficients
coef(model)       # β₀ (intercept) and β₁ (slope)
confint(model)    # 95% confidence intervals for coefficients

# Model diagnostics
cat("R-squared:  ", summary(model)$r.squared, "\n")  # % variance explained
cat("Adj R²:     ", summary(model)$adj.r.squared, "\n")
cat("RMSE:       ", sqrt(mean(residuals(model)^2)), "\n")
cat("F-statistic:", summary(model)$fstatistic[1], "\n")

# Predictions
new_data <- data.frame(Advertising=c(50, 75, 100))
predict(model, new_data)               # Point predictions
predict(model, new_data, interval="confidence")  # 95% CI
predict(model, new_data, interval="prediction")  # 95% PI (wider)

# Visualization
ggplot(data.frame(x, y), aes(x, y)) +
  geom_point(color="#1565C0", size=3) +
  geom_smooth(method="lm", color="red", se=TRUE) +
  labs(title=sprintf("Linear Regression (R² = %.3f)", summary(model)$r.squared),
       x="Advertising Spend ($K)", y="Sales ($K)") +
  theme_minimal()

# ─── HOUSE PRICE PREDICTION MODEL ────────────────────
set.seed(42)
n <- 200
houses <- data.frame(
  size_sqft  = round(runif(n, 800, 3500)),
  bedrooms   = sample(2:6, n, replace=TRUE),
  bathrooms  = sample(1:4, n, replace=TRUE),
  age_years  = sample(1:50, n, replace=TRUE),
  garage     = sample(0:2, n, replace=TRUE),
  location   = sample(c("Urban","Suburban","Rural"), n, replace=TRUE)
)
# Generate realistic prices
houses$price <- round(
  150000 +
  houses$size_sqft * 120 +
  houses$bedrooms * 8000 +
  houses$bathrooms * 12000 -
  houses$age_years * 1500 +
  houses$garage * 15000 +
  ifelse(houses$location=="Urban", 50000, ifelse(houses$location=="Suburban", 20000, 0)) +
  rnorm(n, 0, 15000), -3)

# Train/test split (80/20)
set.seed(42)
train_idx <- sample(1:n, floor(0.8*n))
train <- houses[train_idx, ]
test  <- houses[-train_idx, ]

# Fit MLR model
model_mlr <- lm(price ~ size_sqft + bedrooms + bathrooms + age_years + garage + location,
                 data=train)
summary(model_mlr)

# Evaluate on test set
predictions <- predict(model_mlr, newdata=test)
actuals     <- test$price
rmse <- sqrt(mean((predictions - actuals)^2))
mae  <- mean(abs(predictions - actuals))
r2   <- cor(predictions, actuals)^2

cat("\n=== MODEL EVALUATION ===\n")
cat(sprintf("Train R²: %.4f\n", summary(model_mlr)$r.squared))
cat(sprintf("Test R²:  %.4f\n", r2))
cat(sprintf("RMSE:     $%.0f\n", rmse))
cat(sprintf("MAE:      $%.0f\n", mae))

# Feature importance (standardized coefficients)
cat("\nFeature Importance (coefficient t-values):\n")
coef_summary <- summary(model_mlr)$coefficients
coef_summary <- coef_summary[order(abs(coef_summary[,"t value"]), decreasing=TRUE), ]
print(round(coef_summary, 3))

# Diagnostic plots
par(mfrow=c(2,2))
plot(model_mlr)
par(mfrow=c(1,1))