Simple linear regression models the relationship between a dependent variable \(Y\) and an independent variable \(X\):

\[ Y = \beta_0 + \beta_1 X + \varepsilon \]

• \(\beta_0\): intercept
  
• \(\beta_1\): slope coefficient
  
• \(\varepsilon\): error term

• Predicting sales
  
• Forecasting grades
  
• Estimating house prices
  
• Medical dosage

set.seed(123)
n <- 100
x <- rnorm(n, mean = 50, sd = 10)
y <- 5 + 0.7 * x + rnorm(n, mean = 0, sd = 5)
data <- data.frame(x = x, y = y)
head(data)

##          x        y
## 1 44.39524 32.52464
## 2 47.69823 39.67318
## 3 65.58708 49.67750
## 4 50.70508 38.75585
## 5 51.29288 36.14692
## 6 67.15065 51.78032

ggplot(data, aes(x = x, y = y)) +
  geom_point(color = "blue") +
  geom_smooth(method = "lm", se = TRUE, color = "darkred") +
  labs(title = "Scatterplot with Regression Line", x = "X", y = "Y")

## `geom_smooth()` using formula = 'y ~ x'

model <- lm(y ~ x, data = data)
summary(model)

## 
## Call:
## lm(formula = y ~ x, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -9.5367 -3.4175 -0.4375  2.9032 16.4520 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.79778    2.76324   2.098   0.0385 *  
## x            0.67376    0.05344  12.608   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 4.854 on 98 degrees of freedom
## Multiple R-squared:  0.6186, Adjusted R-squared:  0.6147 
## F-statistic:   159 on 1 and 98 DF,  p-value: < 2.2e-16

data$residuals <- residuals(model)

ggplot(data, aes(x = x, y = residuals)) +
  geom_point(color = "purple") +
  geom_hline(yintercept = 0, linetype = "dashed", color = "gray") +
  labs(title = "Residual Plot", x = "X", y = "Residuals")

z <- rnorm(n, mean = 30, sd = 5)
data$z <- z
data$y3d <- 5 + 0.6 * x + 0.4 * z + rnorm(n)

plot_ly(data, x = ~x, y = ~z, z = ~y3d, type = "scatter3d", mode = "markers",
        marker = list(color = ~y3d, colorscale = "Viridis", size = 4)) %>%
  layout(title = "3D Plot: y ~ x + z",
         scene = list(xaxis = list(title = "x"),
                      yaxis = list(title = "z"),
                      zaxis = list(title = "y")))

\[ \hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x \]

\[ \text{RSE} = \sqrt{\frac{1}{n-2} \sum (y_i - \hat{y}_i)^2} \]

library(ggplot2)
set.seed(123)
x <- rnorm(100, mean = 50, sd = 10)
y <- 5 + 0.7 * x + rnorm(100, mean = 0, sd = 5)
data <- data.frame(x, y)
model <- lm(y ~ x, data = data)
ggplot(data, aes(x = x, y = y)) +
  geom_point() +
  geom_smooth(method = "lm")

• Linear regression overview
  
• R modeling and plots
  
• Plotly 3D integration
  
• Math + residual analysis
  
• Thank you for your attention!