library(ggplot2) library(plotly) library(dplyr) set.seed(1) x <- seq(1,10,length=50) y <- 3 + 2*x + rnorm(50,0,2) data <- data.frame(x,y)
Simple linear regression models the relationship between one predictor variable and one response variable.
Examples include:
Regression helps us predict outcomes and understand relationships in data.
The model is
\[ Y = \beta_0 + \beta_1 X + \epsilon \]
Where
The regression line is estimated using the least squares method.
\[ \sum_{i=1}^{n}(y_i - \hat{y}_i)^2 \]
where
\[ \hat{y} = b_0 + b_1x \]
This minimizes the total squared prediction error.
head(data)
## x y ## 1 1.000000 3.747092 ## 2 1.183673 5.734634 ## 3 1.367347 4.063437 ## 4 1.551020 9.292602 ## 5 1.734694 7.128403 ## 6 1.918367 5.195798
This dataset approximately follows
\[ y = 3 + 2x + error \]
ggplot(data, aes(x=x, y=y)) +
geom_point(size=3) +
theme_minimal() +
labs(title="Scatter Plot of Data",
x="Predictor (X)",
y="Response (Y)")
ggplot(data, aes(x=x, y=y)) + geom_point(size=3) + geom_smooth(method="lm", se=FALSE) + theme_minimal() + labs(title="Linear Regression Fit")
## `geom_smooth()` using formula = 'y ~ x'
z <- 3 + 2*x + rnorm(50) plot_ly( x = ~x, y = ~y, z = ~z, type = "scatter3d", mode = "markers" )
Linear regression is widely used in:
It helps us understand relationships and make predictions.
Simple linear regression models relationships between variables.
The model
\[ Y = \beta_0 + \beta_1 X + \epsilon \]
is one of the most important tools in statistics.