Lecture Notes on Bivariate Data and Regression Analysis

Recap of Previous Lecture

• Overview of using R for calculations and descriptive statistics
  • Entering vectors and basic operations
  • Calculating mean, median, standard deviation
• Introduction to bivariate data and correlation

Bivariate Data

• Definition: Involves two variables, e.g., x and y with values (1, 10), (2, 9), etc.
• Visual representation: Plotting data on x-y plot
• Interpretation: Identifying correlations (inverse in this example)

Correlation

• Correlation coefficient (ρ or r)
  • Formula: ( \rho = \frac{S_{xy}}{S_x \times S_y} )
  • ( S_{xy} ): Covariance of x and y
• Ideal values of ρ:
  • Highly positive correlation → ρ ≈ +1
  • Highly negative correlation → ρ ≈ -1
  • Uncorrelated → ρ ≈ 0

Covariance

• Formula: ( S_{xy} = \frac{\sum{(x - \bar{x})(y - \bar{y})}}{n-1} )
• Simplified Calculation: ( S_{xy} = \frac{\sum{xy} - n\bar{x}\bar{y}}{n-1} )

Example Calculation

• Given values of x and y
  • Calculation of ( S_{xy} ) and correlation coefficient
  • Result: Negative correlation

Regression Analysis

• Goal: Predict y given x using a line of best fit
• Method: Least Squares Regression
  • Minimize the sum of squared errors between actual and predicted values

Linear Regression

• Problem Statement:
  • Minimize error: ( \text{Error} = \sum{(y_i - (a + bx_i))^2} )
• Use of Calculus:
  • Minimize function via partial derivatives
  • Equations derived: ( \frac{\partial E}{\partial a} = 0 ), ( \frac{\partial E}{\partial b} = 0 )

Example of Function Minimization

• Use of partial derivatives to minimize sum of squared deviations

Conclusion

• Discussion on correlation and regression analysis
• Introduction to deriving regression equations using calculus
• Preview of next lecture: Further exploration of regression and prediction methods