Lecture Notes on Correlation, Regression, and Data Fitting

Overview

Bivariate Data: Plotting data to estimate correlation between variables x and y.
Correlation Coefficient (r or ρ):
- Defined as ( \frac{S_{xy}}{S_x \times S_y} ).
- ( S_{xy} ) is the covariance, given by ( \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{n-1} ).
- Bounded between -1 (perfect negative correlation) and 1 (perfect positive correlation).
- Example: For x = y, r = +1; for x = -y, r = -1.
- If data points are scattered, r is close to 0.

Linear Regression: Estimating y given x or vice versa.
- Objective: Approximate data points by a curve, often linear.
- Equation: ( y = a + bx ).
- Finding ( a ) and ( b ):
  - ( \bar{y} = a + b\bar{x} )
  - ( b = \frac{S_{xy}}{S_x^2} = \rho \times \frac{S_y}{S_x} )
- Example calculation with data points (x, y): (1, 1), (2, 2), (3, 2), (4, 2), (5, 3).

R-squared (R²): Measure of goodness of fit.
- Formula: ( R^2 = 1 - \frac{\sum e^2}{\sum (y - \bar{y})^2} ).
- R² value indicates how well the line represents the data.
- Perfect fit: R² = 1; Poor fit: R² close to 0.

Interpolation: Estimating a value within the range of known data points.
- Example: Estimating protein concentration using a standard curve in biology.
Extrapolation: Estimating a value outside the known data range.
- Example: Measuring ligand-receptor binding strength via extrapolation of force measurements.
- Challenges: Assumptions about curve continuation beyond collected data can lead to errors.

Interpolation used in protein concentration estimation for Western blot analysis.
Extrapolation used for estimating binding strength in molecular interactions.
- Example: Varying ligand concentrations and measuring unbinding forces.

Linear regression is useful for data fitting.
Interpolation is generally reliable; extrapolation can be error-prone if assumptions about data trends are incorrect.
Importance of selecting the appropriate function for curve fitting.