L12

Sep 20, 2024

Lecture Notes on Correlation and Regression

Recap of Last Lecture

  • Discussed correlation and regression for bivariate data
  • Focus on finding trends and interpolating/extrapolating to determine unknown values

Introduction to Linear Regression

  • Fit a line to data points: Equation of a line is $y = a + bx$
  • Two unknowns: $a$ (intercept), $b$ (slope)
  • With more than two data points: infinite solutions for $a$ and $b$

Minimizing Deviation

  • Minimize sum of squared deviations: $E = \sum (y_i - (a + bx_i))^2$
  • Avoids cancellation of positive and negative errors

Calculating Derivatives

  • For functions of a single variable: set $f'(x) = 0$
  • For functions of multiple variables: set $\frac{dg}{dx} = 0$ and $\frac{dg}{dy} = 0$

Deriving Equations for $a$ and $b$

  • Use derivatives to find:
    • $\frac{dE}{da} = 0$
    • $\frac{dE}{db} = 0$
  • Derive two main equations:
    • $Y_{bar} = a + bX_{bar}$
    • $\sum x_iy_i = naX_{bar} + b\sum x_i^2$

Solving for $b$

  • Eliminate $a$ by substitution
  • $b$ is related to covariance: $b = \frac{S_{xy}}{S_{x^2}}$
  • Relates to correlation coefficient $\rho$

Solving for $a$

  • Once $b$ is known: $a = Y_{bar} - bX_{bar}$

Example Calculation

  • Given $x$ and $y$ values, calculate $a$ and $b$
  • Use derived formulas to find equations of regression

Error Analysis

  • Error is significant at higher $x$ values
  • Normalize errors to improve model fitting

Polynomial and Non-linear Regression

  • Linear regression isn't always suitable
  • Use polynomials for non-linear trends
  • Choose polynomials based on how rapidly $y$ increases with $x$

Application in Robotics

  • Fit trajectory curves for robots using polynomial functions
  • Given conditions: position and velocity at different times

Conclusion

  • Discussed regression and error calculations
  • Highlighted normalization of errors
  • Emphasized the importance of fitting data appropriately

Note: These notes cover the main topics discussed in the lecture on regression, including linear regression, error minimization, solving for parameters, error analysis, and alternative regression methods.