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Understanding Non-Linear Regression: Exponential and Power Relationships

Explore the distinction between exponential and power relationships in regression analysis. Discover how to distinguish linear, exponential, and power relationships using correlation coefficient, log transformations, and log-linear and log-log graphs. Learn key insights for identifying these relationships visually and utilizing semi-log and log-log plots for analysis.

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Understanding Non-Linear Regression: Exponential and Power Relationships

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  1. Non-linear regression Exponential relationships and power relationships

  2. Does this look linear?

  3. Regression line and correlation coefficient r=0.799

  4. Natural log of y If x and ln(y) are linear, then x and y have an exponential relationship -> y=a*bx

  5. Log-linear graph (or semi-log) We scaled the y-axis (logarithmic scale). On the previous slide scaled the values of y (the result is the same). --here we used log10 scale, but it doesn’t matter!

  6. Proof on chalkboard

  7. Does this look linear? Or Exponential?

  8. Does this look linear? r=0.963

  9. Does this look exponential? Log-linear graph Log-linear (log10) graph is NOT linear. So x and y do NOT have an exponential relationship!!

  10. ln x by ln y If ln(x) and ln(y) are linear, then x and y have an power relationship. y=a*x^b

  11. Log-log graph Here we scaled both the x-axis and the y-axis (both on logarithmic scales). In the previous slide we scaled the values of x and y (result is the same).

  12. Proof on chalkboard

  13. Conclusions • Humans are good at identifying linear relationships visually. • Distinguishing between exponential and power (or other) is very difficult just by looking at data. • So if in doubt, try the semi-log plot and log-log plot to see if either of them are linear.

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