Table of Contents
Exponential Regression
Exponential regression fits a model of the form y = ae^(bx) to data where the dependent variable grows or decays exponentially with the independent variable. It is used for population modeling, radioactive decay, compound interest analysis, and any process with a constant percentage growth rate.
The method works by taking the natural logarithm of both sides (ln(y) = ln(a) + bx), transforming the exponential relationship into a linear one, then applying standard linear regression to find the parameters.
Method
Apply linear regression to (x, ln(y)) pairs to find slope b and intercept ln(a). Then a = e^(intercept).
Model
| Parameter | Meaning |
|---|---|
| a | Y-intercept (value when x=0) |
| b > 0 | Exponential growth rate |
| b < 0 | Exponential decay rate |
| R² | Goodness of fit (0-1) |
Interpretation
The coefficient a represents the initial value. The exponent b represents the continuous growth rate. If b = 0.1, the quantity grows about 10.5% per unit increase in x. Doubling occurs every ln(2)/b units.
Frequently Asked Questions
Why must Y values be positive?
The method requires taking ln(y), which is undefined for zero or negative values. If data contains non-positive values, consider shifting data or using nonlinear least squares instead.
How accurate is linearized regression?
It gives good estimates but is biased because errors in ln(y) space are not equivalent to errors in y space. For precise work, use iterative nonlinear least squares (Levenberg-Marquardt).
When is exponential better than polynomial?
Use exponential when data shows constant percentage growth rather than constant absolute growth. Exponential models extrapolate better for growth/decay phenomena but can produce unrealistically large predictions.