Power Regression Calculator

The power regression model is:

$$y = a \cdot x^b$$

Taking the natural logarithm of both sides:

$$\ln(y) = \ln(a) + b \cdot \ln(x)$$

Defining \(Y' = \ln(y)\) and \(X' = \ln(x)\), this becomes the linear model \(Y' = A + BX'\), where \(A = \ln(a)\) and \(B = b\). Linear regression on the log-log transformed data yields:

$$b = \frac{n\sum \ln(x_i)\ln(y_i) - \sum \ln(x_i)\sum \ln(y_i)}{n\sum [\ln(x_i)]^2 - [\sum \ln(x_i)]^2}$$

$$\ln(a) = \overline{\ln(y)} - b \cdot \overline{\ln(x)} \quad \Rightarrow \quad a = e^{\overline{\ln(y)} - b \cdot \overline{\ln(x)}}$$

The exponent b determines the power law behavior. When b = 1, the relationship is directly proportional (y = ax). When b = 2, y grows as the square of x. When 0 < b < 1, y grows sublinearly (concave curve). When b < 0, y decreases as x increases (inverse relationship). The coefficient a is the Y-value when X = 1, serving as the proportionality constant.

A key property of power laws is scale invariance: multiplying x by any factor k multiplies y by k^b. This means the relative change in Y depends only on the relative change in X, not on the absolute values — a hallmark of fractal and self-similar systems.

Power regression results reveal scaling relationships:

• Coefficient a: The Y-value at X = 1 (since 1^b = 1 for all b). This is the proportionality constant of the power law.
• Exponent b: The power law exponent. b > 1 means superlinear growth (accelerating). 0 < b < 1 means sublinear growth (decelerating). b < 0 means inverse relationship (Y decreases as X increases). b = 2 is a square law, b = 0.5 is a square root law, b = -1 is inverse proportionality, b = -2 is inverse square.
• R²: High R² on the original scale confirms the power law holds. If a log-log plot of your data is approximately linear, the power model is appropriate.

To visually assess whether a power law fits, plot ln(Y) vs ln(X). If the points fall roughly on a straight line, the power model is appropriate — the slope of that line is the exponent b. Power regression is particularly common in allometry (biology), fractal geometry, geophysics, and complex systems research. Both X and Y must be positive; if your data includes zeros, the model is not directly applicable.