Exponential Regression Calculator
Fit an exponential model of the form y = a · e^(b·x) to your data. Paste your X and Y values, and get the best-fit curve, a and b, and the fit quality.
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Results
Enter your data and press Calculate to see the fitted curve and metrics.
Blue dots: data. Purple line: fitted curve.
Exponential Regression: A Friendly Guide
Exponential regression helps you model data that grows or decays at a rate proportional to its current value. It fits a curve of the form y = a · e^(b·x), where:
- a is the starting value (y when x = 0)
- b controls the growth (b > 0) or decay (b < 0) rate
How it works (in simple terms)
We take the natural log of Y and apply ordinary linear regression to ln(Y) = ln(a) + b·X. From that line, we get:
- intercept = ln(a) → a = e^(intercept)
- slope = b
When to use it
- Population or follower growth
- Compound interest or sales growth
- Radioactive decay or cooling processes
A quick example
Suppose X = 0,1,2,3,4 and Y ≈ 2, 2.7, 3.6, 4.9, 6.6. Fitting y = a · e^(b·x) might give something like a ≈ 2.0 and b ≈ 0.3, so every step in X multiplies Y by around e^0.3 ≈ 1.35.
Tips
- All Y values must be positive (we need ln(Y))
- Outliers can bend the curve—double-check unusual points
- R² is computed on the log scale for this method
Want to compare?
You can check the original reference calculator here: Omni Exponential Regression. This app follows the same idea with a simple, mobile-friendly design.
Deep Dive: The Math Behind Exponential Regression
This article walks through the intuition and the math so you can trust and explain your results. We keep it friendly yet precise.
1) Model and Linearization
We assume a curve y = a · e^(b·x). Taking natural logs on both sides gives ln(y) = ln(a) + b·x, which is a straight line with intercept ln(a) and slope b. We then run ordinary least squares on (x, ln(y)).
2) Step-by-step (small example)
- Ensure all y are positive (required for the log).
- Compute Y' = ln(y) for every point.
- Fit a line Y' = β₀ + β₁·x using least squares.
- Translate back: a = e^{β₀} and b = β₁.
- Prediction: ŷ = a · e^{b·x}.
3) R² on the log scale
The goodness-of-fit (R²) shown here is computed on the log-transformed values. It answers “how well does the line fit ln(y)?”—a standard approach for this method.
4) When to prefer power regression
If the relationship is closer to y = a · x^b, use the power model instead. It linearizes as ln(y) = ln(a) + b·ln(x), which requires both x and y to be positive.
5) Practical tips
- Check scatter plots before choosing the model; exponential curves rise/decay multiplicatively.
- Beware of outliers—single extreme points can skew the fit on the log scale.
- Avoid extrapolating too far; exponential curves can explode or vanish quickly.
- Report the model with units and context (e.g., growth per day, per month).
6) FAQs
Why must Y be positive?
Because the natural logarithm is defined only for positive values, and the method relies on ln(y).
Can I fit decreasing trends?
Yes. If the data decays, you’ll typically get a negative b. The curve becomes y = a · e^{b·x} with b < 0.
Is R² directly comparable to linear models on raw Y?
Not exactly—here R² is on ln(y). It’s best compared with models fit using the same transformation.