EXPONENTIAL CURVE FIT

Exponential Regression Calculator

Fit an exponential curve ŷ = a·e^bx to paired X and Y observations using log-space least squares. Every Y value must be positive. The reported RMSE and R² are computed on the original scale so you can compare them directly with other models.

Log-space least squaresPositive Y requiredBrowser-based

YOUR DATA

Enter paired observations

Paste or upload numeric X and Y columns. All Y values must be greater than zero. Use a period as the decimal mark. Up to 5 MB and 5,000 observations are accepted.

Upload a CSV or TSV file

The file is read locally in your browser.

Or paste data

At least two complete pairs with positive Y values are needed.

X variable

Y variable

First row contains column namesTurn this off when the first line is data.

YOUR RESULTS

Exponential fit

Fitted equation—ŷ = a·e^bx

R-squared (R²) on original scale—Proportion of observed variation explained by the back-transformed curve

RMSE on original scale—Typical prediction error in Y units

Parameter a—Back-transformed intercept

Parameter b—Growth or decay rate

Fitting method—Log-space least squares

Observed values and fitted curve

Each point is one input observation.

Observed Fitted

Residuals by observation

Residual = observed Y − predicted Y on the original scale.

Row	Observed Y	Predicted Y	Residual

How to use this calculator

Paste paired X and Y values, or load the worked example.
Choose the X and Y columns. Every Y value must be strictly positive; zero or negative Y values are rejected with a precise message.
Review the fitted equation, curve, original-scale RMSE and R², and row residuals.
Download a JSON summary that records the method and the back-transformation.

Formula and calculation method

The model is ŷ = a·e^bx. Because this is nonlinear in the parameters, the calculator uses log-space least squares: it transforms each observed Y to z = ln(Y), fits the straight line z = α + bx by ordinary least squares, then sets a = e^α. This minimises Σ(ln(observed) − ln(predicted))², not Σ(observed − predicted)².

Displayed RMSE and R² are computed from the back-transformed predictions ŷ = a·e^bx and the original Y values, so they are on the same scale as the observations and can be compared with other models on the same data. The fitting objective itself is squared error in log space.

Worked example: X = [0, 1, 2], Y = [2, 4, 8]. ln(Y) = [ln 2, 2 ln 2, 3 ln 2], so the log-space fit gives α = ln 2 and b = ln 2. Therefore a = e^{ln 2} = 2 and the fitted curve is ŷ = 2·e^{(ln 2)·x} = 2·2^x.

Assumptions and limitations

Every observed Y must be a finite positive number. Zero or negative Y values are rejected because ln(Y) is undefined.
At least two complete paired observations with distinct X values are required.
Back-transformation from log space does not produce an unbiased estimate of the original-scale conditional mean. The calculator reports the exact back-transformed curve and does not apply a bias correction.
R² and RMSE are reported on the original scale for comparability, but the fitting criterion is log-space error. A model with lower log-space error may not have the lowest original-scale RMSE.
Extrapolation far outside the observed X range can grow or shrink without bound; interpret predictions outside the data cautiously.
The current input limit is 5,000 observations and 5 MB per file.

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