Model growth or decay from time series measurements. Choose two or three parameters for flexibility. See fit quality instantly, then download clean outputs today.
| x | y |
|---|---|
| 0 | 10.10 |
| 1 | 7.42 |
| 2 | 5.50 |
| 3 | 4.07 |
| 4 | 3.05 |
| 5 | 2.24 |
Two-parameter model: y = A · e^(B·x)
Taking the natural logarithm gives ln(y) = ln(A) + B·x, which is solved by linear least squares.
Three-parameter model: y = A · e^(B·x) + C is solved by iterative least squares (damped Gauss–Newton).
Exponential curves appear when change is proportional to the current state. You see them in radioactive decay, RC discharge, thermal relaxation, and intensity loss in time-resolved optics. This calculator summarizes measurements with a compact model plus clear fit-quality statistics for reporting.
The sign of B sets the direction: B > 0 for growth and B < 0 for decay. For decay, compute \tau = -1/B (time constant) and t_{1/2} = \ln(2)/|B| (half-life). Both inherit the same unit as x.
For y = A\,e^{Bx}, A is the predicted value at x = 0. It often represents an initial intensity, voltage, concentration, or count rate. Even if measurements start later, A remains a useful back-extrapolated amplitude.
Many datasets include background signals: dark current, ambient light, offsets, or drift. The model y = A\,e^{Bx} + C treats that baseline as C. If late-time residuals lean positive or negative, adding C can reduce bias in A and B.
Fits are most stable when the data span a meaningful change in y (good dynamic range). Keep x and y units consistent, and avoid mixing seconds and milliseconds in one entry. Sampling both early and late regions usually improves the estimate of B.
SSE sums squared residuals, so large outliers matter more. RMSE is an average error scale in the same unit as y, making it easy to compare against instrument noise. R² reports explained variance, but it can be misleading for narrow-range data or strong offsets—check residuals too.
A strong fit shows residuals that scatter randomly around zero. A curved pattern can indicate multiple time constants, saturation, or a different governing process. Since SSE is sensitive to outliers, review the fitted-versus-measured table and consider re-measuring points affected by clipping, drift, or timing errors.
For reproducible reporting, record the model form, parameter values, units, number of points, and at least one error metric (RMSE or SSE). If you publish derived values like \tau or half-life, show the equation used. Exporting CSV/PDF preserves the fitted table for lab notes and reports. Keep raw data alongside outputs for auditability.
The two-parameter model assumes the curve approaches zero. The three-parameter model adds a constant background term C, which is useful when measurements settle to a nonzero baseline.
It uses a logarithm transform (\ln(y)) to solve the regression. Logarithms are only defined for positive values, so any zero or negative y value breaks that method.
For a decaying process, compute \tau = -1/B. If x is in seconds, \tau is in seconds. For growth, 1/B indicates an e-folding time.
A negative B indicates exponential decay: the quantity decreases as x increases. This is common in discharge, relaxation, attenuation, and radioactive decay experiments.
Curved residuals often mean the physics is not a single exponential. Consider a multi-exponential model, a different functional form, or limiting the fit range to the region where one dominant process applies.
Not always. R² can look high even with systematic bias if the data span a large range. Always inspect residuals and compare RMSE to your instrument noise or expected uncertainty.
Enter one x, y pair per line using commas or spaces. You can also use semicolons between pairs. Lines starting with # or // are ignored.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.