Original-scale growth curve paired with a straight log-space auditUpdates with every input
Math & Statistics
Regression Value Table Calculator
Fit an exponential regression y = A exp(kx) to five positive observations and produce a value table with log-space fit, original-scale prediction, multiplicative residual, doubling or half-life, and interpolation rows.
Exponential equation -
Initial scale A -
Continuous rate k -
Factor per x unit -
Log-space R-squared -
Doubling or half-life -
Log residual RMSE -
Prediction at table end -
MULTIPLICATIVE FIT
Original-scale growth curve paired with a straight log-space audit
The upper curve communicates practical scale; the lower panel shows whether logarithms actually follow a line.
PREDICTION VALUE TABLE
Interpolated exponential predictions across the requested x range
Rows include predicted level, log prediction, one-step factor, and whether the x value lies inside observed support.
| Row | x | Predicted y | ln(predicted y) | Factor from prior row | Range status |
|---|
POSITIVE-DATA SETUP
Use exponential regression only for strictly positive outcomes
- Enter y values greater than zero because logarithms of zero or negatives are undefined.
- Keep x units consistent.
- Inspect the log-space panel for approximate linearity.
- Choose table bounds separately from observed bounds and note extrapolation.
- Interpret residuals as ratios rather than constant-unit differences.
MULTIPLICATIVE ERROR
Log fitting changes what the model minimizes
The model minimizes squared errors in ln(y), giving relative deviations more equal weight than original-scale least squares.
Back-transforming the fitted log mean estimates a median-like response under lognormal error unless a smearing correction is applied.
LOG-LINEAR REGRESSION
Fit logarithms, then return predictions to the original scale
Taking natural logs turns y=A exp(kx) into ln(y)=ln(A)+kx. Ordinary least squares is applied to ln(y), so residuals are multiplicative on the original scale.
Detailed calculation process and general formulas
z_i = ln(y_i)k = sum[(x-xbar)(z-zbar)] / sum[(x-xbar)^2]ln(A) = zbar - k xbaryhat(x) = A exp(kx)factor_Delta = exp(k Delta)Symbols, meanings, and units
- A
- model level at x=0y units
- k
- continuous log growth rateper x unit
- z_i
- natural log of observed ylog units
- factor_Delta
- multiplicative change over Delta xratio
- yhat
- back-transformed fitted valuey units
GROWTH INTERPRETATION
Translate k into quantities people can use
The raw log slope is exact for the model but not always intuitive.
Unit growth factor
-exp(k) is the multiplication factor for one x unit.
Doubling or half-life
-ln(2)/|k| translates rate into a characteristic x span.
Observed support
Every value-table row is marked as interpolation or extrapolation.
Decision takeaway: Use the log audit and range status beside every exponential forecast.
Applied decisions
Exponential value-table applications
Early growth phase
Five positive measurements rise by an approximately constant percentage.
What the result clarifies: The table converts the log slope into predicted levels and growth factors.
Decay process
Positive concentrations fall multiplicatively over time.
What the result clarifies: A negative k produces a half-life rather than a doubling time.
Worked current scenario
Substitution, intermediate values, and reconciliation
Method references
Sources for this calculator's specific method
Scope and limitations
All observed y values must be positive. This model minimizes log-space error and does not include a retransformation-bias correction, prediction interval, autocorrelation correction, saturation limit, or structural break.
Regression Value Table Calculator | Exponential Fit and Prediction Ledger FAQ
Why can I not enter zero?
The natural logarithm used by this model is undefined at zero.
Is R-squared measured on the original scale?
No. The reported R-squared is for the linear regression of ln(y) on x.
What does exp(k) mean?
It is the fitted multiplication factor for a one-unit increase in x.