Logarithmic regression#

The Logarithmic trend model is

where $a$ is the value at $x=1$ and $b$ controls how Y changes with the natural logarithm of X. It is a model fitted to GraphX’s numerical coordinates; it is not the same thing as choosing a logarithmic calibrated axis. A dataset digitised from either linear or logarithmic axes can use this trend model when its numerical relationship is appropriate.

Domain#

Every point selected by the Fit range must have $x>0$. Zero and negative X values cannot be used because $\ln(x)$ is undefined there. GraphX does not silently omit incompatible selected points: it rejects the fit and reports how many selected points are incompatible.

Fit-range selection happens before this domain check. An invalid point outside the configured range does not affect the fit; moving the range so that it includes that point will make the fit invalid until the range or data is corrected. The display range must also remain strictly positive for this model.

Fitting method#

GraphX transforms each selected X coordinate to

then performs ordinary least squares on the linear form

The result labels this method Least squares on ln(X). The displayed equation evaluates the fitted model back in the original X coordinate.

Interpretation#

A logarithmic curve changes rapidly near small positive X values and more gradually as X increases when $b>0$; the direction is reversed when $b<0$. Whether that shape is appropriate depends on the process represented by the data, not solely on how well a curve appears to pass through the points.

The primary R² is calculated from original Y-space residuals. See Understanding R² for its definition and limitations. Use Fit and display ranges to avoid accidentally drawing the curve into a range that was not used to determine its coefficients.