Y-Transformed Equations

Conversions For Y-Transformed Equations

TableCurve 2D offers 530 Y-transformed equations. A transformed equation is expressed as f(y)=... where f(y) is one of the following:

ln(y)

1/y

sqrt(y)

y²

Y-transformations enable non-linear equations to be fitted by linear procedures.

Special Weighting of Y-Transformed Equations

TableCurve 2D applies a secondary weighting to compensate in the fitting process for these y-transformations. This weighting is fully transparent to the user. You may, however, observe some unexpected confidence and prediction intervals as a result of this secondary weighting.

The following list summarizes the TableCurve 2D Y-transformed equations which have familiar non-linear forms.

Other Transformed Equations

If you wish to see if a two-parameter or three-parameter non-linear equation has a counterpart within TableCurve 2D’s Y-transformed equations, simply enter the non-linear equation as a calculation, apply it to the data table, and fit the data using only the Y-transformed equations to see if any equation has a perfect r² of 1.0.

For example, entering the calculation Y=10*X^1.5, applying it to a data table, and fitting it only to the Y-transformed equations results in Equation 34 with an r² of 1.0 at the top of the list. The fitted b coefficient is 1.5 and the fitted a is the ln(10).

Limitations Of Y-transformed Equations

The Y-transformations make it possible to fit certain non-linear equations by a linear method. The fitting is thus accomplished in a fast single step matrix solution without initial estimates. Since TableCurve 2D automatically weights the data to compensate for the effects of the transform, it is possible to achieve effective fits which are nearly as good as true iterative non-linear fits.

Equations linearized by Y-transforms are not, however, the original equations. With any degree of noise, a more accurate fit will be achieved using a true non-linear iterative procedure. If the linearized form involves the log or square root transform, no fit will even occur if there are Y-values present less than zero. In such cases, you will have to use the X-Y sectioning or the TableCurve Editor to exclude all points with Y-values less than zero.