Z Transformed Equations

Z-Transformed Equations

TableCurve 3D’s selective subset algorithm offers ln(Z) and 1/(Z) Z-transformed equations. A transformed equation is expressed as f(z)=... where f(z) is. Z-transformations enable non-linear equations to be fitted by linear procedures.

Special Weighting of Z-Transformed Equations

TableCurve 3D applies a secondary weighting to compensate in the fitting process for these z-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. On these equations, this usually appears as narrower intervals at higher z-values and broader intervals at lower z-values.

Limitations of Z-Transformed Equations

The Z-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 estimates. Since TableCurve 3D 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 Z-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 , no fit will even occur if there are Z-values present less than zero. In such cases, you will have to use the sectioning option or the editor to exclude all points with Z-values less than or equal to zero.