Backtransformation

Backtransformation is the inverse process of applying a data transformation. It involves returning data, parameters, or model outputs that were transformed for analysis back to their original scale or coordinates. The feasibility and form of backtransformation depend on the transform being used; if the forward transform has a well-defined inverse, backtransformation is typically straightforward. Some transformations, however, may be non-injective or require special handling, which can complicate or limit back-transformation.

In statistics and data analysis, transformations such as logarithms, square roots, or Box-Cox are commonly used to stabilize variance, normalize distributions, or linearize relationships. Back-transformation converts results on the transformed scale back to the original scale for interpretation. A notable caution is that the expected value on the transformed scale does not generally equal the back-transformed expected value on the original scale, which can introduce bias. Techniques like the smearing estimator (Duan’s method) are sometimes employed to adjust back-transformed predictions to reduce bias.

In predictive modeling, models may be fit on transformed responses, and predictions are often reported on the original scale after back-transformation. In signal and image processing, backtransformation refers to the inverse of forward transforms, such as the inverse Fourier transform or inverse discrete cosine transform, used to reconstruct the original signal or image from its transformed representation.

Overall, backtransformation is a fundamental concept across disciplines, enabling interpretation and practical use of results obtained after applying transformations.