  Model selection based on experimental data is an important challenge in
biological data science. Particularly when collecting data is expensive or time
consuming, as it is often the case with clinical trial and biomolecular
experiments, the problem of selecting information-rich data becomes crucial for
creating relevant models. We identify geometric properties of input data that
result in a unique algebraic model and we show that if the data form a
staircase, or a so-called linear shift of a staircase, the ideal of the points
has a unique reduced Gro \"bner basis and thus corresponds to a unique model.
We use linear shifts to partition data into equivalence classes with the same
basis. We demonstrate the utility of the results by applying them to a Boolean
model of the well-studied lac operon in E. coli.
