As mentioned in a previous post, translinearity is a concept describing the fluidity of the linear and the non-linear, as they are combined in a unified framework. However, linear relationships are still valuable, particularly if you want to develop a robust model. It's just that the rigid classification between linear and non-linear is arbitrary and meaningless when it comes to such a model. To clarify this whole matter I started exploring it further and developed an interesting heuristic to measure the level of non-linearity on a scale that's intuitive and useful.
So, let's start with a single feature or variable. How does it fare by itself in terms of linearity and non-linearity? A statistician will probably tell you that this sort of question is meaningless since the indoctrination he/she has received would make it impossible to ask anything that's not within a Stats course's curriculum. However, the question is meaningful even though it's not as useful as the follow-up questions that can ensue. So, depending on the data in that feature, it can be linear, super-linear, or sub-linear, in various degrees. The Index of Non-Linearity (INL) metric gauges that and through the values it takes (ranging from -1 to 1, inclusive) we can assess what a feature is like on its own. Naturally, these scores can be easily shifted by a non-linear operator (e.g. sqrt(x) or exp(x)) while all linear operators (e.g. standard normalization methods) do not affect these scores. Also, at the current implementation of INL, the value of the heuristic is calculated using three reference points in the variable.
Having established that, we can proceed to explore how a feature fares in relation to another variable (e.g. the target variable in a predictive analytics setting). Usually, the feature is used as the independent variable and the other variable as the dependent one, though you can explore the reverse relationship too, using this same heuristic. Interestingly the problem is not as simple now because the two variables need to be viewed in tandem. That's why all the reference points used shift if we change the order of the variables (i.e. the heuristic is not symmetric). Whatever the case, it is still possible to calculate INL with the same idea but taking into account the reference values of both variables. In the current implementation of the heuristic, the values can go a bit off-limits, which is why they are bound artificially to the [-1, 1] range.
Naturally, metrics like INL are just the tip of the iceberg in this deep concept. However, the existence of INL illustrates that it is possible to devise heuristics for every concept in data science, as long as we are open to the possibilities the data world offers. Not everything has been analyzed through Stats, which despite its indisputable value as a data science tool, it is still just one framework, a singular way of looking at things. Fortunately, the data-scapes of data science can be viewed in many more ways leading to intriguing possibilities worth exploring.
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Zacharias Voulgaris, PhD
Passionate data scientist with a foxy approach to technology, particularly related to A.I.