# MinMax normalization when all elements are same

I'm using min-max normalization to normalize time series which I compare in the following.

My question is, by definition min-max normalization is defined as:

x[i]=(x[i]-min(x))/(max(x)-min(x))


My question is, what should the new value be if all values in the array are same? That is, max=min.

One possibility is to set all elements to 0, the other is to set them all to 1. I could also set all values to half of the range, that is 0.5. But what is the proper approach?

If a variable is actually constant, then it provides no information whatsoever in your model. Saying it in plain English: when you are building a statistical model, or making predictions, you use your data to learn what knowing that $X = x$ tells us about the value of $Y$.

In this case, $X$ is always the same, so values of $Y$ are totally unrelated to it. All humans are mortal, I'm a human, so what can you say about the color of my eyes given this information?

Normalizing it does not change anything, so just drop such a variable from your analysis.

• Ok, thank you for your information, I understand your point of view. But what happens when I want to cluster the data, as I do? How to represent series in such case? Apr 6, 2016 at 10:45
• @Marko it is the same with cluster analysis: you can assume without doing any analysis what values (the same ones) of this variable will be present in each of the clusters. Constant variable would not help you to find differences that make the individual clusters unique...
– Tim
Apr 6, 2016 at 10:50
• +1. Another way to see this is that if constant variables were really useful, then we could add predictors to a model indefinitely with constant values such as 42 or $\pi$. But that wouldn't help one bit. Apr 6, 2016 at 11:00
• @NickCox actually, 42 is not the best example: google.com/… :)
– Tim
Apr 6, 2016 at 11:04
• 42 is the best example! Apr 6, 2016 at 11:07