Continuous values in a dataset

Question

In a data analysis continuous data needs to be discretised, however in a computer it's impossible to have values $\in \mathbb{R}$, so what does this mean?

Answer 1

First of all, it is not a general rule that continuous values should be discretized. Especially in the context of regression, it is actually very much wrong.

Regarding discretization, I believe you may have a wrong definition. Discretizing means transforming a continuous interval, say $[0,10]$ into sub intervals, $[0,1), [1,2),...,[9,10]$, and then attributing the same value to points in the same interval, for instance, $\forall x \in [0,1) \mapsto 0.5$.

I'm not sure what is your point have having value $\in \mathbb{R}$. If it's because you don't know what the interval you have is, then you can simply find the minimum and maximum of your data points, and assume these to be the total interval. Then you can discretize that interval in however many pieces you want.

Answer 2

in a computer it's impossible to have values $\in \mathbb{R}$

Although this is technically right, practically it is practically irrelevant.

First, there are three important facts:

• You can represent any number $q \in \mathbb{Q}$ as a fraction $q = \frac{n}{d}$ where $n$ and $d$ are integers
• There are arbitrary big integers (Explanation, Wikipedia). They are only bound by your memory. Python uses them, for example.
• $\mathbb{Q}$ is dense in $\mathbb{R}$: This means that for any number $r \in \mathbb{R}$ and any distance $\varepsilon > 0$ there is a number $q \in \mathbb{Q}$ such that $|r-q| < \varepsilon$. So for the irrational numbers there is a rational one arbitary close by.