# How to approximate continuous function from discrete uneven interval value?

To clarify, I have a discrete value of an unknown distribution. The value of this distribution domain is from 0 to 1. But the value which the discrete value is sampled has an unknown interval.

For example if I have a function of f(x) = x. And the data I have the f(x) value of [0.1, 0.7, 1.0], so the x value were sampled at [0.1, 0.7, 1.0] respectively. But assume I don't know the x value. So how can I approximate the f function, or at least approximate the x value for each f(x) on an unknown function?

The incorrect way that I can think of is to use x value evenly anyway. So the paired value will be (0, 0.1), (0.5, 0.7), (1.0, 1.0) and then use assumption what kind of f(x) function would be. In this case is a linear function. So I can solve for least squared for that.

Some floating idea is to use neuron network to approximate f(x). And somehow use KL-divergence with the output (but I don't know how).

But is there a theory in statistics that deal with these kind of problem? If so, please let me know

• Love this question. What I'd try to do is to fit a continuous/differentiable gaussian process to your data but also treating $x$ as a parameter. This would undeniably be expensive - so alternatively, you could represent the function as a smooth spline and treat the x variables as parameters. This would be an unidentifiable model, but if you post some data, I'd love to try and see (later) if this approach can work. – InfProbSciX Jan 31 at 10:21
• I am sorry to say that I don't have the actual data yet, so this is more theoretical on how to approach. But I believed that I can generated a mock data with numpy if you want some. – Wakeme UpNow Jan 31 at 10:45