# 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

Thank you in advance

• 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
• Note that there could probably be an infinite number of functions that you can generate with my example above, so the function that you do end up with would be really sensitive to assumptions and initial values for the optimiser – InfProbSciX Jan 31 at 10:52
• I see, the assumption is important to what the model will fit. But the observed data is continuous natural phenomena. For example, let say I have an observation data of a tadpole. which has 6 labels as follows eggs, tadpole with tail, tadpole with hind legs, tadpole with 4 legs, young frog, and adult frog. These were periodically observed as record. But there is no clear define way to tell when the transformation of each stage occurred. In this case what should I do? If we use Gaussian mixture model, I think we can fit all the stage with 6 Gaussians, and then find the boundary. But should I? – Wakeme UpNow Jan 31 at 11:09
• By mean, you have a pool full of tadpole, and you can only count them by scoop them up. But we can estimate the true distribution of the population by using multiple sampling (scooping). The problem with using 6 Gaussian is I cannot tell if the distribution is actually a Gaussian per each stage or not. So what do you think? – Wakeme UpNow Jan 31 at 11:20