# Approximating a 3d function based on 3d scattered points

Hello I have many 3d points (x, y, z). I have averaged every z with a matching (x, y) value and plotted the resulting means as a surface.

The data looks pretty good:

The green indicates where the sample of points was greater than 100 i.e. the more reliable means.

What I want to do is construct a function z = f(x, y) which can take new (x, y) pairs and find where they should lie on the surface.

Having a look at the graph in I can see the changing Y values have similar curves which look like they could be built as a function of Y.

That makes me think that I should go about building the function z = Fy(x) where Fy is a function outputted by some g(y).

Is this possible? Can you create a function which outputs a new function? I suppose it could be a sort of parametric function taking in y and outputting x coefficients?

My brain is fried anyway, am I on the right lines?

• I am not so sure whether I get your question right. But if you are mainly interested in finding a function that maps from x and z to y you should probably think about a nonparametric approach. A possible solution would be a tensor-product smooth that fits a model like y=f(x,z)+u (see stats.stackexchange.com/questions/45446/…). Is this helpful for you? Commented Oct 13, 2017 at 11:38
• Well it's more for mapping x and y to z. I've thought of smoothing but I feel it's all a bit of bodge work. What i'd really like is an approximate but neat formula to produce a 3D surface where the coefficients of f(x) = z (probably up to order 3) are calculated by a parametric function g(y) = [c, x1, x2, x3]. Does this make sense? Commented Oct 13, 2017 at 12:10