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I want to fit my (x,y,z) data points to a function. You can see the data on Fig.1.

Fig. 1. My data

The data is symmetric along the main diagonal. To understand my data I have studied (y,z) curves at different values of x. I have made the following conclusions:

1. When x = 0 the (y,z) curve is a line $$z = k * y^1 + l$$

enter image description here

2. When x = max(x)/2 (in the middle) the (y,z) curve is polynomial of degree 1.5 $$z = k * y^{1.5} + l$$

enter image description here

3. When x = max(x) the (y,z) curve is polynomial of degree 2 $$z = k * y^2 + l$$

enter image description here

Therefore, each "cut" of my data can be explained using a polynomial function where the degree ranges from 1 to 2.

Still I have a problem fitting it all together using a bivariate function.

What I need is: $$z = func(x,y)$$

Could you suggest anything?

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  • $\begingroup$ To match the three conditional relationships in your question, you'd look for something like $E(z)=a_x+b_x y^{c+dx}$, where $a_x$ and $b_x$ vary with $x$. Given that xmin=0, it may be that $c=0$. However, I don't think it's possible to make this functional form match the symmetry condition. $\endgroup$
    – Glen_b
    Commented Apr 13, 2015 at 0:17

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OK, I managed to figure it out myself.

The equation is: $$z = a_1\sqrt{x^2 + y^2}$$

On the plot below you can see how the observed data (black filled points) fit to the model (red unfilled points).

enter image description here
Looking at slices of my data (see my question above) I`ve suggested that the "core" of my function should be a product of x and y that has first degree when x=0 or y=0 and second degree when both x and y are non-negative. This is because at x=0 the (y,z) curve is a line and then with x>0 each next slice of (y,z) becomes parabolic with a higher degree (and the same for y and (x,z)). I`ve tried several options and finally discovered that $$\sqrt{x^2 + y^2}$$ does the trick.

Now I have two questions:

  1. I`ve actually guessed the correct solution. But is there a way to derive it systematically? Some kind of integration over several "slice-functions"?
  2. Does this equation have a name?
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  • $\begingroup$ Your question specifies a y-z relationship for 3 x-values (in each case, $z$ linear in $y^p$, with $p$ linear in $x$) but your solution doesn't match the conditions in your question (e.g. where you say the relationship is linear in $y^{1.5}$ or $y^2$, your solution isn't). That is, you requested a particular, specific thing (the question doesn't appear to admit solutions that are not of the proposed form), yet your solution doesn't do what you asked for. While it seems this is a pretty good fit to the data, how is this an answer to the question that you actually asked? $\endgroup$
    – Glen_b
    Commented Apr 13, 2015 at 0:08
  • $\begingroup$ By the way, answers are not a place to ask followup questions. Questions go in questions spaces, and answer spaces should contain answers. $\endgroup$
    – Glen_b
    Commented Apr 13, 2015 at 0:10

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