0
$\begingroup$

Alright, so I have this table of data.

I want to find the function to calculate the remaining fields marked with a "?" using the constant on the left.

I am assuming (just by looking at these values in a graph) that the function I need is an exponential function.

Now my question is, how would I approach finding the right function for this? I've tried several things the past few days but I just can't get my head around it.

I don't want the full solution, but I'd love to get some kind of kickstart in the right direction.

$\endgroup$
  • $\begingroup$ Functions in 2 variables can be bothersome to compute/guess unless you have some background on the problem. $\endgroup$ – prog_SAHIL Jan 12 '18 at 9:41
0
$\begingroup$

For such kind of problems you may make some experiments using WolframAlpha.
A 'bestfit' for the first line is not very satisfying but considering the reciprocals gives you this.
Even a linear interpolation seems good enough and suggests that $f(1,70)$ could be near $\,90$ !

The first and second line are nearly generated by \begin{align} f(x,70)&\approx\dfrac 1{0.0131215 - 0.00203371 \,x}\\ f(x,100)&\approx\dfrac 1{0.00879286 - 0.00140815 \,x}\\ \end{align}

Your next problem is to find a fit between the values $70, 100$ and the coefficients at the right. Observe that $\,\dfrac{0.00203371}{0.00140815}$ is not too far from $\,\dfrac{100}{70}\,$ and that the ratio of coefficients in the two denominators are near.

We may thus propose the (very) rough approximation : $$f(x,y)\approx \frac y{0.9-0.141\,x}$$

To get something more precise we may minimize the sum of the squares of the differences between $f_{rat}(x,y)=\dfrac{ay}{1-bx-cy}\;$ and the values in your table (I used simulated annealing for that purpose, many other optimization algorithms exist). I obtained : $$f_{rat}(x,y)\approx \frac{1.08386\;y}{1- 0.1449\,x -0.0008513\,y}$$ These approximations will become singular for $x$ a little larger than $6$.
Let's try your exponential suggestion with $\exp(a+b\,x+c\,y)$ but the resulting : $$f_{exp}(x,y)\approx \exp(-1.59+1.129\,x+0.02575\,y)$$ is increasing too fast so let's combine rational and exponential to get : $$f_{mix}(x,y)\approx \frac{\exp(3.2177+0.0248\,x+0.01397\,y)}{1-0.145777\,x-0.0007615\,y}$$

\begin{array} {c|cccc|cccc} x&f(x,70)&f_{rat}(x,70)&f_{exp}(x,70)&f_{mix}(x,70)&f(x,100)&f_{rat}(x,100)&f_{exp}(x,100)&f_{mix}(x,100)\\ \hline 1&?&95.37&3.8&85.0&135&140.8&8.3&133.0\\ 2&111&116.6&11.8&106.5&167&173.4&25.6&167.8\\ 3&142&150.0&36.6&140.4&219&225.7&79.2&223.5\\ 4&200&210.3&113.3&201.6&321&212.3&244.9&327,2\\ 5&342&351.4&349.8&345.0&592&569.3&757.5&586.2\\ 6&1070&1068.5&1081.9&1069.6&2382&2383.7&2342.6&2381.8\\ \end{array}

Extrapolating a third line from only two lines doesn't make much sense without knowledge of the process involved. Anyway for the fun here are the results with the three functions :

\begin{array} {c|ccc} x&f_{rat}(x,170)&f_{exp}(x,170)&f_{mix}(x,170)\\ \hline 1&259.4&50.2&379.7\\ 2&325.8&155.3&487.2\\ 3&438.1&480.3&667.5\\ 4&668.4&1485.5&1031.3\\ 5&1408.9&4594.0&2145.1\\ 6&-13048&14207&-75662\\ \end{array}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.