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I'm trying to find a function to a problem.

I have data points of

Volume (in ml) - Sugar Content(in Grams) - ABV(alcohol by Volume)

Like:

1500    500 16.375
1500    400 13.100
1500    300 9.825
1500    200 6.550
1500    100 3.275
1500    50  1.375
1500    0   0.000
1600    100 2.751
1400    100 3.144
1200    100 3.668
1000    100 4.585
800     100 5.764
600     100 7.860
400     100 12.052
200     100 24.366

I'm using this site to get the values, so I have as many as I need.

What I'm trying to find is the relationship of Volume to ABV in a fixed amount of Sugar.

I managed to find the relationship from Sugar to ABV in a fixed amount of Volume, which is 0.03537% for every Gram of sugar;

I thought that the Volume-ABV was going to be as easy, but I'm stuck. I tried this website for regression from data, but It didn't spit out any good formula.

Do anyone have a clue what can I do to get this formula right?

1 Answers 1

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If we look at the data for $100$ grams of sugar, that is to say $$\left( \begin{array}{cc} \text{Volume} & \text {ABV} \\ 1600 & 2.751 \\ 1500 & 3.275 \\ 1400 & 3.144 \\ 1200 & 3.668 \\ 1000 & 4.585 \\ 800 & 5.764 \\ 600 & 7.860 \\ 400 & 12.052 \\ 200 & 24.366 \end{array} \right)$$ it seems that a model such as $$\text{ABV}=\frac a {V^b}$$ could do a quite good job. You can linearize the model $$\log(\text{ABV})=c+d\log(V)$$ to get estimates of the parameters (standard linear regression) and then use nonlinear regression using as estimates $a=e^c$ anb $b=-d$.

This would give $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a & 5874. & 282 & \{5185,6563\} \\ b & 1.0349 & 0.0084 & \{1.0555,1.0143\} \\ \end{array}$$

Edit

All the above has been done using basic linear and nonlinear regression methods. If you want to combine both effects (what I did not do because a lack of data points, what I should do is to repeat the same process for different amounts of sugar and look how vary the parameters.

You could even simplify the problem assuming (if confirmed) that $b \approx 1$. Taking into your observation, may be model could just be $$\text{ABV}=\frac {\alpha \times \text{sugar}} {V}$$ Using the whole set of data points given in the post, it seems that $\alpha\approx 48.5$ leads to something acceptable.

This would give using the whole set of data points $$\left( \begin{array}{ccc} \text{Volume} & \text {Sugar} & \text {ABV}& \text {predicted}\\ 1500 & 500 & 16.375 & 16.173 \\ 1500 & 400 & 13.1 & 12.938 \\ 1500 & 300 & 9.825 & 9.704\\ 1500 & 200 & 6.55 & 6.469\\ 1500 & 100 & 3.275 & 3.235 \\ 1500 & 50 & 1.375 & 1.617\\ 1500 & 0 & 0. & 0.\\ 1600 & 100 & 2.751 & 3.032\\ 1400 & 100 & 3.144 & 3.466\\ 1200 & 100 & 3.668 & 4.043\\ 1000 & 100 & 4.585 & 4.852\\ 800 & 100 & 5.764 & 6.065\\ 600 & 100 & 7.86 & 8.087\\ 400 & 100 & 12.052 & 12.130\\ 200 & 100 & 24.366 & 24.260 \end{array} \right)$$

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    Could you explain how you came to those values for **a** and **b**? It works great but I can't manage to join both formulas, what I'm trying to do is generalize a formula that takes both **Vol* and **Sugar Cont** – 2017-02-14
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    @f.rodrigues. See my edit. All of that can be done using Excel. – 2017-02-14