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