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I want to analyse the composition of a ternary mixture (a mixture containing of three components) by combining measurements of the mixture's density (denoted $\rho$) and its refractive index (denoted $n_D$).

To do so, I measured both density and refractive index of several samples with known composition. The data I gained should be used to build a model by means of regression. A very basic approach would be as follows:

$$ \rho = a_\rho \cdot w_1 + b_\rho \cdot w_2 + c_\rho \cdot w_3 + d_\rho \\ n_D = a_{n_D} \cdot w_1 + b_{n_D} \cdot w_2 + c_{n_D} \cdot w_3 + d_{n_D} $$

$a_i$ to $d_i$ are regression parameters for the density or the refractive index, respectively. The mixture composition given as mass fractions is described by $w_i$ where $i = 1...3$ (mass fraction for each component in mixture, 3 components in total).

As the mixture fulfills the mass balance, I can extend the system of equations by the following equation: $$ \sum_i w_i = 1 $$

For a ternary mixture, this equation can be rewritten as

$$ 1 = w_1 + w_2 + w_3 $$

As the valid range for mass fractions is $0...1$, I need to meet the following constraint for each mixture component:

$$ 0 \leq w_i \leq 1 $$

I have found a very similar question dealing with a single equation and additional constraints to be fulfilled.

However, how can I perform a regression on my dataset using the set of equations

$$ \rho = a_\rho \cdot w_1 + b_\rho \cdot w_2 + c_\rho \cdot w_3 + d_\rho \\ n_D = a_{n_D} \cdot w_1 + b_{n_D} \cdot w_2 + c_{n_D} \cdot w_3 + d_{n_D} \\ 1 = w_1 + w_2 + w_3 $$

and fulfilling the constraint $0 \leq w_i \leq 1$ for each component $i$?

Any hint would be appreciated as I do not know where to start and definitely need to extend my mathematical knowledge in order to solve this task.

albert
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  • The solution *method* in the referenced equation applies directly to your situation: just incorporate the additional linear constraints. There is no additional numerical, software, or conceptual complication to discuss. – whuber Sep 18 '20 at 12:01
  • Thanks for this hint. However, I am a bit confused about applying constraints to the regression parameters ($a ... d$) and applying constraints to the predicted values ($w_i$). I think I need to perform a two step approach: 1. Perform regression on measured data set while fulfilling the mass fraction constraints. 2. Use regression model to predict mass fractions based on measured values. Do I need to take care about the constraints in the 2nd step as well? – albert Sep 18 '20 at 13:26
  • I am having trouble identifying what your parameters are and what your data are. Could you clarify? – whuber Sep 18 '20 at 15:33
  • I think I to have some problems understanding the general methodology to solve my problem. I want to predict the composition of a three component mixture (ternary mixture) by measuring its density and refractive index (two physical properties of the mixture). In order to generate a predictive model, I measured the before mentioned properties of mixtures with known composition. However, I am not sure about the independance of the mixture composition as the sum up to 1 (Sum(w_i) = 1) and this might reduce the number of independent variables from 3 to 2. – albert Sep 22 '20 at 16:37
  • This reads like a multivariate problem: the response variable is the triple of mixture components and you have two explanatory variables. It would help to clarify this in your post. As far as approaches go, as a first step I would find applying a [CLR or ILR](https://stats.stackexchange.com/questions/259208) to the response to be attractive and using a multivariate linear model for the regression. – whuber Sep 22 '20 at 16:41
  • Thanks for your hint. This this seems to be getting more of a discussion (which I very much appreciate, despite of being a bit chatty and not in the nature of SE) and I think about asking a completely new question after getting a better overview about my type of problem. However, just to clarify: You suggested performing a log transformation on the response. Does response in this case refer to the mixture composition or the measured mixture properties (density and refractive index depending on composition)? – albert Sep 22 '20 at 19:44
  • The response is the mixture and that is what likely would benefit from some kind of transformation. Whether re-expressing the density and r.i. help will depend on how they are related to the response. Typically one addresses these questions sequentially. I outline the method (for an ordinary univariate regression problem) at https://stats.stackexchange.com/a/35717/919. – whuber Sep 22 '20 at 21:00
  • Okay. So, I will address each of the two questions separately. However, as far as I understood ILR, I need to apply the log on the mixture composition given per w_1, w_2, w_3 or ratios of these to be more precise. Since I planned my measurements based on a DoE plan, most of my mixture composition are on the vertices of the simplex (ternary diagram). In these cases at least one of the mass fractions is zero which would lead to some issues when dividing by these values or applying the log to them. – albert Sep 24 '20 at 20:05
  • You aren't limited to the logarithm, fortunately. Zero will not be a problem for any Box-Cox transformation of positive power, for instance. – whuber Sep 24 '20 at 20:36

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