Team:Calgary/Modelling/EmulsionPhaseEquilibriaModel
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Design
Purpose
Our chlorophyll separation system is enabled using water-in-oil emulsions. However, using emulsions can be extremely tricky. What total solution composition of oil, water, and surfactant is optimal? Are there any combinations of variables that we need to avoid? To ensure that we can recover the processed canola oil and remove chlorophyll from it, we sought to rationally design our emulsion system. To do this, we studied machine learning classification algorithms to build models which predicted the microstructure of the emulsion solution as a function of its chemical composition. We combined these results with a set of hydrodynamic models to further understand the physical characteristics of the emulsions we were producing, which allowed us to optimize the performance of our system.
The aim of this model is to find an emulsion which allows for the maximum removal of chlorophyll from the oil by finding the variables of temperature and concentrations of oil, water, and surfactant. Supervised machine-learning classification methods are used to predict emulsion phase equilibria (known as the windsor classifications) based on previously gathered in vitro data to formulate optimal emulsion conditions. Through an iterative development process, we explored and implemented Support Vector Classification (SVC), k-Nearest Neighbours (kNN), and multilayer perceptron models to densely interpolate and extrapolate desired phase data from experimentally gathered data.
Problem Outline
HOW WE VISUALIZE EMULSION DATA
Description of Data
The data given is four dimensional, containing three compositions of oil, water, and surfactant, and its equilibrium phase (classification label). Our model is looking to find a function \(F \; : \; \vec{v} \longrightarrow y \> \) mapping a given vector to a phase class such that, $$ \vec{v} = \begin{pmatrix} r_{oil} \\ r_{water} \\ r_{surfct} \end{pmatrix}$$ $$ r_{oil} + r_{water} + r_{surfct} = 1$$ $$ y \in \{Windsor1, Windsor2, Windsor3, Windsor4\}$$ The training data set was gathered in vitro, where its phase \(y\) is the classification label $$ \{ (\vec{v}_i, y_i) \} \> \> \> i = 0, 1, 2 ... 60 $$ Finding this function \(F\) was accomplished with Support Vector Classification (SVC) and \(k\) - Nearest Neighbours (kNN) formulations (see Theory).
Results
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Sit simul tollit munere ne, dolores plætonēm nō meī, modō eliÞr pri iƿ. Ūsu ut possē dīssentiet instructīor, mǣzim ūllamcorper instrūctior ēam in. Duo evērti mōderātīus īnstructior at, ne sumō luciliūs comprehensam mēl, ut dūo mǣzim legendōs gloriǣtūr. Debet tātion veriÞus an vim. Ad munerē doctūs ēxplicǽrī vim. Eu wīsi noluisse vix, eruditi maƿdamus usu īd. Ne simul tāntas repudiandae hǽs.
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Theory
Detailed formulation of \(k\) -Nearest Neighbours and Support Vector Classification
Support Vector Classification
Support Vector Classification (SVC) provides a classification approach which finds a hyperplane that divides two
classes of vectors within a space. The goal is to find the maximum margin between the labelled data and generate
parameters for a hyperplane that would divide this margin. The optimization problem of generating a separating
hyperplane between two classes holding \(n\) data points can be summarized:
$$ \max_{\beta_0, \beta_1, \beta_2, \beta_3, \epsilon_i, \ldots, \epsilon_n} \mathcal{M} $$
subject to,
$$\beta_0^2 + \beta_1^2 + \beta_2^2 + \beta_3^2 =1 $$
$$ y_i(\beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} \geq \mathcal{M}(1-\epsilon_i)$$
$$ \sum\limits_{i=0}^{n} \epsilon_i \leq \mathcal{C}, \>\>\>\>\> \epsilon \geq 0, \>\>\>\>\> y_i \in \{1, -1\}.$$
Where \(\mathcal{M}\) is the size of the margin, \(\beta_i\) are the parameters defining the hyperplane, \(y_i\) is the label of each vector which
can only be 1 or -1. \(\epsilon_i\) is the error for each vector which is constrained by \(\mathcal{C}\), the cost parameter (James et al. 2017).
Since we have four phase classes to be separated, we applied the one-versus-one approach, where divisions were
constructed for each pair of classes, meaning this optimization was solved 6 times - \( {4}\choose{2} \) is the number of distinct pairs between \(4\) elements.)
Since the data is not linearly separable, a non-linear radial basis function (RBF) was used as a kernel:
$$K(\vec{v_0}, \vec{v_i}) = e^{- \; \gamma \; \vec{v_0} \; \dot \; \vec{v_i}}$$
Where \(\vec{v_0}\) is the vector to be labelled, and the kernel is applied on each training vector \(\vec{v_i}\) for this test observation.
\( \gamma \) is a parameter subject to choice.
The second parameter \(\mathcal{C}\) specifies the amount of errors allowed within the separating hyperplane,
allowing the adjustment of the model’s bias-variance trade off. This trade off is an important consideration in
the approximation of any function. Approximations that are more flexible have greater variance (tend to follow
the data closely) and have low bias. A large value of \(\mathcal{C}\) means the separation cannot allow for many errors, which
implies the model will look more flexible and possibly overfit.
\(k\) -Nearest Neighbours
The aim of a general classification model is to provide the likelihood a new unlabelled vector lies within a class. The \(\mathcal{K}\)-Nearest Neighbours method is a non-parametric approach which looks at the \(\mathcal{K}\) nearest (in terms of distance) vectors within the space and assigns a label based on those closest neighbours. The probability given a vector from described above will be labeled with phase can be calculated with KNN by: $$ Pr( \> Y = y \> | \> X = \vec{v}) = \frac{1}{\mathcal{K}} \sum_{i \in \mathcal{N}}^{} I(\> y_i = y \>)$$ Where \(i\) indexes through the \(\mathcal{K}\) nearest vectors in \(\mathcal{N}\) and I is the identity function which outputs a 1 if the label of the neighbour is equal to and 0 otherwise (James et al. 2017).
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Appendix A
Procedure of data collected in vitro
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