Appendix - Gaussian Mixture Models (GMM) and Expectation Maximization (EM) Algorithm
Gaussian Mixture Models (GMM) assume data points are drawn from a distribution that can be approximated by a mixture of Gaussian distributions. In this regard, assuming Q, the no-show rate of each clinic, is the feature vector, and k is the number of components (clinic clusters), the mixture model can be rewritten as:
$$ p\left( {Q|\Theta } \right) = \sum\nolimits_{{i = 1}}^k {{a_i}prob\left( {Q|{\theta_i}} \right)} $$
(11)
Where \( \left\{ {{a_1},...,{a_k},{\theta_1},...,{\theta_k}} \right\} \) is the collection of parameters with \( 0 \leqslant {a_i} \leqslant 1,\forall i = 1,2,...,k \) and \( \sum\nolimits_{{i = 1}}^k {{a_i} = 1} \) and \( p\left( {Q|{\theta_i}} \right) = \frac{1}{{\sigma \sqrt {{2\pi }} }}\exp \left( { - \frac{{Q - {\mu_i}}}{{2\sigma_i^2}}} \right) \). Having as a set of n, i.i.d samples \( Q = \left\{ {{q^{{(1)}}},{q^{{(2)}}},...,{q^{{(n)}}}} \right\} \) from the above model the log-likelihood function can be rewritten as:
$$ \begin{array}{*{20}{c}} {\log p\left( {Q|{\theta_i}} \right) = } \hfill \\{\log \prod\nolimits_{{j = 1}}^n {p\left( {{q^{{(j)}}}|\Theta } \right)} = \sum\nolimits_{{j = 1}}^n {\log } \sum\nolimits_{{i = 1}}^k {{\alpha_i}p} \left( {{q^{{(j)}}}|{\theta_j}} \right)} \hfill \\\end{array} $$
(12)
Here, the goal is to find Θ that maximizes the log-likelihood function:
$$ {\hat{\Theta }_{{MLE}}} = \arg \;\max \left\{ {\log \,p\left( {Q|\Theta } \right)} \right\} $$
(13)
The surface of the above likelihood function is highly nonlinear, and no closed form solution exists for the above likelihood function. One way to deal with this problem is by introducing a hidden variable Z:
$$ \begin{array}{*{20}{c}} {\log p\left( {Q,Z|{\theta_i}} \right) = } \hfill \\{\sum\nolimits_{{j = 1}}^n {\sum\nolimits_{{i = 1}}^k {z_i^{{(j)}}\log \left[ {{\alpha_i}p\left( {{q^{{(j)}}}|z_i^{{(j)}}{\theta_j}} \right)} \right]} } } \hfill \\\end{array} $$
(14)
and using Expectation Maximization (EM) algorithm as follows [33]:
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i.
Initializing parameters Θ
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ii.
Iterating the following until convergence:
$$ E - Step:\,Q\left( {\Theta |{\Theta^{{(t)}}}} \right) = {E_z}\log \left[ {p\left( {Q,Z|\Theta } \right)|{\Theta^{{(t)}}}} \right] $$
(15)
$$ M - Step:\,{\Theta^{{\left( {t + 1} \right)}}} = \arg \,\max Q\left( {\Theta |{\Theta^{{(t)}}}} \right) $$
(16)