272 J. Zelenty et al. The E step and M step are repeated until convergence,b
Figure 2. The output of k-means for a two-dimensional data set con- taining two clusters. Cluster assignments are shown in red and blue. The implicit requirement of k-means that all atoms must belong to a cluster lead to its failure in modeling solute atoms within the matrix.
where g(xn; μk, σk) is a D-dimensional Gaussian function. In this equation, the numerator is essentially a weighted probability of finding an atom, n, in cluster, k; the denomi- nator normalizes this probability across all clusters. The Gaussian parameters are then optimized given the
newly updated atomic probabilities of being in each cluster. This step, which maximizes the parameter likelihood, is called the M step:
μk = σk =
v u u u u u u t
n = 1 ðÞxn pkjn
P N
P N
n = 1
n = 1 ðÞ xn - μk 2k pkjn k
P N
D
wk = 1 N
P N
n = 1
X N
n = 1 pkjn pkjn ðÞ ðÞ: (5)
where N is the total number of atoms and D the Gaussian dimension. Equations (3) and (4) determine the mean and standard deviation of cluster k, respectively; the contributions of each atom to these values are weighted by the probability of that atom being in the cluster, which was calculated in the previous E step. Similar to equation (2), the denominators of equations (3) and (4) perform normalizations. Equation (5) computes theweight of a given cluster, k. This is accomplished by taking the probability of an atombelonging to cluster k and summing over all atoms.
pkjn : ðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
(3)
to determine the best GMM to fit 2D data (with no back- ground) is shown in Figure 3. In the first panel, (a), the data points are shown in green and the two randomly ini- tialized Gaussians are shown in blue and red. The atoms are probabilistically assigned to the two clusters as shown in panel (b). Atoms are colored based on their probability of belonging to each cluster. Atoms that are equally likely to belong to either the blue or red cluster have been colored purple. This is referred to as the E step. In panel (c), the Gaussian/cluster parameters (mean, weight, and variance) are updated (Mstep) based on the cluster assignments from the previous step. In panels (d), (e), and (f) the E andMsteps are repeated until convergence. A similar process is executed in GEMA, except for a few key differences: GEMA fits an additional background cluster; GEMA assumes the clusters are spherical; and GEMA’s initialization utilizes a kernel density estimate (KDE), which is described in the next section. In summary, the solute clusters within the APT data are
resulting in an estimate of μk, σk, and wk for each cluster, k. An illustration of an EM algorithm being utilized
modelled using 3D Gaussian distributions, one for each cluster. GEMA learns the parameters of these Gaussians and determines which atoms belong to each one via an iterative process. In each iteration GEMA slightly improves its estimate as to which atoms belong to each cluster and the parameters of each Gaussian. These parameters include the location of the cluster (µk), the width of the cluster (σk), and the density of the cluster (wk).
Initialization Using KDEs
EM is an extremely useful tool; however, it is highly sensitive to initialization. Local maxima often prevent the algorithm from finding the global maximum, as convergence can occur at any local maximum. Fortunately, this problem can be solved with careful initialization. Although initialization sensitivity is a thoroughly
: (4)
studied problem for general mixture models, typical ini- tializations, such as k-means or repeated random restarts, were not viable for APT data. This is due to solute fluctua- tions within the matrix: initializations get stuck in these local maxima, which is not a problem for typical GMMs. Therefore, a novel initialization approach was developed specifically adapted to the peculiarities of APT data. First, a KDE (Bishop, 2006) was fit to the empirical
solute distribution. KDEs are smoothed estimates of density, i.e. atoms from high solute density regions, notably from clusters, have high KDE values. The locations of the cluster centers were then initialized using this KDE. In particular, the solute atom with the highest KDE value was identified; this was the first initialization point. Using this point to initialize the cluster center, GEMA was run with K set equal to 1. GEMA’s output was then used to produce a parametric
bEM is guaranteed to converge to a local maximum of the likelihood.
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