270 J. Zelenty et al. In this work, a new cluster search algorithm, which uses a
learning algorithms used to cluster data, is first introduced to illustrate the key ideas underlying iterative clustering algorithms. Next the development of GEMA is described in detail, focusing on the interpretation of GMMs and how they are estimated using EM. The paper is concluded by illustrating the successful application ofGEMAto both simulated and real APT data showing, respectively, that GEMA is often superior to the state-of-the-art and that GEMA provides useful cluster representations of real APT data. Overall, this novel approach provides a powerful, statistically principled, and immensely extensible baseline for cluster detection in APT.
AFIRST APPROACH: K-MEANS
k-means is one of the simplest and most common clustering algorithms in statistics and machine learning (Bishop, 2006). After a random initialization, k-means jointly learns cluster
locations and assigns atoms to clusters by iterating two steps: first, given cluster locations, each atom is assigned to the nearest cluster; second, given atom assignments to clusters, each cluster center is placed at the average location of all atoms assigned to that cluster (Fig. 1). The value of this approach is that it breaks down a complex problem, i.e. jointly determining cluster parameters and assigning atoms, into two simple sub- problems, which are iteratively solved. GEMA uses a similar iterative approach, as do, more generally, all EMalgorithms. However, there are several disadvantages to using
k-means on APT data. Primarily, all atoms must belong to a cluster in k-means, and thus atom assignments to the matrix are not possible (Fig. 2). In addition, atoms must determi- nistically belong to a single cluster, ignoring matrix/cluster and cluster/cluster uncertainty. Near precipitate interfaces, in particular, matrix/cluster uncertainty is a real scientific phenomenon which should be reflected in the modeling assumptions. Currently, all APT clustering algorithms share this undesirable property with k-means.
GEMA
GMMs, which are essentially a more sophisticated alter- native to k-means, can also be used to identify clusters. GEMA, which utilizes GMMs, overcomes the primary problem with k-means by explicitly modeling the matrix
Gaussian mixture model (GMM) to probabilistically distinguish clusters from the matrix, is developed. This unsupervised machine learning algorithm maximizes the data likelihood via expectation maximization (EM). Specifically, using APT data, the algorithm learns the position and size of each cluster. This new clustering algorithm is called the Gaussian mixture model Expectation Maximization Algorithm (GEMA). Arguably, the most important feature of GEMA is that it eliminates the need for external parameter selection while simultaneously improving power, thus providing a key step toward routine reproducibility of quantitative measurements. In this paper, k-means, one of the simplest machine
as an additional Gaussian component with infinite variance. A (bounded) Gaussian with infinite variance is identical to a uniform distribution, which formalizes the assumption that unclustered solute is distributed “randomly” in the matrix.a GEMA addresses the latter shortcoming by probabilistically assigning atoms to clusters and the matrix, thus eliminating the strict cluster assignments imposed by k-means. In regards to cluster assignments, GEMA labels atoms
with a probability of 50% or greater as belonging to a cluster. Although the probability of an atom belonging to the cluster has been made visible to the user via atomic shading, this is not an arbitrary parameter. A threshold of 50% is used by default in the algorithm – as it is Bayes optimal in many contexts – and therefore cluster assignments remain consistent. However, atomic probabilities can be extracted and
utilized for additional analyses, such as cluster composition and cluster size (number of atoms). For instance, cluster composition could be determined by weighting each atom by its probability of belonging to the cluster. In addition, there are frequently potential gradients in
chemistry and crystal structure around the precipitate bound- ary, owing simply to thermodynamics. Therefore, it is import- ant to note that missing data, specifically at the boundary, could potentially affect cluster size. However, this problem is due to thenatureofAPTas atechnique,which lies outsidethe scopeof this paper. In regards to chemical gradients, GEMA does not utilize the chemical identity of an atom when determining the probability of an atom belonging to a cluster. Therefore, chemical uncertainty does not affect cluster definition.
GMMs
GEMA utilizes a GMM (Bishop, 2006) to probabilistically learn cluster locations and assign atoms to clusters and the matrix. As its name suggests, a GMM is a distribution defined by a weighted sum, or mixture, of Gaussians:
pX
ðÞ=X K
k=1 wkNXjμk; σ2 k ; (1)
component, respectively. The goal of GEMA is to find a set of parameters, wk, μk,and σ2
distribution optimallymatches the empirical distribution. In its current state, GEMA assumes that the covariance
k are the weight, mean, and variance of the kth Gaussian k for each k, such that this analytic
where K is the number of Gaussian components and wk, μk,and σ2
matrix is spherical, or a multiple of the identity matrix. This is equivalent to the assumption that all clusters are spherical. However, despite this assumption GEMA works remarkably well on real APT data where this clearly isn’t the case (see Atom Probe Data section). Furthermore, generalizing the covariance matrix to allow ellipsoidal clusters in a future version of GEMA would be relatively straightforward. It should be noted that the current version of GEMA takes only solute atoms into consideration. Solute elements
aIn practice, this assumption is not necessarily true, nor does it need to be true in order for GEMA to be implemented effectively.
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