The goal of clustering is to group together similar local environments with no previous knowledge of the dataset. Clustering is hard to give a numerical score like supervised clustering, so we look to alternate methods of evaluating performance. Visualisation of the dataset like in @fig:order_parameter_overlap is a fantastic method of understanding the data and evaluating how well the clustering has performed. For clustering, each local environment is described by six features, that is, represented as a point in 6D feature space, making the visualisation on a 2D page somewhat more difficult. To evaluate the performance of a clustering algorithm we first need to find an appropriate method for visualising the high dimensional data.
When visualising data on a page, like in this thesis, we are limited to the display of two spatial dimensions at a time. So in visualising a six dimensional feature space, the problem becomes one of reducing the number of dimensions to two, while retaining properties of the six dimensional feature space.
The simplest method for dimensionality reduction is Principal Component Analysis (PCA). [@Jolliffe2013] PCA transforms the data to a new coordinate system such that the greatest variance lies on the first coordinate and the second greatest variance on the second coordinate. A linear PCA is matrix factorisation using Singular Value Decomposition, which gives a transformation that is a linear combination of the original coordinates, allowing the introspection of the contribution of each original dimension to the transformed coordinates. @Fig:dim_reduction_PCA shows the first two dimensions of the PCA with each local structure plotted as a point. The PCA analysis shows the liquid occupying the full range of values, reflecting the liquid state being able to explore the full range of configurations. Additionally the PCA generally groups the crystal configurations together, though there is little separation of the pg crystal from the liquid, while the p2 and the p2gg crystals are inseparable from the liquid. The main result of the PCA analysis, is that the crystal structures are not linearly separable in 2D.
{#fig:dim_reduction_PCA width=85%}
With the linear dimensionality reduction of PCA not providing adequate separation of the crystal structures there are a suite of non-linear methods which could be more suitable for this problem. The Uniform Manifold Approximation and Projection (UMAP) is a technique for dimension reduction with a theoretical foundation in Riemann Geometry and algebraic topology. [@McInnes2018] The UMAP algorithm finds the nearest neighbours of every point in the input feature space, using these neighbours to build a topological representation of the dataset. This topological structure can be projected onto a lower dimensional plane. The UMAP algorithm performs an optimisation such that the chosen representation minimises the difference between the topology of the high and low dimensional representations. The UMAP algorithm gives a low dimensional representation that contains the same local structural elements as the original high dimensional data. The UMAP dimensionality reduction of the 6D feature space (@fig:dim_reduction_UMAP) shows a distinct separation between the liquid state and all the crystal states. This firstly indicates that UMAP is a useful tool for the visualisation of these datasets, and additionally that separation of each of the crystals is possible using a clustering algorithm.
{#fig:dim_reduction_UMAP width=85%}
The UMAP algorithm shows excellent promise for
visualising the clustering within this dataset,
with the drawback that each crystal is separated
into too many smaller clusters.
One of the ways of increasing the complexity of a machine learning algorithm (@sec:supervised_learning)
is to introduce more classes to distinguish between.
Before having to increase the complexity
of a machine learning algorithm to handle many classes,
is there a way of reducing the complexity
of the underlying data?
For any six values of the relative orientation
there are
{#fig:dim_reduction_sorted_UMAP width=85%}
Using the UMAP algorithm is useful for visualisation showing an excellent correlation between the labelled clusters and the visualisations. In this case where we have the opportunity to confirm the clustering with a labelled dataset, we can be confident the clusters from the UMAP dimensionality reduction match the labelled clustering. However, when performing dimensionality reduction the UMAP algorithm can artificially create tears in clusters and increase the density of points for each cluster. These unusual behaviours of the UMAP algorithm are demonstrated in @fig:umap_demo, transforming the two Gaussian functions from @fig:classification_demo. The UMAP transformation creates a large void around the point (0, 0), increasing the distance between points on opposite sides of this void. Additionally, there is an increased density of points, with clumps appearing. The increased density of points is also present in @fig:dim_reduction_sorted_UMAP, where all the liquid configurations are grouped together, contrasted with the PCA dimensionality reduction (@fig:dim_reduction_PCA) where the liquid configurations cover the entire configuration space. These issues which make UMAP unsuitable for clustering have been documented for the similar t-SNE algorithm [@Maaten2008;@Schubert2017;@Wattenberg2016;@Shekhar2016] with a consensus that dimensionality reduction techniques like t-SNE and UMAP should only be used for visualisation and alternative algorithms used for clustering.
{width=80% #fig:umap_demo}
The visualisation of the dataset is important for three reasons. Firstly, the failure of the Principal Components Analysis to separate the structures indicates a non-linear method is required for separating the different structures. Secondly, we have established that the UMAP dimensionality reduction is suitable for performing a dimensionality reduction on the 6D feature space reducing it to 2D space for visualisation. This makes it a suitable tool for a visual analysis of the performance of a clustering algorithm. Thirdly, separating the structures with the UMAP algorithm for visualisation, indicates it is possible to separate each crystal structure using clustering. However, due to the problems with the UMAP algorithm distorting the phase space, an alternative clustering algorithm has to be used. Finally, we have found that reducing the degeneracy of the features is an important simplification which reduces the complexity of the following steps.
With UMAP not being a suitable algorithm for clustering, an alternate choice is required. The non-linear separation allowed by UMAP does provide a hint as to the algorithms which will be suitable. The Hierarchical Density-Based Clustering of Applications with Noise (HDBSCAN) algorithm [@Campello2013;@McInnes2017] was chosen for being similar to UMAP in creating a topological representation of the data. Clusters within the HDBSCAN algorithm are assigned to regions of the input space containing a high density of points, meaning the number of clusters is algorithmically determined. The HDBSCAN algorithm doesn't require all points belong to a cluster; points too far from a cluster are considered noise and left unclustered. Since configurations of an equilibrium liquid should be exploring all the available phase space, that is, all possible relative orientations of the six neighbouring molecules, considering the liquid configurations as noise is appropriate. We are only concerned with configurations with a higher density than the liquid.
To evaluate the results of the clustering algorithm we will use the UMAP dimensionality reduction shown in @fig:dim_reduction_sorted_UMAP. Instead of the colouring being provided by the labels we have assigned to the dataset, they will be applied by the clustering algorithm. These results of using the HDBSCAN algorithm for clustering are shown in @fig:cluster_sorted_hdbscan_vis, where the assignment of clusters from HDBSCAN closely matches the grouping of the UMAP algorithm. The cluster labelled -1 are the points that the HDBSCAN algorithm considered noise, which corresponds to the points we labelled as liquid in @fig:dim_reduction_UMAP. The classification of the liquid as noise showcases the suitability of the HDBSCAN algorithm for identifying recurring local structures within a liquid configuration. The remaining clusters are assigned to the values 1 through 4, these numbers have no meaning other than grouping like items together, however, they can be assigned to each crystal structure manually by comparing with the simulation configurations. @Fig:cluster_sorted_hdbscan uses the same colours as @fig:cluster_sorted_hdbscan_vis so we can assign the clusters to each polymorph. The cluster labelled 0 corresponds to the pg polymorph, the cluster labelled 2 corresponds to the p2 polymorph, while the clusters labelled 1 and 3 are alternating layers of the p2gg crystal. This additional knowledge demonstrates that while the dataset used in the clustering was known and understood, the clustering made no use of that knowledge.
{#fig:cluster_sorted_hdbscan_vis width=80%}
::: {#fig:cluster_sorted_hdbscan class="subfigures"}
{#fig:cluster_reduced_sorted_hdbscan_p2 width=33%}
{#fig:cluster_reduced_sorted_hdbscan_p2gg width=33%}
{#fig:cluster_reduced_sorted_hdbscan_pg width=33%}
Applying the cluster labels from @fig:cluster_sorted_hdbscan_vis to the p2 (a), p2gg (b) and pg (c) crystals using the same colourscheme. These are the configurations used to form the local environments.
:::
Clustering is a tool allowing for the identification of distinct local structures within a simulation having no previous knowledge of the data. These local structures can be manually labelled providing information about the types of structures present. The step of manual labelling is a downside to clustering, with no way to apply the learned information to new datasets.