Minimum Spanning vs. Principal Trees for Structured Approximations of Multi-Dimensional Datasets
2.1. Comparing and Benchmarking Graph-Based Data Approximation Methods Using Data Point Partitioning by Graph Segments
2.1.1. Outlining the Method
- Split each approximating graph into segments. By a segment we mean certain path from one branching point or leaf node to another.
- Create clustering (partitioning) of the dataset by segments. Each data point is associated with the nearest segment of the graph. Thus, in general, if graph is partitioned into N segments at the previous step then the dataset will be partitioned into N clusters.
- Compare the clusterings for the dataset using standard metrics (like adjusted Rand index or adjusted mutual information or any other suitable measure). Since the clusterings are produced from the structure of the approximating graphs, the score shows how two GBDA results are similar to each other.
2.1.2. Details on Step 1: Segmenting the Graphs
2.1.3. Details on Step 2: Clustering (Partitioning) Data Points by Graph Segments
2.1.4. Details on Step 3: Compare the Clustering Measures for the Datasets
2.2. Unsupervised Scores for Comparing GBDA Methods and Parameter Fine-Tuning
- Split each approximating graph into segments
- Create clustering (partitioning) of the dataset by the segments
- Use standard metrics and tools to calculate how good is the clustering:, e.g., calculate silhouette, Calinski–Harabasz, Davies-Bouldin score etc.
2.3. Comparison of Clustering Based Scores with Other GBDA Metrics
2.4. Comparison of MST-Based Graph-Based Approximations and ElPiGraph
- Both MST and ElPiGraph can achieve similar quality of graph-based data approximation by tuning their main parameter, number of nodes. The quality here is estimated by the scores introduced above and in comparison with ground truth for simulated datasets. However, the advantage of ElPiGraph is in its stability with respect to this parameter. Small modifications do not change the quality dramatically, while for MST even small modifications of the parameter may lead to quite a drastic change of the approximating graph structure and, respectively, the quality of ground truth approximation. That means that in practical situation where ground truth is not available, ElPiGraph has an advantage since one should not be afraid of choosing the parameter incorrectly. Therefore, a reasonably wide range of parameters gives similar results, while for MST approach choosing an incorrect parameter would give a significant loss in approximation quality.
- The embedment of approximating graphs produced by ElPiGraph are much smoother comparing to MST. This observation is consistent with the previous one and with ElPiGraph algorithm which penalizes for non-smoothness and hence it produces graphs which would not produce too much branching points, the problem which underlies the MST instability.
- We analyse a possible combination of MST+ElPiGraph methods, where the initial graph approximation is obtained by MST and then ElPiGraph algorithm starts from this initialization. One of our conclusions is that ElPiGraph “forgets” the initial conditions quite fast. Nevertheless the combination of methods may have certain advantages.
- We consider a real-life single cell RNA sequencing dataset and demonstrate that our approach for choosing the parameters of the trajectory inference method based on unsupervised clustering scores provides reasonable results.
2.4.1. Smoothness of ElpiGraph-Based Graph Approximators Comparing to MST
2.4.2. Comparison between ElPiGraph and MST on Large Number of Different Datasets
2.4.3. Initialization of ElPiGraph with MST-Based Tree
2.5. Tuning Parameters of GBDA Algorithms Using Unsupervised Clustering Quality Scores
2.6. Single Cell Data Analysis Example
- ElPiGraph demonstrates much higher stability than MST and choice of main parameter (node number) can be made more easily and reliably.
- Unsupervised metrics for trajectories introduced above (based on silhouette, Calinski–Harabas, Davies-Bouldin clustering scores) provides insights for the choice of parameters for trajectory inference methods.
- Heuristic for MST, to take number of nodes equal to approximately square root of number of datapoints fails significantly for the present example. It would give 21 nodes, however trajectories of MST with 21 nodes show significant “overbranching” (see figures below). Reasonable node numbers for MST is 9 or a little below, but 10 and above leads to biologically incorrect branching.
4. Materials and Methods
4.1. Minimal Spanning Tree-Based Data Approximation Method
- Split the dataset to certain pieces (for example, by applying K-means clustering for some K).
- Construct kNN (k-nearest neighbour) graph with nodes at the centers of clusters.
- Compute an MST (minimal spanning tree) of the kNN graph computed as described above.
4.2. Method of Elastic Principal Graphs (ElPiGraph)
4.3. Benchmark Implementation
Conflicts of Interest
|GBDA||Graph-based data approximations|
|MST||Minimal spanning tree|
|PAGA||Partition-based graph abstraction|
|ElPiGraph||Elastic Principal Graph|
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Chervov, A.; Bac, J.; Zinovyev, A. Minimum Spanning vs. Principal Trees for Structured Approximations of Multi-Dimensional Datasets. Entropy 2020, 22, 1274. https://doi.org/10.3390/e22111274
Chervov A, Bac J, Zinovyev A. Minimum Spanning vs. Principal Trees for Structured Approximations of Multi-Dimensional Datasets. Entropy. 2020; 22(11):1274. https://doi.org/10.3390/e22111274Chicago/Turabian Style
Chervov, Alexander, Jonathan Bac, and Andrei Zinovyev. 2020. "Minimum Spanning vs. Principal Trees for Structured Approximations of Multi-Dimensional Datasets" Entropy 22, no. 11: 1274. https://doi.org/10.3390/e22111274