For many, mathematical modeling is exclusively about algebraic models, based on one form or another of regression or on differential equation modeling in the case of dynamical systems. However, this is too restrictive a point of view. For example, a clustering algorithm can be regarded as a modeling mechanism applicable to data where linear regression simply isn't applicable. Hierarchical clustering can also be regarded as a modeling mechanism, where the output is a dendrogram and contains information about the behavior of clusters at different levels of resolution. Kohonen self-organizing maps can similarly be regarded in this way.

The excited states of polyatomic systems are rather complex, and often exhibit meta-stable dynamical behaviors. Static analysis of reaction pathway often fails to sufficiently characterize excited state motions due to their highly non-equilibrium nature. Here, we proposed a time series guided clustering algorithm to generate most relevant meta-stable patterns directly from ab initio dynamic trajectories. Based on the knowledge of these meta-stable patterns, we suggested an interpolation scheme with only a concrete and finite set of known patterns to accurately predict the ground and excited state properties of the entire dynamics trajectories. As illustrated with the example of sinapic acids, the estimation error for both ground and excited state is very close, which indicates one could predict the ground and excited state molecular properties with similar accuracy. These results may provide us some insights to construct an excited state force field with compatible energy terms as traditional ones.

Dr. Andrea Trevino presents a beginner introduction to the widely-used K-means clustering algorithm in this tutorial. K-means clustering is a type of unsupervised learning, which is used when the resulting categories or groups in the data are unknown. This algorithm finds the groups that exist organically in the data and the results allow the user to label new data quickly. Clustering, in general, is a key tool for understanding your data. This algorithm can be used in a number of applications, including behavioral segmentation, inventory categorization, sorting sensor measurements, and detecting bots or anomalies, to name a few. This tutorial covers the iterative algorithm that determines the clusters and works through a delivery fleet data example in Python.

The majority of medical documents and electronic health records (EHRs) are in text format that poses a challenge for data processing and finding relevant documents. Looking for ways to automatically retrieve the enormous amount of health and medical knowledge has always been an intriguing topic. Powerful methods have been developed in recent years to make the text processing automatic. One of the popular approaches to retrieve information based on discovering the themes in health & medical corpora is topic modeling, however, this approach still needs new perspectives. In this research we describe fuzzy latent semantic analysis (FLSA), a novel approach in topic modeling using fuzzy perspective. FLSA can handle health & medical corpora redundancy issue and provides a new method to estimate the number of topics. The quantitative evaluations show that FLSA produces superior performance and features to latent Dirichlet allocation (LDA), the most popular topic model.

Background-Foreground classification is a fundamental well-studied problem in computer vision. Due to the pixel-wise nature of modeling and processing in the algorithm, it is usually difficult to satisfy real-time constraints. There is a trade-off between the speed (because of model complexity) and accuracy. Inspired by the rejection cascade of Viola-Jones classifier, we decompose the Gaussian Mixture Model (GMM) into an adaptive cascade of classifiers. This way we achieve a good improvement in speed without compensating for accuracy. In the training phase, we learn multiple KDEs for different durations to be used as strong prior distribution and detect probable oscillating pixels which usually results in misclassifications. We propose a confidence measure for the classifier based on temporal consistency and the prior distribution. The confidence measure thus derived is used to adapt the learning rate and the thresholds of the model, to improve accuracy. The confidence measure is also employed to perform temporal and spatial sampling in a principled way. We demonstrate a speed-up factor of 5x to 10x and 17 percent average improvement in accuracy over several standard videos.

It optimizes the number of the cluster when the clustering method is maximizing the variance among the clusters. If you are using for example K-means as clustering algorithm, your method will fail for every number of cluster you try to use! As you can see doesn't exist the right number of clusters, for this problem using the "naive" kmeans. BTW I've seen for kmeans and density based clustering algo, methods based on EM (expectation and maximizazion) and Bayesian information criterion (BIC) that are a little bit more robust than this method. Could you share the table of the points...just to play a little bit with them:)

Includes key concepts of statistical analysis - Probability theory, Types of distribution, Central limit theorem, Hypothesis testing, Statsistical inference.

We present an accelerated algorithm for hierarchical density based clustering. Our new algorithm improves upon HDBSCAN*, which itself provided a significant qualitative improvement over the popular DBSCAN algorithm. The accelerated HDBSCAN* algorithm provides comparable performance to DBSCAN, while supporting variable density clusters, and eliminating the need for the difficult to tune distance scale parameter. This makes accelerated HDBSCAN* the default choice for density based clustering. Library available at: https://github.com/scikit-learn-contrib/hdbscan

One of the longstanding open problems in spectral graph clustering (SGC) is the so-called model order selection problem: automated selection of the correct number of clusters. This is equivalent to the problem of finding the number of connected components or communities in an undirected graph. We propose automated model order selection (AMOS), a solution to the SGC model selection problem under a random interconnection model (RIM) using a novel selection criterion that is based on an asymptotic phase transition analysis. AMOS can more generally be applied to discovering hidden block diagonal structure in symmetric non-negative matrices. Numerical experiments on simulated graphs validate the phase transition analysis, and real-world network data is used to validate the performance of the proposed model selection procedure.

In this work we develop a theory of hierarchical clustering for graphs. Our modeling assumption is that graphs are sampled from a graphon, which is a powerful and general model for generating graphs and analyzing large networks. Graphons are a far richer class of graph models than stochastic blockmodels, the primary setting for recent progress in the statistical theory of graph clustering. We define what it means for an algorithm to produce the "correct" clustering, give sufficient conditions in which a method is statistically consistent, and provide an explicit algorithm satisfying these properties.