Bootstrapping on undirected binary networks via statistical mechanics

From patterned response dependency to structured covariate dependency: Entropy based categorical-pattern-matching

Data generated from a system of interest typically consists of measurements on many covariate features and possibly multiple response features across all subjects in a designated ensemble. Such data is naturally represented by one response-matrix against one covariate-matrix. A matrix lattice is an advantageous platform for simultaneously accommodating heterogeneous data types: continuous, discrete and categorical, and exploring hidden dependency among/between features and subjects. After each feature being individually renormalized with respect to its own histogram, the categorical version of mutual conditional entropy is evaluated for all pairs of response and covariate features according to the combinatorial information theory. Then, by applying Data Could Geometry (DCG) algorithmic computations on such a mutual conditional entropy matrix, multiple synergistic feature-groups are partitioned. Distinct synergistic feature-groups embrace distinct structures of dependency. The explicit details of dependency among members of synergistic features are seen through mutliscale compositions of blocks computed by a computing paradigm called Data Mechanics. We then propose a categorical pattern matching approach to establish a directed associative linkage: from the patterned response dependency to serial structured covariate dependency. The graphic display of such a directed associative linkage is termed an information flow and the degrees of association are evaluated via tree-to-tree mutual conditional entropy. This new universal way of discovering system knowledge is illustrated through five data sets. In each case, the emergent visible heterogeneity is an organization of discovered knowledge.

Bootstrap quantification of estimation uncertainties in network degree distributions

We propose a new method of nonparametric bootstrap to quantify estimation uncertainties in functions of network degree distribution in large ultra sparse networks. Both network degree distribution and network order are assumed to be unknown. The key idea is based on adaptation of the “blocking” argument, developed for bootstrapping of time series and re-tiling of spatial data, to random networks. We first sample a set of multiple ego networks of varying orders that form a patch, or a network block analogue, and then resample the data within patches. To select an optimal patch size, we develop a new computationally efficient and data-driven cross-validation algorithm. The proposed fast patchwork bootstrap (FPB) methodology further extends the ideas for a case of network mean degree, to inference on a degree distribution. In addition, the FPB is substantially less computationally expensive, requires less information on a graph, and is free from nuisance parameters. In our simulation study, we show that the new bootstrap method outperforms competing approaches by providing sharper and better-calibrated confidence intervals for functions of a network degree distribution than other available approaches, including the cases of networks in an ultra sparse regime. We illustrate the FPB in application to collaboration networks in statistics and computer science and to Wikipedia networks.

Integrative Inferences on Pattern Geometries of Grapes Grown under Water Stress and Their Resulting Wines

Multiple datasets of two consecutive vintages of replicated grape and wines from six different deficit irrigation regimes are characterized and compared. The process consists of four temporal-ordered signature phases: harvest field data, juice composition, wine composition before bottling and bottled wine. A new computing paradigm and an integrative inferential platform are developed for discovering phase-to-phase pattern geometries for such characterization and comparison purposes. Each phase is manifested by a distinct set of features, which are measurable upon phase-specific entities subject to the common set of irrigation regimes. Throughout the four phases, this compilation of data from irrigation regimes with subsamples is termed a space of media-nodes, on which measurements of phase-specific features were recoded. All of these collectively constitute a bipartite network of data, which is then normalized and binary coded. For these serial bipartite networks, we first quantify patterns that characterize individual phases by means of a new computing paradigm called "Data Mechanics". This computational technique extracts a coupling geometry which captures and reveals interacting dependence among and between media-nodes and feature-nodes in forms of hierarchical block sub-matrices. As one of the principal discoveries, the holistic year-factor persistently surfaces as the most inferential factor in classifying all media-nodes throughout all phases. This could be deemed either surprising in its over-arching dominance or obvious based on popular belief. We formulate and test pattern-based hypotheses that confirm such fundamental patterns. We also attempt to elucidate the driving force underlying the phase-evolution in winemaking via a newly developed partial coupling geometry, which is designed to integrate two coupling geometries. Such partial coupling geometries are confirmed to bear causal and predictive implications. All pattern inferences are performed with respect to a profile of energy distributions sampled from network bootstrapping ensembles conforming to block-structures specified by corresponding hypotheses.

Unravelling the geometry of data matrices: effects of water stress regimes on winemaking

A new method is proposed for unravelling the patterns between a set of experiments and the features that characterize those experiments. The aims are to extract these patterns in the form of a coupling between the rows and columns of the corresponding data matrix and to use this geometry as a support for model testing. These aims are reached through two key steps, namely application of an iterative geometric approach to couple the metric spaces associated with the rows and columns, and use of statistical physics to generate matrices that mimic the original data while maintaining their inherent structure, thereby providing the basis for hypothesis testing and statistical inference. The power of this new method is illustrated on the study of the impact of water stress conditions on the attributes of 'Cabernet Sauvignon' Grapes, Juice, Wine and Bottled Wine from two vintages. The first step, named data mechanics, de-convolutes the intrinsic effects of grape berries and wine attributes due to the experimental irrigation conditions from the extrinsic effects of the environment. The second step provides an analysis of the associations of some attributes of the bottled wine with characteristics of either the matured grape berries or the resulting juice, thereby identifying statistically significant associations between the juice pH, yeast assimilable nitrogen, and sugar content and the bottled wine alcohol level.

Data mechanics and coupling geometry on binary bipartite networks

We quantify the notion of pattern and formalize the process of pattern discovery under the framework of binary bipartite networks. Patterns of particular focus are interrelated global interactions between clusters on its row and column axes. A binary bipartite network is built into a thermodynamic system embracing all up-and-down spin configurations defined by product-permutations on rows and columns. This system is equipped with its ferromagnetic energy ground state under Ising model potential. Such a ground state, also called a macrostate, is postulated to congregate all patterns of interest embedded within the network data in a multiscale fashion. A new computing paradigm for indirect searching for such a macrostate, called Data Mechanics, is devised by iteratively building a surrogate geometric system with a pair of nearly optimal marginal ultrametrics on row and column spaces. The coupling measure minimizing the Gromov-Wasserstein distance of these two marginal geometries is also seen to be in the vicinity of the macrostate. This resultant coupling geometry reveals multiscale block pattern information that characterizes multiple layers of interacting relationships between clusters on row and on column axes. It is the nonparametric information content of a binary bipartite network. This coupling geometry is then demonstrated to shed new light and bring resolution to interaction issues in community ecology and in gene-content-based phylogenetics. Its implied global inferences are expected to have high potential in many scientific areas.