Defining Bonferroni means over lattices

In the face of mass amounts of information and the need for transparent and fair decision processes, aggregation functions are essential for summarizing data and providing overall evaluations. Although families such as weighted means and medians have been well studied, there are still applications for which no existing aggregation functions can capture the decision makers\u27 preferences. Furthermore, extensions of aggregation functions to lattices are often needed to model operations on L-fuzzy sets, interval-valued and intuitionistic fuzzy sets. In such cases, the aggregation properties need to be considered in light of the lattice structure, as otherwise counterintuitive or unreliable behavior may result. The Bonferroni mean has recently received attention in the fuzzy sets and decision making community as it is able to model useful notions such as mandatory requirements. Here, we consider its associated penalty function to extend the generalized Bonferroni mean to lattices. We show that different notions of dissimilarity on lattices can lead to alternative expressions.<br /

Using linear programming for weights identification of generalized bonferroni means in R

The generalized Bonferroni mean is able to capture some interaction effects between variables and model mandatory requirements. We present a number of weights identification algorithms we have developed in the R programming language in order to model data using the generalized Bonferroni mean subject to various preferences. We then compare its accuracy when fitting to the journal ranks dataset

An Abstraction of Whitney's Broken Circuit Theorem

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of type $\sum_{A\subseteq S} f(A)$ where $S$ is a finite set and $f$ is a mapping from the power set of $S$ into an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the M\"obius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical M\"obius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).Comment: 18 page

Deviations of ergodic sums for toral translations II. Boxes

We study the Kronecker sequence $\{n\alpha\}_{n\leq N}$ on the torus ${\mathbb T}^d$ when $\alpha$ is uniformly distributed on ${\mathbb T}^d.$ We show that the discrepancy of the number of visits of this sequence to a random box, normalized by $\ln^d N$, converges as $N\to\infty$ to a Cauchy distribution. The key ingredient of the proof is a Poisson limit theorem for the Cartan action on the space of $d+1$ dimensional lattices.Comment: 56 pages. This is a revised and expanded version of the prior submission

On extending generalized Bonferroni means to Atanassov orthopairs in decision making contexts

Extensions of aggregation functions to Atanassov orthopairs (often referred to as intuitionistic fuzzy sets or AIFS) usually involve replacing the standard arithmetic operations with those defined for the membership and non-membership orthopairs. One problem with such constructions is that the usual choice of operations has led to formulas which do not generalize the aggregation of ordinary fuzzy sets (where the membership and non-membership values add to 1). Previous extensions of the weighted arithmetic mean and ordered weighted averaging operator also have the absorbent element 〈1,0〉, which becomes particularly problematic in the case of the Bonferroni mean, whose generalizations are useful for modeling mandatory requirements. As well as considering the consistency and interpretability of the operations used for their construction, we hold that it is also important for aggregation functions over higher order fuzzy sets to exhibit analogous behavior to their standard definitions. After highlighting the main drawbacks of existing Bonferroni means defined for Atanassov orthopairs and interval data, we present two alternative methods for extending the generalized Bonferroni mean. Both lead to functions with properties more consistent with the original Bonferroni mean, and which coincide in the case of ordinary fuzzy values.<br /

A penalty-based aggregation operator for non-convex intervals

In the case of real-valued inputs, averaging aggregation functions have been studied extensively with results arising in fields including probability and statistics, fuzzy decision-making, and various sciences. Although much of the behavior of aggregation functions when combining standard fuzzy membership values is well established, extensions to interval-valued fuzzy sets, hesitant fuzzy sets, and other new domains pose a number of difficulties. The aggregation of non-convex or discontinuous intervals is usually approached in line with the extension principle, i.e. by aggregating all real-valued input vectors lying within the interval boundaries and taking the union as the final output. Although this is consistent with the aggregation of convex interval inputs, in the non-convex case such operators are not idempotent and may result in outputs which do not faithfully summarize or represent the set of inputs. After giving an overview of the treatment of non-convex intervals and their associated interpretations, we propose a novel extension of the arithmetic mean based on penalty functions that provides a representative output and satisfies idempotency

Superresolved Three-Dimensional Analysis of the Spatial Arrangement of the Human Immunodeficiency Virus Type-1 (HIV-1) Envelope Glycoprotein at Sites of Viral Assembly

Human Immunodeficiency Virus type 1 (HIV-1) replicates by forcing infected host cells to produce new virus particles, which assemble form protein components on the inner leaflet of the host cell\u27s plasma membrane. This involves incorporation of the essential viral envelope glycoprotein (Env) into a structural lattice of viral Gag proteins. The mechanism of Env recruitment and incorporation is not well understood. To better define this process, we seek to describe the timing of Env-Gag encounters during particle assembly by measuring angular positions of Env proteins about the surfaces of budding particles. Using three-dimensional superresolution microscopy, we show that Env distributions are biased toward the necks of budding particles, indicating incorporation of Env late in the assembly of the lattice. We show that this behavior is dependent on the host cell type and on the long cytoplasmic tail of Env. We propose a model wherein Env incorporation is regulated by opposing mechanisms: Gag lattice trapping of Env cytoplasmic tails, and intracellular sequestering of Env during lattice assembly

A New Framework for Decomposing Multivariate Information

What are the distinct ways in which a set of predictor variables can provide information about a target variable? When does a variable provide unique information, when do variables share redundant information, and when do variables combine synergistically to provide complementary information? The redundancy lattice from the partial information decomposition of Williams and Beer provided a promising glimpse at the answer to these questions. However, this structure was constructed using a much-criticised measure of redundant information, and despite sustained research, no completely satisfactory replacement measure has been proposed. This thesis presents a new framework for information decomposition that is based upon the decomposition of pointwise mutual information rather than mutual information. The framework is derived in two separate ways. The first of these derivations is based upon a modified version of the original axiomatic approach taken by Williams and Beer. However, to overcome the difficulty associated with signed pointwise mutual information, the decomposition is applied separately to the unsigned entropic components of pointwise mutual information which are referred to as the specificity and ambiguity. This yields a separate redundancy lattice for each component. Based upon an operational interpretation of redundancy, measures of redundant specificity and redundant ambiguity are defined which enables one to evaluate the partial information atoms separately for each lattice. These separate atoms can then be recombined to yield the sought-after multivariate information decomposition. This framework is applied to canonical examples from the literature and the results and various properties of the decomposition are discussed. In particular, the pointwise decomposition using specificity and ambiguity is shown to satisfy a chain rule over target variables, which provides new insights into the so-called two-bit-copy example. The second approach begins by considering the distinct ways in which two marginal observers can share their information with the non-observing individual third party. Several novel measures of information content are introduced, namely the union, intersection and unique information contents. Next, the algebraic structure of these new measures of shared marginal information is explored, and it is shown that the structure of shared marginal information is that of a distributive lattice. Furthermore, by using the fundamental theorem of distributive lattices, it is shown that these new measures are isomorphic to a ring of sets. Finally, by combining this structure together with the semi-lattice of joint information, the redundancy lattice form partial information decomposition is found to be embedded within this larger algebraic structure. However, since this structure considers information contents, it is actually equivalent to the specificity lattice from the first derivation of pointwise partial information decomposition. The thesis then closes with a discussion about whether or not one should combine the information contents from the specificity and ambiguity lattices