Markov bases and subbases for bounded contingency tables

In this paper we study the computation of Markov bases for contingency tables whose cell entries have an upper bound. In general a Markov basis for unbounded contingency table under a certain model differs from a Markov basis for bounded tables. Rapallo, (2007) applied Lawrence lifting to compute a Markov basis for contingency tables whose cell entries are bounded. However, in the process, one has to compute the universal Gr\"obner basis of the ideal associated with the design matrix for a model which is, in general, larger than any reduced Gr\"obner basis. Thus, this is also infeasible in small- and medium-sized problems. In this paper we focus on bounded two-way contingency tables under independence model and show that if these bounds on cells are positive, i.e., they are not structural zeros, the set of basic moves of all $2 \times 2$ minors connects all tables with given margins. We end this paper with an open problem that if we know the given margins are positive, we want to find the necessary and sufficient condition on the set of structural zeros so that the set of basic moves of all $2 \times 2$ minors connects all incomplete contingency tables with given margins.Comment: 22 pages. It will appear in the Annals of the Institution of Statistical Mathematic

Assessment of the economic and social impact using SROI: An application to sport companies

This paper evaluates the social impact of a football club and its philanthropic organization on the local community and its stakeholders, namely supporters, sponsors, players, and shopkeepers. The methodology used is the Social Return on Investment (SROI). SROI methodology includes all the beneficiaries that are beyond the scope of the company's accounting and its fiscal and financial statements. The aim is to assess both the benefits and the negative impacts of a company's activities on stakeholders. This type of analysis combines the use of qualitative, quantitative and financial information gathered and analyzes them in order to estimate the amount of "value", including mental health and well-being, created or destroyed by a business activity, by a project or by the overall operation of an organization. The sport club under review in the present analysis is called Virtus Entella, an Italian football club playing in the second division. An SROI indicator was applied in reference to the business activity that took place during the championship season 2017/2018. Results show that the social impact created during the championship amounts to approximately 44 million Euro against a financial investment of 15 million Euro, producing an SROI ratio of 2.98:1. This outcome suggests that for every euro invested by the football club, about 3 Euros of social value is created

Semi-Markov Graph Dynamics

In this paper, we outline a model of graph (or network) dynamics based on two ingredients. The first ingredient is a Markov chain on the space of possible graphs. The second ingredient is a semi-Markov counting process of renewal type. The model consists in subordinating the Markov chain to the semi-Markov counting process. In simple words, this means that the chain transitions occur at random time instants called epochs. The model is quite rich and its possible connections with algebraic geometry are briefly discussed. Moreover, for the sake of simplicity, we focus on the space of undirected graphs with a fixed number of nodes. However, in an example, we present an interbank market model where it is meaningful to use directed graphs or even weighted graphs.Comment: 25 pages, 4 figures, submitted to PLoS-ON

Incidence of prosthesis-patient mismatch in patients receiving mitral Biocor® porcine prosthetic valves

Background: The aim was to assess the incidence of prosthesis-patient mismatch (PPM) after mitral valve replacement (MVR) in patients receiving Biocor® porcine or mechanical valves, and to evaluate the effect of PPM on long-term survival. Methods: All patients undergoing MVR between 2009 and 2013 received either mechanical or bioprosthetic valves (Biocor® porcine). PPM was defined as severe when the indexed effective ori­fice area was < 0.9 cm2/m2, moderate between 0.9 cm2/m2 and 1.2 cm2/m2 or absent > 1.2 cm2/m2. The primary endpoint was all-cause long-term mortality. Results: Among a total of 136 MVR, PPM was severe in 27%, moderate in 44% and absent in 29% of patients. Implanted valves were 57% mechanical and 43% bioprosthetic. Only 3% of patients with mechanical valves had severe PPM vs. 59% with bioprostheses (p < 0.0001). Sixty-month survival with severe mismatch was 0.559 (SE 0.149) and with no mismatch 0.895 (SE 0.058) (p = 0.043). Survival of patients suffering from severe mismatch, or moderate mismatch with pulmonary hypertension (PH) was 0.749 (SE 0.101); while for patients with no mismatch or with moderate mismatch without PH, survival was 0.951 (SE 0.028) (p = 0.016). Conclusions: About one-fourth of patients had severe PPM and almost all of them had received a bioprosthesis. Sixty-month survival was significantly lower in patients with severe mismatch, or moderate mismatch with PH. Specifically, when a bioprothesis is chosen and while further evidence on the impact of PPM on clinical outcomes appears, surgeons are recommended to follow a preoperative strategy to implant a mitral prosthesis of adequate size in order to prevent PP

Influence of the Ion Coordination Number on Cation Exchange Reactions with Copper Telluride Nanocrystals

Cu2-xTe nanocubes were used as starting seeds to access metal telluride nanocrystals by cation exchanges at room temperature. The coordination number of the entering cations was found to play an important role in dictating the reaction pathways. The exchanges with tetrahedrally coordinated cations (i.e. with coordination number 4), such as Cd2+ or Hg2+, yielded monocrystalline CdTe or HgTe nanocrystals with Cu2-xTe/CdTe or Cu2-xTe/HgTe Janus-like heterostructures as intermediates. The formation of Janus-like architectures was attributed to the high diffusion rate of the relatively small tetrahedrally coordinated cations, which could rapidly diffuse in the Cu2-xTe NCs and nucleate the CdTe (or HgTe) phase in a preferred region of the host structure. Also, with both Cd2+ and Hg2+ ions the exchange led to wurtzite CdTe and HgTe phases rather than the more stable zinc-blende ones, indicating that the anion framework of the starting Cu2- xTe particles could be more easily deformed to match the anion framework of the metastable wurtzite structures. As hexagonal HgTe had never been reported to date, this represents another case of metastable new phases that can only be accessed by cation exchange. On the other hand, the exchanges involving octahedrally coordinated ions (i.e. with coordination number 6), such as Pb2+ or Sn2+, yielded rock-salt polycrystalline PbTe or SnTe nanocrystals with Cu2-xTe@PbTe or Cu2-xTe@SnTe core@shell architectures at the early stages of the exchange process. In this case, the octahedrally coordinated ions are probably too large to diffuse easily through the Cu2-xTe structure: their limited diffusion rate restricts their initial reaction to the surface of the nanocrystals, where cation exchange is initiated unselectively, leading to core@shell architectures.Comment: 11 pages, 7 figures in J. Am. Chem. Soc, 13 May 201

Markov basis and Groebner basis of Segre-Veronese configuration for testing independence in group-wise selections

We consider testing independence in group-wise selections with some restrictions on combinations of choices. We present models for frequency data of selections for which it is easy to perform conditional tests by Markov chain Monte Carlo (MCMC) methods. When the restrictions on the combinations can be described in terms of a Segre-Veronese configuration, an explicit form of a Gr\"obner basis consisting of moves of degree two is readily available for performing a Markov chain. We illustrate our setting with the National Center Test for university entrance examinations in Japan. We also apply our method to testing independence hypotheses involving genotypes at more than one locus or haplotypes of alleles on the same chromosome.Comment: 25 pages, 5 figure

Likelihood Geometry

We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded projective variety, known as its maximum likelihood degree. We present an introduction to this theory and its statistical motivations. Many favorite objects from combinatorial algebraic geometry are featured: toric varieties, A-discriminants, hyperplane arrangements, Grassmannians, and determinantal varieties. Several new results are included, especially on the likelihood correspondence and its bidegree. These notes were written for the second author's lectures at the CIME-CIRM summer course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.Comment: 45 pages; minor changes and addition

Efficient and exact sampling of simple graphs with given arbitrary degree sequence

Uniform sampling from graphical realizations of a given degree sequence is a fundamental component in simulation-based measurements of network observables, with applications ranging from epidemics, through social networks to Internet modeling. Existing graph sampling methods are either link-swap based (Markov-Chain Monte Carlo algorithms) or stub-matching based (the Configuration Model). Both types are ill-controlled, with typically unknown mixing times for link-swap methods and uncontrolled rejections for the Configuration Model. Here we propose an efficient, polynomial time algorithm that generates statistically independent graph samples with a given, arbitrary, degree sequence. The algorithm provides a weight associated with each sample, allowing the observable to be measured either uniformly over the graph ensemble, or, alternatively, with a desired distribution. Unlike other algorithms, this method always produces a sample, without back-tracking or rejections. Using a central limit theorem-based reasoning, we argue, that for large N, and for degree sequences admitting many realizations, the sample weights are expected to have a lognormal distribution. As examples, we apply our algorithm to generate networks with degree sequences drawn from power-law distributions and from binomial distributions.Comment: 8 pages, 3 figure

Two factor saturated designs: cycles, Gini index and state polytopes

In this paper we analyze and characterize the saturated fractions of two-factor designs under the simple effect model. Using Li et al.ear algebra, we define a criterion to check whether a given fraction is saturated or not. We also compute the number of saturated fractions, providing an alternative proof of the Cayley's formula. Finally we show how, given a list of saturated fractions, Gini indexes of their margins and the associated state polytopes could be used to classify them

An Operational Definition of a Statistically Meaningful Trend

Linear trend analysis of time series is standard procedure in many scientific disciplines. If the number of data is large, a trend may be statistically significant even if data are scattered far from the trend line. This study introduces and tests a quality criterion for time trends referred to as statistical meaningfulness, which is a stricter quality criterion for trends than high statistical significance. The time series is divided into intervals and interval mean values are calculated. Thereafter, r2 and p values are calculated from regressions concerning time and interval mean values. If r2≥0.65 at p≤0.05 in any of these regressions, then the trend is regarded as statistically meaningful. Out of ten investigated time series from different scientific disciplines, five displayed statistically meaningful trends. A Microsoft Excel application (add-in) was developed which can perform statistical meaningfulness tests and which may increase the operationality of the test. The presented method for distinguishing statistically meaningful trends should be reasonably uncomplicated for researchers with basic statistics skills and may thus be useful for determining which trends are worth analysing further, for instance with respect to causal factors. The method can also be used for determining which segments of a time trend may be particularly worthwhile to focus on