Zero Pattern Research Articles

K-aggregated transformation of discrete distributions improves modeling count data with excess ones

The excess one pattern in count data has been documented in ecology but it has not been explicitly modeled or examined. In this study, we introduce a k-aggregated transformation of discrete distributions to better model count data with excess ones in a Bayesian generalized linear model framework and demonstrate its use with two groups of case studies (group 1: seabird bycatch in longline fisheries and Legionnaires disease incidence; group 2: survey abundance of Leadbeater’s possum and Frigatebird nesting sites). Group 1 examples have a clear excess one data pattern, and these examples are used to demonstrate the concept of the k-aggregation technique. On the other hand, group 2 examples lack a clear excess one pattern, and a modeler may not be motivated enough to use the k-aggregation technique in these cases. Nonetheless, k-aggregated transformation demonstrated better performance for both groups of examples. In all our case studies, the excess zero pattern co-occurred with an excess one pattern, and the excess zeros were modeled thorough either a zero-inflated or hurdle configuration. The better performance of k-aggregated distributions is due to their flexibility of adapting to the relatively high frequency of singletons in the data sets. This new technique has broad applicability and utility in improving modeling count data with potential excess ones.

Polymer collapse transition: a view from the complex fugacity plane

Distributions of zeros of the grand canonical and canonical partition functions in the complex fugacity and in the complex interaction strength plane are examined numerically for a model of rooted self-interacting self-avoiding walks on a hierarchical graph. It is shown that the pattern of zeros of the grand canonical partition function in the complex fugacity plane has a circular-like form, with the exception of zeros lying in the vicinity of the critical point. Exact values of polymer size critical exponents, in the swollen and dense phase, as well as at the point of collapse transition, are obtained by studying asymptotic behavior of zeros lying closest to the positive real axis in the fugacity plane. Distribution of zeros of the canonical partition function in the complex interaction strength plane is found to be more complicated, and an accurate estimate of crossover exponent is obtained from study of asymptotic behavior of zeros lying closest to the positive real axis in this plane. It is also shown that the next-to-leading term in the asymptotic law for canonical partition function in the dense phase has a stretched-exponential rather than the power-law form.

Unions of rank/trace complete preorders

A zero pattern algebra is a matrix algebra determined by a pattern of zeros corresponding to a preorder (i.e., a reflexive transitive digraph). The issue studied in [6] is: under what conditions is a matrix in a zero pattern algebra A a sum of (rank one) idempotents in A or a logarithmic residue in A? Here logarithmic residues are contour integrals of logarithmic derivatives of analytic A-valued functions. As has been established in [6], there is a necessary condition involving certain rank/trace requirements. Algebras for which this necessary condition is also sufficient are said to be rank/trace complete. Several classes of rank/trace complete algebras are identified in [6]. Also operations are described there for producing rank/trace complete algebras out of given ones. A basic operation of that type is taking a disjoint union. In the present article this simple operation is generalized to a more involved one which preserves rank/trace completeness too. The new operation turns out to be a useful tool for establishing rank/trace completeness in situations that up to now could not be handled. One of the positive results is concerned with bouquets, a special type of rooted trees. It provides a partial answer to an issue raised in [7].

Combined matrices of almost strictly sign regular matrices

The combined matrix is a very useful concept for many applications. Almost strictly sign regular (ASSR) matrices form an important structured class of matrices with two possible zero patterns, which are either type-I staircase or type-II staircase. We prove that, under an irreducibility condition, the pattern of zero and nonzero entries of an ASSR matrix is preserved by the corresponding combined matrix. Without the irreducibility condition, it is proved that type-I and type-II staircases are still preserved. Illustrative numerical examples are included.

Multiple Leading Zero Pattern Scheme for Non-volatile Memories

Multiple Leading Zero Pattern Scheme for Non-volatile Memories

Averaging the population projection matrices: Heuristics against uncertainty and nonexistence

In matrix population models, the process of population “projection” through a number of time steps is fundamentally multiplicative, hence the arithmetic mean of the consecutive matrices is of doubtful meaning, while the geometric mean quite corresponds to the multiplication principle. The geometric mean of positive numbers does not bear any problem, but that of matrices does. The “population projection matrices” (PPMs) are rather nonnegative than positive, with the allocation of non-zeros that is predetermined by the life cycle graph (LCG) reflecting the development biology of a given species, and this graph is principally incomplete. The average matrix A should logically have the same fixed pattern of zeros as those to be averaged, and this causes the averaging matrix equation to be overdetermined as a system of element-wise algebraic equations for the unknown positive elements. Therefore, the exact solution to the problem of pattern-geometric averaging does generally not exist, while the classical (least-squares) approximation leads to a significant error. My heuristic approach to finding a better approximate A for the PPMs in the form of L = T(transition) + F(fertility) is to solve the problem in a combined way: the pattern-geometric approximation for the T part and the exact arithmetic mean for F (as population recruitment is an additive process). As a result, the approximation error decreases drastically due to matrix T being always substochastic, while the combined, TF-averaging, method turns out efficient even under ‘reproductive uncertainty’ in data, i.e., for the whole families of feasible matrices F in the sum L = T + {F}. I illustrate the method of TF-averaging with 5 matrices L(t) calibrated for each pair of consecutive years from a 6-year period of observation in a case study of Eritrichium caucasicum, a short-lived perennial herbaceous species. The approximate TF-average enables gaining the ‘age-specific traits from stage-specific models’ (Caswell, 2001, p. 116) that are characteristic of the entire period, and I discuss other motivations/advantages for/of pattern-geometric means in matrix population models.

The critical boundary RSOS M(3,5) model

We consider the critical non-unitary minimal model {\cal M}(3,5) with integrable boundaries. We analyze the patterns of zeros of the eigenvalues of the transfer matrix and then determine the spectrum of the critical theory through the Thermodynamic Bethe Ansatz (TBA) equations. By solving the TBA functional equation satisfied by the transfer matrices of the associated A_{4} RSOS lattice model of Forrester and Baxter in Regime III in the continuum scaling limit, we derive the integral TBA equations for all excitations in the (r=1,s=1) sector then determine their corresponding energies. The excitations are classified in terms of (m,n) systems.

Gaussian covariance graph models accounting for correlated marker effects in genome-wide prediction.

Several statistical models used in genome-wide prediction assume uncorrelated marker allele substitution effects, but it is known that these effects may be correlated. In statistics, graphical models have been identified as a useful tool for covariance estimation in high-dimensional problems and it is an area that has recently experienced a great expansion. In Gaussian covariance graph models (GCovGM), the joint distribution of a set of random variables is assumed to be Gaussian and the pattern of zeros of the covariance matrix is encoded in terms of an undirected graph G. In this study, methods adapting the theory of GCovGM to genome-wide prediction were developed (Bayes GCov, Bayes GCov-KR and Bayes GCov-H). In simulated data sets, improvements in correlation between phenotypes and predicted breeding values and accuracies of predicted breeding values were found. Our models account for correlation of marker effects and permit to accommodate general structures as opposed to models proposed in previous studies, which consider spatial correlation only. In addition, they allow incorporation of biological information in the prediction process through its use when constructing graph G, and their extension to the multi-allelic loci case is straightforward.

Conformal partition functions of critical percolation from D3 thermodynamic Bethe Ansatz equations

Using the planar Temperley–Lieb algebra, critical bond percolation on the square lattice can be reformulated as a loop model. In this form, it is incorporated as in the Yang–Baxter integrable family of logarithmic minimal models . We consider this model of percolation in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite Y-system of functional equations. In this way, we obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a D3 Dynkin diagram. Following the methods of Klümper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge c = 0 and conformal weights for with . For the periodic case, the finite-size corrections agree with the conformal weights , with . These are obtained analytically using Rogers dilogarithm identities. We incorporate all finite excitations by formulating empirical selection rules for the patterns of zeros of all the eigenvalues of the standard modules. We thus obtain the conformal partition functions on the cylinder and the modular invariant partition function (MIPF) on the torus. By applying q-binomial and q-Narayana identities, it is shown that our refined finitized characters on the strip agree with those of Pearce, Rasmussen and Zuber. For percolation on the torus, the MIPF is a non-diagonal sesquilinear form in affine u(1) characters given by the u(1) partition function . The u(1) operator content is for and for and . This result is compatible with the general conjecture of Pearce and Rasmussen, namely with , where the minimal partition function is and the lattice derivation fixes n2,3 = −1.

Modeling Zero Inflation in Count Data Time Series with Bounded Support

Real count data time series often show an excessive number of zeros, which can form quite different patterns. We develop four extensions of the binomial autoregressive model for autocorrelated counts with a bounded support, which can accommodate a broad variety of zero patterns. The stochastic properties of these models are derived, and ways of parameter estimation and model identification are discussed. The usefulness of the models is illustrated, among others, by an application to the monetary policy decisions of the National Bank of Poland.

A parallel multisplitting method with self-adaptive weightings for solving H-matrix linear systems

In this paper, a parallel multisplitting iterative method with the self-adaptive weighting matrices is presented for the linear system of equations when the coefficient matrix is an H-matrix. The zero pattern in weighting matrices is determined in advance, while the non-zero entries of weighting matrices are determined by finding the optimal solution in a hyperplane of α points generated by the parallel multisplitting iterations. Especially, the nonnegative restriction of weighting matrices is released. The convergence theory is established for the parallel multisplitting method with self-adaptive weightings. Finally, a numerical example shows that the parallel multisplitting iterative method with the self-adaptive weighting matrices is effective.

Rank decomposition in zero pattern matrix algebras

For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see H.Bart, A.P.M.Wagelmans (2000). The proof involves elements from integer programming and employs Farkas’ lemma. The algebra of block upper triangular matrices can be viewed as a matrix algebra determined by a pattern of zeros. The present note is concerned with the question whether the decomposition result referred to above can be extended to other zero pattern matrix algebras. It is shown that such a generalization does indeed hold for certain digraphs determining the pattern of zeros. The digraphs in question can be characterized in terms of forests, i.e., disjoint unions of rooted trees.

{0,1} completely positive tensors and multi-hypergraphs

Completely positive graphs have been employed to associate with completely positive matrices for characterizing the intrinsic zero patterns. As tensors have been widely recognized as a higher-order extension of matrices, the multi-hypergraph, regarded as a generalization of graphs, is then introduced to associate with tensors for the study of complete positivity. To describe the dependence of the corresponding zero pattern for a special type of completely positive tensors—the {0,1} completely positive tensors, the completely positive multi-hypergraph is defined. By characterizing properties of the associated multi-hypergraph, we provide necessary and sufficient conditions for any (0,1) associated tensor to be {0,1} completely positive. Furthermore, a necessary and sufficient condition for a uniform multi-hypergraph to be a completely positive multi-hypergraph is proposed as well.

Exploratory tools for outlier detection in compositional data with structural zeros

ABSTRACTThe analysis of compositional data using the log-ratio approach is based on ratios between the compositional parts. Zeros in the parts thus cause serious difficulties for the analysis. This is a particular problem in case of structural zeros, which cannot be simply replaced by a non-zero value as it is done, e.g. for values below detection limit or missing values. Instead, zeros to be incorporated into further statistical processing. The focus is on exploratory tools for identifying outliers in compositional data sets with structural zeros. For this purpose, Mahalanobis distances are estimated, computed either directly for subcompositions determined by their zero patterns, or by using imputation to improve the efficiency of the estimates, and then proceed to the subcompositional and subgroup level. For this approach, new theory is formulated that allows to estimate covariances for imputed compositional data and to apply estimations on subgroups using parts of this covariance matrix. Moreover, the zero pattern structure is analyzed using principal component analysis for binary data to achieve a comprehensive view of the overall multivariate data structure. The proposed tools are applied to larger compositional data sets from official statistics, where the need for an appropriate treatment of zeros is obvious.

Sums of idempotents and logarithmic residues in zero pattern matrix algebras

A zero pattern algebra is a matrix subalgebra of Cn×n determined by a pattern of zeros. The issue in this paper is: under what conditions is a matrix in a zero pattern algebra A a sum of (rank one) idempotents in A or a logarithmic residue in A? Here logarithmic residues are contour integrals of logarithmic derivatives of analytic A-valued functions. It turns out that there is a necessary condition involving certain rank/trace requirements. Although these requirements are generally not sufficient, there are several important cases where they are.

Some more on the history of present-tense -s, do and zero: West Oxfordshire, 1837

AbstractThis paper analyses the present indicative third-person singular markers in a diary kept by a servant from West Oxfordshire in 1837. There are three present-tense markers present in the diary: periphrastic auxiliary verb do, -s, and zero. However, all affirmative declarative do forms are found to be emphatic, so I deduce that periphrastic do was not present. Zero occurs at the surprisingly high rate of 21%. I examine the possibility that zero was the result of hypercorrection (perhaps due to misapplied schooling, or perhaps due to dialect contact as the diarist left West Oxfordshire and took up employment in London), but this is rejected because -s and zero patterns in the same way as in other southern data, based on an analysis of pronoun v. NP subjects, auxiliary v. full-verb have, non-categorical NPPR, and non-categorical Early Modern subjunctive zero. Presumably, these regular patternings would not be present were the zeroes due to an over-applied rule. I hypothesise that as periphrastic do and older -th became abandoned, an empty morpheme slot emerged, which later became filled with generalised -s. It is this temporarily empty morpheme slot, I suggest, which accounts for the 21% zeroes.

Dynamic network signal processing using latent threshold models

We discuss multivariate time series signal processing that exploits a recently introduced approach to dynamic sparsity modelling based on latent thresholding. This methodology induces time-varying patterns of zeros in state parameters that define both directed and undirected associations between individual time series, so generating statistical representations of the dynamic network relationships among the series. Following an overview of model contexts and Bayesian analysis for dynamic latent thresholding, we exemplify the approach in two studies: one of foreign currency exchange rate (FX) signal processing, and one in evaluating dynamics in multiple electroencephalography (EEG) signals. These studies exemplify the utility of dynamic latent threshold modelling in revealing interpretable, data-driven dynamics in patterns of network relationships in multivariate time series.

Parallel multisplitting methods with optimal weighting matrices for linear systems

In this paper, the parallel multisplitting iterative methods with optimal weighting matrices are presented to solve a linear system of equations in which the coefficient matrix is a symmetric positive definite matrix. The zero pattern in weighting matrices is determined by preset set, the non-zero entries of weighting matrices in overlap multisplitting methods are determined optimally by finding the optimal point in hyperplane or convex combination of m points generated by parallel multisplitting iterations. Several schemes of finding the optimal weighting matrices are given. Especially, the nonnegative assumption in the weighting matrices is eliminated. The convergence properties are discussed for these parallel multisplitting methods. Finally, our numerical examples show that these parallel multisplitting methods with the optimal weighting matrices are effective.

Model selection using limiting distributions of second-order blind source separation algorithms

Signals, recorded over time, are often observed as mixtures of multiple source signals. To extract relevant information from such measurements one needs to determine the mixing coefficients. In case of weakly stationary time series with uncorrelated source signals, this separation can be achieved by jointly diagonalizing sample autocovariances at different lags, and several algorithms address this task. Often the mixing estimates contain close-to-zero entries and one wants to decide whether the corresponding source signals have a relevant impact on the observations or not. To address this question of model selection we consider the recently published second-order blind identification procedures SOBIdef and SOBIsym which provide limiting distributions of the mixing estimates. For the first time, such distributions enable informed decisions about the presence of second-order stationary source signals in the data. We consider a family of linear hypothesis tests and information criteria to perform model selection as second step after parameter estimation. In simulations we consider different time series models. We validate the model selection performance and demonstrate a good recovery of the true zero pattern of the mixing matrix.

THE CANONICAL FORM OF INVOLUTARY FUZZY MATRICES

We study special types of matrices. The involutary fuzzy matrices are important in various applications and have many interesting properties. Using the graphical method, we have the zero patterns of involutary fuzzy matrix, that is, involutary Boolean matrices. And we give the construction of all involutary fuzzy matrices for some dimensions and suggest the canonical form of involutary fuzzy matrix.