Average Polynomial Research Articles

Identifiability and estimation of possibly non-invertible SVARMA Models: The normalised canonical WHF parametrisation

This article focuses on the parametrisation, identifiability, and (quasi-) maximum likelihood (QML) estimation of possibly non-invertible structural vector autoregressive moving average (SVARMA) models. SVAR models are routinely adopted due to their well-known implementation strategy. However, for various economic and statistical reasons, multivariate SVARMA settings are often more suitable. These settings introduce complexity in the analysis, primarily due to the presence of the moving average (MA) polynomial. We propose a novel representation of the MA polynomial matrix using the Wiener–Hopf factorization (WHF). A significant advantage of the WHF is its ability to handle possible non-invertibility and thus models with informational asymmetry between economic agents and outside observers. Since solutions of Dynamic Stochastic General Equilibrium (DSGE) models often involve this informational asymmetry, SVARMA models in WHF parametrisation can be considered data-driven alternatives to DSGE models and used for their evaluation. Furthermore, we provide low-level conditions for the asymptotic normality of the (Q)ML estimator and analytic expressions for the score and information matrix. As application, we estimate the Blanchard and Quah model, and compare our results and implied impulse response function with the ones in the SVAR model by Blanchard and Quah and a non-invertible SVARMA model by Gouriéroux and co-authors. Importantly, we have implemented this novel method in a well-documented R-package, making it readily accessible for researchers and practitioners.

Hankel determinants for a Gaussian weight with Fisher–Hartwig singularities and generalized Painlevé IV equation

We study the Hankel determinant generated by a Gaussian weight with Fisher–Hartwig singularities of root type at tj , . It characterizes a type of average characteristic polynomial of matrices from Gaussian unitary ensembles. We derive the ladder operators satisfied by the associated monic orthogonal polynomials and three compatibility conditions. By using them and introducing 2N auxiliary quantities , we build a series of difference equations. Furthermore, we prove that satisfy Riccati equations. From them we deduce a system of second order PDEs satisfied by , which reduces to a Painlevé IV equation for N = 1. We also show that the logarithmic derivative of the Hankel determinant satisfies the generalized σ-form of a Painlevé IV equation.

Nesting the SIRV model with NAR, LSTM and statistical methods to fit and predict COVID-19 epidemic trend in Africa

ObjectiveCompared with other regions in the world, the transmission characteristics of the COVID-19 epidemic in Africa are more obvious, has a unique transmission mode in this region; At the same time, the data related to the COVID-19 epidemic in Africa is characterized by low data quality and incomplete data coverage, which makes the prediction method of COVID-19 epidemic suitable for other regions unable to achieve good results in Africa. In order to solve the above problems, this paper proposes a prediction method that nests the in-depth learning method in the mechanism model. From the experimental results, it can better solve the above problems and better adapt to the transmission characteristics of the COVID-19 epidemic in African countries.MethodsBased on the SIRV model, the COVID-19 transmission rate and trend from September 2021 to January 2022 of the top 15 African countries (South Africa, Morocco, Tunisia, Libya, Egypt, Ethiopia, Kenya, Zambia, Algeria, Botswana, Nigeria, Zimbabwe, Mozambique, Uganda, and Ghana) in the accumulative number of COVID-19 confirmed cases was fitted by using the data from Worldometer. Non-autoregressive (NAR), Long-short term memory (LSTM), Autoregressive integrated moving average (ARIMA) models, Gaussian and polynomial functions were used to predict the transmission rate β in the next 7, 14, and 21 days. Then, the predicted transmission rate βs were substituted into the SIRV model to predict the number of the COVID-19 active cases. The error analysis was conducted using root-mean-square error (RMSE) and mean absolute percentage error (MAPE).ResultsThe fitting curves of the 7, 14, and 21 days were consistent with and higher than the original curves of daily active cases (DAC). The MAPE between the fitted and original 7-day DAC was only 1.15% and increased with the longer of predict days. Both the predicted β and DAC of the next 7, 14, and 21 days by NAR and LSTM nested models were closer to the real ones than other three ones. The minimum RMSEs for the predicted number of COVID-19 active cases in the next 7, 14, and 21 days were 12,974, 14,152, and 12,211 people, respectively when the order of magnitude for was 106, with the minimum MAPE being 1.79%, 1.97%, and 1.64%, respectively.ConclusionNesting the SIRV model with NAR, LSTM, ARIMA methods etc. through functionalizing β respectively could obtain more accurate fitting and predicting results than these models/methods alone for the number of confirmed COVID-19 cases in Africa in which nesting with NAR had the highest accuracy for the 14-day and 21-day predictions. The nested model was of high significance for early understanding of the COVID-19 disease burden and preparedness for the response.

Sphere detector with low complexity Euclidean distance computation based on affine transform modulation

The foreseen scale of connected autonomous and low power consumption devices in the near future is a challenging aspect for the next generation of mobile communication networks. As the number of available connection resources is limited, frequency-time multiplexing may not be sufficient to meet the expected demand for capacity, where scheduling solutions might incur in undesired latency levels and also increase the power expenditure to meet a more stringent synchronisation. In this scenario, spatial multiplexing based on multi-user multiple-input multiple-output can extend the uplink capacity at the cost of inter-antenna interference at the base station, requiring more complex detection schemes. The sphere detector can mitigate the inter-antenna interference, achieving close-to-optimum bit error rate performance with average polynomial complexity that still challenges the field programmable gate array implementation in terms of resources and area. To further reduce the sphere detector complexity, the authors introduce the concept of affine transform modulation, which allows us to describe the received signal in terms of the sequence of transmitted bits per symbol. The authors demonstrate that this approach allows to evaluate the Euclidean distance in the sphere detector algorithm in terms of the transmitted bits, replacing the complex inner product by a complex accumulator in its extensively accessed core function. The approach proposed in this paper leads to a considerable complexity reduction in terms of FLOPs for low-order modulations while attaining the conventional sphere detector performance.

On the unimodality of average edge cover polynomials

The edge cover polynomial of a graph G is the function E(G,x)=∑i≥1e(G,i)xi, where e(G,i) is the number of edge coverings of G with size i. In this paper, we show that the average edge cover polynomial of order n is reduced to the edge cover polynomial of complete graph Kn, based on which we establish that the average edge cover polynomial of order n is unimodal and has at least n−3 non-real roots.

Multiplication operator and average characteristic polynomial associated with exceptional Jacobi polynomials

Studying the multiplication operator associated with exceptional Jacobi polynomials, the zero distribution of the corresponding average characteristic polynomials is determined. Applying this result, the location of zeros of certain self-inversive polynomials is examined.

Variants of Jacobi polynomials in coding theory

In this paper, we introduce the notion of the complete joint Jacobi polynomial of two linear codes of length n over $${\mathbb {F}}_{q}$$ and $${\mathbb {Z}}_{k}$$ . We give the MacWilliams type identity for the complete joint Jacobi polynomials of codes. We also introduce the concepts of the average Jacobi polynomial and the average complete joint Jacobi polynomial over $${\mathbb {F}}_{q}$$ and $${\mathbb {Z}}_{k}$$ . We give a representation of the average of the complete joint Jacobi polynomials of two linear codes of length n over $${\mathbb {F}}_{q}$$ and $${\mathbb {Z}}_{k}$$ in terms of the compositions of n and its distribution in the codes. Further we present a generalization of the representation for the average of the $$(g+1)$$ -fold complete joint Jacobi polynomials of codes over $${\mathbb {F}}_{q}$$ and $${\mathbb {Z}}_{k}$$ . Finally, we give the notion of the average Jacobi intersection number of two codes.

The average domination polynomial of graphs is unimodal

The average domination polynomial of graphs is unimodal

Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality

We compute the average characteristic polynomial of the hermitised product of $M$ real or complex Wigner matrices of size $N\times N$ and the average of the characteristic polynomial of a product of $M$ such Wigner matrices times the characteristic polynomial of the conjugate matrix. Surprisingly, the results agree with that of the product of $M$ real or complex Ginibre matrices at finite-$N$, which have i.i.d. Gaussian entries. For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two characteristic polynomials involves the kernel of complex eigenvalues. This extends the result of Forrester and Gamburd for one characteristic polynomial of a single random matrix and only depends on the first two moments. In the limit $M\to\infty$ at fixed $N$ we determine the locations of the zeros of a single characteristic polynomial, rescaled as Lyapunov exponents by taking the logarithm of the $M$th root. The position of the $j$th zero agrees asymptotically for large-$j$ with the position of the $j$th Lyapunov exponent for products of Gaussian random matrices, hinting at the universality of the latter.

ПРОГНОЗНО-АНАЛИТИЧЕСКАЯ ЧАСТЬ ЦИФРОВОЙ БИЗНЕС-МОДЕЛИ СЕЛЬХОЗОРГАНИЗАЦИИ РЕГИОНА

Profit is the key objective of the business enterprise, and its forecast plays a major role both in the initial business planning stage and in its ongoing work, since it allows the manager to make timely management decisions and adjust the vector of his development. The authors proposed an analytical model for forecasting the production and financial results of the functioning of economic entities of the Saratov region as one of the integral elements of digital business modeling of the activities of the agricultural organization of the region. The article considers the concepts of a modern business model of a commercial enterprise, presents a digital business model of an agricultural organization from the point of view of managing its main constituent elements (processes), shows key approaches to its construction, reveals the structure of its production unit. The proposed predictive and analytical component of the digital business model of the agricultural organization allows to investigate the production and financial results of activities of economic entities of the Saratov region in dynamics over 10 years by applying one of the methods of graphical analysis of data (superimposition of the moving average polynomial line of the second degree), as well as to find the corresponding equation for detecting the state of indicators in the near term. In order to improve the reliability of the predictive and analytical part of the digital business model, a correlation-regression analysis method was used, as a result of which coefficients were identified, reflecting the degree of interaction between individual elements of costs for agricultural production, on the one hand, and factors affecting profits, on the other. The main element of the novelty of this work is the combination of methods of graphical and correlation-regression analysis of data, as a result of which the authors proposed a formula for determining a number of forecast values of factors affecting the profits of agricultural producers, taking into account the coefficients of their relationship with individual cost elements.

Counting points on superelliptic curves in average polynomial time

We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $\mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $m\ge 2$ and $f\in \mathbb Z[x]$ any squarefree polynomial of degree $d\ge 3$, along with a positive integer $N$. It can compute $\#X(\mathbb F_p)$ for all $p\le N$ not dividing $m\mathrm{lc}(f)\mathrm{disc}(f)$ in time $O(md^3 N\log^3 N\log\log N)$. It achieves this by computing the trace of the Cartier--Manin matrix of reductions of $X$. We can also compute the Cartier--Manin matrix itself, which determines the $p$-rank of the Jacobian of $X$ and the numerator of its zeta function modulo~$p$.

Hypergeometric L-functions in average polynomial time

We describe an algorithm for computing, for all primes $p \leq X$, the mod-$p$ reduction of the trace of Frobenius at $p$ of a fixed hypergeometric motive in time quasilinear in $X$. This combines the Beukers--Cohen--Mellit trace formula with average polynomial time techniques of Harvey et al.

A Systematic Review on Lagged Association in Environmental Studies

The delayed association in environment-health studies has been acknowledged in recent years. However, the misspecification of lag dimension in time-series models, particularly distributed lag non-linear models (DLNM), would induce considerable deviation of effect estimate. This study reviewed the existing climate-health English literature with time-series and case-crossover design published during 2000-2019 to summarize the statistical methodologies used and the reported delays of association between meteorological variables and 14 common causes of morbidity and mortality. Generalized linear or additive model was the most widely employed method for regression analysis. Different types of lag design were adopted for infectious disease modeling, including cross-correlation analysis, single lag model, moving average lag model, unconstrained or polynomial distributed lag model, and DLNM, whereas studies on non-communicable diseases predominantly used DLNM to assess the delays of association. For infectious outcomes, the association of daily mean temperature was found to be lagged for one to two weeks for influenza, followed by two to five weeks for diarrhea, and eight to twelve weeks for dengue fever. Meanwhile, the association of both cardiovascular and respiratory diseases with hot temperatures lasted for less than five days, whereas the association of cardiovascular diseases with cold temperatures was observed for ten to twenty days. Additionally, rainfall, as a potential risk factor for infectious diseases, showed a four to eight weeks’ lagged association with diarrheal diseases, while the effect was further delayed to eight to twelve weeks for vector-borne diseases. This is the first systematic review that comprehensively provides epidemiological evidence on the delay of association of common meteorological parameters. Biologically plausible and reasonable definition of effect lag in the modeling process is warranted in further environmental epidemiological studies.

Comparative Results of Regional and Residual Anomalies with the Upward Continuation, Moving Average, and Polynomial Methods for Magnetic Data

Geophysics programing of regional and residual anomaly separation on Magnetic data has been carried out with the results compared with the upward continuation method in the OasisMontaj software. Separation of anomalies with moving average and polynomial methods is processed using Matlab programming. The orders used in the polynomial method are first-order, second-order and third-order. Comparison is done by calculating the match value. The chosen matching method is autocorrelation. Correlation of residual magnetic anomalies resulting from upward continuation (Magpick) to moving averages, 1st-order polynomials, 2nd-order polynomials and 3rd-order polynomials. Correlation values obtained for the moving average method are 0.9604, first order polynomial 0.9072, 2nd order polynomial 0.9482 and third order polynomial 0.6057. The moving average and second order polynomial methods can be used as a substitute method if we do not use the upward continuation method.

The average Laplacian polynomial of a graph

Let G=(V(G),E(G)) be a graph of order n=|V(G)| and size m=|E(G)|, and let Σ(G)={σ:E(G)→{−1,+1}} be the set of sign functions defined on the edges of G. Then, it is well known that μ(G,x)=12m∑σ∈Σ(G)det(xIn−A(Gσ)), where μ(G,x) is the matching polynomial of G, and A(Gσ) is the adjacency matrix of the signed graph Gσ=(G,σ). Motivated by this result, in this paper, we introduce the average Laplacian polynomial of G, which is defined as ψ¯(G,x)=12m∑σ∈Σ(G)det(xIn−L(Gσ)), where L(Gσ) is the Laplacian matrix of the signed graph Gσ. We find that this polynomial is closely related to the structure of the graph. For example, its coefficients can be expressed naturally in terms of the TU-subgraphs of G, and the multiplicity of 0 as a root is equal to the number of tree components in G. The relations between the average Laplacian polynomial of G and other polynomials, especially, the matching polynomial, are also investigated in this paper. Based on the relations, we prove that the roots of the average Laplacian polynomial of any graph are non-negative real numbers.

Asymptotics for recurrence coefficients of X1-Jacobi exceptional polynomials and Christoffel function

ABSTRACTComputing asymptotics of the recurrence coefficients of -Jacobi polynomials, we investigate the limit of the Christoffel function. We also study the relation between the normalized counting measure based on the zeros of the modified average characteristic polynomial and the Christoffel function in limit. The proofs of the corresponding theorems with respect to standard orthogonal polynomials are based on the three-term recurrence relation. The main point is that exceptional orthogonal polynomials possess at least five-term formulae and so Christoffel-Darboux formula also fails. It seems that these difficulties can be handled in a combinatorial way.

High-Precision Modeling for Aerospace TT&C Channels Using Piecewise Osculating Interpolation

Polynomial interpolation is widely used in channel modeling for aerospace tracking, telemetry, and command channel simulators, to accurately regenerate the satellite motion parameters with a high sampling rate from the available low sampling rate motion data. In this paper, a novel piecewise osculating interpolation approach is proposed to improve modeling precision. It uses the available distance and its higher-order derivatives at each pair of adjacent knots to construct the piecewise interpolation polynomial, thus achieving a better precision and continuity of the resulting motion parameters. Moreover, it is efficiently implemented with an interpolation filter based on a modified Farrow structure, and the designed filter can interpolate the motion data obtained both through uniform or nonuniform sampling. Additionally, a sampling optimization algorithm is proposed. It optimizes the motion data sampling by minimizing the maximum interpolation error given the average sampling interval and interpolation polynomial order, and therefore, improves the precision further without increasing the costs. The validity of the proposed approach is verified by numerical simulation, and its advantages are demonstrated by comparison with existing methods.

Identification and Estimation in Non-Fundamental Structural VARMA Models

AbstractThe basic assumption of a structural vector autoregressive moving average (SVARMA) model is that it is driven by a white noise whose components are uncorrelated or independent and can be interpreted as economic shocks, called “structural” shocks. When the errors are Gaussian, independence is equivalent to non-correlation and these models face two identification issues. The first identification problem is “static” and is due to the fact that there is an infinite number of linear transformations of a given random vector making its components uncorrelated. The second identification problem is “dynamic” and is a consequence of the fact that, even if a SVARMA admits a non-invertible moving average (MA) matrix polynomial, it may feature the same second-order dynamic properties as a VARMA process in which the MA matrix polynomials are invertible (the fundamental representation). The aim of this article is to explain that these difficulties are mainly due to the Gaussian assumption, and that both identification challenges are solved in a non-Gaussian framework if the structural shocks are assumed to be instantaneously and serially independent. We develop new parametric and semi-parametric estimation methods that accommodate non-fundamentalness in the MA dynamics. The functioning and performances of these methods are illustrated by applications conducted on both simulated and real data.

Vibrational dependence, temperature dependence, and prediction of line shape parameters for the H2O-H2 collision system

In a recent work on line shape parameters for the H2O-H2 [Renaud et al., Icarus, 306,275, 2018] it was shown that this far off-resonance collision system displays strong vibrational dependence. In Renaud et al., calculations were made for bands with one bend or stretch (symmetric or antisymmetric) vibrational quanta exchanged. Water is seen throughout the universe under a vast range of conditions. The needs of the spectroscopic and astrophysics communities include data for water vapor transitions with multiple quanta exchanged. In this work, calculations were made using the Modified Complex Robert-Bonamy (MCRB) formalism for 0–4 vibrational quanta exchanged in the ν1, ν2, and ν3 bands, and the 3ν1 + ν3 band at 13 temperatures from 200 to 3000 K: calculations for more than 100,000 transitions. From these data, the vibrational dependence of the half-width, line shift, and their temperature dependence are studied as a function of rotational transition. The results show a strong and unusual vibrational dependence. The data were used to develop a prediction routine, which is based on theory [Gamache and Hartmann, JQSRT, 83, 119, 2004] rather than ad hoc fitting to a polynomial or other forms. The resulting prediction algorithm gives line shape parameters with much lower uncertainty compared to those from J″ average values or polynomial methods. A line file, based on HITRAN2016, was created for remote sensing applications that study H2O in hydrogen-rich atmospheres.

Statistical behavior of the characteristic polynomials of a family of pseudo-Hermitian Gaussian matrices

In this paper, we extend previous studies conducted by the authors in a family of pseudo-Hermitian Gaussian matrices. Namely, we further the studies of the two pseudo-Hermitian random matrix cases previously considered, the first of a matrix of order N with two interacting blocks of sizes M and N − M and the second of a chessboard-like structured matrix of order N whose subdiagonals alternate between Hermiticity and pseudo-Hermiticity. Following an average characteristic polynomial approach, we obtain sequences of polynomials whose roots describe the average value of the polynomials of the matrices of the family at hand, for each case considered. We also present numerical results regarding the statistical behavior of the average characteristic polynomial, and contrast that to the spectral behavior of sample matrices of this family.