Algorithmic Differentiation of the MWGS-Based Arrays for Computing the Information Matrix Sensitivity Equations within the Problem of Parameter Identification

The paper considers the problem of algorithmic differentiation of information matrix difference equations for calculating the information matrix derivatives in the information Kalman filter. The equations are presented in the form of a matrix MWGS (modified weighted Gram–Schmidt) transformation. The solution is based on the usage of special methods for the algorithmic differentiation of matrix MWGS transformation of two types: forward (MWGS-LD) and backward (MWGS-UD). The main result of the work is a new MWGS-based array algorithm for computing the information matrix sensitivity equations. The algorithm is robust to machine round-off errors due to the application of the MWGS orthogonalization procedure at each step. The obtained results are applied to solve the problem of parameter identification for state-space models of discrete-time linear stochastic systems. Numerical experiments confirm the efficiency of the proposed solution.

Orthogonal Signals and the Orthogonalization Procedure

Chapter 2 is dedicated to the principle of signal orthogonalization, because orthogonal signals are widely used in telecommunication theory and practice, like the carriers of baseband signals, subcarriers in orthogonal frequency division multiplexing systems, and the spreading sequences in spread-spectrum and code division multiple access (CDMA) systems. The orthonormal basis functions are defined and the procedure of the vector representation of signals is demonstrated. The Gram–Schmidt orthogonalization procedure and construction of the space diagram are presented in detail. Using orthonormal signals, signal synthesizers and analysers that can be used to form discrete-time transmitters and receivers are theoretically founded. Understanding of this chapter is a prerequisite for understanding Chapters 4–10, because the orthonormal signals defined in this chapter will be used throughout the book. The basis harmonic orthonormal functions will define the carriers in the discrete and digital communication systems.

Analysis of Feature Influence on Covid-19 Death Rate Per Country Using a Novel Orthogonalization Technique

We have developed a new technique of Feature Importance, a topic of machine learning, to analyze the possible causes of the Covid-19 pandemic based on country data. This new approach works well even when there are many more features than countries and is not affected by high correlation of features. It is inspired by the Gram-Schmidt orthogonalization procedure from linear algebra. We study the number of deaths, which is more reliable than the number of cases at the onset of the pandemic, during Apr/May 2020. This is while countries started taking measures, so more light will be shed on the root causes of the pandemic rather than on its handling. The analysis is done against a comprehensive list of roughly 3,200 features. We find that globalization is the main contributing cause, followed by calcium intake, economic factors, environmental factors, preventative measures, and others. This analysis was done for 20 different dates and shows that some factors, like calcium, phase in or out over time. We also compute row explainability, i.e. for every country, how much each feature explains the death rate. Finally we also study a series of conditions, e.g. comorbidities, immunization, etc. which have been proposed to explain the pandemic and place them in their proper context. While there are many caveats to this analysis, we believe it sheds light on the possible causes of the Covid-19 pandemic.

Compensation for In-Phase/Quadrature Phase Mismatch in Coherent Free-Space Optical QPSK Communication Systems

The Gram–Schmidt orthogonalization procedure (GSOP) and Löwdin symmetric orthogonalization procedure (SYOP) are the two mainstream algorithms for the compensation of phase mismatch in an imperfect optical 90° hybrid. In this paper, we put forward an algorithm switching orthogonalization procedure (ASOP) according to the quality of in-phase and quadrature signals based on the Q value of the eye diagram with less computation. If the quality of the in-phase and quadrature signals has a significant difference, we use the GSOP and select the signal branch with better quality as the initial reference vector for orthogonalization. If they are of about the same quality, then we use the SYOP. We present computer simulations for a coherent free-space optical (FSO) quadrature phase-shift keying (QPSK) communication system and demonstrate the system improvement that can be achieved using the ASOP. Finally, we also show that the proposed ASOP scheme can contribute to the frequency offset and phase estimation of the FSO system in the environment of atmospheric turbulence.

Numerical and asymptotic flow stability analysis of vortex structures

Stability problem of an axisymmetric swirling flow of a viscous incompressible fluid with respect to nonaxisymmetric perturbations is considered. The system of ordinary differential equations for the amplitude functions is solved numerically by the Runge-Kutta method and orthogonalization procedure. Solutions of equations for perturbations at the neighborhood of singular points are obtained by the Frobenius method. The maximum of amplification coefficients and phase velocities of five unstable modes are calculated.

Influence of orthogonalization procedure on astrophysical S-factor for the direct α + d →6Li + γ capture process in a three-body model

The astrophysical S-factor for the direct [Formula: see text] capture reaction is calculated in a three-body model based on the hyperspherical Lagrange-mesh method. A sensitivity of the E1 and E2 astrophysical S-factors to the orthogonalization method of Pauli forbidden states in the three-body system is studied. It is found that the method of orthogonalising pseudopotentials (OPP) yields larger isotriplet ([Formula: see text]) components than the supersymmetric transformation (SUSY) procedure. The E1 astrophysical S-factor shows the same energy dependence in both cases, but strongly different absolute values. At the same time, the E2 S-factor does not depend on the orthogonalization procedure. As a result, the OPP method yields a very good description of the direct data of the LUNA collaboration at low energies, while the SUSY transformation strongly underestimates the LUNA data.

The study of chaotic and regular regimes of the fractal oscillators FitzHugh-Nagumo

In this paper we study the conditions for the existence of chaotic and regular oscillatory regimes of the hereditary oscillator FitzHugh-Nagumo (FHN), a mathematical model for the propagation of a nerve impulse in a membrane. To achieve this goal, using the non-local explicit finite-difference scheme and Wolf’s algorithm with the Gram-Schmidt orthogonalization procedure and the spectra of the maximum Lyapunov exponents were also constructed depending on the values of the control parameters of the model of the FHN. The results of the calculations showed that there are spectra of maximum Lyapunov exponents both with positive values and with negative values. The results of the calculations were also confirmed with the help of oscillograms and phase trajectories, which indicates the possibility of the existence of both chaotic attractors and limit cycles.

Replication of Goolsbee, Lovenheim, and Slemrod’s “Playing with Fire: Cigarettes, Taxes, and Competition from the Internet” (American Economic Journal: Economic Policy, 2010)

This study replicates the empirical findings of Goolsbee, Lovenheim and Slemrod, henceforth GLS, and performs a variety of robustness checks. Using taxable cigarette consumption, real cigarette excise tax rates, wholesale cigarette prices, per capita income, and other state-level data for the period 1980–2005, GLS report that rising Internet penetration in the presence of cigarette taxes has a significant causal effect on the elasticity of demand for taxable cigarettes. I am able to exactly replicate GLS’s findings, and explore their robustness in three ways. First, I examine sensitivity to the removal of outlier cohorts of states from the data. Second, I use population-unweighted state-year observations in place of GLS’s population-weighted state-year observations. Third, I probe the robustness of GLS’s key interaction term (real state cigarette taxes * Internet penetration) by (i) adding Internet penetration interaction terms to all main effects in the model and (ii) performing an orthogonalization procedure (Balli and Sørensen, 2013) to purge from GLS’s estimate of the key interaction term any spurious correlation that might exist between Internet penetration and the included variables, which would then be loaded on to the interaction between Internet penetration and cigarette taxes. GLS’s results were robust to all but the orthogonalization procedure. This raises the possibility that the effect identified by GLS is an artifact of spurious correlation between Internet penetration and cigarette taxes over time. In sum, GLS’s data may have insufficient power to identify the stand-alone effect of Internet penetration on cigarette tax-sales elasticities in a fully-interacted model.