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We improve the least squares (LS) method for building models of a nonlinear dynamical system given finite time series which are contaminated by observational noise. When the noise level is low, the LS method gives good estimates for the parameters, however, the models selected as the best by information criteria often tend to be over-parameterized or even degenerate. We observe that the correct model is not selected as the best model despite belonging to the chosen model class. To overcome this, we propose a simple but very effective idea to use the LS method more appropriately. We apply the method for model selection. Numerical studies indicate that the method can be used to apply information criteria more effectively, and generally avoid over-fitting and model degeneracy.

In this article we derive rough formulas for the probability distribution functions of sampled higher-order spectra of poly-harmonic signals corrupted by Gaussian noise. These formulas provide a basis for analyzing the performance detection algorithms for finite-sample length.

Multifractal theory provides a new spatial analytical tool for urban studies, but many basic problems remain to be solved. Among various pending issues, the most significant one is how to obtain proper multifractal dimension spectrums. If an algorithm is improperly used, the parameter spectrums will be abnormal. This paper is devoted to investigating two ordinary least squares (OLS)-based approaches for estimating urban multifractal parameters. Using empirical study and comparative analysis, we demonstrate how to utilize the adequate linear regression to calculate multifractal parameters. The OLS regression analysis has two different approaches. One is that the intercept is fixed to zero, and the other is that the intercept is not limited. The results of comparative study show that the zero-intercept regression yields proper multifractal parameter spectrums within certain scale range of moment order, while the common regression method often leads to abnormal multifractal parameter values. A conclusion can be reached that fixing the intercept to zero is a more advisable regression method for multifractal parameters estimation, and the shapes of spectral curves and value ranges of fractal parameters can be employed to diagnose urban problems. This research is helpful for scientists to understand multifractal models and apply a more reasonable technique to multifractal parameter calculations.

A least-squares meshfree collocation method is presented. The method is based on the first-order differential equations in order to result in a better conditioned linear algebraic equations, and to obtain the primary variables (displacements) and the dual variables (stresses) simultaneously with the same accuracy. The moving least-squares approximation is employed to construct the shape functions. The sum of squared residuals of both differential equations and boundary conditions at nodal points is minimized. The present method does not require any background mesh and additional evaluation points, and thus is a truly meshfree method. Unlike other collocation methods, the present method does not involve derivative boundary conditions, therefore no stabilization terms are needed, and the resulting stiffness matrix is symmetric positive definite. Numerical examples show that the proposed method possesses an optimal rate of convergence for both primary and dual variables, if the nodes are uniformly distributed. However, the present method is sensitive to the choice of the influence length. Numerical examples include one-dimensional diffusion and convection-diffusion problems, two-dimensional Poisson equation and linear elasticity problems.

The method of Lin and Huang [Lin and Huang [2004] “Decomposition of incident and reflected higher harmonic waves using four wave gauges,” Coast. Eng.51(5), 395–406.] (LH) is improved for resolution of incident and reflected strongly nonlinear regular waves in shallow waters with measurements of four stationary wave gauges. For first harmonics, wavenumbers, amplitudes and initial phases are obtained by using a nonlinear least squares method. For higher harmonics, wavenumbers of free and bound modes are determined from linear dispersion relation and multiple of first-harmonic wavenumbers, respectively, and the other unknowns are solved by using a linear least squares method. Auto-correlation function is used to determine fundamental wave period for gaining a good performance of Fourier transform. The efficiency and accuracy of the present method are demonstrated by using artificial data and numerical flume data. It is also demonstrated that the present method is less sensitive to signal noise and gauge spacings. Comparison between the present method and the LH method indicates the necessity of employing nonlinear method in determining fundamental wavenumbers of nonlinear shallow-water waves. Finally, the present method is extended to account for obliquely-incident waves. Sensitivity tests indicate the robustness of the extended method with respect to incident angles. Relative position of gauges in the array for avoiding singularity is suggested.

T-wave alternans (TWA) in surface electrocardiograph (ECG) signal is considered a marker of abnormal ventricular function which may be associated with ventricular tachycardia. Several methods have been developed in recent years to evaluate the important feature. One such method is known as modified moving average (MMA) analysis, which performs well for different levels of TWA, but it is sensitive to the noise in T-waves. In this paper we propose an improved MMA algorithm, which adds a stage of T-wave curve fitting for the MMA method before intermediate averaging. The curve fitting is performed by means of least square method technique. Our assessment study demonstrates the improved performance.

In this paper, we study the problem of reconstructing an unknown initial condition for nonlinear diffusion from integral observations which have many practical meanings. We reformulate this problem as a variational one to minimizing objective functional. We prove the existence of this minimized problem in our main result.

Paper is to deal with a general fuzzy regression model with a best response function for the center and the width of the fuzzy output to explain data involving fuzzy numbers. Two numerical examples are presented comparing a performance of the proposed model based on a least squares method with the other fuzzy regression model studied by many authors.