Aufgrund der vernetzten Strukturen und Wirkungszusammenhänge dynamischer Systeme werden die zugrundeliegenden mathematischen Modelle meist sehr komplex und erfordern ein hohes mathematisches Verständnis und Geschick. Bei Verwendung von spezieller Software können jedoch auch ohne tiefgehende mathematische oder informatorische Fachkenntnisse komplexe Wirkungsnetze dynamischer Systeme interaktiv erstellt werden. Als Beispiel wollen wir schrittweise das Modell einer Miniwelt entwerfen und Aussagen bezüglich ihrer Bevölkerungsentwicklung treffen.
We consider wavelet estimation of the time-dependent (evolutionary) power spectrum of a locally stationary time series. Allowing for departures from stationary proves useful for modelling, e.g., transient phenomena, quasi-oscillating behaviour or spectrum modulation. In our work wavelets are used to provide an adaptive local smoothing of a short-time periodogram in the time-freqeuncy plane. For this, in contrast to classical nonparametric (linear) approaches we use nonlinear thresholding of the empirical wavelet coefficients of the evolutionary spectrum. We show how these techniques allow for both adaptively reconstructing the local structure in the time-frequency plane and for denoising the resulting estimates. To this end a threshold choice is derived which is motivated by minimax properties w.r.t. the integrated mean squared error. Our approach is based on a 2-d orthogonal wavelet transform modified by using a cardinal Lagrange interpolation function on the finest scale. As an example, we apply our procedure to a time-varying spectrum motivated from mobile radio propagation.
We derive minimax rates for estimation in anisotropic smoothness classes. This rate is attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present.; As an important application we introduce a new adaptive wavelet estimator of the time-dependent spectrum of a locally stationary time series. Using this model which was resently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, i.e. how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.
With this article we first like to give a brief review on wavelet thresholding methods in non-Gaussian and non-i.i.d. situations, respectively. Many of these applications are based on Gaussian approximations of the empirical coefficients. For regression and density estimation with independent observations, we establish joint asymptotic normality of the empirical coefficients by means of strong approximations. Then we describe how one can prove asymptotic normality under mixing conditions on the observations by cumulant techniques.; In the second part, we apply these non-linear adaptive shrinking schemes to spectral estimation problems for both a stationary and a non-stationary time series setup. For the latter one, in a model of Dahlhaus on the evolutionary spectrum of a locally stationary time series, we present two different approaches. Moreover, we show that in classes of anisotropic function spaces an appropriately chosen wavelet basis automatically adapts to possibly different degrees of regularity for the different directions. The resulting fully-adaptive spectral estimator attains the rate that is optimal in the idealized Gaussian white noise model up to a logarithmic factor.
This paper is devoted to the mathematica l description of the solution of the so-called rainflow reconstruction problem, i.e. the problem of constructing a time series with an a priori given rainflow m atrix. The algorithm we present is mathematically exact in the sense that no app roximations or heuristics are involved. Furthermore it generates a uniform distr ibution of all possible reconstructions and thus an optimal randomization of the reconstructed series. The algorithm is a genuine on-line scheme. It is easy adj ustable to all variants of rainflow such as sysmmetric and asymmetric versions a nd different residue techniques.
In the automotive industry both the loca l strain approach and rainflow counting are well known and approved tools in the numerical estimation of the lifetime of a new developed part especially in the automotive industry. This paper is devoted to the combination of both tools and a new algorithm is given that takes advantage of the inner structure of the most used damage parameters.
The ideas of texture analysis by means of the structure tensor are combined with the scale-space concept of anisotropic diffusion filtering. In contrast to many other nonlinear diffusion techniques, the proposed one uses a diffusion tensor instead of a scalar diffusivity. This allows true anisotropic behaviour. The preferred diffusion direction is determined according to the phase angle of the structure tensor. The diffusivity in this direction is increasing with the local coherence of the signal. This filter is constructed in such a way that it gives a mathematically well-funded scale-space representation of the original image. Experiments demonstrate its usefulness for the processing of interrupted one-dimensional structures such as fingerprint and fabric images.
A survey on continuous, semidiscrete and discrete well-posedness and scale-space results for a class of nonlinear diffusion filters is presented. This class does not require any monotony assumption (comparison principle) and, thus, allows image restoration as well. The theoretical results include existence, uniqueness, continuous dependence on the initial image, maximum-minimum principles, average grey level invariance, smoothing Lyapunov functionals, and convergence to a constant steady state.