A Semi-Parametric Non-linear Neural Network Filter: Theory and Empirical Evidence

In this work, we decompose a time series into trend and cycle by introducing a novel de-trending approach based on a family of semi-parametric artificial neural networks. Based on this powerful approach, we propose a relevant filter and show that the proposed trend specification is a global approximation to any arbitrary trend. Furthermore, we prove formally a famous claim by Kydland and Prescott (1981, 1997) that over long time periods, the average value of the cycles is zero. A simple procedure for the econometric estimation of the model is developed as a seven-step algorithm, which relies on standard techniques, where all relevant measures may be computed routinely. Next, using relevant DGPs, we compare and show by means of Monte Carlo simulations that our approach is superior to Hodrick–Prescott (HP) and Baxter and King (BK) regarding the generated distortionary effects and the ability to operate in various frequencies, including changes in volatility, amplitudes and phase. In fact, while keeping the structure of the model relatively simple, our approach is perfectly capable of addressing the case of stochastic trend, in the sense that the generated distortionary effects in the near unit root case are minimal and, by all means, considerably fewer than those generated by HP and BK. Application to EU15 business cycles clustering is presented and the empirical results are consistent with the rigorous theoretical framework developed in this work.     

Three-Dimensional Diffraction Tomography Using Filtered Backpropagation and Multiple Illumination Planes

In this paper, a three-dimensional (3-D) extension of the well-known filtered-backpropagation (FBP) algorithm is presented with the aim of taking into account scattered-field-data measurements obtained using incident directions not restricted in a single plane. The FBP algorithm has been extensively used to solve the two-dimensional inverse-scattering problem under the first-order Born and Rytov approximations for weak scatterers. The extension of this algorithm in three dimensions is not straightforward, because the task of collecting the data needed to obtain a low-pass filtered version of the scattering object, taking into account all spatial frequencies within a radius of radic2k₀, and of incorporating these data to the FBP algorithm, needs to be addressed. A simple extension using incident field directions restricted to a single plane (illumination plane) leaves a region of spatial frequencies of the sphere of radius radic2k ₀ undetermined. The locus of these spatial frequencies may be crucial for the accurate reconstruction of objects which do not vary slowly along the axis perpendicular to the illumination plane. The proposed 3-D FBP algorithm presented here is able to incorporate the data collected from more than one illumination plane and to ensure the reliability of the reconstruction results     

Investigating the enhancement of three-dimensional diffraction tomography by using multiple illumination planes

The three dimensional (3-D) extension of the two well-known diffraction tomography algorithms, namely, direct Fourier interpolation (DFI) and filtered backpropagation (FBP), are presented and the problem of the data needed for a full 3-D reconstruction is investigated. These algorithms can be used efficiently to solve the inverse scattering problem for weak scatterers in the frequency domain under the first-order Born and Rytov approximations. Previous attempts of 3-D reconstruction with plane-wave illumination have used data obtained with the incident direction restricted at the xy plane. However, we show that this restriction results in the omission of the contribution of certain spatial frequencies near the omegaz axis for the final reconstruction. The effect of this omission is studied by comparing the results of reconstruction with and without data obtained from other incident directions that fill the spatial frequency domain. We conclude that the use of data obtained for incident direction in only the xy plane is sufficient to achieve a satisfactory quality of reconstruction for a class of objects presenting smooth variation along the z axis, while abrupt variations along the z axis cannot be imaged. This result should be taken into account in the process of designing the acquisition geometry of a tomography scanner.