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parametric-nonparametric identification of Hammerstein systems. — IEEE Trans. Automat. Contr. , Vol. 49, No. 8, pp. 1370-1376. Hasiewicz Z. and Mzyk G. (2004b): Nonparametric instrumental variables for Hammerstein system identification. — Int. J. Contr., (submitted). Hasiewicz Z., Pawlak M. and Śliwiński P. (2005): Nonparametric identification of nonlinearities in block-oriented systems by orthogonal wavelets with compact support. — IEEE Trans. Circ. Syst. I: Fund. Theory Applic. , Vol. 52, No. 2, pp. 427-442. Härdle W. (1990): Applied Nonparametric Regression

esti- mation by local linear smoothing, Bernoulli 4 (1998), 3-14. [4] FARAWAY, J. J.: Sequential design for the nonparametric regression of curves and sur- faces, in: Proc. of the 22nd Symposium on the Interface, Comput. Sci. Statist., Vol. 22, Springer, 1990, pp. 104-110. [5] FEDOROV, V. V.-MONTEPIEDRA,G.-NACHTSHEIM, C. J.: Design of experiments for locally weighted regression, J. Statist. Plann. Inference 81 (1999), 363-382. [6] GASSER, T.-M¨ULLER, H.-G.: Estimating regression functions and their derivatives by the kernel method, Scand. J. Stat. 11 (1984), 171

Nonlinear Image Processing and Filtering: A Unified Approach Based on Vertically Weighted Regression

A class of nonparametric smoothing kernel methods for image processing and filtering that possess edge-preserving properties is examined. The proposed approach is a nonlinearly modified version of the classical nonparametric regression estimates utilizing the concept of vertical weighting. The method unifies a number of known nonlinear image filtering and denoising algorithms such as bilateral and steering kernel filters. It is shown that vertically weighted filters can be realized by a structure of three interconnected radial basis function (RBF) networks. We also assess the performance of the algorithm by studying industrial images.

Abstract

The idea of worm tracking refers to the path analysis of Caenorhabditis elegans nematodes and is an important tool in neurobiology which helps to describe their behavior. Knowledge about nematode behavior can be applied as a model to study the physiological addiction process or other nervous system processes in animals and humans. Tracking is performed by using a special manipulator positioning a microscope with a camera over a dish with an observed individual. In the paper, the accuracy of a nematode’s trajectory reconstruction is investigated. Special attention is paid to analyzing errors that occurred during the microscope displacements. Two sources of errors in the trajectory reconstruction are shown. One is due to the difficulty in accurately measuring the microscope shift, the other is due to a nematode displacement during the microscope movement. A new method that increases path reconstruction accuracy based only on the registered sequence of images is proposed. The method Simultaneously Localizes And Tracks (SLAT) the nematodes, and is robust to the positioning system displacement errors. The proposed method predicts the nematode position by using NonParametric Regression (NPR). In addition, two other methods of the SLAT problem are implemented to evaluate the NPR method. The first consists in ignoring the nematode displacement during microscope movement, and the second is based on a Kalman filter. The results suggest that the SLAT method based on nonparametric regression gives the most promising results and decreases the error of trajectory reconstruction by 25% compared with reconstruction based on data from the positioning system

] N. Ling, Q. Xu, Asymptotic normality of conditional density estimation in the single index model for functional time series data, Statistics & Probability Letters, 82 (12), (2012), 2235–2243. [8] Nengxiang Ling, Zhihuan Li, Wenzhi Yang (2014) Conditional Density Estimation in the Single Functional Index Model for a-Mixing Functional Data, Communications in Statistics - Theory and Methods , 43:3, 441–454, DOI: 10.1080/03610926.2012.664236 [9] E. Masry, Nonparametric regression estimation for dependent functional data: Asymptotic normality. Stoch. Proc. and

-1604. Brown, L.D. and Levine, M. (2007). Variance estimation in nonparametric regression via the difference sequence method, Annals of Statistics 35(5): 2219-2232. Carroll, R.J. and Ruppert, D. (1988). Transformation and Weighting in Regression, CRC Press, Boca Raton, FL. Dai, W., Ma, Y., Tong, T. and Zhu, L. (2015). Difference-based variance estimation in nonparametric regression with repeated measurement data, Journal of Statistical Planning and Inference 163: 1-20. Diggle, P.J. and Verbyla, A.P. (1998). Nonparametric estimation of covariance structure in longitudinal

Analysis, Theory and Applications 30 (3): 1343-1354. Rafajłowicz E. and Skubalska-Rafajłowicz E. (1993). FFT in calculating nonparametric regression estimate based on trigonometric series, International Journal of Applied Mathematics and Computer Science 3 (4): 713-720. Sher C. F., Tseng C. S. and Chen C. (2001). Properties and performance of orthogonal neural network in function approximation, International Journal of Intelligent Systems 16 (12): 1377-1392. Tseng C. S. and Chen C. S. (2004). Performance comparison between the training method and the numerical method

Identification , Cambridge University Press, New York, NY. Györfi, L., Kohler, M., Krzyżak, A. and Walk, H. (2002). A Distribution-Free Theory of Nonparametric Regression , Springer-Verlag, New York, NY. Haber, R. and Keviczky, L. (1999). Nonlinear System Parameter Identification , Kluwer Academic Publishers, Dordrecht/Boston/London. Hasiewicz, Z., Pawlak, M. and Śliwiński, P. (2005). Nonparametric identification of non-linearities in block-oriented complex systems by orthogonal wavelets with compact support, IEEE Transactions on Circuits and Systems I: Regular Papers   52

-77. Greblicki, W. and Pawlak, M. (1987). Necessary and sufficient consistency conditions for a recursive kernel regression estimate, Journal of Multivariate Analysis 23 (1): 67-76. Greblicki, W. and Pawlak, M. (1989). Recursive nonparametric identification of Hammerstein systems, Journal of the Franklin Institute 326 (4): 461-481. Greblicki, W. and Pawlak, M. (2008). Nonparametric System Identification , Cambridge University Press, New York, NY. Györfi, L., Kohler, M., Krzyżak, A. and Walk, H. (2002). A Distribution-Free Theory of Nonparametric Regression , Springer

nonparametric conditional mode estimator for functional time series data, Statist. Neerlandica. , 64 (2010), 171–201. [14] F. Ferraty, A. Laksaci, P. Vieu, Estimating some characteristics of the conditional distribution in nonparametric functional models. Statist. Inference Stochastic Process , 9 (2006), 47–76. [15] F. Ferraty, A. Laksaci, A. Tadj, P. Vieu, Rate of uniform consistency for nonparametric estimates with functional variables, J. Statist. Plann. and Inf. , 140 (2010), 335–352. [16] F. Ferraty, A. Mas, P. Vieu, Advances in nonparametric regression for