Doğrusal süzgeçleme ve doğrusal olmayan öngörü için yeni uyarlanır algoritmalar

In this thesis, we consider two adaptive filtering tasks: Linear Adaptive Filtering andNonlinear Adaptive Prediction. We handle system identification and sequential (online)nonlinear prediction problems for these tasks respectively. 3 novel adaptive algorithms (2in linear filtering and 1 in nonlinear prediction) are presented in this thesis.For linear adaptive filtering, Least Mean Squares (LMS) is a fundamental, simple yetnot fast enough converging algorithm. Proportionate update idea that is proposed by Dut-tweiler in [1] achieves a significant development in the convergence speed of LMS for sparsesystems. We develop the Proportionate Normalized Least Mean Fourth (PNLMF) algo-rithm by minimizing mean fourth error (MFE) in the same way as [2] produces the LeastMean Fourth (LMF) algorithm from LMS. Yukawa's implementation of a Krylov subspaceprojection technique into the problem extends the use of proportionate update idea to non-sparse systems. We exploit the same Krylov subspace projection technique and introducethe Krylov-Proportionate Normalized Least Mean Fourth (KPNLMF) algorithm by againminimizing MFE. The Krylov-Proportionate Normalized Least Mean Squares (KPNLMS)algorithm minimizes mean square error (MSE) and our introduced KPNLMF algorithm isthe MFE counterpart of KPNLMS. We observe the same relation between KPNLMS andKPNLMF as the one between LMS and LMF that is presented in [2]. Simulations showthat KPNLMF attains a much lower mismatch (filter weight error power) than KPNLMSwhen the system noise has a probability density function among certain types that are ofpractical importance. It is also shown in the simulations that KPNLMF converges fasterthan the Normalized LMF (NLMF) algorithm. Another contribution of this thesis is thatthe steady-state MSE analysis is performed both for KPNLMS and KPNLMF. They areboth shown theoretically to converge to the desired solution according to the steady-stateMSE criterion.In the second main part of the thesis, we deal with the sequential nonlinear prediction ofan arbitrary, deterministic and bounded signal from its noise-corrupted past samples undersquare error loss. We present a novel randomized sequential prediction algorithm and nameit Competitive Randomized Noisy Nonlinear Predictor (CRNNP). The main contribution ofthis thesis in this topic is that the CRNNP algorithm achieves a high prediction performanceunder additive noise. Since our CRNNP algorithm works for an arbitrary deterministicsignal, we introduce a competitive framework in order to define a meaningful performancemeasure. CRNNP works in a competition class of algorithms that hypothetically workin parallel. We show that CRNNP achieves the performance of the best algorithm thatcan both select the best partition of the past observations space and the affine modelparameters based on the desired clean signal in hindsight. The competition class is theclass of certain nonlinear models, i.e. piecewise affine models represented on a context-tree.So, we employ a context-tree structure to model the nonlinearity that exists in the problem.The CRNNP algorithm serves for sequential decision problem and it makes its decision bychoosing a strategy from several number of strategies at each time in a randomized fashion.Randomization weights are determined according to the prediction performances of thestrategies.