58,691 research outputs found
Stochastic Behavior Analysis of the Gaussian Kernel Least-Mean-Square Algorithm
The kernel least-mean-square (KLMS) algorithm is a popular algorithm in nonlinear adaptive filtering due to its
simplicity and robustness. In kernel adaptive filters, the statistics of the input to the linear filter depends on the parameters of the kernel employed. Moreover, practical implementations require a finite nonlinearity model order. A Gaussian KLMS has two design parameters, the step size and the Gaussian kernel bandwidth. Thus, its design requires analytical models for the algorithm behavior as a function of these two parameters. This paper studies the steady-state behavior and the transient behavior of the
Gaussian KLMS algorithm for Gaussian inputs and a finite order nonlinearity model. In particular, we derive recursive expressions for the mean-weight-error vector and the mean-square-error. The model predictions show excellent agreement with Monte Carlo simulations in transient and steady state. This allows the explicit analytical determination of stability limits, and gives opportunity
to choose the algorithm parameters a priori in order to achieve prescribed convergence speed and quality of the estimate. Design examples are presented which validate the theoretical analysis and illustrates its application
Application of Multivariate Selective Bandwidth Kernel Density Estimation for Data Correction
This paper presents an intuitive application of multivariate kernel density
estimation (KDE) for data correction. The method utilizes the expected value of
the conditional probability density function (PDF) and a credible interval to
quantify correction uncertainty. A selective KDE factor is proposed to adjust
both kernel size and shape, determined through least-squares cross-validation
(LSCV) or mean conditional squared error (MCSE) criteria. The selective
bandwidth method can be used in combination with the adaptive method to
potentially improve accuracy. Two examples, involving a hypothetical dataset
and a realistic dataset, demonstrate the efficacy of the method. The selective
bandwidth methods consistently outperform non-selective methods, while the
adaptive bandwidth methods improve results for the hypothetical dataset but not
for the realistic dataset. The MCSE criterion minimizes root mean square error
but may yield under-smoothed distributions, whereas the LSCV criterion strikes
a balance between PDF fitness and low RMSE.Comment: 16 pages, 6 figure
Quaternion Information Theoretic Learning Adaptive Algorithms for Nonlinear Adaptive
Information Theoretic Learning (ITL) is gaining popularity for designing adaptive filters for a non-stationary or non-Gaussian environment [1] [2] . ITL cost functions such as the Minimum Error Entropy (MEE) have been applied to both linear and nonlinear adaptive filtering with better overall performance compared with the typical mean squared error (MSE) and least-squares type adaptive filtering, especially for nonlinear systems in higher-order statistic noise environments [3].
Quaternion valued data processing is beneficial in applications such as robotics and image processing, particularly for performing transformations in 3-dimensional space. Particularly the benefit for quaternion valued processing includes performing data transformations in a 3 or 4-dimensional space in a more convenient fashion than using vector algebra [4, 5, 6, 7, 8]. Adaptive filtering in quaterion domain operates intrinsically based on the augmented statistics which the quaternion input vector covariance is taken into account naturally and as a result it incorporates component-wise real valued cross-correlation or the coupling within the dimensions of the quaternion input [9].
The generalized Hamilton-real calculus (GHR) for the quaternion data simplified product and chain rules and allows us to calculate the gradient and Hessian of quaternion based cost function of the learning algorithms eciently [10][11] . The quaternion reproducing kernel Hilbert spaces and its uniqueness provide a mathematical foundation to develop the quaternion value kernel learning algorithms [12]. The reproducing property of the feature space replace the inner product of feature samples with kernel evaluation. In this dissertation, we first propose a kernel adaptive filter for quaternion data based on minimum error entropy cost function. The new algorithm is based on error entropy function and is referred to as the quaternion kernel minimum error entropy (QKMEE) algorithm [13]. We apply generalized Hamilton-real (GHR) calculus that is applicable to quaternion Hilbert space for evaluating the cost function gradient to develop the QKMEE algorithm. The minimum error entropy (MEE) algorithm [3, 14, 15] minimizes Renyis quadratic entropy of the error between the lter output and desired response or indirectly maximizing the error information potential. ITL methodology improves the performance of adaptive algorithm in biased or non-Gaussian signals and noise enviorments compared to the mean squared error (MSE) criterion algorithms such as the kernel least mean square algorithm.
Second, we develop a kernel adaptive filter for quaternion data based on normalized minimum error entropy cost function [14]. We apply generalized Hamilton-real GHR) calculus that is applicable to Hilbert space for evaluating the cost function gradient to develop the quaternion kernel normalized minimum error entropy (QKNMEE) algorithm [16]. The new proposed algorithm enhanced QKMEE algorithm where the filter update stepsize selection will be independent of the input power and the kernel size.
Third, we develop a kernel adaptive lter for quaternion domain data, based on information theoretic learning cost function which could be useful for quaternion based kernel applications of nonlinear filtering. The new algorithm is based on error entropy function with fiducial point and is referred to as the quaternion kernel minimum error entropy with fiducial point (QKMEEF) algorithm [17]. In our previous work we developed quaternion kernel adaptive lter based on minimum error entropy referred to as the QKMEE algorithm [13]. Since entropy does not change with the mean of the distribution, the algorithm may converge to a set of optimal weights without having zero mean error. Traditionally, to make the zero mean output error, the output during testing session was biased with the mean of errors of training session. However, for non-symmetric or heavy tails error PDF the estimation of error mean is problematic [18]. The minimum error entropy criterion, minimizes Renyi\u27s quadratic entropy of the error between the filter output and desired response or indirectly maximizing the error information potential [19]. Here, the approach is applied to quaternions. Adaptive filtering in quaterion domain intrinsically incorporates component-wise real valued cross-correlation or the coupling within the dimensions of the quaternion input. We apply generalized Hamilton-real (GHR) calculus that is applicable to Hilbert space for evaluating the cost function gradient to develop the Quaternion Minimum Error Entropy Algorithm with Fiducial point. Simulation results are used to show the behavior of the new algorithm (QKMEEF) when signal is non-Gaussian in presence of unimodal noise versus bi-modal noise distributions. Simulation results also show that the new algorithm QKMEEF can track and predict the 4-Dimensional non-stationary process signals where there are correlations between components better than quadruple real-valued KMEEF and Quat-KLMS algorithms.
Fourth, we develop a kernel adaptive filter for quaternion data, using stochastic information gradient (SIG) cost function based on the information theoretic learning (ITL) approach. The new algorithm (QKSIG) is useful for quaternion-based kernel applications of nonlinear ltering [20]. Adaptive filtering in quaterion domain intrinsically incorporates component-wise real valued cross-correlation or the coupling within the dimensions of the quaternion input. We apply generalized Hamilton-real (GHR) calculus that is applicable to quaternion Hilbert space for evaluating the cost function gradient. The QKSIG algorithm minimizes Shannon\u27s entropy of the error between the filter output and desired response and minimizes the divergence between the joint densities of input-desired and input-output pairs. The SIG technique reduces the computational complexity of the error entropy estimation. Here, ITL with SIG approach is applied to quaternion adaptive filtering for three different reasons. First, it reduces the algorithm computational complexity compared to our previous work quaternion kernel minimum error entropy algorithm (QKMEE). Second, it improves the filtering performance by considering the coupling within the dimensions of the quaternion input. Third, it performs better in biased or non-Gaussian signal and noise environments due to ITL approach. We present convergence analysis and steady-state performance analysis results of the new algorithm (QKSIG). Simulation results are used to show the behavior of the new algorithm QKSIG in quaternion non-Gaussian signal and noise environments compared to the existing ones such as quadruple real-valued kernel stochastic information gradient (KSIG) and quaternion kernel LMS (QKLMS) algorithms.
Fifth, we develop a kernel adaptive filter for quaternion data, based on stochastic information gradient (SIG) cost function with self adjusting step-size. The new algorithm (QKSIG-SAS) is based on the information theoretic learning (ITL) approach. The new algorithm (QKSIG-SAS) has faster speed of convergence as compared to our previous work QKSIG algorithm
A stochastic behavior analysis of stochastic restricted-gradient descent algorithm in reproducing kernel Hilbert spaces
This paper presents a stochastic behavior analysis of a kernel-based
stochastic restricted-gradient descent method. The restricted gradient gives a
steepest ascent direction within the so-called dictionary subspace. The
analysis provides the transient and steady state performance in the mean
squared error criterion. It also includes stability conditions in the mean and
mean-square sense. The present study is based on the analysis of the kernel
normalized least mean square (KNLMS) algorithm initially proposed by Chen et
al. Simulation results validate the analysis
Improving Sparsity in Kernel Adaptive Filters Using a Unit-Norm Dictionary
Kernel adaptive filters, a class of adaptive nonlinear time-series models,
are known by their ability to learn expressive autoregressive patterns from
sequential data. However, for trivial monotonic signals, they struggle to
perform accurate predictions and at the same time keep computational complexity
within desired boundaries. This is because new observations are incorporated to
the dictionary when they are far from what the algorithm has seen in the past.
We propose a novel approach to kernel adaptive filtering that compares new
observations against dictionary samples in terms of their unit-norm
(normalised) versions, meaning that new observations that look like previous
samples but have a different magnitude are not added to the dictionary. We
achieve this by proposing the unit-norm Gaussian kernel and define a
sparsification criterion for this novel kernel. This new methodology is
validated on two real-world datasets against standard KAF in terms of the
normalised mean square error and the dictionary size.Comment: Accepted at the IEEE Digital Signal Processing conference 201
- âŠ