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The random Fourier features (RFFs) method is a powerful and popular technique in kernel approximation for scalability of kernel methods. The theoretical foundation of RFFs is based on the Bochner theorem (Bochner in Harmonic Analysis and the Theory of Probability, University of California Press, 1995) that relates symmetric, positive definite (PD) functions to probability measures. This condition naturally excludes asymmetric functions with a wide range applications in practice, e.g., directed graphs, conditional probability, and asymmetric kernels. Nevertheless, understanding asymmetric functions (kernels) and its scalability via RFFs is unclear both theoretically and empirically. In this paper, we introduce a complex measure with the real and imaginary parts corresponding to four finite positive measures, which expands the application scope of the Bochner theorem. By doing so, this framework allows for handling classical symmetric, PD kernels via one positive measure; symmetric, non-positive definite kernels via signed measures; and asymmetric kernels via complex measures, thereby unifying them into a general framework by RFFs, named AsK-RFFs. Such approximation scheme via complex measures enjoys theoretical guarantees in the perspective of uniform convergence. In algorithmic implementation, to speed up the kernel approximation process, which is expensive due to the calculation of total masses, we propose a subset-based fast estimation method. This method focuses on optimizing total masses within a sub-training set, effectively transforming the numerical integration for total masses into quadratic programming in three-dimension with low time complexity. AsK-RFFs provides two explicit feature mappings to approximate an asymmetric kernel. These mappings can be utilized in classifiers within the framework of AsK-LS (He et al. in IEEE Trans Pattern Anal Machine Intell 45(8):10044–10054), fulfilling the purpose of using asymmetric kernels in machine learning applications. Our AsK-RFFs method is empirically validated on several typical large-scale datasets and achieves promising kernel approximation performance, which demonstrates the effectiveness of AsK-RFFs.

Kernel density estimation is a nonparametric procedure making use of the smoothing power of the convolution operation. Yet, it performs poorly when the density of a positive variable is estimated, due to boundary issues. So, various extensions of the kernel estimator allegedly suitable for R + -supported densities, such as those using asymmetric kernels, abound in the literature. Those, however, are not based on any valid smoothing operation. By contrast, in this paper a kernel density estimator is defined through the Mellin convolution, the natural analogue on R + of the usual convolution. From there, a class of asymmetric kernels related to Meijer G -functions is suggested, and asymptotic properties of this ‘Mellin–Meijer kernel density estimator’ are presented. In particular, its pointwise- and L 2 -consistency (with optimal rate of convergence) are established for a large class of densities, including densities unbounded at 0 and showing power-law decay in their right tail.

In this paper, we complement a study recently conducted in a paper of H.A. Mombeni, B. Masouri and M.R. Akhoond by introducing five new asymmetric kernel c.d.f. estimators on the half-line [0,∞), namely the Gamma, inverse Gamma, LogNormal, inverse Gaussian and reciprocal inverse Gaussian kernel c.d.f. estimators. For these five new estimators, we prove the asymptotic normality and we find asymptotic expressions for the following quantities: bias, variance, mean squared error and mean integrated squared error. A numerical study then compares the performance of the five new c.d.f. estimators against traditional methods and the Birnbaum–Saunders and Weibull kernel c.d.f. estimators from Mombeni, Masouri and Akhoond. By using the same experimental design, we show that the LogNormal and Birnbaum–Saunders kernel c.d.f. estimators perform the best overall, while the other asymmetric kernel estimators are sometimes better but always at least competitive against the boundary kernel method from C. Tenreiro.

The idea behind density-based clustering is to associate groups to the connected components of the level sets of the density of the data to be estimated by a nonparametric method. This approach claims some advantages over both distance- and model-based clustering. Some researchers developed this technique by proposing a graph theory–based method for identifying local modes of the underlying density being estimated by the well-known kernel density estimation (KDE) with normal and t kernels. The present work proposes a semi-parametric KDE with a more flexible family of kernels including skew-normal (SN) and skew- t (ST). We show that the proposed estimator not only reduces boundary bias but it is also closer to the actual density compared to that of the usual estimator employing the Gaussian kernel. Finding optimal bandwidth for one-dimensional and multidimensional cases under the mentioned asymmetric kernels is another main result of this paper where we shrink the bandwidth more than the one obtained under the normal assumption. Finally, through a comprehensive numerical study, we will illustrate the application of the proposed semi-parametric KDE on the density-based clustering using some simulated and real data sets.

An accurate prediction of the machining tool condition during the cutting process is crucial for enhancing the tool life, improving the production quality and productivity, optimizing the labor and maintenance costs, and reducing workplace accidents. Currently, tool condition monitoring is usually based on machine learning algorithms, especially deep learning algorithms, to establish the relationship between sensor signals and tool wear. However, deep mining of feature and fusion information of multi-sensor signals, which are strongly related to the tool wear, is a critical challenge. To address this issue, in this study, an integrated prediction scheme is proposed based on deep learning algorithms. The scheme first extracts the local features of a single sequence and a multi-dimensional sequence from DenseNet incorporating a heterogeneous asymmetric convolution kernel. To obtain more perceptual historical data, a “dilation” scheme is used to extract features from a single sequence, and one-dimensional dilated convolution kernels with different dilation rates are utilized to obtain the differential features. At the same time, asymmetric one-dimensional and two-dimensional convolution kernels are employed to extract the features of the multi-dimensional signal. Ultimately, all the features are fused. Then, the time-series features hidden in the sequence are extracted by establishing a depth-gated recurrent unit. Finally, the extracted in-depth features are fed to the deep fully connected layer to achieve the mapping between features and tool wear values through linear regression. The results indicate that the average errors of the proposed model are less than 8%, and this model outperforms the other tool wear prediction models in terms of both accuracy and generalization.

The modified (or second version) gamma kernel of Chen [Probability density function estimation using gamma kernels, Annals of the Institute of Statistical Mathematics 52 (2000), pp. 471–480] should not be automatically preferred to the standard (or first version) gamma kernel, especially for univariate convex densities with a pole at the origin. In the multivariate case, multiple combined gamma kernels, defined as a product of univariate standard and modified ones, are here introduced for nonparametric and semiparametric smoothing of unknown orthant densities with support [0,∞)d. Asymptotical properties of these multivariate associated kernel estimators are established. Bayesian estimation of adaptive bandwidth vectors using multiple pure combined gamma smoothers, and in semiparametric setup, are exactly derived under the usual quadratic function. The simulation results and four illustrations on real datasets reveal very interesting advantages of the proposed combined approach for nonparametric smoothing, compare to both pure standard and pure modified gamma kernel versions, and under integrated squared error and average log-likelihood criteria.

Deep learning has made tremendous advances in the domains of image segmentation and object classification. However, real-time lane line detection and departure estimates in complex traffic conditions have proven to be hard in autonomous driving research. Traditional lane line detection methods require manual parameter modification, but they have some limitations that are still susceptible to interference from obscuring objects, lighting changes, and pavement deterioration. The development of accurate lane line detection and departure estimate algorithms is still a challenge. This article investigated a convolutional neural network (CNN) for lane line detection and departure estimate in a complicated road environment. CNN includes a weight-sharing function that lowers the training parameters. CNN can learn and extract features frequently in image segmentation, object detection, classification, and other applications. The symmetric kernel convolution of classical CNN is upgraded to the structure of asymmetric kernel convolution (AK-CNN) based on lane line detection and departure estimation features. It reduces the CNN network's computational load and improves the speed of lane line detection and departure estimates. The experiment was carried out on the CULane dataset. The lane line detection results have high accuracy in a complex environment by 80.3%. The detection speed is 84.5 fps, which enables real-time lane line detection.

Nonparametric density estimation for nonnegative data is considered in a situation where a random sample is not directly available but the data are instead observed from the length-biased sampling. Due to the so-called boundary bias problem of the location-scale kernel, the approach in this paper is an application of asymmetric kernel. Some nonparametric density estimators are proposed. The mean integrated squared error, strong consistency, and asymptotic normality of the estimators are investigated. Simulation studies and a real data analysis illustrate the estimators.

In the nonparametric setup, smooth density estimators are obtained by using a kernel function and it is used to account for contribution of each observation to the estimator. Generally, these kernel functions are symmetric about zero. We propose an estimator by using data dependent asymmetric kernels being generated by an unimodal symmetric kernel. To account for the contribution of the ith order statistic to the estimator, a suitable asymmetric kernel is used. Asymptotic Bias, Mean Square Error (MSE) and Mean Integrated Square Error (MISE) of the estimator are obtained with higher accuracies by using some properties of the densities of order statistics. The convergence rates depend on both n, the sample size and hn , the band width. Expressions for the optimum band width minimizing the asymptotic MISE and the rate of convergence of the optimum MISE are obtained. Simulation study and data analysis indicate that the proposed estimator not only reduces bias outside the support significantly but also it is closer to the true density function as compared to that of the usual estimator based on a common symmetric kernel. Scope for the multivariate generalization of the method is given. Some methods of generating multivariate skew symmetric kernels are described. As an illustration one of them has been used in the simulation study and estimation of density in bivariate case.

The joint analysis of non-stationary and high frequency financial data poses theoretical challenges due to that such massive data varies with time and possesses no fixed density function. This paper proposes the local linear smoothing to estimate the unknown volatility function in scalar diffusion models based on Gamma asymmetric kernels for high frequency financial big data. Under the mild conditions, we obtain the asymptotic normality for the estimator at both interior and boundary design points. Besides the standard properties of the local linear estimator such as simple bias representation and boundary bias correction, the local linear smoothing using Gamma asymmetric kernels possesses some extra advantages such as variance reduction and resistance to sparse design, which is validated through finite sample simulation study and empirical analysis on 6-month Shanghai Interbank Offered Rate (abbreviated as Shibor) in China.