According to a new mean-value theorem, under the conditions of a function f ensuring the existence and uniqueness of Lagrange’s mean, there exists a unique mean M such that f(x)− f(y) x− y = M ( f ′(x), f ′(y) ) . The main result says that, in this equality, M is a power mean if, and only if, M is either geometric, arithmetic or harmonic. A Cauchy relevant type result is also presented. Introdu...

Crossover rate (CR) is a key parameter affecting the operation of differential evolution (DE). According to the different status appear in CR adaptive process, the present paper employs power mean averaging operators to improve the value of CR in appropriate chance and propose a Power Mean based Crossover Rate Adaptive Differential Evolution (PMCRADE). The performance of PMCRADE is evaluated on...

As real industrial processes have measurement samples with noises of different statistical characteristics and obtain the sample one by one usually, on-line sequential learning algorithms which can achieve better learning performance for systems with noises of various statistics are necessary. This paper proposes a new online Extreme Learning Machine (ELM, of Huang et al.) algorithm, namely rec...

Correntropy is a second order statistical measure in kernel space, which has been successfully applied in robust learning and signal processing. In this paper, we define a nonsecond order statistical measure in kernel space, called the kernel mean-p power error (KMPE), including the correntropic loss (C-Loss) as a special case. Some basic properties of KMPE are presented. In particular, we appl...

1. Institute of Artificial Intelligence and Robotics, Xi'an Jiaotong University, Xi'an, 710049, China 2. School of Electronic and Information Engineering, South China University of Technology, Guangzhou, 510640, China 3. School of Automation & Information Engineering, Xi'an University of Technology, Xi'an 710048, China 4. Department of Electrical and Computer Engineering, University of Florida,...

In this paper, bounds on the mean power-weighted nearest neighbor distance are derived. Previous work concentrates mainly on the infinite sample limit, whereas our bounds hold for any sample size. The results are expected to be of importance for example in statistical physics, nonparametric statistics and computational geometry, where they are related to the structure of matter as well as prope...

For a large class of statistical systems a geometric mean value of the ob-servables is constrained. These observables are characterized by a power-law statistical distribution. In everyday life we find events of very different nature often to follow similar statistical distributions. These distributions arise because the same stochastic process is at work, and this process can be understood bey...

Recently much attention has been paid to the nonextensive canonical distributions: the αfamilies. Such distributions have been found in many real-world systems such as fully developed turbulence and financial markets. In this paper, a generalized mean-field method to approximate the expectations of the α-families is proposed. We calculate the α′-projection of a probability distribution to find ...

A selfadjoint involutive matrix J endows C with an indefinite inner product [·, ·] given by [x, y] := 〈Jx, y〉, x, y ∈ C. We present some inequalities of indefinite type involving the α−power mean and the chaotic order. These results are in the vein of those obtained by E. Kamei [6, 7].