Quasi-arithmetic Means Research Articles

Weighted squared envelope nonlinear Gini index and its application to rotating machinery condition monitoring

Abstract Sparsity measures are effective tools for rotating machinery condition monitoring. However, under complex operating conditions, existing sparsity measures often exhibit significant fluctuations, making it difficult to accurately detect early faults and monotonically represent the degradation processes. To tackle these challenges, a novel health indicator named the weighted squared envelope nonlinear Gini index is proposed in this paper. Firstly, by introducing a nonlinearly increasing weight sequence inspired by the sigmoid function and the quadratic function-based quasi-arithmetic mean, a novel sparsity measure, the nonlinear Gini index, is developed based on the ratio of different quasi-arithmetic means framework. Building upon this foundation, the weighted squared envelope nonlinear Gini index is further constructed for condition monitoring by incorporating the weighted squared envelope into the nonlinear Gini index. When applied to two bearing run-to-failure datasets, the proposed health indicator shows improved sensitivity to early fault features and is able to depict the degradation processes monotonically, demonstrating notable advantages in condition monitoring.

Extremal values-based aggregation functions

We introduce and study aggregation functions based on extremal values, namely extended (l,u)-aggregation functions whose outputs only depend on a fixed number l of extremal lower input values and a fixed number u of extremal upper input values, independently of the arity of the input n-tuples (n≥l+u). We discuss several general properties of (l,u)-aggregation functions and we study special (l,u)-aggregation functions with neutral element, including t-conorms, t-norms and uninorms. We also study (l,u)-aggregation functions defined by means of integrals with respect to discrete fuzzy measures, as well as (l,u)-ordered weighted quasi-arithmetic means based on appropriate weighting vectors. We also stress some generalizations based on recently introduced new types of monotonicity. Some possible applications are sketched, too.

A generalized form of the distance-induced OWA operators – Demonstrating its use for evaluation indicator system in China

Using multi-criteria decision making (MCDM) technique to rank alternatives is a well-known area of study in which aggregation operators, such as ordered weighted averaging (OWA) and induced OWA (IOWA) operators, play an important role in merging information and producing an overall ranking. The distance measures from ideal argument values in aggregation operators have gained attention in recent literature. Distance measures are traditionally used as argument variables, which leads to the depiction that the attribute cannot be aggregated directly. In this paper, a generalized form of distance-induced OWA (DIOWA) operators is proposed with distance measures used as order-inducing variables. A distinctive benefit of DIOWA operators is that they permit us to consider ideal argument values while simultaneously also taking the attribute values as argument variables. Three variants of DIOWA operators are proposed and investigated, namely a) the Hamming distance-induced OWA (HDIOWA) operator, b) the normalized Hamming distance-induced OWA (NHDIOWA) operator, and c) the weighted Hamming distance-induced OWA (WHDIOWA) operator. We highlight their important properties and provide proofs to necessary theorems, and also suggest the determination methods for calculating their associated weights. We discuss further extensions of the proposed DIOWA operators with the help of generalized and quasi-arithmetic means. We discuss the use of our proposed family of operators for two different decision making situations, and demonstrate their validity by an illustrative numerical example. Finally, we apply the proposed operators to a real-life problem of ranking Chinese provinces for their science and technology (S&T) development levels. The proposed operators are shown to be a useful addition to the aggregation toolbox for decision analysts.

Necessary and sufficient conditions for discrete inequalities of Jensen–Steffensen type with applications

AbstractIn this paper we give a necessary and sufficient condition for the discrete Jensen inequality to be satisfied for real (not necessarily nonnegative) weights. The result generalizes and completes the classical Jensen–Steffensen inequality. The validity of the strict inequality is studied. As applications, we first give the form of our result for strongly convex functions, then we study discrete quasi-arithmetic means with real (not necessarily nonnegative) weights, and finally we refine the inequality obtained.

Weak log-majorization and inequalities of power means

As noncommutative versions of the quasi-arithmetic mean, we consider the Lim-Pálfia's power mean, Rényi right mean, and Rényi power means. We prove that the Lim-Pálfia's power mean of order $t \in [-1,0)$ is weakly log-majorized by the log-Euclidean mean and fulfills the Ando-Hiai inequality. We establish the log-majorization relationship between the Rényi relative entropy and the product of square roots of given variables. Furthermore, we show the norm inequalities among power means and provide the boundedness of Rényi power mean in terms of the quasi-arithmetic mean.

A representation of a class of quasi-arithmetic means using a unary modifier operator

In this short communication, we will show that a quasi-arithmetic mean induced by an additive generator of a strict t-norm (strict t-conorm, respectively) can be represented by the composition of a unary operator (called the tau function) and the strict t-norm (strict t-conorm, respectively), both induced by the same generator function as the quasi arithmetic mean. Here, we will also state a connection between the idempotency of a transformed strict t-norm (strict t-conorm, respectively) and the tau function.

Refinements of discrete and integral Jensen inequalities with Jensen’s gap

Motivated by a paper of Dragomir, we give new refinements for both discrete and integral Jensen inequalities using the Jensen’s gap. As applications, we give refinements of various inequalities verifiable by Jensen’s inequality. Topics covered: norms, quasi-arithmetic means, Hölder’s inequality and f-divergences in information theory.

New cyclic Jensen type estimations VIA weighted Hadamard inequalities under the effect of Fink's identity

We take into account differences resulting from the discrete and integral cyclic Jensen's inequalities and provide upper and lower bounds using weighted Hermite-Hadamard inequalities with the support of Fink's identity. The theory of n-times differentiable convex functions is used to demonstrate this case. Our findings are applicable to any n ​≥ ​2, and we present specific examples to demonstrate the accuracy of the bounds found for unique situations. We also formulate results for power means and quasi-arithmetic means for higher order convexity. Last but not least, we provide applications in information theory by offering new uniform estimations of the generalized Csiszár, Rényi, Shannon, Kullback-Leibler, χ2 ​− ​divergence, Bhattacharyya, Zipf, and Hybrid Zipf-Mandelbrot entropies.

On the Jensen convexity of quasideviation and Bajraktarević means

AbstractMotivated by the characterization theorem about the Jensen convexity of quasiarithmetic means obtained by the authors in 2021, our main goal is to establish a characterization of the Jensen convexity of quasideviation as well as of Bajraktarević means without any additional and unnatural regularity assumptions.

Sugeno Integral for Hermite–Hadamard Inequality and Quasi-Arithmetic Means

Abstract In this paper, we present the Sugeno integral of Hermite–Hadamard inequality for the case of quasi-arithmetically convex (q-ac) functions which acts as a generator for all quasi-arithmetic means in the frame work of Sugeno integral.

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some numerical experiments are performed to determine the significance and correctness of the intended estimates. Several outcomes of the main results are discussed for the Hölder inequality and the power and quasi-arithmetic means. Furthermore, some applications are presented in information theory, which provide some bounds for the divergences, Bhattacharyya coefficient, Shannon entropy, and Zipf–Mandelbrot entropy.

A dichotomy result for strictly increasing bisymmetric maps

In this paper we show some remarkable consequences of the method which proves that every bisymmetric, symmetric, reflexive, strictly monotonic binary map on a proper interval is continuous, in particular it is a quasi-arithmetic mean. Now we demonstrate that this result can be refined in the way that the symmetry condition can be weakened by assuming symmetry only for a pair of distinct points of an interval.

Power function-based Gini indices: New sparsity measures using power function-based quasi-arithmetic means for bearing condition monitoring

The Gini index (GI), GI II, and GI III are proven to be effective sparsity measures in the fields of machine condition monitoring and fault diagnosis, and they can be reformulated as the ratio of different quasi-arithmetic means (RQAM). Under this framework, generalized Gini indices (GGIs) have been developed for sparse quantification by applying nonlinear weights to GI, and another generalized form of GI, referred to here as power function-based Gini indices I (PFGI1s), has been introduced by using power function as the generator of quasi-arithmetic means. The GGIs with different weight parameters exhibit reliable sparse quantization capability for repetitive transient features, while their repetitive transient discriminability is lower than kurtosis and negentropy under noise contamination. PFGI1 achieves enhanced repetitive transient discriminability with increasing power exponent, showing the advantage of the generalization approach. In this paper, based on RQAM, a single-parameter generalization method for generating PFGI1s is introduced into GI II and GI III from the perspective of the quasi-arithmetic mean generator, which leads to the power function-based Gini indices II and III (PFGI2s and PFGI3s) constructed from GI II and GI III, respectively. Mathematical derivation proves that PFGI2s and PFGI3s satisfy at least five of six typical attributes of sparsity measures and are two new families of sparsity measures. Simulation analysis shows that, similar to PFGI1s, PFGI2s and PFGI3s can monotonically estimate the sparsity of the data sequence and can simultaneously achieve strong random transient resistibility and high repetitive transient discriminability compared with traditional sparsity measures. The experimental results of bearing run-to-failure demonstrate that PFGI1s, PFGI2s, and PFGI3s with appropriate power exponents can effectively quantify the repetitive transient features caused by bearing faults and can accurately characterize the bearing degradation status compared with the state-of-the-art sparsity measures.

ON THE INVARIANCE OF GENERALIZED QUASIARITHMETIC MEANS

The generalized quasiarithmetic mean is generated by two functions and one probability measure, and includes quasiarithmetic, Cauchy and Bajraktarević meas. In this paper, we investigate the invariance of the arithmetic mean with respect to generalized quasiarithmetic means and get some solutions of it under high-order differentiability assumptions.

Limit Theorems for Deviation Means of Independent and Identically Distributed Random Variables

We derive a strong law of large numbers, a central limit theorem, a law of the iterated logarithm and a large deviation theorem for so-called deviation means of independent and identically distributed random variables. (For the strong law of large numbers, we suppose only pairwise independence instead of (total) independence.) The class of deviation means is a special class of M-estimators or more generally extremum estimators, which are well studied in statistics. The assumptions of our limit theorems for deviation means seem to be new and weaker than the known ones for M-estimators in the literature. In particular, our results on the strong law of large numbers and on the central limit theorem generalize the corresponding ones for quasi-arithmetic means due to de Carvalho (Am Stat 70(3):270–274, 2016) and the ones for Bajraktarević means due to Barczy and Burai (Aequ Math 96(2):279–305, 2022).

Generalizing the Alpha-Divergences and the Oriented Kullback–Leibler Divergences with Quasi-Arithmetic Means

The family of α-divergences including the oriented forward and reverse Kullback–Leibler divergences is often used in signal processing, pattern recognition, and machine learning, among others. Choosing a suitable α-divergence can either be done beforehand according to some prior knowledge of the application domains or directly learned from data sets. In this work, we generalize the α-divergences using a pair of strictly comparable weighted means. Our generalization allows us to obtain in the limit case α→1 the 1-divergence, which provides a generalization of the forward Kullback–Leibler divergence, and in the limit case α→0, the 0-divergence, which corresponds to a generalization of the reverse Kullback–Leibler divergence. We then analyze the condition for a pair of weighted quasi-arithmetic means to be strictly comparable and describe the family of quasi-arithmetic α-divergences including its subfamily of power homogeneous α-divergences. In particular, we study the generalized quasi-arithmetic 1-divergences and 0-divergences and show that these counterpart generalizations of the oriented Kullback–Leibler divergences can be rewritten as equivalent conformal Bregman divergences using strictly monotone embeddings. Finally, we discuss the applications of these novel divergences to k-means clustering by studying the robustness property of the centroids.

IGIgram: An Improved Gini Index-Based Envelope Analysis for Rolling Bearing Fault Diagnosis

The transient impulse features caused by rolling bearing faults are often present in the resonance frequency band which is closely related to the dynamic characteristics of the machine structure. Informative frequency band identification is a crucial prerequisite for envelope analysis and thereby accurate fault diagnosis of rolling bearings. In this paper, based on the ratio of quasi-arithmetic means and Gini index, improved Gini indices (IGIs) are proposed to quantify the transient impulse features of a signal, and their effectiveness and advantages in sparse quantification are confirmed by simulation analysis and comparisons with traditional sparsity measures. Furthermore, an IGI-based envelope analysis method named IGIgram is developed for fault diagnosis of rolling bearings. In the new method, an IGI-based indicator is constructed to evaluate the impulsiveness and cyclostationarity of the narrow-band filtered signal simultaneously, and then a frequency band with abundant fault information is adaptively determined for extracting bearing fault features. The performance of the IGIgram method is verified on the simulation signal and railway bearing experimental signals and compared with typical sparsity measures-based envelope analysis methods and log-cycligram. The results demonstrate that the proposed IGIs are efficient in quantifying bearing fault-induced transient features and the IGIgram method with appropriate power exponent can effectively achieve the diagnostics of different axle-box bearing faults.

Limit theorems for quasi-arithmetic means of random variables with applications to point estimations for the Cauchy distribution

We establish some limit theorems for quasi-arithmetic means of random variables. This class of means contains the arithmetic, geometric and harmonic means. Our feature is that the generators of quasi-arithmetic means are allowed to be complex-valued, which makes considerations for quasi-arithmetic means of random variables which could take negative values possible. Our motivation for the limit theorems is finding simple estimators of the parameters of the Cauchy distribution. By applying the limit theorems, we obtain some closed-form unbiased strongly-consistent estimators for the joint of the location and scale parameters of the Cauchy distribution, which are easy to compute and analyze.

Gini Indices II and III: Two new Sparsity Measures and Their Applications to Machine Condition Monitoring

Machine condition monitoring (MCM) uses signal processing and machine learning methods to analyze monitoring data and perform timely condition-based maintenance. Since monitoring data usually have a sparsity property, sparsity measures (SMs) are naturally considered to quantify the sparsity of signals and they serve as the objective functions of many signal processing and machine learning methods. Although Gini index, kurtosis, smoothness index, negative entropy, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Lp</i> / <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Lq</i> norm have been considerably investigated for MCM, the design of new SMs for enhancing MCM is rarely reported. In this article, based on the ratio of different quasi-arithmetic means (RQAM), two new SMs, coined as Gini index Ⅱ (GI2) and Gini index Ⅲ (GI3), are designed. New proofs show that the GI2 and GI3 satisfy all six sparsity attributes. Subsequently, the GI2 and GI3 of the square envelope of Gaussian white noise are theoretically investigated and their theoretical values are, respectively, equal to 2/3 and 1/3, which can be used as baselines for machine abnormality detection. Once GI2 and GI3 exceed the baselines, abnormal health conditions can be detected without needing historical data and prior fault knowledge. Finally, simulated and experimental case studies showed that the proposed GI2 and GI3 have competitive performance with Gini index and that they are better than kurtosis, negative entropy, and smoothness index, in characterizing the sparsity of signals. This article demonstrates that the RQAM is a potential framework to design new SMs.

Refinement of Discrete Lah–Ribarič Inequality and Applications on Csiszár Divergence

In this paper we give a new refinement of the Lah–Ribarič inequality and, using the same technique, we give a refinement of the Jensen inequality. Using these results, a refinement of the discrete Hölder inequality and a refinement of some inequalities for discrete weighted power means and discrete weighted quasi-arithmetic means are obtained. We also give applications in the information theory; namely, we give some interesting estimations for the discrete Csiszár divergence and for its important special cases.