Norm Entropy Research Articles

Optimal guessing under nonextensive framework and associated moment bounds

We consider the problem of guessing the realization of a random variable but under more general Tsallis' non-extensive entropic framework rather than the classical Maxwell-Boltzman-Gibbs-Shannon framework. We consider both the conditional guessing problem in the presence of some related side information, and the unconditional one where no such side-information is available. For both types of the problem, the non-extensive moment bounds of the required number of guesses are derived; here we use the $q$-normalized expectation in place of the usual (linear) expectation to define the non-extensive moments. These moment bounds are seen to be a function of the logarithmic norm entropy measure, a recently developed two-parameter generalization of the Renyi entropy, and hence provide their information theoretic interpretation. We have also considered the case of uncertain source distribution and derived the non-extensive moment bounds for the corresponding mismatched guessing function. These mismatched bounds are interestingly seen to be linked with an important robust statistical divergence family known as the relative $(\alpha,\beta)$-entropies; similar link is discussed between the optimum mismatched guessing with the extremes of these relative entropy measures.

Detection of Parkinson’s disease from EEG signals using discrete wavelet transform, different entropy measures, and machine learning techniques

Early detection of Parkinson’s disease (PD) is very important in clinical diagnosis for preventing disease development. In this study, we present efficient discrete wavelet transform (DWT)-based methods for detecting PD from health control (HC) in two cases, namely, off-and on-medication. First, the EEG signals are preprocessed to remove major artifacts before being decomposed into several EEG sub-bands (approximate and details) using DWT. The features are then extracted from the wavelet packet-derived reconstructed signals using different entropy measures, namely, log energy entropy, Shannon entropy, threshold entropy, sure entropy, and norm entropy. Several machine learning techniques are investigated to classify the resulting PD/HC features. The effects of DWT coefficients and brain regions on classification accuracy are being investigated as well. Two public datasets are used to verify the proposed methods: the SanDiego dataset (31 subjects, 93 min) and the UNM dataset (54 subjects, 54 min). The results are promising and show that four entropy measures: log energy entropy, threshold entropy, sure entropy, and modified-Shannon entropy (TShEn) lead to high classification accuracy, indicating they are good biomarkers for PD detection. With the SanDiego dataset, the classification results of off-medication PD versus HC are 99.89, 99.87, and 99.91 for accuracy, sensitivity, and specificity, respectively, using the combination of DWT + TShEn and KNN classifier. Using the same combination, the results of on-medication PD versus HC are 94.21, 93.33, and 95%. With the UNM dataset, the obtained classification accuracy is around 99.5% in both cases of off-and on-medication PD using DWT + TShEn + SVM and DWT + ThEn + KNN, respectively. The results also demonstrate the importance of all DWT coefficients and that selecting a suitable small number of EEG channels from several brain regions could improve the classification accuracy.

Seismic Damage Identification Method for Curved Beam Bridges Based on Wavelet Packet Norm Entropy

Curved beam bridges, whose line type is flexible and beautiful, are an indispensable bridge type in modern traffic engineering. Nevertheless, compared with linear bridges, curved beam bridges have more complex internal forces and deformation due to the curvature; therefore, this type of bridge is more likely to suffer damage in strong earthquakes. The occurrence of damage reduces the safety of bridges, and can even cause casualties and property loss. For this reason, it is of great significance to study the identification of seismic damage in curved beam bridges. However, there is currently little research on curved beam bridges. For this reason, this paper proposes a damage identification method based on wavelet packet norm entropy (WPNE) under seismic excitation. In this method, wavelet packet transform is adopted to highlight the damage singularity information, the norm entropy of wavelet coefficient is taken as a damage characteristic factor, and then the occurrence of damage is characterized by changes in the damage index. To verify the feasibility and effectiveness of this method, a finite element model of Curved Continuous Rigid-Frame Bridges (CCRFB) is established for the purposes of numerical simulation. The results show that the damage index based on WPNE can accurately identify the damage location and characterize the severity of damage; moreover, WPNE is more capable of performing damage location and providing early warning than the method based on wavelet packet energy. In addition, noise resistance analysis shows that WPNE is immune to noise interference to a certain extent. As long as a series of frequency bands with larger correlation coefficients are selected for WPNE calculation, independent noise reduction can be achieved.

Convergence Rates for the Quantum Central Limit Theorem

Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state rho with finite second moments, converges to the Gaussian state with the same first and second moments as rho . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate mathcal {O}left( n^{-1/2}right) in the Hilbert–Schmidt norm whenever the third moments of rho are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities lambda ^{1/n} fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate mathcal {O}Big (n^{-frac{1}{2(m+1)}}Big ). This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function chi _rho is uniformly bounded by some eta _rho <1 outside of any neighbourhood of the origin; also, eta _rho can be made to depend only on the energy of the state rho .

Correlation minor norms, entanglement detection and discord

In this paper we develop an approach for detecting entanglement, which is based on measuring quantum correlations and constructing a correlation matrix. The correlation matrix is then used for defining a family of parameters, named Correlation Minor Norms, which allow one to detect entanglement. This approach generalizes the computable cross-norm or realignment (CCNR) criterion, and moreover requires measuring a state-independent set of operators. Furthermore, we illustrate a scheme which yields for each Correlation Minor Norm a separable state that maximizes it. The proposed entanglement detection scheme is believed to be advantageous in comparison to other methods because correlations have a simple, intuitive meaning and in addition they can be directly measured in experiment. Moreover, it is demonstrated to be stronger than the CCNR criterion. We also illustrate the relation between the Correlation Minor Norm and entanglement entropy for pure states. Finally, we discuss the relation between the Correlation Minor Norm and quantum discord. We demonstrate that the CMN may be used to define a new measure for quantum discord.

On the entropy norm on $${\text {Ham}}(S^2)$$

In this note we prove that for each positive integer m there exists a bi-Lipschitz embedding $${\mathbf{Z}}^m\rightarrow {\text {Ham}}(S^2)$$ , where $${\text {Ham}}(S^2)$$ is equipped with the entropy metric. In particular, the same result holds when the entropy metric is replaced with the autonomous metric.

Flow Regime, Slug Frequency and Wavelet Analysis of Air/Newtonian and Air/non-Newtonian Two-Phase Flow

This study focused on gas/Newtonian and gas/non-Newtonian two-phase horizontal fluid flow behavior by analyzing their flow regime identification and flow structural analysis on a horizontal flow loop apparatus. This involved the recognition of two-phase flow regimes for this flow loop and validation with existing flow maps in the literature. In addition, the study included flow pattern identification via wavelet analysis for gas/Newtonian and gas/non-Newtonian two-phase fluid flow in a horizontal flow loop apparatus. Furthermore, the study was extended to the detailed examination of slug frequency in the presence of air/Newtonian and air/non-Newtonian fluid flow, and the predicted slug frequency model was applied to the studied systems. The obtained results suggest that the flow regime maps and slug frequency analysis have a significant impact. The obtained pressure sensor results indicate that the experimental setup could not provide high-frequency and high-resolution data; nevertheless, wavelet decomposition and wavelet norm entropy were calculated. It offered recognizable flow characteristics for bubble, bubble-elongated bubble, and slug flow patterns. Therefore, this study can provide deep insight into intricate multiphase flow patterns, and the wavelet could potentially be applied for flow analysis in oil and gas pipelines.

Quantum renormalization of $${l_1}$$-norm and relative entropy of coherence in quantum spin chains

We investigate quantum-phase transitions (QPT) in the Ising transverse field model, the XY-Heisenberg model with staggered Dzyaloshinskii–Moriya (DM) interaction, and the bond alternating Ising model with DM interaction, on a one-dimensional periodic chain using the quantum renormalization group (QRG) method. Based on $${l_1}$$-norm and relative entropy, we employ two quantum coherence measures to identify QPT. Our results show that the QPT can be characterized by the coherence measure calculated for each QRG step. It is found that after large number of QRG iterations, the coherence in the ground state develops two saturated values corresponding to two different phases and changes abruptly at a quantum critical point (QCP). The scaling behavior of the system is investigated by computing the derivative of the coherence measure around the QCP that shows a logarithmic dependence on the length of the chain. The critical exponent of the correlation length is also calculated.

Automated Alcoholism Detection Using Fourier-Bessel Series Expansion Based Empirical Wavelet Transform

In this paper, the Fourier-Bessel series expansion based empirical wavelet transform (FBSE-EWT) is proposed for automated alcoholism detection using electroencephalogram (EEG) signals. The FBSE-EWT is applied to decompose EEG signals into narrow sub-band signals using a boundary detection approach. The accumulated line length, log energy entropy, and norm entropy features are extracted from different frequency scales of narrow sub-band signals. A total of twenty features are extracted from each attribute and out of which ten features are from low to high frequency sub-band signals and other ten features are from high to low frequency sub-band signals. In order to reduce the classification model complexity, the most significant features are selected using feature selection techniques. Six feature ranking methods such as Relief-F, t-test, Chi-test, relief attribute evaluation, correlation attribute evaluation, and gain ratio are used to select the most common features based on the majority voting technique. Experiments are performed by considering top ranked 5, 10, 15, and 20 features and classification methods such as least square support vector machine (LS-SVM), support vector machine (SVM), and k nearest neighbor (k-NN) classifiers. The training and testing is done using leave-one out cross-validation (LOOCV) in order to avoid over-fitting. The performances of classifiers are evaluated using accuracy, sensitivity, and specificity measures. The results suggest that LS-SVM with radial basis function (RBF) kernel achieves a highest average accuracy of 98.8%, sensitivity of 98.3%, and specificity of 99.1% with top 20 significant features.

A novel fault diagnostic method in power converters for wind power generation system

In order to decrease the downtime, enhance maintenance efficiency, and reduce the management cost of the wind power system, fault diagnosis technologies are considered as powerful tools for its good operation and maintenance. For example, high fault diagnosis accuracy can ensure effective fault-tolerant control and improve the durability of wind turbines under fault conditions. However, fault diagnosis accuracy can be easily affected by nonlinear and noise of measured signals under different working conditions. In this paper, a novel fault diagnostic method of power converters is proposed for the wind power generation system. In the proposed method, the measured output voltage is firstly processed by ensemble empirical mode decomposition (EEMD), and a series of intrinsic mode functions (IMF) can be obtained. Norm entropy (NE) is then calculated based on the statistical characteristics of the IMFs, and the extracted IMF-NE information is used to describe the diagnostic features. The effectiveness and reliability of the proposed method are then validated in a simulated 1.5 MW doubly-fed wind power system. The results show that the final diagnostic accuracy of 22 fault modes is 99.57% for wind speed variation, and the diagnosis accuracy can be maintained above around 70% for different signal to noise ratio. Compared with the other advanced fault diagnosis methods, the proposed method shows outstanding performance in terms of robustness, high accuracy, and simple implementation without complex parameter tuning.

Comparing of K-Means, K-Medodis and Fuzzy C Means Cluster Method for Analog Modulation Recognition

A modulation process is required to transmit analog signals with higher quality. Modulation is the process of transporting the signal by another carrier signal. This study aims to process analog signals. Using 200 samples of each of the six types of analog modulation modules. Nowadays these are Amplitude Modulation (AM), Double Side Band (DSB), Upper Side Band (USB), Lower Side Band (LSB), Frequency Modulation (FM) and Phase Modulation(PM) respectively. In the study an intelligent clustering method has been developed. The 5th level Discrete Wavelet Transform (DWT), Norm Entropy and Energy properties of AM, DSB, USB, LSB, FM and PM analog modulated signals have been removed during feature extraction phase. The results have been compared using K-Means, k-Medoid and Fuzzy C Mean (FCM) algorithms using a feature vector of 6x2x1200 obtained at the feature extraction stage and carrying out smart intelligent clustering for recognition. The most successful result has been obtained with FCM of 85.75%.00

A new edge detection approach via neutrosophy based on maximum norm entropy

A new edge detection approach via neutrosophy based on maximum norm entropy

On the entropy norm on the group of diffeomorphisms of closed oriented surface

We prove that the entropy norm on the group of diffeomorphisms of a closed orientable surface of positive genus is unbounded.

A new edge detection approach via neutrosophy based on maximum norm entropy

It is a quite important step to find object edges in applications such as object recognition, classification and segmentation. Therefore, the edge detection algorithm to be used directly influences the performance of these applications. In this study, a new edge detection method based on Neutrosophic Set (NS) structure via using maximum norm entropy (EDA-NMNE) is proposed. Many experts and intelligence systems, including the fuzzy system, do not satisfactorily succeed in resolving indeterminacies and deficiencies. However, in the NS approach, problems are solved by dividing them into True (T), False (F) and Indeterminacy (I) subsets. In addition, because the approach has a powerful algorithmic structure, NS's conditions with indeterminate and missing situations can be solved successfully. In this study, object edges can be found successfully with the proposed approach because edges of the object are considered as indeterminate. Thus, using the proposed EDA-NMNE, a strong intelligent expert system-based edge finding software was designed. In our study, 5 different object edge detection results were obtained by converting 5 different types of entropy into NS-based edge detection software. Thus, edge detection analysis has been performed by experimenting many types of entropy that have not been previously used for NS edge detection. The Metric Figure of Merit (FOM), Peak Signal-to-Noise Ratio (PSNR) analysis, mean square error (MSE) were used in this experimental study. Based on these results, highest FOM and PSNR values are obtained by using NORM entropy. In addition, the proposed edge detection approach were compared with Edge Detection Approach via Canny (EDA-C), Edge Detection Approach via Sobel (EDA-S), Edge Detection Approach via Variation-Adaptive Ant Colony Optimization (EDA-VAACO) and Edge Detection Approach via Active Contour without Edge (EDA-ACWE). FOM and PSNR tests have been used to evaluate the edge detection results obtained via 5 different methods. The findings demonstrated that the performance of the proposed edge detection approach is more successful compared to other methods.

A new weighted [formula omitted]-norm information measure with application in coding theory

In the present communication, we introduce a quantity which is called weighted (α,β)-norm entropy and discuss its some major properties with Shannon and other entropies in the literature. Corresponding to the proposed entropy, a new weighted directed divergence measure has been introduced and its validity is established. Further, we give the application of (α,β)-norm entropy in coding theory and a coding theorem analogous to the ordinary coding theorem for a noiseless channel has been proved. The theorem states that the proposed entropy is the lower bound of mean code word length.

An intuitionistic fuzzy $$(\delta , \gamma )$$ ( δ , γ ) -norm entropy with its application in supplier selection problem

In this communication, based on the concept of $$(\delta , \gamma )$$ -norm entropy, an $$(\delta , \gamma )$$ -norm intuitionistic fuzzy entropy measure is introduced in the settings of intuitionistic fuzzy set theory. Some of its major properties are also proved. The performance of the proposed entropy measure is demonstrated using numerical examples. Based on the proposed intuitionistic fuzzy (IF) entropy, a new multiple-attribute decision-making method (MADM) considering the merits of subjective and objective weights has been introduced. Two methods of finding the weights of attributes are discussed. The proposed MADM method is effectively explained with the help of supplier selection example.

A Novel (R,S)-Norm Entropy Measure of Intuitionistic Fuzzy Sets and Its Applications in Multi-Attribute Decision-Making

The objective of this manuscript is to present a novel information measure for measuring the degree of fuzziness in intuitionistic fuzzy sets (IFSs). To achieve it, we define an ( R , S ) -norm-based information measure called the entropy to measure the degree of fuzziness of the set. Then, we prove that the proposed entropy measure is a valid measure and satisfies certain properties. An illustrative example related to a linguistic variable is given to demonstrate it. Then, we utilized it to propose two decision-making approaches to solve the multi-attribute decision-making (MADM) problem in the IFS environment by considering the attribute weights as either partially known or completely unknown. Finally, a practical example is provided to illustrate the decision-making process. The results corresponding to different pairs of ( R , S ) give different choices to the decision-maker to assess their results.

Vehicle speed measurement by on-board acoustic signal processing

Estimation of vehicle speed by analysis of drive-by noise is a known technique. The methods used in this kind of practice generally estimate the velocity of the vehicle with respect to the microphone(s), so they rely on the relative motion of the vehicle to the microphone(s). There are also other methods that do not rely on this technique. For example, recent research has shown that there is a statistical correlation between vehicle speed and drive-by noise emissions spectra. This does not rely on the relative motion of the vehicle with respect to the microphone(s) so it inspires us to consider the possibility of predicting velocity of the vehicle using an on-board microphone. This has the potential for the development of a new kind of speed sensor. For this purpose we record sound signal from a vehicle under speed variation using an on-board microphone. Sound emissions from a vehicle are very complex, which is from the engine, the exhaust, the air conditioner, other mechanical parts, tires, and air resistance. These emissions carry both stationary and non-stationary information. We propose to make the analysis by wavelet packet analysis, rather than traditional time or frequency domain methods. Wavelet packet analysis, by providing arbitrary time-frequency resolution, enables analyzing signals of stationary and non-stationary nature. It has better time representation than Fourier analysis and better high-frequency resolution than Wavelet analysis. Subsignals from the wavelet packet analysis are analyzed further by Norm Entropy, Log Energy Entropy, and Energy. These features are evaluated by feeding them into a multilayer perceptron. Norm entropy achieves the best prediction with 97.89% average accuracy with 1.11 km/h mean absolute error which corresponds to 2.11% relative error. Time sensitivity is ±0.453 s and is open to improvement by varying the window width. The results indicate that, with further tests at other speed ranges, with other vehicles and under dynamic conditions, this method can be extended to the design of a new kind of vehicle speed sensor.

Engine Speed–Independent Acoustic Signature for Vehicles

A vehicle, when running, makes a complex sound emission from the engine, the exhaust, the air conditioner, and other mechanical parts. Analysis of this sound for the purpose of vehicle identification is an interesting practice which has security- and transportation-related applications. Engine speed variation, which causes shifts in the frequency content of the emissions, makes Fourier-based methods ineffective in terms of providing a stable signature for the vehicle. We search for an engine speed–independent acoustic signature for the vehicle, and for this purpose, we propose wavelet packet analysis rather than traditional time- or frequency-domain methods. Wavelet packet analysis, by providing arbitrary time–frequency resolution, enables analyzing signals of stationary and non-stationary nature. It has better time representation than Fourier analysis and better high-frequency resolution than wavelet analysis. Under varying engine speed, sound emissions are recorded from four cars and analyzed by wavelet packet analysis. Wavelet packet analysis subimages are further analyzed to obtain feature vectors in the form of log energy entropy, norm entropy, and energy. These feature vectors are fed into a classifier, multilayer perceptron, for evaluation. While norm entropy achieves a classification rate of 100%, log energy entropy and energy achieves classification rates of 99.26% and 97.79%, respectively. These results indicate that, wavelet packet analysis along with norm entropy and multilayer perceptron provides an accurate vehicle-specific acoustic signature independent of the engine speed.

Flow Measurement by Wavelet Packet Analysis of Sound Emissions

Fluid, when running through pipes, makes a complex sound emission whose parameters change nonlinearly with respect to flow speed. Especially, in household pipe systems, there may be spraying effects and resonance effects which make the emission more complex. We present a novel approach for predicting flow speed based on wavelet packet analysis of sound emissions rather than traditional time and frequency domain methods. Wavelet packet analysis, by providing arbitrary time–frequency resolution, enables analyzing signals of stationary and non-stationary nature. It has better time representation than Fourier analysis and better high-frequency resolution than wavelet analysis. Wavelet packet analysis subimages are further analyzed to obtain feature vectors of norm entropy. These feature vectors are fed into a multilayer perceptron for prediction. Prediction accuracy of 98.62%, with 3.99E−04 L/s mean absolute error and its corresponding 1.85% relative error is achieved. Time sensitivity is ±0.453 s and is open to improvement by varying window width. The result indicates that the proposed method is a good candidate for flow measurement by acoustic analysis.