The Correlation Coefficient of Hesitancy Fuzzy Graphs in Decision Making

The hesitancy fuzzy graphs(HFGs),an extension of fuzzy graphs,are useful tools for dealing with ambiguity and uncertainty in issues involving decision-making(DM).This research implements a correlation coefficient measure(CCM)to assess the strength of the association between HFGs in this article since CCMs have a high capacity to process and interpret data.The CCM that is proposed between the HFGs has better qualities than the existing ones.It lowers restrictions on the hesitant fuzzy elements’length and may be used to establish whether the HFGs are connected negatively or favorably.Additionally,a CCMbased attribute DM approach is built into a hesitant fuzzy environment.This article suggests the use of weighted correlation coefficient measures(WCCMs)using the CCM concept to quantify the correlation between two HFGs.The decisionmaking problems of hesitancy fuzzy preference relations(HFPRs)are considered.This research proposes a new technique for assessing the relative weights of experts based on the uncertainty of HFPRs and the correlation coefficient degree of each HFPR.This paper determines the ranking order of all alternatives and the best one by using the CCMs between each option and the ideal choice.In the meantime,the appropriate example is given to demonstrate the viability of the new strategies.

Mathematical Proof of the Synthetic Running Correlation Coefficient and Its Ability to Reflect Temporal Variations in Correlation

The running correlation coefficient(RCC)is useful for capturing temporal variations in correlations between two time series.The local running correlation coefficient(LRCC)is a widely used algorithm that directly applies the Pearson correlation to a time window.A new algorithm called synthetic running correlation coefficient(SRCC)was proposed in 2018 and proven to be rea-sonable and usable;however,this algorithm lacks a theoretical demonstration.In this paper,SRCC is proven theoretically.RCC is only meaningful when its values at different times can be compared.First,the global means are proven to be the unique standard quantities for comparison.SRCC is the only RCC that satisfies the comparability criterion.The relationship between LRCC and SRCC is derived using statistical methods,and SRCC is obtained by adding a constraint condition to the LRCC algorithm.Dividing the temporal fluctuations into high-and low-frequency signals reveals that LRCC only reflects the correlation of high-frequency signals;by contrast,SRCC reflects the correlations of high-and low-frequency signals simultaneously.Therefore,SRCC is the ap-propriate method for calculating RCCs.

Narrowband time delay estimation based on correlation coefficient

The problems of time delay estimation of narrowband signals are presented. The disadvantages of the existing algorithms are analyzed, and a new narrowband time delay estimating algorithm based on correlation coefficient is proposed. The original time delay information is transfered into the delay between the autocorrelation and cross-correlation function, and the precise estimating result by wave-comparison is given. The algorithm proposed here is also compared with other algorithms and its advantages over other algorithms are proved. The theoretical analysis and simulation show the effectiveness of the proposed algorithm.

Reliability Sensitivity-based Correlation Coefficient Calculation in Structural Reliability Analysis

The correlation coefficients of random variables of mechanical structures are generally chosen with experience or even ignored,which cannot actually reflect the effects of parameter uncertainties on reliability.To discuss the selection problem of the correlation coefficients from the reliability-based sensitivity point of view,the theory principle of the problem is established based on the results of the reliability sensitivity,and the criterion of correlation among random variables is shown.The values of the correlation coefficients are obtained according to the proposed principle and the reliability sensitivity problem is discussed.Numerical studies have shown the following results：（1） If the sensitivity value of correlation coefficient ρ is less than（at what magnitude 0.000 01）,then the correlation could be ignored,which could simplify the procedure without introducing additional error.（2） However,as the difference between ρs,that is the most sensitive to the reliability,and ρR,that is with the smallest reliability,is less than 0.001,ρs is suggested to model the dependency of random variables.This could ensure the robust quality of system without the loss of safety requirement.（3） In the case of ｜Eabs｜ρ0.001 and also ｜Erel｜ρ0.001,ρR should be employed to quantify the correlation among random variables in order to ensure the accuracy of reliability analysis.Application of the proposed approach could provide a practical routine for mechanical design and manufactory to study the reliability and reliability-based sensitivity of basic design variables in mechanical reliability analysis and design.

LARGE DEVIATION FOR THE EMPIRICAL CORRELATION COEFFICIENT OF TWO GAUSSIAN RANDOM VARIABLES

In this article, the author obtains the large deviation principles for the empirical correlation coefficient of two Gaussian random variables X and Y. Especially, when considering two independent Gaussian random variables X, Y with the means EX, EY （both known）, wherein the author gives two kinds of different proofs and gets the same results.

Correlation Degree and Correlation Coefficient of Multi-Output Functions

We present definitions of the correlation degree and correlation coefficient of multi-output functions. Two relationships about the correlation degree of multi-output functions are proved. One is between the correlation degree and independency, the other is between the correlation degree and balance. Especially the paper discusses the correlation degree of affine multioutput functions. We demonstrate properties of the correlation coefficient of multi-output functions. One is the value range of the correlation coefficient, one is the relationship between the correlation coefficient and independency, and another is the sufficient and necessary condition that two multi-output functions are equivalent to each other.

A Correlation Coefficient Approach for Evaluation of Stiffness Degradation of Beams Under Moving Load

This paper presents a new approach using correlation and cross-correlation coefficients to evaluate the stiffness degradation of beams under moving load.The theoretical study of identifying defects by vibration methods showed that the traditional methods derived from the vibration measurement data have not met the needs of the actual issues.We show that the correlation coefficients allow us to evaluate the degree and the effectiveness of the defects on beams.At the same time,the cross-correlation model is the basis for determining the relative position of defects.The results of this study are experimentally conducted to confirm the relationship between the correlation coefficients and the existence of the defects.In particular,the manuscript shows that the sensitivity of the correlation coefficients and cross-correlation is much higher than the parameters such as changes in stiffness(EJ)and natural frequency values(Δf).This study suggests using the above parameters to evaluate the stiffness degradation of beams by vibration measurement data in practice.

Weighted Parameterized Correlation Coefficients of Indeterminacy Fuzzy Multisets and Their Multicriteria Group Decision Making Method with Different Decision Risks

Real-life data introduce noise,uncertainty,and imprecision to statistical projects;it is advantageous to consider strategies to overcome these information expressions and processing problems.Neutrosophic(indeterminate)numbers can flexibly and conveniently represent the hybrid information of the partial determinacy and partial indeterminacy in an indeterminate setting,while a fuzzy multiset is a vital mathematical tool in the expression and processing of multi-valued fuzzy information with different and/or same fuzzy values.If neutrosophic numbers are introduced into fuzzy sequences in a fuzzy multiset,the introduced neutrosophic number sequences can be constructed as the neutrosophic number multiset or indeterminate fuzzy multiset.Motivated based on the idea,this study first proposes an indeterminate fuzzy multiset,where each element in a universe set can be repeated more than once with the different and/or identical indeterminate membership values.Then,we propose the parameterized correlation coefficients of indeterminate fuzzy multisets based on the de-neutrosophication of transforming indeterminate fuzzy multisets into the parameterized fuzzy multisets by a parameter(the parameterized de-neutrosophication method).Since indeterminate decision-making issues need to be handled by an indeterminate decision-making method,a group decision-making method using the weighted parameterized correlation coefficients of indeterminate fuzzy multisets is developed along with decision makers’different decision risks(small,moderate,and large risks)so as to handle multicriteria group decision-making problems in indeterminate fuzzy multiset setting.Finally,the developed group decision-making approach is used in an example on a selection problem of slope design schemes for an open-pit mine to demonstrate its usability and flexibility in the indeterminate group decision-making problem with indeterminate fuzzy multisets.

ON TESTING THE EQUALITY OF K MULTIPLE AND PARTIAL CORRELATION COEFFICIENTS

Coutsourides (1980) derives an ad hoc nuisance parameter removal test for testing the equality of two multiple correlation coefficients of two independent p variate normal populations, under the assumption that a sample of size n is available from each population. He also extends his ad hoc nuisance parameter removal test to the testing of the equality of two multiple correlation matrices. This paper presents likelihood ratio tests for testing the equality of k multiple correlation coefficients, and also k partial correlation coefficients.

Improvement of a new rotation function for molecular replacement by designing new scoring functions and dynamic correlation coefficient

A previously published new rotation function has been improved by using a dynamic correlation coefficient as well as two new scoring functions of relative entropy and mean-square-residues to make the rotation function more robust and independent of a specific set of weights for scoring and ranking. The previously described new rotation function calculates the rotation function of molecular replacement by matching the search model directly with the Patterson vector map. The signal-to-noise ratio for the correct match was increased by averaging all the matching peaks. Several matching scores were employed to evaluate the goodness of matching. These matching scores were then combined into a single total score by optimizing a set of weights using the linear regression method. It was found that there exists an optimal set of weights that can be applied to the global rotation search and the correct solution can be ranked in the top 100 or less. However, this set of optimal weights in general is dependent on the search models and the crystal structures with different space groups and cell parameters. In this work, we try to solve this problem by designing a dynamic correlation coefficient. It is shown that the dynamic correlation coefficient works for a variety of space groups and cell parameters in the global search of rotation function. We also introduce two new matching scores： relative entropy and mean-square-residues. Last but not least, we discussed a valid method for the optimization of the adjustable parameters for matching vectors.

ASSESSMENT OF LOCAL INFLUENCE FOR GENERALIZED CORRELATION COEFFICIENTS

In this paper, we study local influence analysis for Zhang's generalized correlation coefficients and Hotelling's generalized correlation coefficient by using approach. of local influence analysis suggested by Shi (1991), i.e., generalized influence function (GIF) and generalized Cook distance (GCD). An example is given to illustrate our results.

Decomposition Method of Genetic Correlation Coefficient Based on NC Ⅱ Mating Design

There are different degrees of correlation between crop traits. The phenotypic correlation is decomposed into genetic and environmental correlation in quantitative genetics. In this paper,according to stochastic model of variance and covariance analysis,we calculate different genetic components,bring up a decomposition method of genetic correlation coefficient based on NC II mating design,and use examples to show analytic steps and interpret results.

A Simulation Based Comparison of Correlation Coefficients with Regard to Type I Error Rate and Power

In this simulation study, five correlation coefficients, namely, Pearson, Spearman, Kendal Tau, Permutation-based, and Winsorized were compared in terms of Type I error rate and power under different scenarios where the underlying distributions of the variables of interest, sample sizes and correlation patterns were varied. Simulation results showed that the Type I error rate and power of Pearson correlation coefficient were negatively affected by the distribution shapes especially for small sample sizes, which was much more pronounced for Spearman Rank and Kendal Tau correlation coefficients especially when sample sizes were small. In general, Permutation-based and Winsorized correlation coefficients are more robust to distribution shapes and correlation patterns, regardless of sample size. In conclusion, when assumptions of Pearson correlation coefficient are not satisfied, Permutation-based and Winsorized correlation coefficients seem to be better alternatives.

Regression models of Pearson correlation coefficient

We propose two simple regression models of Pearson correlation coefficient of two normal responses or binary responses to assess the effect of covariates of interest.Likelihood-based inference is established to estimate the regression coefficients,upon which bootstrap-based method is used to test the significance of covariates of interest.Simulation studies show the effectiveness of the method in terms of type-I error control,power performance in moderate sample size and robustness with respect to model mis-specification.We illustrate the application of the proposed method to some real data concerning health measurements.

Correlation Study of Operational Data and System Performance of District Cooling System with Ice Storage

The district cooling system (DCS) with ice storage can reduce the peak electricity demand of the business district buildings it serves, improve system efficiency, and lower operational costs. This study utilizes a monitoring and control platform for DCS with ice storage to analyze historical parameter values related to system operation and executed operations. We assess the distribution of cooling loads among various devices within the DCS, identify operational characteristics of the system through correlation analysis and principal component analysis (PCA), and subsequently determine key parameters affecting changes in cooling loads. Accurate forecasting of cooling loads is crucial for determining optimal control strategies. The research process can be summarized briefly as follows: data preprocessing, parameter analysis, parameter selection, and validation of load forecasting performance. The study reveals that while individual devices in the system perform well, there is considerable room for improving overall system efficiency. Six principal components have been identified as input parameters for the cold load forecasting model, with each of these components having eigenvalues greater than 1 and contributing to an accumulated variance of 87.26%, and during the dimensionality reduction process, we obtained a confidence ellipse with a 95% confidence interval. Regarding cooling load forecasting, the Relative Absolute Error (RAE) value of the light gradient boosting machine (lightGBM) algorithm is 3.62%, Relative Root Mean Square Error (RRMSE) is 42.75%, and R-squared value (R2) is 92.96%, indicating superior forecasting performance compared to other commonly used cooling load forecasting algorithms. This research provides valuable insights and auxiliary guidance for data analysis and optimizing operations in practical engineering applications. .

Analyses on schedule-cost coefficient correlation of spaceflight project based on historical statistics and its application

When analyze the uncertainty of the cost and the schedule of the spaceflight project, it is needed to know the value of the schedule-cost correlation coefficient. This paper deduces the schedule distribution, considering the effect of the cost, and proposes the estimation formula of the correlation coefficient between the in（schedule） and the cost. On the basis of the fact and Taylor expansion, the relation expression between the schedule-cost correlation coefficient and the in-schedule-cost correlation coefficient is put forward. By analyzing the value features of the estimation formula of the in-schedule-cost correlation coefficient, the general rules are proposed to ascertain the value of the schedule-cost correlation coefficient. An example is given to demonstrate how to approximately amend the schedule-cost correlation coefficient based on the historical statistics, which reveals the traditional assigned value is inaccurate. The universality of this estimation method is analyzed.

Study on Probability Plot Correlation Coefficient of the Log-Weibull Distribution

The log-Weibull distribution is a variant of the three-parameter Weibull distribution. The probability plot of a distribution model is desired since it can help to decide on whether the model is appropriate for fitting a given dataset and can provide the initial estimate of the model parameters. The decision on the appropriateness of a distribution is somehow subjective. This paper presents a probability plot of the log-Weibull distribution(LWPP). The distribution of the probability plot correlation coefficient is studied. From this distribution, a lower confidence limit is determined for determining whether the probability plot correlation coefficient derived from a given data set is large enough. The appropriateness and usefulness of this study are illustrated by two real-world examples.

ATMOSPHERIC WAVE AND ITS PHYSICAL ESSENCE AS REVEALED BY CHANGED CORRELATION COEFFICIENTS AND CORRELATION FIELD

Formal change is made of the correlation coefficient(COCOEF)expression,leading to a new formula consisting of Fourier spectral coefficients that indicates a direct connection between the new form and atmospheric dynamic equations,thus resulting in a dynamic equation represented by COCOEF,which is meaningful in exploring a large-scale dynamic process in terms of the correlation field because the connection revealed by the field can have dynamic explana- tion with the aid of the new formula.

Prediction model for cost data of a power transmission and transformation project based on Pearson correlation coefficient-IPSO-ELM

In view of the difficulty in predicting the cost data of power transmission and transformation projects at present,a method based on Pearson correlation coefficient-improved particle swarm optimization(IPSO)-extreme learning machine(ELM)is proposed.In this paper,the Pearson correlation coefficient is used to screen out the main influencing factors as the input-independent variables of the ELM algorithm and IPSO based on a ladder-structure coding method is used to optimize the number of hidden-layer nodes,input weights and bias values of the ELM.Therefore,the prediction model for the cost data of power transmission and transformation projects based on the Pearson correlation coefficient-IPSO-ELM algorithm is constructed.Through the analysis of calculation examples,it is proved that the prediction accuracy of the proposed method is higher than that of other algorithms,which verifies the effectiveness of the model.

Simulation on hydrodynamics of non-spherical particulate system using a drag coefficient correlation based on artificial neural network

Fluidization of non-spherical particles is very common in petroleum engineering.Understanding the complex phenomenon of non-spherical particle flow is of great significance.In this paper,coupled with two-fluid model,the drag coefficient correlation based on artificial neural network was applied in the simulations of a bubbling fluidized bed filled with non-spherical particles.The simulation results were compared with the experimental data from the literature.Good agreement between the experimental data and the simulation results reveals that the modified drag model can accurately capture the interaction between the gas phase and solid phase.Then,several cases of different particles,including tetrahedron,cube,and sphere,together with the nylon beads used in the model validation,were employed in the simulations to study the effect of particle shape on the flow behaviors in the bubbling fluidized bed.Particle shape affects the hydrodynamics of non-spherical particles mainly on microscale.This work can be a basis and reference for the utilization of artificial neural network in the investigation of drag coefficient correlation in the dense gas-solid two-phase flow.Moreover,the proposed drag coefficient correlation provides one more option when investigating the hydrodynamics of non-spherical particles in the gas-solid fluidized bed.