Grey Differential Equation Research Articles

Grey System Model GM (1,1) for Predicting Student Performance

The purpose of this study is to predict student quiz assignments in the General Physics I course during one semester using the Grey system GM (1,1) model. The prediction model integrates the first-order two-variable grey differential equation model from the Grey system theory. Additionally, a MATLAB program for the GM (1,1) prediction model was developed. The predicted results were compared with the original GM (1,1) model, showing good agreement.

Application of Grey Prediction Models for Efficient Traffic Flow Prediction

Traffic flow prediction is crucial in transportation management and optimizing control strategies and infrastructure planning. However, traditional prediction methods often need help handling traffic data's inherent uncertainty and complexity. This paper explores the application of Grey prediction models as a promising approach for efficient traffic flow prediction. Grey prediction models, rooted in Grey systems theory, offer a robust framework for modelling systems with incomplete and uncertain information. The paper provides a comprehensive overview of Grey prediction models, including the Grey Verhulst, Grey-Markov, and Grey differential equation models. It discusses the process of data collection, pre-processing, and model application in traffic flow prediction. Case studies and experimental results demonstrate the effectiveness of Grey prediction models in accurately forecasting traffic flow patterns. Despite challenges such as data sparsity and model complexity, Grey prediction models offer significant advantages over traditional methods, including improved prediction accuracy and adaptability to dynamic traffic conditions. The paper concludes with recommendations for practitioners and researchers interested in leveraging Grey prediction models for efficient traffic management.

Prediction method of building inclination around foundation pit based on Grey Model

In order to improve the prediction accuracy and effect of building inclination around foundation pit, a prediction method of building inclination around foundation pit based on grey model is designed in this paper. When calculating the parameters of the prediction model, the number of the bottom layer of the foundation pit excavation, the stress change value and the excavation load value are taken as the basis. At the same time, the grey derivative defined by the smooth discrete function is combined with the grey differential equation to build the grey model. The determined parameters are input into the grey prediction model to output the slope value of the buildings affected by the foundation pit. Complete the design of building inclination prediction method around the foundation pit. The experimental results show that the proposed method can improve the prediction of building inclination around the foundation pit, and clarify the direct relationship between foundation pit and building inclination, which verifies the effectiveness of the proposed method.

Forecasting the development trend of new energy vehicles in China by an optimized fractional discrete grey power model

As an effective technology to reduce traffic pollution emissions, the new energy vehicle industry has developed rapidly in recent years, and the sales of new energy vehicles have doubled in 2021. Accurately forecasting methods provide important references for industrial policy deployment, infrastructure construction and energy demand estimation. In order to predict the development trend of China's new energy vehicle industry under the limited data samples, this paper proposes an optimized discrete grey power model to fit the nonlinear relationship between grey information factor and temporal factor. Firstly, the grey differential equation and its discrete form are directly constructed by using power accumulated data, which simplifies the integral solution process of the traditional grey power model. Secondly, fractional accumulation operator is introduced into the discrete model to ensure the new information priority of the original data. Then, the heuristic algorithm is used to accurately estimate the new parameters of the proposed model. Thirdly, the proposed discrete model is the unified form of two existing discrete grey power models, which expands the scope of modeling and has higher adaptability. Simulation experiments and two numerical cases are used to verify the effectiveness of the proposed method. Finally, the proposed method is used to predict the annual sales and ownership of new energy vehicles in China. The predicted results show that by 2025, China's new energy vehicle sales will reach 8.84 million, accounting for about 24% of the total vehicle sales, surpassing the industry development target (20%) set by the Chinese government.

An Improved Gray Neural Network Method to Optimize Spatial and Temporal Characteristics Analysis of Land-Use Change.

In this article, the principles of the gray model and BP neural network model are analyzed, and the characteristics of land-use change and spatial and temporal distribution are studied in-depth, and at the same time, to explore the influence of land-use change on ESV, the relationship between the two is analyzed using gray correlation degree, and a mathematical model is constructed to maximize the benefits of the regional system, coupling economic and ecological benefits, combined with Geo SOS-FLUS model to achieve the optimization of land use. This article constructs a combined prediction model of a gray neural network. The gray differential equation parameters correspond to the weights and thresholds of the neural network, and the optimized parameters are determined by training the neural network to make it stable. Then the training results of the BP neural network are fitted with the results obtained from the gray GM (1.1) model. Finally, the prediction results of the three models, gray GM (1.1), BP God Meridian, and gray neural network model, are compared and analyzed. The global spatial autocorrelation and local spatial aggregation patterns of regional soil erosion and its erosion factors are analyzed using the Exploratory Spatial Data Analysis (ESDA) method in spatial measurement theory.

Grey FNN control and robustness design for practical nonlinear systems

To ensure asymptomatic stability and improve vehicle ride comfort, this paper develops a fuzzy neural network (FNN) based on the evolved bat algorithm (EBA) to design adaptive backstepping controllers with gray signal predicators. A recoil method is used to evaluate the nonlinearity of the controlled systems and to derive the control law which is evolved for the tracking of the signals. A group of grey differential equations are applied for the grey model (GM) (n, h), which is an active model where h is the number of considered variables and n is the order of the grey differential equations. In the article, the Discrete GM (2.1) is used to obtain the advanced motion of the nonlinear system, so that the command controller can prove the Lyapunov stability and feasibility of the entire scheme through the Lyapunov-like lemma. The controller design criteria are demonstrated for mechanical elastic wheels (MEW) to establish a viable mathematical framework for the new wheels.

Forecasting carbon dioxide emissions in BRICS countries by exponential cumulative grey model

Grey models should give different weights to old and new information when dealing with different cases, whereas existing grey models always give greater weight to new information. With this in mind, a novel grey model is proposed in this paper. The model introduces an exponential parameter that regulates the weight of information by adjusting the size of the parameter. This model is used to study the carbon dioxide emissions of BRICS (Brazil, Russia, India, China and South Africa) countries. Firstly, the intelligent algorithm is used for multiple iterations to find the exponential order that minimizes the simulation error. Secondly, the raw data on CO2 emissions from BRICS countries are processed and grey differential equations are developed. Finally, the least squares method is used to solve the coefficients of the differential equation and to predict the CO2 emissions of these countries based on the corresponding time series. The results show that carbon dioxide emissions in Brazil and Russia are on a downward trend. China and South Africa show a slight upward trend in carbon dioxide emissions. India has a large trend of increasing carbon dioxide emissions. The results can provide reference value for the relevant authorities.

A systematic analysis of water flow reduction and water-quality deterioration of the Niangziguan Spring area in northern China

The Niangziguan Spring is one of the largest karst springs in northern China and water in this spring area is the main water-supply source for people in Yangquan City for daily life, industrial processes and agricultural production. However, decreasing groundwater levels and growing waterpollution issues have been observed recently. Therefore, the objective of this study was to conduct a systematic analysis of water-flow reduction and water-quality deterioration of the Niangziguan Spring. The results show that various wastewater sources upstream of the spring, such as coal mining, and urban industrial and domestic effluents, have contributed to the deterioration of the Niangziguan Spring water quality. A mathematical model known as the one-order one-variable grey differential equation model GM(1,1) was employed to evaluate the contribution of climate change versus human activities potentially affecting the reduction of spring-water flow. The calculations for the period from 1979 to 2006 suggest that a contribution of human activities to water-flow reduction was 2.5 times greater than those that can be attributable to climate change. The research results obtained as presented in this paper can provide a scientific basis for rational exploitation of karst-water resources and environmental protection in this karst area of northern China.

Energy recovery potential from incineration using municipal solid waste based on multi-scenario analysis in Beijing.

With the rapid growth of municipal solidwaste (MSW) generation, waste-to-energy (WTE) incineration has gained popularity due to its potential for electricity generation and swiftly reducing the volume and mass of MSW. Recognizing the urgency of MSW management, Beijing's government launched the "MSW Source Separation Regulation" on May 1, 2020. This paper attempts to take into account the different levels of MSW source separation as well as the impact of different MSW compositions on energy recovery potential from incineration. We assume that the MSW management level in Beijing in 2025 will reach that of developed countries (e.g., Australia), which have abundant experience and mature MSW management systems. The amount of MSW generation is predicted by a first-order and one-variable grey differential equation model, GM (1,1), and a kernel-based nonlinear multivariate grey model, KGM (1,n). We establish six scenarios and distinguish them by the composition and lowerheating values (LHV) of MSW, considered with and without recycling and source separation to varying degrees. The predicted amount of MSW generation in 2025 is 11,505,400 tons with 2.255% mean absolute percentage error (MAPE). With increasing source separation and removal of food waste, the volume of electricity generated per ton of waste ranges from 0.336 to 1.114 MWh/ton. At its lowest, the cost of electricity generation is 977.60 yuan/MWh. The results not only provide implications for balancing the benefits and costs of implementing WTE incineration strategies but also shed light on government management of MSW source separation.

Parameter estimation of modified gray model GM (1, N) and model application

The GM (1, N) model is an important prediction model among gray models, but the conventional GM (1, N) model shows big errors sometimes mainly due to the structural inconsistency between the gray differential equation for parameter estimation and the whitening equation for prediction. The paper derives the modified model of conventional gray differential equation based on the whitening equation, and the gray differential equation obtained in this way has the structure basically consistent with the structure of whitening equation. The paper estimates parameters with the gray differential equation of modified GM (1, N) model and gives three parameter estimation methods with different background values. The time response equation for conventional GM (1, N) model generally has no analytical solution, but the paper gives a specific solving method for the time response equation of modified model. The example proves that the modified GM (1, N) model given and the models built with the methods proposed improve the simulation and prediction accuracy significantly compared with the conventional model.

Assessment of Mechanical Crack Fault Dependent on Wavelet Finite Element Model

Given that cracks in the forging and heat treatment process of switch actuator rod can affect its service life, damage mechanics is combined with the finite element analysis method to calculate the fatigue crack propagation life by using the Paris equation. The wavelet finite element model is used to establish the GM (1,1) grey differential equation. The identification parameter of the model is determined by experimental data calculation to assess the fatigue crack life of the switch actuator rod and compare it with the measured data in the experiment. The results show that the application of the wavelet finite element model to assess the fault of switch actuator rod is simple, feasible, and highly accurate, with excellent application value.

Forecasting China’s per capita energy consumption using dynamic grey model based on ARIMA model

Energy is the cornerstone of social progress, and energy consumption is an important reference data when the government makes energy policies. Therefore, prediction of energy consumption can provide a basis for the formulation of policies. In the traditional grey system theory, the grey input is always regarded as an invariant after determined by the least square method. However, the grey input would be changed along with the change of the internal information in the grey system. As a result, the performance and accuracy of the grey model would be affected by it.Therefore, this paper proposes the ARIMA-b-GM (1,1) model, which uses the ARIMA prediction model to fit and predict the grey input and realize the dynamic change of the grey input. Then the time response sequence of the ARIMA-b-GM (1,1) model is deduced by changing the form of grey differential equation. Finally, the ARIMA-b-GM (1,1) model is applied to predict the per capita energy consumption in China. The results exhibit that ARIMA-b-GM (1,1) model has higher prediction accuracy than GM (1,1) model, NGM (1,1) model and SAIGM (1,1) model.

Forecasting Clean Energy Consumption in China by 2025: Using Improved Grey Model GM (1, N)

Forecasting China’s clean energy consumption has great significance for China in making sustainably economic development strategies. Because the main factors affecting China’s clean energy consumption are economic scale and population size, and there are three variables in total, this paper tries to simulate and forecast China’s clean energy consumption using the grey model GM (1, 3). However, the conventional grey GM (1, N) model has great simulation and forecasting errors, the main reason for which is the structural inconsistency between the grey differential equation for parameter estimation and the whitening equation for forecasting. In this case, this paper improves the conventional model and provides an improved model GM (1, N). The modeling results show that the improved grey model GM (1, N) built with the method proposed improves simulation and forecasting precision greatly compared with conventional models. To compare the model with other forecasting models, this paper builds a grey GM (1, 1) model, a regression model and a difference equation model. The comparison results show that the improved grey model GM (1, N) built with the method proposed shows simulation and forecasting precision superior to that of other models as a whole. In the final section, the paper forecasts China’s clean energy consumption from 2019 to 2025 using the improved grey model GM (1, N). The forecasting results show that, by 2025, China’s clean energy consumption shall reach the equivalent of 1.504976082 billion tons of standard coal. From 2019 to 2025, clean energy consumption shall increase by 11.32% annually on average, far above the economic growth rate, indicating China’s economic growth shall have a great demand for clean energy in the future. Studies have shown that China’s clean energy consumption shall increase rapidly with economic growth and population increase in the next few years.

A nonlinear grey forecasting model with double shape parameters and its application

The solution of Bernoulli differential equation can be described as a generalized Logistic curve function. Weibull cumulative distribution function is also an analytic solution of one variable coefficient differential equation. But, however because of the exact analytical solution problem of the equation, up to now, the developing coefficient of all the grey forecasting models is still defined as a constant. So, the aim of this paper is just to construct a novel grey differential equation model by combining NGBM(1,1) and Weibull cumulative distribution function. The proposed model(WBGM(1,1)) has the advantages of NGBM(1,1) and Weibull cumulative distribution. Where there are double shape parameters, the developing coefficient of the grey forecasting model is extended to be a variant. Property analysis of WBGM(1,1) shows that the fitting accuracy is higher and the applicable confines are wider. Finally, this paper gives an optimization method for the parameters of WBGM(1,1). A classic example and a practical case are studied for confirming the effectiveness of WBGM(1,1). The case study is the prediction for the number of invention patents of integrated circuit(IC) filed in China from 2007 to 2017. Results of the example and case study are compared to other forecasting models, including GM(1,1), NGBM(1,1), Holt exponential smoothing and ARIMA. Results show that WBGM(1,1) is a more general and more efficient model in grey prediction theory.

Grey Mathematics Model for Atmospheric Pollution Based on Numerical Simulation

Urban air environment is a complex system with multiple objectives, multi levels and multiple factors. In this paper, the grey analysis of atmospheric environmental system is carried out, and the grey differential equation of air pollution is established, and the calculation method analysis and model verification of the grey numerical model are carried out. The relationship between the model parameters and the numerical solution is mainly analyzed, and the concentration field of the pollutant diffusion in the unsteady state is plotted. In this paper, the atmospheric environment system is analyzed from the grey system theory, and a more objective grey mathematical model is set up to carry out the Grey Numerical Simulation of the air pollution diffusion in urban area. It is of great practical significance to the flexible management and decision of air pollution control.

On the improvement of the parameter estimation of the grey model GM(1,1) and model application

The GM(1,1) model has been applied widely for grey prediction. However, traditional GM(1,1) model prediction sometimes results in large errors. In recent years, the grey model GM(1,1) has been improved, and the research mainly focuses on the modification, extension, and optimization of it. Most of the methods change the basic structure of the model. The study aims to improve the model’s parameter estimation without changing the model’s structure by using the following four methods: (1) optimizing the generating coefficient and estimating the model’s parameters in the case of the optimal generating coefficient; (2) using the approximation by polynomial to estimate the model’s parameters as a whole, including the parameter estimation of initial value; (3) translating the grey differential equation into the difference equation to estimate the model’s parameters with the difference equation; and (4) using the modern intelligent algorithm (that is, the improved particle swarm optimization algorithm) for parameter estimation. The study builds the GM(1,1) model of the total consumption of China’s industrial crude oil using the four improved methods. The simulation and prediction results show that these methods all significantly improve the prediction accuracy of model.

The NLS-based nonlinear grey Bernoulli model with an application to employee demand prediction of high-tech enterprises in China

PurposeThe purpose of this paper is to propose a new method based on nonlinear least squares (NLS) for solving the parameters of nonlinear grey Bernoulli model (NGBM(1,1)) and to verify the proposed model using the case of employee demand prediction of high-tech enterprises in China.Design/methodology/approachFirst of all, minimising the square sum of fitting error of grey differential equation of NGBM(1,1) is taken as the optimisation target and the parameters of classic grey model (GM(1,1)) are set as the initial value of parameter vector. Afterwards, the structural parameters and power exponents are solved by using the Gauss-Newton iteration algorithm so as to calculate the parameters of NGBM(1,1) under given rules for ceasing the algorithm. Finally, by taking the employee demand of high-tech enterprises in the state-level high-tech industrial development zone in China as examples, the validity of the new method is verified.FindingsThe results show that the parameter estimation algorithm based on the NLS method can effectively identify the power exponents of NGBM(1,1) and therefore can favourably adapt to the nonlinear fluctuations of sequences. In addition, the algorithm is superior to the GM(1,1) model, grey Verhulst model, and Quadratic-Exponential smoothing algorithm in terms of the simulation and prediction accuracy.Research limitations/implicationsUnder the framework of solving parameters based on NLS, various aspects of NGBM(1,1) remain to be further investigated including background value, initial condition and variable structural modelling methods.Practical implicationsThe parameter estimation algorithm based on NLS can effectively identify the power exponent of NGBM(1,1) and therefore it can favourably adapt to the nonlinear fluctuation of sequences.Originality/valueAccording to the basic principle of NLS, a new method for solving the parameters of NGBM(1,1) is proposed by using the Gauss-Newton iteration algorithm. Moreover, by conducting the modelling case about employees demand in high-tech enterprises in China, the effectiveness and superiority of the new method are verified.

Mathematical modeling - food price analysis

This paper mainly uses the idea of pedigree clustering analysis, gray prediction and principal component analysis. The clustering analysis model, GM (1,1) model and principal component analysis model were established by using SPSS software to analyze the correlation matrices and principal component analysis. MATLAB software was used to calculate the correlation matrices. In January, The difference in price changes of major food prices in cities is calculated, and had forecasted the various food prices in June 2016. For the first issue, the main food is classified and the data are processed. After that, the SPSS software is used to classify the 27 kinds of food into four categories by using the pedigree cluster analysis model and the system clustering. The four categories are made by EXCEL. The price of food changes over time with a line chart that analyzes the characteristics of food price volatility. For the second issue, the gray prediction model is established based on the food classification of each kind of food price. First, the original data is cumulated, test and processed, so that the data have a strong regularity, and then establish a gray differential equation, and then use MATLAB software to solve the model. And then the residual test and post-check test, have C <0.35, the prediction accuracy is better. Finally, predict the price trend in June 2016 through the function. For the third issue, we analyzed the main components of 27 kinds of food types by celery, octopus, chicken (white striped chicken), duck and Chinese cabbage by using the data of principal given and analyzed by principal component analysis. It can be detected by measuring a small amount of food, this predict CPI value relatively accurate. Through the study of the characteristics of the region, select Shanghai and Shenyang, by looking for the relevant CPI and food price data, using spss software, principal component analysis, the impact of the CPI on several types of food, and then calculated by matlab algorithm weight, and then the data obtained by the analysis and comparison, different regions should be selected for different types of food for testing.

Study on distance measuring and sorting method of general grey number

PurposeThe purpose of this paper is to study distance measuring and sorting method of general grey number.Design/methodology/approachFirst, the concept of generalised grey number based on grey system theory is given in this paper. Second, from the perspective of kernel and degree of greyness of general grey number, the method of measuring the distance of general grey number and its properties are given. At the same time, the concepts of the kernel expectation and the kernel variance of the general grey number are proposed.FindingsUp to now, the method of measuring the distance and sorting of general grey number is established. Thus, the difficult problem for set up sorting of general grey number has been solved to a certain degree.Research limitations/implicationsThe method exposed in this paper can be used to integrate information form a different source. Distance measuring and sorting method of general grey number could be extended to the case of grey algebraic equation, grey differential equation and grey matrix which includes general grey numbers, etc.Originality/valueThe concepts of the kernel expectation and the kernel variance of the general grey number are proposed for the first time in this paper; the novel sorting rules of general grey numbers were also constructed.

A Motion Heart-Rate Monitor Watch with Improved Grey Differential Equation Model Based on Reflective Photoplethysmography

With the advantage of noninvasive measurement, reflective photoplethysmography (PPG) has been adopted by a group of gadgets to estimate heart rate (HR) from the wrists. Challenges include PPG artifacts reduction and comfortable type design. Aimed at improving the accuracy of measurement during wrist movements such as handshake or exercises, in this paper, a motion HR monitor watch has presented. The work mainly focused on two categories: the HR extraction algorithm derived from grey differential equation model (abbreviated as GM(1, 1) model) combined with acceleration model and the enhanced hardware structure designed with 523 nm green leds. The tests have been carried out and results analysis show high correlation among the values measured respectively from this watch, HR chest belt (taken as HR gold standard) and commercial motion HR watch in various intensities of exercises.