Parameter estimation for grey system models: A nonlinear least squares perspective

Wei, Baolei; Xie, Naiming

doi:10.1016/j.cnsns.2020.105653

Cited by 22 publications

(9 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is clear that ɛ ( k ), u 1 ( k ) and u 2 ( k ) in Equation (20), and ζ ( k ), v 1 ( k ) and v 2 ( k ) in Equation (21), explicitly containing the measurement error e ( k ), makes the resultant estimates biased and inconsistent and also indicates the main source of errors causing bias. The pseudo linear regression Equation (20) and Equation (21) are, in fact, error-in-variables models and thus can be solved by using advanced methods, such as the two-stage least squares (Dattner, 2020) and prediction error minimization-based non-linear least squares (Wei and Xie, 2021).…”

Section: Simulationsmentioning

confidence: 99%

Parameter estimation for grey system models: gradient matching versus integral matching

Wei

Xie²,

Yang³

2022

View full text Add to dashboard Cite

PurposeThe cumulative sum (Cusum) operator, also referred to as accumulating generation operator, is the fundamental of grey system models and proves to be successful in various real-world applications. This paper aims to uncover the advantages of the Cusum operator from a parameter estimation perspective, i.e. comparing integral matching with classical gradient matching.Design/methodology/approachGrey system models are represented as a state space form to investigate the effect of measurement errors on estimation performance; subsequently, gradient matching and integral matching are respectively formulated to estimate parameters from noisy observations and, then, their quantitative relationships are established by using matrix computation tricks.FindingsExtensive simulations, which are conducted on both linear and non-linear models under different sample size and noise level combinations, show that integral matching is superior to gradient matching, and, also the former is less sensitive to measurement error.Originality/valueThis paper explains why the Cusum operator is widely utilized in grey system models, thereby further solidifying the mathematical fundamentals of grey system models.

show abstract

Section: Simulationsmentioning

confidence: 99%

Parameter estimation for grey system models: gradient matching versus integral matching

Wei

Xie²,

Yang³

2022

View full text Add to dashboard Cite

show abstract

“…These are the most common in real applications, as most variables can be influenced by external factors such as energy consumption is influenced by many economic factors and the economy is susceptible to emergencies like the COVID-19 crisis raging throughout the world and the ongoing conflict between Russia and Ukraine. Focusing on cumulative transformation [ 37 ], parameter estimation [ 38 – 40 ], grey differential equation [ 41 – 43 ], and the expression for response function [ 44 – 46 ], many novel multivariate grey models have also been presented. These models are based on multivariate GM(1,N) [ 9 ].…”

Section: Introductionmentioning

confidence: 99%

Two novel nonlinear multivariate grey models with kernel learning for small-sample time series prediction

Wang

Xie

et al. 2023

Nonlinear Dyn

View full text Add to dashboard Cite

For many applications, small-sample time series prediction based on grey forecasting models has become indispensable. Many algorithms have been developed recently to make them effective. Each of these methods has a specialized application depending on the properties of the time series that need to be inferred. In order to develop a generalized nonlinear multivariable grey model with higher compatibility and generalization performance, we realize the nonlinearization of traditional GM(1,N), and we call it NGM(1,N). The unidentified nonlinear function that maps the data into a better representational space is present in both the NGM(1,N) and its response function. The original optimization problem with linear equality constraints is established in terms of parameter estimation for the NGM(1,N), and two different approaches are taken to solve it. The former is the Lagrange multiplier method, which converts the optimization problem into a linear system to be solved; and the latter is the standard dualization method utilizing Lagrange multipliers, that uses a flexible estimation equation for the development coefficient. As the size of the training data increases, the estimation results of the potential development coefficient get richer and the final estimation results using the average value are more reliable. The kernel function expresses the dot product of two unidentified nonlinear functions during the solving process, greatly lowering the computational complexity of nonlinear functions. Three numerical examples show that the LDNGM(1,N) outperforms the other multivariate grey models compared in terms of generalization performance. The duality theory and framework with kernel learning are instructive for further research around multivariate grey models to follow. Supplementary Information The online version contains supplementary material available at 10.1007/s11071-023-08296-y.

show abstract

“…In contrast to commonly used time-series models, grey predictions have attracted more and more attention in the last decade because they can use limited numbers of samples to realize unknown systems (Liu et al , 2017; Wei and Xie, 2021). In addition, conformity of data to statistical assumptions is not required for grey prediction (Liu et al , 2017).…”

Section: Introductionmentioning

confidence: 99%

Constructing interval models using neural networks with non-additive combinations of grey prediction models in tourism demand

Jiang

2022

View full text Add to dashboard Cite

PurposeIn contrast to point forecasts, interval forecasts provide information on future variability. This research thus aimed to develop interval prediction models by addressing two significant issues: (1) a simple average with an additive property is commonly used to derive combined forecasts, but this unreasonably ignores the interaction among sequences used as sources of information, and (2) the time series often does not conform to any statistical assumptions.Design/methodology/approachTo develop an interval prediction model, the fuzzy integral was applied to nonlinearly combine forecasts generated by a set of grey prediction models, and a sequence including the combined forecasts was then used to construct a neural network. All required parameters relevant to the construction of an interval model were optimally determined by the genetic algorithm.FindingsThe empirical results for tourism demand showed that the proposed non-additive interval model outperformed the other interval prediction models considered.Practical implicationsThe private and public sectors in economies with high tourism dependency can benefit from the proposed model by using the forecasts to help them formulate tourism strategies.Originality/valueIn light of the usefulness of combined point forecasts and interval model forecasting, this research contributed to the development of non-additive interval prediction models on the basis of combined forecasts generated by grey prediction models.

show abstract

Parameter estimation for grey system models: A nonlinear least squares perspective

Cited by 22 publications

References 19 publications

Parameter estimation for grey system models: gradient matching versus integral matching

Parameter estimation for grey system models: gradient matching versus integral matching

Two novel nonlinear multivariate grey models with kernel learning for small-sample time series prediction

Constructing interval models using neural networks with non-additive combinations of grey prediction models in tourism demand

Contact Info

Product

Resources

About