Accurate prediction serves as a critical foundation for formulating and optimizing industrial development strategies, particularly in projecting the output of the high-tech industry through a scientific approach.
A novel nonlinear time-varying delay grey multivariable model (NTVDGM(1,N)) is designed in this work. First, we propose a new time-varying delay function, which can overcome the constraint of a fixed time delay in the existing grey time-delay models. On this basis, a new cumulative time-delay function combined with the Gamma function is constructed, indicating a decreasing trend and an inverted U-shaped trend. Then, the power parameter is designed to express the nonlinear relationship between the inputs and outputs, and the Grey Wolf Optimization (GWO) algorithm is selected to determine the hyperparameter.
Simulations validate the robustness and stability of the (NTVDGM(1,N)) model, demonstrating that the proposed model possesses superior forecasting accuracy.
The proposed model is applied to project the output of the high-tech industry in Guangdong province, the grey and non-grey prediction models are selected for comparison to the new model, and the results show that the (NTVDGM(1,N)) model achieves a satisfactory prediction accuracy in forecasting the high-tech industry's output.
This work proposes a novel nonlinear time-varying delay grey multivariable model with a cumulative function and variable time-delay parameters to capture the dynamic time-delay effect.
1. Introduction
Currently, the traditional mathematical and econometric models, machine learning models, and grey models are attracting significant attention. Many econometric models have played significant roles in the field, such as distributed lag models (Pang, 2019), autoregressive moving average models, and regression models (Ávila et al., 2019). Machine learning models also have a significant involvement in the field, such as artificial neural networks (Bonyadi and Jalilian, 2022) and support vector machines (Kouziokas, 2020). However, these methods require large amounts of data, which is not always available in the real world. Thus, as a tool working with limited data, grey models are receiving more and more attention. Grey prediction models, being a portion of grey theory, possess a relatively simple structure and fewer parameters that can handle problems with insufficient samples (Deng, 1982). As the grey models are valuable, a series of improvement models have been proposed and widely applied in various filed (Wang and Zhang, 2022; Wang et al., 2022; Mao et al., 2020; Xu and Li, 2023). Multivariable grey prediction models have gained popularity in recent years due to their capacity to characterize the connection between inputs and outputs.
Many scholars have made lots of improvements from various perspectives for the grey multivariable model to broaden its applicability (Zeng et al., 2019; Yan et al., 2023;Li et al., 2023a, b, c). For instance, Xie and Liu (2008) developed a discrete grey multivariable model (). Based on the , Zeng et al. (2019) created a grey multivariable model with a flexible structure (). To enhance the precision of predicting the values in the marine emerging industry, Li et al. (2023a, b, c) proposed a grey multivariable model incorporating interactive effects. Moonchai and Chutsagulprom (2020) put forward a modified grey multivariable model and used Kalman filtering to optimize the parameters. Ma et al. (2023) established a novel multivariable prediction model by integrating wavelet kernel learning with the grey model. Kiran et al. (2021) incorporated econometric methods into the grey model to propose an enhanced grey multivariable model.
Moreover, to address the issue of predicting nonlinearity characteristics, some researchers have incorporated a power exponent into the grey multivariable model to elucidate the nonlinear connection between influencing factors and outputs (Wang and Lv, 2021). For example, a discrete grey power multivariable model () was constructed by Ding et al. (2018), through a driver control function to identify the parameters. Zhang et al. (2022) presented a new grey multivariable method () with power exponent and correct term, which was proven to be able to provide a satisfactory prediction accuracy in projecting energy consumption. Additionally, Ma et al. (2019) developed an enhanced Bernoulli grey model () and utilized the convolution integral approach for calculating the time response function. Besides, Duman et al. (2019) used Particle Swarm Optimization to further improve the model's performance and applied the optimized model in electronic waste estimation.
An observed delay is a common phenomenon in economic systems. Policy effects, technological innovation, and investment require time before their effects manifest. Furthermore, the time-delay in control systems reflects the feedback delay, effect delay, and so on (Li et al., 2023a, b, c, 2024a, b). Considering the presence of a potential time-delay effect between the influencing factors and outputs, many scholars have proposed grey forecasting models and incorporated this time-delay effect. Ye et al. proposed a series of grey time-delay models (Ye et al., 2021a, b). Shi et al. (2020) combined the time delay parameters and interval grey numbers to construct a new time delay , and then used it to forecast air pollution. In addition, Zhou et al. (2023) designed an optimized discrete grey time-delay multivariable model () and utilized the traversal technique to determine the time delay parameters., and then, considering the time-varying parameters, a novel time-delay model () is proposed (Zhou et al., 2024). Duan et al. (2022) developed a dynamic grey model incorporating time delay factors to investigate the interaction between energy supply and demand. Meanwhile, they adjusted the background value through the simulated annealing method. Ding et al. (2023) introduced a novel model by considering the time-delay and interaction effects simultaneously (). Besides, Wan et al. (2022) established a mixed-frequency time-delay grey model () to deal with the different frequency data. To explore the relationships between the marine and land economy-energy-environment, Li et al. (2023a, b, c) designed a grey prediction model () involving fractional order and time-delay effect. In addition, Yan et al. (2023) built a new grey time-delay multivariable model () containing fractional order for predicting online public opinion. To accurately forecast the agricultural drought hazard, Zhang and Luo (2023) coupled the time-delay with the existing grey multivariable model, and put forward a new grey prediction approach with a time-delayed driving term ().
Parameter estimation plays an essential role in model optimization, and various researchers have used intelligent algorithms to find the optimal parameters. The Grey wolf optimizer (GWO) (Mirjalili et al., 2014) is s swarm intelligence optimization algorithm, which has received wide attention. It is easy to use, and many scholars have used GWO to solve their research issues. Both Rajakumar et al. (2021) and Pan et al. (2023) have demonstrated its superiority to other algorithms. The GWO algorithm has the advantages of not requiring complex parameter settings, simple implementation, and strong global search capability. Thus, in this work, we will select the GWO to search for the parameters in the proposed model, and the detailed comparison will be discussed in the empirical study.
As presented in previous studies, significant progress has been made by the developed grey time-delay models recently. However, certain limitations remain in addressing the time-delay prediction issues. For the model construction, although current grey models account for time-delay effects, they all consider the lag period to be a fixed constant and ignore the time-varying characteristics of the time-delay, which may lead to unsatisfactory prediction results. Besides, some studies have focused on developing a nonlinear framework for modeling processes, but with limited attention to their combined effect. Moreover, the existing grey projection research does not consider both the cumulative time-delay effect with varying parameters and the nonlinear relationship between factors and outputs.
To address these issues, an improved grey time-delay multivariable model is proposed, which integrates a flexible time-varying function with a cumulative time-delay effect and nonlinear characteristics. This new model can explore the varying accumulative time-delay effect and its nonlinear feature simultaneously, and extend the model's flexibility and applicability in real-world scenarios. The major innovations of this study can be stated as the following aspects:
A novel time-varying delay grey multivariable model with power exponent, abbreviated as , is developed. The new model not only takes into account the time-delay effect with variable parameters but also incorporates its nonlinear relationship between inputs and outputs into the proposed model, which further improves the applicability of . Furthermore, this newly-designed model not only improves prediction accuracy but also offers a unified framework for incorporating time-delay and discrete grey models, enhancing the generalizability of the approach.
The time-varying delay function is incorporated into the proposed model so that can capture the dynamic time-delay. The new-design function with dynamic lag parameters in the proposed model overcomes the constraint of a fixed time delay in the existing grey time-delay models, thereby strikingly enhancing the flexibility of this new model. In addition, in light of the nonlinear connection between the inputs and outputs, the power exponent is also integrated to further improve the adaptability of . Overall, the new grey prediction model can simultaneously explore the variable time delay parameters and nonlinear interaction between influencing factors and outputs.
The Grey Wolf Optimization (GWO) algorithm is employed to determine the time-varying lag parameters and power exponents, which further augments its prediction capability. Simulations validate the robustness and stability of . Consequently, this new model is validated with feasible forecasting capacity.
To demonstrate the reliability of , this work utilizes the new model to forecast the high-tech industry's output in Guangdong province, exhibiting satisfactory performance over the compared models.
The remainder of this work is organized as follows. Section 2 presents the methodology of the proposed model. Simulations with two types of random data, including stochastic fluctuating and exponential data, are conducted to illustrate the stability of the new approach in Section 3. Subsequently, a case study on forecasting the high-tech industry's output in Guangdong province is conducted to illustrate the practicability of the new model in Section 4. Lastly, Section 5 summarizes the conclusion and future work.
2. Methodology
2.1 Defect of the existing grey time-delay prediction model
Xie and Liu (2008) Let be the dependent sequence, and the sequences
represent the independent variables. is first order sequence, which can be defined by the first-order accumulation generation operation (1-AGO): Then
is the discrete grey multivariable model (), where the parameter vector can be determined by the least square method (LSM). Notably, can describe the multivariable system with independent variables. However, the independent factors have an instantaneous effect on the dependent factor, and the effects between inputs and outputs are synchronized, ignoring the lag relationship in the system. Nevertheless, the outputs may not immediately experience the effects of the inputs. Considering the delay relationship between inputs and outputs would be more reasonable to explore the time-delay effect in the system. Besides, does not consider the time-varying characteristic of the model's parameters. Particularly, the parameters might not be fixed values with the dynamic development of the system. In addition, can not explore the nonlinear relationship between independent and dependent variables. Given some data that exhibits a nonlinear characteristic, it might fail to process such a series.
Therefore, when dealing with the time series characterized by time-delay effect, accumulative effect, and the nonlinear relationship between inputs and outputs, we will construct a new grey time-delay multivariable model, considering the accumulative effect, time-varying lag parameters, and nonlinear characteristic between variables.
2.2 The proposed model
Assuming that are defined as Definition 1, then the following equation
is called the nonlinear grey multivariable discrete model with time-varying delay function, denoted as . In this model, represents the driving term with an accumulative time-delay effect. , and , which is the time-varying delay function, indicating that the independent variable lags behind the dynamics of the dependent variable , and denotes the strength of the cumulative time delay effect of the variable on the output. In addition, is the power item of the variable, representing the nonlinear effect of variable on the dependent variable. is the linear term, and is the action grey item.
The time-varying delay function is an essential component of . In this work, is constructed as
Besides, is a lag of static time-delay, and . The time-varying lag function proposed in this paper is dynamic over time and is a decreasing function of the time variable, indicating that the delay effect of the drivers on the system's output gradually decreases over time. The construction of the time-varying lag function enhances the fit of the model to the series, and the trend of the series subject to the time-delay effect can be further investigated by analyzing the properties of the time-varying delay function.
By establishing the time-varying delay function , reflecting the dynamic change process of the time delay function over time, the static time delay in the existing grey multivariate time-delay model is expanded to dynamic time-delay, and the value of the lag parameter is expanded from integer to non-integer, which expands the scope of application of the grey time-delay forecasting model. When , the proposed model decreases the static grey time-delay model. Thus, enables the conversion of dynamic time delay to static time delay.
In accordance with Eq. (3), the values of the time-varying delay function are determined by the static time delay parameters and index parameters . In this work, we will use the traversal algorithm (Zhou et al., 2023) to obtain the , and the can be determined by intelligent algorithms, which will be introduced in Section 2.3.
Due to the time-varying delay function, the is a non-integer, thus, the value of in Eq. (2) requires to be determined. In this paper, we will utilize the cubic spline interpolation (Li et al., 2018) to ascertain the value of . The spline function generated by cubic spline interpolation has better smoothness than linear spline interpolation, and the sequence with time delay parameters is fitted by using cubic spline interpolation.
In the model, represents the cumulative time-delay effect of independent variables. The commonly used cumulative time-delay effects can be divided into two categories:
The declining type: The influence of lagged variables on the system's outputs is weighted more heavily on the trend change in the early years and tends to decrease gradually over time.
Inverted U type: This type indicates that the effect of the lag factors on the dependent variable is low in the early stages, but the cumulative time-delay effect gradually increases over time, and when it peaks at a certain point, its effect gradually decreases.
The existing grey time-delay model generally constructs two functions to characterize the above two different cumulative time-delay effects. This work constructs a new type of cumulative time-delay function, which unifies the characterization of decreasing and inverted U-type cumulative time-delay, and reduces the computational complexity of the new model. can be expressed as follows:
where represents the Gamma function, and is the shape parameter and unknown. The curves of the cumulative time-delay function are shown as in Figure 1. When , is a monotonically decreasing function. When , is the single-peak function. The designed cumulative time-delay function can adaptively calculate the cumulative time delay intensity under different time-delay types with different parameters , which strengthens the adaptive prediction ability of the model for complex uncertain systems.
A line graph showing the curves of the cumulative time-delay function with different shape parameters. The x-axis represents time (t) ranging from 0 to 20, and the y-axis represents the time-delay function ranging from 0 to 1.8. The graph includes six lines, each representing a different shape parameter (gamma) with values 0.5, 1, 2, 3, 4, and 5. The blue line represents gamma equals 0.5, the black line represents gamma equals 1, the red line represents gamma equals 2, the green line represents gamma equals 3, the magenta line represents gamma equals 4, and the cyan line represents gamma equals 5. All values are approximated.The curves of the cumulative time-delay function with different shape parameters. Source: Author’s own creation/work
A line graph showing the curves of the cumulative time-delay function with different shape parameters. The x-axis represents time (t) ranging from 0 to 20, and the y-axis represents the time-delay function ranging from 0 to 1.8. The graph includes six lines, each representing a different shape parameter (gamma) with values 0.5, 1, 2, 3, 4, and 5. The blue line represents gamma equals 0.5, the black line represents gamma equals 1, the red line represents gamma equals 2, the green line represents gamma equals 3, the magenta line represents gamma equals 4, and the cyan line represents gamma equals 5. All values are approximated.The curves of the cumulative time-delay function with different shape parameters. Source: Author’s own creation/work
Suppose that and are defined as in Definition 1, is the parameter vector of , then the structure parameters satisfy
where
Proof. Substituting the into Eq. (2), we can obtain the equations
Eq. (3) can be expressed as
As the matrix is a column-full rank matrix, Eq. (7) can be presented as , and then we can get .
Assuming that and are described as Definition 1, is shown as Theorem 1. Given the initial value , the model's solution can be determined in the following equation
Proof. Theorem 2 can be proven by using the mathematical induction method. When , by Eq. (2), we can obtain
Eq. (9) can be expressed as
Based on Eq. (2), when , we have
According to Eq. (13), we get that Eq. (8) holds true when . Thus, providing proof for the conclusion of Theorem 2. As for the first-order inverse accumulation generating operation, we have
2.3 Parameter estimation
This section focuses on determining the unknown parameters of the proposed model. model includes the parameters , and the structure parameters . Among them, the structure parameters can be obtained by Theorem 1, the parameter can be calculated by the traversal algorithm, and then the other parameters will be solved by the GWO algorithm. This this paper, the mean absolute percentage error is used as the objective function. Based on the constraints, the objective function of is proposed as
The GWO algorithm is a search method for optimization designed based on the predation behavior of grey wolves. This algorithm demonstrates notable convergence performance, requires minimal parameters, and may be easily implemented. In GWO, the social hierarchy is organized from highest to lowest using the symbols: , and from high to low. The hunting behavior of the grey wolf primarily involves the pursuit and capture of prey by searching, encircling, and engaging in predatory attacks.
To formulate the encircling behaviors of wolves during a hunt, the following equations are presented:
where represents the current iteration, and stand for coefficient vectors, is the position of the prey. Adjustments are made to the positions of each search agent in accordance with potential locations “”, “”, and “” of the prey. The subsequent equations are used to determine the optimal search space.
where are the three best positions of the best search agents and are the distances from the search agents to the three best solutions.
2.4 Criteria for assessment and modeling steps
To assess the predictive capabilities of , four critical indicators are presented in Table 1, where and denote the raw data and estimated results.
Comparison of prediction model performance
| Metric | Definition | Formula |
|---|---|---|
| APE | Absolute percentage error | |
| MAPE | Mean absolute percentage error | |
| RMSE | Root mean squared error | |
| MAE | Mean absolute error |
The model's modeling steps can be described as follows.
Step 1. Collect the raw data and , and then obtain the AGO sequences and by processing the original data.
Step 2. Establish the model based on Definition 2, based on Eq. (15), perform integer traversal of static time delay , and determine the optimal parameters by using the GWO algorithm.
Step 3. Substitute the optimal parameters and into Theorem 1 for solving .
Step 4. Determine the fitted and predicted values by using Theorem 2.
Step 5. The model's prediction accuracy can be compared with the benchmark models by using the critical indicators in Table 1.
2.5 Properties of model
Assume that sequences are the multiple transformations of , where . Then, the estimated and predicted values corresponding to and satisfy . Then, the vector corresponding to satisfies
Proof. Assume that
The matrix can be expressed as
Due to , is an invertible matrix, thus the inverse matrix of can be calculated:
Based on Theorem 1, we can obtain
Substituting and into Eq. (17), the parameter vector corresponding to the multiple transformation can be described as
End of proof.
Assume that the sequences are multiple transformations of . Then, the predicted value of is satisfied
Proof. Based on Theorem 1, we can obtain that the predicted values of the sequence are
Thus, we can conclude that
3. Simulations
To demonstrate the robustness of the new model, two simulation experiments are designed with two types of random data, including stochastic fluctuating and exponential data. Based on the proposed model, the output data can be obtained by , where represents the input data. Besides, is a random error, which is generated by the normal distribution with mean 0 and variance . For the experimental simulation, we randomly generate 20 datasets with the characteristics of random fluctuations and exponential laws. The first 18 observations are used to establish the prediction models, and the rest 2 samples for testing the forecasting accuracy. In addition, to illustrate the simulation results, the experiments are repeated 200 times. Figure 2 shows the boxplots of the fitted and predicted errors in 200 times with two prediction models ( (Wan et al., 2022; Luo et al., 2021), and the average MAPE values are tabulated in Tables 2 and 3.
The image contains six sets of boxplots, each set showing the Mean Absolute Percentage Error (MAPE) values for three different models: NTVDGM, AMTGM, and TDAGM. The boxplots are divided into two columns: the left column shows the fitted MAPE values, while the right column shows the predicted MAPE values. Each row corresponds to a different variance level: 0.01, 0.8, and 1.8. The boxplots illustrate the distribution of MAPE values across 200 experiments for each model and variance level. The models are compared based on their performance in fitting and predicting data with varying levels of variance. All values are approximated.Boxplots for the MAPE values in fitted and predicted periods generated by 200 experiments with different variances . Source: Author’s own creation/work
The image contains six sets of boxplots, each set showing the Mean Absolute Percentage Error (MAPE) values for three different models: NTVDGM, AMTGM, and TDAGM. The boxplots are divided into two columns: the left column shows the fitted MAPE values, while the right column shows the predicted MAPE values. Each row corresponds to a different variance level: 0.01, 0.8, and 1.8. The boxplots illustrate the distribution of MAPE values across 200 experiments for each model and variance level. The models are compared based on their performance in fitting and predicting data with varying levels of variance. All values are approximated.Boxplots for the MAPE values in fitted and predicted periods generated by 200 experiments with different variances . Source: Author’s own creation/work
The average MAPE values are generated by exponential data with 200 experiments
| Variance | 0.01 | 0.8 | 1.8 | |||
|---|---|---|---|---|---|---|
| MAPE-fitted | MAPE-pred | MAPE-fitted | MAPE-pred | MAPE-fitted | MAPE-pred | |
| NTVDGM | 0.724 | 0.279 | 1.705 | 0.876 | 0.589 | 2.114 |
| AMTGM | 1.412 | 1.510 | 4.649 | 2.536 | 5.258 | 3.275 |
| TDAGM | 3.411 | 3.357 | 7.212 | 4.628 | 8.538 | 7.446 |
Note(s): The optimum value of the difference variance for six predictions is in italic
The average MAPE values are generated by random fluctuation data with 200 experiments
| Variance | 0.01 | 0.8 | 1.8 | |||
|---|---|---|---|---|---|---|
| MAPE-fitted | MAPE-pred | MAPE-fitted | MAPE-pred | MAPE-fitted | MAPE-pred | |
| NTVDGM | 3.717 | 4.159 | 8.333 | 10.134 | 7.699 | 10.432 |
| AMTGM | 1.652 | 4.311 | 9.837 | 14.862 | 9.541 | 15.227 |
| TDAGM | 7.629 | 9.465 | 11.026 | 15.938 | 9.540 | 20.939 |
Note(s): The optimum value of the difference variance for six predictions is in italic
As noted in Figure 2(a) and Table 2, for the exponential data, the MAPE values of and are superior to the compared models with the difference variance. In the predicted stage, the results of the three models stay within 5%, and when the variance increases to 1.8, the MAPE values generated by , , and remain in the range of [3.40%,3.43%], [2.85%,13.42%], and [8.41%,8.59%], indicating that the proposed model still maintains a stable prediction accuracy when the variance increases. Overall, as the variance increases, the fitting and prediction accuracy of demonstrates significant superiority over the two comparative models. This finding suggests that when the data follow a quasi-exponential distribution pattern, the proposed model in this study exhibits enhanced fitting performance, improved predictive capability, and greater stability compared to the other two grey time-delay models.
Moreover, from the perspective of the random fluctuation data, as depicted in Figure 2(b) and Table 3, the predicted values generated by are predominant among the projection models, indicating that the proposed model yields remarkable forecasting ability with the fluctuation data. As the variance increases, maintains stable fitting accuracies of approximately 3.7%, 8.2%, and 7.6%, while its prediction accuracies stabilize at around 4.1%, 10.1%, and 10.4%, respectively. In contrast, the two comparative models exhibit significantly lower fitting and prediction accuracies under fluctuating data conditions. Furthermore, their error distributions are broader across all three variance levels, particularly at a variance of 0.01, where the dispersion is most pronounced. These results indicate that the computational stability of the comparative models is inferior to that of the proposed model. Consequently, the robustness of the proposed model is proven with the different data features.
4. Application
The model is implemented to forecast the output of the high-tech industry in Guangdong province. In order to evaluate its performance, the model is compared with several other existing models. To assess model performance, we conduct a comparative analysis with several established benchmark models. This evaluation is particularly relevant given China's 15th Five-Year Plan, which emphasizes the strategic importance of high-tech industries as a cornerstone of national economic development. Accurate forecasting of high-tech industry trends becomes crucial, as it provides policymakers with evidence-based insights for strategic decision-making. Consistent with economic theory, industrial output demonstrates direct dependence on input factors. Empirical studies consistently identify Research and Development (R&D) intensity as a primary determinant of high-tech industry performance. Accordingly, our analysis incorporates R&D expenditure and R&D personnel as key input variables, while accounting for time-delay effects in the input-output relationship.
To assess the practicability of , five existing prediction models are chosen as the compared models for the performance comparison. It is worth noting that the grey models consist of two grey time-delay models (; ), and one grey power model ( power model), the non-grey models include distributed-lag model () and Back Propagation Neural Network (BPNN). The experiment utilizes the data from 2006 to 2020 to construct the prediction models, while data from 2021 to 2022 are employed for projection. The errors generated by and the above compared models are measured by four error indices. The results are displayed in Table 4 and Figure 3.
Simulation and prediction results of the proposed model and the compared models
| Year | Actual | NTVDGM(1,N) | TDAGM(1,N,t) | AMTGM(1,N) | GM(1,N) power | DLM | BPNN | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| value | Fitted value | APE (%) | Fitted value | APE (%) | Fitted value | APE (%) | Fitted value | APE (%) | Fitted value | APE (%) | Fitted value | APE (%) | |
| 2006 | 12794.13 | – | – | 12794.13 | 0.00 | – | – | 12794.13 | 0.00 | – | – | 10987.49 | 14.12 |
| 2007 | 14582.9 | – | – | 13584.36 | 6.85 | – | – | 15069.76 | 3.34 | – | – | 10853.08 | 25.58 |
| 2008 | 16070.8 | – | – | 16172.71 | 0.63 | 16070.80 | 0.00 | 16497.03 | 2.65 | 28912.40 | 79.91 | 14302.12 | 11.01 |
| 2009 | 16758.05 | 16758.05 | 0.00 | 18497.84 | 10.38 | 16314.99 | 2.64 | 18103.91 | 8.03 | 32124.61 | 91.70 | 15640.29 | 6.67 |
| 2010 | 20952.8 | 20933.10 | 0.09 | 20556.22 | 1.89 | 21316.55 | 1.74 | 20045.56 | 4.33 | 34470.53 | 64.52 | 18377.26 | 12.29 |
| 2011 | 23227.6 | 23088.36 | 0.60 | 22936.83 | 1.25 | 23476.34 | 1.07 | 22352.64 | 3.77 | 35063.80 | 50.96 | 21693.88 | 6.60 |
| 2012 | 25046.6 | 26287.11 | 4.95 | 25252.25 | 0.82 | 25426.53 | 1.52 | 25037.89 | 0.03 | 38165.09 | 52.38 | 24747.53 | 1.19 |
| 2013 | 27871.1 | 27419.71 | 1.62 | 27875.66 | 0.02 | 27574.48 | 1.06 | 27953.64 | 0.30 | 41905.48 | 50.35 | 25229.09 | 9.48 |
| 2014 | 30328.9 | 30391.44 | 0.21 | 30632.44 | 1.00 | 30308.37 | 0.07 | 30844.44 | 1.70 | 47222.50 | 55.70 | 26905.37 | 11.29 |
| 2015 | 33308.1 | 33048.71 | 0.78 | 33856.88 | 1.65 | 33016.13 | 0.88 | 34056.85 | 2.25 | 47869.52 | 43.72 | 30857.48 | 7.36 |
| 2016 | 37765.2 | 36289.68 | 3.91 | 37454.46 | 0.82 | 37018.81 | 1.98 | 37447.36 | 0.84 | 52481.24 | 38.97 | 36156.64 | 4.26 |
| 2017 | 42,256 | 41144.27 | 2.63 | 41056.67 | 2.84 | 42521.51 | 0.63 | 40613.69 | 3.89 | 58095.55 | 37.48 | 40949.55 | 3.09 |
| 2018 | 46,747 | 46036.48 | 1.52 | 44512.43 | 4.78 | 46530.15 | 0.46 | 44184.55 | 5.48 | 59814.31 | 27.95 | 44191.88 | 5.47 |
| 2019 | 46,723 | 50929.15 | 9.00 | 47877.37 | 2.47 | 48716.57 | 4.27 | 47606.51 | 1.89 | 53697.39 | 14.93 | 45899.99 | 1.76 |
| 2020 | 50,185 | 50902.97 | 1.43 | 51839.49 | 3.30 | 48889.05 | 2.58 | 52180.19 | 3.98 | 60285.47 | 20.13 | 57539.80 | 14.66 |
| MAPE (%) | 2.43 | 2.58 | 1.57 | 3.03 | 48.36 | 8.99 | |||||||
| RMSE | 1475.83 | 1010.28 | 765.75 | 1118.86 | 13528.37 | 2839.34 | |||||||
| MAE | 944.97 | 742.91 | 545.19 | 853.20 | 13297.52 | 2333.02 | |||||||
| 2021 | 53,915 | 54674.68 | 1.41 | 57220.70 | 6.13 | 49223.02 | 8.70 | 57888.17 | 7.37 | 56593.73 | 4.97 | 68053.27 | 26.22 |
| 2022 | 56,285 | 58738.37 | 4.36 | 63221.21 | 12.32 | 50798.45 | 9.75 | 64290.71 | 14.22 | 65273.89 | 15.97 | 69183.37 | 22.92 |
| MAPE (%) | 2.88 | 9.23 | 9.23 | 10.80 | 10.47 | 24.57 | |||||||
| RMSE | 1816.06 | 5433.17 | 5104.75 | 6319.71 | 6632.34 | 13532.53 | |||||||
| MAE | 1606.53 | 5120.96 | 5089.26 | 5989.44 | 5833.81 | 13518.32 | |||||||
A box-and-whisker plot comparing the errors of APE values generated by six prediction models. The plot features six vertical box plots labeled as NTVDGM, TDAGM, AMTGM, GM(1,N) power, DLM, and BPNN. The x-axis represents the prediction models, while the y-axis represents the APE percentage ranging from 0 to 90. Each box plot shows the distribution of APE values for the respective model. The NTVDGM, TDAGM, AMTGM, and GM(1,N) power models have lower APE values with minimal spread, indicating better performance. The DLM model shows a significantly higher median and wider spread, suggesting higher errors. The BPNN model has a moderate spread with a higher median compared to the first four models. Outliers are present in all models, with DLM having the most. All values are approximated.The errors of APE values generated by the six prediction models. Source: Author’s own creation/work
A box-and-whisker plot comparing the errors of APE values generated by six prediction models. The plot features six vertical box plots labeled as NTVDGM, TDAGM, AMTGM, GM(1,N) power, DLM, and BPNN. The x-axis represents the prediction models, while the y-axis represents the APE percentage ranging from 0 to 90. Each box plot shows the distribution of APE values for the respective model. The NTVDGM, TDAGM, AMTGM, and GM(1,N) power models have lower APE values with minimal spread, indicating better performance. The DLM model shows a significantly higher median and wider spread, suggesting higher errors. The BPNN model has a moderate spread with a higher median compared to the first four models. Outliers are present in all models, with DLM having the most. All values are approximated.The errors of APE values generated by the six prediction models. Source: Author’s own creation/work
As shown in Table 4, in the training stage, the MAPE values of , , , and power models are approximately 3%, indicating that and the three benchmark grey models achieve satisfactory computational accuracy during the fitted stage. Among them, demonstrates the best fitting performance, with MAPE, RMSE, and MAE values of 1.57%, 765.75, and 545.19, respectively. However, the fitting accuracy of is slightly inferior to that of . In the testing stage, the MAPE, RMSE, and MAE values are 2.88%, 1816.06, and 1606.53, respectively, which are the lowest values among the six forecasting models, demonstrating superior predictive accuracy. Overall, , , , and power models demonstrate close agreement with the original series during the fitting phase. However, in the prediction stage, all three comparative grey models exhibit varying degrees of deviation, while maintains the closest proximity to the original series, indicating their superior modeling performance.
Compared with the five benchmark models, during the training stage, achieves the lowest MAPE (1.57%) among all six forecasting models. In contrast, shows the poorest performance with maximum values of MAPE (48.36%), RMSE (13528.37), and MAE (13,297.52) among all prediction models, indicating its limited adaptability to raw data and constrained predictive capability for complex nonlinear systems. The three grey prediction models demonstrate superior fitting accuracy compared to the two non-grey models, suggesting that grey forecasting methods are more suitable for predicting operating income in Guangdong's high-tech industry. In addition, during the testing stage, , , and the power models all exhibit prediction errors exceeding 8% for both 2021 and 2022 years, showing significant deviations from . While the non-grey compared models performed reasonably well in the fitting stage, their predictive performance deteriorated markedly during the testing stage, with MAPE values reaching 24.57%.
Additionally, as evidenced by Table 2 and Figure 3, exhibits the most stable APE distribution with minimal fluctuations. The APE values of range within [0.09%, 9.00%]. In comparison, the , , and power models, demonstrate APE distributions of [0.02%, 12.32%], [0.07%, 9.75%], and [0.03%, 14.22%] respectively, while models and show wider ranges of [4.97%, 91.70%] and [1.19%, 26.22%]. Comparative analysis of APE values across all six forecasting models indicates that maintains the highest computational stability.
In summary, the model incorporating both time-varying characteristics of the delay parameter and the cumulative time-delay effect demonstrates optimal modeling performance, accurately capturing the development trend of operating income in Guangdong's high-tech industry.
5. Conclusion
In light of the system's inherent time-varying delay and nonlinearity, this paper develops an advanced nonlinear grey time-delay multivariable model with time-varying lag parameters and accumulative delay effect (). What's more, the properties of are discussed in the multiple transformations, indicating that the multiple transformations do not influence the modeling performance.
The model can significantly explore the dynamic time-delay, different delay effects, and fluctuation features within the system. The newly developed time delay function has the capability to accurately represent both static and dynamic time delays over time, which is more flexible than the existing grey time-delay model with fixed time delay parameters. Moreover, a new dynamic time-delay function is constructed, which adaptively adjusts its parameters according to different observed lag effects. Besides, the power exponent has been integrated into enhancing the practicality of addressing the complexity associated with the time series' multiple characteristics. Furthermore, the variable lag parameters, shape parameter, and power exponent are calculated using the GWO algorithm, resulting in further improvement in the forecasting accuracy of the newly-designed approach.
The predominant performance derives from the newly-designed model, which takes into account the cumulative time-delay effect and time-varying characteristics of the time delay parameters simultaneously. The time delay function with time-varying parameters can explore the dynamic time-delay effect, which can broaden the model's application. Additionally, the power exponent is incorporated into for capturing the nonlinear influence of the input factors.
The simulation experiments are used to analyze the robustness and stability of . The results generated by simulation experiments highlight the excellent reliability of the proposed model.
The empirical analysis of high-tech industry's output in Guangdong Province demonstrates that incorporates dynamic variations in time-lag parameters and system nonlinearity, outperforms both grey and non-grey comparative models in terms of prediction accuracy and stability. These results indicate that the proposed is more suitable for forecasting analysis of high-tech industries under time-delay effects.
The model can provide satisfactory prediction performance. We will optimize this model to further broaden its application in the future. Further discussions on the complex interactions in the system would be considered in the prediction model, such as the interaction effect between inputs and outputs. The time-delay function will be optimized with a simple form to reduce the computational complexity of the model and improve the stability of the optimal values of the parameters.
