含虚拟变量的新结构多变量灰色预测模型及其应用

doi:10.16381/j.cnki.issn1003-207x.2024.0686

Abstract

Abstract:

In contrast to traditional manufacturing enterprises， research and development （R&D） activities play a more pivotal role in product development， manufacturing， and decision-making processes within manufacturing high-tech enterprises （abbreviated as MH_TE）. Accurately forecasting R&D expenditures in MH_TE is essential for mitigating financial risks and ensuring a stable supply of resources. The factors influencing R&D expenditures can be broadly classified into two categories： Entity variables and Dummy variables. Entity variables are quantifiable metrics， such as the full-time equivalent of R&D personnel， new product development expenditures， patent applications， and effective invention patents. Dummy variables， on the other hand， are qualitative indicators that reflect attributes such as policy influences， typically assigned binary values of 0 or 1 based on their presence or absence. In this study， Dummy variables are incorporated into the multivariate grey prediction model NSGM（1，N）， and its driving term structure is extended to develop a novel model DVSGM（1，N）， specifically designed for predicting R&D expenditures in MH_TE. The model is optimized using a particle swarm optimization （PSO） algorithm to minimize relative simulation errors， with constraints applied to the time response and cumulative reduction types. The results demonstrate that DVSGM（1，N） achieves a significantly lower integrated error （1.74%） compared to other grey prediction models （2.59%， 7.48%， and 16.05%）， highlighting its superior predictive accuracy. The DVSGM（1，N） model is subsequently applied to forecast R&D expenditures for MH_TE in China， providing valuable insights for policy formulation. The findings indicate that R&D expenditures are projected to reach 3，263，879 million yuan within a decade， underscoring both the substantial financial support for technological advancement and the potential financial strain on enterprises. To address these challenges， it is recommended that enterprises enhance the efficiency of R&D outcomes， while the government should implement measures to alleviate financial burdens and foster effective collaboration between the public and private sectors. Not only a robust tool is provided for predicting R&D expenditures but also the methodological framework of multivariate grey prediction models is advanced by incorporating dummy variables. This innovation enhances the model's completeness and practical applicability， offering a more comprehensive approach to forecasting in the context of MH_TE.

Key words: multivariate grey forecasting model, structure model expansion, dummy variable driving term, R&D expenditures prediction of MH_TE

CLC Number:

N941.5

Bo Zeng,Lingbo Zhang,Sifeng Liu, et al. A New Structural Multi-variable Grey Forecasting Model with Virtual Variables and Its Application[J]. Chinese Journal of Management Science, 2026, 34(7): 84-94.

Figures/Tables 10

符号	含义	计算方法
$ε k$	残差	$ε k = x ̂ 10 k - x 10 k$
$Δ k$	相对模拟/预测百分误差	$ε k / x 10 k × 100 %$
$R M S E$	均方根误差	$1 / n - 1 × ∑ k = 2 n ε k 2$
$R 2$	拟合系数	$1 - ∑ k = 2 n ε k 2 / ∑ k = 2 n x ¯ 10 - x 10 k 2$
$M A P E$	平均绝对百分比误差	$∑ k = 2 n ε k / x 10 k × 100 % / n - 1$

序号	研发经费 $X 10$	R&D人员全时当量 $X 20$	新产品开发经费支出 $X 30$	专利申请数 $X 40$	有效发明专利数 $X 50$	虚拟变量 $D 60$
1	1.0000	1.0000	1.0000	1.0000	1.0000	0
2	1.2591	1.0919	1.2267	1.4445	1.2227	0
……	……	……	……	……	……	……
18	17.9523	7.2486	20.6654	25.8003	121.6317	1

$x 1 (0) (k)$		$G M (1, N)$		$D V C G M (1, N)$		$N S G M (1, N)$		$D V S G M (1, N)$
$x 1 (0) (k)$		$x ̂ 1 (0) k$	$Δ (k)$	$x ̂ 1 (0) k$	$Δ (k)$	$x ̂ 1 (0) k$	$Δ (k)$	$x ̂ 1 (0) k$	$Δ (k)$
训练集数据
1	1.0000	1.0000	-	1.0000	-	1.0000	-	1.0000	-
2	1.2591	0.7282	0.4216	0.8274	0.3429	1.3523	0.0740	1.2591	0.0000
3	1.5043	1.6153	0.0737	1.9246	0.2794	1.3331	0.1138	1.5404	0.0240
4	1.8074	2.1049	0.1646	2.0241	0.1199	1.7790	0.0157	1.6900	0.0650
5	2.1352	2.5924	0.2141	2.8115	0.3167	2.2855	0.0704	2.2580	0.0575
6	2.6698	2.9931	0.1211	3.4911	0.3076	2.6698	0.0000	2.6698	0.0000
7	3.4146	3.4556	0.0120	3.6667	0.0738	3.3024	0.0329	3.2666	0.0434
8	4.1145	4.3677	0.0615	4.9464	0.2022	4.2712	0.0381	4.2796	0.0401
9	4.7846	4.8109	0.0055	5.2898	0.1056	4.7713	0.0028	4.7925	0.0017
10	5.3026	5.3187	0.0030	5.8845	0.1097	5.1889	0.0215	5.2119	0.0171
11	6.1233	6.1044	0.0031	6.4759	0.0576	6.1150	0.0014	6.1205	0.0005
12	6.7244	6.8494	0.0186	7.1026	0.0562	6.7457	0.0032	6.7377	0.0020
13	7.2957	7.5804	0.0390	7.9548	0.0903	7.3130	0.0024	7.3020	0.0009
14	8.0345	8.2824	0.0309	8.8598	0.1027	8.0302	0.0005	8.0413	0.0009
15	8.4905	8.7951	0.0359	8.8538	0.0428	8.5674	0.0091	8.5217	0.0037
16	12.8251	13.3767	0.0430	15.8154	0.2332	12.7339	0.0071	12.7820	0.0034
测试集数据
17	15.6817	15.7903	0.0069	17.9808	0.1466	15.1357	0.0348	15.2259	0.0291
18	17.9523	18.2677	0.0176	20.4782	0.1407	17.7214	0.0129	17.8292	0.0069

年份	$X 1 (0)$	$X 2 (0)$	$X 3 (0)$	$X 4 (0)$	$X 5 (0)$	$D 6 (0)$
2023	19.7824	6.5089	24.8669	29.4201	151.8813	1
2024	23.0226	7.0552	30.4110	36.3184	185.9386	1
2025	27.0495	7.6464	37.3478	45.0241	227.4617	1
2026	31.8847	8.2862	46.0382	56.0189	278.0885	1
2027	37.6805	8.9788	56.9361	69.9131	339.8161	1
2028	44.6388	9.7284	70.6133	87.4788	415.0798	1
2029	53.0081	10.5400	87.7890	109.6940	506.8487	1
2030	63.0916	11.4185	109.3685	137.7966	618.7437	1
2031	75.2592	12.3697	136.4915	173.3540	755.1795	1
2032	89.9624	13.3994	170.5920	218.3507	921.5392	1

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年份	研发经费/亿元	R&D人员全时当量/万人年	新产品开发经费支出/亿元	专利申请数/件	有效发明专利数/件	虚拟变量
2005	362.5	17.3	415.7	16,823	6,658	0
2006	456.44	18.89	509.95	24,301	8,141	0
2007	545.32	24.82	652.03	34,446	13,386	0
2008	655.2	28.5	798.4	39,656	23,915	1
2009	774.05	35.94	925.08	51,989	31,830	1
2010	967.8	39.9	1,006.9	59,683	50,166	1
2011	1,237.8	42.7	1,528.0	77,725	67,428	1
2012	1,491.5	52.6	1,827.5	97,200	97,878	1
2013	1,734.4	55.9	2,069.5	102,532	115,884	1
2014	1,922.2	57.3	2,350.6	120,077	147,927	1
2015	2,219.7	59.0	2,574.6	114,562	199,728	1
2016	2,437.6	58.0	3,000.4	131,680	257,234	1
2017	2,644.7	59.0	3,421.3	158,354	306,431	1
2018	2,912.5	66.1	3,768.4	179,600	327,965	1
2019	3,077.8	63.9	4,389.3	199,660	365,523	1
2020	4,649.1	99.0	6,152.4	348,522	570,905	1
2021	5,684.6	112.0	7,510.0	397,524	685,428	1
2022	6,507.7	125.4	8,590.6	434,039	809,824	1

模型	训练集MAPE(%)	预测集MAPE(%)	综合RMSE	综合R²	综合MAPE(%)
GM(1,N)	8.32	1.22	0.29	0.9963	7.48
DVCGM(1,N)	16.27	14.37	1.213	0.9359	16.05
NSGM(1,N)	2.62	2.38	0.1677	0.9988	2.59
DVSGM(1,N)	1.73	1.8	0.1358	0.9992	1.74

A New Structural Multi-variable Grey Forecasting Model with Virtual Variables and Its Application

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