三参数区间灰数序列的矩阵型自回归时滞灰色多变量模型

doi:10.16381/j.cnki.issn1003-207x.2021.0610

摘要/Abstract

摘要：

已有的灰色预测模型只能间接地预测三参数区间灰数序列，为进一步提高灰色模型对三参数区间灰数的预测精度，在传统灰色模型的基础上考虑时滞效应对系统特征的影响，把系统特征的自回归时滞项引进传统灰色模型，构建了自回归时滞灰色多变量模型。然后进一步把模型变量看作列向量，把参数设置为矩阵的形式，提出了矩阵型自回归时滞灰色多变量模型。该模型保持了三参数区间灰数的完整性，接连了灰数各界点之间的内在关系，能直接对三参数区间灰数序列进行建模和预测。对第二产业总值和第三产业总值的三参数区间灰数序列进行拟合和预测，结果表明，新模型具有比竞争模型更高的精度，具有一定的有效性和实用性。

关键词: 三参数区间灰数, 灰色多变量模型, 自回归, 时滞效应

Abstract:

The existing grey models can only predict three-parameter interval grey number sequences indirectly， and the influence of time-delay effect is seldom considered. In order to improve the prediction accuracy of grey model for three-parameter interval grey numbers， a new model considering time-delay effect is proposed. Firstly， the autoregressive time-delay terms of system characteristic are introduced into the traditional grey models. Further， a matrix autoregressive time-delay grey multivariable model （MARGM（1， N）） is proposed by taking the variables as three-dimensional column vectors and setting the parameters as matrices. The integrity of the three-parameter interval grey numbers is maintained and the relations between the boundaries are connected， so that the three-parameter interval grey number sequences can be modeled and predicted directly by MARGM（1， N）. MARGM（1， N） is used to forecast the output of China's secondary industry and tertiary industry. The results show that the models considering the time-delay effect have better fitting effect， the matrix models have higher prediction accuracy， and MARGM（1， N） has the best performance in both fitting and forecasting compared with the competition models. Overall， MARGM（1， N） is feasible and effective. In application， it is helpful for relevant departments to make correct policies to coordinate economic development.

Key words: three-parameter interval grey numbers, grey multivariable model, autoregression, time-delay effect

中图分类号:

N941.5

梅韵杰,曾祥艳,闫书丽. 三参数区间灰数序列的矩阵型自回归时滞灰色多变量模型[J]. 中国管理科学, 2024, 32(9): 48-58.

Yunjie Mei,Xiangyan Zeng,Shuli Yan. A Matrix Autoregressive Time-delay Grey Multivariable Model for Three-parameter Interval Grey Number Sequences[J]. Chinese Journal of Management Science, 2024, 32(9): 48-58.

图/表 11

图1

表1

MARGM（1，N）在第二产业总值预测中的参数估计结果"

下界参数	$a 11$	$a 12$	$a 13$	$b 11 (1)$	$b 12 (1)$	$b 13 (1)$	$b 11 (2)$	$b 12 (2)$	$b 13 (2)$	$q 11 (2)$	$q 12 (2)$	$q 13 (2)$	$c 1$
下界参数	-3.09	6.03	-2.85	-0.75	0.71	0.85	-0.76	1.47	-0.64	-1.11	1.89	-0.76	-1547.67
中界参数	$a 21$	$a 22$	$a 23$	$b 21 (1)$	$b 22 (1)$	$b 23 (1)$	$b 21 (2)$	$b 22 (2)$	$b 23 (2)$	$q 21 (2)$	$q 22 (2)$	$q 23 (2)$	$c 2$
中界参数	0.25	0.29	-0.28	0.61	-1.43	1.50	0.04	-0.04	0.06	-0.43	0.07	0.34	1622.50
上界参数	$a 31$	$a 32$	$a 33$	$b 31 (1)$	$b 32 (1)$	$b 33 (1)$	$b 31 (2)$	$b 32 (2)$	$b 33 (2)$	$q 31 (2)$	$q 32 (2)$	$q 33 (2)$	$c 3$
上界参数	3.34	-3.17	0.27	1.30	-1.90	0.85	0.51	-0.70	0.26	0.44	-1.56	1.06	9691.73

表1

表2

第二产业总值的原始序列、MARGM（1，N）和MTIGM（1，1）的拟合和预测结果"

年份	原序列	MARGM(1，N)	MTIGM(1，1)
年份	拟合值
2000	[9547.8, 11415.9, 13102.0]	[9547.8, 11415.9, 13102.0]	[9547.8, 11415.9, 13102.0]
2001	[10641.4, 12414.9, 13915.4]	[10691.4, 12469.0, 13991.0]	[11539.7, 13808.1, 15814.1]
2002	[11319.7, 13526.0, 15460.6]	[11848.8, 13932.3, 15812.6]	[12267.4, 14667.5, 16800.9]
2003	[13146.2, 15674.0, 18031.6]	[12790.2, 15643.7, 18221.3]	[13450.2, 16145.9, 18553.5]
2004	[15292.1, 18571.3, 21535.5]	[15104.9, 18531.5, 21350.4]	[15274.5, 18420.1 21230.1]
2005	[18159.0, 22020.6, 25723.9]	[18040.7, 21911.2, 25362.1]	[17843.0, 21648.0, 25041.4]
2006	[21418.2, 26089.8, 30429.8]	[21705.2, 26375.1, 30693.7]	[20974.5, 25609.3, 29738.8]
2007	[25983.1, 31657.7, 36861.1]	[26043.7, 31881.9, 37165.8]	[25031.1, 30630.1, 35593.4]
2008	[31613.2, 37488.3, 41196.4]	[31041.0, 36229.9, 40175.9]	[29273.5, 35182.1, 40309.3]
2009	[32594.0, 40042.2, 46928.2]	[21503.2, 39845.5, 46699.5]	[34679.8, 41146.1, 46636.9]
2010	[39364.2, 47906.6, 55933.6]	[39846.4, 48476.1, 56528.1]	[41302.2, 49370.3, 56242.3]
2011	[47194.0, 56758.8,64334.6]	[47128.3, 56617.4, 64177.5]	[48055.9, 57299.3, 56242.3]
2012	[52316.2, 61159.8, 68633.5]	[52408.1, 61509.3, 68972.0]	[52394.5, 62066.7, 70240.1]
2013	[55861.2, 65487.9, 74211.4]	[55463.0, 65170.2, 73822.2]	[54032.6, 64124.5, 72800.5]
2014	[59127.4, 69320.7, 77725.9]	[59232.5, 69441.3, 77820.6]	[56203.6, 66712.5, 75868.0]
2015	[60505.9, 70334.7, 78356.3]	[60861.3, 70824.3, 78824.3]	[57966.5, 68461.7, 77683.5]
2016	[61106.8, 73857.0, 85504.1]	[61005.3, 73512.1, 85288.3]	[62766.0, 74196.9, 84292.9]
2017	[69315.5, 82895.2, 95368.0]	[68985.7, 82509.1, 94923.8]	[70664.4, 84067.0, 95940.9]
2018	[76598.2, 91208.8, 104023.9]	[76811.6, 91534.3, 104344.7]	[77792.1, 92858.0, 106259.6]
2019	[80596.7, 95167.7, 107730.1]	[84142.8, 99794.1, 114270.2]	[85939.5, 102731.5, 117692.3]
2020	[72533.4, 96063.8, 113939.9]	[93147.8, 10598.8, 126121.1]	[95934.1, 114912.6, 131831.1]

表2

表3

表4

表5

表6

MARGM（1，N）的参数"

上界参数	$a 11$	$a 12$	$a 13$	$b 11 (1)$	$b 12 (1)$	$b 13 (1)$	$q 11 (2)$	$q 12 (2)$	$q 13 (2)$	$q 11 (3)$	$q 11 (3)$	$q 11 (3)$	$c 1$
上界参数	2.01	-3.04	0.93	1.31	-3.53	2.54	-1.80	2.01	-0.18	1.27	-1.15	-0.36	5330.60
中界参数	$a 21$	$a 22$	$a 23$	$b 21 (1)$	$b 22 (1)$	$b 23 (1)$	$q 21 (2)$	$q 22 (2)$	$q 23 (2)$	$q 21 (3)$	$q 22 (3)$	$q 23 (3)$	$c 2$
中界参数	0.96	-1.59	0.45	1.59	-3.95	2.49	-3.86	4.05	-0.31	1.19	-0.99	-0.32	8056.04
下界参数	$a 31$	$a 32$	$a 33$	$b 31 (1)$	$b 32 (1)$	$b 33 (1)$	$q 31 (2)$	$q 32 (2)$	$q 33 (2)$	$q 31 (3)$	$q 32 (3)$	$q 33 (3)$	$c 3$
下界参数	-1.17	3.19	-2.04	1.39	-2.91	1.20	-3.10	3.39	-0.30	1.04	0.77	-1.47	13234.91

表6

表7

表8

表9

图2

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年份	ARGM(1，N)	GM(1，N)	GM(1，1)
年份	拟合值
2000	[9547.8, 11415.9, 13102.0]	[9547.8, 11415.9, 13102.0]	[9547.8, 11415.9, 13102.0]
2001	[10688.5, 12864.7, 14740.5]	[-150, -336.7, -124.7]	[15226.8, 18253.0, 20822.8]
2002	[13643.8, 15744.9, 17488.9]	[1525.8, 1541.3, 1999.2]	[16327.4, 19543.1, 22279.9]
2003	[14076.3, 16615.5, 18947.7]	[3261.9, 3629.2, 4362.8]	[17553.6, 20995.4, 23941.2]
2004	[13643.8, 14076.3, 15868.8]	[5310.3, 6035.4, 7090.4]	[18978.9, 22698.5, 25903.8]
2005	[18174.7, 21952.7, 25461.1]	[7764.4, 9172.4, 10620.6]	[20655.3, 24717.3, 28247.9]
2006	[21194.7, 25579.1, 29900.2]	[10860.9, 13053.0, 14974.1]	[22638.8,27110.1, 31033.3]
2007	[24798.2, 30169.2, 35222.2]	[14378.8, 17288.3, 19759.1]	[25014.5, 29982.1, 34371.0]
2008	[29954.5, 36427.7, 42229.8]	[18673.0, 22622.0, 25762.2]	[27901.0, 33421.0, 38242.8]
2009	[35987.1, 42575.9, 46519.3]	[24271.3, 29198.3, 33114.1]	[31116.7, 37276.9, 42614.0]
2010	[36303.8, 44544.8, 52155.5]	[29415.0, 35599.9, 40350.9]	[34720.7, 41651.0,47716.1]
2011	[43114.5, 52736.6, 61704.4]	[35816.2, 43376.7, 49128.3]	[39058.8, 46856.4, 53681.7]
2012	[51695.1, 62381.0, 70501.6]	[43852.0, 52841.7, 59750.4]	[44046.0, 52721.0, 60277.2]
2013	[56767.2, 66309.8, 74184.6]	[52126.4, 62091.3, 70203.9]	[49467.5, 59019.7, 67362.6]
2014	[59417.7, 69615.5, 79055.9]	[60142.6, 71154.9, 80515.9]	[55230.4, 65724.3, 74899.0]
2015	[62152.0,73206.1, 82475.4]	[67325.4, 79145.8, 89741.1]	[61226.1, 72669.9, 82640.9]
2016	[63970.3, 74408.7, 83180.7]	[72463.7, 85177.1, 96952.7]	[67321.0, 79841.2, 90768.7]
2017	[65808.3, 7981.5, 92408.0]	[76076.9, 90432.1, 103435.2]	[73857.4, 87637.1, 99740.4]
2018	[77895.7, 92840.9, 106262.0]	[80473.1, 96575.4, 110956.2]	[81170.2, 96296.0, 109630.6]
2019	[88301.0, 104172.9, 117668.4]	[87898.2, 106890.7, 158062.3]	[89494.3, 106109.5, 120781.4]
2020	[101913.3, 119368.7, 133729.5]	[97976.7, 122487.4, 227064.4]	[98938.0,117216.4, 133388.7]

模型	MARGM(1，N)	MTIGM(1，1)	ARGM(1，N)	GM (1，N)	GM (1，1)
拟合MAPE（%）	0.86	3.23	4.14	37.12	14.81
拟合MAE	296.37	1141.53	1582.19	10406.69	4714.17
拟合RMSE	87.07	316.00	470.66	2596.35	1234.52
预测MAPE（%）	11.60	15.23	18.40	38.33	18.36
预测MAE	10340.55	13834.91	16520.35	39058.00	16649.46
预测RMSE	8332.77	10798.40	12917.05	29865.64	12516.75
总体MAPE（%）	1.94	4.43	5.56	37.24	15.16
总体MAE	1300.78	2410.87	3076.01	13271.82	5907.50
总体RMSE	837.30	1116.74	1360.30	3959.48	1678.79

年份	流通中现金供应量	社会消费品零售总额
2019	[71152.7, 73133.1, 81476.4]	[30874.5, 34867.4, 38579.2]
2020	[72321.0, 73341.6,76495.0]	[32357.2, 36607.0, 39980.4]
MAPE(%)	8.40	5.64

年份	原序列	MARGM(1，N)	MTIGM(1，1)
年份	拟合值
2002	[12502.1, 12855.8, 13584.2]	[12502.1, 12855.8, 13584.2]	[12502.1, 12855.8, 13584.2]
2003	[13870.6, 14439.0, 15481.4]	[13693.1, 14329.7, 15245.5]	[13325.9, 13974.8, 15267.1]
2004	[16124.3, 16662.7, 17524.6]	[16475.4, 16895.4, 18016.3]	[16163.9, 16833.6, 18215.4]
2005	[18746.0, 19357.5, 20344.5]	[18792.3, 19391.4, 20431.7]	[19423.3, 20052.4, 21044.9]
2006	[22190.4, 22940.6, 24575. 3]	[22201.3, 23040.2, 24773.9]	[22882.7, 23661.7, 25066.1]
2007	[27703.2, 28946.9, 31469.8]	[27462.7, 28671.0, 30877.9]	[27198.0, 28140.8, 29810.7]
2008	[33392.9, 34206.9, 36062.5]	[33312.7, 34143.6, 35874.0]	[32455.5, 33742.9, 36326.0]
2009	[37065.4, 38691.3, 41541.7]	[36952.5, 38571.2, 41440.8]	[38262.4, 39623.5, 42116.9]
2010	[43311.1, 45515.5, 49187.8]	[43526.4, 45723.6, 49428.1]	[43808.0, 45413.1, 48114.8]
2011	[51683.2, 54030.9, 57487.8]	[51555.4, 53929.5, 57490.9]	[50434.1, 52661.8, 56580.9]
2012	[58595.5, 61214.1, 65107.2]	[58686.2, 61260.1, 65045.0]	[58157.3, 60946.9, 65232.1]
2013	[66719.3, 69495.9, 73696.6]	[66773.5, 69595.4, 73985.3]	[66288.1, 69432.5, 73571.8]
2014	[74492.4, 77663.5, 82580.9]	[74326.4, 77514.2, 82247.7]	[75111.2, 78706.5, 83881.3]
2015	[83258.8, 87436.2, 92840.6]	[83575.8, 87715.0, 93353.3]	[83754.3, 87736.5, 93639.0]
2016	[92990.5, 97707.0, 104334.0]	[92871.9, 97556.4, 104073.6]	[93369.5, 97897.9, 103750.6]
2017	[104346.3, 109589.0, 117067.8]	[104237.7, 109516.8, 116934.0]	[103940.5, 109021.2, 116356.0]
2018	[116861.8, 122425.2, 129846.2]	[116908.3, 122468.5, 129920.8]	[116776.9, 122476.0, 130174.9]
2019	[127803.3, 133842.8, 141603.5]	[129161.8, 134103.6, 136560.5]	[131135.7, 137154.6, 144606.1]
2020	[123008.5, 138494.2, 152727.6]	[123691.6, 130557.4, 151822.6]	[148255.4, 154384.5, 160888.3]

年份	ARGM(1，N)	GM(1，N)	GM(1，1)
年份	拟合值
2002	[12502.1, 12855.8, 13584.2]	[12502.1, 12855.8, 13584.2]	[12502.1, 12855.8, 13584.2]
2003	[13325.9, 13974.8, 15267.1]	[7565.1, 8304.7, 8479.5]	[17580.0, 18241.2, 19508.1]
2004	[16163.9, 16833.6, 18215.4]	[11425.3, 12152.1, 12622.8]	[19527.5, 20273.8, 21659.6]
2005	[19423.3, 20052.4, 21044.9]	[15540.1, 16380.0, 17127.7]	[21791.4, 22627.8, 24128.1]
2006	[22882.7, 23661.7, 25066.1]	[20187.9, 21210.5, 22480.5]	[24449.2, 25392.1, 27056.2]
2007	[27198.0, 28140.8, 29810.7]	[25190.9, 26210.1, 27518.6]	[27688.5, 28783.1, 30709.5]
2008	[32455.5, 33742.9, 36326.0]	[30559.1, 31585.3, 33486.9]	[31655.2, 32910.4, 35111.5]
2009	[38262.4, 39623.5, 42116.9]	[36376.0, 37515.4, 40160.9]	[36229.6, 37674.5, 40170.1]
2010	[43808.0, 45413.1, 48114.8]	[43343.9, 44931.4, 47271.7]	[41448.1, 43177.6, 46084.3]
2011	[50434.1, 52661.8, 56580.9]	[51296.7, 53324.5, 56984.8]	[47615.5, 49683.2, 53037.9]
2012	[58157.3, 60946.9, 65232.1]	[59721.0, 62167.3, 66348.9]	[54775.5, 49683.2, 53037.9]
2013	[66288.1, 69432.5, 73571.8]	[68769.3, 71606.2, 75600.9]	[62911.4, 65757.0, 70077.2]
2014	[75111.2, 78706.5, 83881.3]	[77770.7, 81187.9, 87559.9]	[72079.5, 75374.3, 80264.1]
2015	[83754.3, 87736.5, 93639.0]	[86493.7, 90854.1, 97537.1]	[82321.4, 86164.0, 91698.9]
2016	[99369.5, 109021.2, 116356.0]	[95338.8, 100408.7, 106189.4]	[93764.4, 98263.6, 104551.6]
2017	[103940.5, 109021.2, 116356.0]	[104178.0, 109896.8, 116943.1]	[106575.4, 111810.9, 118983.7]
2018	[116776.9, 122476.0, 130174.9]	[111815.1, 116788.0, 124128.5]	[120938.3, 126973.7, 135078.7]
2019	[131135.7, 137154.6, 144606.1]	[122046.7, 128355.3, 134240.4]	[137339.3, 144412.2, 153551.9]
2020	[148255.4, 154284.5, 160888.3]	[128810.3, 134295.6, 138785.6]	[156536.3, 164607.4, 174966.3]