定量和定性因子混合的罚盲克里金模型

doi:10.12305/j.issn.1001-506X.2025.01.16

Abstract

Abstract:

Kriging model is a very effective spatial interpolation method, which is widely studied and applied to surrogate models for observation in engineering fields such as geographical science, environmental science and atmospheric science. For high-dimensional observation samples with both quantitative and qualitative factors as input, a penalized blind Kriging (PBK) model with a mix of quantitative and qualitative factors is proposed. On the basis of the PBK model, for the case that the input data contains a mixture of quantitative and qualitative factors, the Gaussian correlation model with a mixture of quantitative and qualitative factors is used to establish the PBK model with mixed factors, and the penalty function is used to select factors for the mean function. The numerical experiments of linear and nonlinear piecewise functions verify that the PBK model with quantitative and qualitative factors has a high accuracy. The results show that the first-order PBK model and the second-order PBK model with limited samples both have less relative root mean square error, normalized root mean squared error, and root mean square percentage error.

Key words: penalized blind Kriging (PBK), quantitative and qualitative factor, Lasso penalty, least angle regression

CLC Number:

N945.12

Dahao CHEN, Zhijun CHENG, Jian ZHONG, Zhengqiang PAN. Penalized blind Kriging model with mix quantitative and qualitative factors[J]. Systems Engineering and Electronics, 2025, 47(1): 153-163.

Figures/Tables 12

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References 31

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函数名称	R_k(θ_k, d_i)
指数	exp(－θ_kd_k)
幂指数	exp(－θ_kd_k^δ), 0<δ≤2
高斯	exp(－θ_kd_k²)
线性	max{0, 1－θ_kd_k}
球形	1－1.5ξ_i+0.5ξ_i³, ξ_i=min{1, θ_kd_k}
立方	1－3ξ_i+2ξ_i³, ξ_i=min{1, θ_kd_k}

样本数	指标	线性分段函数f₁(x)
样本数	指标	OK	UK1	UK2	PBK1	PBK2
90	RRMSE	0.109 76	0.045 89	0.042 21	0.029 85	0.026 43
	NRMSE	0.021 68	0.009 06	0.008 34	0.005 90	0.005 22
	RMSPE	0.000 15	0.000 06	0.000 05	0.000 04	0.000 04
150	RRMSE	0.063 31	0.028 42	0.025 60	0.013 69	0.012 24
	NRMSE	0.012 52	0.005 63	0.005 07	0.002 71	0.002 42
	RMSPE	0.000 09	0.000 04	0.000 03	0.000 02	0.000 02
210	RRMSE	0.040 52	0.018 21	0.016 13	0.010 99	0.010 28
	NRMSE	0.007 96	0.003 57	0.003 16	0.002 16	0.002 02
	RMSPE	0.000 06	0.000 02	0.000 02	0.000 01	0.000 01
样本数	指标	非线性分段函数f₂(x)
样本数	指标	OK	UK1	UK2	PBK1	PBK2
90	RRMSE	0.087 47	0.077 61	0.062 69	0.045 21	0.040 72
	NRMSE	0.020 20	0.017 94	0.014 50	0.010 46	0.009 42
	RMSPE	0.000 86	0.000 64	0.000 71	0.000 39	0.000 42
150	RRMSE	0.050 01	0.043 55	0.034 53	0.024 94	0.021 83
	NRMSE	0.011 64	0.010 13	0.008 04	0.005 80	0.005 08
	RMSPE	0.000 36	0.000 28	0.000 30	0.000 16	0.000 18
210	RRMSE	0.032 84	0.029 56	0.023 65	0.016 18	0.014 93
	NRMSE	0.007 60	0.006 84	0.005 47	0.003 74	0.003 45
	RMSPE	0.000 25	0.000 19	0.000 20	0.000 11	0.000 13

模型	1	x₁	x₂	x₃	x₄	x₅	x₁²	x₁x₂	x₁x₃	x₁x₄	x₁x₅
OK	0.104	-	-	-	-	-	-	-	-	-	-
UK1	0.024	0.669	0.347	0.357	0.623	-0.030	-	-	-	-	-
UK2	-0.027	0.653	0.352	0.369	0.626	-0.005	0.017	-0.001	0.006	-0.009	-0.088
PBK1	0	0.654	0.310	0.339	0.562	0	-	-	-	-	-
PBK2	0	0.647	0.314	0.345	0.578	0	0.010	0.002	0	0	-0.064
模型	x₂²	x₂x₃	x₂x₄	x₂x₅	x₃²	x₃x₄	x₃x₅	x₄²	x₄x₅	x₅²	-
OK	-	-	-	-	-	-	-	-	-	-	-
UK1	-	-	-	-	-	-	-	-	-	-	-
UK2	0.010	0.003	-0.004	0.117	0.002	0.008	0.132	0.000	-0.158	-0.020	-
PBK1	-	-	-	-	-	-	-	-	-	-	-
PBK2	0	0.004	0	0.085	-0.004	0.006	0.113	0	-0.126	0	-

模型	1	x₁	x₂	x₃	x₄	x₅	x₁²	x₁x₂	x₁x₃	x₁x₄	x₁x₅
OK	0.074	-	-	-	-	-	-	-	-	-	-
UK1	0.111	0.166	0.155	0.274	0.052	-0.134	-	-	-	-	-
UK2	1.159	0.194	0.165	0.252	0.082	-0.139	-0.005	0.054	0.017	0.008	-0.070
PBK1	0	0.145	0.112	0.259	0.034	-0.086	-	-	-	-	-
PBK2	0	0.139	0.102	0.220	0.036	0	0	0.045	0.008	0	-0.016
模型	x₂²	x₂x₃	x₂x₄	x₂x₅	x₃²	x₃x₄	x₃x₅	x₄²	x₄x₅	x₅²	-
OK	-	-	-	-	-	-	-	-	-	-	-
UK1	-	-	-	-	-	-	-	-	-	-	-
UK2	0.050	0.060	0.016	-0.052	-0.027	0.057	-0.065	0.050	-0.064	-1.236	-
PBK1	-	-	-	-	-	-	-	-	-	-	-
PBK2	0.055	0.043	0.010	-0.016	-0.002	0.052	-0.015	0.060	-0.016	-0.429	-

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Penalized blind Kriging model with mix quantitative and qualitative factors

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