A two-step method for cusp catastrophe model construction based on the selection of important variables
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摘要: 突變是工程實踐過程中廣泛存在的現象。當系統的狀態發生跳躍性變化時，基于微積分的傳統數學建模方法精度較低，人工神經網絡等機器學習算法無法對突變現象作出合理的解釋。基于突變理論的尖點突變模型可以用來解釋系統狀態的不連續變化，然而在輸入變量維度較大的情況下，傳統的尖點突變模型復雜度高且精度較差。為了解決這一問題，提出了一種基于變量選擇的尖點突變模型的兩步構建方法。第一步，利用多模型集成重要變量選擇算法(MEIVS)量化待選變量的重要性并提取重要變量；第二步，基于極大似然法(MLE)利用所提取的重要變量構建尖點突變模型。仿真結果表明，在具有突變特征的數據集上，通過MEIVS降維后的尖點突變模型在評價指標上優于線性模型、Logistic模型和通過其他方法降維的尖點突變模型，并且可以用來解釋研究對象的不連續變化。Abstract: Sudden transition is a widely existing phenomenon in engineering practice. When the state of the system experiences sudden abrupt transition, calculus-based traditional mathematical modeling methods has low accuracy. Although theoretically, machine learning algorithms, such as artificial neural networks, can approximate any nonlinear function, this type of black-box method makes no reasonable explanation for the sudden transition phenomenon. The cusp catastrophe model based on the catastrophe theory can be applied to explain the discontinuous changes in the system’s state. However, the construction of traditional cusp catastrophe models is often based on large amounts of prior knowledge to select the input variables for modeling. On the condition that there is a lack of prior knowledge and comparatively large dimensions of input variables, the model has high complexity and poor accuracy. In this paper we have put forward a two-step method for constructing a cusp catastrophe model based on the selection of variables to solve the abovementioned problems. The first step was to apply multimodel ensemble important variable selection (MEIVS) to quantify the importance of the variables to be selected and extract important variables. The second step was to use the extracted important variables to construct a cusp catastrophe model based on the framework of maximum likelihood estimation (MLE). Results indicate that on a dataset with characteristics of catastrophe, the cusp catastrophe model is simple in form using the MEIVS dimensionality reduction algorithm and outperforms the unreduced cusp catastrophe model and reduced cusp catastrophe model using other dimensionality reduction algorithms in terms of evaluation indicators. This shows that the algorithm proposed in this paper have improved the accuracy and reduced the complexity of the cusp catastrophe model. At the same time, the cusp catastrophe model exhibits higher accuracy compared with the linear and logistic models. Thus, it can be used to explain the discontinuous changes of the research object, and it has a practical engineering significance.
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Key words:
- catastrophe theory /
- catastrophe flag /
- cusp catastrophe model /
- variable selection /
- model integration
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圖 1 尖點突變模型的平衡曲面和控制平面
Figure 1. Equilibrium surface and control plane of the cusp catastrophe model
圖 2 MEIVS算法流程圖. (a) MEIVS算法主流程; (b) 排列算法流程
Figure 2. MEIVS algorithm flowchart: (a) main process steps of the MEIVS algorithm; (b) process steps of the permutation algorithm
圖 3 每日住宿價格的概率密度非參數估計
Figure 3. Nonparametric estimation of the probability density of the daily accommodation price
圖 4 歐洲旅館住宿價格數據集待選變量重要性得分
Figure 4. Importance score of the variables to be selected in the European hotel accommodation price dataset
圖 5 歐洲旅館住宿價格數據在控制平面 (a) 和平衡曲面(b) 上的分布
Figure 5. Distribution of the European hotel accommodation price dataset on the control plane (a) and equilibrium surface (b)
圖 7 北京大氣腐蝕數據集待選變量重要性得分
Figure 7. Importance score of the variables to be selected in the Beijing atmospheric corrosion dataset
圖 8 北京大氣腐蝕數據在控制平面(a)和平衡曲面(b)上的分布
Figure 8. Distribution of the Beijing atmospheric corrosion dataset on the control plane (a) and equilibrium surface (b)
表 1 歐洲旅館住宿價格數據集建模結果評價
Table 1. Evaluation of the modeling results of the European hotel accommodation price dataset
Model Number of parameters R2 AIC BIC Linear 0.549 1306 1323 Logistic 0.626 1294 1324 Cusp (based on the two-step method) 12 0.727 195 228 Cusp (based on the traditional method) 16 0.697 190 235 Cusp (based on SCC) 10 0.572 204 232 Cusp (based on MIC) 6 0.421 234 251 Cusp (based on RFVIM) 12 0.565 210 243 
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表 2 北京大氣腐蝕數據集建模結果評價
Table 2. Evaluation of the modeling results of the Beijing atmospheric corrosion dataset
Model Number of parameters R2 AIC BIC Linear 0.668 2180 2203 Logistic 0.755 1970 2011 Cusp (based on the two-step method) 10 0.778 670 716 Cusp (based on the traditional method) 20 0.775 672 764 Cusp (based on SCC) 10 0.719 816 862 Cusp (based on MIC) 8 0.725 820 857 Cusp (based on RFVIM) 18 0.765 673 755
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