规范变换与误差修正结合的环境系统的前向网络和投影寻踪预测模型
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摘要
为了建立适用于环境系统的结构简洁、形式统一、程序规范、应用普适的神经网络和投影寻踪回归预测模型,针对传统的神经网络和投影寻踪回归用于多因子、大样本预测建模,存在模型结构复杂、学习效率低的局限,提出设置环境系统预测量及其影响因子参照值和规范变换式的原则和方法,使规范变换后的影响因子皆"等效"于同一个规范影响因子,从而将多因子的的预测建模简化为等效规范因子的预测建模,使模型结构得到极大地简化,提高了学习效率;此外,为了提高预测模型的预测精度,还提出了对预测样本的模型输出值的误差修正公式.在对环境系统的预测量及其影响因子进行规范变换的基础上,将m个规范影响因子的每个建模样本组成m个"等效"训练样本,应用免疫进化算法优化模型参数,分别建立适用于环境系统的2个或3个规范影响因子的前向神经网络和投影寻踪回归两类预测模型;并依据误差理论,对误差修正公式修正后的模型预测精度的提高进行了严格的数学论证.将基于规范变换与相似样本误差修正相结合的两类预测模型,用于某市5个点位的SO_2浓度预测,并与6种传统预测模型和方法的预测结果进行了比较.结果表明:对同一个预测样本,同类模型的两种不同结构的的预测值及其相对误差都几乎完全相同或彼此相差甚小;此外,两种不同结构的两类预测模型用于5个样本预测,其相对误差绝对值的平均值分别为2.59%、2.67%;2.18%、2.62%,均远小于传统BP神经网络模型的25.72%、传统PPR模型的14.20%、传统SVR模型的22.13%、模糊识别模型的21.57%、组合算子模型的18.36%和多元回归模型的25.31%;而两类模型预测的最大的相对误差绝对值分别为4.11%和3.57%,更加远远小于传统的6种预测模型的37.18%、56.07%、27.40%、32.14%、38.38%和60.26%.实例分析结果证实了误差修正公式对提高模型预测精度具有切实可行性.基于规范变换与误差修正相结合的前向神经网络和投影寻踪回归两类预测模型不仅避免了"维数灾难",提高了学习效率和模型的预测精确度,而且具有简洁、普适、规范、统一和稳定的特点,对其他预测建模也有借鉴作用.
The purpose of this study was to establish prediction models of neural network and projection pursuit regression for environmental system, which have the charactertctics of simple structure, unified form, standardized procedures and universal application. The predictive models of traditional neural network and projection pursuit regression, which are used in multi-factor and large sample numbers, have limits of complex model structure, low learning efficiency. Therefore,the design principles and methods of reference values and the standard transformation formula used predicting variable and its influencing factors were proposed. The normalized influence factors were equivalent to the same normative influence factor. Thus, the predictive modeling of multiple factors is simplified as the predictive modeling of the "equivalent" norm factor, which greatly simplifies the model structure and improves the learning efficiency. In addition, in order to improve the prediction accuracy of the prediction model, the error correction formula for the model output value of the prediction sample was also proposed. On the basis of standard transformations for the predictive variable and its influencing factors of environment system, each modeling sample with m canonical influence factors were formed m "equivalent" training samples. Then, the immune evolutionary algorithm was used to optimize the model parameters, two different structures of prediction models of forward neural network and projection pursuit regression for environment system prediction were built: the case of 2-2-1 structure, which was used to any 2 normative impact factors and the case of 3-2-1 structure was, which was used to any 3 normative impact factors. Furthermore, based on the error theory, a rigorous mathematical demonstration was made for the improvement of the prediction accuracy of the model by the error correction formula. Two kinds of prediction models based on standard transformation and similar sample error correction were applied to predict the SO_2 concentration of 5 spots in a city. Results were compared with the prediction results of six traditional prediction models and methods. The results show that for the same forecast sample, the predicted values and relative errors of two different structures of the same model(forward neural networks or projection pursuit regression) were almost identical or very small. In addition, two kinds of prediction models with two different structures were used for the prediction of 5 samples, and the means of relative error absolute values were 2.59%, 2.67% and 2.18%, 2.62%, respectively. They were far less than the results of 22.13%, 25.72%,14.20%,21.57%, 18.36% and 25.31% of the prediction models of traditional BP neural network(BP), traditional projection pursuit regression(PPR), traditional support vector machine(SVM), fuzzy recognition, combination operator and multiple regression respectively. The maximum relative error absolute values of samples of the two prediction models were 4.11% and 3.57%,respectively. They were smaller than the results of 56.07%, 27.40%, 37.18%, 32.14%, 38.38% and 60.26% of six traditional forecasting models. The example analysis results confirm that the error correction formula is feasible for improving the prediction accuracy of the model. Two prediction models of forward neural network and projection pursuit regression based on the combination of standard transformation and error correction can avoid the "dimension disaster", improve the learning efficiency and model prediction accuracy. They have the characteristics of simplicity, universality, standardization, unity and stability. They can also be used for reference in other forecasting models.
The purpose of this study was to establish prediction models of neural network and projection pursuit regression for environmental system, which have the charactertctics of simple structure, unified form, standardized procedures and universal application. The predictive models of traditional neural network and projection pursuit regression, which are used in multi-factor and large sample numbers, have limits of complex model structure, low learning efficiency. Therefore,the design principles and methods of reference values and the standard transformation formula used predicting variable and its influencing factors were proposed. The normalized influence factors were equivalent to the same normative influence factor. Thus, the predictive modeling of multiple factors is simplified as the predictive modeling of the "equivalent" norm factor, which greatly simplifies the model structure and improves the learning efficiency. In addition, in order to improve the prediction accuracy of the prediction model, the error correction formula for the model output value of the prediction sample was also proposed. On the basis of standard transformations for the predictive variable and its influencing factors of environment system, each modeling sample with m canonical influence factors were formed m "equivalent" training samples. Then, the immune evolutionary algorithm was used to optimize the model parameters, two different structures of prediction models of forward neural network and projection pursuit regression for environment system prediction were built: the case of 2-2-1 structure, which was used to any 2 normative impact factors and the case of 3-2-1 structure was, which was used to any 3 normative impact factors. Furthermore, based on the error theory, a rigorous mathematical demonstration was made for the improvement of the prediction accuracy of the model by the error correction formula. Two kinds of prediction models based on standard transformation and similar sample error correction were applied to predict the SO_2 concentration of 5 spots in a city. Results were compared with the prediction results of six traditional prediction models and methods. The results show that for the same forecast sample, the predicted values and relative errors of two different structures of the same model(forward neural networks or projection pursuit regression) were almost identical or very small. In addition, two kinds of prediction models with two different structures were used for the prediction of 5 samples, and the means of relative error absolute values were 2.59%, 2.67% and 2.18%, 2.62%, respectively. They were far less than the results of 22.13%, 25.72%,14.20%,21.57%, 18.36% and 25.31% of the prediction models of traditional BP neural network(BP), traditional projection pursuit regression(PPR), traditional support vector machine(SVM), fuzzy recognition, combination operator and multiple regression respectively. The maximum relative error absolute values of samples of the two prediction models were 4.11% and 3.57%,respectively. They were smaller than the results of 56.07%, 27.40%, 37.18%, 32.14%, 38.38% and 60.26% of six traditional forecasting models. The example analysis results confirm that the error correction formula is feasible for improving the prediction accuracy of the model. Two prediction models of forward neural network and projection pursuit regression based on the combination of standard transformation and error correction can avoid the "dimension disaster", improve the learning efficiency and model prediction accuracy. They have the characteristics of simplicity, universality, standardization, unity and stability. They can also be used for reference in other forecasting models.
引文
Chen S Y,Ji H L.2005.Fuzzy optimization neural network approach for ice forecast in the Inner Mongolia reach of the Yellow River[J].Hydrological Sciences Journal,50(2):319-329
Chen C I.2008.Application of the novel nonlinear grey Bernoulli model for forecasting unemployment rate[J].Chaos,Solitons & Fractals,37(1):278-287
Chen S Y,Xue Z C,Li M.2013.Variable sets principle and method for flood classification[J].Science China Technological Sciences,56(9):2343-2348
Comrie A C.2013.Comparing neural networks and regression models for ozone forecasting[J].Journal of the Air & Water Mangement Association,47(6):653-663
Diaz Robles L A,Ortega J C,Fu J S,et al.2008.A hybrid ARIMA and artificial neural networks model to forecast particulate matter in urban areas:The case of Temuco Chile[J].Atmospheric Environment,42(35):8331-8340
笪英云,汪晓东,赵永刚,等.2015.基于关联向量机回归的水值预测模型[J].环境科学学报,35(11):3730-3735
Grivas G,Chaloulakon A.2006.Artificial neural network models for prediction of PM10 hourly concentrations,in the Greater Area of Athens,Greece[J].Atmospheric Environment,40(7):1216-1229
Ghose D K,Panda S S.2010.Prediction of water table depth in western region,Orissa using BPNN and RBFN neural networks[J].Journal of Hydrology,39(4):296-304
黄思,唐晓,徐文帅,等.2015.利用多模式集合和多元线性回归改进北京PM10 预报[J].环境科学学报,35(1):56 -64
Jubas I,Makra L,Toth B.2008.Forecasting of traffic origin NO and NO2 concentrations by support vector machines and neural networks using principal component analysis[J].Simulation Modelling Prictice and Theory,16(9):1488-1502
姜庆华.2006.大气污染预测的参数化组合算子方法[J].山东大学学报(理学版),41(4):76-79
金菊良,魏一鸣,王文圣.2009.基于集对分析的水资源相似预测模型[J].水力发电学报,28(1):72-77
Lange I A,Steel E A.2007.Using wavelet analysis to detect changes in water temperature regimes at multiples scales:Effects of multi-purpose dams in the Willamette River basin[J].River Research and Application,23(4):351-359
李祚泳,汪嘉杨,金相灿,等.2007.基于进化算法的湖泊富营养化投影寻踪回归预测模型[J].四川大学学报(工程科学版),39(2):1-8
李祚泳,徐源蔚,汪嘉杨,等.2015.基于前向神经网络的广义环境系统评价普适模型 [ J ].环境科学学报,35(9):2996-3005
李祚泳,汪嘉杨,徐源蔚.2018.基于规范变换与误差修正的回归支持向量机的环境系统预测[J].环境科学学报,38(3):1235-1244
刘永,郭怀成.2004.城市大气污染物浓度预测方法研究[J].安全与环境学报,4(4):59-62
Moazami S,Noori R,Amiri B J,et al.2016.Reliable prediction of carbon monoxide using development support vector machine[J].Atmospheric Pollution Research,7(3):412-418
Noori R,Barbassi A,Ashrafi K,et al.2012.Active and online prediction of BOD5 in river systems using reduced-order support vector machine[J].Environmental Earth Sciences,67(1):141-149
倪长健,丁晶,李祚泳.2003.免疫进化算法[J].西南交通大学学报,38(1):87-91
Ortiz Garcia E G,Salcedo Sanz S,Perez Bellido A M,et al.2010.Predic of hourly O3 concentrations using support vector regression algorithms[J].Atmospheric Environment,44(35):4481-4488
Park Y,Cho K H,Park J,et al.2015.Development of early-warning protocol for predicting chlorophyll-a concentration using machine learning models in freshwater and estuarine reservois,Korea[J].Science of the Total Environment,502:31-41
Paschalidou A K,Karakitsios S,Kleanthous S,et al.2011.Forecasting hourly PM10 concentration in Cyprus through artificial neural networks and multiple regression models implications to local environmental management[J].Environmental Science and Pollution Research,18(2):316-327
彭荔红,李祚泳,郑文教,等.2002.环境污染的投影寻踪回归预测模型[J].厦门大学学报(自然科学版),41(1):79-83
Qi X N,Liu Z G,Li D D.2008.Prediction of the performation of a shower cooling tower based on projection pursuit regression[J].Application Thermal Engineering,28(10):1031-1038
Shaban K B,Kadri A,Rezk E,et al.2016.Urban air pollution monitoring system with forecasting models[J].IEEE Sensors Journal,16(8):2598-2606
孙宝磊,孙蒿,张朝能,等.2017.基于BP神经网络的大气污染物浓度预测[J].环境科学学报,37(5):1864-1871
Wang X G,Tang Z,Tamura H,et al.2004.An improved bad - propagation algorithm to avoid the local minima problem[J].Neuro Computing,56(1):455-460
王保良,范昊,冀海峰,等.2016.基于分段线性表示的最近邻的水质预测方法[J].环境工程学报,10(2):1005-1009
肖鸣,李卫明,刘德富,等.2017.基于多重优化灰色模型的三峡库区香溪河支流回水区水华变化趋势预测研究[J].环境科学学报,37(3):1153-1161
熊德琪,陈守煜.1993.城市大气污染物浓度预测模糊识别理论与模型[J].环境科学学报,13(4):482 -490
杨云,付彦丽.2016.关于空气中PM2.5质量浓度预测研究[J].计算机仿真,33(3):413-418
邹志红,王乐娟.2009.湖泊富营养化趋势的灰色马尔柯夫预测[J].环境科学学报,29(2):427-432
郑彤,陈春云编.2003.环境系统数学模型[M].北京:化学工业出版社.31
Chen C I.2008.Application of the novel nonlinear grey Bernoulli model for forecasting unemployment rate[J].Chaos,Solitons & Fractals,37(1):278-287
Chen S Y,Xue Z C,Li M.2013.Variable sets principle and method for flood classification[J].Science China Technological Sciences,56(9):2343-2348
Comrie A C.2013.Comparing neural networks and regression models for ozone forecasting[J].Journal of the Air & Water Mangement Association,47(6):653-663
Diaz Robles L A,Ortega J C,Fu J S,et al.2008.A hybrid ARIMA and artificial neural networks model to forecast particulate matter in urban areas:The case of Temuco Chile[J].Atmospheric Environment,42(35):8331-8340
笪英云,汪晓东,赵永刚,等.2015.基于关联向量机回归的水值预测模型[J].环境科学学报,35(11):3730-3735
Grivas G,Chaloulakon A.2006.Artificial neural network models for prediction of PM10 hourly concentrations,in the Greater Area of Athens,Greece[J].Atmospheric Environment,40(7):1216-1229
Ghose D K,Panda S S.2010.Prediction of water table depth in western region,Orissa using BPNN and RBFN neural networks[J].Journal of Hydrology,39(4):296-304
黄思,唐晓,徐文帅,等.2015.利用多模式集合和多元线性回归改进北京PM10 预报[J].环境科学学报,35(1):56 -64
Jubas I,Makra L,Toth B.2008.Forecasting of traffic origin NO and NO2 concentrations by support vector machines and neural networks using principal component analysis[J].Simulation Modelling Prictice and Theory,16(9):1488-1502
姜庆华.2006.大气污染预测的参数化组合算子方法[J].山东大学学报(理学版),41(4):76-79
金菊良,魏一鸣,王文圣.2009.基于集对分析的水资源相似预测模型[J].水力发电学报,28(1):72-77
Lange I A,Steel E A.2007.Using wavelet analysis to detect changes in water temperature regimes at multiples scales:Effects of multi-purpose dams in the Willamette River basin[J].River Research and Application,23(4):351-359
李祚泳,汪嘉杨,金相灿,等.2007.基于进化算法的湖泊富营养化投影寻踪回归预测模型[J].四川大学学报(工程科学版),39(2):1-8
李祚泳,徐源蔚,汪嘉杨,等.2015.基于前向神经网络的广义环境系统评价普适模型 [ J ].环境科学学报,35(9):2996-3005
李祚泳,汪嘉杨,徐源蔚.2018.基于规范变换与误差修正的回归支持向量机的环境系统预测[J].环境科学学报,38(3):1235-1244
刘永,郭怀成.2004.城市大气污染物浓度预测方法研究[J].安全与环境学报,4(4):59-62
Moazami S,Noori R,Amiri B J,et al.2016.Reliable prediction of carbon monoxide using development support vector machine[J].Atmospheric Pollution Research,7(3):412-418
Noori R,Barbassi A,Ashrafi K,et al.2012.Active and online prediction of BOD5 in river systems using reduced-order support vector machine[J].Environmental Earth Sciences,67(1):141-149
倪长健,丁晶,李祚泳.2003.免疫进化算法[J].西南交通大学学报,38(1):87-91
Ortiz Garcia E G,Salcedo Sanz S,Perez Bellido A M,et al.2010.Predic of hourly O3 concentrations using support vector regression algorithms[J].Atmospheric Environment,44(35):4481-4488
Park Y,Cho K H,Park J,et al.2015.Development of early-warning protocol for predicting chlorophyll-a concentration using machine learning models in freshwater and estuarine reservois,Korea[J].Science of the Total Environment,502:31-41
Paschalidou A K,Karakitsios S,Kleanthous S,et al.2011.Forecasting hourly PM10 concentration in Cyprus through artificial neural networks and multiple regression models implications to local environmental management[J].Environmental Science and Pollution Research,18(2):316-327
彭荔红,李祚泳,郑文教,等.2002.环境污染的投影寻踪回归预测模型[J].厦门大学学报(自然科学版),41(1):79-83
Qi X N,Liu Z G,Li D D.2008.Prediction of the performation of a shower cooling tower based on projection pursuit regression[J].Application Thermal Engineering,28(10):1031-1038
Shaban K B,Kadri A,Rezk E,et al.2016.Urban air pollution monitoring system with forecasting models[J].IEEE Sensors Journal,16(8):2598-2606
孙宝磊,孙蒿,张朝能,等.2017.基于BP神经网络的大气污染物浓度预测[J].环境科学学报,37(5):1864-1871
Wang X G,Tang Z,Tamura H,et al.2004.An improved bad - propagation algorithm to avoid the local minima problem[J].Neuro Computing,56(1):455-460
王保良,范昊,冀海峰,等.2016.基于分段线性表示的最近邻的水质预测方法[J].环境工程学报,10(2):1005-1009
肖鸣,李卫明,刘德富,等.2017.基于多重优化灰色模型的三峡库区香溪河支流回水区水华变化趋势预测研究[J].环境科学学报,37(3):1153-1161
熊德琪,陈守煜.1993.城市大气污染物浓度预测模糊识别理论与模型[J].环境科学学报,13(4):482 -490
杨云,付彦丽.2016.关于空气中PM2.5质量浓度预测研究[J].计算机仿真,33(3):413-418
邹志红,王乐娟.2009.湖泊富营养化趋势的灰色马尔柯夫预测[J].环境科学学报,29(2):427-432
郑彤,陈春云编.2003.环境系统数学模型[M].北京:化学工业出版社.31
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