基于Copula的海洋生态稳态转换及Lyapunov指数回归树估计
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摘要
覆盖地球表面积72%的海洋是我们赖以生存的全球生态系统的重要组成部分。在过去10多年中，海洋生态系统研究在国内外都受到较多的关注，取得重大进展。在国际上，联合国和IGBP/SCOR/IOC/UNEP/UNDP等许多重要国际组织、强调海洋生态系统功能研究的重要意义。在国内，国家实施了一系列有关的重大海洋科学计划，如国家973计划、基金重大和重点项目以及有关的国家专项等。我国科学家关注并积极参与国内外的研究发展，开始探讨海洋多学科交叉与整合研究的科学思路。本文首次利用统计学Copula理论计算海洋生态稳态转移概率，提出时间序列Lyapunov指数的回归树估计方法，整合各种检验方法选择预测模型。所做的创新性工作如下：
将生态模型分岔参数(如总氮和捕捞强度)看作随机变量，基于历史数据以Copula理论计算得到分岔参数的概率分布。将传统动力学每一个分岔区域看成动力系统的一个状态，建立状态转移的齐次马尔科夫链模型。运用蒙特卡罗方法计算一个区域转移到其它区域的概率，得到转移概率矩阵，再计算多稳态转换的平稳概率，给出分岔参数调整的明确建议值，使理想状态的平稳概率(停留时间越长)最大化，从而为海洋生态系统的管理提供理论依据。
基于相空间重构，运用随机梯度Boosting算法，从回归树的线性组合中直接计算时间序列的最大Lyapunov指数。该方法不用计算估计函数的雅可比矩阵，继承了随机梯度Boosting算法的许多优点，如较高的预测精度，容易处理高维数据等。随机模拟结果表明该方法非常接近真值，而且具有较小的标准误，对于噪声和嵌入维数都很稳健，在实际运用中具有很大的优点。
采用BDS、Box-Pierce和Ljung-Box独立性检验1995—2005年渤海COD浓度历史数据存在相关性，考虑非线性检验代替数据检验IAAFT、White和Terasvirta人工神经网络弱非线性检验、Hinich双谱检验以及无Fourier变换检验。统计量选取本文最大Lyapunov指数回归树估计，都不能判断出渤海COD存在非线性相关。综合比较线性ARMA、局部线性、规则集成、随机森林、随机梯度Boosting、支持向量、人工神经网络、自适应样条8种预测方法。结果表明线性ARMA模型误差均值和方差最小，证明线性ARMA模型比其它方法更适合渤海COD数据，确实证明前面各种非线性检验结果的可靠性。
Marine, covered 72 percent surface of the earth, is the key part of the globalecosystem. In the past ten years, more attention has been paid to marine ecosystemresearch and great progress was achieved. Many famous international organizations,like The United Nations, GBP, SCOR, IOC, UNEP and UNDP, are emphasizing theimportance of marine ecosystem function study. Major programs about marine science,such as national 973 plan, foundation projects and national special fund, were carriedout in China. Our scientist noticed and took part in the development and began to seekcross-disciplinary and integration about marine system. This paper originally calcu-lated regime shift probability by statistical method-Copula, proposed regression treeestimation of Lyapunov exponent from time series and unified several statistical tests tochoose forecast model. Original work is listed as below:
Taking bifurcation parameter of ecological model like total nitrogen and fishingintensity as stochastic variable, their probability distribution were estimated by Copulabased on yearly data. State transfer was modeled by homogeneous Markov chain ifevery traditional bifurcation region was regarded as a state. Transfer probability matrixwas calculated by Monte Carlo. Stable probability was clear sequently. The adjustmentof bifurcation parameter was suggested to make ideal state probability maximization,which provided theory evidence for marine ecosystem manager.
Based on phase space reconstruction, largest Lyapunov exponent of time series wascomputed from linear combination of regression tree using stochastic gradient boosting.Our method, without using Jacobin matrix, inherit many virtues of stochastic gradientboosting like perfect prediction precision and easily dealing with higher dimension.Simulation showed that the method got very close to the true value, had smaller standarderror and was robust to noise and embedding dimension, so it is convenient in practice.
Dependence was tested in Bohai COD from 1995 to 2005 by BDS, Box-pierceand Ljunng-Box independence test. Considering surrogate data test IAAFT for nonlin-earity, White’s and Teravirta’s artificial neuron network test for neglected nonlinearity,Hinich’s bispectrum test and a test without Fourier transfer, no nonlinearity was madeout. The statistic was Lyapunov exponent from regression tree. The mean error andstandard error of linear ARMA model were the minimum comparing local linear, ruleensemble, random forest, stochastic gradient boosting, support vector, artificial neuron network and multivariate adaptive regression splines. So linear ARMA model fittedbetter, which testified former tests result.
将生态模型分岔参数(如总氮和捕捞强度)看作随机变量，基于历史数据以Copula理论计算得到分岔参数的概率分布。将传统动力学每一个分岔区域看成动力系统的一个状态，建立状态转移的齐次马尔科夫链模型。运用蒙特卡罗方法计算一个区域转移到其它区域的概率，得到转移概率矩阵，再计算多稳态转换的平稳概率，给出分岔参数调整的明确建议值，使理想状态的平稳概率(停留时间越长)最大化，从而为海洋生态系统的管理提供理论依据。
基于相空间重构，运用随机梯度Boosting算法，从回归树的线性组合中直接计算时间序列的最大Lyapunov指数。该方法不用计算估计函数的雅可比矩阵，继承了随机梯度Boosting算法的许多优点，如较高的预测精度，容易处理高维数据等。随机模拟结果表明该方法非常接近真值，而且具有较小的标准误，对于噪声和嵌入维数都很稳健，在实际运用中具有很大的优点。
采用BDS、Box-Pierce和Ljung-Box独立性检验1995—2005年渤海COD浓度历史数据存在相关性，考虑非线性检验代替数据检验IAAFT、White和Terasvirta人工神经网络弱非线性检验、Hinich双谱检验以及无Fourier变换检验。统计量选取本文最大Lyapunov指数回归树估计，都不能判断出渤海COD存在非线性相关。综合比较线性ARMA、局部线性、规则集成、随机森林、随机梯度Boosting、支持向量、人工神经网络、自适应样条8种预测方法。结果表明线性ARMA模型误差均值和方差最小，证明线性ARMA模型比其它方法更适合渤海COD数据，确实证明前面各种非线性检验结果的可靠性。
Marine, covered 72 percent surface of the earth, is the key part of the globalecosystem. In the past ten years, more attention has been paid to marine ecosystemresearch and great progress was achieved. Many famous international organizations,like The United Nations, GBP, SCOR, IOC, UNEP and UNDP, are emphasizing theimportance of marine ecosystem function study. Major programs about marine science,such as national 973 plan, foundation projects and national special fund, were carriedout in China. Our scientist noticed and took part in the development and began to seekcross-disciplinary and integration about marine system. This paper originally calcu-lated regime shift probability by statistical method-Copula, proposed regression treeestimation of Lyapunov exponent from time series and unified several statistical tests tochoose forecast model. Original work is listed as below:
Taking bifurcation parameter of ecological model like total nitrogen and fishingintensity as stochastic variable, their probability distribution were estimated by Copulabased on yearly data. State transfer was modeled by homogeneous Markov chain ifevery traditional bifurcation region was regarded as a state. Transfer probability matrixwas calculated by Monte Carlo. Stable probability was clear sequently. The adjustmentof bifurcation parameter was suggested to make ideal state probability maximization,which provided theory evidence for marine ecosystem manager.
Based on phase space reconstruction, largest Lyapunov exponent of time series wascomputed from linear combination of regression tree using stochastic gradient boosting.Our method, without using Jacobin matrix, inherit many virtues of stochastic gradientboosting like perfect prediction precision and easily dealing with higher dimension.Simulation showed that the method got very close to the true value, had smaller standarderror and was robust to noise and embedding dimension, so it is convenient in practice.
Dependence was tested in Bohai COD from 1995 to 2005 by BDS, Box-pierceand Ljunng-Box independence test. Considering surrogate data test IAAFT for nonlin-earity, White’s and Teravirta’s artificial neuron network test for neglected nonlinearity,Hinich’s bispectrum test and a test without Fourier transfer, no nonlinearity was madeout. The statistic was Lyapunov exponent from regression tree. The mean error andstandard error of linear ARMA model were the minimum comparing local linear, ruleensemble, random forest, stochastic gradient boosting, support vector, artificial neuron network and multivariate adaptive regression splines. So linear ARMA model fittedbetter, which testified former tests result.
引文
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