Indistinguishable states II - The imperfect model scenario
Indistinguishable states II - The imperfect model scenario
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DOI：
10.1016/j.physd.2004.03.020
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发表时间：
2004-09-15
影响因子：
4
通讯作者：
Smith, LA
中科院分区：
文献类型：
--
作者：
Judd, K;Smith, LA
给定一个混沌系统的完美模型和一组任意持续时间的噪声观测，不可能精确地确定该系统的状态，而是必须考虑一组在给定观测[K]的情况下彼此无法区分的状态。洛杉矶贾德Smith，不可区分的状态I，Physica D 151（2001）]。然而，完美的模型场景是虚构的;实际上，所有模型都是不完美的。在不完美模型下，完美模型情景的结果如何变化？它被证明是必不可少的，即使是小的模型缺陷考虑：不这样做可以系统地降低状态估计或预测的非线性系统。对于不完美的模型，系统状态空间和模型状态空间很少（如果有的话）是等价的，因此必须考虑系统状态到模型状态空间的投影。此外，几乎可以肯定的是，没有模型的轨迹是一致的一个无限系列的观察，因此没有一致的方法来估计系统状态的投影使用轨迹。然而，存在与观测一致的伪轨道，并且这些伪轨道可以用于估计系统状态的投影。使用伪轨道，人们发现，在完美模型的情况下，有一组状态是无法区分的系统状态的投影。讨论了这组不可分辨态的估计和这些态的概率密度。主要结论是：（i）没有模型的状态可以与系统的状态相识别;（ii）当使用不完美的模型来预测系统时，必须非常小心，因为从观测值初始化模型状态可以为系统提供一个很差的模拟。即使人们能够获得系统状态的无噪声投影，预测也不会很长时间地影响系统的未来行为。概率预测的最终目标应该根据这些结果重新审视。(C)2004年由Elsevier B. V.出版
Given a perfect model of a chaotic system and a set of noisy observations of arbitrary duration, it is not possible to determine the state of this system precisely, rather one must consider a set of states which are indistinguishable from one another given the observations [K. Judd, L.A. Smith, Indistinguishable states I, Physica D 151 (2001)]. Yet the perfect model scenario is a fiction; in practice all models are imperfect. How do the results from the perfect model scenario change under imperfect models? It is shown to be essential to take even small model imperfections into account: failure to do so can systematically degrade state estimation or prediction of nonlinear systems. With an imperfect model, the system state space and model state space are rarely (if ever) equivalent, and so one must consider a projection of the system state into the model state space. Furthermore, it is almost certain that no trajectory of the model is consistent with an infinite series of observations, thus there is no consistent way to estimate the projection of system state using trajectories. There are pseudo-orbits, however, that are consistent with observations and these can be used to estimate the projection of the system state. Using pseudo-orbits one finds that, as in the perfect model scenario, there is a set of states that are indistinguishable from the projection of the system state. Estimation of this set of indistinguishable states and the probability density on these states is discussed. The main conclusions are (i) that there is no state of the model that can be identified with the state of the system; and (ii) that great care must be taken when using an imperfect model to forecast the system, because the initialization of the model state from observations can provide a poor analogue for the system. The forecast may not shadow the future behaviour of the system for very long, even if one were able to obtain a noise-free projection of the system state. The ultimate aims of probability forecasts should be re-examined in light of these results. (C) 2004 Published by Elsevier B.V.