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第1行： − 此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。+ − In the study of [[dynamical systems]], a '''delay embedding theorem''' gives the conditions under which a [[chaos theory|chaotic]] [[dynamical system]] can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., [[diffeomorphism|diffeomorphisms]]), but it does not preserve the geometric shape of structures in phase space.+ − In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., diffeomorphisms), but it does not preserve the geometric shape of structures in phase space.+ − 在动力系统的研究中，延迟嵌入定理给出了一个条件，在这个条件下，一个混沌动力系统可以从一系列对动力系统状态的观测中重构出来。这种重构保留了动力系统的性质，即在光滑的坐标变化(即，微分同胚)下不会改变，但是它不能在相空间中保留结构的几何形状。+ − '''Takens' theorem''' is the 1981 delay embedding theorem of [[Floris Takens]]. It provides the conditions under which a smooth [[attractor]] can be reconstructed from the observations made with a [[Baire space|generic]] function. Later results replaced the smooth attractor with a set of arbitrary [[box counting dimension]] and the class of generic functions with other classes of functions.+ − − Takens' theorem is the 1981 delay embedding theorem of Floris Takens. It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic function. Later results replaced the smooth attractor with a set of arbitrary box counting dimension and the class of generic functions with other classes of functions. − − 塔肯斯定理是1981年弗洛里斯 · 塔肯斯的延迟嵌入定理。它提供了一个条件，在这个条件下，一个光滑的吸引子可以从一个一般函数的观察重建。后面的结果用一组任意的盒计数维数代替了光滑吸引子，用其他类的函数代替了一类泛函。 − − − − Delay embedding theorems are simpler to state for − − Delay embedding theorems are simpler to state for − − 延迟嵌入定理更容易描述 − − [[Dynamical system#Types of dynamical systems|discrete-time dynamical system]]s. − − discrete-time dynamical systems. − − 离散时间动力系统。 − − The state space of the dynamical system is a <math>\nu</math>-dimensional [[manifold]] <math>M</math>. The dynamics is given by a [[smooth map]] − − The state space of the dynamical system is a <math>\nu</math>-dimensional manifold <math>M</math>. The dynamics is given by a smooth map − − 动力系统的状态空间是一个数学 / 数学维的流形数学 / 数学。动态是由一个平滑的地图给出的 − − − − :<math>f: M \to M.</math> − − <math>f: M \to M.</math> − − 数学 f: m to m / math − − − − Assume that the dynamics <math>f</math> has a [[strange attractor]] <math>A</math> with [[box counting dimension]] <math>d_A</math>. Using ideas from [[Whitney's embedding theorem]], <math>A</math> can be embedded in <math>k</math>-dimensional [[Euclidean space]] with − − Assume that the dynamics <math>f</math> has a strange attractor <math>A</math> with box counting dimension <math>d_A</math>. Using ideas from Whitney's embedding theorem, <math>A</math> can be embedded in <math>k</math>-dimensional Euclidean space with − − 假设动态数学 f / math 有一个奇怪的吸引子数学 a / math 和盒计数维数学 d / math。利用惠特尼嵌入定理的思想，数学 a / math 可以嵌入到数学 k / 数学维欧几里德空间中 − − − − :<math>k > 2 d_A.</math> − − <math>k > 2 d_A.</math> − − 数学 k 2 d a. / 数学 − − − − That is, there is a [[diffeomorphism]] <math>\phi</math> that maps <math>A</math> into <math>\R^k</math> such that the [[derivative]] of <math>\phi</math> has full [[rank (linear algebra)|rank]]. − − That is, there is a diffeomorphism <math>\phi</math> that maps <math>A</math> into <math>\R^k</math> such that the derivative of <math>\phi</math> has full rank. − − 也就是说，有一个区分同胚数学 phi / math 把数学 a / math 映射成数学 r ^ k / math，这样数学的导数 φ / math 就有满秩了。 − − − − A delay embedding theorem uses an ''observation function'' to construct the embedding function. An observation function <math>\alpha</math> must be twice-differentiable and associate a real number to any point of the attractor <math>A</math>. It must also be [[Baire space|typical]], so its derivative is of full rank and has no special symmetries in its components. The delay embedding theorem states that the function − − A delay embedding theorem uses an observation function to construct the embedding function. An observation function <math>\alpha</math> must be twice-differentiable and associate a real number to any point of the attractor <math>A</math>. It must also be typical, so its derivative is of full rank and has no special symmetries in its components. The delay embedding theorem states that the function − − 延迟嵌入定理利用观察函数构造嵌入函数。观测函数的数学 α / 数学必须是可二次微分的，并且将实数与吸引子数学 a / 数学中的任意点联系起来。它也必须是典型的，所以它的导数是满秩的，它的分量没有特殊的对称性。延迟嵌入定理证明了函数的一些性质 − − − − :<math>\phi_T(x)=\left(\alpha(x), \alpha\left(f(x)\right), \dots, \alpha\left(f^{k-1}(x)\right)\right)</math> − − <math>\phi_T(x)=\left(\alpha(x), \alpha\left(f(x)\right), \dots, \alpha\left(f^{k-1}(x)\right)\right)</math> − − Math phi t (x) left ( alpha (x) ， alpha left (f (x)右) ，dots， alpha 左(f ^ k-1}(x)右)右) / math − − − − is an embedding of the strange attractor <math>A</math>. − − is an embedding of the strange attractor <math>A</math>. − − 是奇怪吸引子数学 a / 数学的嵌入。 − − − − ==Simplified, slightly inaccurate version== − − − − Suppose the <math>d</math>-dimensional − − Suppose the <math>d</math>-dimensional − − 假设数学 d / math-dimensional − − state vector <math>x_t</math> evolves according to an unknown but continuous − − state vector <math>x_t</math> evolves according to an unknown but continuous − − 状态向量数学 x t / math 根据一个未知但是连续的 − − and (crucially) deterministic dynamic. Suppose, too, that the − − and (crucially) deterministic dynamic. Suppose, too, that the − − 以及(至关重要的)确定性动力学。假设，也是这样 − − one-dimensional observable <math>y</math> is a smooth function of <math>x</math>, and “coupled” − − one-dimensional observable <math>y</math> is a smooth function of <math>x</math>, and “coupled” − − 一维可观测数学 y / math 是数学 x / math 的光滑函数，“耦合” − − to all the components of <math>x</math>. Now at any time we can look not just at − − to all the components of <math>x</math>. Now at any time we can look not just at − − 数学 x / 数学的所有组成部分。现在任何时候我们都不能只看 − − the present measurement <math>y(t)</math>, but also at observations made at times − − the present measurement <math>y(t)</math>, but also at observations made at times − − 目前的测量数学 y (t) / 数学，但也有时在观察 − − removed from us by multiples of some lag <math>\tau: y_{t-\tau}, y_{t-2\tau} </math>, etc. If we use − − removed from us by multiples of some lag <math>\tau: y_{t-\tau}, y_{t-2\tau} </math>, etc. If we use − − 被一些滞后数学的倍数从我们身上移除: y { t- tau } ，y { t-2 tau } / math，等等。如果我们使用 − − <math>k</math> lags, we have a <math>k</math>-dimensional vector. One might expect that, as the − − <math>k</math> lags, we have a <math>k</math>-dimensional vector. One might expect that, as the − − 数学 k / 数学滞后，我们有一个数学 k / 数学维向量。人们可能会这样认为，因为 − − number of lags is increased, the motion in the lagged space will become − − number of lags is increased, the motion in the lagged space will become − − 滞后数目增加，滞后空间中的运动将成为 − − more and more predictable, and perhaps in the limit <math> k \to \infty </math> would become − − more and more predictable, and perhaps in the limit <math> k \to \infty </math> would become − − 越来越容易预测，也许在有限的情况下，数学 k to infty / math 会变成 − − deterministic. In fact, the dynamics of the lagged vectors become − − deterministic. In fact, the dynamics of the lagged vectors become − − 确定性的。实际上，滞后向量的动力学问题已经成为一个复杂的问题 − − deterministic at a finite dimension; not only that, but the deterministic − − deterministic at a finite dimension; not only that, but the deterministic − − 有限维度上的确定性; 不仅如此，还有确定性 − − dynamics are completely equivalent to those of the original state space (More exactly, they are related by a smooth, invertible change of coordinates, − − dynamics are completely equivalent to those of the original state space (More exactly, they are related by a smooth, invertible change of coordinates, − − 动力学完全等价于原始状态空间的动力学(更确切地说，它们通过平滑的、可逆的坐标变化相关联, − − or diffeomorphism.) The magic embedding dimension <math>k</math> is − − or diffeomorphism.) The magic embedding dimension <math>k</math> is − − 或者差异同晶。)魔法嵌入维数数学 k / math 是 − − at most <math>2d+1</math>, and often less.<ref>{{cite book|last1=Shalizi|first1=Cosma R.|editor1-last=Deisboeck|editor1-first=ThomasS|editor2-last=Kresh|editor2-first=J.Yasha|title=Complex Systems Science in Biomedicine|date=2006|publisher=Springer US|isbn=978-0-387-30241-6|pages=33–114|chapter=Methods and Techniques of Complex Systems Science: An Overview|doi=10.1007/978-0-387-33532-2_2|series=Topics in Biomedical Engineering International Book Series}}</ref> − − at most <math>2d+1</math>, and often less. − − 最多是数学2d + 1 / 数学，通常更少。 − − − − == See also == − − − − * [[Whitney embedding theorem]] − − * [[Nonlinear dimensionality reduction]] − − − − ==References== − − {{Reflist}} − − {{Refimprove|date=November 2014}} − − − − == Further reading == − − * {{cite journal − − |journal = Physical Review Letters − − |journal = Physical Review Letters − − 物理评论快报 − − |year = 1980 − − |year = 1980 − − 1980年 − − |title = Geometry from a time series − − |title = Geometry from a time series − − | 标题来自时间序列的几何 − − |pages = 712–716 − − |pages = 712–716 − − 712-- 716 − − |author = [[Norman Packard|N. Packard]], [[James P. Crutchfield|J. Crutchfield]], [[James Doyne Farmer|D. Farmer]] and [[Robert Shaw (physicist)|R. Shaw]] − − |author = N. Packard, J. Crutchfield, D. Farmer and R. Shaw − − 作者 n. Packard，j. Crutchfield，d. Farmer and r. Shaw − − |volume = 45 − − |volume = 45 − − 第45卷 − − |doi = 10.1103/PhysRevLett.45.712 − − |doi = 10.1103/PhysRevLett.45.712 − − 10.1103 / physrvlett. 45.712 − − |bibcode=1980PhRvL..45..712P − − |bibcode=1980PhRvL..45..712P − − 1980 / phrvl. 45. . 712 p − − |issue = 9 − − |issue = 9 − − 第九期 − − }} − − }} − − }} − − * {{cite conference − − |booktitle = Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898 − − |booktitle = Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898 − − 动力系统和湍流，数学讲义，第卷。898 − − |year = 1981 − − |year = 1981 − − 1981年 − − |title = Detecting strange attractors in turbulence − − |title = Detecting strange attractors in turbulence − − 在湍流中检测到奇怪的吸引子 − − |pages = 366–381 − − |pages = 366–381 − − 366381页 − − |author = [[Floris Takens|F. Takens]] − − |author = F. Takens − − 作者 f. Takens − − |publisher = Springer-Verlag − − |publisher = Springer-Verlag − − | 出版商 Springer-Verlag − − |editor = [[D. A. Rand]] and [[L.-S. Young]] − − |editor = D. A. Rand and L.-S. Young − − 编辑: d. a. Rand and L.-S. Young − − }} − − }} − − }} − − * {{cite conference − − |booktitle = Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898 − − |booktitle = Dynamical Systems and Turbulence, Lecture Notes in Mathematics, vol. 898 − − 动力系统和湍流，数学讲义，第卷。898 − − |year = 1981 − − |year = 1981 − − 1981年 − − |title = On the dimension of the compact invariant sets of certain nonlinear maps − − |title = On the dimension of the compact invariant sets of certain nonlinear maps − − 关于某些非线性映射的紧不变集的维数 − − |pages = 230–242 − − |pages = 230–242 − − 230-- 242页 − − |author = [[Ricardo Mañé|R. Mañé]] − − |author = R. Mañé − − |author = R. Mañé − − |publisher = Springer-Verlag − − |publisher = Springer-Verlag − − | 出版商 Springer-Verlag − − |editor = D. A. Rand and L.-S. Young − − |editor = D. A. Rand and L.-S. Young − − 编辑: d. a. Rand and L.-S. 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Yorke，Martin castagli − − |volume = 65 − − |volume = 65 − − 第65卷 − − |doi = 10.1007/BF01053745 − − |doi = 10.1007/BF01053745 − − 10.1007 / BF01053745 − − |bibcode = 1991JSP....65..579S − − |bibcode = 1991JSP....65..579S − − 1991JSP... 65. . 579 s − − |issue = 3–4 }} − − |issue = 3–4 }} − − 第三季第四集 − − * {{cite journal − − |journal = Phil. Trans. R. Soc. Lond. A − − |journal = Phil. Trans. R. Soc. Lond. A − − 日记菲尔。反式。R. Soc.隆德。答: − − |year = 1994 − − |year = 1994 − − 1994年 − − |title = Nonlinear forecasting for the classification of natural time series − − |title = Nonlinear forecasting for the classification of natural time series − − 自然时间序列分类的非线性预测 − − |pages = 477–495 − − |pages = 477–495 − − 477-- 495 − − |author = [[G. Sugihara]] − − |author = G. Sugihara − − 作者: g. Sugihara − − |volume = 348 − − |volume = 348 − − 第348卷 − − |doi = 10.1098/rsta.1994.0106 − − |doi = 10.1098/rsta.1994.0106 − − 10.1098 / rsta. 1994.0106 − − |bibcode = 1994RSPTA.348..477S − − |bibcode = 1994RSPTA.348..477S − − | bibcode 1994RSPTA. 348. . 477 s − − |issue = 1688 }} − − |issue = 1688 }} − − 第1688期 − − * {{cite journal − − |journal = Science − − |journal = Science − − 科学》杂志 − − |year = 1999 − − |year = 1999 − − 1999年 − − |title = Episodic fluctuations in larval supply − − |title = Episodic fluctuations in larval supply − − 幼虫供应的周期性波动 − − |pages = 1528–1530 − − |pages = 1528–1530 − − 15281530页 − − |author = [[P.A. Dixon]], [[M.J. Milicich]], and [[G. Sugihara]] − − |author = P.A. Dixon, M.J. Milicich, and G. Sugihara − − 作者 p.a。和 g. Sugihara − − |volume = 283 − − |volume = 283 − − 第283卷 − − |doi = 10.1126/science.283.5407.1528 − − |doi = 10.1126/science.283.5407.1528 − − 10.1126 / science. 283.5407.1528 − − |pmid=10066174 − − |pmid=10066174 − − 10066174 − − |bibcode = 1999Sci...283.1528D − − |bibcode = 1999Sci...283.1528D − − 1999 / sci... 283.1528 d − − |issue=5407}} − − |issue=5407}} − − 第5407期 − − * {{cite journal − − |journal = PNAS − − |journal = PNAS − − 美国科学院院刊 − − |year = 1999 − − |year = 1999 − − 1999年 − − |title = Residual delay maps unveil global patterns of atmospheric nonlinearity and produce improved local forecasts − − |title = Residual delay maps unveil global patterns of atmospheric nonlinearity and produce improved local forecasts − − 残余延迟图揭示全球大气非线性模式并产生改进的局部预报 − − |pages = 210–215 − − |pages = 210–215 − − 210-- 215页 − − |author = [[G. Sugihara]], [[M. Casdagli]], [[E. Habjan]], [[D. Hess]], [[P. Dixon]] and [[G. Holland]] − − |author = G. Sugihara, M. Casdagli, E. Habjan, D. Hess, P. 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Holland − − |volume = 96 − − |volume = 96 − − 第96卷 − − |pmid=10588685 − − |pmid=10588685 − − 10588685 − − |issue = 25 − − |issue = 25 − − 第25期 − − |pmc = 24416 − − |pmc = 24416 − − 24416 − − |doi=10.1073/pnas.96.25.14210 − − |doi=10.1073/pnas.96.25.14210 − − 10.1073 / pnas. 96.25.14210 − − |bibcode = 1999PNAS...9614210S }} − − |bibcode = 1999PNAS...9614210S }} − − 1999PNAS... 9614210S } − − * {{cite journal − − |journal = Nature − − |journal = Nature − − 自然》杂志 − − |year = 2005 − − |year = 2005 − − 2005年 − − |title = Distinguishing random environmental fluctuations from ecological catastrophes for the North Pacific Ocean − − |title = Distinguishing random environmental fluctuations from ecological catastrophes for the North Pacific Ocean − − 区分北太平洋的随机环境波动和生态灾难 − − |first4 = G − − |first4 = G − − | first4 g − − |last4 = Sugihara − − |last4 = Sugihara − − | last 4 Sugihara − − |first3 = AJ − − |first3 = AJ − − | first3 AJ − − |last3 = Lucas − − |last3 = Lucas − − 最后3个卢卡斯 − − |first2 = SM − − |first2 = SM − − | first2 SM − − |pages = 336–340 − − |pages = 336–340 − − 336340页 − − |last2 = Glaser − − |last2 = Glaser − − 2 Glaser − − |pmid = 15902256 − − |pmid = 15902256 − − 15902256 − − |issue = 7040 − − |issue = 7040 − − 第7040期 − − |author = [[C. Hsieh]] − − |author = C. Hsieh − − 作者 c. Hsieh − − |volume = 435 − − |volume = 435 − − 第435卷 − − |doi = 10.1038/nature03553 − − |doi = 10.1038/nature03553 − − 10.1038 / nature03553 − − |bibcode = 2005Natur.435..336H }} − − |bibcode = 2005Natur.435..336H }} − − 2005 / natur. 435. . 336 h } − − * {{cite journal − − |journal = Remote Sensing of Environment − − |journal = Remote Sensing of Environment − − 环境遥感杂志 − − |year = 2015 − − |year = 2015 − − 2015年 − − |title = Estimating determinism rates to detect patterns in geospatial datasets − − |title = Estimating determinism rates to detect patterns in geospatial datasets − − 估计确定性比率以检测地理空间数据集中的模式 − − |pages = 11–20 − − |pages = 11–20 − − 11-- 20页 − − |author = [[R. A. Rios]], [[L. Parrott]], [[H. Lange]] and [[R. F. de Mello]] − − |author = R. A. Rios, L. Parrott, H. Lange and R. F. de Mello − − 作者 r. a. Rios，l. Parrott，h. Lange 和 r. f. de Mello − − |volume = 156 − − |volume = 156 − − 第156卷 − − |doi = 10.1016/j.rse.2014.09.019|bibcode = 2015RSEnv.156...11R − − |doi = 10.1016/j.rse.2014.09.019|bibcode = 2015RSEnv.156...11R − − 10.1016 / j.rse. 2014.09.019 | bibcode 2015RSEnv. 156... 11R − − }} − − }} − − − − − − ==External links== − − * [http://www.scholarpedia.org/article/Attractor_Reconstruction Attractor Reconstruction (scholarpedia)] − − * [https://web.archive.org/web/20130917012451/http://www.scientio.com/Products/ChaosKit] Scientio's ChaosKit product uses embedding to create analyses and predictions. Access is provided online via a web service and graphic interface. − − − − [[Category:Theorems in dynamical systems]] − − Category:Theorems in dynamical systems − − 范畴: 动力系统中的定理 − +
Moved page from wikipedia:en:Takens' theorem (history)
此词条暂由彩云小译翻译，翻译字数共3，未经人工整理和审校，带来阅读不便，请见谅。
#REDIRECT [[Takens's theorem]]
REDIRECT Takens's theorem
重定向塔肯斯定理
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<small>This page was moved from [[wikipedia:en:Takens's theorem]]. Its edit history can be viewed at [[塔肯斯定理/edithistory]]</small></noinclude>
<small>This page was moved from [[wikipedia:en:Takens' theorem]]. Its edit history can be viewed at [[塔肯斯定理/edithistory]]</small></noinclude>
[[Category:待整理页面]]
[[Category:待整理页面]]