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More informationNONLINEAR TIME SERIES ANALYSISThis book represents a modern approach to time series analysis which is based onthe theory of dynamical systems.It starts from a sound outline of the underlyingtheory to arrive at very practical issues,which are illustrated using a large number ofempirical data sets taken from variousfields.This book will hence be highly usefulfor scientists and engineers from all disciplines who study time variable signals,including the earth,life and social sciences.The paradigm of deterministic chaos has influenced thinking in manyfields ofscience.Chaotic systems show rich and surprising mathematical structures.In theapplied sciences,deterministic chaos provides a striking explanation for irregulartemporal behaviour and anomalies in systems which do not seem to be inherentlystochastic.The most direct link between chaos theory and the real world is the anal-ysis of time series from real systems in terms of nonlinear dynamics.Experimentaltechnique and data analysis have seen such dramatic progress that,by now,mostfundamental properties of nonlinear dynamical systems have been observed in thelaboratory.Great efforts are being made to exploit ideas from chaos theory where-ver the data display more structure than can be captured by traditional methods.Problems of this kind are typical in biology and physiology but also in geophysics,economics and many other sciences.This revised edition has been significantly rewritten an expanded,includingseveral new chapters.In view of applications,the most relevant novelties will be thetreatment of non-stationary data sets and of nonlinear stochastic processes insidethe framework of a state space reconstruction by the method of delays.Hence,non-linear time series analysis has left the rather narrow niche of strictly deterministicsystems.Moreover,the analysis of multivariate data sets has gained more atten-tion.For a direct application of the methods of this book to the reader’s own datasets,this book closely refers to the publicly available software package TISEAN.The availability of this software will facilitate the solution of the exercises,so thatreaders now can easily gain their own experience with the analysis of data sets.Holger Kantz,born in November1960,received his diploma in physics fromthe University of Wuppertal in January1986with a thesis on transient chaos.InJanuary1989he obtained his Ph.D.in theoretical physics from the same place,having worked under the supervision of Peter Grassberger on Hamiltonian many-particle dynamics.During his postdoctoral time,he spent one year on a Marie Curiefellowship of the European Union at the physics department of the University ofMore informationFlorence in Italy.In January1995he took up an appointment at the newly foundedMax Planck Institute for the Physics of Complex Systems in Dresden,where heestablished the research group‘Nonlinear Dynamics and Time Series Analysis’.In1996he received his venia legendi and in2002he became adjunct professorin theoretical physics at Wuppertal University.In addition to time series analysis,he works on low-and high-dimensional nonlinear dynamics and its applications.More recently,he has been trying to bridge the gap between dynamics and statis-tical physics.He has(co-)authored more than75peer-reviewed articles in scien-tific journals and holds two international patents.For up-to-date information seehttp://www.mpipks-dresden.mpg.de/mpi-doc/kantzgruppe.html.Thomas Schreiber,born1963,did his diploma work with Peter Grassberger atWuppertal University on phase transitions and information transport in spatio-temporal chaos.He joined the chaos group of Predrag Cvitanovi´c at the Niels BohrInstitute in Copenhagen to study periodic orbit theory of diffusion and anomaloustransport.There he also developed a strong interest in real-world applications ofchaos theory,leading to his Ph.D.thesis on nonlinear time series analysis(Univer-sity of Wuppertal,1994).As a research assistant at Wuppertal University and duringseveral extended appointments at the Max Planck Institute for the Physics of Com-plex Systems in Dresden he published numerous research articles on time seriesmethods and applications ranging from physiology to the stock market.His habil-itation thesis(University of Wuppertal)appeared as a review in Physics Reportsin1999.Thomas Schreiber has extensive experience teaching nonlinear dynamicsto students and experts from variousfields and at all levels.Recently,he has leftacademia to undertake industrial research.NONLINEAR TIME SERIES ANALYSIS HOLGER KANTZ AND THOMAS SCHREIBERMax Planck Institute for the Physics of Complex Systems,DresdenMore informationMore informationpublished by the press syndicate of the university of cambridgeThe Pitt Building,Trumpington Street,Cambridge,United Kingdomcambridge university pressThe Edinburgh Building,Cambridge CB22RU,UK40West20th Street,New York,NY10011–4211,USA477Williamstown Road,Port Melbourne,VIC3207,AustraliaRuiz de Alarc´o n13,28014Madrid,SpainDock House,The Waterfront,Cape Town8001,South AfricaC Holger Kantz and Thomas Schreiber,2000,2003This book is in copyright.Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published2000Second edition published2003Printed in the United Kingdom at the University Press,CambridgeTypeface Times11/14pt.System L A T E X2ε[tb]A catalogue record for this book is available from the British LibraryLibrary of Congress Cataloguing in Publication dataKantz,Holger,1960–Nonlinear time series analysis/Holger Kantz and Thomas Schreiber.–[2nd ed.].p.cm.Includes bibliographical references and index.ISBN0521821509–ISBN0521529026(paperback)1.Time-series analysis.2.Nonlinear theories.I.Schreiber,Thomas,1963–II.TitleQA280.K3552003519.5 5–dc212003044031ISBN0521821509hardbackISBN0521529026paperbackThe publisher has used its best endeavours to ensure that the URLs for external websites referred to in this bookare correct and active at the time of going to press.However,the publisher has no responsibility for the websites and can make no guarantee that a site will remain live or that the content is or will remain appropriate.More informationContentsPreface to thefirst edition page xiPreface to the second edition xiiiAcknowledgements xvI Basic topics11Introduction:why nonlinear methods?32Linear tools and general considerations132.1Stationarity and sampling132.2Testing for stationarity152.3Linear correlations and the power spectrum182.3.1Stationarity and the low-frequency component in thepower spectrum232.4Linearfilters242.5Linear predictions273Phase space methods303.1Determinism:uniqueness in phase space303.2Delay reconstruction353.3Finding a good embedding363.3.1False neighbours373.3.2The time lag393.4Visual inspection of data393.5Poincar´e surface of section413.6Recurrence plots434Determinism and predictability484.1Sources of predictability484.2Simple nonlinear prediction algorithm504.3Verification of successful prediction534.4Cross-prediction errors:probing stationarity564.5Simple nonlinear noise reduction58vMore informationvi Contents5Instability:Lyapunov exponents655.1Sensitive dependence on initial conditions655.2Exponential divergence665.3Measuring the maximal exponent from data696Self-similarity:dimensions756.1Attractor geometry and fractals756.2Correlation dimension776.3Correlation sum from a time series786.4Interpretation and pitfalls826.5Temporal correlations,non-stationarity,and space timeseparation plots876.6Practical considerations916.7A useful application:determination of the noise level using thecorrelation integral926.8Multi-scale or self-similar signals956.8.1Scaling laws966.8.2Detrendedfluctuation analysis1007Using nonlinear methods when determinism is weak1057.1Testing for nonlinearity with surrogate data1077.1.1The null hypothesis1097.1.2How to make surrogate data sets1107.1.3Which statistics to use1137.1.4What can go wrong1157.1.5What we have learned1177.2Nonlinear statistics for system discrimination1187.3Extracting qualitative information from a time series1218Selected nonlinear phenomena1268.1Robustness and limit cycles1268.2Coexistence of attractors1288.3Transients1288.4Intermittency1298.5Structural stability1338.6Bifurcations1358.7Quasi-periodicity139II Advanced topics1419Advanced embedding methods1439.1Embedding theorems1439.1.1Whitney’s embedding theorem1449.1.2Takens’s delay embedding theorem1469.2The time lag148More informationContents vii9.3Filtered delay embeddings1529.3.1Derivative coordinates1529.3.2Principal component analysis1549.4Fluctuating time intervals1589.5Multichannel measurements1599.5.1Equivalent variables at different positions1609.5.2Variables with different physical meanings1619.5.3Distributed systems1619.6Embedding of interspike intervals1629.7High dimensional chaos and the limitations of the time delayembedding1659.8Embedding for systems with time delayed feedback17110Chaotic data and noise17410.1Measurement noise and dynamical noise17410.2Effects of noise17510.3Nonlinear noise reduction17810.3.1Noise reduction by gradient descent17910.3.2Local projective noise reduction18010.3.3Implementation of locally projective noise reduction18310.3.4How much noise is taken out?18610.3.5Consistency tests19110.4An application:foetal ECG extraction19311More about invariant quantities19711.1Ergodicity and strange attractors19711.2Lyapunov exponents II19911.2.1The spectrum of Lyapunov exponents and invariantmanifolds20011.2.2Flows versus maps20211.2.3Tangent space method20311.2.4Spurious exponents20511.2.5Almost two dimensionalflows21111.3Dimensions II21211.3.1Generalised dimensions,multi-fractals21311.3.2Information dimension from a time series21511.4Entropies21711.4.1Chaos and theflow of information21711.4.2Entropies of a static distribution21811.4.3The Kolmogorov–Sinai entropy22011.4.4The -entropy per unit time22211.4.5Entropies from time series data226More informationviii Contents11.5How things are related22911.5.1Pesin’s identity22911.5.2Kaplan–Yorke conjecture23112Modelling and forecasting23412.1Linear stochastic models andfilters23612.1.1Linearfilters23712.1.2Nonlinearfilters23912.2Deterministic dynamics24012.3Local methods in phase space24112.3.1Almost model free methods24112.3.2Local linearfits24212.4Global nonlinear models24412.4.1Polynomials24412.4.2Radial basis functions24512.4.3Neural networks24612.4.4What to do in practice24812.5Improved cost functions24912.5.1Overfitting and model costs24912.5.2The errors-in-variables problem25112.5.3Modelling versus prediction25312.6Model verification25312.7Nonlinear stochastic processes from data25612.7.1Fokker–Planck equations from data25712.7.2Markov chains in embedding space25912.7.3No embedding theorem for Markov chains26012.7.4Predictions for Markov chain data26112.7.5Modelling Markov chain data26212.7.6Choosing embedding parameters for Markov chains26312.7.7Application:prediction of surface wind velocities26412.8Predicting prediction errors26712.8.1Predictability map26712.8.2Individual error prediction26812.9Multi-step predictions versus iterated one-step predictions27113Non-stationary signals27513.1Detecting non-stationarity27613.1.1Making non-stationary data stationary27913.2Over-embedding28013.2.1Deterministic systems with parameter drift28013.2.2Markov chain with parameter drift28113.2.3Data analysis in over-embedding spaces283More informationContents ix13.2.4Application:noise reduction for human voice28613.3Parameter spaces from data28814Coupling and synchronisation of nonlinear systems29214.1Measures for interdependence29214.2Transfer entropy29714.3Synchronisation29915Chaos control30415.1Unstable periodic orbits and their invariant manifolds30615.1.1Locating periodic orbits30615.1.2Stable/unstable manifolds from data31215.2OGY-control and derivates31315.3Variants of OGY-control31615.4Delayed feedback31715.5Tracking31815.6Related aspects319A Using the TISEAN programs321A.1Information relevant to most of the routines322A.1.1Efficient neighbour searching322A.1.2Re-occurring command options325A.2Second-order statistics and linear models326A.3Phase space tools327A.4Prediction and modelling329A.4.1Locally constant predictor329A.4.2Locally linear prediction329A.4.3Global nonlinear models330A.5Lyapunov exponents331A.6Dimensions and entropies331A.6.1The correlation sum331A.6.2Information dimension,fixed mass algorithm332A.6.3Entropies333A.7Surrogate data and test statistics334A.8Noise reduction335A.9Finding unstable periodic orbits336A.10Multivariate data336B Description of the experimental data sets338B.1Lorenz-like chaos in an NH3laser338B.2Chaos in a periodically modulated NMR laser340B.3Vibrating string342B.4Taylor–Couetteflow342B.5Multichannel physiological data343More informationx ContentsB.6Heart rate during atrialfibrillation343B.7Human electrocardiogram(ECG)344B.8Phonation data345B.9Postural control data345B.10Autonomous CO2laser with feedback345B.11Nonlinear electric resonance circuit346B.12Frequency doubling solid state laser348B.13Surface wind velocities349References350Index365More informationPreface to thefirst editionThe paradigm of deterministic chaos has influenced thinking in manyfields of sci-ence.As mathematical objects,chaotic systems show rich and surprising structures.Most appealing for researchers in the applied sciences is the fact that determinis-tic chaos provides a striking explanation for irregular behaviour and anomalies insystems which do not seem to be inherently stochastic.The most direct link between chaos theory and the real world is the analysis oftime series from real systems in terms of nonlinear dynamics.On the one hand,experimental technique and data analysis have seen such dramatic progress that,by now,most fundamental properties of nonlinear dynamical systems have beenobserved in the laboratory.On the other hand,great efforts are being made to exploitideas from chaos theory in cases where the system is not necessarily deterministicbut the data displays more structure than can be captured by traditional methods.Problems of this kind are typical in biology and physiology but also in geophysics,economics,and many other sciences.In all thesefields,even simple models,be they microscopic or phenomenological,can create extremely complicated dynamics.How can one verify that one’s model isa good counterpart to the equally complicated signal that one receives from nature?Very often,good models are lacking and one has to study the system just from theobservations made in a single time series,which is the case for most non-laboratorysystems in particular.The theory of nonlinear dynamical systems provides new toolsand quantities for the characterisation of irregular time series data.The scope ofthese methods ranges from invariants such as Lyapunov exponents and dimensionswhich yield an accurate description of the structure of a system(provided thedata are of high quality)to statistical techniques which allow for classification anddiagnosis even in situations where determinism is almost lacking.This book provides the experimental researcher in nonlinear dynamics with meth-ods for processing,enhancing,and analysing the measured signals.The theorist willbe offered discussions about the practical applicability of mathematical results.ThexiMore informationxii Preface to thefirst editiontime series analyst in economics,meteorology,and otherfields willfind inspira-tion for the development of new prediction algorithms.Some of the techniquespresented here have also been considered as possible diagnostic tools in clinical re-search.We will adopt a critical but constructive point of view,pointing out ways ofobtaining more meaningful results with limited data.We hope that everybody whohas a time series problem which cannot be solved by traditional,linear methodswillfind inspiring material in this book.Dresden and WuppertalNovember1996More informationPreface to the second editionIn afield as dynamic as nonlinear science,new ideas,methods and experimentsemerge constantly and the focus of interest shifts accordingly.There is a continuousstream of new results,and existing knowledge is seen from a different angle aftervery few years.Five years after thefirst edition of“Nonlinear Time Series Analysis”we feel that thefield has matured in a way that deserves being reflected in a secondedition.The modification that is most immediately visible is that the program listingshave been be replaced by a thorough discussion of the publicly available softwareTISEAN.Already a few months after thefirst edition appeared,it became clearthat most users would need something more convenient to use than the bare libraryroutines printed in the book.Thus,together with Rainer Hegger we prepared stand-alone routines based on the book but with input/output functionality and advancedfeatures.Thefirst public release was made available in1998and subsequent releasesare in widespread use now.Today,TISEAN is a mature piece of software thatcovers much more than the programs we gave in thefirst edition.Now,readerscan immediately apply most methods studied in the book on their own data usingTISEAN programs.By replacing the somewhat terse program listings by minuteinstructions of the proper use of the TISEAN routines,the link between book andsoftware is strengthened,supposedly to the benefit of the readers and users.Hencewe recommend a download and installation of the package,such that the exercisescan be readily done by help of these ready-to-use routines.The current edition has be extended in view of enlarging the class of data sets to betreated.The core idea of phase space reconstruction was inspired by the analysis ofdeterministic chaotic data.In contrast to many expectations,purely deterministicand low-dimensional data are rare,and most data fromfield measurements areevidently of different nature.Hence,it was an effort of our scientific work over thepast years,and it was a guiding concept for the revision of this book,to explore thepossibilities to treat other than purely deterministic data sets.xiiiMore informationxiv Preface to the second editionThere is a whole new chapter on non-stationary time series.While detectingnon-stationarity is still briefly discussed early on in the book,methods to deal withmanifestly non-stationary sequences are described in some detail in the secondpart.As an illustration,a data source of lasting interest,human speech,is used.Also,a new chapter deals with concepts of synchrony between systems,linear andnonlinear correlations,information transfer,and phase synchronisation.Recent attempts on modelling nonlinear stochastic processes are discussed inChapter12.The theoretical framework forfitting Fokker–Planck equations to datawill be reviewed and evaluated.While Chapter9presents some progress that hasbeen made in modelling input–output systems with stochastic but observed inputand on the embedding of time delayed feedback systems,the chapter on mod-elling considers a data driven phase space approach towards Markov chains.Windspeed measurements are used as data which are best considered to be of nonlinearstochastic nature despite the fact that a physically adequate mathematical model isthe deterministic Navier–Stokes equation.In the chapter on invariant quantities,new material on entropy has been included,mainly on the -and continuous entropies.Estimation problems for stochastic ver-sus deterministic data and data with multiple length and time scales are discussed.Since more and more experiments now yield good multivariate data,alternativesto time delay embedding using multiple probe measurements are considered at var-ious places in the text.This new development is also reflected in the functionalityof the TISEAN programs.A new multivariate data set from a nonlinear semicon-ductor electronic circuit is introduced and used in several places.In particular,adifferential equation has been successfully established for this system by analysingthe data set.Among other smaller rearrangements,the material from the former chapter“Other selected topics”,has been relocated to places in the text where a connectioncan be made more naturally.High dimensional and spatio-temporal data is now dis-cussed in the context of embedding.We discuss multi-scale and self-similar signalsnow in a more appropriate way right after fractal sets,and include recent techniquesto analyse power law correlations,for example detrendedfluctuation analysis.Of course,many new publications have appeared since1997which are potentiallyrelevant to the scope of this book.At least two new monographs are concerned withthe same topic and a number of review articles.The bibliography has been updatedbut remains a selection not unaffected by personal preferences.We hope that the extended book will prove its usefulness in many applicationsof the methods and further stimulate thefield of time series analysis.DresdenDecember2002More informationAcknowledgementsIf there is any feature of this book that we are proud of,it is the fact that almost allthe methods are illustrated with real,experimental data.However,this is anythingbut our own achievement–we exploited other people’s work.Thus we are deeplyindebted to the experimental groups who supplied data sets and granted permissionto use them in this book.The production of every one of these data sets requiredskills,experience,and equipment that we ourselves do not have,not forgetting thehours and hours of work spent in the laboratory.We appreciate the generosity ofthe following experimental groups:NMR laser.Our contact persons at the Institute for Physics at Z¨u rich University were Leci Flepp and Joe Simonet;the head of the experimental group is E.Brun.(See AppendixB.2.)Vibrating string.Data were provided by Tim Molteno and Nick Tufillaro,Otago University, Dunedin,New Zealand.(See Appendix B.3.)Taylor–Couetteflow.The experiment was carried out at the Institute for Applied Physics at Kiel University by Thorsten Buzug and Gerd Pfister.(See Appendix B.4.) Atrialfibrillation.This data set is taken from the MIT-BIH Arrhythmia Database,collected by G.B.Moody and R.Mark at Beth Israel Hospital in Boston.(See Appendix B.6.) Human ECG.The ECG recordings we used were taken by Petr Saparin at Saratov State University.(See Appendix B.7.)Foetal ECG.We used noninvasively recorded(human)foetal ECGs taken by John F.Hofmeister as the Department of Obstetrics and Gynecology,University of Colorado,Denver CO.(See Appendix B.7.)Phonation data.This data set was made available by Hanspeter Herzel at the Technical University in Berlin.(See Appendix B.8.)Human posture data.The time series was provided by Steven Boker and Bennett Bertenthal at the Department of Psychology,University of Virginia,Charlottesville V A.(SeeAppendix B.9.)xvMore informationxvi AcknowledgementsAutonomous CO2laser with feedback.The data were taken by Riccardo Meucci and Marco Ciofini at the INO in Firenze,Italy.(See Appendix B.10.)Nonlinear electric resonance circuit.The experiment was designed and operated by M.Diestelhorst at the University of Halle,Germany.(See Appendix B.11.)Nd:YAG laser.The data we use were recorded in the University of Oldenburg,where we wish to thank Achim Kittel,Falk Lange,Tobias Letz,and J¨u rgen Parisi.(See AppendixB.12.)We used the following data sets published for the Santa Fe Institute Time SeriesContest,which was organised by Neil Gershenfeld and Andreas Weigend in1991:NH3laser.We used data set A and its continuation,which was published after the contest was closed.The data was supplied by U.H¨u bner,N.B.Abraham,and C.O.Weiss.(SeeAppendix B.1.)Human breath rate.The data we used is part of data set B of the contest.It was submitted by Ari Goldberger and coworkers.(See Appendix B.5.)During the composition of the text we asked various people to read all or part of themanuscript.The responses ranged from general encouragement to detailed technicalcomments.In particular we thank Peter Grassberger,James Theiler,Daniel Kaplan,Ulrich Parlitz,and Martin Wiesenfeld for their helpful remarks.Members of ourresearch groups who either contributed by joint work to our experience and knowl-edge or who volunteered to check the correctness of the text are Rainer Hegger,Andreas Schmitz,Marcus Richter,Mario Ragwitz,Frank Schm¨u ser,RathinaswamyBhavanan Govindan,and Sharon Sessions.We have also considerably profited fromcomments and remarks of the readers of thefirst edition of the book.Their effortin writing to us is gratefully appreciated.Last but not least we acknowledge the encouragement and support by SimonCapelin from Cambridge University Press and the excellent help in questions ofstyle and English grammar by Sheila Shepherd.。

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population 母体sample 样本census 普查sampling 抽样quantitative 量的qualitative/categorical质的discrete 离散的continuous 连续的population parameters 母体参数sample statistics 样本统计量descriptive statistics 叙述统计学inferential/inductive statistics 推论... 抽样调查（sampliing survey单纯随机抽样（simple random sampling 系统抽样（systematic sampling分层抽样（stratified sampling整群抽样（cluster sampling多级抽样（multistage sampling常态分配(Parametric Statistics)无母数统计学(Nonparametric Statistics) 实验设计(Design of Experiment)参数(Parameter)Data analysis 资料分析Statistical table 统计表Statistical chart 统计图Pie chart 圆饼图Stem-and-leaf display 茎叶图Box plot 盒须图Histogram 直方图Bar Chart 长条图Polygon 次数多边图Ogive 肩形图Descriptive statistics 叙述统计学Expectation 期望值Mode 众数Mean 平均数Variance 变异数Standard deviation 标准差Standard error 标准误Covariance matrix 共变异数矩阵Inferential statistics 推论统计学Point estimation 点估计Interval estimation 区间估计Confidence interval 信赖区间Confidence coefficient 信赖系数Testing statistical hypothesis 统计假设检定Regression analysis 回归分析Analysis of variance 变异数分析Correlation coefficient 相关系数Sampling survey 抽样调查Census 普查Sampling 抽样Reliability 信度Validity 效度Sampling error 抽样误差Non-sampling error 非抽样误差Random sampling 随机抽样Simple random sampling 简单随机抽样法Stratified sampling 分层抽样法Cluster sampling 群集抽样法Systematic sampling 系统抽样法Two-stage random sampling 两段随机抽样法Convenience sampling 便利抽样Quota sampling 配额抽样Snowball sampling 雪球抽样Nonparametric statistics 无母数统计The sign test 等级检定Wilcoxon signed rank tests 魏克森讯号等级检定Wilcoxon rank sum tests 魏克森等级和检定Run test 连检定法Discrete uniform densities 离散的均匀密度Binomial densities 二项密度Hypergeometric densities 超几何密度Poisson densities 卜松密度Geometric densities 几何密度Negative binomial densities 负二项密度Continuous uniform densities 连续均匀密度Normal densities 常态密度Exponential densities 指数密度Gamma densities 伽玛密度Beta densities 贝他密度Multivariate analysis 多变量分析Principal components 主因子分析Discrimination analysis 区别分析Cluster analysis 群集分析Factor analysis 因素分析Survival analysis 存活分析Time series analysis 时间序列分析Linear models 线性模式Quality engineering 品质工程Probability theory 机率论Statistical computing 统计计算Statistical inference 统计推论Stochastic processes 随机过程Decision theory 决策理论Discrete analysis 离散分析Mathematical statistics 数理统计统计学: Statistics母体: Population样本: Sample资料分析: Data analysis统计表: Statistical table统计图: Statistical chart圆饼图: Pie chart茎叶图: Stem-and-leaf display 盒须图: Box plot直方图: Histogram长条图: Bar Chart次数多边图: Polygon肩形图: Ogive叙述统计学: Descriptive statistics 期望值: Expectation众数: Mode平均数: Mean变异数: Variance标准差: Standard deviation标准误: Standard error共变异数矩阵: Covariance matrix 推论统计学: Inferential statistics 点估计: Point estimation区间估计: Interval estimation信赖区间: Confidence interval信赖系数: Confidence coefficient统计假设检定: Testing statistical hypothesis回归分析: Regression analysis变异数分析: Analysis of variance相关系数: Correlation coefficient抽样调查: Sampling survey普查: Census抽样: Sampling信度: Reliability效度: Validity抽样误差: Sampling error非抽样误差: Non-sampling error随机抽样: Random sampling简单随机抽样法: Simple random sampling分层抽样法: Stratified sampling群集抽样法: Cluster sampling系统抽样法: Systematic sampling两段随机抽样法: Two-stage random sampling 便利抽样: Convenience sampling配额抽样: Quota sampling雪球抽样: Snowball sampling无母数统计: Nonparametric statistics等级检定: The sign test魏克森讯号等级检定: Wilcoxon signed rank tests 魏克森等级和检定: Wilcoxon rank sum tests连检定法: Run test离散的均匀密度: Discrete uniform densities二项密度: Binomial densities超几何密度: Hypergeometric densities卜松密度: Poisson densities几何密度: Geometric densities负二项密度: Negative binomial densities连续均匀密度: Continuous uniform densities常态密度: Normal densities指数密度: Exponential densities伽玛密度: Gamma densities贝他密度: Beta densities多变量分析: Multivariate analysis主因子分析: Principal components区别分析: Discrimination analysis群集分析: Cluster analysis因素分析: Factor analysis存活分析: Survival analysis时间序列分析: Time series analysis线性模式: Linear models品质工程: Quality engineering机率论: Probability theory统计计算: Statistical computing统计推论: Statistical inference随机过程: Stochastic processes决策理论: Decision theory离散分析: Discrete analysis数理统计: Mathematical statistics统计名词市调辞典众数(Mode) 普查(census)指数(Index) 问卷(Questionnaire)中位数(Median) 信度(Reliability)百分比(Percentage) 母群体(Population)信赖水准(Confidence level) 观察法(Observational Survey)假设检定(Hypothesis Testing) 综合法(Integrated Survey)卡方检定(Chi-square Test) 雪球抽样(Snowball Sampling)差距量表(Interval Scale) 序列偏差(Series Bias)类别量表(Nominal Scale) 次级资料(Secondary Data)顺序量表(Ordinal Scale) 抽样架构(Sampling frame)比率量表(Ratio Scale) 集群抽样(Cluster Sampling)连检定法(Run Test) 便利抽样(Convenience Sampling)符号检定(Sign Test) 抽样调查(Sampling Sur)算术平均数(Arithmetic Mean) 非抽样误差(non-sampling error) 展示会法(Display Survey)调查名词准确效度(Criterion-Related Validity)元素(Element) 邮寄问卷法(Mail Interview)样本(Sample) 信抽样误差(Sampling error)效度(Validity) 封闭式问题(Close Question)精确度(Precision) 电话访问法(Telephone Interview)准确度(Validity) 随机抽样法(Random Sampling)实验法(Experiment Survey)抽样单位(Sampling unit) 资讯名词市场调查(Marketing Research) 决策树(Decision Trees)容忍误差(Tolerated erro) 资料采矿(Data Mining)初级资料(Primary Data) 时间序列(Time-Series Forecasting) 目标母体(Target Population) 回归分析(Regression)抽样偏差(Sampling Bias) 趋势分析(Trend Analysis)抽样误差(sampling error) 罗吉斯回归(Logistic Regression)架构效度(Construct Validity) 类神经网络(Neural Network)配额抽样(Quota Sampling) 无母数统计检定方法(Non-Parametric Test) 人员访问法(Interview) 判别分析法(Discriminant Analysis)集群分析法(cluster analysis) 规则归纳法(Rules Induction)内容效度(Content Validity) 判断抽样(Judgment Sampling)开放式问题(Open Question) OLAP(Online Analytical Process)分层随机抽样(Stratified Random sampling) 资料仓储(Data Warehouse) 非随机抽样法(Nonrandom Sampling) 知识发现(Knowledge Discovery Absolute deviation, 绝对离差Absolute number, 绝对数Absolute residuals, 绝对残差Acceleration array, 加速度立体阵Acceleration in an arbitrary direction, 任意方向上的加速度Acceleration normal, 法向加速度Acceleration space dimension, 加速度空间的维数Acceleration tangential, 切向加速度Acceleration vector, 加速度向量Acceptable hypothesis, 可接受假设Accumulation, 累积Accuracy, 准确度Actual frequency, 实际频数Adaptive estimator, 自适应估计量Addition, 相加Addition theorem, 加法定理Additive Noise, 加性噪声Additivity, 可加性Adjusted rate, 调整率Adjusted value, 校正值Admissible error, 容许误差Aggregation, 聚集性Alpha factoring,α因子法Alternative hypothesis, 备择假设Among groups, 组间Amounts, 总量Analysis of correlation, 相关分析Analysis of covariance, 协方差分析Analysis Of Effects, 效应分析Analysis Of Variance, 方差分析Analysis of regression, 回归分析Analysis of time series, 时间序列分析Analysis of variance, 方差分析Angular transformation, 角转换ANOVA （analysis of variance）, 方差分析ANOVA Models, 方差分析模型ANOVA table and eta, 分组计算方差分析Arcing, 弧/弧旋Arcsine transformation, 反正弦变换Area 区域图Area under the curve, 曲线面积AREG , 评估从一个时间点到下一个时间点回归相关时的误差ARIMA, 季节和非季节性单变量模型的极大似然估计Arithmetic grid paper, 算术格纸Arithmetic mean, 算术平均数Arrhenius relation, 艾恩尼斯关系Assessing fit, 拟合的评估Associative laws, 结合律Asymmetric distribution, 非对称分布Asymptotic bias, 渐近偏倚Asymptotic efficiency, 渐近效率Asymptotic variance, 渐近方差Attributable risk, 归因危险度Attribute data, 属性资料Attribution, 属性Autocorrelation, 自相关Autocorrelation of residuals, 残差的自相关Average, 平均数Average confidence interval length, 平均置信区间长度Average growth rate, 平均增长率Bar chart, 条形图Bar graph, 条形图Base period, 基期Bayes' theorem , Bayes定理Bell-shaped curve, 钟形曲线Bernoulli distribution, 伯努力分布Best-trim estimator, 最好切尾估计量Bias, 偏性Binary logistic regression, 二元逻辑斯蒂回归Binomial distribution, 二项分布Bisquare, 双平方Bivariate Correlate, 二变量相关Bivariate normal distribution, 双变量正态分布Bivariate normal population, 双变量正态总体Biweight interval, 双权区间Biweight M-estimator, 双权M估计量Block, 区组/配伍组BMDP(Biomedical computer programs), BMDP统计软件包Boxplots, 箱线图/箱尾图Breakdown bound, 崩溃界/崩溃点Canonical correlation, 典型相关Caption, 纵标目Case-control study, 病例对照研究Categorical variable, 分类变量Catenary, 悬链线Cauchy distribution, 柯西分布Cause-and-effect relationship, 因果关系Cell, 单元Censoring, 终检Center of symmetry, 对称中心Centering and scaling, 中心化和定标Central tendency, 集中趋势Central value, 中心值CHAID -χ2 Automatic Interaction Detector, 卡方自动交互检测Chance, 机遇Chance error, 随机误差Chance variable, 随机变量Characteristic equation, 特征方程Characteristic root, 特征根Characteristic vector, 特征向量Chebshev criterion of fit, 拟合的切比雪夫准则Chernoff faces, 切尔诺夫脸谱图Chi-square test, 卡方检验/χ2检验Choleskey decomposition, 乔洛斯基分解Circle chart, 圆图Class interval, 组距Class mid-value, 组中值Class upper limit, 组上限Classified variable, 分类变量Cluster analysis, 聚类分析Cluster sampling, 整群抽样Code, 代码Coded data, 编码数据Coding, 编码Coefficient of contingency, 列联系数Coefficient of determination, 决定系数Coefficient of multiple correlation, 多重相关系数Coefficient of partial correlation, 偏相关系数Coefficient of production-moment correlation, 积差相关系数Coefficient of rank correlation, 等级相关系数Coefficient of regression, 回归系数Coefficient of skewness, 偏度系数Coefficient of variation, 变异系数Cohort study, 队列研究Collinearity, 共线性Column, 列Column effect, 列效应Column factor, 列因素Combination pool, 合并Combinative table, 组合表Common factor, 共性因子Common regression coefficient, 公共回归系数Common value, 共同值Common variance, 公共方差Common variation, 公共变异Communality variance, 共性方差Comparability, 可比性Comparison of bathes, 批比较Comparison value, 比较值Compartment model, 分部模型Compassion, 伸缩Complement of an event, 补事件Complete association, 完全正相关Complete dissociation, 完全不相关Complete statistics, 完备统计量Completely randomized design, 完全随机化设计Composite event, 联合事件Composite events, 复合事件Concavity, 凹性Conditional expectation, 条件期望Conditional likelihood, 条件似然Conditional probability, 条件概率Conditionally linear, 依条件线性Confidence interval, 置信区间Confidence limit, 置信限Confidence lower limit, 置信下限Confidence upper limit, 置信上限Confirmatory Factor Analysis , 验证性因子分析Confirmatory research, 证实性实验研究Confounding factor, 混杂因素Conjoint, 联合分析Consistency, 相合性Consistency check, 一致性检验Consistent asymptotically normal estimate, 相合渐近正态估计Consistent estimate, 相合估计Constrained nonlinear regression, 受约束非线性回归Constraint, 约束Contaminated distribution, 污染分布Contaminated Gausssian, 污染高斯分布Contaminated normal distribution, 污染正态分布Contamination, 污染Contamination model, 污染模型Contingency table, 列联表Contour, 边界线Contribution rate, 贡献率Control, 对照, 质量控制图Controlled experiments, 对照实验Conventional depth, 常规深度Convolution, 卷积Corrected factor, 校正因子Corrected mean, 校正均值Correction coefficient, 校正系数Correctness, 正确性Correlation coefficient, 相关系数Correlation, 相关性Correlation index, 相关指数Correspondence, 对应Counting, 计数Counts, 计数/频数Covariance, 协方差Covariant, 共变Cox Regression, Cox回归Criteria for fitting, 拟合准则Criteria of least squares, 最小二乘准则Critical ratio, 临界比Critical region, 拒绝域Critical value, 临界值Cross-over design, 交叉设计Cross-section analysis, 横断面分析Cross-section survey, 横断面调查Crosstabs , 交叉表Crosstabs 列联表分析Cross-tabulation table, 复合表Cube root, 立方根Cumulative distribution function, 分布函数Cumulative probability, 累计概率Curvature, 曲率/弯曲Curvature, 曲率Curve Estimation, 曲线拟合Curve fit , 曲线拟和Curve fitting, 曲线拟合Curvilinear regression, 曲线回归Curvilinear relation, 曲线关系Cut-and-try method, 尝试法Cycle, 周期Cyclist, 周期性D test, D检验Data acquisition, 资料收集Data bank, 数据库Data capacity, 数据容量Data deficiencies, 数据缺乏Data handling, 数据处理Data manipulation, 数据处理Data processing, 数据处理Data reduction, 数据缩减Data set, 数据集Data sources, 数据来源Data transformation, 数据变换Data validity, 数据有效性Data-in, 数据输入Data-out, 数据输出Dead time, 停滞期Degree of freedom, 自由度Degree of precision, 精密度Degree of reliability, 可靠性程度Degression, 递减Density function, 密度函数Density of data points, 数据点的密度Dependent variable, 应变量/依变量/因变量Dependent variable, 因变量Depth, 深度Derivative matrix, 导数矩阵Derivative-free methods, 无导数方法Design, 设计Determinacy, 确定性Determinant, 行列式Determinant, 决定因素Deviation, 离差Deviation from average, 离均差Diagnostic plot, 诊断图Dichotomous variable, 二分变量Differential equation, 微分方程Direct standardization, 直接标准化法Direct Oblimin, 斜交旋转Discrete variable, 离散型变量DISCRIMINANT, 判断Discriminant analysis, 判别分析Discriminant coefficient, 判别系数Discriminant function, 判别值Dispersion, 散布/分散度Disproportional, 不成比例的Disproportionate sub-class numbers, 不成比例次级组含量Distribution free, 分布无关性/免分布Distribution shape, 分布形状Distribution-free method, 任意分布法Distributive laws, 分配律Disturbance, 随机扰动项Dose response curve, 剂量反应曲线Double blind method, 双盲法Double blind trial, 双盲试验Double exponential distribution, 双指数分布Double logarithmic, 双对数Downward rank, 降秩Dual-space plot, 对偶空间图DUD, 无导数方法Duncan's new multiple range method, 新复极差法/Duncan新法Error Bar, 均值相关区间图Effect, 实验效应Eigenvalue, 特征值Eigenvector, 特征向量Ellipse, 椭圆Empirical distribution, 经验分布Empirical probability, 经验概率单位Enumeration data, 计数资料Equal sun-class number, 相等次级组含量Equally likely, 等可能Equivariance, 同变性Error, 误差/错误Error of estimate, 估计误差Error type I, 第一类错误Error type II, 第二类错误Estimand, 被估量Estimated error mean squares, 估计误差均方Estimated error sum of squares, 估计误差平方和Euclidean distance, 欧式距离Event, 事件Event, 事件Exceptional data point, 异常数据点Expectation plane, 期望平面Expectation surface, 期望曲面Expected values, 期望值Experiment, 实验Experimental sampling, 试验抽样Experimental unit, 试验单位Explained variance （已说明方差）Explanatory variable, 说明变量Exploratory data analysis, 探索性数据分析Explore Summarize, 探索-摘要Exponential curve, 指数曲线Exponential growth, 指数式增长EXSMOOTH, 指数平滑方法Extended fit, 扩充拟合Extra parameter, 附加参数Extrapolation, 外推法Extreme observation, 末端观测值Extremes, 极端值/极值F distribution, F分布F test, F检验Factor, 因素/因子Factor analysis, 因子分析Factor Analysis, 因子分析Factor score, 因子得分Factorial, 阶乘Factorial design, 析因试验设计False negative, 假阴性False negative error, 假阴性错误Family of distributions, 分布族Family of estimators, 估计量族Fanning, 扇面Fatality rate, 病死率Field investigation, 现场调查Field survey, 现场调查Finite population, 有限总体Finite-sample, 有限样本First derivative, 一阶导数First principal component, 第一主成分First quartile, 第一四分位数Fisher information, 费雪信息量Fitted value, 拟合值Fitting a curve, 曲线拟合Fixed base, 定基Fluctuation, 随机起伏Forecast, 预测Four fold table, 四格表Fourth, 四分点Fraction blow, 左侧比率Fractional error, 相对误差Frequency, 频率Frequency polygon, 频数多边图Frontier point, 界限点Function relationship, 泛函关系Gamma distribution, 伽玛分布Gauss increment, 高斯增量Gaussian distribution, 高斯分布/正态分布Gauss-Newton increment, 高斯-牛顿增量General census, 全面普查Generalized least squares, 综合最小平方法GENLOG (Generalized liner models), 广义线性模型Geometric mean, 几何平均数Gini's mean difference, 基尼均差GLM (General liner models), 通用线性模型Goodness of fit, 拟和优度/配合度Gradient of determinant, 行列式的梯度Graeco-Latin square, 希腊拉丁方Grand mean, 总均值Gross errors, 重大错误Gross-error sensitivity, 大错敏感度Group averages, 分组平均Grouped data, 分组资料Guessed mean, 假定平均数Half-life, 半衰期Hampel M-estimators, 汉佩尔M估计量Happenstance, 偶然事件Harmonic mean, 调和均数Hazard function, 风险均数Hazard rate, 风险率Heading, 标目Heavy-tailed distribution, 重尾分布Hessian array, 海森立体阵Heterogeneity, 不同质Heterogeneity of variance, 方差不齐Hierarchical classification, 组内分组Hierarchical clustering method, 系统聚类法High-leverage point, 高杠杆率点High-Low, 低区域图Higher Order Interaction Effects，高阶交互作用HILOGLINEAR, 多维列联表的层次对数线性模型Hinge, 折叶点Histogram, 直方图Historical cohort study, 历史性队列研究Holes, 空洞HOMALS, 多重响应分析Homogeneity of variance, 方差齐性Homogeneity test, 齐性检验Huber M-estimators, 休伯M估计量Hyperbola, 双曲线Hypothesis testing, 假设检验Hypothetical universe, 假设总体Image factoring,, 多元回归法Impossible event, 不可能事件Independence, 独立性Independent variable, 自变量Index, 指标/指数Indirect standardization, 间接标准化法Individual, 个体Inference band, 推断带Infinite population, 无限总体Infinitely great, 无穷大Infinitely small, 无穷小Influence curve, 影响曲线Information capacity, 信息容量Initial condition, 初始条件Initial estimate, 初始估计值Initial level, 最初水平Interaction, 交互作用Interaction terms, 交互作用项Intercept, 截距Interpolation, 内插法Interquartile range, 四分位距Interval estimation, 区间估计Intervals of equal probability, 等概率区间Intrinsic curvature, 固有曲率Invariance, 不变性Inverse matrix, 逆矩阵Inverse probability, 逆概率Inverse sine transformation, 反正弦变换Iteration, 迭代Jacobian determinant, 雅可比行列式Joint distribution function, 分布函数Joint probability, 联合概率Joint probability distribution, 联合概率分布K-Means Cluster逐步聚类分析K means method, 逐步聚类法Kaplan-Meier, 评估事件的时间长度Kaplan-Merier chart, Kaplan-Merier图Kendall's rank correlation, Kendall等级相关Kinetic, 动力学Kolmogorov-Smirnove test, 柯尔莫哥洛夫-斯米尔诺夫检验Kruskal and Wallis test, Kruskal及Wallis检验/多样本的秩和检验/H检验Kurtosis, 峰度Lack of fit, 失拟Ladder of powers, 幂阶梯Lag, 滞后Large sample, 大样本Large sample test, 大样本检验Latin square, 拉丁方Latin square design, 拉丁方设计Leakage, 泄漏Least favorable configuration, 最不利构形Least favorable distribution, 最不利分布Least significant difference, 最小显著差法Least square method, 最小二乘法Least Squared Criterion，最小二乘方准则Least-absolute-residuals estimates, 最小绝对残差估计Least-absolute-residuals fit, 最小绝对残差拟合Least-absolute-residuals line, 最小绝对残差线Legend, 图例L-estimator, L估计量L-estimator of location, 位置L估计量L-estimator of scale, 尺度L估计量Level, 水平Leveage Correction，杠杆率校正Life expectance, 预期期望寿命Life table, 寿命表Life table method, 生命表法Light-tailed distribution, 轻尾分布Likelihood function, 似然函数Likelihood ratio, 似然比line graph, 线图Linear correlation, 直线相关Linear equation, 线性方程Linear programming, 线性规划Linear regression, 直线回归Linear Regression, 线性回归Linear trend, 线性趋势Loading, 载荷Location and scale equivariance, 位置尺度同变性Location equivariance, 位置同变性Location invariance, 位置不变性Location scale family, 位置尺度族Log rank test, 时序检验Logarithmic curve, 对数曲线Logarithmic normal distribution, 对数正态分布Logarithmic scale, 对数尺度Logarithmic transformation, 对数变换Logic check, 逻辑检查Logistic distribution, 逻辑斯特分布Logit transformation, Logit转换LOGLINEAR, 多维列联表通用模型Lognormal distribution, 对数正态分布Lost function, 损失函数Low correlation, 低度相关Lower limit, 下限Lowest-attained variance, 最小可达方差LSD, 最小显著差法的简称Lurking variable, 潜在变量Main effect, 主效应Major heading, 主辞标目Marginal density function, 边缘密度函数Marginal probability, 边缘概率Marginal probability distribution, 边缘概率分布Matched data, 配对资料Matched distribution, 匹配过分布Matching of distribution, 分布的匹配Matching of transformation, 变换的匹配Mathematical expectation, 数学期望Mathematical model, 数学模型Maximum L-estimator, 极大极小L 估计量Maximum likelihood method, 最大似然法Mean, 均数Mean squares between groups, 组间均方Mean squares within group, 组内均方Means (Compare means), 均值-均值比较Median, 中位数Median effective dose, 半数效量Median lethal dose, 半数致死量Median polish, 中位数平滑Median test, 中位数检验Minimal sufficient statistic, 最小充分统计量Minimum distance estimation, 最小距离估计Minimum effective dose, 最小有效量Minimum lethal dose, 最小致死量Minimum variance estimator, 最小方差估计量MINITAB, 统计软件包Minor heading, 宾词标目Missing data, 缺失值Model specification, 模型的确定Modeling Statistics , 模型统计Models for outliers, 离群值模型Modifying the model, 模型的修正Modulus of continuity, 连续性模Morbidity, 发病率Most favorable configuration, 最有利构形MSC（多元散射校正）Multidimensional Scaling (ASCAL), 多维尺度/多维标度Multinomial Logistic Regression , 多项逻辑斯蒂回归Multiple comparison, 多重比较Multiple correlation , 复相关Multiple covariance, 多元协方差Multiple linear regression, 多元线性回归Multiple response , 多重选项Multiple solutions, 多解Multiplication theorem, 乘法定理Multiresponse, 多元响应Multi-stage sampling, 多阶段抽样Multivariate T distribution, 多元T分布Mutual exclusive, 互不相容Mutual independence, 互相独立Natural boundary, 自然边界Natural dead, 自然死亡Natural zero, 自然零Negative correlation, 负相关Negative linear correlation, 负线性相关Negatively skewed, 负偏Newman-Keuls method, q检验NK method, q检验No statistical significance, 无统计意义Nominal variable, 名义变量Nonconstancy of variability, 变异的非定常性Nonlinear regression, 非线性相关Nonparametric statistics, 非参数统计Nonparametric test, 非参数检验Nonparametric tests, 非参数检验Normal deviate, 正态离差Normal distribution, 正态分布Normal equation, 正规方程组Normal P-P, 正态概率分布图Normal Q-Q, 正态概率单位分布图Normal ranges, 正常范围Normal value, 正常值Normalization 归一化Nuisance parameter, 多余参数/讨厌参数Null hypothesis, 无效假设Numerical variable, 数值变量Objective function, 目标函数Observation unit, 观察单位Observed value, 观察值One sided test, 单侧检验One-way analysis of variance, 单因素方差分析Oneway ANOVA , 单因素方差分析Open sequential trial, 开放型序贯设计Optrim, 优切尾Optrim efficiency, 优切尾效率Order statistics, 顺序统计量Ordered categories, 有序分类Ordinal logistic regression , 序数逻辑斯蒂回归Ordinal variable, 有序变量Orthogonal basis, 正交基Orthogonal design, 正交试验设计Orthogonality conditions, 正交条件ORTHOPLAN, 正交设计Outlier cutoffs, 离群值截断点Outliers, 极端值OVERALS , 多组变量的非线性正规相关Overshoot, 迭代过度Paired design, 配对设计Paired sample, 配对样本Pairwise slopes, 成对斜率Parabola, 抛物线Parallel tests, 平行试验Parameter, 参数Parametric statistics, 参数统计Parametric test, 参数检验Pareto, 直条构成线图（又称佩尔托图）Partial correlation, 偏相关Partial regression, 偏回归Partial sorting, 偏排序Partials residuals, 偏残差Pattern, 模式PCA（主成分分析）Pearson curves, 皮尔逊曲线Peeling, 退层Percent bar graph, 百分条形图Percentage, 百分比Percentile, 百分位数Percentile curves, 百分位曲线Periodicity, 周期性Permutation, 排列P-estimator, P估计量Pie graph, 构成图,饼图Pitman estimator, 皮特曼估计量Pivot, 枢轴量Planar, 平坦Planar assumption, 平面的假设PLANCARDS, 生成试验的计划卡PLS（偏最小二乘法）Point estimation, 点估计Poisson distribution, 泊松分布Polishing, 平滑Polled standard deviation, 合并标准差Polled variance, 合并方差Polygon, 多边图Polynomial, 多项式Polynomial curve, 多项式曲线Population, 总体Population attributable risk, 人群归因危险度Positive correlation, 正相关Positively skewed, 正偏Posterior distribution, 后验分布Power of a test, 检验效能Precision, 精密度Predicted value, 预测值Preliminary analysis, 预备性分析Principal axis factoring,主轴因子法Principal component analysis, 主成分分析Prior distribution, 先验分布Prior probability, 先验概率Probabilistic model, 概率模型probability, 概率Probability density, 概率密度Product moment, 乘积矩/协方差Profile trace, 截面迹图Proportion, 比/构成比Proportion allocation in stratified random sampling, 按比例分层随机抽样Proportionate, 成比例Proportionate sub-class numbers, 成比例次级组含量Prospective study, 前瞻性调查Proximities, 亲近性Pseudo F test, 近似F检验Pseudo model, 近似模型Pseudosigma, 伪标准差Purposive sampling, 有目的抽样QR decomposition, QR分解Quadratic approximation, 二次近似Qualitative classification, 属性分类Qualitative method, 定性方法Quantile-quantile plot, 分位数-分位数图/Q-Q图Quantitative analysis, 定量分析Quartile, 四分位数Quick Cluster, 快速聚类Radix sort, 基数排序Random allocation, 随机化分组Random blocks design, 随机区组设计Random event, 随机事件Randomization, 随机化Range, 极差/全距Rank correlation, 等级相关Rank sum test, 秩和检验Rank test, 秩检验Ranked data, 等级资料Rate, 比率Ratio, 比例Raw data, 原始资料Raw residual, 原始残差Rayleigh's test, 雷氏检验Rayleigh's Z, 雷氏Z值Reciprocal, 倒数Reciprocal transformation, 倒数变换Recording, 记录Redescending estimators, 回降估计量Reducing dimensions, 降维Re-expression, 重新表达Reference set, 标准组Region of acceptance, 接受域Regression coefficient, 回归系数Regression sum of square, 回归平方和Rejection point, 拒绝点Relative dispersion, 相对离散度Relative number, 相对数Reliability, 可靠性Reparametrization, 重新设置参数Replication, 重复Report Summaries, 报告摘要Residual sum of square, 剩余平方和residual variance (剩余方差)Resistance, 耐抗性Resistant line, 耐抗线Resistant technique, 耐抗技术R-estimator of location, 位置R估计量R-estimator of scale, 尺度R估计量Retrospective study, 回顾性调查Ridge trace, 岭迹Ridit analysis, Ridit分析Rotation, 旋转Rounding, 舍入Row, 行Row effects, 行效应Row factor, 行因素RXC table, RXC表Sample, 样本Sample regression coefficient, 样本回归系数Sample size, 样本量Sample standard deviation, 样本标准差Sampling error, 抽样误差SAS(Statistical analysis system ), SAS统计软件包Scale, 尺度/量表Scatter diagram, 散点图Schematic plot, 示意图/简图Score test, 计分检验Screening, 筛检SEASON, 季节分析Second derivative, 二阶导数Second principal component, 第二主成分SEM (Structural equation modeling), 结构化方程模型Semi-logarithmic graph, 半对数图Semi-logarithmic paper, 半对数格纸Sensitivity curve, 敏感度曲线Sequential analysis, 贯序分析Sequence, 普通序列图Sequential data set, 顺序数据集Sequential design, 贯序设计Sequential method, 贯序法Sequential test, 贯序检验法Serial tests, 系列试验Short-cut method, 简捷法Sigmoid curve, S形曲线Sign function, 正负号函数Sign test, 符号检验Signed rank, 符号秩Significant Level, 显著水平Significance test, 显著性检验Significant figure, 有效数字Simple cluster sampling, 简单整群抽样Simple correlation, 简单相关Simple random sampling, 简单随机抽样Simple regression, 简单回归simple table, 简单表Sine estimator, 正弦估计量Single-valued estimate, 单值估计Singular matrix, 奇异矩阵Skewed distribution, 偏斜分布Skewness, 偏度Slash distribution, 斜线分布Slope, 斜率Smirnov test, 斯米尔诺夫检验Source of variation, 变异来源Spearman rank correlation, 斯皮尔曼等级相关Specific factor, 特殊因子Specific factor variance, 特殊因子方差Spectra , 频谱Spherical distribution, 球型正态分布Spread, 展布SPSS(Statistical package for the social science), SPSS统计软件包Spurious correlation, 假性相关Square root transformation, 平方根变换Stabilizing variance, 稳定方差Standard deviation, 标准差Standard error, 标准误Standard error of difference, 差别的标准误Standard error of estimate, 标准估计误差Standard error of rate, 率的标准误Standard normal distribution, 标准正态分布Standardization, 标准化Starting value, 起始值Statistic, 统计量Statistical control, 统计控制Statistical graph, 统计图Statistical inference, 统计推断Statistical table, 统计表Steepest descent, 最速下降法Stem and leaf display, 茎叶图Step factor, 步长因子Stepwise regression, 逐步回归Storage, 存Strata, 层（复数）Stratified sampling, 分层抽样Stratified sampling, 分层抽样Strength, 强度Stringency, 严密性Structural relationship, 结构关系Studentized residual, 学生化残差/t化残差Sub-class numbers, 次级组含量Subdividing, 分割Sufficient statistic, 充分统计量Sum of products, 积和Sum of squares, 离差平方和Sum of squares about regression, 回归平方和Sum of squares between groups, 组间平方和Sum of squares of partial regression, 偏回归平方和Sure event, 必然事件Survey, 调查Survival, 生存分析Survival rate, 生存率Suspended root gram, 悬吊根图Symmetry, 对称Systematic error, 系统误差Systematic sampling, 系统抽样Tags, 标签Tail area, 尾部面积Tail length, 尾长Tail weight, 尾重Tangent line, 切线Target distribution, 目标分布Taylor series, 泰勒级数Test(检验)Test of linearity, 线性检验Tendency of dispersion, 离散趋势。

傅里叶变换和transformer 结合

傅里叶变换和transformer 结合

傅里叶变换（Fourier Transform）作为一种信号处理和频域分析的重要方法，在通信、音频处理、图像处理等领域得到了广泛的应用。

而近年来，随着深度学习的发展，transformer模型在自然语言处理和语音识别等领域也取得了巨大的成功。

本文将探讨傅里叶变换和transformer模型的结合，探讨其在信号处理和语音识别等领域的应用。

一、傅里叶变换概述傅里叶变换是一种将时域信号转换成频域信号的数学工具，它能够将任意时域信号分解成不同频率的正弦和余弦信号的叠加。

通过傅里叶变换，我们可以获得信号的频谱信息，进而进行频域分析和滤波处理。

二、transformer模型概述transformer是一种基于注意力机制的深度学习模型，它在自然语言处理如机器翻译和语音识别等任务中取得了突出的成绩。

transformer 模型将输入序列通过多层自注意力机制和前向神经网络进行编码和解码，能够捕捉长距离依赖关系，并在序列建模任务中表现优异。

三、傅里叶变换与transformer的结合在某些领域，信号处理和深度学习的任务需要同时考虑时域和频域的信息，传统的信号处理方法往往无法充分利用时频信息，而transformer模型可以有效地捕捉时域信息和频域信息，因此结合傅里叶变换和transformer模型能够在一定程度上提高信号处理和语音识别的性能。

四、傅里叶变换与transformer的应用案例1. 语音信号的频谱建模传统的傅里叶变换能够将语音信号从时域转换到频域，但无法建模长时序依赖关系，而transformer模型可以有效地捕捉长距离依赖关系，因此可以结合傅里叶变换和transformer模型来建模语音信号的频谱。

2. 图像信号的频域分析图像处理领域常常需要对图像进行频域分析和滤波处理，传统的傅里叶变换能够将图像信号转换到频域，但无法对频域信息进行进一步的处理，而transformer模型可以在频域上进行自注意力计算，因此可以结合傅里叶变换和transformer模型进行图像信号的频域分析。

第四章 混沌时间序列分析及相空间重构

第四章 混沌时间序列分析及相空间重构


Lyapunov Exponents
f
• Quantifies separation in time between trajectories, assuming rate of growth (or decay) is exponential in time, as: n
1 i lim ln( eig J(p)) n n p 0
估计吸引子维数的算法，需要大量的数据点作为输入，当这些点的 输入被选择为最大化的包含吸引子信息情况下，输入数据点的数量可以减 少。（由Holzfuss和Mayer—kress 1986年提出） 重构相空间所需要解决的关键问题，就是确定重构维数m。 在重构相空间维数未知的情况下，可用以下方法获得： 令 nr 为重构空间的维数。首先把nr （或m）设置为1，计算重构吸引子 的维数Dcap，然后增加 nr （或m）的大小，并重复计算重构吸引子的维数 Dcap,直到Dcap不再改变为止（如曹书p103），最后的Dcap是正确的相 关维数，产生正确的Dcap的最小 nr (m) 即重构空间的最小维数m.
Time delay embedding
Differs from traditional experimental measurements
Provides detailed information about degrees of freedom beyond the scalar measured Rests on probabilistic assumptions - though not guaranteed to be valid for any particular system Reconstructed dynamics are seen through an unknown “smooth transformation” Therefore allows precise questions only about invariants under “smooth transformations” It can still be used for forecasting a time series and “characterizing essential features of the dynamics that produced it”
Chaos_19_033137

Chaos_19_033137

Complex network from time series based on phase space reconstruction Zhongke Gao and Ningde JinSchool of Electrical Engineering and Automation,Tianjin University,Tianjin300072,People’s Republic of China͑Received23April2009;accepted24August2009;published online17September2009͒We propose in this paper a reliable method for constructing complex networks from a time serieswith each vector point of the reconstructed phase space represented by a single node and edgedetermined by the phase space distance.Through investigating an extensive range of networktopology statistics,wefind that the constructed network inherits the main properties of the timeseries in its structure.Specifically,periodic series and noisy series convert into regular networks andrandom networks,respectively,and networks generated from chaotic series typically exhibit small-world and scale-free features.Furthermore,we associate different aspects of the dynamics of thetime series with the topological indices of the network and demonstrate how such statistics can beused to distinguish different dynamical regimes.Through analyzing the chaotic time series cor-rupted by measurement noise,we also indicate the good antinoise ability of our method.©2009American Institute of Physics.͓DOI:10.1063/1.3227736͔Characterizing complicated dynamics from time series (or measured signals)is a fundamental problem of con-tinuing interest in a wide variety offields.Different mea-sures have been proposed to analyze these dynamics,for example,Lyapunov exponent,entropies,and fractal plex networks have established themselves in recent years as being particularly suitable andflexible for representing and modeling many complex natural and artificial systems.Investigation of nonlinear time series in terms of complex networks has begun only recently.In this paper,we focus on one basic question:how to inves-tigate the nonlinear dynamics of time series through com-plex network.First,we propose a reliable method for constructing complex network from time series with the purpose of encoding the underlying time series dynamics in the network topology.Then we associate different as-pects of the dynamics of the time series with the topologi-cal indices of the network and demonstrate how such sta-tistics can be used to distinguish different dynamical regimes.Our result can provide useful clues as how to properly construct the network to effectively investigate the complicated dynamics of different time series.I.INTRODUCTIONThe past few years have witnessed dramatic advances in thefield of complex networks1–9since the publication of the seminal works of Watts and Strogatz10as well as Barabási and Albert.11Complex networks,which provide us with a new viewpoint for understanding a complex system from the relations between the elements in a global way,may be a powerful tool for investigating nonlinear time series in prac-tice.But how to construct a network from a time series͑or measured signals͒is still an essential problem to be solved. Zhang and Small12first introduced a transformation from pseudoperiodic time series to complex network by represent-ing each cycle of pseudoperiodic series as a basic node. Lacasa et al.13proposed an algorithm to characterize fractal time series using the visibility graph.Yang and Yang14built network from the correlation matrix of a time series.Al-though the network construction algorithms mentioned above have the advantage of being simple,which means that the embedding step is avoided,they may have difficulties in getting enough information from a high dimensional system. An important advantage of utilizing a phase space recon-struction is that if the embedding is chosen appropriately,the topological distribution of the set of vector points in phase space will accurately reflect the underlying dynamics of the original system.Therefore,the network inferred from that phase space reconstruction can be related directly back to the evolution operator of the underlying dynamical system.In this paper,we introduce a reliable and effective method for constructing complex network from time series based on phase space reconstruction.That is,we use false nearest neighbors͑FNNs͒method15and C-C method16re-construct phase space from time series and construct network with each vector point of the reconstructed phase space rep-resented by a single node and connection determined by the phase space distance.Due to the fact that this method can encode the underlying time series dynamics in the network topology,we characterize the complicated dynamics of dif-ferent time series through exploring a number of network topology statistics and demonstrate that some statistics not only can reflect different aspects of specific dynamics but also can be used to distinguish different dynamical regimes. In addition,the antinoise ability of our method is also inves-tigated by analyzing the chaotic time series corrupted by measurement noise.II.CONSTRUCTION OF NETWORKFROM TIME SERIESWe start from the phase space reconstruction.Takens’embedding theorem,17which is very often invoked as the motivation for applying a time delay embedding to recon-struct phase space from a time series,can be described as following,for arbitrary time series z͑it͒,i=1,2,...,M͑t isCHAOS19,033137͑2009͒1054-1500/2009/19͑3͒/033137/12/$25.00©2009American Institute of Physics19,033137-1sampling interval,M is the sample size ͒,if the embedding delay time is selected as ␶,and the embedding dimension as m ,the vector point in phase space can be represented as follows:X k ៝=͕x k ͑1͒,x k ͑2͒,...,x k ͑m ͖͒=͕z ͑kt ͒,z ͑kt +␶͒,...,z ͑kt +͑m −1͒␶͖͒,͑1͒where k =1,2,...,N and N =M −͑m −1͒ء␶/t denotes the total vector points of the reconstructed phase space.C-C method 16is the frequent method that can be used to determine the delay time ␶.The general idea of C-C method is that we can subdivide a single time series ͕z 1,z 2,...,z M ͖into t disjoint time series,where t is the index lag,and then calculate S ͑m ,M ,r ,t ͒from these time series as follows:S ͑m ,M ,r ,t ͒=1t ͚s =1t͓C s ͑m ,M /t ,r ,t ͒−C sm͑1,M /t ,r ,t ͔͒,͑2͒⌬S ͑m ,t ͒=max ͑S ͑m ,r j ,t ,M ͒͒−min ͑S ͑m ,r j ,t ,M ͒͒,͑3͒where C ͑m ,M ,r ,␶͒=2/N ͑N −1͚͒1Յi Ͻj ՅN ␪͑r −͉x i −x j ͉͒,N =M −͑m −1͒␶,r Ͼ0,and ␪͑a ͒is the Heaviside function.The general range of m ,r is 2Յm Յ5,␴/2Յr Յ2␴,␴is the standard deviation of data sets,S ¯͑t ͒=116͚m =25͚j =14S ͑m ,r j ,t ͒,͑4͒⌬S ¯͑t ͒=14͚m =25⌬S ͑m ,t ͒.͑5͒The corresponding time of the first local minimum of ⌬S ¯͑t ͒is the optimal delay time ␶.More details about C-C method can be found in Ref.16.Through studying four algorithms that can be used to select ␶,i.e.,mutual information I,18mutual information II,19autocorrelation function method,and C-C method,we have argued in Ref.20that C-C method shows good antinoise ability.Hence,in this paper we use C-C method to determine the delay time ␶and use FNN method to determine the embedding dimension m ,that is,we use C-C method and FNN method reconstruct phase space from time series.After reconstructing phase space from a given time se-ries,denoted as ͕z ͑1͒,z ͑2͒,...,z ͑M ͖͒,we proceed to con-struct complex network by considering each vector point in reconstructed phase space as a basic node and using the phase space distance to determine network connection.Thephase space distance between vector points X i ៝and X j ៝in this study is defined asd ij =͚n =1mʈX i ͑n ͒−X j ͑n ͒ʈ,͑6͒where X i ͑n ͒=z ͑i +͑n −1͒␶͒and X j ͑n ͒=z ͑j +͑n −1͒␶͒is the n th element of X i ៝and X j ៝,m and ␶are the embedding dimen-sion and delay time,respectively.The constructed network containing N =M −͑m −1͒ء␶/t nodes can be studied either as a weighted network or as an unweighted one.We first con-struct weighted network by choosing the distance between each pair of vector points in reconstructed phase space as the weight between corresponding nodes and define the node strength S of weighted network asS i =1N ͚j =1NW ij ,͑7͒where W ij is the weight between nodes i and j and W ij =d ij .We use a chaotic time series from Lorenz system and a white noise time series ͓see Figs.1͑a ͒and 1͑d ͒for details ͔inves-tigates the S distribution and find that,as shown in Fig.2,the S distribution of the chaotic time series has significant dif-ference from that of noisy time series.As can been seen,since the S distribution of the chaotic time series is rather random when S is over 25,compared to a more clear power law distribution,which can be seen in Figs.7and 9of our previous paper,9the overall S distribution of the chaotic time series is not a power law distribution.By contrast,the overall S distribution of noisy time series is bell-shaped distribution.Since the weighted network is fully connected and may contain redundant information,we now convert the weighted network to its unweighted counterpart by setting a threshold on the weight.That is,choosing a critical threshold r c ,the distance matrix D =͑d ij ͒can be converted into adjacent ma-trix A =͑a ij ͒,the rules of which read a ij =1if d ij Յr c and a ij =0if d ij Ͼr c .The topological structure of the network can be described with the adjacent matrix A ,and the conditions a ij =1and a ij =0correspond to connection and disconnection,respectively.An appropriate threshold,therefore,should be chosen to fully preserve the main property of the network,but not to be too large that may obscure or conceal the local property by overconnecting the nodes.In this paper,we em-ploy the network density 21to study the threshold,which is defined as the number of edges divided by the largest number of edges possible.In order to show how to select a suitable threshold,a chaotic time series from Lorenz system ͓see Fig.1͑a ͔͒is studied by our method,and Fig.3shows the density of the constructed network versus the threshold r .As can be seen in Fig.3͑b ͒,the increase in the whole network degree reaches the maximum rate at about r c =11.2,where we set the critical threshold.We here explain why we set the thresh-old r at this critical point.Since the nodes inside a cluster are adjacent to each other,the degree will increase very rapidly as the threshold changes within the cluster radius.For a net-work from a chaotic system that has many clusters differing in sizes,as shown in Figs.4͑a ͒and 4͑b ͒,the edge increase will reach the maximum rate when the threshold approaches the mean radius of all the clusters,i.e.,the critical point r c .The unweighted network obtained at r c will maintain the clustering property,and thresholds beyond this value will result in a much slower edge increase,causing redundant connections among nodes.Consequently we choose r c to de-rive the unweighted network.Furthermore,it should be noted that the size of the network ͑i.e.,the length of the time series ͒will influence r c ,and that is why we choose r c =11.2to derive network of 200nodes,in contrast to r c =10to derive network of 600nodes from the Lorenz system.033137-2Z.Gao and N.Jin Chaos 19,033137͑2009͒100020003000-30-20-100102030x (t )t10002000-10-5051015x (t )t200400600-1.0-0.50.00.51.0x (t )t010*******0.00.20.40.60.81.0x (t )t(b)(a)(c)(d)FIG.1.Four different scalar time series that are used to construct complex networks:͑a ͒chaotic time series from Lorenz system ͑␴=16,␳=45.92,␤=4͒,͑b ͒chaotic time series from Rossler system ͑a =0.2,b =0.2,c =5.7͒,͑c ͒periodic time series,and ͑d ͒white noise time series.C o u n tSC o u n tS(b)(a)FIG.2.Node strength distribution for weighted network of 1000nodes from ͑a ͒Lorenz system ͑log-log plot ͒and ͑b ͒white noise time series ͑regular plot ͒.204060800.00.20.40.60.81.0r c =11.2D e n s i t yThreshold204060800.0000.0050.0100.0150.0200.0250.030r c =11.2D e r i v a t i v eThreshold(b)(a)FIG.3.͑Color online ͒Complex network from the Lorenz system with 200nodes.͑a ͒Density vs threshold.͑b ͒Derivative of density vs threshold with m =3,␶=10,r c =11.2.033137-3Complex network from time seriesChaos 19,033137͑2009͒We use the method presented in this paper to generate networks from Lorenz system,Rossler system,periodic time series,and white noise time series,respectively ͑details of the four different time series can be seen in Fig.1͒,and find that the networks constructed from different kinds of timeseries demonstrate fundamentally different structures.The chaotic time series is the data from the x component of Lo-renz systemdx /dt =␴͑y −x ͒,FIG.4.͑Color online ͒Complex network of 200nodes from ͑a ͒Lorenz system with m =3,␶=10,r c =11.2,͑b ͒Rossler system with m =3,␶=13,r c =6.2,͑c ͒pe-riodic time series with m =4,␶=3,r c =0.15,and ͑d ͒white noise time series with m =4,␶=5,r c =0.41.The structure of network drawn by software UCINET and NET-DRAW is shown in the left panel and the corresponding network connectivity ͑adjacency ͒matrix is shown in the right panel.033137-4Z.Gao and N.Jin Chaos 19,033137͑2009͒dy /dt =x ͑␳−z ͒−y ,͑8͒dz /dt =xy −␤z ,where ␴=16,␳=45.92␤=4,as shown in Fig.1͑a ͒,and the data from the x component of Rossler systemdx /dt =−y −z ,dy /dt =x +ay ,͑9͒0.00.20.40.60.81.0Lorenz system Rossler system Periodic time series Wihite Noise time seriesC l u s t e r i n g C o e f f i c i e n t24681012141618Lorenz system Rossler system Periodic time series Wihite Noise time seriesS h o r t e s t P a t h(b)(a)FIG. 5.Small-world characteristics of the networks from chaotic time series.<K n n (k )>K<K n n (k )>K(b)(a)FIG.6.The regular plots of the nearest neighbor average connectivity of nodes with respect to connectivity of the networks containing 600nodes from ͑a ͒Lorenz system with m =3,␶=23,r c =10and ͑b ͒white noise time series with m =4,␶=3,r c =0.8.(a)(b)FIG.7.The bitmap of sorted adjacent matrix of network containing 600nodes from ͑a ͒Lorenz system with m =3,␶=23,r c =10and ͑b ͒white noise time series with m =4,␶=3,r c =0.8.033137-5Complex network from time seriesChaos 19,033137͑2009͒dz /dt =b +z ͑x −c ͒,where a =0.2,b =0.2,c =5.7,as shown in Fig.1͑b ͒.It should be mentioned here that the structure of constructed network,i.e.,Fig.4,is drawn by the software NETDRAW ,and the algorithm used is the Kamada–Kawai 22spring embed-ding algorithm.The general idea of Kamada–Kawai spring embedding algorithm is as follows.First,the algorithm cal-culates the energy for each node.It then loops over all the nodes to find the one with the highest energy and begins iterating the Newton–Raphson stage to compute new posi-tions for the node until its energy is below epsilon.At this point,it again looks for the node with the highest energy and begins moving it ͑for more details,see Ref.22͒.This process continues until there is no node with energy above epsilon,and then the algorithm is completed.As can be seen in Figs.4͑a ͒and 4͑b ͒,the network structure corresponding to chaotic time series has nodes congregating at different locations,and some regions are highly clustered with nodes and others are rather sparse.The network generated from periodic time se-ries in Fig.4͑c ͒is obviously a regular network.In compari-son,the network from the white noise time series looks like a random network with the nodes entangled with each other and their edges intersecting,as shown in Fig.4͑d ͒.WORK STATISTICS VERSUS DYNAMICAL REGIMESUnstable periodic orbits ͑UPOs ͒21,23,24embedded in the chaotic attractor are fundamental to the understanding of the chaotic dynamics.For a chaotic attractor,its trajectory will typically switch or hop among different UPOs.Specifically,the trajectory will approach an UPO along its stable mani-fold.This approach can last for several loops during which the orbit remains close to the UPO.Eventually,the orbit is ejected along the unstable manifold and proceeds until it is captured by the stable manifold of another UPO.A UPO of order n contains n loops lying in different locations in phase space.Each loop that belongs to a certain UPO-n has many other loops in its vicinity due to the attraction of the stable manifold associated with the UPO-n .It then becomes a cen-ter of a cluster and the density of the neighbors is related to its stability decided by the vector field along its trajectory.Since the stability of each center loop may vary,we can see sparse as well as dense regions in the structure of constructed complex network,as shown in Figs.4͑a ͒and 4͑b ͒.Due to the fact that the loops in phase space are spatially clustered around the UPOs and we use phase space to determine the network connection,the complex network generated from chaotic time series shows the small-world characteristics,i.e.,small shortest path and large clustering coefficient,which can be seen in Fig.5.C o u n tBetweennessC o u n tBetweenness(b)(a)FIG.8.Betweenness distribution in log-log scale for complex network of 600nodes from ͑a ͒Lorenz system with m =3,␶=23,r c =10and ͑b ͒white noise time series with m =4,␶=3,r c =0.8.B e t w e e n n e s sClustering coefficientB e t w e e n n e s sClustering coefficient(b)(a)FIG.9.Clustering coefficient-betweenness correlations in log-log scale for complex network of 600nodes from ͑a ͒Lorenz system with m =3,␶=23,r c =10and ͑b ͒white noise time series with m =4,␶=3,r c =0.8.033137-6Z.Gao and N.JinChaos 19,033137͑2009͒We now demonstrate how to discriminate different dy-namical regimes of the time series through investigating the degree correlations,Pearson coefficient,betweenness distri-bution,and clustering coefficient-betweenness correlationsof two distinct complex networks from the chaotic time se-ries ͑Lorenz system ͒and white noise time series.An impor-tant way of capturing the degree correlations is to examine the average degree of the nearest neighbors of nodes with degree k ,25which is defined ask nn ͑k ͒=͚k Јk ЈP ͑k Ј͉k ͒,͑10͒where P ͑k Ј͉k ͒denotes the conditional probability that an edge of degree k connects a node with degree k Ј.If there are no degree correlations,Eq.͑10͒gives k nn ͑k ͒=͗k 2͘/͗k ͘,i.e.,k nn ͑k ͒is independent of k .Correlated networks are classified as assortative mixing if k nn ͑k ͒is an increasing function of k ,whereas they are referred to as disassortative mixing when k nn ͑k ͒is a decreasing function of k .In other words,in assor-tative networks the nodes tend to connect to their connectiv-ity peers,while in disassortative networks nodes with low degree are more likely connected with highly connected ones.As can be seen in Fig.6,k nn ͑k ͒increases with k for chaotic Lorenz system,while it decreases with k for noisy time series.In order to quantify such a correlation,we cal-culate the Pearson coefficient,26which is defined as follows.Let e kl ͑=e lk ͒be the joint probability distribution for an edge to be with a node with connectivity k at one end and a node with connectivity l at the other.Its marginal q k =͚l e kl obeys the normalization condition ͚k q k =1.Then,the Pearson cor-relation coefficient is given byr =1␴q2͚k ,l kl ͑e kl −q k q l ͒,͑11͒where ␴q 2=͚k k 2q k −͚͑k kq k ͒2is the variance of q k .r ෈͓−1,1͔and r is positive ͑negative ͒for assortative ͑disas-100020003000-30-20-100102030x (t )t50010001500200025003000-40-30-20-10010203040x (t )t(b)(a)FIG.10.Two different chaotic time series that are corrupted by measure-ment Gaussian white noise:͑a͒␩=0.1͑SNR=10dB ͒and ͑b ͒␩=0.3͑SNR=5dB ͒.FIG.11.͑Color online ͒Complex network of 200nodes from chaotic time series that are corrupted by measure-ment Gaussian white noise.͑a ͒Noise intensity ␩=0.1͑SNR=10dB ͒.͑b ͒Noise intensity ␩=0.3͑SNR =5dB ͒.The structure of network drawn by software UCINET and NETDRAW is shown in the left panel and the corresponding network connectivity ͑adjacency ͒matrix is shown in the right panel.033137-7Complex network from time series Chaos 19,033137͑2009͒sortative ͒mixing.The Pearson coefficient of the network containing 600nodes from chaotic Lorenz system and noisy time series is 0.1009and Ϫ0.0037,respectively.Consistent with the one obtained from the analysis of the nearest neigh-bor average connectivity,the Pearson correlation coefficient is positive,confirming that the network from chaotic time series has assortative mixing.On the other hand,the network from noisy time series is of no assortative mixing.We can explain why the network corresponding to chaotic time se-ries possesses the property of assortative mixing in terms of UPOs.For a complex network generated from chaotic time series,there are multiple clusters and most nodes connected to each other within the same cluster have a roughly similar number of connected neighbors or degrees.Because of the different stability of the center loop of UPOs,the common degree shared by the nodes from one cluster may differ from that of another.Due to the fact that the nodes within the same cluster usually have similar degrees and such nodes are al-ways interconnected to each other,chaotic time series pos-sesses the property of assortative mixing.Another way of capturing the clustering property of cha-otic time series is the sorted adjacent matrix proposed in this paper.That is,for a given complex network,we first sort nodes in ascending order of the degree.If two or more than two nodes have the same degree,we then arrange them inascending order of the sum of neighbor degree,which is the sum of degree of a node’s neighbor.Finally,the above node sorting rules produce a sorted adjacent matrix,which can be represented as a black-and-white bitmap,i.e.,if nodes i and j are connected,a black pixel is placed at the coordinate of ͑i ,j ͒;otherwise a white pixel is placed there.Figure 7de-picts the bitmap of sorted adjacent matrix of two networks from different time series,i.e.,chaotic series and noisy se-ries.As can be seen,multiple black square blocks could be clearly found along the principal diagonal in the sorted adja-cent matrix bitmap of the network from chaotic time series,which implies the fact that there are multiple clusters in the network and the nodes with similar degrees are intercon-nected to each other ͑i.e.,assortative mixing ͒.On the con-trary,no obvious black square block could be detected along the principal diagonal in the sorted adjacent matrix bitmap corresponding to the noisy time series.K<K n n(k )><K n n (k )>K(b)(a)FIG.12.The regular plots of the nearest neighbor average connectivity of nodes with respect to connectivity of the networks containing 600nodes from chaotic time series that are corrupted by measurement Gaussian white noise.͑a ͒Noise intensity ␩=0.1͑SNR=10dB ͒.͑b ͒Noise intensity ␩=0.3͑SNR=5dB ͒.(b)(a)FIG.13.The bitmap of sorted adjacent matrix of network containing 600nodes from chaotic time series that are corrupted by measurement Gaussian white noise.͑a ͒Noise intensity ␩=0.1͑SNR=10dB ͒.͑b ͒Noise intensity ␩=0.3͑SNR=5dB ͒.033137-8Z.Gao and N.JinChaos 19,033137͑2009͒From the above analysis,we can see that the clustering property of chaotic time series can be well characterized by the degree correlations,Pearson coefficient,as well as the sorted adjacent matrix,and for networks from the noisy time series,no assortative mixing is found at different thresholds.Assortative mixing provides the information about the inter-actions of all node pairs.Sometimes we may also want to know how important or how central a single node is in a network.A concept that fulfills this requirement is between-ness or betweenness centrality.Together with the degree and the closeness of a node,the betweenness is one of the stan-dard measures of node centrality,originally introduced to quantify the importance of an individual in a social network.27,28More precisely,the betweenness b i of a node i ,sometimes referred to also as load,is defined asb i =͚j ,k ෈N ,j kn jk ͑i ͒n jk,͑12͒where n jk is the number of shortest paths connecting j and k ,while n jk ͑i ͒is the number of shortest paths connecting j and k and passing through i .Through exploring the betweenness distribution and clustering coefficient-betweenness correla-tions,we not only show in Fig.8that the overall between-ness of the network constructed from chaotic time series is different from that of the network generated from noisy timeseries,but also demonstrate in Fig.9that a large number of nodes with small clustering coefficient have high between-ness in the network from chaotic time series,in contrast to the high betweenness only for the nodes with large clustering coefficient in the network from noisy time series.The dis-tinct distributions show that the vector points are structured with different mechanisms in the phase space.As can be seen in Fig.8,compared to a more clear power law distribution which can be found in Figs.7and 9of our previous paper,9the overall betweenness distribution of the chaotic time se-ries is not an exactly power law distribution.But the overall betweenness distribution of the network constructed from noisy time series is an exponential distribution.From Figs.8and 9,we find that for the chaotic time series,the number of nodes with high betweenness greatly exceeds that from the noisy time series.This is essentially a reflection of the clus-tering property associated with the UPOs embedded in the chaotic attractor.The high betweenness nodes correspond to the nodes in between adjacent clusters that act as bridges,and such nodes usually with small clustering coefficient compared to the nodes within the clusters.Since a chaotic attractor contains infinitely UPOs,there will be many clus-ters in the corresponding network,which implies the exis-tence of numerous high betweenness nodes.C o u n tBetweennessC o u n tBetweenness(b)(a)FIG.14.Betweenness distribution in log-log scale for complex network of 600nodes from chaotic time series that are corrupted by measurement Gaussian white noise.͑a ͒Noise intensity ␩=0.1͑SNR=10dB ͒.͑b ͒Noise intensity ␩=0.3͑SNR=5dB ͒.B e t w e e n n e s sClustering coefficientB e t w e e n n e s sClustering coefficient(b)(a)FIG.15.Clustering coefficient-betweenness correlations in log-log scale for complex network of 600nodes from chaotic time series that are corrupted by measurement Gaussian white noise.͑a ͒Noise intensity ␩=0.1͑SNR =10dB ͒.͑b ͒Noise intensity ␩=0.3͑SNR=5dB ͒.033137-9Complex network from time seriesChaos 19,033137͑2009͒IV.ANALYSIS OF THE ANTINOISE ABILITYSince most real data are corrupted by noise,in this sec-tion we will discuss the antinoise ability of our method by analyzing two chaotic time series that are corrupted by mea-surement noise.The time series we used are generated from the following equation:z i =L i +␩␴␧i ,͑13͒where L i is the noise-free time series from chaotic Lorenz system ͓see Fig.1͑a ͔͒,␴is the standard deviation,␧i is the Gaussian independent identically distributed ͑iid ͒random variable with zero mean and a standard deviation of 1,and ␩is the strength of noise.Noise levels of 0.1and 0.3were added to the Lorenz time series to generate two time series whose signal-to-noise ratio ͑SNR ͒is 10and 5dB ͓see Figs.10͑a ͒and 10͑d ͒for details ͔.We construct networks from these two time series and find that when SNR Ͼ5dB,the topological structure of constructed network attractor is very similar to the one constructed from deterministic chaotic Lo-renz system,as shown in Figs.11͑a ͒and 4͑a ͒;when the noise intensity becomes very large,i.e.,SNR Յ5dB,because of the strong influence of Gaussian white noise,the correspond-ing topological structure distorts to a certain extent,as shown in Fig.11͑b ͒.To cast light into an important question of whether the network statistics mentioned in Sec.III can be used to distinguish chaotic time series corrupted by Gaussian white noise from white noise time series,we investigate thedegree correlations,betweenness distribution,and clustering coefficient-betweenness correlations of the two series pre-sented in Fig.10.The increase in k nn ͑k ͒with k in Fig.12and the multiple black square blocks along principal diagonal in Fig.13both indicate the fact that the chaotic time series corrupted by Gaussian white noise also possesses the prop-erty of assortative mixing.As can be seen in Figs.14and 15,the clustering property associated with the UPOs of the cha-otic time series corrupted by Gaussian white noise also can be reflected by the betweenness distribution and clustering0501001502000.00.20.40.60.81.0a=4,x(0)=0.55X (n )n501001502000.20.40.60.81.0a=3.785,x(0)=0.55X (n )n501001502000.40.60.8a=3.5,x(0)=0.55X (n )n0501001502000.20.40.60.81.0a=3.679,x(0)=0.55X (n )n(b)(a)(c)(d)FIG.16.Four different time series from logistic function.N e t w o r k L a m i n a r i t yaFIG.17.The NLAM for networks from logistic function with different bifurcation parameters.033137-10Z.Gao and N.Jin Chaos 19,033137͑2009͒。