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PDF | In and , two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of. Holger Kantz, Max-Planck-Institut für Physik komplexer Systeme, Dresden, Thomas Schreiber, Max-Planck-Institut für Physik komplexer Systeme, Dresden. Subjects: Physics And Astronomy, Nonlinear Science and Fluid Dynamics, Statistics and Probability, Statistics for Physical. NONLINEAR TIME SERIES ANALYSIS. HOLGER KANTZ AND THOMAS SCHREIBER. Max Planck Institute for the Physics of Complex Systems, Dresden .

After a slow start, its development has been rapid recently. The following events have provided timely and important momentum to the development of the subject. Important specialist monographs have also appeared steadily over the past twenty years or so, notably Priestley , Tong , , Chan and Tong , Fan and Yao and others. The need to break away from linear models was recognized almost as early as when Udny Yule introduced the first linear time series model, namely the autoregressive model. The fact that it took statisticians more than half a century to accomplish the first breakthrough might seem strange at first sight.

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A comprehensive resource that draws a balance between theory and applications of nonlinear time series analysis Nonlinear Time Series Analysis offers an important guide to both parametric and nonparametric methods, nonlinear state-space models, and Bayesian as well as classical approaches to nonlinear time series analysis.

This vital guide: Markov Chains 2. Discrete Distributions for Count Data 5. Propagation 8. After the initial breakthroughs in the early s, progress was still slow over the decade, due to the enormity of the theoretical and methodological challenges. Fortunately, with successful applications in diverse fields, e. As a result, publications have grown exponentially.

Just as an illustration, according to the ISI Web of Knowledge, the citations of a threshold time series model have reached about 1, in since its birth in The corresponding figures in and were approximately 10 and respectively!

Many exciting developments are indeed taking place. The relatively recent entry of non-parametric and semi-parametric techniques into the time series analysis has given modern time series analysts powerful tools to explore the time series data from many different perspectives, before deciding on an appropriate parametric model for more in-depth exploration.

Nonlinear dimension reduction methodology is being developed and exploited to handle high-dimensional time series data. Various factor models are being developed to cope with high frequency data, so often found in financial applications. The interplay begins at the beginning, of course, with talented individuals coming up with imaginative conjectures on which others build and which are subject to the scrutiny of the whole relevant community, and it is present at every stage.

The warrant of any empirical proposition depends in part on experimental evidence, i.

Data provide an evidentiary portal to the real world to which there is only limited access. For example, Charles Darwin consolidated several months of studying numerous marine and terrestrial samples collected in the Galapagos into his famous sketch of the evolutionary tree Berra, In another example, Leonardo da Vinci compiled a lifetime of observing nature into notebooks explaining diverse behaviours, including water and sediment movement in river systems, waves in ponds, sound waves in air, and even whether spirits can speak da Vinci, The contributions of both scientists are especially renowned because they detected patterns that were not obvious in the data, and processed that information into astounding knowledge of systematic natural behaviour that has withstood the test of time.

The results of scientific inquiry must agree with reality if they are to be put to reliable use. For example, policymakers reasonably expect that theory supporting high-stakes public policy ad- equately represents the real world that they are charged with regulating. Recently, a US Congressional special hearing, Building a Science of Economics for the Real World US Congress Subcommittee on Science and Technology, , criticized the performance of macroeconomic models for failing to reproduce temporal patterns of booms and busts observed in the financial crisis.

An economist empanelled at the hearing recommended that policy models be formally audited by the National Science Foundation.

An earlier recommendation by Oreskes et al. NLTS facilitates well-conducted evidentiary scientific inquiry by providing a col- lection of mathematically rigorous procedures that help practitioners to extract infor- mation on real-world dynamics from observed data that often have a complex, highly variable and random appearance.

Applied science disciplines conventionally presume that apparent randomness of volatile data must result from exogenous shocks to inher- ently stable dynamic systems Feder, ; Uusitalo et al. Alternatively, the theory of randomness teaches us that mathematically random output can be generated by both physically random indeterministic and physically nonrandom deterministic processes Horan, , and breakthroughs in nonlinear dynamics demonstrate that parsimonious determin- istic models can produce surprisingly irregular and complex behaviour Glendinning, The possibility of deterministic volatility should not be surprising.

Many essential biophysical processes exhibit strong patterns of dynamic behaviour to align with en- vironmental regularities.

Eating and sleeping in animals including humans , as well as leaf movements and photosynthetic reactions in plants, exhibit built-in circadian roughly hour rhythms that sunlight adjusts to the local environment. Tidal tran- sitions exhibit a tidal Climate exhibits regular diurnal and seasonal cycles, quasi-periodic cycles e.

El Nino and highly irregular cycles e. NLTS allows us to replace presumption of stochasticity with rigorous empirical evidence. The data themselves can serve as our initial guide for ascertaining whether observed volatility is driven by stochastic or deterministic nonlinear real-world dynam- ics. This distinction matters critically both for theory and its practical application. For example, the efficient-markets hypothesis in economics is based on the presumption that market instability results from exogenous shocks to otherwise stable markets.

The hypothesis holds that markets tend to equilibrate in response to these shocks as economic agents process all available information in adjusting supply to demand conditions Fama, Observed volatility reflects corrective supply and demand ad- justments.

The hypothesis supports laissez faire market policies that do not interfere with corrective adjustments. The failure of the efficient-markets hypothesis to predict the financial crisis awoke the profession to another possibility: real-world markets do not naturally equi- librate but may be inherently unstable. In striving to understand the economics of market instability, economists returned to the earlier work of Minsky , who developed the financial instability hypothesis explaining how financial booms systematically breed their own busts see also Economist, b.

According to this hypothesis, markets do not provide a natural corrective mechanism, and public intervention should be geared to smoothing systematic boom and bust cycles, and buffering their negative impacts on consumers and producers Huffaker et al.

But problems arise when they mistake the map for the territory. We investigate how complex behaviour can arise from a simple deterministic nonlinear specification Chapter 2 , how system dynamics can be reconstructed from a single solution variable with phase space reconstruction techniques Chapter 3 and how tools from nonlinear dynamics can be used to measure characteristics of reconstructed dynamics Chapters 4 and 5.

Phase space records the level of system state variables at each point in time. For example, assume that a dynamic system is fully expressed with two state variables xt and yt that change with time t. Phase space plots yt against xt , so that one point could be their values in the year x , y Running a line through this point and past and future points results in a phase space trajectory depicting the evolution of the system through time.

In dissipative dynamic systems, trajectories converge to a subset of phase space where they oscillate aperiodically along an attractor — a geomet- ric structure with noticeable regularity — forever after Brown, Consequently, an attractor encapsulates the long-term dynamic behaviour of the system. Prior to the s, researchers assumed that data on all system variables were needed to construct phase space representations of system dynamics.

This is problematic in practice be- cause we cannot reasonably identify all of the variables at work in real-world dynamic systems, and we might not be able to adequately measure some variables that we can identify. Researchers then discovered that phase space dynamics could be recon- structed from time series data on a single variable using delay coordinates Breeden and Hubler, As a result, we can potentially reconstruct real-world system dy- namics from time series data on a single observed variable.

Consider a simple preliminary example of phase space reconstruction using data on snowshoe hares prey and lynx predator collected by the Hudson Bay company in Canada from to Odum, The time series for each population cycles through time Figure 1.

System dynamics are portrayed in phase space by plotting lynx against hares at each point in time Figure 1.

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