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Using Time Series to Analyze Long-Range Fractal Patterns
Using Time Series to Analyze Long Range Fractal Patterns presents methods for describing and analyzing dependency and irregularity in long time series. Irregularity refers to cycles that are similar in appearance, but unlike seasonal patterns more familiar to social scientists, repeated over a time scale that is not fixed. Until now, the application of these methods has mainly involved analysis of dynamical systems outside of the social sciences, but this volume makes it possible for social scientists to explore and document fractal patterns in dynamical social systems. Author Matthijs Koopmans concentrates on two general approaches to irregularity in long time series: autoregressive fractionally integrated moving average models, and power spectral density analysis. He demonstrates the methods through two kinds of examples: simulations that illustrate the patterns that might be encountered and serve as a benchmark for interpreting patterns in real data; and secondly social science examples such a long range data on monthly unemployment figures, daily school attendance rates; daily numbers of births to teens, and weekly survey data on political orientation. Data and R-scripts to replicate the analyses are available on an accompanying website at

Series Editor Introduction
About the Author
Chapter 1: Introduction
A. Limitations of Traditional Approaches

B. Long-Range Dependencies

C. The Search for Complexity

D. Plan of the Book

Chapter 2: Autoregressive Fractionally Integrated Moving Average or Fractional Differencing
A. Basic Results in Time Series Analysis

B. Long-Range Dependencies

C. Application of the Models to Real Data

D. Chapter Summary and Reflection

Chapter 3: Power Spectral Density Analysis
A. From the Time Domain to the Frequency Domain

B. Spectral Density in Real Data

C. Fractional Estimates of Gaussian Noise and Brownian Motion

D. Chapter Summary and Reflection

Chapter 4: Related Methods in the Time and Frequency Domains
A. Estimating Fractal Variance

B. Spectral Regression

C. The Hurst Exponent Revisited

D. Chapter Summary and Reflection

Chapter 5: Variations on the Fractality Theme
A. Sensitive Dependence on Initial Conditions

B. The Multivariate Case

C. Regular Long-Range Processes and Nested Regularity

D. The Impact of Interventions

Chapter 6: Conclusion
A. Benefits and Drawbacks of Fractal Analysis

B. Interpretation of Parameters in Terms of Complexity Theory

C. A Note About the Software and Its Use



This is coherent treatment of fractal time-series methods that will be exceptionally useful.

Courtney Brown
Emory University

Each analysis is explained, and also the differences between the analyses are explained in a systematic way.

Mustafa Demir
State University of New York, Plattsburgh

This volume offers a nice introduction to the various methods that can be used to discuss long range dependencies in univariate time series data. Koopmans makes a compelling case for these methods and offers clear exposition

Clayton Webb
University of Kansas

This amazing book provides a concise and solid foundation to the study of long-range process. In a short volume, the author successfully summarizes the theory of fractal approaches and provides many interesting and convincing examples. I highly recommend this book.

I-Ming Chiu
Rutgers University-Camden

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