ISC 6450, Spring 2004, Dr. Larry S. Liebovitch

Methods in Complex Systems

18426 M...F 1:00 PM - 2:50 PM (Complex Systems & Brain Sciences Classroom, Innovation Centre II, Boca Raton

Introduction to Linear and Nonlinear Statistics

Required for the Ph.D. program in Complex Systems and Brain Sciences.
It can also be used as the first semester course needed to fulfill the statistics requirement for graduate Psychology majors. (The LAB is also required for Psych students.)

Larry S. Liebovitch, Ph.D.
Florida Atlantic University
Center for Complex Systems and Brain Sciences
777 Glades Road, Boca Raton, FL 33431
telephone: 561.297.2239, fax: 561.297.2223

If you want to speak with me please telephone, DO NOT SEND E-MAIL (I am overwhelmed with e-mail which I do not have time to read.)


Experimental design and statistical analysis of linear and nonlinear systems. Presents the classical statistical analysis and inference of linear systems that have a small number of noninteracting pieces and how those statistical methods and analysis procedures are different for nonlinear complex systems with many pieces that interact strongly with each other, such as fractals and chaos.


M. R. Spiegel and L. J. Stephens. Schaum's Outlines: Statistics, 3rd Edition. McGraw Hill, New York

M. Hollander and D. A. Wolf. Nonparametric Statistical Nethods. John Wiley and Sons, New York, 1973.
L. S. Liebovtich. Fractals and Chaos Simplified for the Life Sciences. Oxford University Press, New York, 1998.


Attendance: Students are expected to attend all scheduled classes. Should it become necessary for a student to miss a class, the student is responsible for the material covered during that class. It is the responsibility of the student to withdraw from this class, should that status be desired - the instructor cannot withdraw students from the course.

Reading the Textbook: PLEASE read the chapters assigned prior to the class session in which the material will be presented.

Homework Problems: Please try to turn in the homework problems on time. NONE will be accepted after the last meeting of the class.

Grading: The grade will be determined entirely from the homework problems. There will be no exams.


(Chapters in Spiegel textbook)

  1. What is science? (paradigms, observation, theory, and experiments)
  2. Why measure more than once? (mathematical basis, distributions) (Ch. 1-2,6,7)
  3. Characterizing data (moments, central tendency, dispersion) (Ch. 3-5)
  4. Fitting functional relationships to data (least squares, correlation, Hogdes-Lehman, linear, log) (Ch. 13-14)
  5. Review of Homework 1: mean, variance, Gaussian distributions
  6. Designing Experiments (controls, blind, double blind, surrogate end points)
  7. Review of Homework 2: least squares and correlation coefficients
  8. Statistical tests - Gaussian (z, t, F, Chi Square) (Ch. 8-12)
  9. Statistical tests - Nonparametric (rank order, Hodges-Lehman) (Ch. 17)
  10. Review of Homework 3: parametric statistical tests
  11. Analysis of factorial experiment (main effects, interactions, ANOVA) (Ch. 16)
  12. Review of Homework 4: non-parametric tests
  13. Introduction to fractals
  14. Review of Homework 5: ANOVA
  15. Fractal methods of analysis (power spectra, Hurst, F(n), variance, Fano, coeff. var.)
  16. Introduction to chaos (nonlinear dyanmaics)
  17. Chaos methods of analysis (phase space, dimension, Liapunov, power spectra)
  18. Analysis of data from student experiments


Homework #1
1) Compute the pdf (probability density function) of the data in Spiegel problem 2.2 (page 41).
2) Compute the mean, standard deviation of the data in #1.
3) Plot a Gaussian curve, using the mean and standard deviation of #2 on the same pdf as obtained in #1.
4) Compute the 3rd and 4th moments of this data. Compare the 4th moment with 3s4.
5) Splatter ink on a piece of paper. Measure the diameter of 100 ink spots. Compute the pdf of the diameters.

Homework #2
1) Use the data in Spiegel, Table 13.4, Problem 13.10, Page 291. Write a computer program that uses the least squares method to determine the best straight line that represents the function of: a) Weight as a function of Height as well as b) Height as a function of Weight
2) Compute the correlation function r for the two lines in problem #1.
(OPTIONAL) 3) Write a computer program that uses the nonparametric Hodges-Lehmann estimator for the slope and intercept for the two lines in problem #1.

Homework #3
1) Use the Chi-Square test to determine if the data from the first homework (Spiegel Problem 2.2, page 41) is a Gaussian or not.
2) Spiegel Problem 10.9, page 226.
3) Spiegel Problem 10.18, page 233.
4) Spiegel Problem 11.6, page 248-9.
5) Spiegel Problem 11.8, page 250-1.

Homework #4
1) When a series of numbers is uncorrelated the fractal parameter called the Hurst H coefficient is approximately equal to 0.5. We found that H=0.61 in the data from one patient. If the data from the patient was correlated, then when we shuffle it, those correlations should be removed and the value of H should be closer to 0.5. To determine if the H measured from the patient is different from 0.5, we shuffled that data and then measured H. We did this 16 times. In these shuffled data sets we found that H=0.55, 0.56, 0.64, 0.54, 0.51, 0.63, 0.55, 0.48, 0.62, 0.55, 0.54, 0.51, 0.53, 0.54, 0.54, 0.50. Is the H from the patient statistically significantly higher than that of uncorrelated data? (HINT: We found that 3 out of 16 times the value of H from the patient was greater than the uncorrelated randomized data. Use a binomial test to determine if 3 out of 16 is statistically significant).
(OPTIONAL) 2) Perform a rank-sum (Wilcoxon) test on the data from Table 1. page 69 in Hollander and Wolfe to determine if there is a difference in the water permeability between the two groups.

Homework #5
1) Spiegel Problems 16.4, 16.5, and 16.6, pages 374-7.
2) Spiegel Problem 16.11, pages 380-383.
3) Spiegel Problem 16.13, pages 383-7.