Complejidad. Geofisica Biologí a. MacroEconomía. Meteorología Ecología. Psicologia. Dante R. Chialvo

Transcription

1 Complejidad Geofisica Biologí a MacroEconomía Meteorología Ecología Psicologia Dante R. Chialvo dchialvo@ucla.edu

2 La estadistica que aprendimos describe la uniformidad (gaussianas) La naturaleza es NO HOMOGENEA!!!, por donde se la mire Ejemplo: distribución de peso versus distribución de pesos $ una forma muchas formas

3 Fractals Que son? Como estudiarlos? por que nos pueden importar? Material extraido del libro Introduction to Fractals Larry S. Liebovitch Lina A. Shehadeh

4 How fractals CHANGE the most basic ways we analyze and understand experimental data.

5 Non-Fractal

6 Fractal

7 El universo es Fractal

8 Non - Fractal Size of Features 1 cm 1 characteristic scale

9 Fractal Size of Features 2 cm 1 cm 1/2 cm 1/4 cm many different scales

10 Fractals Self-Similarity Auto-similaridad

11 Self-Similarity Pieces resemble the whole. Water Water Water Land Land Land

12 Sierpinski Triangle

13 Branching Patterns blood vessels in the retina Family, Masters, and Platt 1989 Physica D38: Mainster 1990 Eye 4: air ways in the lungs West and Goldberger 1987 Am. Sci. 75:

14 Blood Vessels in the Retina

15 PDF - Probability Density Function HOW OFTEN there is THIS SIZE Straight line on log-log plot = Power Law

16 Statistical Self-Similarity The statistics of the big pieces is the same as the statistics of the small pieces.

17 Currents Through Ion Channels

18 Currents Through Ion Channels

19 Currents Through Ion Channels ATP sensitive potassium channel in cell from the pancreas Gilles, Falke, and Misler (Liebovitch 1990 Ann. N.Y. Acad. Sci. 591: ) F C = 10 Hz 5 sec F C = 1k Hz 5 pa 5 msec

20 Closed Time Histograms Number of closed Times per Time Bin in the Record potassium channel in the corneal endothelium Liebovitch et al Math. Biosci. 84:37-68 Closed Time in ms

21 Closed Time Histograms Number of closed Times per Time Bin in the Record potassium channel in the corneal endothelium Liebovitch et al Math. Biosci. 84:37-68 Closed Time in ms

22 Closed Time Histograms Number of closed Times per Time Bin in the Record potassium channel in the corneal endothelium Liebovitch et al Math. Biosci. 84:37-68 Closed Time in ms

23 Closed Time Histograms Number of closed Times per Time Bin in the Record potassium channel in the corneal endothelium Liebovitch et al Math. Biosci. 84:37-68 Closed Time in ms

24 Fractals Scaling

25 Scaling The value measured depends on the resolution used to do the measurement.

26 How Long is the Coastline of Britain? Richardson 1961 The problem of contiguity: An Appendix to Statistics of Deadly Quarrels General Systems Yearbook 6: Log 10 (Total Length in Km) AUSTRIALIAN COAST CIRCLE SOUTH AFRICAN COAST GERMAN LAND-FRONTIER, 1900 WEST COAST OF BRITIAN LAND-FRONTIER OF PORTUGAL LOG 10 (Length of Line Segments in Km)

27 Genetic Mosaics in the Liver P. M. Iannaccone FASEB J. 4: Y.-K. Ng and P. M. Iannaccone Devel. Biol. 151:

28 Fractal Kinetics Kinetic Rate Constant: k = Prob. to change states in the next t. Effective Kinetic Rate Constant: k eff = Prob. to change states in the next t, given that we have already remained in the state for a time k eff. age-specific failure rate k eff = Pr ( T=t, t+ t T > t eff ) / t = d ln P(t) dt P(t) = cumulative dwell time distribution

29 k eff 70 ps K + Channel Corneal Endothelium effective kinetic rate constant Liebovitch et al Math. Biosci. 84: in Hz 1000" 100" 10" 1-D k eff = A t eff 1" 1" 10" 100" 1000" effective time scale t eff in msec

30 Fractal Approach New viewpoint: Analyze how a property, the effective kinetic rate constant, k eff, depends on the effective time scale, t eff, at which it is measured. This Scaling Relationship: We are using this to learn about the structure and motions in the ion channel protein.

31 Logarithm of the measuremnt one measurement: not so interesting one value Scaling Logarithm of the measuremnt scaling relationship: much more interesting slope Logarithm of the resolution used to make the measurement Logarithm of the resolution used to make the measurement

32 Fractals Statistics

33 Not Fractal

34 Not Fractal

35 Gaussian Bell Curve Normal Distribution

36 Fractal

37 Fractal

38 Non - Fractal Mean pop More Data

39 The Average Depends on the Amount of Data Analyzed

40 The Average Depends on the Amount of Data Analyzed each piece

41 Ordinary Coin Toss Toss a coin. If it is tails win $0, If it is heads win $1. The average winnings are: = 0.5 1/2 Non-Fractal

42 Ordinary Coin Toss

43 Ordinary Coin Toss

44 St. Petersburg Game (Niklaus Bernoulli) Toss a coin. If it is heads win $2, if not, keep tossing it until it falls heads. If this occurs on the N-th toss we win $2 N. With probability 2 -N we win $2 N. The average winnings are: = H $2 TH $4 TTH $8 TTTH $ = Fractal

45 St. Petersburg Game (Niklaus Bernoulli)

46 St. Petersburg Game (Niklaus Bernoulli)

47 Non-Fractal Log avg density within radius r Log radius r

48 Fractal Meakin 1986 In On Growthand Form: Fractal and Non-Fractal Patterns in Physics Ed. Stanley & Ostrowsky, Martinus Nijoff Pub., pp Log avg density within radius r Log radius r

49 Electrical Activity of Auditory Nerve Cells Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52 voltage action potentials time

50 Electrical Activity of Auditory Nerve Cells Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52 Divide the record into time windows: Count the number of action potentials in each window: Firing Rate = 2, 6, 3, 1, 5,1

51 Electrical Activity of Auditory Nerve Cells Teich, Johnson, Kumar, and Turcott 1990 Hearing Res. 46:41-52 Repeat for different lengths of time windows: Firing Rate = 8, 4, 6

52 Electrical Activity of Auditory Nerve Cells FIRING RATE Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46: T = 50.0 sec T = 5.0 sec T = 0.5 sec The variation in the firing rate does not decrease at longer time windows. SAMPLE NUMBER (each of duration T sec)

53 Fractals Power Law PDFs

54 Heart Rhythms

55 Inter-event Times Cardioverter Defibrillator Episodes of Ventricular Tachycardia (v-tach) t 1 t 2 t 3 t 4 t 5 time ->

56 Patient # Relative Frequency =! Relative! Frequency (9.8581) Interval " Interval (in days)

57 Patient #53 Relative! Frequency Relative Frequency =! (3.2545) Interval " Interval (in days)

58 6 Patients Liebovitch et al Phys. Rev. E59:

59 Behavior 59

60 Behavior: is there an average rate for animal motion? Anteneodo and Chialvo, Chaos(2009)

61 Wish a scale free life? stop working hard, scaling is due to inactivity pauses! Anteneodo and Chialvo, Chaos(2009)

62 More on scaling and inactivity pauses Inactivity pauses Anteneodo and Chialvo, Chaos(2009)

63 Inter-arrival Times of Viruses Liebovitch and Schwartz 2003 Phys. Rev. E68: t 1 t 2 t 3 t 4 t 5 time -> AnnaKournikova "Hi: Check This! AnnaKournikova.jpg vbs." Magistr Subject, body, attachment from other files: erase disk, cmos/bios. Klez from its own phrases: infect by just viewing in Outlook Express." Sircam I send you this file in order to have your advice.

64 Viruses 20,884 viruses 153,519 viruses

65 Viruses 413,183 viruses 781,626 viruses

66 Determining the PDF from a Histogram Bins t Small Good at small t. BAD at large t. Bins t Large BAD at small t. Good at large t. PDF t

67 Determining the PDF Liebovitch et al Phys. Rev. E59: Solution: Make ONE PDF From SEVERAL Histograms of DIFFERENT Bin Size PDF(t) = N(t) N t t N(t) = number in [t+ t, t] N t = total number t = bin size Choose t = 1, 2, 4, 8, 16 seconds

68 Determining the PDF PDF y = e-2 * x^ R^2 = LSL algorithm Constant Bins New multi-histogram 10-2 Standard fixed t Values

69 Fractals Summary

70 Summary of Fractal Properties Self-Similarity Pieces resemble the whole.

71 Summary of Fractal Properties Scaling The value measured depends on the resolution.

72 Summary of Fractal Properties Statistical Properties Moments may be zero or infinite.

73 Statistics is NOT a dead science. 400 years ago: Gambling Problems Probability Theory 200 years ago: Statistics How we do experiments. 100 years ago: Student s t-test, F-test, ANOVA Now: Still changing

74 Fractals CHANGE the most basic ways we analyze and understand experimental data. Fractals No Bell Curves No Moments No mean ± s.e.m. Measurements over many scales. What is real is not one number, but how the measured values change with the scale at which they are measured (fractal dimension).