advertisement

Fractals overview

55 %
45 %
advertisement
Information about Fractals overview
Education

Published on September 14, 2007

Author: Rajinder

Source: authorstream.com

advertisement

Slide 1: Introduction to Fractals Larry S. Liebovitch Florida Atlantic University Center for Complex Systems and Brain Sciences Center for Molecular Biology and Biotechnology Department of Psychology Department of Biomedical Sciences Lina A. Shehadeh Copyright 2003 by Larry S. Liebovitch How fractals CHANGE the most basic ways we analyze and understand experimental data. : How fractals CHANGE the most basic ways we analyze and understand experimental data. Slide 3: Non-Fractal Slide 4: Fractal Slide 5: Non - Fractal Size of Features 1 cm 1 characteristic scale Slide 6: Fractal Size of Features 2 cm 1 cm 1/2 cm 1/4 cm many different scales Slide 7: Fractals Self-Similarity Slide 8: Water Land Water Land Water Land Self-Similarity Pieces resemble the whole. Slide 9: Sierpinski Triangle Slide 10: Branching Patterns blood vessels Family, Masters, and Platt 1989 Physica D38:98-103 Mainster 1990 Eye 4:235-241 in the retina air ways in the lungs West and Goldberger 1987 Am. Sci. 75:354-365 Slide 11: Blood Vessels in the Retina Slide 12: PDF - Probability Density Function HOW OFTEN there is THIS SIZE Straight line on log-log plot = Power Law Slide 13: Statistical Self-Similarity The statistics of the big pieces is the same as the statistics of the small pieces. Slide 14: Currents Through Ion Channels Slide 15: Currents Through Ion Channels Slide 16: Currents Through Ion Channels ATP sensitive potassium channel in cell from the pancreas Gilles, Falke, and Misler (Liebovitch 1990 Ann. N.Y. Acad. Sci. 591:375-391) 5 sec 5 msec 5 pA FC = 10 Hz FC = 1k Hz Slide 17: Closed Time Histograms potassium channel in the corneal endothelium Number of closed Times per Time Bin in the Record Liebovitch et al. 1987 Math. Biosci. 84:37-68 Closed Time in ms Slide 18: Closed Time Histograms potassium channel in the corneal endothelium Number of closed Times per Time Bin in the Record Liebovitch et al. 1987 Math. Biosci. 84:37-68 Closed Time in ms Slide 19: Closed Time in ms Number of closed Times per Time Bin in the Record Closed Time Histograms potassium channel in the corneal endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68 Slide 20: Closed Time Histograms potassium channel in the corneal endothelium Number of closed Times per Time Bin in the Record Liebovitch et al. 1987 Math. Biosci. 84:37-68 Closed Time in ms Slide 21: Fractals Scaling Slide 22: Scaling The value measured depends on the resolution used to do the measurement. Slide 23: How Long is the Coastline of Britain? Richardson 1961 The problem of contiguity: An Appendix to Statistics of Deadly Quarrels General Systems Yearbook 6:139-187 Log10 (Total Length in Km) AUSTRIALIAN COAST CIRCLE SOUTH AFRICAN COAST GERMAN LAND-FRONTIER, 1900 WEST COAST OF BRITIAN LAND-FRONTIER OF PORTUGAL 4.0 3.5 3.0 1.0 1.5 2.0 2.5 3.0 3.5 LOG10 (Length of Line Segments in Km) Slide 24: Genetic Mosaics in the Liver P. M. Iannaccone. 1990. FASEB J. 4:1508-1512. Y.-K. Ng and P. M. Iannaccone. 1992. Devel. Biol. 151:419-430. Fractal Kinetics : Kinetic Rate Constant: k = Prob. to change states in the next ?t. Effective Kinetic Rate Constant: keff = Prob. to change states in the next ?t, given that we have already remained in the state for a time keff. k = Pr ( T=t, t+?t | T > t ) / ?t eff eff age-specific failure rate = – d dt ln P(t) P(t) = cumulative dwell time distribution Fractal Kinetics 70 pS K+ ChannelCorneal Endothelium : 70 pS K+ ChannelCorneal Endothelium Liebovitch et al. 1987 Math. Biosci. 84:37-68. effective time scale t eff in msec 100 1000 10 1 1 10 100 1000 k eff = A t eff 1-D Fractal Approach : Fractal Approach New viewpoint: Analyze how a property, the effective kinetic rate constant, keff, depends on the effective time scale, teff, at which it is measured. This Scaling Relationship: We are using this to learn about the structure and motions in the ion channel protein. Slide 28: one measurement: not so interesting slope Scaling Logarithm of the measuremnt Logarithm of the measuremnt one value Logarithm of the resolution used to make the measurement Logarithm of the resolution used to make the measurement scaling relationship: much more interesting Slide 29: Fractals Statistics Slide 30: Not Fractal Slide 31: Not Fractal Slide 32: Gaussian Bell Curve “Normal Distribution” Slide 33: Fractal Slide 34: Fractal Slide 35: Mean Non - Fractal More Data pop Slide 36: The Average Depends on the Amount of Data Analyzed Slide 37: The Average Depends on the Amount of Data Analyzed each piece Slide 38: Ordinary Coin Toss Toss a coin. If it is tails win $0, If it is heads win $1. The average winnings are: 2-1.1 = 0.5 1/2 Non-Fractal Slide 39: Ordinary Coin Toss Slide 40: Ordinary Coin Toss Slide 41: 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 $2N. With probability 2-N we win $2N. H $2 TH $4 TTH $8 TTTH $16 The average winnings are: 2-121 + 2-222 + 2-323 + . . . = 1 + 1 + 1 + . . . = Fractal Slide 42: St. Petersburg Game (Niklaus Bernoulli) Slide 43: St. Petersburg Game (Niklaus Bernoulli) Slide 44: Non-Fractal Log avg density within radius r Log radius r Slide 45: Fractal Log avg density within radius r Log radius r .5 -1.0 -2.0 -1.5 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 0 -2.5 0 Meakin 1986 In On Growthand Form: Fractal and Non-Fractal Patterns in Physics Ed. Stanley & Ostrowsky, Martinus Nijoff Pub., pp. 111-135 Slide 46: Electrical Activity of Auditory Nerve Cells Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52 voltage time action potentials Slide 47: Electrical Activity of Auditory Nerve Cells Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52 2 Count the number of action potentials in each window: 6 3 1 5 1 Firing Rate = 2, 6, 3, 1, 5,1 Divide the record into time windows: Slide 48: Electrical Activity of Auditory Nerve Cells Teich, Johnson, Kumar, and Turcott 1990 Hearing Res. 46:41-52 Repeat for different lengths of time windows: 8 4 6 Firing Rate = 8, 4, 6 Slide 49: Electrical Activity of Auditory Nerve Cells Teich, Jonson, Kumar, and Turcott 1990 Hearing Res. 46:41-52 0 The variation in the firing rate does not decrease at longer time windows. 4 8 12 16 20 24 28 70 60 80 90 100 120 130 140 110 150 T = 50.0 sec T = 5.0 sec T = 0.5 sec FIRING RATE SAMPLE NUMBER (each of duration T sec) Slide 50: Fractals Power Law PDFs Heart Rhythms : Heart Rhythms Inter-event Times : Inter-event Times Episodes of Ventricular Tachycardia (v-tach) t 1 t 2 t 3 t 4 t 5 time -> Cardioverter Defibrillator Patient #33 : Interval (in days) Relative Frequency Patient #33 Patient #53 : Interval (in days) Relative Frequency Relative Frequency = (3.2545) Interval-1.3664 10 3 10 2 10 1 10 0 10 -1 10 -2 10 -3 10 -4 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 Patient #53 6 Patients : 6 Patients Liebovitch et al. 1999 Phys. Rev. E59:3312-3319. Inter-arrival Times of E-mail Viruses : Inter-arrival Times of E-mail Viruses t 1 t 2 t 3 t 4 t 5 time -> Liebovitch and Schwartz 2003 Phys. Rev. E68:017101. AnnaKournikova "Hi: Check This!” AnnaKournikova.jpg vbs. Magistr Subject, body, attachment from other files: erase disk, cmos/bios. Klez E-mail from its own phrases: infect by just viewing in Outlook Express. Sircam “I send you this file in order to have your advice.” E-mail Viruses : E-mail Viruses 20,884 viruses 153,519 viruses E-mail Viruses : E-mail Viruses 413,183 viruses 781,626 viruses Determining the PDFfrom a Histogram : Determining the PDFfrom a Histogram Bins ?t Small Good at small t. BAD at large t. Bins ?t Large BAD at small t. Good at large t. Determining the PDF : Determining the PDF Liebovitch et al. 1999 Phys. Rev. E59:3312-3319. Solution: Make ONE PDF From SEVERAL Histograms of DIFFERENT Bin Size Choose ?t = 1, 2, 4, 8, 16 … seconds ?t = bin size Determiningthe PDF : Determiningthe PDF New multi-histogram Standard fixed ?t Slide 62: Fractals Summary Summary of Fractal Properties : Summary of Fractal Properties Self-Simialrity Pieces resemble the whole. Summary of Fractal Properties : Summary of Fractal Properties Scaling The value measured depends on the resolution. Summary of Fractal Properties : Summary of Fractal Properties Statistical Properties Moments may be zero or infinite. Slide 66: 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 Statistics is NOT a dead science. Fractals CHANGE the most basic ways we analyze and understand experimental data. : Fractals CHANGE the most basic ways we analyze and understand experimental data. Fractals 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). No Bell Curves No Moments No mean ± s.e.m. References: : References: Fractals and Chaos and Simplified for the Life Sciences Larry S. Liebovitch Oxford Univ. Press, 1998 The Mathematics and Science of Fractals Larry S. Liebovitch and Lina Shehadeh www.ccs.fau.edu/~liebovitch/larry.html CD ROM NSF DUE-9752226 DUE-9980715

Add a comment

Related presentations

Related pages

Fractal - Wikipedia

A fractal is a mathematical set that exhibits a repeating pattern that displays at every ... Images of fractals can be created by fractal generating programs.
Read more

Fractal Science Kit - Mandelbrot Fractal Overview

Discussion of Mandelbrot fractals, Julia fractals, Convergent fractals, Newton fractals, Orbit Traps, Apollonian Gasket, Circle Inversion, Schottky Group ...
Read more

Fractal Science Kit - Orbital Fractal Overview

Orbital Fractal Overview. The Fractal Science Kit fractal generator Orbital fractals collect statistics during the orbit of a fractal formula and use these ...
Read more

Fractal Gallery Friedrich Lohmueller/ Fraktal-Galerie ...

fractals, 3D computer graphics, fractal images, gallery by Friedrich A. Lohmueller
Read more

Fractals: A Brief Overview - Brothers Technology

Fractals demonstrate a fourth type of symmetry; they possess “self-similarity.” Self-similar objects appear the same under magnification. They
Read more

Fractals of the Mists - Guild Wars 2 Wiki (GW2W)

Fractals of the Mists is a special type of dungeon that consists of an array of mini-dungeons called fractals, where each fractal has its own story and ...
Read more

Overview | Fractal Analytics

Overview. Analytics that creates Insight, Impact and Innovation. Today’s global leaders understand analytics is the new secret weapon to drive ...
Read more

FRACTALS | F6S

Innovative ideas for the farm of the future. FRACTALS's Mentors, Startups, Funding and Jobs. Located in Novi Sad, Serbia
Read more

What are Fractals? – Fractal Foundation

A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating ...
Read more

Industry: CPG, Financial Services, Insurance, Retail ...

Industry. Overview; Consumer Packaged Goods; Financial Services; Healthcare; Insurance; Retail; Technology; ... Industry Leading companies leverage Big ...
Read more