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Home , Cantor set, Fractal dimension

LINDENMAYER SYSTEMS

November 22, 2013 Rolf Pfeifer Rudolf M. Füchslin

RECAP HIDDEN MARKOV MODELS What Letter Is Written Here? What Letter Is Written Here? What Letter Is Written Here? The Idea Behind Hidden Markov Models

First letter: Maybe „a“, maybe „q“ Second letter: Maybe „r“ or „v“ or „u“ Take the most probable combination as a guess! Hidden Markov Models

Sometimes, you don‘t see the states, but only a mapping of the states. A main task is then to derive, from the visible mapped of states, the actual underlying sequence of „hidden“ states. HMM: A Fundamental Question

What you see are the observables. But what are the actual states behind the observables? What is the most probable sequence of states leading to a given sequence of observations? The Viterbi-Algorithm

We are looking for indices M1,M2,...MT, such that P(qM1,...qMT) = Pmax,T is maximal. 1. Initialization 1()ib i i k 1  (i ) 0 1

2. (1  t  T-1) t1(j ) max( t ( i ) a i  j ) b j  k i t1 (j ) i : ( i ) a max. t1 t i j

3. Termination Pimax,TT max( ( ))

qmax,T q i:  T ( i ) max.

4. Backtracking MMt t11() t Efficiency of the Viterbi Algorithm

• The brute force approach takes O(TNT) steps. This is even for N = 2 and T = 100 difficult to do. • The Viterbi – algorithm in contrast takes only O(TN2) which is easy to do with todays computational means. Applications of HMM

• Analysis of handwriting. • Speech analysis. • Construction of models for prediction.

Only few processes are really Markov processes (neither writing nor speech is), but often, models based on Markov processes are good approximations. RECAP FRACTALS Natural

Geometry in text books Geometry in nature Fractals: Informal Definition

• Termed coined by • Geometry without smoothness  Structure on all scales (detail persists when zoomed arbitrarily) • Geometrical objects generally with non-integer • Self-similarity (contains infinite copies of itself)

Fractals in the Human Body

Kidney

Lung

Cortical (?) The Length of Borders

• : of war between two adjacent countries proportional to length of border? • Checking the theory required gathering data about border lengths. • Surprising finding: There are strongly varying numbers in the literature. The Border of Great Britain The Border of Great Britain

The closer you look, the longer the border. And the growth doesn‘t stop! A Slightly Different View on Dimension One-Dimensional Objects

 1  Nc 1      : Diameter of disk N : Number of disks

c1 : Constant Two-dimensional Objects

N  ?  : Diameter of disk N : Number of disks

c2 : Constant Two-dimensional Objects

2  1  Nc 2      : Diameter of disk N : Number of disks

c2 : Constant Definition of the Dimension

D  1  Nc  The number of disks    necessary for covering  : Diameter of disk a structure grows with shrinking λ. N : Number of disks c : Constant D:

log(N ( )) log( c ) log( N (  )) log( N (  )) D lim  lim   lim 011   0   0 log( ) log( ) log( )  Example: The Sierpinski Triangle

Construction of the Sierpinski-triangle

A Sierpinski triangle contains a whole copy of itself in its parts.

Example: The Sierpinski Triangle

Construction of the Sierpinski-triangle

log(N ( )) D lim 0 log( ) Example: The Sierpinski Triangle

Hausdorff Dimension: log(3n ) log(3) D  lim   1.585 n 1 log( ) log(2) 2n of Sierpinski triangle

2 2 n  1  AN lim3  n  0 n  2  Boundary length of ST: 1 L3 KN  lim3 K 3n   n 2n Example: Cantor Dust

Take out the middle third! log(N ( )) D lim 0 log( ) Example: Cantor Dust

Take out the middle third! log(2n ) log(2) D  lim  n 1 log( ) log(3) 3n 1 L lim(no elements = 2n ) (length element = ) n 3n Example: The Mandelbrot Example: The

x x22  y  a z z2 c n1 n n nn1 y2 x y b zc0, C n1 n n 0 z x  iy, c  a  ib The Mandelbrot Set Three-Dimensional L-Systems

Compressibility

• The Mandelbrot looks complex. • The algorithm describing the Mandelbrot-set is very short. •  Procedures generating fractal structures give a very compressed form of storing complex-looking shapes. • Directly storing these pictures is actually impossible.

2 znn1  z c

zc0 0, C Self-Similarity and Scale-Invariance

• “When each piece of a shape is geometrically similar to the whole, both the shape and the cascade that generate it are called self-similar.” (B. Mandelbrot) • Contains infinite copies of itself • Scaling/Scale = The value measured for a property does not depend on the resolution at which it is measured • Two types of invariance: - Geometrical - Statistical Fractals in Reality

• Strict self-similarity is mostly found in mathematical examples. • Statistical interpretation of self-similarity: If a part of system can be zoomed up, and this part shows the same statistical properties as the whole system, one speaks of self-similarity. • In reverse (and more important), if coarse-graining does not change the statistical properties of a system, one speaks of self-similarity. Non-Fractal

Random distribution of spheres with uniform radius. Fractal

Random distribution of spheres with varying random radius ( distribution). Self-Similarity and Coarse-Graining

Coarse graining = or formation of blocks with averaged properties Self-Similarity in Reality

ρ= 0.5 ρ= 0.55 SELF SIMILARITY

ρ= 0.6 Physically important in the description of ρ= 0.7 phase transition. of Time Series

Univ. Zürich: G. Wieser, P. F. Meier, Y. Shen, HR. Moser. R. Füchslin

EEG EEG during epileptic seizure

Some statistical measures such as the fractal D2 decrease shortly before and during an epiliptic seizure. D2 can be used as a diagnostic measure. Random Numbers

• Question: Is a random number self similar? LINDENMAYER-SYSTEMS: STUDYING DEVELOPMENT USING FORMAL LANGUAGES A Real Puzzle

• Nature is full of well-structured objects. • These objects are not assembled using global control and a blue-print, but emerge from local behavior.

External control Self-organization Self-Assembly is Powerful, but ….

Even if self-assembly processes may lead to non-trivial and finite structures with global shape and mesoscopic induced by microscopic interactions, it is not the way how nature works.

Paul. W. Rothemund Developmental Representation vs. Blueprint

• All higher living organism develop from a fertilized egg (a zygote) into their adult form. This process is, at least to a large degree, controlled by their respective genome. • Is it that the genome contains a sort of "blueprint" of the organism?

Developmental Representation vs. Blueprint

• All higher living organism develop from a fertilized egg (a zygote) into their adult form. This process is, at least to a large degree, controlled by their respective genome. • It is NOT TRUE that the genome contains a sort of "blueprint" of the organism. Rather, the genome contains instructions which lead to molecules that in the interaction with the environment lead to organisms.

Developmental Representation vs. Blueprint

Developmental process is influence by: • An initial seed. • The (probably time-dependent) interactions of the building blocks of a body • The environment and the physical and chemical laws ruling this environment. Developmental representations are iterative in the sense that they tell you how to proceed if there is already something there. Developmental Representation vs. Blueprint

The genome does not contain all the information it needs to build your body.

Development requires embodiment!

Instructions for construction = Developmental representation + Laws of the environment Formal Languages Are Not Enough

• How to describe development by a formal system? • Problem: The languages we know do not necessarily lead to globally structured outcomes with repetitive . • Reason: External decision of location where a replacement rule is applied. On Growth and Form: L-Systems

• The patterns observed in multicellular algae are the result of developmental processes • Mathematical formalism introduced in 1968 by Aristid Lindenmayer. • Productions are rewriting rules which state how new symbols (or cells) can be produced from old symbols (or cells)

L-Systems / Rewriting Systems

• Lindenmayer systems belong to the general class of parallel grammars or parallel rewriting systems. • Difference to grammars as we know them: In a parallel rewriting system, rules are applied to all possible instances simultaneously.  L-systems are subsets of languages. • Most practical L-systems are related to context-free languages. • Context-free grammars suit the emulation of maturation and division. A   Context-free language AVV,()     The as an L-System

Non-Terminals (variables): AB , Terminals (constants) : none Start : A Rules : A ABA B BBB The Cantor Set as an L-System

Non-Terminals (variables): AB , Terminals (constants) : none Start : A Rules : A ABA B BBB

1. A 2. ABA 3. ABABBBABA 4. ABABBBABABBBBBBBBBABABBBABA 5......

Anabaena Catenula

A Bio-Inspired L-System

Anabeana catenula: Two types of polar cells, photosynthesis and nitorgen fixation

Variables : AABB , , , Constants : none Start : A Rules : A AB A BA BA BA Visualization: Turtle Graphics

+ -

F

Move forward by Rotate by angle δ Rotate by angle -δ distance F

Bracket notation: [ : Store position and direction ] : Go back to position and direction of matching [ Visualization; Turtle Graphics

Simple system:

Axiom (Start): F Rule: F→F[-F][+F] Angle: 30° More Plants

Professional Visualization Generalizations

? Generalizations

• Stochasticity • Contex sensitive rules • Delay times • Reaction-diffusion systems Homology of Structure Potential Advantages of Development

• Compact description • Supports symmetry • Supports modularity • Supports reuse of mechanisms in different contexts. • Supports scalability • Decentralized control by self-orgnaization and parallelism. • Turns out to be robust • Enables adaptivity • Change of structure can be achieved by small changes. • Change of structure by change of timing (heterochrony). • Evolvability