User:Salix alba/Bios theory

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Bios theory, little quoted theory developed by H. Sabelli and colleagues, attempts to characterise the behavior of certain nonlinear dynamical systems that are sensitive to initial conditions and generate novelty. The theory states that causal processes can generate new patterns, not only random or stochastic processes. Biotic patterns are found in physical, biological and psychological processes ^[1], including heartbeat intervals and other physiological processes, the distribution of galaxies along the z (time-space) axis, the wave function as described by Schrödinger equation, meteorological time series, economic data. Biotic patterns are aperiodic cycles demonstrating creativity (diversification, novelty, asymmetry and hence lower entropy) and causation (pattern in the series of differences between consecutive terms, partial autocorrelation, sensitivity to initial conditions) and which may be bounded (e.g. heartbeat intervals, shorelines) or diffusive, and may or may not show high frequency chaotic oscillations. Many aperiodic processes suspected to be the product of chance or the signature of natural chaos appear to be biotic. Bios seems to be a generic, widespread pattern of natural processes, extending from physics to psychobiology. Bios is generated mathematically by feedback processes ^[2],^[3].

1 History
2 Mathematical bios
3 Biotic motion
4 Criticisms of bios theory
5 Application
6 See also
7 References
- 7.1 Academic papers
- 7.2 Semitechnical and popular works
8 External links

[edit] History

RRI (heart rate intervals), bios and chaos (generated with equation written above)

Biotic pattern was first discovered in the study of the heart rate variation, and it was defined when it was possible to reproduce it with mathematical recursions ^[4]. Bios has been often identified as chaos because data is often detrended before analysis, and difference of biotic data is chaotic. This elementary but unexpected fact prompted Louis H. Kauffman of the Department of Mathematics, University of Illinois at Chicago, and Hector Sabelli of the Rush University, to define a mathematical pattern that they call bios. Also, many biotic series were regarded as random walks because there was no deterministic mathematical model that could describe them. Much of the mathematics of bios theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand.

The term bios comes from the greek: bio-, comb. form of bios "life, course or way of living" (as opposed to zoe "animal life, organic life"). It was adopted by physician Hector Sabelli for this theory. Stuart Kauffman, who also works in complex systems, has also used the name BIOS for a company he founded, but the use is completely independant.

[edit] Mathematical bios

Mathematical bios was first generated with the recursion

$A(t+1) = A(t) + g*sin(A(t)) \,.$

When g is kept constant, then, depending on its value, this recursion generates either a steady state, periodicity, chaos, bios or infinitation (output increases in size toward infinity). Without a conserved term, A(t), this recursion cannot produce bios. This recurrence relation is related to the circle map.

Steady state, bifurcations, chaotic phase, and bios in development of recursive equation. Note that y-axis does not show expanding series, so that bifurcations and chaos could be seen.

Transition from chaos to bios (detail from the image above with y-axis showing the full range of the series and x-axis focusing on the transition/expansion of the series).

[edit] Biotic motion

In order to classify the behavior of a system as biotic, the system must exhibit the following properties:

it must be sensitive to initial conditions
it must generate temporal patterning
it must generate complexity
it must generate diversity
it must generate asymmetry
it must generate novelty

Sensitivity to initial conditions means that two points in such a system may move in vastly different trajectories in their phase space even if the difference in their initial configurations is very small. This is explained in more detail in Butterfly effect.

Time-limited patterns (that some other authors call drift, disrupted large-scale structures, or complexes) detected in biotic series by recurrence plots and wavelet plots. These patterns are similar to those observed for 1/f noise, in contrast to stationary random, periodic and chaotic patterns.

Diversity means increase in Standard Deviation with time.

Asymmetry is evident in the distribution of biotic series.

For the understanding of novelty, we have to understand recurrence plots first.

Recurrence plot enables us to investigate the $m$ -dimensional phase space trajectory through a two-dimensional representation of its recurrences. Such recurrence of a state at time $i$ at a different time $j$ is pictured within a two-dimensional squared matrix with black and white dots, where black dots mark a recurrence, and both axes are time axes. This representation is called recurrence plot.

Novelty was first measured with the recurrence methods of Zbilut and Webber^[5]. Due to change in the definition of recurrence in the RQA 5.2^[6], Sabelli made a distinction between the two definitions, calling former isometry recurrence (or shorter isometry) and later similarity recurrence. For the definition of novelty, definition of isometry recurrence is necessary.

[edit] Isometry recurrence

Recurrence (isometry) plot can be mathematically expressed as

$\mathbf{R}(i,j) = \Theta(\varepsilon - |(||\vec{x}(i)|| - ||\vec{x}(j)||)|), \quad \vec{x}(i) \in \Bbb{R}^m, \quad i, j=1, \dots, N,$

where $N$ is the number of considered states $\vec{x}(i)$ , $\varepsilon$ is a threshold distance, $|| \cdot ||$ a norm (Euclidean norm), $\Theta( \cdot )$ the Heaviside step function, and

$\vec{x}(i) = (u(i), u(i), \ldots, u(i(m-1)),$

where $u (i)$ is the time series, $m$ the embedding dimension.

The difference between isometry and similarity recurrences can be expressed as follows:

in similarity recurrence $\vec{x}(i)\approx \vec{x}(j),\,$

in isometry recurrence $||\vec{x}(i)||\approx ||\vec{x}(j)||,\,$

which means that for the similarity recurrence both the direction and length of vectors are being calculated, while for isometry recurrence only length of vectors is being calculated.

[edit] Novelty

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Novelty can be detected in the series for m > 1. The time series x where total number of isometric points is smaller than the total number of isometric points in the series x after shuffling, has a feature called novelty.

Novelty can be measured with recurrence and embedding plots.

[edit] Isometry Recurrence plots

Different series show different patterns when presented in recurrence plots. Random and chaotic series produce uniform plots. Periodic series produce periodic plots. Biotic series produce plots with distinct complexes that are lost after shuffling the data.

Recurrence plot (embedding 10) of longitude of a part of a British coastline

Recurrence plot (embedding 10) of mathematically generated bios.

Image:Recurrence plot of mathematicaly generated chaos.JPG

Recurrence plot (embedding 10) of mathematically generated chaos.

Total number of calculated isometries as percent of the number of total possible isometries is a useful quantification that is used in embedding plots.

[edit] Embedding plots

Embedding plots display quantifications of recurrence plots^[7] of the original series (bold line) and its shuffled copy (thin line) on many embedding dimensions. Each point in embedding plot represents the number of isometries (as percent of all possible isometries) calculated in the recurrence plot for the corresponding embedding dimension. When this is done for many embeddings, and also for the shuffled series, few patterns become evident. In general, number of isometries can: decrease with shuffling as observed with chaotic series; increase with shuffling (novelty) as observed with biotic series; periodically increase and decrease in original series reaching the maximum when the number of embedding corresponds with the period of the series, as observed with periodic series.

Embedding plot for time series generated with the process equation with g=4.43 (chaos) r=10% of the range. Shuffling the series decreases the number of isometries.

Embedding plot for time series generated with the process equation with g=4.65 (bios) r=10% of the range. Shuffling the series increases the number of isometries (novelty).

Embedding plot of the sine wave. Shuffling the series eliminates periodicity.

[edit] Criticisms of bios theory

One criticism of bios theory is that it represents simply a diffusion of a chaotic system. This criticism is disputed by proponents of the theory, on the ground that bios can also be bounded (as is demonstrated with some recursions) and with finding properties that define a bounded bios in the heart-rate interval series (RRI).

What characterizes bios is not diffusion but the generation of novelty, which is absent in non-diffusive chaos but is present in non-diffusive bios (it is independent from diffusion). Novelty can be easily observed in embedding plots.

Bios is distinguished from chaos by its creative features: temporal patterning, novelty^[8], complexity, diversity, and asymmetry.

[edit] Application

Bios theory is being applied in mathematics, medicine^[9], psychology, economy, etc.

Rush University is implementing this theory in medicine and psychology.

[edit] See also

Bifurcation theory
Complexity
Dynamical system
Fractal
Chaos theory
Edge of chaos
Recurrence plot
Mitchell Feigenbaum
Predictability
Stuart Kauffman, un-connected to this project. Also works in complex systems and founded the unrelated BIOS Group.

[edit] References

[edit] Academic papers

↑ Sabelli H., Sugerman A., Kovacevic L., Kauffman L., Carlson-Sabelli L., Patel M., and Konecki J. (2005) Bios Data Analyzer. Nonlinear Dynamics, Psychology and the Life Sciences. 9(4):505-38.
↑ Chirikov B. V., Lieberman M. A., Shepelyansky D.L., and Vivaldi F. (1985) A Theory of Modulational Diffusion. Physica 14D: 289-304.
↑ Kauffman, L. and Sabelli, H. (2003) Mathematical Bios. Kybernetes 31: 1418-1428.
↑ Kauffman, L. and Sabelli, H. (1998) The Process equation. Cybernetics and Systems 29: 345-362
↑ Zbilut, J.P. and C.L. Webber. (1992). Embeddings and delays as derived from quantification of recurrence plots. Physics letters A 171: 199-203
↑ Sabelli, H. (2001). Novelty, a Measure of Creative Organization in Natural and Mathematical Time Series. Nonlinear Dynamics, Psychology, and Life Sciences. 5: 89-113.
↑ A. Sugerman and H. Sabelli. Novelty, Diversification And Nonrandom Complexity Define Creative Processes. Kybernetes 32: 829-836, 2003
↑ Hector C. Sabelli, Linnea Carlson-Sabelli, Minu K. Patel, Joseph P. Zbilut, Joseph V. Messer, and Karen Walthall, Psychocardiological Portraits: A Clinical Application of Process Theory. In Chaos theory in Psychology (1995), F. D. Abraham and A. R. Gilgen (Eds). Greenwood Publishing Group, Inc., Westport, CT. pp 107-125.