Creative Feedback in Natural and Mental Systems
Hector Sabelli, M.D., Ph. D.
Chicago Center for Creative Development
2400 Lakeview Avenue, Chicago, Illinois, U.S.A. 60614.
Hector_Sabelli@rush.edu
Text of conference intended for a 2001 meeting of
the Indian System Society (cancelled) and delivered at several universities in
India.
Abstract: Here I describe a new
cybernetic concept, bipolar feedback (i.e. feedback that is both positive and
negative). It is a creative process
present in many natural and human systems.
Mathematical recursions that embody bipolar feedback with trigonometric
functions, such as the process equation At+1 = At + k * t
* sin(At), generate a sequence of patterns of increasing complexity:
steady state, periodicity, chaos and bios.
Bios is aperiodic, nonstationary, and highly complex. Biotic patterns are characteristic of many
biological, economic, and meteorological processes. Biotic patterns can be distinguished
from randomness and chaos with newly developed methods that measure
diversification, novelty, nonrandom complexity in time series. The generation of patterns of increasing
complexity by feedback suggests how synergic and conflictual interactions in
nature may contribute to creative evolution at all levels of organization. At
the most basic level, the dialectic of mathematical certainty and quantum
uncertainty may create the physical universe.
This co-creation theory supports novel methods for time series analysis,
and strategies to foster creative behavior.
‘“Who
truly knows, who could here declare whence was born, whence comes this
creation?” Rigveda
Natural
and mental processes are creative. They
produce original patterns and structures with limited life tenure, not only
repetitive forms or ephemeral transients.
Cosmological evolution, human history, and personal development are
exemplary, but simpler biological processes are more readily investigated. In order to study natural creativity, we
have found useful to examine heartbeat
interval series, systematically
examining their properties, and noting how they differ from simpler physical
processes Cardiac timing portrays a complex, creative,
and vital process. Its variation is a
significant indicator of cardiac health; excessive regularity predicts death
within 24 hours. Sequences of heartbeat
intervals never repeat; as language, cardiac timing is productive. Heartbeats
represent the unit of cardiac action. Beat to beat intervals provide a portrait
of a fundamental timing in the organism.
This timing can be precisely measured by the interval between R waves in
the electrocardiogram (RR interval, RRI). It is modulated mainly by the
opposite actions of the accelerating sympathetic nerves and the decelerating
parasympathetic nerves, so patterns can and should be interpreted in terms of
complementary opposites. Action and
opposition are fundamental components of processes at all levels of organization. Heartbeat intervals are influenced by both
simple and complex factors, ranging from temperature to emotions, behavior, and
interactions. Thus heart rate variation reflects our interactions with
the universe that surrounds us.
Multiple
interactions, synergistic and antagonistic, create complex patterns. Indeed, creative processes are so complex
that their pattern is not evident. They
appear erratic, random or chaotic (figure 1 top). One morning I said to myself that if creative processes are
generated by the interaction of opposites, as postulated by dialectics, I
should be able to find them in the wavering time series of heartbeat intervals
by calculating the sine and cosine of each interval. Sine and cosine are complementary opposites, waxing and waning
reciprocally. Five minutes later I was
staring at a Mandala image in the computer screen (figure 1 bottom). Plotting the data in the trigonometric plane
reveals regularity not observable in the Cartesian plane of phase portraits.
Figure 1: Cardiac Mandala and Indian Mandalas. Top: Erratic time series of heart beat
intervals (measured as the interval between R waves in the
electrocardiogram with a precision of 1/128th of a second). Middle: Complement plot of the same data, generated by
calculating the sine and the cosine of each R to R interval (RRI) and
plotting them in a Cartesian plane. The circular form imposed on the data
reveals patterns of transition in the time series, and thereby of
opposition in the process. This
plot of RRIs reveals a surprising regularity. that reminds us of Mandala
archetypes such as Asoka’s chakra in the Indian national flag.
Mandala
symbols are found in almost all civilizations.
Mandalas are spontaneously drawn by young children, and appear in dreams
and doodles [Jung]. Finding a Mandala
in both psychosocial symbols and heartbeat series points to a deep isomorphism
between physical and mental processes.
The Mandala is an archetype, i.e. a widely occurring form present in
both nature and mind. Finding a Mandala
in complement plots of heartbeat series indicates the importance of
complementary opposites as an organizing principle in natural processes.
Correspondingly, Mandalas can be generated by mathematically generated series
that involve positive and negative changes, such as integer biotic series
generated by bipolar feedback. Mandala
forms are not generated by random or chaotic series.
This conjunction of empirical research, mathematical
methodology, and philosophical theory exemplifies the concept of co-creation as
a scientific strategy. The intercourse
of medicine, mathematics and philosophy gave birth to science. Historically, the first numerical law of science
was Pythagoras’ relation between the length of strings and musical
harmony. This indicates a profound
isomorphism between mathematics and nature, and also points to psychobiology as
a foundation for both logic and natural science. The study of complex psychobiological processes offers
perspectives to the understanding of creative natural processes not available
from consideration of non-biological physics.
Pythagoras appears to have learned from Indian mathematicians his notion
of number as a portrait of form, not only quantity. Concepts of nothingness,
unity, duality, triads, and tetrads have been developed by Indian philosophers,
as befits a civilization that has distinguished itself by its mathematical
achievements. In writing these lines,
I am aware of what a great honor it is for me to speak about our mathematical
experiments in India, the land of mathematics, the motherland of Aryabhata and
Ramanujan, and to present trigonometric feedback as a model for dialectic
processes in the country where trigonometry was born, and dialectics achieved
such a high degree of development. I
must ask you to overlook my mathematical naiveté, as I come not only from
another country but also from another field.
I began my work as a physician when the discovery of pharmacological
treatments for mental illness, pioneered by the Indian investigators Sen and
Bose (1931), demonstrated that mind is a creative process originating from the
function of the brain. Mind is matter,
not maya; reality, not illusion. This explains
why psychological archetypes and mathematical science do portray reality.
Our aim is to define, measure, and account for
creative processes as natural and spontaneous, rather than being the result of
chance or of extraterrestrial intervention. There also is a practical significance to this project. Dynamic and statistical analyses demonstrate
creative patterns in biological, economic and meteorological time series. In the case of heartbeat series, there are
marked differences in recordings obtained from healthy subjects, cardiac
patients, and psychiatric patients.
These findings indicate that biotic patterns may be widespread in
natural processes, and point to significant practical applications for their
analysis.
Section I introduces the concepts of bios and
co-creation through empirical studies.
It describes newly developed methods that characterize creative
processes (empirical or computer generated bios) and distinguish them from
random, periodic and chaotic series.
Section II describes mathematical models of bipolar feedback that
generate biotic patterns. Section III
explores how bipolar feedback may contribute to creative evolution.
I. Creative Processes in
Medicine and Economics
Operational definition of creative processes:
We define a process as creative when it generates
new and more complex organization with a limited life span. In the case of time series, this implies
episodic patterns (Sabelli et al., 1995), diversity, novelty (Sabelli, 2001),
nonrandom complexity [Sabelli, accepted for publication), and
irreversibility. None of these
properties are observed in periodic, chaotic, or random series (Table 1).
Table
1. Process Causation and Pattern
|
|
Probabilistic |
Deterministic |
Natural
|
Time
series characteristics |
Conser-vative
|
Uniform
|
Mechanical |
Electron
orbitals |
Unchanging
phase portrait, otherwise as attractive processes |
Attractive
|
Mean
of Gaussian (normal) random |
Point Periodic Chaotic |
Resting
pendulum Energy
driven pendulum Chemical
oscillations |
Uniform
pattern Converging
phase portrait Recurrent Low
nonrandom complexity |
Creative
|
Statistical
noise: Brownian
motion Pink
(1/f ) noise |
Biotic Parabiotic |
Biotic
heartbeat interval series Parabiotic
economic series |
Episodic
pattern (complexes) Diversifying
phase portrait Novelty High
nonrandom complexity |
Creative processes diversify, generating new
patterns, and expanding their phase space volume. Mechanical as well as random processes are conservative, i.e.
they maintain their pattern, phase space volume, and dimensionality. Attractive processes converge to an
attractor, which can be equilibrium (point attractor), a regular cycle
(periodic attractor), or an aperiodic fluctuation (strange attractor, often
called chaotic). Diversification
differentiates creative processes from random, mechanical, and chaotic
processes (figure 2).
Random, periodic, chaotic, and biotic series differ
also in many other tests, to be described later, and summarily presented in
figure 3. Simple time graphs (left
column show marked differences among these series. Recurrence (plotted in the second column for a range of
embeddings from 1 to 200) is lower for cardiac and mathematical bios than for
other series. Recurrence and wavelet
plots show the temporal organization of biotic series as contrasted to the
uniform plots of chaos and periodic series.
Figure 2: Change in phase
space volume over time – a comparison of creative, conservative and attractive
dynamics. Random walk as well as
deterministic bios display the features characteristic of creative processes.
Creativity thus differs not
only from mechanical change and random fluctuation, but also from homeostasis,
autopoiesis, or convergence to periodic or chaotic attractors. Mechanical determinism cannot account for
creativity, so the emergence of novelty, diversity and complexity is attributed
to chance events. But to postulate from
the outset that there is no explanation for creativity is uncalled for. Randomness by itself provides no tools to
predict, modify, or generate creative change.
Creativity requires a deterministic component. Creative patterns can be generated by the addition of random
changes, as exemplified by random walk, but addition implies deterministic
conservation. Natural processes
generate complex patterns and structures in a consistent manner that belies the
notion of random accident. Complex
organization may be created by deterministic interactions between simpler
processes. May [] and others have shown that simple deterministic processes
can generate complex time series. Creative patterns can be
generated by bipolar feedback. This is
bios (figure 4).
Bios is a phenomenon that we
have discovered through examining both empirical data and mathematical
recursions. The exemplar of bios is
heart rate variation, just as the exemplar of chaos is turbulence. While chaos is a deterministic equivalent to
random, bios is a deterministic equivalent to random walk. Bios may be regarded
as a nonstationary form of chaos; however, there are stationary forms of bios.
Moreover, the definition of chaos varies from author to author. [There is no rigorous definition of chaos,
nor empirical methods to demonstrate it unequivocally in empirical data,
differentiating it from random. For
instance, Smale’s definition of chaos by the presence of a transversal
homoclinic point is based on the proof that homoclinic points imply chaos, but
the converse has not been formalized; further, Smale allows for “some
borderline cases in which this may not be exactly right”.] At the present time we identify bios, and
distinguish it from chaos, through a series of measurable properties of the
time series-using novel methods that we have recently developed, to be
described in section I. In brief, we currently
define bios as nonstationary aperiodic series that are (1) globally sensitive
to initial conditions (as contrasted to chaos that is only locally sensitive),
(2) less recurrent than their randomized copy, (3) increase in variance with
the duration of the sample, and (4) consist of episodic patterns with a
beginning and end (“complexes”).
Figure 4:
Venn diagram representation of aperiodic time series patterns. The three circles are defined by empirically
measurable properties of the time series.
Bios shows all three types of properties, while random exhibits none of
them. Bios and random walk share the
descriptive characteristics of creative processes.
Co-creation
Deterministic interactions between complementary opposite actions can generate new bifurcations into opposites and thereby novelty, diversity, and complexity. This co-creation hypothesis [Sabelli et al, 1997; Sabelli, in press] updates the notion of complementary opposites advanced in antiquity. In modern times catastrophe theory [Thom, 1983] and bifurcation cascades [Feigenbaum, 1983] model the generation of opposites in natural processes. Differentiation through bifurcation necessarily precedes chemical combination, dialectic synthesis, system formation, or any other type of creative interaction. The concept of creative development discussed here was inspired by Waddington, who described embryological development as a creative process, not fully determined by genetic information. Thom proposed that the “choices” made in such processes could be described by relatively simple, elementary “catastrophes”. He regarded catastrophes as archetypes embodying the tension of complementarity of opposites[1]. Catastrophes are the result of pull by opposite attractors. Thus opposites are the cause and the result of catastrophes. Subsequently Feigenbaum [1983] and others described sequences of bifurcations into opposites as a route to chaos, and therefore exemplary of the generation of complexity. Bifurcation cascades occur in many processes; they also should be regarded as archetypes present in both the physical world and the mental world of mathematics.
From
a static perspective, opposites are classes made of different substances. From a process perspective, opposition is a
process made of the same stuff, i.e. the flow of energy in time (action). Bifurcation is a process that generates
pairs of opposites (figure 5).
Opposites are synergic, i.e. increasing together in time but in opposite
direction, and thereby generating one unit of information.
Figure 5: A bifurcation is a forking of a time series into two complementary opposites that grow together in time but in opposite directions. Left: the initial bifurcation generated by the process equation. As the energy of the feedback gain increases, there is a greater separation of opposites, and further bifurcations leading to chaos and bios. Right: Schematic relation between complexity and the energy and symmetry of opposite actions. Note also the relation between abstract and concrete concepts of opposition.
How does the interaction of opposites generate
complexity? Obvious examples of creative interactions are sexual procreation, and the
formation of hydrogen by a proton and its asymmetric opposite, the electron. Let
me explore the co-creation process thorough a clinical example. A person is asked to portray his positive
and negative feelings regarding certain choice (e.g., a partner for a given
task, an occupation, or a purchase).
Clearly harmony and conflict often go together. We are more likely to love and
to have conflicts with those persons who are close to us than with strangers;
similarly, desirability and undesirable cost often grow together regarding
purchases. Attraction and repulsion, harmony and conflict, like and
dislike are complementary opposites, that often wax and wane together. The difference between opposite motivations
is the information that underlies the choice (selection or rejection), but the
intensity of positive and negative feelings add to provide energy to the
issue. Thus opposites are not polar
opposites in a linear continuum. They
are complementary actions that are in some ways similar, synergic, and mutually
reinforcing, and in another way different, antagonistic, and inversely proportional. In other words, opposites are orthogonal to
each, but not independent from each other.
We thus measure attraction and
repulsion in separate scales, which are combined in a two-dimensional Cartesian
scale (figure 6). Measuring opposites on orthogonal
scales, rather than as extremes of a single linear scale, allows one to portray
how opposites coexist, and vary together.
Data on either the left or the right quadrants represent
cases in which one opposite clearly dominates over the other, either attraction
or repulsion. The bottom quadrant
signifies that both forces are of low intensity (neutrality). The top quadrant
is occupied when opposites are both of high intensity. In this case, one observes both choices and
rejections, and catastrophic switches from one to the other. The study of choices as a function of
motivation reveals a simple linear relation between motivation and choice when the energy
is low or one opposite clearly predominates over the other, and nonlinearity
reminiscent of a fold or a cusp catastrophe when the intensity is high and the
opposites are symmetric [Carlson- Sabelli et al, 1992].
Figure 6:
Complementary opposites. Top: The Diamond of Opposites, a
two-dimensional scale which we use clinically to study opposite motivations and
feelings. Bottom: The complexity of
patterns generated by bipolar feedback as a function of the intensity and
symmetry of opposites (to be described in Section II).
Catastrophes are tridimensional
forms regulated
by two parameters, bifurcating (b) and asymmetric (a). In our example, the sum of opposite
motivations functions as a bifurcating control variable: at high intensities,
choices and rejections obtain, and sudden switches often occur. The difference between opposites functions as
the asymmetric factor. An asymmetry
between opposite motivations determines choice or rejection; when the opposite
motivations are of similar intensity, both outcomes are equally possible. The sum of opposite motivations gives energy
to the issue in question; the difference between opposites provides information
regarding its resolution (figure 6 top).
This suggests a general relation between energy E and information I, the
sum and the difference of opposites O and O-1, and the bifurcating
(b) and asymmetric (a) parameters governing the generation of complexity such
as a catastrophe (figure 6).
In actuality, the coexistence of high-energy
opposites not only produces catastrophic switches, but also the most intense,
most vital, most creative situations from which new behaviors emerge. We regard catastrophes as simple examples of
the generation of higher dimensional organization by the interaction of
opposites. The generation of a third
dimension from the interaction of opposite forces is also evident in physical
processes. For instance, opposing flows
generate tridimensional eddies. Complex
patterns are created by the interference of light rays. Tridimensional matter emerges from
radiation. We speculate that the
interaction of opposites is a generic process for the production of patterns of
higher dimensionality.
Catastrophes
constitute the simplest case of self-organization. Catastrophe theory may guide our attempts to understand more
complex forms of co-creation. Its
principles may apply beyond simple catastrophes, which are the initial step in
mathematical recursions as the series (see section II), to cascades of
bifurcations, chaos and bios. Thom
[1983] explicitly formulated his notion of catastrophes in the context of
Heraclitus’ union of opposite.
Expanding catastrophe theory, the energy provided by the sum of
opposites generates bifurcation when the difference between opposites is small
(symmetry). Note that asymmetry
provides information regarding a catastrophic separation of opposites, while
the (quasi) symmetry of opposites underlies the generation of higher dimensions
(co-creation). This is indeed found in
mathematical recursions of bipolar feedback to be described in Section II.
Process
Method of Time Series Analysis
The
characteristic features of creative processes are evaluated with new techniques
for time series analysis that measure action, information, and dimension (Table
2). Action is the unit of
process. Information (change
in action) is the unit of communication.
Dimension is the unit of
complexity.
Action
is the flow of energy in time; energy and time are inseparable quantum
conjugates. A process is a directed
sequence of actions; it is portrayed by the time graph of the intervals or
amplitude of the action units.
Actions
generate interactions that increase or decrease them. Changes in action embody information. Relations seldom are purely synergic or solely conflictual
(linear opposition); they often are both synergic and conflictual
(complementary opposition). . In
a time series, interactions are embodied in the transitions from one term to
another. The time series of the
differences between consecutive terms (At versus At+1) shows a chaotic pattern for biotic series
generated by the process equation, demonstrating that some forms of bios
contain chaos –just as chaos contain periodicity. Note a practical implication regarding time series analysis:
differencing prior to analysis can indicate chaos when the series is biotic. The return map (At versus At+1)
reveals the sine wave generator in biotic series generated by the process
equation. The presence of opposition
can be revealed more clearly by portraying these transitions in complement
plots; they show Mandala patterns for both cardiac and mathematical bios.
Table 2. Time series analysis of primary features of processes
|
Global |
Process |
Communication |
System organization |
|
Unit |
Action
=
energy * time |
Interaction
= Information |
Dimension
= type of relation |
|
Essential
features |
1.
Quantic (units), unidirectional (linear, asymmetric) |
2.
Bipolarity and bidimensionality of complementary opposites (synergic and
antagonistic) |
3.
Three or more orthogonal dimensions |
|
Time
series analysis |
Time
graph of series and vectors
(diversification and recurrence plots) |
Complement
plot |
Embedding
plots |
|
Coordinates |
Cartesian
plane: At
vs. t, f(At) vs. t |
Circular
plot: sin
(At) vs. cos(At) |
Recurrence
measures vs. embedding |
A relation is a repetitive interaction (i.e. a process of interaction), as contrasted to independent
interaction events. A relation is a
tension of opposites –the union and separation of two entities. Relations form
and organize systems. Relations
imply mutual feedback that generates complexity of organization. Each type
of relation determines a qualitatively different, orthogonal dimension.
Dimensionality (i.e. the number of
qualitatively different, orthogonal dimensions) measures the complexity of
organization. All processes include
physical dimensions (e.g. energy, time, spatial extension, electrical charge),
but there are features of organization, prominent in biological and
psychological processes, and also relevant to planetary and astronomical
morphology, that are not captured by the physical dimensions known so far. One must then devise ways to investigate
these additional dimensions of form and information. Whenever possible, one measures dimensions separately. In the case of time series, we have
available only one-dimensional data.
Yet we can evaluate additional dimensions of organization by methods
such as differencing and embedding. It
is mathematically proven that a vector of time-delayed copies of the observable
data (recorded for a sufficiently long period of time) will generate a
trajectory in the dimensional space so created that is similar to the original
process [Takens, 1993]. Temporal
organization may be observed in recurrence plots, and dimensionality can be
measured in embedding plots of recurrence.
These embedding dimensions must be a function of the actual dimensions
of the process under consideration, but how these two types of dimension relate
to each other is a subject to be explored.
Diversification
and Dispersion
Creative processes generate
variation (diversification). In the
case of time series, this implies an expansion of the phase space volume in
phase portraits that plot each term of the series vs. the difference with the
previous value, that is, At vs. At - At-1.
As the sample size increases, heartbeat series and biotic trajectories
occupy a greater volume, while the chaotic trajectory does not (figure 7
top). We have developed a number of
measures of diversification that quantify the change in phase space volume over
time. The simplest methods measure a
statistical measure of variation while increasing the duration of the
sample. For instance, the standard
deviation (S.D.) is measured for data points 1 to 100, then 1 to 200, 1 to 300,
etc., up to N data points, where N
is the entire time series. A plot of
S.D. vs. time reveals that S.D. increases with the duration of the sample
for heart rate intervals, as well as for biotic data generated by the process
equation (figure 7 bottom). This
indicates an expanding phase space volume.
By contrast, S.D. decreases or remains flat for chaotic and uniform random
data.
Figure 7: Diversification. Top: Return map of heartbeat intervals (RRI) from a patient with
coronary artery disease (CAD). The
phase space volume increases with the number of data points. Bottom:
A simple measure of diversification Left: Increased S.D. with N for bios
but not for chaos or for randomized copy of bios. Right: Increased S.D. with N for heartbeat intervals (RRI) in a
healthy subject.
For smaller data sets,
diversification can be estimated by measuring dispersion with a large number of embeddings. The S.D. is computed for sets of increasing
duration (from 2 to 200 terms) starting from each term in the series. We compute the average S.D. over all 2-dimensional vectors, all
3-dimensional vectors, etc, and then plot the averages as a function of the
number of embeddings (figure 8). The
correlation of these S.D. with the
embedding dimension provides two measures of dispersion. Low embeddings (2 to 25) portray divergence
such as measured by the Lyapunov exponent [West et al., 2000]. High embeddings (5 to 200) measure
diversification. For large embeddings,
the correlation is positive (S.D. increases with the number of embeddings) for
heartbeat data and for bios generated by the process equation. The correlation is zero or negative (S.D.
remains flat or decreases) for chaos and uniform random data.
Figure 8: Dispersion.
Complexes
Figure
9: Recurrence plot of time series
of corn prices. The horizontal and
vertical axes represent the vectors A1, A2, …. AN, formed by M consecutive terms starting
with each term in the series. (M is the embedding). These vectors are compared to each
other, and if they are sufficiently alike (isometry = difference between
the Euclidean norms of the vectors falls below a chosen cutoff radius), a
point is plotted at coordinates (i,
j) to indicate a recurrence. As many random pairs of points
may be tested as necessary in order to portray the pattern.
Creative processes show episodic patterns with a beginning
and an end, in contrast to random, periodic or chaotic series that generate
uniform time series. This difference
often cannot be detected in simple time graphs, but can be revealed by
statistical analysis by epochs, and by wavelet or recurrence plots (figure 3).
Episodic patterning is most
clearly revealed by the recurrence method developed by Eckmann et al. [1987] and Webber and Zbilut [1994]. A recurrence represents a repetition of a
previous pattern. We measure recurrence
isometry, i.e. the recurrence of vectors of equal length (see figure 9
legend). Recurrences are distributed
nearly uniformly throughout the time series for random, periodic, or chaotic
series. By contrast, creative processes
produce fewer recurrences that are arranged in distinct groupings, which we
named complexes when
we first identified them in heartbeat interval series [Sabelli et al.,
1995]. Complexes are separated by brief recurrence-free epochs
(“interruptions”), representing significant shifts in the moving average.
Clearly, the observation of
episodic patterns depends on examination of actual data. Empirical studies cannot detect patterns
excluded by the methods used for data collection and analysis. The selection of stationary subsamples
eliminates the difference between creative and noncreative processes. Global statistics cannot portray the
temporal organization of the original series that can be noted by the analysis
of epochs. Differencing, a common
manipulation to detrend data before dynamic analysis, also obscures the
temporal organization. In fact, the
time series of the differences between successive terms in a random walk is
purely random, and in a biotic series is chaotic. Thus chaos may appear to take place in empirical series with a
biotic pattern when the investigator differentiates them prior to
analysis. Selecting stationary
subsamples, examining global distributions, differencing, and detrending,
represent transformations of the data driven by the unwarranted assumption that
only what is stable matters. Creative
features are obscured or erased by manipulating the data to fit this
Procrustean bed. Being faithful to the
actual data, and focusing on processes of change, represents an opposite, and I
claim more scientific, philosophy.
Novelty
Recurrence
measures the more stable or static component of a process. Recurrence is high in periodic data, low in
random and chaotic series, and extremely low in heartbeat series and in
bios. This reflects an essential
difference between crystal-like order, that is stable, and living organization
that generates more variation than would be expected from purely random
events. For instance, the mixing of
chromosomes in sexual reproduction produces faster variations than random
mutation. From these considerations, we
develop a technical definition of novelty
in time series analysis based on randomization of the data. We measure the ratio of percentage of
isometries in a shuffled[2]
copy of the data over percentage of isometries in the original time series
[Sabelli, 2001]. An increase in
isometry by shuffling defines novelty.
Novelty is one of the defining features of biotic patterns present in
natural processes such as heartbeat interval series and economic time series
(figure 10), or generated by the process equation and related recursions. Novelty is also present in statistical noise
generated by random walk. Biological processes, bios, and random walk are more
variable than random. In contrast,
shuffling reduces recurrence in periodic data.
Random and chaotic data are neither novel nor recurrent (figure
11).
Determination
The
number of recurrences that are consecutive, meaning that Ai is
recurrent with Aj, and Ai+1 is recurrent with Aj+1,
reflects a crucial difference between causally determined processes and
aleatory change. The percentage of
consecutive recurrences is greater for periodic, chaotic and biotic series than
for random series, such as generated by shuffling the data. Consecutive recurrences indicate continuity,
inherent in causal action, as contrasted to random fluctuation; thus Webber and
Zbilut [1994] regard consecutive recurrence as a measure of determination.
Periods,
chaos and bios have more consecutive recurrence than random, but differ from
each other in recurrence rate, and in its change by shuffling. Plotting the percentage of recurrences and
of consecutive recurrences in the Cartesian plane shows that stabilizing
ordering and creative organization may be regarded as opposite departures from
random (figure 11). This is at variance
with the standard identification of organization with the order.
Figure
10. Recurrence plots of time series of
Treasury bill (left) and its randomized copy (right) illustrate the increase in
recurrence by randomization characteristic of creative processes.
Figure 11:
Characterization of processes by recurrence analysis. The X axis represents the percentage of
recurrence (isometry) and the Y axis the percentage of consecutive isometries.
Presumably they also represent energy and information (see figures 5 and 6). The results obtained for biotic, chaotic,
periodic and random series suggests that organization and ordering are
complementary opposite processes.
Shuffling increases recurrence for bios (novelty), reduces it for
periods, and does not change it for chaos. Shuffling reduces consecutive
recurrences for all three series,
Complexity
Essential to creativity is the nonrandom generation
of complexity. Many measures of
complexity equate it with the number of dimensions. In recurrence analysis, one may measure the median embedding
dimension (M.E.D.), i.e. the embedding at which 50% of the recurrences are
consecutive (figure 12) [Sabelli et al., 1995]. Random and chaotic series have high
M.E.D. while natural processes and bios have an intermediate number of
dimensions (Table 3). Thus the M.E.D.,
as many other measures of complexity, assigns the highest value to
random series. This is clearly at variance with our sense of complexity. It is evident that biological processes are
more complex than random change. True complexity has a U shaped relation with dimensionality:
simple processes have either low (period 1, 2, etc) or high (random series)
dimensions. This is consistent with the
fact that largest complexity obtains at moderate temperature (370C),
pressure, volume, duration, etc.
Figure 12.
Embedding plots for a time series generated with the process
equation.
Comparing the recurrence parameters of RRI series
with those of random, chaotic, and periodic series, suggested to us a measure
of nonrandom complexity that we call arrangement [Sabelli et al.,
1995; Sabelli, in press]. Arrangement
is calculated as the ratio of the percentage of consecutive recurrences to the
percentage of total recurrences; we measure it at the M.E.D. (figure 12). Arrangement is high in time series of
physiological recordings and of economic processes, and in biotic series
generated by recursions of bipolar feedback (Table 3). Arrangement is low in random, periodic and
chaotic series. This correlates with an
intuitive notion of complexity.
Table 3.
Complexity: Median Embedding Dimension (M.E.D.) and Arrangement. Isometry measured with
delay 1, cutoff radius 0.1. N > 1000. M.E.D. is the embedding at which 50% of recurrences are
consecutive.
|
|
M.E.D. |
Arrangement |
|
Random (uniform) |
214 |
16 |
|
1/f pink noise |
120 |
97 |
|
Logistic chaos (logistic equation, g = 4) |
363 |
15 |
|
Process chaos (process equation, g = 4.3) |
345 |
12 |
|
Bios (process equation, g = 4.65) |
21 |
57 |
|
Bios (process equation, g = 4.7) |
78 |
289 |
|
Dow Jones Industrial Average |
10 |
54 |
|
Corn prices |
35 |
98 |
|
Heartbeat intervals (31 samples, mean + S.E.) |
60 + 4 |
61 + 4 |
Complex
temporal form
Creative
organization implies complex temporal form, while periodic order represents
simplicity. Between these two extremes
lies unformed randomness. Embedding plots, i.e. the graph recurrence measurements
as a function of the number of embeddings (1, 2 ... N) used in their
calculation [Sabelli et al, 1995] portrays both periodic and aperiodic
components of a process –in contrast, power spectrum analysis decompose
a time series into sinusoidal waves that do not corresponds to real components
of the process. For random data, the
number of recurrences is very low at 1 embedding, and increases with the number
of embeddings. Periodic series generate
periodic embedding plots, in which the percentage of recurrences is high only
when the embedding coincides with the period (figure 3); coexisting
periodicities appear as distinct peaks.
Natural processes often show a diversity of patterns, including peaks
denoting periodicities and a gradual increase in recurrence indicating
random-like aperiodic components. For
instance, heartbeat intervals show a respiratory periodicity as well as a daily
cycle related to activity and sleep, in addition to the biotic pattern that
occurs between these two temporal dimensions.
Similarly, weather variables (e.g. temperature in the American Midwest,
but not precipitation) have seasonal as well as daily periodicity, but between
these two periodicities there are aperiodic patterns that appear to be biotic
rather than chaotic [Sabelli, 2000].
Heartbeat interval series, many economic time series, and bios have high
recurrence at low embeddings, a subsequent decrease, and a later increase in
recurrence with the number of embeddings
(figure 12). Generalizing from
these observations, a natural process does not have a single pattern with a
unique dimension at which should be analyzed, as commonly considered. The time series of creative
processes may be expected to contain both complex patterns and the simple
processes that generate them. In
contrast, random data and random walks lack simple components of
variation.
In summary, a set of newly
developed mathematical measurements unequivocally identifies biotic patterns in
time series, and distinguishes them from other aperiodic deterministic patterns
such as chaos. More difficult is to
differentiate bios from statistical noise, and chaos from random data. These methods indicate that heartbeat
interval series have biotic patterns –a point of considerable interest given
the clinical significance of heart rate variation. Similarly, the time series of economic data (daily fluctuations of 10
different currencies, prices of 6 different commodities, and the Dow-Jones
Index Average) have biotic patterns with an added trend. We call these patterns parabiotic; as we
shall discuss in Section II, parabiotic series are generated by asymmetric
bipolar feedback. Biotic patterns also appear in some time series of
meteorological variables. Creative
patterns, rather than chaos, equilibrium, or randomness appear to characterize
natural processes.