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# Dimensionality of Market Trading: Open (State-Input-Output)/Closed (State-Input-State), from Douglas Dimick

January 28, 2010 |

Dimensionality of Market Trading: Open (State-Input-Output)/Closed (State-Input-State)

Query, Comments, and the Issue

What is the equivalent of an open versus a closed game in market trading, and under what conditions is each better or worse?

Upon reading the comments, one may conclude that there is significant variation in the interpretation of the meanings of open and closed relative game theory applications to market trading. Ball and bat indicates baseball or cricket. The Black Monday events related to futures distinguishes valuation tools and calculation of market support. A two-dimensional answer also may encompass both (option spreads and directional futures) issuances and (high/low volatility) market conditions. Fees then VIX, cash balance, and price-volume-spread applications may be linear processes.

My presupposition here is that Victor’s query originates from his recent recommendation of Nigel Davies' book on chess. As the ecology of chess may function as a multi-dimensional system (e.g., 3d chess), and as open (1 e4 e5) and closed (1 d4 d5) moves operate as tactical or game strategies, we may address the query by researching the issue:

If market trading functions as a multi-dimensional system, how may open and closed strategies operate for optimal performance?

Excerpts — A Survey of Research on Open/Closed Concepts

Consider the following excerpts from research on open and closed concepts. Note that the direction of this research leads to theory of chaos and logistic mapping.

Ecology (definition): The branch of sociology that is concerned with studying the relationships between human groups and their physical and social environments: also called human ecology.

Note that both games and markets are human constructs constituting physical/social environments that function as rules-based systems.

Systems and states (terms): A system is a combination of interacting elements that performs a function not possible with any of the individual elements. In a dynamic system, outputs depend on present and past values of inputs and must define the concept of a state.

The state of a system makes the system’s history irrelevant. The state of the system contains all of the information needed to calculate responses to present and future inputs without reference to the past history of inputs and outputs; present inputs and the sequence of future inputs allow computation of all future states (and outputs). Some dynamic systems are modeled best with state equations while others are modeled best with state machines.

Dimensionality of dynamical systems: Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. However, the Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system (specified by differential equations) if it has three or more dimensions. Linear systems are never chaotic; for a dynamical system to display chaotic behavior it has to be nonlinear.

Open/Closed Principle: The term Open/Closed Principle is that once completed, the implementation of a class could only be modified to correct errors. New or changed features would require that a different class be created; that class could reuse coding from the original class through inheritance. The derived subclass might or might not have the same interface as the original class.

Implementation can be reused through inheritance but interface specifications need not be. The existing implementation is closed to modifications, and new implementations need not implement the existing interface.

Polymorphic Open/Closed Principle advocates inheritance from abstract base classes. Interface specifications can be reused through inheritance but not implementation.

The existing interface is closed to modifications and new implementations must, at a minimum, implement that interface. Thus, the Principle became popularly redefined to refer to the use of abstracted interfaces, where the implementations can be changed and multiple implementations could be created and polymorphically substituted for each other.

The concept of an “open system” was formalized within the framework of thermodynamics. This concept was expanded upon with the advent of information theory and subsequently systems theory.

In the social sciences, an open system process exchanges material, energy, people, capital and information with its environment. In the natural sciences, an open system is one whose border is permeable to both energy and mass. By contrast in physics, a closed system is permeable to energy but not to matter.

Open systems have a number of consequences. A closed system contains limited energies. Open system assumes that supplies of energy cannot be depleted from a surrounding environment, being infinite for the purposes of study. For instance, the radiant energy system receives its energy from solar radiation, which may be considered inexhaustible.

A closed system is a system in a state isolated from its surrounding environment. An idealized system is where closure is perfect, yet no system can be completely closed (only varying degrees of closure).

In thermodynamics, a closed system can exchange heat and work (aka energy) but not matter with its surroundings. In contrast, an open system can exchange all of heat, work and matter.

Note that Victor and Laurel analogize market exchanges with energy related to thermodynamics (see Chapter 13 in Practical Speculation and Page 32 of my current book project, Theory of Quantitative Relativity for Program Trading and Portfolio Management Systems Architecture).

Syllable: A syllable is a unit of organization for a sequence of speech sounds and is typically made up of a syllable nucleus (most often a vowel) with optional initial and final margins (typically, consonants). Syllables are often considered the phonological “building blocks” of words. The general structure of a syllable consists of the following segments:

Onset (obligatory in some languages, optional or even restricted in others) Rime Nucleus (obligatory in all languages) Coda (optional in some languages, highly restricted or prohibited in others)

In some theories of phonology, these syllable structures are displayed as tree diagrams (similar to the trees found in some types of syntax). The syllable nucleus is typically a sonorant, usually making a vowel sound, in the form of a monophthong, diphthong, or triphthong, but sometimes sonorant consonants like [l] or [r].

The syllable onset is the sound or sounds occurring before the nucleus, and the syllable coda (literally ‘tail’) is the sound or sounds that follow the nucleus. The term rime covers the nucleus plus coda.

Generally, every syllable requires a nucleus. Onsets are extremely common, and some languages require all syllables to have an onset. (That is, a CVC syllable like cat is possible, but a VC syllable such as at is not.) A coda-less syllable of the form V, CV, CCV, etc. is called an open syllable (or free syllable), while a syllable that has a coda (VC, CVC, CVCC, etc.) is called a closed syllable (or checked syllable). Note that they have nothing to do with open and close vowels.

Dynamical system (concept): The dynamical system concept is a mathematical formalization for any fixed “rule” that describes the time dependence of a point’s position in its ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake.

At any given time a dynamical system has a state given by a set of real numbers (a vector) which can be represented by a point in an appropriate state space (a geometrical manifold). Small changes in the state of the system correspond to small changes in the numbers.

The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time interval only one future state follows from the current state.

The concept of a dynamical system has its origins in Newtonian mechanics. The evolution rule gives the state of the system only a short time into the future based on a relation that is either a differential equation, difference equation, or other time scale.

To determine the state for all future times requires iterating the relation many times – each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system. Once the system can be solved, given an initial point it is possible to determine all its future points, a collection known as a trajectory or orbit.

Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The difficulties arise because:

Trajectories may be periodic and wander through different states of a system, so applications often require enumerating these classes or maintaining the system within one class. Classifying all possible trajectories has led to the qualitative study of dynamical systems, that is, properties that do not change under coordinate changes. Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood.

The behavior of trajectories as a function of a parameter may be what is needed for an application. As a parameter is varied, the dynamical systems may have bifurcation points where the qualitative behavior of the dynamical system changes. For example, it may go from having only periodic motions to apparently erratic behavior, as in the transition to turbulence of a fluid.

The trajectories of the system may appear erratic, as if random. In these cases it may be necessary to compute averages using one very long trajectory or many different trajectories. The averages are well defined for ergodic systems and a more detailed understanding has been worked out for hyperbolic systems. Understanding the probabilistic aspects of dynamical systems has helped establish the foundations of statistical mechanics and of chaos.

Dynamical system (definition): A dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions that for any element of the time, map a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold.

Differential equations can be used to define the evolution rule: an equation that arises from the modeling of mechanical systems with complicated constraints. Many of the concepts in dynamical systems can be extended to infinite-dimensional manifolds—those that are locally Banach spaces—in which case the differential equations are partial differential equations.

Linear dynamical systems: Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers.

Flows: For a flow, the vector field is a linear function of the position in the phase space with A a matrix, b a vector of numbers and x the position vector. The solution to this system can be found by using the superposition principle (linearity).

The distance between two different initial conditions in the case A ? 0 will change exponentially in most cases by either converging exponentially fast towards a point or diverging exponentially fast. Linear systems display sensitive dependence on initial conditions in the case of divergence.

Maps: A discrete-time, affine dynamical system has the form with A a matrix and b a vector. In the new coordinate system, the origin is a fixed point of the map and the solutions are of the linear system. The solutions for the map are no longer curves, but points that hop in the phase space. The orbits are organized in curves, or fibers, which are collections of points that map into themselves under the action of the map. There are also many other discrete dynamical systems.

Local dynamics: The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative). A singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop.

It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well understood. In the qualitative study of dynamical systems, the approach is to show that there is a change of coordinates (usually unspecified, but computable) that makes the dynamical system as simple as possible.

Rectification: A flow in most small patches of the phase space can be made very simple. If y is a point where the vector field v(y) ? 0, then there is a change of coordinates for a region around y where the vector field becomes a series of parallel vectors of the same magnitude. This is known as the rectification theorem.

The rectification theorem says that away from singular points the dynamics of a point in a small patch is a straight line. The patch can sometimes be enlarged by stitching several patches together, and when this works out in the whole phase space M the dynamical system is integrable. In most cases the patch cannot be extended to the entire phase space.

Bifurcation theory: When the evolution map (or the vector field it is derived from) depends on a parameter, the structure of the phase space will also depend on this parameter. Small changes may produce no qualitative changes in the phase space until a special value is reached. At this point the phase space changes qualitatively and the dynamical system is said to have gone through a bifurcation.

Bifurcation theory considers a structure in phase space (typically a fixed point, a periodic orbit, or an invariant torus) and studies its behavior as a function of the parameter. At the bifurcation point the structure may change its stability, split into new structures, or merge with other structures. By using Taylor series approximations of the maps and an understanding of the differences that may be eliminated by a change of coordinates, it is possible to catalog the bifurcations of dynamical systems.

Ergodic systems: In many dynamical systems it is possible to choose the coordinates of the system so that the volume (really a ?-dimensional volume) in phase space is invariant. This happens for mechanical systems derived from Newton’s laws as long as the coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum).

The ergodic hypothesis turned out not to be the essential property needed for the development of statistical mechanics and a series of other ergodic-like properties were introduced to capture the relevant aspects of physical systems. Koopman approached the study of ergodic systems by the use of functional analysis.

Nonlinear dynamical systems and chaos: Simple nonlinear dynamical systems and even piecewise linear systems can exhibit a completely unpredictable behavior, which might seem to be random – within completely deterministic systems. This seemingly unpredictable behavior has been called chaos.

Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. In hyperbolic systems the tangent space perpendicular to a trajectory can be well separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold).

This branch of mathematics deals with the long-term qualitative behavior of dynamical systems. Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions, for example:

• Will the system settle down to a steady state in the long term?

• If so, what are the possible attractors?

• Does the long-term behavior of the system depend on its initial condition?

Note that the chaotic behavior of complicated systems is not the issue.

Meteorology has been known for years to involve complicated—even chaotic—behavior. Chaos theory has been so surprising because chaos can be found within almost all trivial systems.

Chaos: Although there is no universally accepted mathematical definition of chaos, a commonly-used definition says that, for a dynamical system to be classified as chaotic, it must have the following properties:

a) it must be sensitive to initial conditions;

b) it must be topologically mixing, and;

c) its periodic orbits must be dense.

Sensitivity to initial conditions: Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.

Sensitivity to initial conditions is popularly known as the “butterfly effect,” so called because of the title of a paper given by Edward Lorenz in 1972 entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?

The Lyapunov exponent characterizes the extent of the sensitivity to initial conditions. Quantitatively, two trajectories in phase space with initial separation diverge. The rate of separation can be different for different orientations of the initial separation vector. Thus, there is a whole spectrum of Lyapunov exponents — the number of them is equal to the number of dimensions of the phase space.

Topological mixing: Topological mixing (or topological transitivity) means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. This mathematical concept of “mixing” corresponds to the standard intuition, and the mixing of colored dyes or fluids is an example of a chaotic system.

Topological mixing is often omitted from popular accounts of chaos, which equate chaos with sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos.

Density of periodic orbits: Density of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. Topologically mixing systems failing this condition may not display sensitivity to initial conditions, and hence may not be chaotic. For example, an irrational rotation of the circle is topologically transitive, but does not have dense periodic orbits, and hence does not have sensitive dependence on initial conditions.

Lorenz and butterflies: An early pioneer was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a simple digital computer to run his weather simulation. He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation, which he had calculated last time.

To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 was printed as 0.506. This difference is tiny and the consensus at the time would have been that it should have had practically no effect.

However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome. Lorenz’s discovery, which gave its name to Lorenz attractors, proved that meteorology could not reasonably predict weather beyond a weekly period (at most).

Mandelbrot and snowflakes: The year before, Benoît Mandelbrot found recurring patterns at every scale in data on cotton prices. Beforehand, he had studied information theory and concluded noise was patterned like a Cantor set: on any scale the proportion of noise-containing periods to error-free periods was a constant – thus errors were inevitable and must be planned for by incorporating redundancy.

Mandelbrot described to effects. The “Noah effect” in which sudden discontinuous changes can occur (e.g., in a stock’s prices after bad news), thus challenging normal distribution theory in statistics (aka Bell Curve). The “Joseph effect” is where persistence of a value can occur for a while yet suddenly change afterwards.

An object whose irregularity is constant over different scales (”self-similarity”) is a fractal (for example, the Koch curve or “snowflake”, which is infinitely long yet encloses a finite space and has fractal dimension equal to circa 1.2619, the Menger sponge and the Sierpinski gasket). In 1975 Mandelbrot published The Fractal Geometry of Nature, which became a classic of chaos theory. Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model.

Distinguishing random from chaotic data: It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in practice no time series consists of pure ’signal.’ There will always be some form of corrupting noise, even if it is present as round-off or truncation error. Thus any real time series, even if mostly deterministic, will contain some randomness.

Statistical tests attempting to separate noise from the deterministic skeleton or inversely isolate the deterministic part risk failure. Things become worse when the deterministic component is a non-linear feedback system. In presence of interactions between nonlinear deterministic components and noise, the resulting nonlinear series can display dynamics that traditional tests for nonlinearity are sometimes not able to capture.

Logistic mapping: In the case of the logistic map, the quadratic difference equation (1) describing it may be thought of as a stretching-and-folding operation on the interval (0,1). This stretching-and-folding does not just produce a gradual divergence of the sequences of iterates, but an exponential divergence (see Lyapunov exponents) as evidenced also by the complexity and unpredictability of the chaotic logistic map.

In fact, exponential divergence of sequences of iterates explains the connection between chaos and unpredictability: a small error in the supposed initial state of the system will tend to correspond to a large error later in its evolution. Hence, predictions about future states become progressively (and exponentially) worse when there are even very small errors in our knowledge of the initial state..

It is often possible, however, to make precise and accurate statements about the likelihood of a future state in a chaotic system. If a (possibly chaotic) dynamical system has an attractor, then there exists a probability measure that gives the long-run proportion of time spent by the system in the various regions of the attractor.

Conclusion

Based on Quantitative Relativity, if market trading functions as a multi-dimensional system, distinguishing between market situation processing and market strategy operation for order execution may provide the optimal performance of systemic applications of open and closed strategies.

The Theory of Quantitative Relativity distinguishes between control rules (or proof planning anticipatory systems) for correlating that “nexus” of the matter observed as energy transformation and transference, constituting nonrandom sequencing of combination structures within electronic exchange markets of financial instruments.

In a dynamic system, outputs depend on present and past values of inputs and must define the concept of a state. Therefore, an open system would be indicated for state-input-output processing.

As the state of a system makes the system’s history irrelevant, some dynamic systems are modeled best with state equations while others are modeled best with state machines. Therefore, market situation processing indicates open systematics, whereas market strategy indicates closed systematics.

A strange attractor can only arise in a continuous dynamical system (specified by differential equations) if it has three or more dimensions. Linear systems are never chaotic; for a dynamical system to display chaotic behavior, it has to be nonlinear. In that market trading may be multi-dimensional, it can be chaotic; therefore, quantification of strange attractors requires open systematics for state-input-output processing.

The term Open/Closed Principle is that once completed, the implementation of a class could only be modified to correct errors. New or changed features would require that a different class be created. Therefore, to achieve a linear system for order execution, a closed system is required within a rules-based subsystem of a state machine that quantitatively defines each state.

The dynamical system concept is a mathematical formalization for any fixed “rule” that describes the time dependence of a point’s position in its ambient space, such as the flow of water in a pipe. Therefore, closed systematics for (rules-based) function integration is required to quantify space and time correlation of price action (as a form of energy).

As (a) a dynamical system has a state given by a set of real numbers (a vector) which can be represented by a point in an appropriate state space (a geometrical manifold), and (b) small changes in the state of the system correspond to small changes in the numbers, therefore, open systematics are indicated to quantify state transitioning (or state-input-state), whereas a closed system is required to define rules-based excitation for state transitioning.

Iteration to determine the state for all future times is solving the system or integrating the system. Once the system can be solved, given an initial point, it is possible to determine all its future points, a collection known as a trajectory or orbit. This system operates as the (state-input-output) processing of the market situation; therefore, based on Quantitative Relativity, the solving of the system indicates open systematics, whereas the collection as a trajectory requires a closed system to establish linear processing for nonrandom storage and recall – being pattern recognition and connection of intelligence.

Linear dynamical systems can be solved in terms of simple functions and the behavior of all orbits classified. In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers. Therefore, as solutions for the map are no longer curves but points that hop in the phase space, a closed system is supported for binary processing of order execution protocol.

The qualitative properties of dynamical systems do not change under a smooth change of coordinates (this is sometimes taken as a definition of qualitative). Therefore, as the incongruency of averaging is minimized during market strategy processing, efficient operation of a closed system is viable for state transitioning.

If coordinates are the position and the momentum and the volume is measured in units of (position) × (momentum), phase space may be invariant. Therefore, as open systematics are required to quantify price action as energy coordinates of a market situation, a closed system may achieve state transitioning of patterns during unpredictable behavior, which might seem to be random – within completely deterministic systems (or being the chaos of market exchange behavior).

Hyperbolic systems are dynamical systems that exhibit properties of chaotic systems and may be separated into two parts: one with the points that converge towards the orbit (the stable manifold) and another of the points that diverge from the orbit (the unstable manifold). Chaotic behavior of complicated systems is not the issue but for defining rules-based applications of market trading; therefore, quantification of the stable manifold for convergence of market strategy supports closed systematics, whereas divergence of market situation requires open systematics to define and quantify state transitioning.

Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories; therefore, open systematics are indicated for market situation processing to both define and quantify state transitioning.

Topological mixing (or topological transitivity) means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. Therefore, to effect binary processing of synchronous-oriented state transition functions, closed systematics are indicated.

Exponential divergence of sequences of iterates explains the connection between chaos and unpredictability: a small error in the supposed initial state of the system will tend to correspond to a large error later in its evolution. It is often possible, however, to make precise and accurate statements about the likelihood of a future state in a chaotic system.

Therefore, open and closed systematics are required to both define rules-based quantification and operate function integration for optimal performance of market trading programs.

Attribution: Please see wikipedia for research excerpted herein.

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