Forecasting, When Power Law Distributions Apply

<p>Whilst a lot of our strategic focus in the public sector is on linear policy approaches, many systems/ phenomena of importance are defined as non-linear or far from equilibrium. Traditional approaches to linear forecasting have not proved effective for non-linear systems, since non-linear systems follow a different set of rules. Historically, non-linear systems were too hard to forecast, but over recent decades some rules and approaches are starting to emerge. One important and clearly defined category of non-linear systems are those that follow a ‘power-law’ distribution rather than the ‘normal’ distribution, which is often associated with linear systems or systems in equilibrium. My research collects, analyses, and does a comparative analysis of the different power law populations, as well as the main strategic forecasting techniques that can be applied to those populations/ systems. Overall Conclusions and observations. Just as in science and mathematics, there is now a clearly defined separation and understanding of linear and non-linear systems and the rules that apply to each. My thesis has as its central theme, the idea that strategy as a subject also fits this same philosophical separation of approaches, which I have called the strategic planning versus the strategic thinking divide. Strategic planning is essentially the linear approach – being rational and assuming relatively stable conditions. Strategic thinking assumes the world is effectively non-linear and ‘far from equilibrium’. Non-linear approaches mean acknowledging concepts like; punctuated equilibrium, power law ‘log-log’ graphs, ‘scale-free’ characteristics, ‘self organising criticality’, accepting only pattern prediction (including 1/f formulas) and not precise prediction etc. Understanding non-linearity is essential to understand such things as ‘Black Swans’. Luck, serendipity and ‘bounded rationality’ are always involved in non-linear complex adaptive systems, whereas linear systems tend to comply with the so called ‘rational’ traditions in science and economics. Power law statistical distributions can be seen in a wide variety of non-linear natural and man-made phenomena, from earthquakes and solar flares to populations of cities and sales of books. This sheer diversity of effects that have power law distributions is actually an amazing fact that has only become evident over the last decade or so. Since the world contains aspects that are clearly linear and other aspects that are clearly non-linear, it is essential for someone interested in strategy to be able to understand both systems and be able to apply the correct techniques to each approach. The two parts of ‘punctuated equilibrium’ effectively link the two strategic approaches together as there is only one world and not two separate realities. It therefore follows that a strategist needs a good understanding of both strategic planning and strategic thinking, since both are needed for different phases or periods, and perhaps both are needed for any period when you can't tell what phase you are in, which can also happen. I suggest that under a linear phase, the strategic planning approach should be dominant, but supported by strategic thinking (since you never know when events will turn abruptly); whereas in a turbulent non-linear period the strategic thinking approach should be dominant, but supported by strategic planning (since you know that great turbulence will not last). This is a sort of a swapping dominant/ recessive situation, which has a loose parallel in the theory of the left/ right brain split, where it is not wise to use only one style of thinking, since there are two styles which suit different situations. The key is to pick the right thinking style for the right situation. Just as we have one brain, but two thinking styles, so in the strategy toolbox we also have two valid, useful and complimentary general strategic approaches. However for this thesis, I have focused on the non-linear power law aspects of life which have strong implications for strategic thinking, since that is the new area for me as well as one of the new knowledge frontiers for strategy as a subject (and for leadership, politics and many other areas).</p>

Forecasting, When Power Law Distributions Apply

<p>Whilst a lot of our strategic focus in the public sector is on linear policy approaches, many systems/ phenomena of importance are defined as non-linear or far from equilibrium. Traditional approaches to linear forecasting have not proved effective for non-linear systems, since non-linear systems follow a different set of rules. Historically, non-linear systems were too hard to forecast, but over recent decades some rules and approaches are starting to emerge. One important and clearly defined category of non-linear systems are those that follow a ‘power-law’ distribution rather than the ‘normal’ distribution, which is often associated with linear systems or systems in equilibrium. My research collects, analyses, and does a comparative analysis of the different power law populations, as well as the main strategic forecasting techniques that can be applied to those populations/ systems. Overall Conclusions and observations. Just as in science and mathematics, there is now a clearly defined separation and understanding of linear and non-linear systems and the rules that apply to each. My thesis has as its central theme, the idea that strategy as a subject also fits this same philosophical separation of approaches, which I have called the strategic planning versus the strategic thinking divide. Strategic planning is essentially the linear approach – being rational and assuming relatively stable conditions. Strategic thinking assumes the world is effectively non-linear and ‘far from equilibrium’. Non-linear approaches mean acknowledging concepts like; punctuated equilibrium, power law ‘log-log’ graphs, ‘scale-free’ characteristics, ‘self organising criticality’, accepting only pattern prediction (including 1/f formulas) and not precise prediction etc. Understanding non-linearity is essential to understand such things as ‘Black Swans’. Luck, serendipity and ‘bounded rationality’ are always involved in non-linear complex adaptive systems, whereas linear systems tend to comply with the so called ‘rational’ traditions in science and economics. Power law statistical distributions can be seen in a wide variety of non-linear natural and man-made phenomena, from earthquakes and solar flares to populations of cities and sales of books. This sheer diversity of effects that have power law distributions is actually an amazing fact that has only become evident over the last decade or so. Since the world contains aspects that are clearly linear and other aspects that are clearly non-linear, it is essential for someone interested in strategy to be able to understand both systems and be able to apply the correct techniques to each approach. The two parts of ‘punctuated equilibrium’ effectively link the two strategic approaches together as there is only one world and not two separate realities. It therefore follows that a strategist needs a good understanding of both strategic planning and strategic thinking, since both are needed for different phases or periods, and perhaps both are needed for any period when you can't tell what phase you are in, which can also happen. I suggest that under a linear phase, the strategic planning approach should be dominant, but supported by strategic thinking (since you never know when events will turn abruptly); whereas in a turbulent non-linear period the strategic thinking approach should be dominant, but supported by strategic planning (since you know that great turbulence will not last). This is a sort of a swapping dominant/ recessive situation, which has a loose parallel in the theory of the left/ right brain split, where it is not wise to use only one style of thinking, since there are two styles which suit different situations. The key is to pick the right thinking style for the right situation. Just as we have one brain, but two thinking styles, so in the strategy toolbox we also have two valid, useful and complimentary general strategic approaches. However for this thesis, I have focused on the non-linear power law aspects of life which have strong implications for strategic thinking, since that is the new area for me as well as one of the new knowledge frontiers for strategy as a subject (and for leadership, politics and many other areas).</p>

Power law distributions in a new toy model with interacting N agents

A new toy model with interacting N agents is proposed in this paper. The agents in this model possess quantity, and have an interaction radius depending on the quantity. They exchange a part of the quantity with agents belonging to within their interaction radii. The cumulative distribution function about observing the quantity in a stationary state exhibits a power law, and the exponent is universally close to –1 if the density of agents is sufficiently small.

Econophysics and the Entropic Foundations of Economics

This paper examines relations between econophysics and the law of entropy as foundations of economic phenomena. Ontological entropy, where actual thermodynamic processes are involved in the flow of energy from the Sun through the biosphere and economy, is distinguished from metaphorical entropy, where similar mathematics used for modeling entropy is employed to model economic phenomena. Areas considered include general equilibrium theory, growth theory, business cycles, ecological economics, urban–regional economics, income and wealth distribution, and financial market dynamics. The power-law distributions studied by econophysicists can reflect anti-entropic forces is emphasized to show how entropic and anti-entropic forces can interact to drive economic dynamics, such as in the interaction between business cycles, financial markets, and income distributions.

Local Analysis of Heterogeneous Intracellular Transport: Slow and Fast Moving Endosomes

Trajectories of endosomes inside living eukaryotic cells are highly heterogeneous in space and time and diffuse anomalously due to a combination of viscoelasticity, caging, aggregation and active transport. Some of the trajectories display switching between persistent and anti-persistent motion, while others jiggle around in one position for the whole measurement time. By splitting the ensemble of endosome trajectories into slow moving subdiffusive and fast moving superdiffusive endosomes, we analyzed them separately. The mean squared displacements and velocity auto-correlation functions confirm the effectiveness of the splitting methods. Applying the local analysis, we show that both ensembles are characterized by a spectrum of local anomalous exponents and local generalized diffusion coefficients. Slow and fast endosomes have exponential distributions of local anomalous exponents and power law distributions of generalized diffusion coefficients. This suggests that heterogeneous fractional Brownian motion is an appropriate model for both fast and slow moving endosomes. This article is part of a Special Issue entitled: “Recent Advances In Single-Particle Tracking: Experiment and Analysis” edited by Janusz Szwabiński and Aleksander Weron.

Power laws in the Roman Empire: a survival analysis

The Roman Empire shaped western civilization, and many Roman principles are embodied in modern institutions. Although its political institutions proved both resilient and adaptable, allowing it to incorporate diverse populations, the Empire suffered from many conflicts. Indeed, most emperors died violently, from assassination, suicide or in battle. These conflicts produced patterns in the length of time that can be identified by statistical analysis. In this paper, we study the underlying patterns associated with the reign of the Roman emperors by using statistical tools of survival data analysis. We consider all the 175 Roman emperors and propose a new power-law model with change points to predict the time-to-violent-death of the Roman emperors. This model encompasses data in the presence of censoring and long-term survivors, providing more accurate predictions than previous models. Our results show that power-law distributions can also occur in survival data, as verified in other data types from natural and artificial systems, reinforcing the ubiquity of power-law distributions. The generality of our approach paves the way to further related investigations not only in other ancient civilizations but also in applications in engineering and medicine.

Onset of runaway fragmentation of salt marshes

Salt marshes are valuable but vulnerable coastal ecosystems that adapt to relative sea level rise (RSLR) by accumulating organic matter and inorganic sediment. The natural limit of these processes defines a threshold rate of RSLR beyond which marshes drown, resulting in ponding and conversion to open waters. We develop a simplified formulation for sediment transport across marshes to show that pond formation leads to runaway marsh fragmentation, a process characterized by a self-similar hierarchy of pond sizes with power-law distributions. We find the threshold for marsh fragmentation scales primarily with tidal range and that sediment supply is only relevant where tides are sufficient to transport sediment to the marsh interior. Thus the RSLR threshold is controlled by organic accretion in microtidal marshes regardless of the suspended sediment concentration at the marsh edge. This explains the observed fragmentation of microtidal marshes and suggests a tipping point for widespread marsh loss.