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The Acceleration of Variability
Extreme weather events are becoming more extreme and more frequent; 100-year rain events are now occurring multiple times per decade. The recent record heat dome over the Pacific northwest and western Canada, the alternation of drought and flooding in the Mississippi/Missouri basin and the earlier, longer Atlantic hurricane season are all linked to the increase in average atmospheric temperature. The increased variability in weather associated with rising temperatures, including higher frequency of extreme events was predicted by climatologists in the 1970s and 1980s. The greenhouse gasses carbon dioxide and methane trapping solar energy in the atmosphere are “forcing” agents, literally heating up atmospheric processes and speeding up circulation.
The atmosphere is not the only system that becomes more variable as its average value increases through natural or artificial forcing. In fact this phenomenon is well known and seen in many systems natural and man-made. The increase in variance with average was first described in regards to population ecology and published in a Letter to Nature in 1962. This analysis of variability data of two dozen species showed the variance of population density to be a power function of average density and has since become known as Taylor’s power law. Thousands of other examples have now been published. The distribution and abundance of organisms are intimately related, and together define a population’s size and structure in space and time. Locally, the frequency distribution of population number per unit area may be described by a known statistical distribution. Different locations and/or times are described by different distributions that are linked by the power law of mean and variance in a continuum of shapes in which the distributions themselves change in a density dependent fashion. Under similar sampling conditions, this continuum of frequency distributions is common to populations in different places and times.
In the nearly six decades since the 1962 paper, examples have been documented in disciplines as diverse as microbiology, genetics, economics, sociology and psychology, computer science, meteorology, geology, astronomy and physics where it is better known as fluctuation scaling. For example, the distributions of frequency and severity of earthquakes, the size of tropical storms and the number of tornados in a tornado outbreak all obey Taylor’s power law. It also describes the distribution of website accesses and internet traffic through routers as well as the magnitude of commodity, currency and equity price changes. One of the most remarkable data streams described by the law is the distribution of prime numbers. In the biological realm, anatomy and physiology and human demography, psychology and sociology variables have also been shown to obey Taylor’s power law. For example, different crime categories (hooliganism and vandalism, burglary, violent crime, criminal damage, arson and drug offenses) have different Taylor’s power law scalings and are spatially stable at regional and local scales. These examples are among the hundreds described in detail in Taylor’s Power Law: Order and Pattern in Nature.
Taylor’s power law has never been satisfactorily deduced from first principles, although many attempts have been made; so for now it must be considered strictly phenomenological. However, deductions bordering on predictions can be derived. Commodity prices tend to be correlated over long time spans leading to large price changes more likely to be followed by large changes than small and small changes being followed by small changes. This is also the case for equity and foreign exchange prices. Such time series or spectra are said to display “brown noise” distinct from the “white noise” of Gaussian random walks. These spectra look similar regardless of the time scale. The frequency distributions of price changes have long right-hand tails and when sampled in a series of windows or bins obey Taylor’s power law. Similar long-range correlations have been found in the time series of discharge of rivers such as the Nile. River discharge is analysed by the Hurst equation whose characteristic exponent is linearly related to Taylor’s exponent.
One important deduction from Taylor’ power law, reinforced by analyses of the spectra of financial instruments and river discharge, is that the normal (Gaussian) distributed is rare in natural systems and many man-made systems. While a number of independent as well as interacting financial events contributed to the financial meltdown that resulted in the Great Recession of 2007-9, a model used to evaluate and manage risk in the financial and derivative markets, the Gaussian copula function, played an important role.
Taylor’s power law predicts that as the value of a variable (price of the dollar, price of a company’s shares, price of a commodity) increases, the variance about that value will accelerate (the volatility increases). Although broad market indices like the Dow-Jones and S&P indices have been steadily rising, history shows the rises are punctuated by periods of increased volatility. This is expected in systems that, like the flow of the Nile, display brown spectra with Hurst or Taylor exponent.
The Gaussian copula function allowed hugely complex risks to be modelled easily, making it possible for traders to create and sell new types of securities, amongst them “mortgage-backed securities “ and “collateralized debt obligations” in which financial risk is based on a probability of default. To minimize risk, assets (bonds, mortgages, etc.) are divided into “tranches” with the expectation that if defaults should occur, losses can be restricted to the tranches receiving the highest rates of compensation in exchange for the higher default risk. Implicit in the model is the assumption that correlations between assets are normally distributed. Taylor’s power law teaches us however, that few processes are normally distributed and most have long right hand “fat tails”. These fat tails contain the rare exceptional events (sometimes called black swans) that are way outside the normal bell curve assumed by the Gaussian copula models of risk.
As the average value of an equity or debt increases, the variation about that average accelerates, just like population variation around its mean, leading to very long right hand tails, much longer than the tails assumed in Gaussian models. Thus, as the valuation of securitized debts increased, so too did the risk of default, but much faster than assumed in the Gaussian copulas. As valuations increased, the risk of default increased disproportionately and the longer it took for a major default to occur, the greater the financial damage done by the defaulting financial instrument as it triggered other collateralized instruments to default in a chain reaction. The failure of these risk models contributed to the near global financial meltdown in 2007. Their failure was triggered by the unanticipated large number of sub-prime mortgage defaults occurring simultaneously in several markets – an event the Gaussian copula function and similar models implicitly assumed could not happen.
Processes producing data streams that are described by Taylor’s power law always produce distributions that are skewed right; they are never symmetrical about their mean like Gaussian distributions. At low averages the distributions are very pointed with the mode close to zero and with comparatively short tails. But as the mean increases the mode shifts to the right making the distribution appear more symmetrical, but occasional extremely high values lengthen the right-hand tail and dramatically increase the variance. This seems to be a very common phenomenon which is described for many systems with analysis of data from multiple disciplines in Taylor’s Power Law: Order and Pattern in Nature. In it I offer a tentative explanation for this ubiquitous relationship. In a synthesis of the examples I suggest that the ubiquity of Taylor’s power law in diverse natural and man-made systems may be the result of a statistical property of numbers in a data stream generated by recursion of a small number of comparatively simple rules. In this view Taylor’s power law is an emergent property that summarizes complex patterns such as those that emerge from simple cellular automata.
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