by David Orrell
3. See Wolfram 2002, pp. 149–53.
4. Lorenz 1963.
5. See also Orrell 2003 for other examples.
6. Orrell 2001.
7. This moral is, of course, different from the original “butterfly effect” moral, which implies that forecast error is primarily the result of sensitivity to initial condition.
8. Watson and Lovelock 1983.
9. As Lovelock wrote, “Neither Lynn Margulis nor I have ever proposed that planetary self-regulation is purposeful. . . . Yet we have met persistent, almost dogmatic, criticism that our hypothesis is teleological.” Lovelock 1991.
10. See, for example, http://www.acad.carleton.edu/curricular/GEOL/DaveSTELLA/Daisyworld/daisyworld_model.htm orhttp://gingerbooth.com/courseware/pages/demos.html#daisy.
11. Heraclitus, fragment 54.
12. Hinkle 1996.
GLOSSARY
AMINO ACID:One of the twenty different molecules that are combined in a linear fashion to form proteins.
ANALYSIS: In meteorology, this word refers to a weather centre’s estimate of the state of the atmosphere. It is obtained by compiling observed data from a variety of sources—such as ground stations, weather balloons, commercial airplanes, and satellites—and reconciling it with model predictions from an earlier time. The analysis then forms the initial condition for a new forecast.
ATTRACTOR: The trajectories of a dynamical system can be drawn to one of three basic types of attractors. A point, or steady-state, attractor is a single fixed point. In a periodic attractor, trajectories are drawn into a repeating cycle. The third class is the strange attractor, which is characteristic of chaotic systems (see Appendix II). Small changes in parameter values can sometimes cause a sudden bifurcation from one kind of attractor to another. Attractors can be thought of as imposing a kind of constraint on a system’s behaviour. They are properties of sets of mathematical equations, not real physical systems, so we can talk about attractors for climate models, but not for the climate itself.
AVIAN FLU: A type of influenza virus that is hosted by birds but may also infect other animals, including humans, where it can lead to acute respiratory distress and pneumonia. At the time of writing, the strain known as H5N1 has been identified by many scientists as the most likely source for the next pandemic. For that to happen, though, the strain will have to genetically mutate so that it can be easily transmitted from person to person without losing its virulence.
BASE: One of the four substances (adenine, cytosine, guanine, and thymine) that make up DNA.
BELL CURVE: See normal distribution.
BIFURCATION: In a dynamical system, a bifurcation occurs when a small change in a parameter causes a qualitative change in the system solution. A jump from a periodic attractor to a chaotic attractor would be an example of a bifurcation.
BUTTERFLY EFFECT: This is the theory that the small atmospheric disturbance caused by a butterfly flapping its wings can cause a storm on the other side of the world. It was inspired by the sensitivity to initial condition of chaotic systems like the Lorenz system (see Appendix II). While the butterfly effect is an attractive idea, it doesn’t account for the local damping of small perturbations, and as the science writer Matt Ridley (2003) notes, it probably says less about the atmosphere than it does about the human desire “to preserve linear causality in such systems” (compare emergence). The butterfly effect is also used to describe the sensitivity to initial condition of weather models (as opposed to the weather itself ), which is a real but rather weak effect.
CELLULAR AUTOMATON: This is a mathematical system consisting of a grid of coloured cells that evolve in discrete time steps according to local rules. The simplest grid is a one-dimensional line (see Appendix I for an example). In two dimensions, the grid can be square, triangular, or hexagonal. Higher-dimensional systems can also be constructed. The cells can be black or white (binary), or a discrete set of colours, or a continuous range. The simplest rules specify the colour of the cell based on the colour of adjacent cells.
CHAOS THEORY: This refers to the study of non-linear dynamical systems characterized by aperiodic behaviour and sensitivity to initial condition. See also attractor and butterfly effect.
CHROMOSOME: A cell’s DNA is divided and tightly folded into separate chromosomes. These earned their name (from the Greek words chromo, for “colour,” and somos, for “body”), well before their function was understood, because they were easily made visible under the microscope by staining.
CLIMATE: This refers to properties of a region’s long-term weather. Quantities of interest include average and maximum/minimum rainfall and temperature for a particular time of year.
CLIMATOLOGICAL FORECAST: This is a naïve forecast for a system’s future based on long-term statistics. An example would be predicting that the temperature next week will be equal to the average temperature for that time of year.
COMPLEX SYSTEM: There is no shortage of mathematical definitions for this expression. In the context of cellular automata, a complex system is generally one in which the behaviour is neither entirely ordered nor entirely random, but somewhere in between — in other words, interesting. Such systems, though based on simple local rules, have emergent properties (see emergence) that cannot be understood in terms of those rules.
CORRELATION: This statistical term defines a relationship between two sets of randomly varying data. A strong correlation implies that the two tend to vary together, while no correlation means that they are completely independent. For example, lung cancer correlates with smoking—that is, those who smoke are more likely to develop the disease. Many correlations are falsely reported. Chance alone ensures that if enough experiments are performed to test for a relationship between two factors, there is a high probability that at least one will show a positive correlation. Of all the experiments, it will be the only one that is deemed interesting and makes it into a scientific report.
DNA (DEOXYRIBONUCLEIC ACID): A long molecule in the form of a double helix, DNA is composed of the four bases (A, C, G, and T) that encode genetic traits.
Drift: The model drift is one technique for estimating the component of forecast error owing to both errors in a mathematical model and uncertainty in the observations of the underlying system. To calculate model drift, one performs a number of short forecasts at a sequence of observation times. The drift can be compared with the results from shadow experiments.
DYNAMICAL SYSTEM: This mathematical term refers to a set of equations that specifies the rate of change of a number of variables with time. A solution of a dynamical system is a trajectory starting from a specified initial condition. The equations modelling the fall of a stone in Chapter 2 make up a simple dynamical system. Note that the falling stone itself is not a dynamical system—it is a stone. The mathematical equations are not the same as the underlying reality. See also attractor.
EMH (EFFICIENT MARKET HYPOTHESIS): There are different versions of this theory, but an efficient market was originally defined by Fama (1965) as “a market where there are large numbers of rational profit maximizers actively competing, with each trying to predict future market values of individual securities, and where important current information is almost freely available to all participants.” The market price for an asset is a “good estimate of its intrinsic value.” Any deviations will be small and random, so “actual prices of securities will wander randomly about their intrinsic values.” It follows that in an efficient market, no investor can exploit price discrepancies based on fundamental analysis or any other method.
EMERGENCE: A property or behaviour of a complex system that arises from basic laws or principles but cannot be predicted from them. The word was used to describe living systems by philosophers such as Samuel Alexander (1920), who wrote: “Physical and chemical processes of a certain complexity have the quality of life. The new quality life emerges with this constellation of such processes, and therefore life is at once a physico-chemical complex and is not merely physical and
chemical. . . . The existence of emergent qualities thus described is something to be noted, as some would say, under the compulsion of brute empirical fact. . . . It admits no explanation.” Emergent phenomena do not have a single cause (compare butterfly effect) but are a product of the system as a whole. See also uncomputable.
ENSEMBLE FORECAST: This is a technique that makes many separate forecasts using slightly perturbed initial conditions, different models, or a combination thereof. The results are then interpreted using statistical techniques to make a probabilistic forecast. In weather forecasting, ensemble-forecast schemes were originally designed to account for the butterfly effect.
EXPONENTIAL GROWTH: A quantity grows exponentially if the rate of growth is proportional to the size of the quantity. It is often an indicator of positive feedback. Money in a bank account grows exponentially if the interest is reinvested. Note that exponential growth need not mean rapid growth (a deposit held in a chequing account, for example, grows slowly).
FEEDBACK: This term from the branch of mathematics known as control theory is used also in a variety of fields, including engineering, biology, and economics. It refers to a situation in which a portion of a signal is fed back into the source, thus modifying the signal. Feedback can be positive or negative. An example of positive feedback is a microphone pointed at a speaker—any noise is picked up by the microphone, amplified, and sent to the speaker. The speaker’s output is picked up by the microphone, sent back to the speaker, and so on (until the speaker explodes or someone grabs the microphone). Positive feedback can also lead to collapse instead of growth. When a business loses customers, it has less money to invest in its products, which means it loses more customers, and so on. An example of negative feedback is the flyball governor of Chapter 3, which kept the speed of steam engines at a steady level. In general, positive feedback tends to accentuate a change to a signal, while negative feedback reduces change.
FRACTAL: his term (from the Latin fractus, for “broken”) was coined in 1975 by Benoit Mandelbrot. Unlike the figures of classical geometry, such as circles and squares, a fractal is an irregular figure that shows a self-similar structure over a range of scales, so the same visual motifs appear no matter how close you zoom in. The figures can be generated using extremely simple mathematical procedures, but often they have a remarkably complex appearance. Fractals are better viewed than described; many examples, such as the Mandelbrot set, can be found on the Internet.
GAIA THEORY: This is the holistic theory that the earth can usefully be viewed as having the self-regulatory, homeostatic properties of a living organism.
GDP (GROSS DOMESTIC PRODUCT): The GDP is an estimate of the monetary value of goods and services produced within a nation. It does not include the so-called invisible economy, such as unpaid labour in the home, voluntary work, or the black market. Nor does it account for the costs of environmental damage to natural systems. It can therefore be highly misleading as a measure of a society’s true condition.
GENE: This word (from the Greek genos, for “origin”) refers to a rather fuzzy concept. In functional terms, it is used to describe a basic unit of inheritance that passes certain traits from generation to generation. In physical terms, it is a portion or portions of DNA that codes for a particular RNA or protein. A gene can produce different proteins if the intermediate RNA is modified.
GENOTYPE: This is the sum of the genetic information in an organism’s DNA.
HOLISM: The opposite of reductionism, holism (from the Greek holos, meaning “whole”) is the belief that certain systems cannot be understood in their constituent parts, but instead must be treated as a unified, organic whole. A holistic philosophy was espoused by artists such as Goethe in reaction to mechanistic science (and got a bad rap when the Nazi party in Germany promoted the holistic ideal of the state as a living organism).
HOMEOSTASIS: Homeostasis (from the Greek words homeo and stasis, meaning “to remain the same”) is an expression coined by the American physiologist Walter Cannon in 1932. It refers to the ability of living organisms to maintain certain critical aspects of their internal state within a narrow range (as opposed to a precisely fixed level). Examples in the human body include temperature and salinity.
INITIAL CONDITION: This is the initial value of the variables at time zero for a mathematical model. In the example of the falling stone in Chapter 2, the initial condition is the starting height and velocity of the stone.
Linear equation: An equation of the form y = ax + b, where a and b are constant parameters, is linear in the variable x. Linear equations are so named because when y is plotted versus x, it gives a straight line. In a linear dynamical system, the rates of change of variables are given by linear equations.
MATHEMATICAL MODEL: A mathematical model is a simulation of a physical, biological, economic, or other system using a set of equations, which typically make up a dynamical system. Mathematical models are useful for the scientific study of natural systems. Note that the dynamical system should not be confused with the underlying system, which is generally far more complicated and cannot be expressed using equations. Models can therefore provide valuable insights, but they must be used with care, especially when making predictions.
MECHANISTIC SCIENCE: This is the view that all processes, including the behaviour of living things, are machine-like and amenable to a reductionist approach. According to mechanistic science, it is possible to accurately simulate a system by building a mathematical model based on fundamental laws.
METABOLISM: This is the sum of all biochemical processes in a living organism.
MODEL ERROR: The error in a forecast that arises because of a discrepancy between the mathematical model and the underlying system is called the model error. It can be evaluated using techniques such as model drift and shadow orbits.
NAIVE FORECAST: This is a forecast based on simple rules of thumb rather than a mathematical model (like the climatological forecast or the persistence forecast). A forecast is said to have skill if it can beat naïve forecasts.
NON-LINEAR EQUATION: Any equation that is not linear (i.e., most equations) is called non-linear. The difficulties in solving non-linear dynamical systems meant that they were little studied until the development of fast computers in the 1960s.
NORMAL DISTRIBUTION: This is a particular probability distribution; it’s also known as the bell curve, for its shape. The distribution is symmetric, with 68 percent within one standard deviation of the mean and 95 percent within two standard deviations. It has many useful mathematical properties, including the central limit theorem, which states that the sum of independent, identically distributed (i.i.d.) random variables will approach a normal distribution. This means, for example, that if an error in a calculation is the result of the cumulative sum of many small, independent errors of a similar type, then it will tend to follow a normal distribution. The distribution was used by Laplace in 1783 to study measurement errors, but it has also been adopted in areas such as finance, where its use is less justified.
ODE (ORDINARY DIFFERENTIAL EQUATION): This class of equations involves mathematical derivatives, which specify the rate of change of one variable with respect to another. In the example of the falling stone in Chapter 2, the two ODEs involve the rates of change of position and velocity with time.
PANDEMIC: This is an epidemic that spreads so that it infects people over a large geographical area. In modern times, this effectively means the whole world.
Parameter: A parameter is a number in an equation that doesn’t change with time. In the example of the falling stone in Chapter 2, the force of gravity was treated as a fixed parameter, even though in reality it varies with distance from the earth. See also variable.
PARAMETERIZATION: In mathematical models, parameterization refers to the representation of a complex process by an approximate equation. An example is the equations for the formation and dissipation of clouds in GCMs. A property of many models discussed in this book, including GCMs, is their
sensitivity to parameterization. As a result, the models can be tuned to fit past data, but they still fail to predict the future.
PERSISTENCE FORECAST: This is a naïve forecast for a system’s future based on its current state. An example of a persistence forecast would be predicting that the temperature next week will be equal to the temperature today.
PHENOTYPE: The phenotype refers to an organism’s traits, such as size or skin colour; the phenotype is influenced by the genotype and the environment, whose effects work in concert.
POWER LAW: Two quantities (x and y) are said to be related by a power law if y = cxk where c and the exponent k are constant numbers. For example, if y represents the area of a circle of radius x, then x and y are related by a power law y = �x2x2 with exponent 2. A power law probability distribution is one in which the probability y of a state x obeys a power law. If x represents wealth and y represents the number of people with that amount of wealth, then in many countries y approximately follows a power law relationship with exponent around -2. The negative sign means that the chances of having low wealth are much greater than the chances of having great wealth.
PROBABILISTIC FORECAST: This is a forecast expressed in terms of a probability— saying, for example, that there is an 80 percent chance it will rain tomorrow. It can be obtained by performing an ensemble forecast. Alternatively, a forecaster can modify a single model forecast into a statement of likelihood— turning a model prediction for rain into “a strong chance of rain,” for example— either by mathematically accounting for expected errors in some plausible fashion or more simply by using his or her own judgment. (Of course, this is subjective, but so are many of the choices that lie behind the development of the model and any ensemble scheme.)
PROBABILITY DISTRIBUTION: This is any equation or table that assigns a probability y for any number x. For example, IQ scores were designed so that the chances of a randomly chosen person having a certain IQ x follows a normal distribution with mean 100 and standard deviation of about 15.
