By Roger Reed,2014-11-20 20:28
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    Seek simplicity, and distrust it.

    -Alfred North Whitehead

    In James R. Newman, The World of Mathematics, Vol. II,

    p. 1055, Tempus, WA: Redmond (1988).




    2.1 Mathematical Representations and Theoretical Thinking

    Measurement cannot be separated from theory. Theoretical thinking is better

    through a mathematical representation. The choice of mathematical

    representations depends on the essential features in empirical observation and

    theoretical perspective. A mathematical representation should be powerful enough

    to display stylized features to be explained and simple enough to manage its

    mathematical solution to be solved. In the history of science, theoretical

    breakthrough often introduces radical changes in mathematical representation. For

    example, physicists once considered the Euclid geometry as an intrinsic nature of

    space. Einstein's work on gravitation theory offered a better alternative of a specific non-Euclid geometry in theoretical physics.

    Mathematical representation is an integrated part of theoretical thinking. New mathematical representations are introduced under new perspectives. Newtonian

    mechanics was developed by means of deterministic representation. Probability

    representation made its way through kinetic theory of gas, statistical mechanics,

    and quantum mechanics. The study of deterministic chaos in Hamiltonian and

    dissipative systems reveals a complementary relation between these two


    There are different motives in choosing mathematical representation. For some scientists, the choice of mathematical representation is a choice of belief. Einstein refused the probability explanation of quantum mechanics because of his belief that God did not throw dice. Equilibrium economists reject economic chaos because of

    a fear that the existence of a deterministic pattern implies a failure of the perfect market. For some scientists, the issue of mathematical representation is a matter of taste and convenience. Hamiltonian formulation in theoretical economics has

    tremendous appeal because of its theoretical beauty and logical elegance. The

    discrete-time framework is dominated in econometrics because of its computational

    convenience in regression practice. For us, empirical relevance and theoretical

    generality are main drives in seeking new mathematical representations.

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    The equilibrium feature of economic movements is characterized by the Gaussian distribution with a finite mean and variance. The disequilibrium features can be described by a unimodal distribution deviated from the Gaussian distribution. During a bifurcation or transition process, a U-shaped or multimodal distribution may occur under nonequilibrium conditions. We will study the deterministic and probabilistic representations for equilibrium, disequilibrium, and nonequilibrium conditions.

    2.2 Trajectory and Probability Representation of Dynamical Systems

    Both trajectory and probability representation are mathematical abstractions of

    the real world. In physics, a trajectory of a planet is an abstraction and approximation when we ignore the size of the planet and its perturbation during movement. In biologic and social science, the trajectory representation can be perceived as an average behavior over repeated observations. The same procedure of averaging can be applied to the probability representation. The probability representation holds for large ensembles with identical properties.

    People may think that deterministic and stochastic approaches are conflicting representations. One strong argument in favor of stochastic modeling in economics is a human being's free will against determinism. However, this belief ignores a simple fact that these two representations coexist in theoretical literature. For example, the wave equation in quantum mechanics is a deterministic equation. However, its wave function has a probability interpretation. Traffic flow could be described by deterministic and stochastic models that were verified by extensive experiments (Prigogine and Herman 1971).

    For a given deterministic equation, we can have both trajectory representation and probability representation.

    The choice of mathematical representation depends on the question asked in your research. If your goal is forecasting a time path, you need the trajectory representation. If your interest is their average properties such as mean and variance, you need the probability representation.

    2.2.1 Time Averaging, Ensemble Averaging, and Ergocity

    Trajectory representation can be easily visualized by a time path of an observable such as a moving particle. Probability representation is more difficult because it contains more information.

    There are two approaches to introduce the concept of probability distribution. From a repeated experiment such as the case of coin tossing, a static approach can define a probability distribution as the average outcome. Probability represents the expectation from an event or experiment. A dynamic approach can construct a histogram from a time series. A dynamic probability distribution can be revealed from a histogram if the underlying dynamic is not changing over time. The question is does the time average represent the true possibility in future events. In statistical physics, the probability distribution is described in an ensemble

    that consists of a large number of identical systems. The probability distribution is considered as an average behavior of these identical systems. In mathematical literature, if the time averaging is equal to the ensemble averaging, this property is

Mathematical Representations31

    called ergocity. In mathematical economics, ergodic behavior is often assumed

    for stochastic models in time series analysis (Granger and Teresvirta 1993,

    Hamilton 1994). In statistical physics, it is hard to establish the ergotic behavior

    for physical systems. For example, non-ergotic behavior was found from the

    anharmonic oscillators in Hamiltonian systems (Reichl 1998). Contrary to previous

    belief, ergocity and the approach to equilibrium do not hold for most Hamiltonian

    systems. Chaos in Hamiltonian systems plays an important role in approaching

    equilibrium and ergocity. Almost all dynamic systems exhibit chaotic orbits.

    Nonlinearity and chaos appear to be the rule rather than exception in dynamical


    Before discussing a probability distribution, we introduce the concept of a mean

    µ which is a numerical measurement of the mathematical or average value

    Xexpectation E[x]. If there are N measurements of a variable , and n

    n=1,2,...,N, we have

    XXX+++...12Ne = (2.2.1a)N

    Px()Px()?0Pxdx()=1For a probability distribution where and , we ?

    may have:

    <>=xxPxdx()E[x] = µ = (2.2.1b)?

    We also introduce the variance VAR[x] and the standard deviation σ:

    22VAR[x] = (2.2.2)<(x?µ)>=σ

    In general, we can define a central moment about the mean:


    The most important distribution in probability theory is the Gaussian or normal

    distribution. A Gaussian distribution is completely determined by its first two

    moments: the mean µ and the standard deviation σ.

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    (b)Figure 2.1 Deterministic and probabilistic representation of

    Gaussian white noise.



    300Histogram of Gaussian white noise









    X2Time path of Gaussian noise









Mathematical Representations33

    We may simply denote the Gaussian distribution as N.(,)µσ

    As a numerical example, the time path and histogram of the Gaussian random

    noise is shown in Figure 2.1.

    In principle, a stochastic system can be uniquely determined if its infinite

    moments are known. In empirical science, the critical issue is to determine the

    minimum moments for some characteristic behavior. In equilibrium statistical

    physics, the first two moments are good enough for many applications. In

    nonequilibrium statistical physics, higher moments may be needed. For example,

    the theory of non-Gaussian behavior in strong turbulence predicts up to the seventh

    moments observed in experiments. In economics, the first two to four moments are

    studied in empirical analysis.

    2.2.2 The Law of Large Numbers, The Central Limit Theorem,

    and Their Breakdown

    For a large number of events, the Gaussian distribution provides a good

    description of its distribution. We have a set of N independent stochastic variables

    X, X, . . . X with a common distribution. If their mean µ exist, the law of large 12N

    numbers states that (Feller 1968):


    We denote its sum S= X+X+ . . .+X. Therefore, S average (S/N) N 12NNN

    approaches µ, and S approaches Nµ.N

    If the first two moments exist for the above stochastic variables, the central limit

    theorem states that the probability distribution of S approaches a Gaussian N

    distribution with a mean of Nµ and a standard deviation of σ (van Kampen N



    The Gaussian distribution is widely applied in statistics and econometrics

    because of the power of the law of large numbers and the central limit theorem.

    Therefore, the limitation of the Gaussian distribution can be seen when the law of

    large numbers and the central limit theorem break down.

    One notable case is the non-existence of variance. For example, the Levy-Pareto

    distribution has infinite variance. The Levy distribution L(x) has an inverse power

    tail for large | x | (Montroll and Shlesinger 1984):


    where 0 < α < 2.

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    When α = 1, the Levy distribution has a special case of the Cauchy distribution

    which has finite mean b but infinite variance (Feller 1968).

    af(x)=(2.2.8)22π[(x?b)+a)]Empirical evidence of the Levy distribution is found in a broad distribution of

    commodity prices long-range correlations in turbulent flow (Mandelbrot 1963,

    Cootner 1964, Bouchaud and Georges 1990, Klafter, Shlesinger, and Zumofen

    1996, Reichl 1998).

    Figure 2.2. Gaussian and Cauchy Distribution. The N(0, 1) is the

    tallest in solid line. The Cauchy(1, 0) distribution is in the middle in

    dashed line, and Cauchy(π, 0) is the lowest and fattest distribution in

    dotted line.

    A comparison of the Cauchy distribution and the Gaussian distribution is shown

    in Figure 2.2. Both are unimodal distributions with zero mean. The variance of the

    standard Gaussian distribution is 1, and the variance of Cauchy distribution is

    infinite caused by its long tails.

    2.2.3 U-Shaped Distribution

    Another interesting case is the U-shaped distribution such as the polarization in

    ferromagnetism and public opinion (Haken 1977, Chen 1991). For a U-shaped

    distribution, the concept of the expectation is meaningless, because the mean is the

    most unlikely event.

    In probability theory, there is a case of the arc sine law for a last visits that

    indicates a startling result in chance fluctuation (Feller 1968). Consider an ideal


    035.Gaussian and Cauchy distributions









Mathematical Representations35

    coin-tossing in 2n trials. The chances of tossing a head or a tail are equal. If the

    accumulated numbers of heads and tails are 2k. The probability is related to the

    following U-shaped distribution function:


    kwhere and 0 < x < 1.x=n

    Figure 2.3 The Arc Sine Distribution for Accumulated Events in

    Coin Tossing.

    Its probability distribution is shown in Figure 2.3. Its mean is 0.5. Its variance is 0.125. Its skewness is zero. Its kurtosis is 93. We must know that even the

    distribution of the sample mean approaches the Gaussian distribution, but the

    mean of the arc sine distribution represents the least likely event! This is a case

    that the central limit theorem is valid for a U-shaped distribution but the mean

    value may be quite misleading!

    People may think the most likely event should be anyone leading half the trial numbers. It is not! The most probable values for k are the extremes 0 and n. It is

    quite likely that in a long coin-tossing game, one of the players remains the whole

    time on the winning side, the other on the losing side. For example, in 20 tossings,

    the probability of leading 16 times or more is about 0.685; but the probability of

    leading 10 times is only 0.06! Therefore, the intuition of time averaging may lead

    to an erroneous picture of the probable effects of chance fluctuations. In other

    words, the expected value is misleading under a U-shaped distribution.


    Arc sine distribution10







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    2.2.4 Delta Function and Deterministic Representation

    Deterministic representation can be considered as a special case of probability

    representation when the probability distribution is a delta function. The delta

    function is very useful in quantum mechanics (Merzbacher 1970).

    δ(x?x')=0when (2.2.10a)xx?'δ(0)=?(2.2.10b)


    ? (2.2.10d)f(x')=f(x)δ(x?x')dx???

    There are some useful properties for the delta function:




    1δ(x?a)δ(x?b)=[δ(x?a)+δ(x?b)](2.2.11d)|a?b|The delta function may have the following representations in terms of a limiting

    process (Merzbacher 1970, Stremler 1982):

    (a) A harmonic wave:


    where .i=?1

    (b) A Gaussian pulse:


    (c) A two-sided exponential:

Mathematical Representations37


    (d) A Cauchy distribution pulse:


    (e) A Sinc function:


    (f) A rectangular pulse function:

    1εε(2.2.11f)δ(x)=lim[u(x+)?u(x?)]ε?0ε22where the unit step function is:

    uxx(')?=1 when xx?'uxx(')?=0 when (2.2.11g)xx<'


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    Figure 2.4 The Relationship between Deterministic and Probabilistic

    Representation. (a) A bifurcation tree of a deterministic system. (b) The

    trajectory and probability distribution representationIn economic dynamics, both a deterministic equation and a stochastic equation can be considered as an average description of a large number of systems. If the

    probability has a unimodal distribution, the time-path of its average position can be described by a trajectory.

    For a nonlinear deterministic system, bifurcation may occur like a bifurcation tree in Figure 2.4a. During a bifurcation point of a deterministic equation, the

    corresponding probability distribution must have a polarized distribution as shown in Figure 2.4b.

    2.3 Linear Representations in the Time and Frequency Domain

    In theoretical and empirical analysis, the time domain representation is applied in correlation analysis and the frequency domain representation is used in spectral

    analysis. They are useful tools for time series analysis for deterministic and

    stochastic systems.

    Theoretically, we can consider the spectral analysis as a representation in a functional space whose base function is harmonic waves, and the base function of

    correlation analysis is delta functions. Therefore, linear representations of a time series have only two building blocks: harmonic cycle and white noise. These two

    extreme models have a remarkable similarity: both of them can be described by

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