| San José State University Department of Economics |
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of a Levy-Stable Probability Distribution for the JISAO Arctic Oscillation Index |
The Joint Institute for the Study of the Atmosphere and Ocean (JISAO) of the University of Washington, in collaboration with the National Oceanic and Atmospheric Administration (NOAA), compiled an index for the Arctic Oscillation. This index is based upon differences in sea level pressure in the region. This index was compiled as monthly values from January 1899 to June 2002.
The histogram of the values of the index for January 1899 to December 2001 appears below.
The most notable characteristic of the data is the occurrence of extreme values. The data values are pressure differences in millibars multiplied by 100. The typical values are less than 100 in magnitude. There are however some months for the which the values are about -300 and others for which the values are about +300. (There is one month, December of 1944, for which the value is not available and for that month an interpolation between the preceding and following month was used.) The intervals for the histogram are 50 units wide; i.e., from -350 to -300, etc.
The distribution looks remarkably like a normal distribution. There is good reason for natural phenomena to have normal distributions. The Central Limit Theorem says roughly that the more independent influences on a variable the closer its distribution is to a normal distribution. There is a qualification that the separate influences must have finite variance. An extension of the central limit theorem removes the qualification and expands the limit distribution to the Lévy stable distributions. Some non-normal Lévy stable distributions look similar to normal distributions but result in more extreme cases than do normal distributions. Therefore the above distribution needs to be tested for strict normality.
The mean value for the JISAO AO index data set used is -3.95 and the standard deviation is 99.9. The maximum value of the data set is 406, which is over four standard deviation units away from the mean. The probability of a deviation this large or larger is about 1 in 31,000. The data set contained only 1296 observations. The minimum value for the data is -523. The probability of getting a negative deviation from the mean of this magnitude or larger is less than one out of 1.7 million. Thus there is a strong indication that the distribution of the AO index is not really a normal distribution. Instead it is what is sometimes called a fat-tailed distribution. There will be more on this later.
As explained in the reading on characteristic functions the characteristic function for a distribution is the expected value of exp(izω). The characteristic function will generally be a complex function; i.e., Γ(ω) = X(ω) + iY(ω).
Since exp(iωz} = cos(ωz) + isin(ωz} the components of the characteristic function are given by:
X(ω) = E{cos(ωz)} = ∫-∞∞cos(ωz)p(z)dz.
and
Y(ω) = E{sin(ωz)} = ∫-∞∞sin(ωz)p(z)dz.
As a practical matter the probability distribution is in the form of a histogram; i.e., the frequencies or probabilities for specified ranges of the random variable. The computation of the components of the characteristic function is based upon the midpoints of the ranges for the histogram. In effect, this replaces the probability distribution which is defined over a continuous variable with one defined for a discrete variable. Instead of the distribution curve being a smooth curve it is taken to be structured like a comb. This means that there is a limit for the frequency variable ω. If the frequency is so high that the wavelength of the trigometric functions is comparable to the interval between the midpoints of the histogram ranges then the estimates for the characteristic function are no longer valid. For this reason the characteristic function components are computed for values of ω no higher than 8. The results of the computations are shown below.
The components of the characteristic function can be used to compute the components of the logarithm of the characteristic function. This process is not simply a matter of the taking the logarithm of the components.
The logarithm of the characteristic function will also be a complex function with real and imaginary components. The logarithm of a variable W is defined as as w if:
exp(w) = W.
For a complex variable X+iY we must find x+iy such that
exp(x+iy) = X+iY.
Since
exp(x+iy)=exp(x)exp(iy)
= exp(x)(cos(y)+isin(y)
= exp(x)cos(x)+iexp(x)sin(x)
it follows that
X = exp(x)cos(y)
and
Y = exp(x)sin(y)
Thus the imaginary component y can be determined from:
tan(y) = Y/X and hence y = tan-1(Y/X)
The real component x can then be found from:
x = ln(|X|) - ln(|cos(y)|).
The reason for wanting the components of the logarithm of the characteristic function is that these components provide a way to estimate the parameters of a Lévy stable distribution. If the distribution is a Lévy stable distribution the plot of ln(-x(ω)) versus ln(ω) should be a straight line whose slope is equal to the alpha parameter of a stable distribution.
The real component of the log-characteristic function for a stable distribution is
Ψ(ω) = - |νω|α = - να|ω|α
and therefore
να|ω|α = - Ψ(ω)
This last relationship implies that:
ln(-Ψ(ω)) = αln(|ω|) + αln(ν)
Thus for a stable distribution the graph of the logarithm of the real component of the log-characteristic function as a function of the logarithm of ω is a straight line, the slope of which is the stability index of the distribution, α. The graph for the distribution of the AO index is below.
The data is remarkably close to a straight line over the range of ω=0.25 to ω=2. The deviation from a straight line over the higher frequencies can be attributed to wavelength approaching that of the separation between the midpoints of the categories of the histogram.
The slope of the straight line is 1.78. This is the value of the alpha parameter for the distribution of the AO index. A normal distribution has an alpha value of 2.0.
The value of the logarithm of ln(-Ψ(ω)) when ω=1 is the intercept of the straight line and is equal to αln(ν). Thus a knowledge of the intercept and the value of α determines the value of ν, the dispersion parameter of the distribution; i.e.,
ln(ν) = ln(-Ψ(1))/α
For the AO index distribution the value of the dispersion parameter nu is 0.327.
The imaginary component of the log-characteristic function for stable distributions is:
Φ(ω) = δω + βν|ω|αF(ω,α,ν)
where F(ω,α,ν) for α≠1 and ω>0 is tan(πα/2). With α and ν known the values of δ and β can be determined from the imaginary component of the log-characteristic function. The values of δ and β can be found from any two points on the curve; i.e., by solving the linear equations in the two unkowns δ and β:
Φ(ω1) = δω1 + βν|ω1|αF(ω1,α,ν)
Φ(ω2) = δω2 + βν|ω2|αF(ω2,α,ν)
The data for the AO index are plotted below.
The values of δ and β can be computed from the two points, for ω=0.25 and ω=2. The values are
δ = -0.0.01888
β = -0.0270806
The values for the higher frequencies which did not fit on a smooth curve were not shown. The failure of these values to fit into the pattern can be attributed to the problems associated with using the midpoints of the ranges in computing the characteristic function.
Conclusion
The general conclusion is that distribution for the AO index can be represented as Lévy stable function. The values of the alpha parameter of 1.78 indicates that the distribution has higher probabilities of extremely large deviations from the mean and also higher probabilities of small deviations from the mean than a best-fitting normal distribution would have. On the other hand it has lower probabilities of moderate deviations from the mean than a best-fitting normal distribution would have.
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