San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

Some Statistical Characteristics of the
Arctic Oscillation Index 1899 to 2001

Meteorologist have identified some oscillations over oceanic regions that have significant impacts over wide areas. The most famous of these is the Southern Oscillation over the South Pacific between Indonesia and the western coast of South America. This gives rise to the El Niño condition which affect weather conditions over a global region for a year or so. The Southern Oscillation was discovered by the British meteorologist Gilbert Walker in his investigation of the causes for the occasional failures of the monsoons of India. For more on this oscillation see ENSO.

Another important oceanic oscillation is the North Atlantic Oscillation which affects weather in Europe and the Mediterranean regions. When there is a strong pressure difference between the Azores High and the Iceland Low the winds flowing southeast from the subarctic of North America get diverted to northern Europe bringing warmer temperatures and more precipitation. When there is not a strong pressure differential those winds reach the Mediterranean area with their moisture and northern Eurpope is left dry and cold. This is a multidecadal oscillation. For more on this oscillation see NAO.

A more rececently identified oceanic oscillation is the Pacific Multidecadal Oscillation. There is evidence that the cycle in this oscillation is the major driving force of a cycle in average global temperature that is falsely being taken a evidence for catastrophic global warming. For more on this see AGT and the PDO.

Another recently identified oscillation is the Artic Oscillation (AO) which offers the potential of explaining a great deal of climate and weather variations. For example, in the record of average global temperature (AGT) a remarkable amount of the increase has come as a result of temperature increases in central Siberia and northwest Canada. These are areas with a very low content of water vapor in the air due to the temperatures and their distance from oceans. This lack of water vapor reaches its minimum at night in the winter. When the water vapor content is negligible then the increase in carbon dioxide can have a significant effect whereas in other places where the water vapor content is 1 to 2 percent an increase in carbon dioxide content from 0.03 of 1 percent to 0.04 of 1 percent is inconsequential. The increase in the temperature in central Siberia and northwest Canada at night in the wintertime could be taken to be evidence of the impact of increase carbon dioxide in the atmosphere. However the image below from NASA suggests that the temperature change in central Siberia and northwest Canada may be a consequence of the Artic Oscillation.

The existence of the Arctic Oscillation would also explain the decline in the Arctic sea ice in the summer. This phenomenon could not be due to a global factor as it is not occurring in Antarctica. For more on this matter see Arctic sea ice.

The Arctic Oscillation Index is based upon spatial differences in sea level pressure. Two tabulations of an Arctic Oscillation Index are used for the statistical analysis here. One tabulation is available from the Joint Institute for the Study of the Atmosphere and Ocean (JISAO) of the University of Washington. The JISAO functions in collaboration with the National Oceanic and Atmospheric Administration (NOAA). The data set from JISAO covers monthly values from January of 1899 to June of 2002. For the statistical analysis the data for 2002 was left out.

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.)

A frequency distibution was tabulated and is shown below.

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. The extension of the central limit theorem removes the qualification and expands the limit distribution to the Lévy stable distributions. Some L&eactute;vy 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 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.

One thing to check is the variation in the AO index by months. The monthly average values and the monthly standard deviations are shown below.

There are several notable aspects of the monthly statistics. First there is a seasonal pattern with positive values for January through April and negative values for May through December. Second there is a surprisingly drastic shift in the values from December to January. Third, the variation is more extreme in the winter and least in the summer, as measured by the standard deviation.

The maximum and minimum values of +406 and -523 are less extreme in comparison to the standard deviation of the months in which they occur, January and February. However they are still 2.5 and 3.35 standard deviation units away from the means for their months. The probability of getting a value as high as 406 in January is about 1 out of 2500 and the probability of getting a value as low as -523 in February is about 1 out of a half million. When the other month maxima and minima are compared to their monthly standard deviations there are some that are even less likely than values of 406 and -523 noted above. Therefore it is likely that the monthly distributions of the Arctic Oscillation index are not normal.

If the distributions are not normal then they are likely to be Lévy stable distributions. Stable in this sense means that if two variables have stable distributions then their sum also has a stable distribution. The normal distribution is one instance of such stability.

The general family of stable distributions, identified by the French mathematician Paul Lévy, is characterized by four parameters.

For a normal distribution α=2, β=0, ν is equal to the standard deviation and δ is equal to the mean. For other stable distributions, while the dispersion parameter has a finite value the standard deviation per se is not finite. The sample standard deviations for variable having non-normal stable distributions are therefore meaningless.

For a procedure for estimating the values of the parameters of a stable distribution from sample observations see Stable Parameter Estimation.

Symmetric stable distributions with α less than 2 are called fat-tailed distributions because there is a higher probability of getting extreme values than there is for a normal distribution. They are also called leptokurtic. Such distributions not only have a higher probability of extreme deviations from the mean but, surprisingly, also a higher probability of getting very low deviations from the mean. What they have a low probability of getting is moderate deviations from the mean. The graph below explains the situation.

A plethora of small deviations from the mean for such distributions often leads observers into thinking that the volatility of the phenonenon is low, then comes some event that seems completely out of line. Such extreme cases are not truly out of line, they are part of the distribution.

(To be continued.)

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