Atmospheric Power Law BehaviorA Look at Southeastern US Temperatures

James Duncan

Motivation & Introduction

Extreme climatic events are weather phenomena

that occupy the tails of a dataset‟s probability density

function (PDF).

Advanced stochastic theory asserts that power law

distributions should exist in the tail ends of our data.

Questions to Answer:

Show That Power Law Distributions Are Evident within

Temperature Data.

Analyze how power law distributions change with varying

weather and climatic patterns (seasons, ENSO, etc.).

What is a Power Law

Distribution?Mathematically, a power law probability distribution of

quantity x may be written as:

Where α is the exponent or scaling parameter and C is

the normalization constant.

p(x) =a -1

xmin

x

xmin

æ

èç

ö

ø÷

-a

p x( ) =Cx-a

[Neelin et al. 2011]

Data & Methods

Data

Methods

Daily observed maximum and minimum temperatures across the

southeastern United States (AL, FL, GA, NC, SC) spanning 1960-

2009.

Measures of quality control have been put in place resulting in the

omission of 20 stations.

Trends have been removed from the data. If data is to follow a power law distribution, it does so above some

lower bound xmin.

To find our lower bound, we employ the Kolmogorov-Smirnov or KS

Statistic which calculates the maximum difference between the CDF of the

observed data and estimated power law distribution.

To calculate our scaling parameter α, we employ the “method of

maximum likelihood”.

 

ˆ a =1+ n lnx i

xmini=1

n

åé

ë ê

ù

û ú

-1

 

D = maxx³xmin

|F(x) -P(x) |

Significance Testing

Employ the use of a goodness of fit test which will

measure and analyze the KS distance of our power

law distribution with that of other synthetically

derived power law distributions.

From this goodness of fit test, we are able to derive

a „p-value‟ which expresses the probability that the

estimated power law distribution is a good fit to the

observed data.

Presence of Power Law

Distributions

Power Law Fit &

Significance Skewness

KurtosisP-Value Tests

Criteria Is Power Law Fit Significant?

Ppower>0.10 and Pgauss<0.10 YES

Ppower<0.10 and Pgauss>0.10 NO

Ppower>0.10 and Pgauss>0.10 but Ppower>Pgauss Both Fits Are Significant, But Can Say Power Law is Better Fit (YES)

Ppower<0.10 and Pgauss<0.10 NO

Xmin & Alpha

More analysis is needed to adequately note whether patterns exist in the spatial

distributions of Xmin and Alpha.

Distinguished Power Laws

Tamiami, FL

Distinguished Criteria (ppower>0.90 & pgauss~0)

Hialeah, FL

Asheville, NC

Henderson, NC

Maximum

Temperatures

Minimum

Temperature

s

Seasonal Shifts in

Power Law Distributions

Now that we have established that power law

distributions are existent, how are they

modulated by changes in the seasonal cycle?

Fall Power Law Fit &

Significance Skewness

KurtosisP-Value Tests

Criteria Is Power Law Fit Significant?

Ppower>0.10 and Pgauss<0.10 YES

Ppower<0.10 and Pgauss>0.10 NO

Ppower>0.10 and Pgauss>0.10 but Ppower>Pgauss Both Fits Are Significant, But Can Say Power Law is Better Fit (YES)

Ppower<0.10 and Pgauss<0.10 NO

Future Work & Conclusions

There does appear to be a dynamic link between

areas of significant power law fit and areas of distinct

skewness and kurtosis.

Further examine how power law distributions change

with respect to season, ENSO, and other climatic

cycles.

Look to see if these modulations in the power law

distribution may be explained by any specific physical

processes.

Look into more ways to objectively characterize

changes in the power law parameters (Xmin and

Alpha) and distribution.

ReferencesClauset, A., C. R. Shalizi, and M. E. J. Newman, 2009: Power-law distributions in empirical data, SIAM Rev., 51, 661-

703.

Neelin, D., and T. W. Ruff, 2011: Long tails in regional surface temperature probability distributions with implications for

extremes under global warming. Geophys. Res. Lett., 39, l04704, doi: 10.1029/2011GL05061.

Newman, M. E. J., 2005: Power laws, Pareto distributions and Zipf‟s law, Contemp. Phys., 46, 323-351.

Sura, P., 2011: A general perspective of extreme events in weather and cliamte. Atmos. Res., 101, 1-21.

Stefanova, L., P. Sura, and M. Griffin, 2012: Quantifying the non-Gaussianity of wintertime daily maximum and

minimum temperatures in the Southeast United States. J. Climate, in press.

Winter Power Law Fit &

SignificanceSkewness KurtosisP-Value Tests

Criteria Is Power Law Fit Significant?

Ppower>0.10 and Pgauss<0.10 YES

Ppower<0.10 and Pgauss>0.10 NO

Ppower>0.10 and Pgauss>0.10 but Ppower>Pgauss Both Fits Are Significant, But Can Say Power Law is Better Fit (YES)

Ppower<0.10 and Pgauss<0.10 NO

Values of Xmin

Appears to be several more distinct regions of behavior than Annual Behavior;

however, more analysis and comparison is need to adequately depict the

potential patterns developing spatially.