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Statistical analysis of global temperature and precipitation data. Imre Bartos, Imre Jánosi Department of Physics of Complex Systems Eötvös University. The GDCN database Correlation properties of temperature data Short-term Long-term Nonlinear Cumulants Extreme value statistics - PowerPoint PPT Presentation
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Statistical analysis of global temperature and precipitation
data
Imre Bartos, Imre JánosiDepartment of Physics of Complex Systems
Eötvös University
Outline
• The GDCN database
• Correlation properties of temperature data
• Short-term
• Long-term
• Nonlinear
• Cumulants
•Extreme value statistics
• Recent results
• Degrees of Freedom estimation
Global Daily Climatology Network
Temperature stations
Precipitation stations
32857 stations…
1950-2000
Correlation properties
Short-term correlation
Long-term correlation
Ti
Correlation properties
Short-term correlation
Long-term correlation
Ti
ai+1 = Ti+1 - Ti+1 = F(ai) + i
Short term memory: exponential decay
Autoregressive process:
Linear case: AR1ai+1 = A ai + i
C1() = aiai+ ~ A
Short-term correlation
ai+1 = A ai + i
in terms of temperature change:
ai+1 = ai+1 – ai ~ Ti+1 – Ti = (A-1) ai + i
thus the response function one measures:
ai+1 = (A-1) ai + 0
The fitted curve:
ai+1 = c1 ai + c0
Király, Jánosi, PRE (2002).
Short-term correlation
ai+1 = c1 ai + c0
c0
c1
Bartos, Jánosi, Geophys. Res. Lett. (2005).
• |c1| it increases to the South-East• c0 != 0 significantly
ai - asymmetric distribution
Short-term correlation
• more warming steps (Nm) then cooling (Nh)
Bartos, Jánosi, Geophys. Res. Lett. (2005).
• the average cooling steps (Sh) are bigger then the average warming steps (Sm)
• Warming index:
W = (Nm Sm) / (Nh Sh)
Do these two effects compensate each other?
asymmetric distribution
Global warming (?)
Correlation properties
Short-term correlation
Long-term correlation
Ti
C() = aiai+ ~ -
Long term memory: power decay
Long-term correlation
Measurement: Detrended Fluctuation Analysis (DFA)
F(n) ~ n
= 2 (1 - )
C() = aiai+ ~ -
DFA curve:
Initial gradient (0)
Asymptotic gradient ()
~ long-term memory0 ~ short term memory
Detrended Fluctuation Analysis (DFA)
Király, Bartos, Jánosi, Tellus A (2006).
All time series are long term correlated
Nonlinear correlation
Linear (Gauss) process: Cq>2 = f(C2) (3rd or higher cumulants are 0)
Two-point correlation: C2 = aiaj, q-point correlation: Cq = F(aiajak…)
C2 completely describes the process
Nonlinear (multifractal) process: 3rd or higher cumulants are NOT 0
the 2-point correlation doesn’t give the full picture
One needs to measure the nonlinear correlations for the full description
Nonlinear correlation
The 2-point correlation of the volatility time series features the nonlinear correlation properties of the anomaly time series
ai |ai+1 - ai| „volatility” time series:
volatility - DFA exponent
Nonlinear correlation
There is also short- and long-term memory for the volatility time series
volatility - initial DFA exponent
In short…
Daily temperature values are correlated in both short and long terms and both linearly and nonlinearly.
We constructed the geographic distributions for these properties, and described or explained some of them in details.
volatility - initial DFA exponent
Cumulants
skewness
kurtosis
- nonuniform can affect the EVS
Extreme value statistics
• we want to use temperature time series
• temperature
• anomaly
• normalized anomaly
Extreme value statistics
• we try to get rid of the spatial correlation
lets use one station in every 4x4 grid
Dangers in filtering for extreme value statistics
• after filtering out the flagged (bad) data:
cutoff at 3.5
Daily normalized distribution
seems exactly like a Weibull distribution
Explanation: preliminary filtering of „outliers”
Then how can we filter out bad data??
Extreme value statistics
There are certainly bad data in the series. The usual way to filter them out is to flag the suspicious ones, but it seems we cannot use the flags.
One try to find real outliers:
Temperature difference distribution
Impossible to validate
Another possible way:
try to isolate unreliable stations
Extreme value statistics
Now we use all the data without filtering spatial correlations
Also notice the two peaks
New problem: the two peaks
Extreme value statistics
What makes the average maximum values differ for some stations?
Why two peaks?
skewness kurtosis correlation
depends doesn’t depend doesn’t depend
New problem: the two peaks
Extreme value statistics
Average yearly maximum
One can spatially separate the different peaks
Separate one peak by using US stations only:
Extreme value statistics
Finally we get to the Gumbel distribution
Degrees of Freedom
Why does the average maximum value not depend on the correlation exponent?
One can calculate the degrees of freedome of N variables with
long time correlation characterized by correlation exponent
DOF = N^2 / i^2
Where i is the ith eigenvalue of the covariance matrix, containing the covariance of each pair of days of the year.
Long term correlation: C(|x-y|) = c * |x-y|^
Short term correlation: Ti+1 = A * Ti + noise
Variables determining the DOF: c, , A.
Degrees of Freedom – Dependence on correlation
C = 1 C = 0.25
C = 0.0001
Short-term
Degrees of Freedom – measurement and calculation
Estimation with with c=1
Measurement: Chi square method
(underestimation)
(underestimation)
Degrees of Freedom – difficulties
c = 1 estimation: this causes the difference
It is hard to measure anything due to the bad signal to noise rato
To say something about c: correlation between consequtive years
Imre Bartos, Imre Jánosi
Department of Physics of Complex Systems, Eötvös University
Statistical analysis of global temperature and precipitation data
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