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