yoden shigeo dept. of geophysics, kyoto univ., japan march 3-4, 2005: sparc temperature trend...
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![Page 1: YODEN Shigeo Dept. of Geophysics, Kyoto Univ., JAPAN March 3-4, 2005: SPARC Temperature Trend Meeting at University of Reading 1.Introduction 2.Statistical](https://reader035.vdocuments.mx/reader035/viewer/2022062516/56649d565503460f94a34b9e/html5/thumbnails/1.jpg)
YODEN ShigeoYODEN ShigeoDept. of Geophysics, Kyoto Univ., JAPAN
March 3-4, 2005: SPARC Temperature Trend Meeting at University of Reading
1. Introduction2. Statistical considerations3. Internal variability in a numerical
model4. Spurious trend experiment5. Concluding remarks
Spurious TrendSpurious Trend
in Finite Length Datasetin Finite Length Dataset
with Natural Variabilitywith Natural Variability
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Causes of interannual variations of the stratosphere-troposphere coupled system
Yoden et al. (2002; JMSJ )
1. Introduction
monotonic change response
(linear) trend
ramdom process (asumption)
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Labitzke Diagram (Seasonal Variation of Histograms
of the Monthly Mean Temperature; at 30 hPa)
South Pole(NCEP)
North Pole(NCEP)
North Pole(Berlin)
Length of the observed dataset is at most 5050 years.
Separation of the trend fromnatural variations is a big problem.
Observed variations
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Linear Trend of the Monthly Mean Temperature
( Berlin, NCEP )
A spurious trend may exist infinite length dataset with natural variability.
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Nishizawa and Yoden (2005, JGR in press) Linear trend
We assume a linear trend in a finite-length dataset with random
variability
Spurious trend We estimate the linear trend by the least square method
We define a spurious trend as
( ) ( ), 1, ,i iX n an b n n N
ˆˆ ˆ( )i i iX k a k b
1
ˆ'
12 1' ( )
( 1)( 1) 2
i i
N
i in
a a a
Na n n
N N N
( )i n
2. Statistical considerations
N = 5 10 20
N = 50
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Moments of the spurious trend Mean of the spurious trend is 0
Standard deviation of the spurious trend is
Skewness is also 0
Kurtosis is given by
3
2'
1212
( 1)( 1)a NN N N
standard deviation ofnatural variability
kurtosis ofnatural variability
+ Monte Carlo simulation with Weibull (1,1) distribution
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Probability density function (PDF) of the spurious trend
When the natural variability is Gaussian distribution
When it is non-GaussianEdgeworth expansion of the PDF Cf. Edgeworth expansion of sample mean (e.g., Shao 2003)
2
'2
'
1
2' 0,
'
1( ) ( ) .
2a
a
x
a Na
f x f x e
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Edgeworth expansion of the cumulative distribution function, of is written by
and and is the PDF and the distribution function of , respectively.
where is k - th Hermite polynomialand is k - th cumlant ( ).
Non-Gaussian distribution
(0,1)N
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Errors of t -test, Bootstrap test, and Edgeworth test for a non-Gaussian distribution of for a finite data length N
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But the length of observed datasets is at most 5050 years.
Only numerical experimentsnumerical experiments can supply much longer datasets
to obtain statistically significant results,although they are not real but virtual.
We need accurate values ofthe moments of natural internal variability
for accurate statistical text.
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3D global Mechanistic Circulation Model: Taguchi, Yamaga and Yoden(2001)
simplified physical processes
Taguchi & Yoden(2002a,b) parameter sweep exp. long-time integrations
Nishizawa & Yoden(2005) monthly mean T(90N,2.6hPa) based on 15,200 year data reliable PDFs
3. Internal variability in a numerical model
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stratosphere troposphere
Labitzke diagram for normalized temperature (15,200 years)
Different dynamical processes
produce these seasonally
dependent internal variabilities
↓“Annual mean” may introduce
extra uncertainty or danger
into the trend argument
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Estimation error of sample moments depends on deta length N and PDF of internal variability
Normalized sample mean: (mN - μ)/σε
Standard deviation of sample meanThe distribution converges to a normal distribution as N becomes large (the central limit theorem)
sample variance [ skewness, kurtosis, ... ]
stratosphere troposphere
1
2Nm N es s
-=
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Spatial and seasonal distribution of moments 10 ensembles of 1,520-year integrations without external trend
65
More informationmoments of variations → moments of spurious trends
Zonal mean temperature
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How many years do we need to get statistically significant trend ?
- 0.5K/decade in the stratosphere 0.05K/decade in the troposphere
Max value of the needed length Month for the max value
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Necessary length for 99% statistical significance [years]
87N 47N
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50-year data 20-year data
[K/decade] [K/decade]
How small trend can we detect in finite length data with statistical
significance ?
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Cooling trend run 96 ensembles of 50-year integration with external linear trend
-0.25K/year around 1hPa
Normal (present) Cooled (200 years) Difference
[K/50years]
4. Spurious trend experiment
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JAN (large internal variation)
JUL (small internal variation)
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Standard deviation of internal variability
25298.0)20/50( ,12 2
3
2
3
'
sNsaTheoretical result
Ensemble mean of estimated trend and standard deviation of spurious trend
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Edgeworth test
Comparison of significance tests Edgeworth test: true The worst case in 96
runs but both test look good
t-test
Bootstrap test
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Application to real data 20-year data of NCEP/NCAR reanalysis
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t-test
Bootstrap test
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5. Concluding remarks
Statistical considerations on spurious trend in general non-Gaussian cases:
Edgeworth expansion of the spurious trend PDF detectability of “true” trend for finite data length enough length of data, enough magnitude of trend evaluation of t-test and bootstrap test
Very long-time integrations (~15,000 years) give reliable PDFs (non-Gaussian, bimodal, …. ), which give nonlinear perspectives on climatic variations and trend.
Recent progress in computing facilities has enabled us to do parameter sweep experiments with 3D Mechanistic Circulation Models.
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Ensemble transient exp.(e.g., Hare et al., 2004) vs. Time slice (perpetual) exp.(e.g., Langematz, 200x)
assumption:internal variability is independent of time
m - member ensembles of N - year transient runsestimated trend in a run:
mean of the estimated trends:
two L-year time slice runsestimated mean in each run:
estimated trend:
comparison under the same cost: mN = 2L
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New Japan reanalysis data JRA-25 now internal evaluation is ongoing
Statistics of internal variations of the atmosphere could be well estimated by long time integrations of state-of-the-art GCMs. Those give some characteristics of the nature of trend.
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Time series of monthly averaged zonal-mean temperature
January
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Estimated trend [K/decade]
90N
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Normalized estimated trend and significance
90N
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Thank you !
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Estimated trend [K/decade]
90N
50N
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Normalized estimated trend and significance
50N
90N
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1. Introduction
Difference of the time variations between the two hemispheres
annual cycle: periodic response to the solar forcing
intraseasonal variations: mostly internal processes
interannual variations: external and internal causes
Daily Temperature at 30 hPa[K] for 19 years (1979-1997)
North Pole
South Pole
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Difference of Gaussian distribution and Edgeworth for a non-Gaussian distribution of for a finite data length N
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3. Spurious trends due to finite-length datasets with internal variability Nishizawa, S. and S. Yoden, 2005:
Linear trend IPCC the 3rd report (2001) Ramaswamy et al. (2001)
Estimation of sprious trend Weatherhead et al. (1998)
Importance of variability with non-Gaussian PDF
SSWs extreme weather events
We do not know PDF of spurious trend significance of the estimated value
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stratosphere troposphere
Normalized sample variance
The distribution is similar to χ2distribution in the troposphere, where internal variability has nearly a normal distribution
Standard deviation of sample variance Nss
/222
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stratosphere troposphere
Sample skewness
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stratosphere troposphere
Sample kurtosis
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Years needed for statistically significant trend -0.5K/decade in the stratosphere 0.05K/decade in the troposphere
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Significance test of the estimated trend
t-testIf the distribution of is Gaussian, then the test statistic
follows the t-distribution with the degrees of freedom n -2
22
1
' 1 ˆˆ, ( )212
( 1)( 1)
i
i
Ni
i i i in
at s X n a n b
Ns
N N N
i
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2. Trend in the real atmosphere
Datasets ERA40
1958-2002 1000-1 hPa
NCEP/NCAR 1948-2003 1000-10 hPa
JRA25 1979-1985,1991-1997 1000-1 hPa
Berlin Stratospheric data 1963-2000 100-10 hPa
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Time series of monthly averaged
zonal-mean temperatureJanuary
90N 50N
EQ
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90N 50N
EQ
July
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90N 50N
Same period (1981-2000)January
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90N 50NJuly
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90N 50N
Same vertical factorJanuary
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90N 50NJuly
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Mean90N
Mean difference from ERA40
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50N
Mean difference from ERA40
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standard deviation90N
stddev difference from ERA40
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50N
stddev difference from ERA40