Transcript Numerical Experiments on Two-dimensional Turbulence on a

March 3-4, 2005: SPARC Temperature Trend Meeting at University of Reading
Spurious Trend
in Finite Length Dataset
with Natural Variability
YODEN Shigeo
Dept. of Geophysics, Kyoto Univ., JAPAN
1.
2.
3.
4.
5.
Introduction
Statistical considerations
Internal variability in a numerical model
Spurious trend experiment
Concluding remarks
1. Introduction

Causes of interannual variations of
the stratosphere-troposphere coupled system
Yoden et al. (2002; JMSJ )
ramdom process
(asumption)
monotonic change
response
(linear) trend

Observed variations
Labitzke Diagram (Seasonal Variation of Histograms
Length
of Mean
the observed
dataset
of the
Monthly
Temperature;
at 30 is
hPa)
South Pole
(NCEP)
atNorth
most
Pole50 years.North Pole
(NCEP)
(Berlin)
Separation of the trend from
natural variations is a big problem.
Linear Trend of the Monthly Mean Temperature
( Berlin, NCEP )
A spurious trend may exist in
finite length dataset with natural variability.
2. Statistical considerations
Nishizawa and Yoden (2005, JGR in press)
 Linear trend



We assume a linear trend
in a finite-length dataset with random variability  i (n)
X i (n)  an  b   i (n),
n  1, , N
Spurious trend


We estimate the linear trend
by the least square method
Xˆ i (k )  aˆi k  bˆi
N = 5 10
We define a spurious trend as
a 'i  aˆi  a
N
12
N 1

a 'i 
n



 i (n)
N ( N  1)( N  1) n 1 
2 
20
N = 50

Moments of the spurious trend

Mean of the spurious trend is 0

Standard deviation of the spurious trend is
a' 
3

12
   12 N 2  
N ( N  1)( N  1)

Skewness is also 0

Kurtosis is given by
kurtosis of
natural variability
standard deviation of
natural variability
+ Monte Carlo simulation
with Weibull (1,1) distribution

Probability density function (PDF)
of the spurious trend

When the natural variability is Gaussian distribution
fa ' ( x)  f N

0, 
2
a'
( x) 
1
2 a '
e
1 x 
 

2 a' 
2
.
When it is non-Gaussian
 Edgeworth expansion of the PDF
 Cf. Edgeworth expansion of sample mean (e.g., Shao 2003)

Non-Gaussian distribution
Edgeworth expansion of the cumulative distribution function,
of
is written by
and
and
is the PDF and the distribution function
of N (0,1) , respectively.
where
and
is k - th Hermite polynomial
is k - th cumlant (
).

Errors of t -test, Bootstrap test, and Edgeworth test for a
non-Gaussian distribution of
for a finite data length N
We need accurate values of
the moments of natural internal variability
for accurate statistical text.
But the length of observed datasets is
at most 50 years.
Only numerical experiments
can supply much longer datasets
to obtain statistically significant results,
although they are not real but virtual.
3. Internal variability in a numerical model

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

Labitzke diagram for normalized temperature (15,200 years)
stratosphere
troposphere
Different dynamical processes
produce these seasonally
dependent internal variabilities
↓
“Annual mean” may introduce
extra uncertainty or danger
into the trend argument

Estimation error of sample moments

depends on deta length N and PDF of internal variability

Normalized sample mean: (mN －μ)/σε
stratosphere
troposphere
-
1
2
 Standard deviation of sample mean s m = N s e
N
 The distribution converges to a normal distribution
as N becomes large (the central limit theorem)

sample variance [ skewness, kurtosis, ... ]

Spatial and seasonal distribution of moments


10 ensembles of 1,520-year integrations
without external trend
Zonal mean temperature
65

More information
 moments of variations → moments of spurious trends

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
Necessary length for 99% statistical significance [years]
87N
47N

How small trend can we detect
in finite length data with statistical significance ?
50-year data
[K/decade]
20-year data
[K/decade]
4. Spurious trend experiment

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]
JAN
(large internal variation)
JUL
(small internal variation)
Ensemble mean of estimated trend
and standard deviation of spurious trend

3
2
Standard deviation
of internal variability

3
2
Theoretical result sa '  12 N s , (50 / 20)  0.25298

Comparison of
significance tests



t-test
Edgeworth test: true
The worst case in 96 runs
but both test look good
Edgeworth test
Bootstrap test

Application to real data

20-year data of NCEP/NCAR reanalysis
Bootstrap test
t-test
5. Concluding remarks

Recent progress in computing facilities has
enabled us to do parameter sweep experiments
with 3D Mechanistic Circulation Models.

Very long-time integrations (~15,000 years)
give reliable PDFs (non-Gaussian, bimodal, …. ),
which give nonlinear perspectives
on climatic variations and trend.

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

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 runs
estimated trend in a run:
mean of the estimated trends:
two L-year time slice runs
estimated mean in each run:
estimated trend:
comparison under the same cost: mN = 2L

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.

New Japan reanalysis data JRA-25
 now internal evaluation is ongoing
Time series of monthly averaged zonal-mean temperature
January

Estimated trend [K/decade]
90N

Normalized estimated trend and significance
90N
Thank you !

Estimated trend [K/decade]
90N
50N

Normalized estimated trend and significance
90N
50N
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

Difference of Gaussian distribution and Edgeworth for a
non-Gaussian distribution of
for a finite data length N
3. Spurious trends due to
finite-length datasets with internal variability

Nishizawa, S. and S. Yoden, 2005:

Linear trend



Estimation of sprious trend


Weatherhead et al. (1998)
Importance of variability
with non-Gaussian PDF



IPCC the 3rd report (2001)
Ramaswamy et al. (2001)
SSWs
extreme weather events
We do not know


PDF of spurious trend
significance of the estimated
value
Normalized sample variance
stratosphere


troposphere
The distribution is similar to χ2distribution in the
troposphere, where internal variability has nearly a
normal distribution
Standard deviation of sample variance ss 2   2  2 / N 
Sample skewness
stratosphere
troposphere
Sample kurtosis
stratosphere
troposphere

Years needed for statistically significant trend


-0.5K/decade in the stratosphere
0.05K/decade in the troposphere

Significance test of the estimated trend

t-test
If the distribution of  i is Gaussian,
then the test statistic
ti 
ai '
,
12
si
N ( N  1)( N  1)


1 N
si 
X i (n)  aˆi n  bˆi

N  2 n 1
2

2
follows the t-distribution with the degrees of freedom n -2
2. Trend in the real atmosphere
Datasets

ERA40




1948-2003
1000-10 hPa
JRA25

1958-2002
1000-1 hPa
NCEP/NCAR




1979-1985,1991-1997
1000-1 hPa
Berlin Stratospheric data


1963-2000
100-10 hPa

Time series
of monthly averaged
zonal-mean temperature
January
90N
EQ
50N
EQ
July
90N
50N

Same period (1981-2000)
January
90N
50N
July
90N
50N

Same vertical factor
January
90N
50N
July
90N
50N

Mean
90N
Mean
difference from ERA40
50N
Mean
difference from ERA40

standard deviation
90N
stddev
difference from ERA40
50N
stddev
difference from ERA40