YODEN%20Shigeo

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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. Introduction
2. Statistical considerations
3. Internal variability in a numerical model
4. Spurious trend experiment
5. Concluding remarks

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1. Introduction

• Causes of interannual variations of
• the stratosphere-troposphere coupled system

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• Observed variations

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Linear Trend of the Monthly Mean
Temperature ( Berlin, NCEP )
A spurious trend may exist in finite length
dataset with natural variability.
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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
• Spurious trend
• We estimate the linear trend
• by the least square method
• We define a spurious trend as

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

standard deviation of natural variability
Monte Carlo simulation with Weibull (1,1)
distribution
kurtosis of natural variability
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• Probability density function (PDF)
• of the spurious trend
• When the natural variability is Gaussian
  distribution
• When it is non-Gaussian
• Edgeworth expansion of the PDF
• Cf. Edgeworth expansion of sample mean (e.g.,
  Shao 2003)

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• Non-Gaussian distribution

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 polynomial and is k - th cumlant (
).
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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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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.
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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

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• Labitzke diagram for normalized temperature
  (15,200 years)

stratosphere
troposphere
Different dynamical processes produce these
seasonally dependent internal variabilities ? An
nual 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)/se
• Standard deviation of sample mean
• The distribution converges to a normal
  distribution

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• Spatial and seasonal distribution of moments
• 10 ensembles of 1,520-year integrations
• without external trend
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• More information
• moments of variations ? moments of spurious trends

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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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• How small trend can we detect
• in finite length data with statistical
  significance ?

50-year data
20-year data K/decade
K/decade
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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
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JAN (large internal variation)
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Ensemble mean of estimated trend and standard
deviation of spurious trend
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• Comparison of significance tests
• Edgeworth test true
• The worst case in 96 runs
• but both test look good

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• Application to real data
• 20-year data of NCEP/NCAR reanalysis

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Bootstrap test
t-test
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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

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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 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

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• 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

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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
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• Normalized estimated trend and significance

90N
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Daily Temperature at 30 hPa K for 19 years
(1979-1997)
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

North 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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Normalized sample variance
stratosphere
troposphere

• The distribution is similar to ?2distribution in
  the troposphere, where internal variability has
  nearly a normal distribution
• Standard deviation of sample variance

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Sample skewness
stratosphere
troposphere
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Sample kurtosis
stratosphere
troposphere
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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-test
• If the distribution of is Gaussian,
• then the test statistic
• follows the t-distribution with the
  degrees of freedom n -2

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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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EQ

• Time series
• of monthly averaged zonal-mean temperature
• January

90N
50N
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EQ
July
90N
50N
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• Same period (1981-2000)
• January

90N
50N
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July
90N
50N
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• Same vertical factor
• January

90N
50N
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July
90N
50N
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• Mean
• 90N

Mean difference
from ERA40
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50N
Mean difference
from ERA40
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• standard deviation
• 90N

stddev difference
from ERA40
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50N
stddev difference
from ERA40