Piet de Jong

Transcription

Piet de Jong

Inferring and predicting global temperature trends
Craig Ansley, NZ & Piet de Jong , Macquarie University,
Sydney
June 29, 2012
Craig Ansley, NZ & Piet de Jong , Macquarie University, Sydney
Temperature records – GISS, CRU and UAH
15
C
14
13
12
1840
1860
1880
1900
1920
1940
1960
1980
2000
2020
year
1562 + 1922 + 375 = 3859 monthly observations
Many other time series like this available of differing lengths,
frequency, relatedness (tree thickness), missing data
properties, validity (urban heat islands), comprehensiveness
etc. (Selection bias)
Temperature trends
Monthly – with northern/southern hemisphere seasonality
issues removed
Different starts for the 3 series – all end at 2010
All “standardized” on same (arbitrary) level – even though
they are centred differently (see GLS later)
Noisy – but each appears to suggest (global) warming
Decreasing volatility?
Evidence of this sort is basis for $trillions proposed
taxes/expenditure on climate change mitigation strategies
What really is the trend/slope out into the future and
associated confidence intervals?
Overview
Time series analysis done on such “climatic” time series is
typically inadequate an open to much criticism
Many, many people “laying in” on such analyses – similar to
stock market “chartism”
See for example the various websites: climateaudit.org,
realclimate.org etc etc for the bitterness of the controversies.
Most “statistical” analyses (even by the “climate scientists”)
ignore the time series literature in particular as well as the
broader statistical literature (significance, selection biases,
preprocessing issues often ignored/swept away).
We aim to partly critique existing approaches and provide a
more “proper” approach.
“Proper” approach is based on the ideas and methods
initiated, motivated, inspired and developed by Andrew
Harvey.
Area is controversial, politicised, at times bitter. Warrants
attention from proper time series analysts.
“Mannian” smoothing – after Mann (2004)
This may be typical what is goes in “climatic time series analysis.”
To smooth a climate series:
Use a “Butterworth” filter to smooth the time series
Where the filter runs into either end of the series just “mirror”
the time series “horizontally” about t = 0 or t = n.
To “preserve trend/slope,” mirror the mirrored series
“vertically” about y = yn . (“Double mirroring”)
Result is a smoothing procedure, unhampered by end effects,
that “properly resolves” the trend
Why Butterworth?
What, are the statistical properties of such a procedure?
Voodoo smoothing? ỹn = yn always!
How do these smoothing procedures relate to “proper” time
series procedures?
Example series: Northern Hemisphere Temperature Index –
Mann(2008)
Annual: 1850 – Current
1
correlation
0.5
C
0
-0.5
1850
1900
1950
0.5
0
-0.5
2000
0
5
year
15
20
10
15
20
1
correlation
1
correlation
10
lag
0.5
0
-0.5
0
5
10
15
20
lag
0.5
0
-0.5
0
5
lag
Harvey & Trimbur (2003)
Connection between Butterworth and Components Model
– summary
Components Model
m – order of differencing
q = σζ2 /σ�2
Butterworth
m ”Slope” of the gain function
Determine window half–width: 40 years?
λ0 equal to reciprocal of window half width
Determine q = {2 sin(λ0 /2)}2m
Note: End effects handled properly above but “improperly” by
Mann etc
Busetti Harvey (2007) Model
Suppose stochastic trend (m = 2) + AR(1)+WN
yt = µ t + a t + λ � t ,
st+1 = st + ηt ,
µt+1 = µt + st ,
at+1 = ρ at + θ νt ,
�t ∼ (0, σ 2 )
ηt , νt ∼ (0, σ 2 )
Note:
λ and θ are “noise–to–signal” parameters.
σ 2 is scale parameter: not of interest
Imagine model holds for 3 series with µt , at “common” and �t
r
r
“specific” with λ equal to θ times e 1 , e a + b t or e 3
µt to be inferred/predicted: inference/prediction is “pooled”
Each series “shifted” with “starting conditions” to “level”
each series on the common µt . How?
Forecasts + Error Bounds based on KF/SF
slope
level
16
C
C/month
17
0.005
0
15
14
13
1850
1900
1950
2000
2050
year
1850
1900
1950
2000
2050
year
Pooled estimator: > 3 series unlikely to narrow bounds
No distinction between “inferring” and “prediction”
End effects handled “properly” through use of KF/SF
MIT Professor Richard Lindzen: “no statistically significant
warming since 1995”
MIT President Susan Hockfield: climate change
“accelerating.”
Technicalities: starts and shifts
KF/SF dealt with all estimation/prediction issues including
the problem of “end effects” (vs Mannian smoothing)
KF/SF dealt with different starting points through use of
different starting conditions:
Estimates of shifts and initial conditions
estimate
std. dev
Shift
GISS
UAH
14.0753 14.3381
0.0035
0.0070
CRU
Jan. 1850 level Jan. 1850 slope
13.8225
-0.0004
0.0630
0.0006
Estimation in terms of ln(θ) and ln(λi ) = ln(θ) + ri , i = 1, 3
and r2 = a + bt allowing for changing volatility.
Model parameter estimates
estimate
std dev.
correlation
matrix
AR
ln(θ)
7.96
0.21
1.00
-0.08
-0.09
0.04
-0.06
0.06
GISS
r1
-0.86
0.06
-0.08
1.00
0.54
-0.32
0.16
0.43
parameter vector ψ
CRU
UAH
a
b
r3
0.83 -0.87 -0.22
0.04
0.03
0.03
-0.09 0.04 -0.06
0.54 -0.32 0.16
1.00 -0.75 0.28
-0.75 1.00 -0.11
0.28 -0.11 1.00
0.22 -0.10 0.06
AR coeff.
ρ
0.63
0.02
0.06
0.43
0.22
-0.10
0.06
1.00
Estimates and cov matrix derived from KF/SF based
likelihood evaluation
Parameter estimate uncertainty used to simulate future
scenarios
2060 temperature
Predicted 2060 global temperature
18
18
17
17
16
16
15
15
14
14
13
13
1.02
1.04
1.06
1.08
1.1
1.12
600 c : change in temp
0
0.2
0.4
0.6
0.8
1
exceedence probability
Figure: Forecast February 2060 temperature. The left panel plots the
forecast versus the estimate of the 2010 slope. Different points
correspond to different choices of ψ. The right panel indicates
exceedence probabilities with the flatter curve assuming ψ = ψ̂ and the
steeper curve factoring in the uncertainty regarding ψ.
Is the model adequate? – Stress tests
slope
1
1
0.8
0.8
p-value
p-value
level
0.6
0.4
0.2
0.6
0.4
0.2
0
0
1850
1900
1950
2000
year
1850
1900
1950
2000
year
Figure: Left panel display p–values of estimated shocks introduced to the
level while the right hand side is p–values for the slope.
Largest p–values for shocks to st all occur in the period
1963–1977 indicating the model is under most stress from
rapidly increasing temperatures during this period.
Little evidence that shocks are warranted to the slope post
1977.
Conclusions – “real” time series analysts should step up to
the plate
Professor Andrew Harvey has been instrumental in inspiring,
developing, applying, exploring, justifying a very useful and
powerful approach to time series analysis rooted in the
statistical paradigm
This approach is largely if not completely ignored by the
“climate science community”
The approach is completely applicable and useful for the
extensive set of climate time series
More should be done to “muscle in” on this territory.

Piet de Jong

Transcription

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