Statistics of Multiple Constraints: Analysis of Global ...inez/MSRI-NCAR_CarbonDA/... · Statistics...
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Statistics of Multiple
Constraints:
Analysis of Global Change
and the Carbon Cycle
Ian G. Enting
MASCOS
The University of Melbourne
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Acknowledgments
The ARC funds the Center of Excellence for
Mathematics and Statistics (MASCOS).
My fellowship at MASCOS is supported by CSIRO
through a sponsorship agreement.
Collaborators:
Cathy Trudinger and YingPing Wang of CSIRO Marine
and Atmospheric Research.
Roger Francey, Denis O’Brien, Peter Rayner, formerly
of CSIRO Atmospheric Research.
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Summary
Theme: reviewing time series analysis for:
• improvement of modelling
• understanding issues of spatial analysis
Topics:
• interpreting the carbon cycle
• inversions as statistics
• digital filtering — resolution
• possible problems — beyond Gaussian
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Data for Carbon Cycle Studies
• Air sampling networks interpreted by inverse
modelling;
• Satellite data, for quantities such as leaf-area index
and phenology
• Terrestrial biosphere models;
• Convective boundary layer measurements;
• Stand-level flux networks;
• Ecosystem experiments;
• Small cuvettes.
Also, potentially satellite concentration data.
From Canadell et al. 2000.Workshop on Data Assimilation for the Carbon Cycle
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Key characteristics of statistics
• magnitude;
• degree of correlation between components;
• temporal correlation structure;
• spatial correlation structure;
• distribution;
• mismatches in averaging;
• contribution from model representativeness error.
From Raupach et al. 2005
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Interpretation
Interpretation is an inverse problem, working
backwards from results to causes.
Two main inverse problems: calibration and
data assimilation (deconvolution).
C(t) = C(0) =∫ t
0R(t− t′)S(t′) dt′
C(t) = C(0) =∫ t
0R(t′′)S(t− t′′) dt′′
The problems of deducing model response,
R(t) and forcing term S(t) are formally
equivalent, but in practice differ in the
characteristics of the statistics.
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Inverse Problems as Statistics
• Any uncertainty analysis needs to be based
on statistics.
• Any statistical analysis is assuming (either
implicitly or explicitly) some statistical
model.
• any variability that cannot be modelled
deterministically . . . must be . . . modelled
statistically (Enting, 2002).
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Aims of statistical analysis
• Compact characterisation of variability
• Design of techniques for data processing:
statistical efficiency, robustness etc
• Formalism for propagating uncertainties
through chain of calculations
• Design of new experiments:
How much is uncertainty reduced?
• Testing statistical assumptions underlying
data analysis techniques
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Origins of Uncertainty
For empirical quantities: ‘model’ and ‘data’
• Statistical variation
• Variability
• Inherent randomness
• Subjective judgement
• Linguistic imprecision
• Disagreement
• Approximation
From Morgan and Henrion (my split), Uncertainty (CUP, 1990).
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Combining information
Bayesian: Pr(x|z) ∝ Pr(z|x)Pr0(x)
Multiple constraints:
Pr(x|z1, z2) ∝ Pr(z2|x)Pr(z1|x)Pr0(x)
∝ Pr(z2|x)Pr1(x)
For linear relations with multivariate normal
distributions, constraints zj = Gjx + ej, with
inverse covariances, Xj, combine to give
inverse covariance, W , as:
W = Wprior +∑
GjTXjGj for estimate
x = [Wprior +∑
GjTXjGj]
−1[W x0 +∑
j GjTXjzj]
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Digital filtering
Model is zk = sk + nk where signal has power
spectrum, fs(θ) and noise has spectrum fn(θ).
Estimate signal as sk =∑
j Φjzk−j
Mean square error of estimate:
E[(sk − sk)2] = bias + variance:
MSE =∫ π
−π
[|1− φ(θ)|2fs(θ) + |φ(θ)|2fn(θ)
]dθ
Optimal filter: φ(θ) = fs(θ)/[fn(θ) + fs(θ)]
with
MSE =fs(θ)fn(θ)
fs(θ) + fn(θ)=
1
fs(θ)+
1
fn(θ)
−1
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Characterising resolution
Use characteristic numbers:
Nobs How many observations?
Ndata How many effectively independent observations?
Kcomp How many components used in the calculations?
Msignal How many components needed to specify
signal?
Ksynth How many components used to fit the signal?
Ms:n How many signal components exceed the noise
level?
Ktarget How many signal components is one trying to
estimate?
Expanded from Enting (2002: Section 8.3), including Msignal and
distinguishing Nobs from Ndata.Workshop on Data Assimilation for the Carbon Cycle
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More data: Ms:n < Ndata
fsig(θ)
fnoise(θ)
φopt(θ)
Integrand
of MSE
0 2012345
0 20
1
0 20
1
0 2 4 6012345
0 2 4 60
1
0 2 4 60
1
0 2 4 6 8 10 12012345
0 2 4 6 8 10 120
1
0 2 4 6 8 10 120
1
MSE ∼ 1/Ndata due to reduced aliasing in noise.Workshop on Data Assimilation for the Carbon Cycle
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Correlated data: Ndata < Nobs
fsig(θ)
fnoise(θ)
φopt(θ)
Integrand
of MSE
0 2012345
0 20
1
0 20
1
0 2 4 6012345
0 2 4 60
1
0 2 4 60
1
0 2 4 6 8 10 12012345
0 2 4 6 8 10 120
1
0 2 4 6 8 10 120
1
No change in MSE unless increasing Nobs increases Ndata.Workshop on Data Assimilation for the Carbon Cycle
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Aliasing: Ndata < Msignal
fsig(θ)
fnoise(θ)
φopt(θ)
Integrand
of MSE
0 2012345
0 20
1
0 20
1
0 2 4 6012345
0 2 4 60
1
0 2 4 60
1
0 2 4 6 8 10 12012345
0 2 4 6 8 10 120
1
0 2 4 6 8 10 120
1
MSE too low, due to ignoring aliasing (truncation error).Workshop on Data Assimilation for the Carbon Cycle
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Aliasing: Ndata < Msignal
fsig(θ)
fnoise(θ)
φopt(θ)
Integrand
of MSE
0 2 4 6 8 10 12012345
0 2 4 6 8 10 120
1
0 2 4 6 8 10 120
1
0 2 4 6012345
0 2 4 60
1
0 2 4 60
1
0 2012345
0 20
1
0 20
1
Treating aliased signal as an error contribution.Workshop on Data Assimilation for the Carbon Cycle
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Aliasing from Truncation error
(i) objective, (ii) risk, (iii) hope, (iv) Wunsch, (v)
strong priors, (vi) correction.
(i)
desired solution
a
b (ii)
solution from biased data
a
b
unifo
rm
basis
Constraintfrom data
(iii)
solution from unbiased data
a
b
(iv)
project from full space
a
b (v)
work in target space
a
b shap
edba
sis
(vi)
apply truncation error to data
a
b
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Smoothing splines
Fit a set of data, zn, with a smooth curve,
f(t) chosen to minimise
Θ =∑j[zj − f(tj)]
2 + λ∫ tNt1
[f ′′(t)]2 dt
Spline acts as approximate digital filter withφ(θ) = 1/[1 + (θ/θ0.5)
4]
where 50% attenuation occurs at:θ0.5 = 2π/T0.5 = [λ∆t]−1/4
Fit is linear in data, so data uncertainties can
be linearly propagated through calculations.
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Spline example: Law Dome CH4
Spline fit, f(t), with ±2 s.d.
data uncertainty.
Growth rate, ˙f(t), and
source estimate,˙f(t) + ˙f(t)/τ , also ±2 s.d.
Lower pre-1500 data density
implies smoother spline.
Uncertainties are uncertainty
in spline (smooth part of
source), not uncertainty in
complete source function.
Have chosen Ktarget !!
1000 1200 1400 1600 1800600
700
800
900
CH
4 (p
pb)
1000 1200 1400 1600 1800-5
0
5
10
CH
4 in
crea
se (
ppb/
yr)
1000 1200 1400 1600 180060
70
80
90C
H4
inpu
t (pp
b/yr
)
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Kalman filter paradigm
Mixed deterministic-stochastic model assumes evolution
of a state, x by: x(n + 1) = F (n)x(n) + u(n) + w(n)
with indirect noisy observations: z(n) = H(n)x(n) + e(n)
Where F , H and u are taken as known, and w and e are
zero-mean multivariate normal with known covariances Q
and R.
Kalman filter formalism gives the optimal estimates, x(n)
of state, given z(1) to z(n).
Combines mulitvariate normal distributions of
observations, z(n), and projection x(n|z1 . . . zn−1).
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Modelling for Kalman filter
State-space model for methane from ice cores:
x1 is methane concentration, x2 is source.
F =
1−∆t/τ a∆t
0 1
Q =
0 00 Q
H = [1,0] (or ([0,0] if no data).
Data uncertainty R, unit conversion factor, a.
Thus, source is modelled as a ‘random walk’.
Simplified from Trudinger et al, 2002.
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Kalman filter response
Frequency domain characterisation of stationary
case of Kalman filter
E.g. noise as white
noise, fn = R/2π
Random walk model of
forcing
fs ∝ 1/(1− cos θ) 0.00 0.05 0.10 0.15 0.20 0.25frequency (y-1)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
freq
uenc
y re
spon
se
20.0 10.0 6.7 5.0 4.0period (y)
R/Q ratio (9/81, 25/81, 49/36 for ∆t = 2, and
49/1 for ∆t = 20, right to left) changes the
relative weights of ‘prior’ and observations in
frequency-dependent manner.Workshop on Data Assimilation for the Carbon Cycle
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Kalman filter on CO2
Combined model, including
CO2 and 13CO2, using
concentration values
corrected for firn diffusion.
Case on left uses published
δ13C uncertainties.
Case on right multiplies
these by four.
Again, scale of uncertainty
affects the optimal
smoothing.
(From Trudinger et al., 2002).
1850 1900 1950 2000280
300
320
340
360
CO
2 (
ppm
)
a)
1850 1900 1950 2000Year
-8.0
-7.5
-7.0
-6.5
-6.0
δ13C
(pe
rmil)
b)
1850 1900 1950 2000Year
-3
-2
-1
0
1
Sour
ce (
Gt C
/yr)
c)
Deduced sources
oceanbiosphere
1850 1900 1950 2000280
300
320
340
360
CO
2 (
ppm
)
d)
1850 1900 1950 2000Year
-8.0
-7.5
-7.0
-6.5
-6.0
δ13C
(pe
rmil)
e)
1850 1900 1950 2000Year
-3
-2
-1
0
1
Sour
ce (
Gt C
/yr)
f)
Deduced sources
oceanbiosphere
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Time-dependent CO2 inversion
As with simple Kalman filter case, statistical
assumptions about structure of errors in time
can greatly influence what is estimated:
• Synthesis in terms of independent monthly
pulses (Rayner et al 1999), effectively
assumes no long-term systematic error.
• Representing prior information as
’mean-plus-anomaly’ avoids artificially-low
uncertainty on posterior mean.
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The hard stuff: spatial statistics
• Two-dimensional, (or 2 + 1 when time
involved)
– more modes to consider
– more complex statistics are possible
• Non-stationary (c.f. Kalman filter)
• Response (atmospheric transport) involved
for concentration data (c.f. Kalman filter)
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Spatial statistics:
In more than one
dimension, more complex
behaviour can occur.
Simple local conditional
dependence can lead to
long-range (fractal)
behaviour.
Closed-form expressions
for model statistics are
seldom known.
Identifying appropriate
statistical models is
correspondingly hard.Example from statistical physics, showing multiple length-scales.
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Terrestrial distribution
In process inversions, biome-specific distributions will
be modulated by climatic variations.
Map data from Matthews 1983.Workshop on Data Assimilation for the Carbon Cycle
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Grassland footprint
Distribution of grasslands reflects a 13C signal due to
C3-C4 differences.
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Number of modes
In an ill-conditioned inverse problem, only a
limited number of modes will be resolved by
the data.
• these may not be the modes that you wantto know about
• you don’t get to choose which modes areresolved
• c.f. biased estimates by Fan et al. byputting fine source discretisation in region(Nth America) without corresondingly finedata set.
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Semi-quantitative
An exploratory signal-to-noise analysis doesn’t
need to be as precise as actual inversion.
x =∑j
ajzj
typically with lots on near-cancellations in an
ill-conditioned problem.
E[(x− x)2] =∑j|aj|2Rj
and so the variance calculation is much less
sensitive to errors in the aj (i.e. model error).
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Toy model
• Model atmospheric transport as pure
anisotropic diffusion
• Eigenmodes are spherical harmonics.
Amplification factor for relating surface
sources to surface concentration response is:
γnmk tanh(γnmk) where
γ2nmk
(p0 − p1)2=
κθ[n(n + 1) + αM2]
κpR2earth
+2πik
κpT
Vertical, N-S and E-W diffusion: κp, κθ, ακθ
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Reality check
• Amplitude of responses for full 3-D model
are approximated well by the 1/n response
to latitudinal variation
• Inversion factor ∝ n comparable to
numerical differentiation,
• Suggests that latitudinally-integrated flux
could have much less correlation, than the
actual flux estimates – confirmed by results
from actual inversions
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Generalities
• Satellite data don’t have the
1/√n(n + 1) + αm2 attenuation factor
(but vertical averages are still attenuated
relative to surface distribution)
Low precision can give valuable constraints.
• For comparable spatial resolution, E-W
sampling density possibly needs to be
greater than N-S density (however, consider
re-visiting this, with ‘toy model’ including
‘solid-body rotation’, for advective term).
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From point to globe
Observations:Fluxes, forcings and proxies
↓Local parametric model
↓← Global distribution
↓ of forcing and/or proxy
Global model
Contributions to uncertainty in global model are:
global distribution * parametric uncertainty +
parametric sensitivity * uncertainty in distribution
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What if it ain’t so?
If Pr(z|x) ∝ exp(−λ|z − h(x)|) and not
Pr(z|x) ∝ exp(−λ|z − h(x)|2) then
• 90% of the ‘nice’ mathematics goes away
• Results of inversions are terrible
It’s not the things you don’t know that get you into
trouble — it’s the things you know for sure that just
ain’t so. (Mark Twain)
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Central value
For Pr(z|θ) = exp(−|z − θ|2/σ2) then
maximum likelihood estimate is
θN = 1N
∑n zn batch mode
θn+1 = θn + 1n+1
[zn+1 − θn
]recursive
Each estimate θn contains all information about
all previous data values needed for optimal
estimator for the next step (given new data).
For Pr(z|θ) = exp(−|z − θ|/σ) the maximum
likelihood estimate θn is median of all previous
zm ( m ≤ n) — no compact stepwise algorithm.
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Implications for inversions
If Pr(z|x) ∝ exp(−λ|z − h(x)|) then
1: 90% of the ‘nice’ mathematics goes away
• you don’t have compact ‘sufficient
statistics’
• multi-variate cases are really messy
• automatic differentiation isn’t a lot of use
2: Results of inversions are terrible
• major features depend on how much you
believe Samoa
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Emerging future directions
Inversion methodology
Shrinkage estimators :
Estimating variances :
Better statistics
Errors-in variables : e.g. using proxy source
data.
Hidden Markov models :
Financial time series models : e.g.
Stochastic volatility.
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Concluding thoughts:
Take-home questions:
• What are your calculations really
estimating?
• Can what you want really be estimated
given limits to resolution imposed by
model, data, ill-conditioning and
signal-to-noise ratio?
• Are the results, including residuals,
consistent with the statistical assumptions?
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Further Information
• I. Enting: Characterising the Temporal Variability of
the Global carbon Cycle. CSIRO Atmospheric
Research, Technical paper 40.
• I. Enting: Inverse Problems in Atmospheric
Constituent Transport. 2002, CUP.
• C. Rodenbeck: Estimating CO2 sources and sinks
. . . . MPI-BGC Technical Report 6.
See also:
• Trudinger et al (2002a,b): Kalman filter analysis of
ice-core data: 1 and 2. JGR
• Enting, Trudinger and Etheridge: Propagating data
uncertainty through smoothing spline fits. Tellus:
(in press, 2006).
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