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Estimation and Prediction of Complex Systems:Progress in Weather and Climate
Gregory J. Hakim
University of Washington
Math Across Campus Lecture—3 December 2009
Thanks: David Battisti, Karin Bumbaco, Seb Dirren, Josh Hacker,
Jim Hansen, Helga Huntley, Rahul Mahajan, Cliff Mass, Guillaume Mauger, Phil Mote,
Angie Pendergrass, Gerard Roe, Chris Snyder, & Ryan Torn
Gregory J. Hakim Estimation and Prediction of Complex Systems
Estimation & Prediction of Complex Systems
GoalsOpportunities for fusing models & observationsReach both non-specialists & practioners in this areaMeet people from other fields—opportunity to organize?
Gregory J. Hakim Estimation and Prediction of Complex Systems
Outline
Fusing Observations & Models
Motivation: Challenges in complex systemsEstimation primer: least squares & Bayesian methodsPrediction I: Weather
methods to deal with complexitysuccess: decreasing forecast uncertainty (obs & estimation)
Prediction II: Climatemethods compound complexityfailure: forecast uncertainty not decreasing (feedbacks)
Rethinking models: complexification vs. simplification
Gregory J. Hakim Estimation and Prediction of Complex Systems
Motivation
Estimation & prediction are ubiquitous activities
Randomly chosen examplessocial: social trends; agents (Sugarscape); electionsmedicine: diagnosis; disease spread; flu vaccinesbiology: protein structure; neuron models; populationsfinance: portfolios; markets; inventory managementnetworks: web traffic; highways; electrical gridengineering: autonomous vehicles; robotics; trackingenvironment: weather; hydrology; ecosystems
Lots of field-specific jargon. Is there a lingua franca?
Yes: Math!
Gregory J. Hakim Estimation and Prediction of Complex Systems
Gregory J. Hakim Estimation and Prediction of Complex Systems
Estimation primer
Simple scalar example: xEstimate x (e.g. true temperature at a point)Single observation: y1
Uncertainty = error variance (σ2) of the observation
Gregory J. Hakim Estimation and Prediction of Complex Systems
Estimation primer: Least squares
Simple scalar example: x
Two observations: y1 and y2; error variance σ21 and σ2
2
Least squares: minimize, e.g., error variance in x
x̂ =σ2
2σ2
1+σ22y1 +
σ21
σ21+σ2
2y2 σ2
x̂ =σ2
1σ22
σ21+σ2
2< σ2
1, σ22
Gregory J. Hakim Estimation and Prediction of Complex Systems
Multivariate probability density
marginal probability densityjoint probability density & likelihoodconditional probability density
Gregory J. Hakim Estimation and Prediction of Complex Systems
Estimation primer: Bayesian perspective
Conditional probability
P(x |y) “probability of x , given y ”
Bayes rule: P(x |y) = P(y |x)P(x)P(y)
P(y |x): “likelihood function” for measurement errors.P(x): “prior distribution”; e.g. the first observation.P(y): “marginal distribution”—a normalizing constant.
If P(y |x) ∼ N(x , σ2y ) and P(x) ∼ N(x̄ , σ2
x )
Then P(x |y) ∼ N(
σ2y
σ2x +σ2
yx̄ + σ2
xσ2
x +σ2yy , σ2
xσ2y
σ2x +σ2
y
)Same as least squares if x̄ is the first observation!
x̂ = x̄ + K (y − x̄) σ2x̂ = (1− K )σ2
x K = σ2x
σ2x +σ2
y
Gregory J. Hakim Estimation and Prediction of Complex Systems
Estimation primer: Bayesian example #1
Gregory J. Hakim Estimation and Prediction of Complex Systems
Estimation primer: Bayesian example #2
Gregory J. Hakim Estimation and Prediction of Complex Systems
Estimation primer: Bayesian example #3
Gregory J. Hakim Estimation and Prediction of Complex Systems
A more complete Bayesian framework I
Large numbers of state elements and observations: vectors
x = [x1 x2 · · · xn]T “state vector”P(xt |Yt ) “P(x) at time t , given all current and past obs”Yt = [yt ,Yt−1]
If observation errors are uncorrelated in time, thenP(xt |Yt ) ∝ P(yt |xt ) P(xt |Yt−1)
P(yt |xt ): observation likelihoodP(xt |Yt−1): prior, given all past observations: modelRecursive if we can update P(xt |Yt−1)!
Gregory J. Hakim Estimation and Prediction of Complex Systems
A more complete Bayesian framework II
State dynamics & conservation of probability
State dynamics:dxdt
= F(x)
Liouville equation:dPdt
= −P∇ · F(x)
Solution:
P(xt+1|Yt ) = P(xt |Yt ) e−R t+1
t ∇·F(x)dt ′
probability density decrease exponentially whenstate-space trajectories diverge. chaos!completely impractical for complex systems: weathermodels ∼ (108)n for n moments.
Gregory J. Hakim Estimation and Prediction of Complex Systems
Example for the Lorenz attractor
from: T. N. Palmer (2006)
Gregory J. Hakim Estimation and Prediction of Complex Systems
Weather Prediction
Now I don’t care what the weatherman saysWhen the weatherman says it’s rainingYou’ll never hear me complaining, I’m certain the sun will shine.
—Louis Armstrong (“Jeepers Creepers,” 1938)
“Numerical” weather prediction began in the 1950s.application of laws of physics to atmosphere.mathematics made the problem tractable.
Gregory J. Hakim Estimation and Prediction of Complex Systems
Weather Prediction
The numbers (European Centre for Medium Range Weather Forecasting)
analyses and forecasts every 6-12 hours.model: ∼128 million degrees of freedomobservations: ∼9 million per cyclecomputers: 2 x 156 TFlops (US: 2 x 94 TFlops)
Gregory J. Hakim Estimation and Prediction of Complex Systems
Satellites dominate the observation data stream
Gregory J. Hakim Estimation and Prediction of Complex Systems
Weather forecast: A long-term success story
Lines show the forecast day at which skill is lost.from: ECMWF (2009)
Gregory J. Hakim Estimation and Prediction of Complex Systems
Increasingly realistic forecasts
from: ECMWF (2009)
Gregory J. Hakim Estimation and Prediction of Complex Systems
Simplified Estimation for Weather Prediction I
Assumption #1: Gaussian statisticsP(xt |Yt ) ∝ P(yt |xt ) P(xt |Yt−1)
−lnP(xt |Yt ) ∝(xt − x̄)T B−1 (xt − x̄)︸ ︷︷ ︸ + [y−H(x̄)]T R−1 [y−H(x̄)]︸ ︷︷ ︸
Jb Jo
x̄ prior. B and R error covariance matrices.Note: −lnP ∝ Shannon’s information metric.
Two solution paths(1) let J = −lnP and search for the minimum. Variational.(2) direct solve: ∂J
∂x = 0. Kalman filter.
Gregory J. Hakim Estimation and Prediction of Complex Systems
(1) Variational approach
3DVAR
descent algorithm based on ∂J∂x
assumes fixed B (empirical)this is a sequential filter
4DVARextension into time domainJo =
∑Jo(t)
this is a smoother
Gregory J. Hakim Estimation and Prediction of Complex Systems
(2) Kalman Filter approach
analytical derivation of posterior distributionxa
t = xbt + K
[yt −H(xb
t )]
K = f (B,R)
Need to advance B in time; too big (108 × 108)Approximate with an ensemble of forecasts
Ensemble filtersEstimate x̄ and B with O(100) forecastsWhy does this work? Strong, state-dependent, correlations
I.e., problem is not as high dimensional as it seems
Ensemble predictionApproximation to Liouville equationAdvance analysis ensemble in timeNote: variational approach is deterministic
Gregory J. Hakim Estimation and Prediction of Complex Systems
Ensemble forecast spread: t = 0
Gregory J. Hakim Estimation and Prediction of Complex Systems
Ensemble forecast spread: t = 3 days
Gregory J. Hakim Estimation and Prediction of Complex Systems
Ensemble forecast spread: t = 5 days
Gregory J. Hakim Estimation and Prediction of Complex Systems
Ensemble forecast spread: t = 10
Gregory J. Hakim Estimation and Prediction of Complex Systems
Applications
Sensitivity analysisWhat structure most affects the forecast?
Observation impactWhich observations contain the most information?Filter redundant observations.Optimally design observing networks.
Adaptive samplingTarget observations to reduce forecast errors.
System dynamicsUse ensemble statistics for hypothesis testing.
Gregory J. Hakim Estimation and Prediction of Complex Systems
Forecast sensitivity example: short-time forecast
red = greater forecast sensitivity.
Gregory J. Hakim Estimation and Prediction of Complex Systems
Forecast sensitivity example: medium-time forecast
red = greater forecast sensitivity.
Gregory J. Hakim Estimation and Prediction of Complex Systems
Forecast sensitivity example: long-time forecast
red = greater forecast sensitivity.
Gregory J. Hakim Estimation and Prediction of Complex Systems
Real weather exampleWith Ryan Torn (SUNY Albany)
Gregory J. Hakim Estimation and Prediction of Complex Systems
Typhoon Tokage (2004): Large forecast errors
Gregory J. Hakim Estimation and Prediction of Complex Systems
Typhoon Tokage (2004): Observation Impact
greatest impact well removed from the storm!
Gregory J. Hakim Estimation and Prediction of Complex Systems
Perturbed forecast for a single observation
Gregory J. Hakim Estimation and Prediction of Complex Systems
Climate Prediction
Gregory J. Hakim Estimation and Prediction of Complex Systems
Climate Prediction
IPCC Projected global temperature change
Range of +1.4–5.8◦C (2100−1990).This large range of uncertainty has changed little with time.
Gregory J. Hakim Estimation and Prediction of Complex Systems
Earth’s Energy Balance
Ts =[
S(1−A)2σ(2−ε)
]1/4ε A Ts
no atmosphere 0 0.3 −18◦C20th century 0.75 0.3 14◦Cglobal warming #1 0.85 0.3 20◦Cglobal warming #2 0.85 0.35 15◦C
Use climate models to be more precise...
Gregory J. Hakim Estimation and Prediction of Complex Systems
History of projections based on climate simulations
2100−1990 global-mean surface temperature change (A1B scenario)
report mean rangeNAS (1979) N/A 1.6+◦CAR1 (1992) 2.9◦C 1.9–4.4◦CAR3 (2001) 3.0◦C 2.0–4.0◦CAR4 (2007) 2.8◦C 1.7–4.4◦C
Why hasn’t the uncertainty (range) changed?
Gregory J. Hakim Estimation and Prediction of Complex Systems
Climate Feedbacks and Uncertainty
climate sensitivity: equilibrium change in surfacetemperature due to “forcing.”forcing: anything that changes the net energy (e.g.absorbing gases)feedbacks: changes that reinforce forcing.
e.g. “water-vapor” feedback: T ↑⇒ H2O vapor↑⇒ T ↑
Roe & Baker (2007):linear approximation: ∆T = λ∆R
without feedbacks: λ ≡ λ0 ≈ 0.3 K/ (W/m2)∆T0 = 1.2◦C for 2 x CO2.
feedback: ∆T = λ0∆R + λ0C∆T=⇒ ∆T = ∆T0
1−f f = λ0C feedback factor
Gregory J. Hakim Estimation and Prediction of Complex Systems
Climate Feedbacks and Uncertainty
Gaussian forcing uncertainty→ skewed response in ∆T .Roe & Baker (2007)
Gregory J. Hakim Estimation and Prediction of Complex Systems
Rethinking models of complex systems (A math challenge)
“first principles” models“faithful” representation of basic processesMoore’s Law enables better representation; more processes(“complexification”)parameterized processes often do not go awaycomplexification + parameterization + feedbacks = trouble
“simplified” modelsobjective model formulation; e.g., information theorylet complexity emerge from simple building blocks (e.g. agents)
careful calibration before increasing complexity
Gregory J. Hakim Estimation and Prediction of Complex Systems
Summary
estimation & predictionpowerful “fusion” of models with observationsequivalent Bayesian & least-squares approachesvariational & ensemble methods for complex systems
application to weather & climateweather: success due to better estimation & observations
huge increase in observationsbetter estimation & faster computerssensitivity theory: observation targeting & selection
climate: failure due to model “complexification” & feedbacksNo argument: world is warming, and we know whyUncertainty in projections resistant to CPU and $Uncertainty in feedbacks; complexification does not help
Gregory J. Hakim Estimation and Prediction of Complex Systems
Thank You!
Gregory J. Hakim Estimation and Prediction of Complex Systems