Estimation and Prediction of Complex Systems: Progress in Weather and Climate

transcript

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?

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

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!

Estimation primer

Simple scalar example: xEstimate x (e.g. true temperature at a point)Single observation: y1

Uncertainty = error variance (σ2) of the observation

Estimation primer: Least squares

Simple scalar example: x

Two observations: y1 and y2; error variance σ21 and σ2

Least squares: minimize, e.g., error variance in x

x̂ =σ2

1+σ22y1 +

σ21+σ2

2y2 σ2

x̂ =σ2

σ21+σ2

2< σ2

1, σ22

Multivariate probability density

marginal probability densityjoint probability density & likelihoodconditional probability density

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

Then P(x |y) ∼ N(

σ2x +σ2

yx̄ + σ2

x +σ2yy , σ2

σ2x +σ2

)Same as least squares if x̄ is the first observation!

x̂ = x̄ + K (y − x̄) σ2x̂ = (1− K )σ2

x K = σ2x

σ2x +σ2

Estimation primer: Bayesian example #1

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)!

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.

Example for the Lorenz attractor

from: T. N. Palmer (2006)

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.

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)

Satellites dominate the observation data stream

Weather forecast: A long-term success story

Lines show the forecast day at which skill is lost.from: ECMWF (2009)

Increasingly realistic forecasts

from: ECMWF (2009)

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)]︸︷︷︸

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.

(1) Variational approach

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

(2) Kalman Filter approach

analytical derivation of posterior distributionxa

t = xbt + K

[yt −H(xb

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

Ensemble forecast spread: t = 0

Ensemble forecast spread: t = 3 days

Ensemble forecast spread: t = 5 days

Ensemble forecast spread: t = 10

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.

Forecast sensitivity example: short-time forecast

red = greater forecast sensitivity.

Forecast sensitivity example: medium-time forecast

Forecast sensitivity example: long-time forecast

Real weather exampleWith Ryan Torn (SUNY Albany)

Typhoon Tokage (2004): Large forecast errors

Typhoon Tokage (2004): Observation Impact

greatest impact well removed from the storm!

Perturbed forecast for a single observation

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.

Earth’s Energy Balance

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...

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?

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

Climate Feedbacks and Uncertainty

Gaussian forcing uncertainty→ skewed response in ∆T .Roe & Baker (2007)

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

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

Thank You!

Estimation and Prediction of Complex Systems: Progress in Weather and Climate

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