Hierarchical Bayesian Modeling and Analysis: Why and How

Transcription

1 Hierarchical Bayesian Modeling and Analysis: Why and How Mark Berliner Department of Statistics The Ohio State University IMA June 10, 2011 Outline I. Bayesian Hierarchical Modeling II. Examples: Glacial dynamics; Paleoclimate III. Using Large-scale Models

2 Why: General Points Goal: Combine information sources Uncertainty Quantification (UQ) Physical-Statistical Modeling (Berliner 2003 JGR): Build in science (qual. & quant.) Physics and (not versus) Statistics Model-data combination Not just data assimilation Relationships among physical processes; crossing scales

3 How: Probability & Statistics Background Probability density function (PDF) for 2 variables X,Y: [x, y] Marginal PDF of X: [x] = [x, y]dy Conditional PDF of Y given X = x: [y x] [x, y]: [y x] = [x, y]/[x] Hence, [x, y] = [y x][x] (= [x y][y]) Bayes Theorem: [x y] = Bases of hierarchical modeling [y x] [x] [y] factoring joints: [x, y, z] = [y x, z] [x z] [z], etc. forming models: [x] = [x y] [y] dy

4 Basic Stat Problem: Inference about X and its distribution Parameterized Model: [x θ] where θ is unknown Obs. Data: Observe Y = y with conditional PDF [y x, θ] Classical Stat: Observed y provides indirect info about X, θ Bayesian Stat: 1. Inverse probability problem: from [y x, θ] to [x, θ y] [x, θ y] = [y x, θ] [x, θ] [y] 2. Bayesian learning: update [x, θ] having observed y [x, θ y] = [x, θ] [y x, θ] [y]

5 Bayesian Hierarchical Modeling (BHM) HM: Sequence of conditional probability distributions corresponding to a joint distribution BHM (example) Parameterized Model: [x 1,..., x p θ 1,..., θ p ] and [θ 1,..., θ p η] Framework for modeling Observations Y; Processes (state variables) X; Parameters θ 1. Data Model [Y X, θ] 2. Process Model [X θ] 3. Parameter Model [ θ ] Bayes Theorem: [X, θ Y] Implied Marginal: [X Y] = [X, θ Y] dθ

6 BHM: Thinking Conditionally Data Models [Y X, θ ] Combining diverse, complex datasets Multiscale datasets: up- & down-scaling Process Models [X θ ] Incorporate PDE (or SPDE): From D(X) = 0 to [X θ] Interrelate processes, B.C., & I.C. via conditional models [X 1, X 2 θ] = [X 1 X 2, θ] [X 2 θ] Parameterization Parameter Models [ θ ] Incorporate science Enrich structure: multiscale, space-time varying parameters

7 II. Glacial Dynamics (Berliner et al J. Glaciol) Steady Flow of Glaciers and Ice Sheets Flow: gravity moderated by drag (base & sides) & stuff Simple models: flow from geometry Data Program for Arctic Climate Regional Assessments Radarsat Antarctic Mapping Project surface topography (laser altimetry) basal topography (radar altimetry) velocity data (interferometry)

8 North East Ice Stream, Greenland

9 Physical Modeling: Surface: s, Thickness: H, Velocity: u Basal Stress: τ = ρ g H ds/dx (+ stuff ) Velocities: u = u b + b 0 H τ n where u b = k τ p + (ρgh) q Our Model Random Basal Stress: τ = ρ g H ds/dx + η where η is a corrector process Random H: wavelet smoothing prior Random s: parameterized model from literature Random Velocities: u = u b + b H τ n + e where u b = kτ p + (ρgh) q or a constant b is unknown parameter, e is a noise process Process Model : [u s, H, η] [η] [s, H]

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13 II. Paleoclimate: Temperature Reconstruction (Brynjarsdóttir & Berliner 2010 Ann Applied Stat) Use of proxies: Inverse problem: proxy f(climate) climate g(proxy) Inverse probability problem: [proxy data climate] [climate proxy data] Boreholes: earth stores info about surface temp s Inverse: borehole data f(surface temp s) Model: Heat equation Infer boundary condition

14 Relative temperature [ºC] San Rafael Desert San Rafael Swell Depth [m] ºC SRD 3 SRD 2 SRD 7 SRD SRD SRS SRS 4 SRS 5 WSR 1

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16 Data Model : Y T r, q N( T r + T q R(k), σ 2 y I) T r : reduced temperatures (literature: T r = T T 0 1 q R(k)) T 0 : reference surface temperature q: surface heat flow R: thermal resistances ( k 1, thermal conductivities, adjusted for rock types, etc.) Process Model : Heat equation applied to T r B.C.: Surface temp history T h (assumed piecewise constant) Easy to solve the heat equation T r T h, q N(A T h, σ 2 I) Parameter Model : next T h q N( 0, σ 2 h I)

17 Spatial Hierarchy Nine boreholes: 5 in desert region, 4 in swell region Extend the hierarchy: for j = 1,..., 9 Data Model : Yj T rj, q j N( T rj + T 0j 1 + q j Rj (k j ), σ 2 yj I) Process Model : T rj T hj, q j N(A j Thj, σ 2 j I) Parameter Model T hj q j N( 0, σ 2 hj I) T h1,..., T h5 conditionally independent N( µ d, γ 2 d I) T h6,..., T h9 conditionally independent N( µ s, γ 2 s I) µ d & µ s N( µ 0, γ 2 0 I) q 1,..., q 5 conditionally independent N(ν d, η 2 d) q 6,..., q 9 conditionally independent N(ν s, η 2 s) etc.

18 SRD 1, SR Desert SRD 2, SR Desert SRD 3, SR Desert SRD 4, SR Desert SRD 7, SR Desert SRS 3, SR Swell SRS 4, SR Swell SRS 5, SR Swell WSR 1, SR Swell

19 µ S µ D

20 SRD 1, SR Desert SRD 2, SR Desert SRD 3, SR Desert SRD 4, SR Desert SRD 7, SR Desert SRS 3, SR Swell SRS 4, SR Swell SRS 5, SR Swell WSR 1, SR Swell Multi site Single site

21 SRD 2, SR Desert SRS 5, SR Swell Multi site Single site Multi site Single site

22 III. Incorporating Large-Scale Computer Models Recall Data, Process, and Parameter Models 1. Model output O = O 1,..., O n as observations Data Model: [Y, O X, θ] [ O X, θ] to include bias, offset,.. Convenient for design 2. Process Model from model output: Data analysis on O to estimate a prob. dist. Use result (with modifications) as a prior for X e.g., kernel density estimate Σ i α i k(x O i ) 3. Parameter Model from model output. eg: Prior on coefficients of fingerprints (Berliner, Levine, & Shea 2003 J. Climate) 4. Combinations and another approach in prep

23 A Bayesian Multi-model Climate Projection (Berliner & Kim 2008 J. Climate) Applied to Decadal Forecasting: Kim & Berliner 2011 Themes climate = parameters of prob. dist. of weather build or parameterize scales into dynamic model for X Future climate depends on future, but unknown, inputs. IPCC: construct plausible future inputs, SRES Scenarios (CO 2 etc.) Assume a scenario and find corresponding projection

24 Hemispheric Monthly Surface Temperatures (X) Observations (Y) for Two models ( O): PCM (n=4), CCSM (n=1) for SRES scenarios (B1,A1B,A2).

25 Hierarchical Data Model for Model Output Scalar climate variable X; m = 1,..., M models (time fixed) O m : ensemble of size n m of estimates of X from model m. 1. Given means µ m and variances σ 2 Y m, m = 1,..., M; O m are independent and O m µ m Gau(µ m 1 nm, σ 2 Y m I nm ) 2. Given β, model biases b m and variances σ 2 µ m ; µ m are independent and 3. Given X, µ m β, b m Gau(β + b m, σ 2 µ m ) β X Gau(X, σ 2 β ) and b m X Gau(b 0m, σ 2 b m )

26 Remarks Implied marginal: O given X : Integrating out β induces dependence within & across ensembles Modify intuition about value of increasing ensemble size Infinite ensembles do not give perfect forecasts: If all biases = 0, infinite ensembles give the value of β, not X Extensions to different model classes (more β s) and richer models are feasible.

27 Model Overview 1. [Y X, θ]: measurement error model Gaussian with mean = true temp. & unknown variance (with a change-point) 2. [X θ]: Time series models with time varying parameters X t = µ i(t) + ηn j(t) 0 0 η s j(t) (X t 1 µ i(t 1) ) + e t 3. [θ] Climate = parameters of distribution of weather Climate-weather: multiscale phenomena Time evolution: µ i(t) slow; η j(t) moderate; e t fast, but variances of e t are slow µ i = A + B CO 2i + noise Obs period: η j = C + D SOI j + noise Fore. period: AR model (i.e., SOI not observed) Variances of e t : AR-like prior

28 mean year annual temperature year

29 mean year annual temperature year

30 system variance year

31 Model Adapted to Decadal Forecasting 16 Temperature(Celcius) Temperature(Celcius)