bayesian hierarchical modeling of hydroclimate problems

Bayesian Hierarchical Modeling of Hydroclimate Problems

Balaji Rajagopalan Department of Civil, Environmental and Architectural

EngineeringAnd

Cooperative Institute for Research in Environmental Sciences

(CIRES)

University of ColoradoBoulder, CO, USA

Bayes by the Bay Conference, PondicherryJanuary 7, 2013

Co-authors & Collaborators

Upmanu Lall and Naresh Devineni – Columbia University, NY

Hyun-Han Kwon, Chonbuk National University, South Korea

Carlos Lima, Universidade de Brasila, Brazil Pablo Mendoza James McCreight & Will

Kleiber – University of Colorado, Boulder, CO Richard Katz – NCAR, Boulder, CO

NSF, NOAA, USBReclamation and Korean Science Foundation

Outline Bayesian Hierarchical Modeling

Introduction from GLM

Hydroclimate Applications BHM Contrast with near Bayesian models currently in vogue

Stochastic Rainfall Generator BHM (Lima and Lall, 2009, WRR) Latent Gaussian Process Model (Kleiber et al., 2012, WRR)

Riverflow Forecasting (Kwon et al., 2009, Hydrologic Sciences)

Seasonal Flow Flow extremes

Paleo Reconstruction of Climate (Devineni and Lall, 2012, J. Climate)

Linear Regression Models

Suppose the model relating the regressors to the response is

In matrix notation this model can be written as


where


We wish to find the vector of least squares estimators that minimizes:

The resulting least squares estimate is

12-1 Multiple Linear Regression Models

12-1.4 Properties of the Least Squares Estimators

Unbiased estimators:

Covariance Matrix:

12-1 Multiple Linear Regression Models

12-1.4 Properties of the Least Squares Estimators

Individual variances and covariances:

In general,

Generalized Linear Model (GLM) Bayesian Perspective

• Linear Regression is not appropriate • when the dependent variable y is not Normal• Transformations of y to Normal are not possible • Several situations (rainfall occurrence; number of wet/dry days; etc.)

• Hence, GLM• Linear model is fitted to a ‘suitably’ transformed variable of y • Linear model is fitted to the ‘parameters’ of the assumed distribution of y

Likelihood


• Noninformative prior on β • Assuming Normal distribution for Y, g (.) is identity Linear Regression

Exponential family PDF, parameters

All distributionsArise from thisNormal, Exponential,GammaBinomial,Poisson, etc


• Log and logit – Canonical Link Functions


InverseChi-Square

• GLM is hierarchical• Specific Distribution• Link function• With a simple step – i.e., Providing priors

and computing likelihood/posterior BHM

• Assuming Normal distribution of dependent variable and uninformative priors• BHM collapses to a standard Linear

Regression Model

• Thus BHM is a generalized framework

• Uncertainty in the model parameters and model

Structure are automatically obtained.

Summary

Generalized Linear Model (GLM) Example - Bayesian Hierarchical Model

• Hard to sample from posterior - Use MCMC

Stochastic Weather Generators

Precipitation Occurrence, Rain Onset Day (Lima and

Lall, 2009)Precipitation Occurrence and Amounts (Kleiber,

2012)

28.5

………

12.4

23.1

………

10.2

29.1

………

11.4

25.8

………9.7

…

HistoricalData

Synthetic series – Conditional onClimate Information

Process model

Frequency distribution of

outcomes

• Users most interested in sectoral/process outcomes (streamflows, crop yields, risk of disease X, etc.)

• Need for a robust spatial weather generator

Need for Downscaling

Seasonal climate forecasts and future climate model projections often have coarse scales:

Spatial: regional Temporal: seasonal, monthly

Process models (hydrologic models, ecological models, crop growth models) often require daily weather data for a given location

There is a scale mismatch! Stochastic Weather Generators can

help bridge this scale gap.

Precipitation Occurrence

504 stations in Brazil (Latitude & Longitude

shown in figure) Lima and Lall (WRR, 2009)

Modeling of rainfall occurrence (0 = dry, 1 = rain, P = 0.254mm threshold) using a probabilistic model (logistic regression):

Modeling Occurrence at a Site

where yst(n) is a non-homegeneous Bernoulli random variable for station s, day n and year t, being either 1 for a wet state or 0 for a dry state. • pst(n) is the rainfall probability for station s and day n of year t. The seasonal cycle is modeled through Fourier harmonics:

Results from Site #3

Outlier?

Bayesian Hierarchical Model (BHM)But rainfall occurrence is correlated in space – how to model? - partial

BHM

•Shrinks paramters towards a common mean, reduce uncertainty since we are use more information to estimate model parameters;

•Parameter uncertainties are fully accounted during simulations

Bayesian Hierarchical Model (BHM)Likelihood Function

Posterior Distribution – Bayes theorem

MCMC to obtain posterior distribution

Results for Station #3 – Yearly Probability of Rainfall

Results Station #3 - Average Probability of Rainfall

T

tsts

T

tsts

T

tsts

cT

C

bT

B

aT

A

1

1

1

1

1

1

))cos()sin((itlog)( 1 nCnBAnP ssss

Clusters on average day of max probability

221logit sssmas

s CBAP

Max Probability of Rainfall

Day of Max Probability of Rainfall

• Max Probability of rainfall correlatedWith climate variables – ENSO, etc.• Characterize rainfall ‘onset’• Prediction of ‘onset’• Lima and Lall (2009, WRR)

Space-time Precipitation Generator

Latent Gaussian Process(Kleiber et al., 201, WRR)

Latent Gaussian Process Fit a GLM for Precipitation Occurrence and amounts at each

location independently Occurrence logistic regression-based Amounts Gamma link function

Spatial Process to smooth the GLM coefficients in space Almost Bayesian Hierarchical Modeling Alpha, gamma – shape and scale parameter of Gamma

Latent Gaussian ProcessOccurrence Model

Latent Gaussian Process

Parameter Estimation MLE, two step

GLM + Latent GaussianProcess

Kleiber et al. (2012)

For Max and Min Temperature ModelsConditioned on Precipitation Model- Using Latent Gaussian Process

Kleiber et al. (2013, Annals of App. Statistics, in press)

Outline Bayesian Hierarchical Modeling

Introduction from GLM

Hydroclimate Applications BHM Contrast with near Bayesian models currently in vogue

Stochastic Rainfall Generator BHM (Lima and Lall, 2009, WRR) Latent Gaussian Process Model (Kleiber et al., 2012, WRR)

Riverflow Forecasting (Kwon et al., 2009, Hydrologic Sciences)

Seasonal Flow Flow extremes

Paleo Reconstruction of Climate (Devineni and Lall, 2012, J. Climate)

Seasonal average and maximum Streamflow

Forecasting(Kwon et al.,2009,

Hydrologic Sciences)

Streamflow Forecasting at Three Gorges Dam

Yichanghydrological station (YHS)

Yichanghydrological station (YHS)

Identify Predictors•Correlate seasonal streamflow with large scale climate variables from preceding seaons

•JJA flow with MAM climate

•Select regions of strong (Grantz et al., 2005) correlation

• predictors


Climate predictors JJA Seasonal Flow Annual Peak Flow

SST1 -10°N~10°N 150°E~180°E -0.27 ‡ -0.28 ‡SST2 -20°N~0° 75°E~110°E 0.51 † 0.20 †SST3 10°N~30° N 130°E~150°E 0.38 † 0.45 †Snow -10°N~0°N 200°E~230°E 0.42 † 0.42 †

Zone selected

†: Significant at 95% confidence; ‡: Significant at 90% confidence

a) SST Vs Mean JJA Flow(1970-2001)

b) Snow Vs Mean JJA Flow(1970-2001)

c) SST Vs Peak Flow(1970-2001)

d) Snow Vs Peak Flow(1970-2001)

BHM for Seasonoal Streamflow Model

is distributed as half-Cauchy with parameter 25 “mildly informative”Gelman (2006, Bayesian Analysis)

MCMC is used to obtain the posterior distributions

Data showed mild nonlinearity Quadraticterms in the model


0 10 20 30 400

100

200

300

400

Histogram of tau

1.8 2 2.2 2.4 2.60

100

200

300

400

Histogram of Beta1

-0.4 -0.2 0 0.2 0.4 0.60

100

200

300

400

Histogram of Beta2

-0.4 -0.2 0 0.2 0.4 0.60

100

200

300

400

Histogram of Beta3

-0.4 -0.2 0 0.2 0.4 0.60

100

200

300

400

Histogram of Beta4

-0.4 -0.2 0 0.2 0.4 0.60

100

200

300

400

Histogram of Beta5

0 10 20 30 400

100

200

300

400

Histogram of tau

1.8 2 2.2 2.4 2.60

100

200

300

400

Histogram of Beta1

-0.4 -0.2 0 0.2 0.4 0.60

100

200

300

400

Histogram of Beta2

-0.4 -0.2 0 0.2 0.4 0.60

100

200

300

400

Histogram of Beta3

-0.4 -0.2 0 0.2 0.4 0.60

100

200

300

400

Histogram of Beta4

-0.4 -0.2 0 0.2 0.4 0.60

100

200

300

400

Histogram of Beta5

Description Node Mean Standard Dev. 2.50% Median 97.50%

Interceptor Beta1 2.273 0.074 2.129 2.273 2.420SST1 Beta2 -0.111 0.050 -0.209 -0.111 -0.011

SST12 Beta3 0.130 0.048 0.035 0.130 0.224

SST2 Beta4 0.276 0.051 0.176 0.276 0.377

Snow2 Beta5 0.083 0.025 0.034 0.083 0.132

Performance Measure R CoE IoA Bias RMSE0.802 0.643 0.886 0.001 0.231Seasonal (JJA)

Predictors2, 3, 4 and 5Show tighterBounds

Uncertaintyin predictors(i.e. model) isobtained andpropogated in the forecacsts

You can usePCA or stepwiseetc. to reducethe number ofpredictors(this can becrude)



Interceptor Beta1 2.273 0.074 2.129 2.273 2.420SST1 Beta2 -0.111 0.050 -0.209 -0.111 -0.011

SST12 Beta3 0.130 0.048 0.035 0.130 0.224

SST2 Beta4 0.276 0.051 0.176 0.276 0.377

Snow2 Beta5 0.083 0.025 0.034 0.083 0.132

Performance Measure R CoE IoA Bias RMSE0.802 0.643 0.886 0.001 0.231Seasonal (JJA)

Maximum Seasonal Streamflow

Extreme Value Analysis – Floods

(Kwon et al.,2010, Hydrologic Sciences)

020,00040,00060,00080,000

100,000120,000140,000160,000180,000

1900 1920 1940 1960 1980 2000

An

n M

ax

Flo

w

Year

American River at Fair Oaks - Ann. Max. Flood

100 yr flood estimated from 21 & 51 yr moving windows

Floods

The time varying (nonstationary) nature of hydrologic (flood) frequency (few examples)

Climate Variability and Climate Change Climate Mechanisms that lead to changes in flood statistics

Adaptation Strategy ‘Adaptive’ Flood Risk Estimation

Nonstationary Flood Frequency Estimation Seasonal to Inter-annual Forecasts & Climate

Change

Improved Infrastructure Management

Summary / Climate Questions and Issues related to Hydrologic Extremes

Flood mean given DJF NINO3 and PDO

NINO3 PDO

Flood Variance given DJFNINO3 and PDO

NINO3PDO

Derived using weighted local regression with 30 neighbors

Correlations:

Log(Q) vs DJF NINO3 -0.34 vs DJF PDO -0.32

Jain & Lall, 2000

IWV(cm)

Atmospheric Rivergenerates flooding

CZD

Russian River flooding in Monte Rio, California

18 February 2004

photo courtesy of David Kingsmill

Russian River, CA Flood Eventof 18-Feb-04

GPS IWV data from near CZD: 14-20 Feb 2004

Bodega Bay

Cloverdale

Atmospheric river

10” rain at CZD

in ~48 hours

IWV

(c

m)

IWV

(i

nch

es)

Slide from Paul Neiman’s talk

Flood Estimation Under Nonstationarity

Significant interannual/interdecadal variability of floods

Stationarity assumptions (i.i.d) are invalid Large scale climate features in the Ocean-

Atmosphere-Land system orchestrate floods at all time scales

Need tools that can capture the nonstationarity Incorporate large scale climate information Year-to-Year time scale (Climate Variability)

Flood mitigation planning, reservoir operations Interdecadal time scale (Climate Variability and Change)

Facility design, planning and management

Exponential (light, shape = 0), Pareto (heavy, shape > 0) and Beta (bounded, shape< 0)

Generalized extreme value (GEV) can be used to characterize extreme flow distribution(Katz et al., 2002)

3 Model parameters

Location parameter: (where distribution is centered) Scale parameter: (spread of the distribution) 0Shape parameter: (behavior of distribution tail)

Gumbell, Frischet, Weibull

/1

1exp)(z

zG

(Coles 2001)“Unconditional” GEV

Incorporate covariates into GEV parameters to account for nonstationarity

x10

Could apply to any parameter, but location is most intuitive:

GLM Framework

Hierarchical Bayesian Modeling natural and attractive alternative

GEV fit using extRemes toolkit in R(Gilleland and Katz, 2011) http://www.isse.ucar.edu/extremevalues/extreme.html

(Gilleland and Katz 2005)

http://www.isse.ucar.edu/extremevalues/extreme.html




Zone selected


a) SST Vs Mean JJA Flow(1970-2001)

b) Snow Vs Mean JJA Flow(1970-2001)

c) SST Vs Peak Flow(1970-2001)

d) Snow Vs Peak Flow(1970-2001)

BHM for Seasonal Maximum Flow Model

is distributed as half-Cauchy with parameter 25 “mildly informative”Gelman (2006, Bayesian Analysis)

MCMC is used to obtain the posterior distributions

Data showed mild nonlinearity Quadraticterms in the model


Predictors3 and 5Show tighterBounds

0 0.5 1 1.50

100

200

300

400

Histogram of tau

3 3.5 4 4.5 5 5.50

100

200

300

400

Histogram of Beta1

-0.5 0 0.5 1 1.50

100

200

300

400

Histogram of Beta2

-0.5 0 0.5 1 1.50

100

200

300

400

Histogram of Beta3

-0.5 0 0.5 1 1.50

100

200

300

400

Histogram of Beta4

0 0.5 1 1.50

100

200

300

400

Histogram of tau

3 3.5 4 4.5 5 5.50

100

200

300

400

Histogram of Beta1

-0.5 0 0.5 1 1.50

100

200

300

400

Histogram of Beta2

-0.5 0 0.5 1 1.50

100

200

300

400

Histogram of Beta3

-0.5 0 0.5 1 1.50

100

200

300

400

Histogram of Beta4



Zone selected




Interceptor Beta1 4.174 0.195 3.791 4.171 4.548

SST12 Beta2 0.198 0.119 -0.055 0.203 0.423

SST3 Beta3 0.699 0.148 0.410 0.706 0.986

SST32 Beta4 -0.089 0.079 -0.264 -0.085 0.053

Snow2 Beta5 0.302 0.098 0.091 0.310 0.473

Performance Measure R CoE IoA Bias RMSE

0.729 0.531 0.828 -0.001 0.602Annual Peak Flow

Nonstationary Flood Risk at Three Gorges Dam

020,00040,00060,00080,000

100,000120,000140,000160,000180,000

1900 1920 1940 1960 1980 2000

An

n M

ax

Flo

w

Year

Dynamic 50-year flood from BHM and Stationary 50-year flood

Conditional (nonstationary) Extremes in Water Quality(Towler et al., 2009, WRR)

Case study location: PWB Towler et al. (2009)

“Forest to Faucet”

- Rain -Runoff

-Storage (2 reservoirs)

-Chemical Disinfection (Cl2, NH3)

-No physical filtration (“unfiltered”)-Distribution

Case study location: PWB

Exceedances (SWTR criterion: turbidity < 5 NTU)

Precipitation events

High Flows

Back-up groundwater source(Pumping $$)

GEV Model

Uncond CondT CondR CondRT CondR+T

Variable β0 β0+β1T β0+β1R β0+β1(RT) β0+β1R+β2T

β0 (se) 1924 (120) 1930 (1000) 1739 (410) 611.4 (150) 1911 (880)

β1 (se) - -0.8914 (27) 61.08 (32) 3.716 (0.36) 141.2 (14)

β2 (se) - - - - -36.45 (24)

σ (se) 1245 (84) 1220 (81) 1246 (160) 923.7 (69) 968.5 (74)

ξ (se) -0.02246 (0.065) -0.01286 (0.065) -0.06180 (0.084) 0.07009 (0.082) 0.01619 (0.075)

llh -1289 -1289 -1274 -1250 -1250

K 1 2 2 2 3

AIC 2580 2582 2552 2504 2506

M0* - Uncond Uncond Uncond CondR

D - 0 30 78 48

Sig** - No (0.635) Yes (0.000) Yes (0.000) Yes (0.000)

ρ*** - - 0.5516 0.5989 0.5918

* Nested model to which model is compared in likelihood ratio test** Significance is tested at α=0.05 level, and ( ) indicates p-value. *** Correlation between the cross-validated z90 estimates and the observed maximum values

Conditional quantiles correspond well to observed record

1970 1980 1990 2000

Year

Max

imum

Str

eam

flow

(cf

s)

0

2000

4000

6000

8

000

1970 1980 1990 2000

02

00

04

00

06

00

08

00

0

Year

Ma

xim

um

Str

ea

mflo

w (

cfs)

Uses concurrent climate, but

could also be used with seasonal forecast

Maximum Streamflow (cfs)

PD

F

0 2000 4000 6000 8000

0e

+0

01

e-0

42

e-0

43

e-0

44

e-0

4

GEV distribution can be compared for specific historic times

P and T climate change projections from IPCC AR4 are readily available

12 km2 resolution (1/8 of a grid cell)

Bias correct P & T to historic data for PWB watershed area

http://gdo-dcp.ucllnl.org/downscaled_cmip3_projections/#Welcome

Results indicate increasing maximum streamflow anomalies

Observed

16 GCM models

GCM model average

Year

1950 2000 2050 2100

Max

imum

Str

eam

flow

Ano

mal

y (%

)

-2

5

0

25

50

75

Streamflow quantiles shift higher under CC projections

Observed

16 GCM models

Probability of a turbidity spike given a certain maximum flow

Likelihood of Turbidity Spike

(Ang and Tang 2007)

)()|()(0

SPSEPEP

Maximum Flow (CFS)

Con

ditio

nal P

(E)

Likelihood of a turbidity spike increases under CC projections

Observed

16 GCM models

Percentile 1950-2007 2070-2099

95th (top whisker) 13 28

75th (box top) 6.3 11

50th (box middle) 4.2 5.9

Likelihood of a turbidity spike increases

P(E

)

23 24

10

4041

75

16

115

62

0

20

40

60

80

100

120

140

2010-2039 2040-2069 2070-2099

Per

cen

t In

crea

se i

n E

xpec

ted

Lo

ss R

elat

ive

to 1

950-

2007

P

erio

d

50th

75th

95th

Small shifts in risk can result in high expected loss

Expected loss can be high, especially for the risk averse

Summary

• Bayesian Hierarchical Modeling •Powerful tool for all functional (regression) estimation problems(which is most of forecasting/simulation)

•Provides model and parameter uncertainties•Obviates the need for discarding covariates•Enables incorporation of expert opinions•Enables modeling a rich variety of variable types

•Continuous, skewed, bounded, categorical, discrete etc.•And distributions (Binomial, Poisson, Gammma, GEV)

•Generalized Framework •Traditional linear models are a subset

Paleo Hydrology Reconstruction

Devineni and Lall, 2012, J. Climate accepted

0

2

4

6

8

10

12

14

16

18

20

1914

1918

1922

1926

1930

1934

1938

1942

1946

1950

1954

1958

1962

1966

1970

1974

1978

1982

1986

1990

1994

1998

2002

2006

Calnder Year

Ann

ual Flo

w (

MA

F)

Total Colorado River Use 9-year moving average.

NF Lees Ferry 9-year moving average

Colorado River Demand - Supply

UC CRSS stream gaugesLC CRSS stream gauges

MotivationPaleo Hydrology

Colorado River Example

Streamflow and Tree Ring Data

#* #*

#* #*

#*̂

^ ^

^ ^

Ulster

Delaware

Otsego

Pike

Sullivan

Orange

Wayne

Berkshire

Litchfield

Chenango

Dutchess

Greene

Broome

Luzerne

Monroe

Fairfield

Sussex

Madison

Albany

Morris

Columbia

Schoharie

Susquehanna

Rensselaer

Suffolk

Saratoga

Fulton

Carbon

Warren

OneidaHerkimer

Cortland

Lehigh

Westchester

Bennington

Nassau

Lackawanna

Bergen

Putnam

Wyoming

Montgomery

New Haven

Onondaga

NorthamptonSchuylkill

Passaic

Hunterdon

Berks

Essex

Rockland

Washington

SomersetUnion

Schenectady

Hampden

Queens

Tioga

Hartford

Windham

Kings

Hampshire

Hudson

Bucks

Bronx

Middlesex

Hartford

Franklin

Richmond

Bradford

New York

Queens

Batavia Kill

West Branch

Delaware Riv er

Schoharie Creek

Rondout Cree

k

East Bra

nch D

elaw

are R

iver

Neversink River

y5y4

y3y2

y1

Roundout

Pepacton

Schoharie

Neversink

Canonsville MPPMSB

MRHMLQ

MHH

MiCO

MoTPMoCO

New York

Pennsylvania

New Jersey

Connecticut

New York

Massachusetts

Vermont

Streamflow and Tree Ring Data

variable length streamflow record (Yt)(5 sites)

246 years chronology (Xt)(8 tree ring chronologies)

1754

1937

1999

199919501903

Average Summer (JJA) Flows as Predictand

Annual Tree Ring Growth Index (Chronology) as Predictor – 246 years common dataAbbreviation Site Species Number of Trees Number of Series Data Record # of years

MHH Mohonk, NY Humpty Dumpty Helmlock 43 25 1754 - 1999 246MLQ Mohonk, NY Long, QUSP 20 34 1754 - 1999 246MRH Mohonk, NY Rock Rift Hemlock 18 25 1754 - 1999 246MSB Mohonk, NY Sweet Birch, BELE 17 27 1754 - 1999 246MPP Mohonk, NY Pitch Pine 23 45 1754 - 1999 246

MoCO Montplace, NY Chestnut Oak, QUPR 21 34 1754 - 1999 246MoTP Montplace, NY Tulip Popular, LITU 20 32 1754 - 1999 246MiCO Middleburgh, NY Chestnut Oak, QUPR 23 42 1754 - 1999 246

Reservoir System Feed Creek Stream Gauge Data Record # of years Drainage Area (mi2)

Schoharie Schoharie 1350000 1903 – 1999 97 237

Neversink Neversink 1435000 1937 – 1999 63 67

Roundout Roundout 1365000 1937 – 1999 63 38

Canonsville West branch Delaware River 1423000 1950 – 1999 50 332

Pepacton East Branch Delaware River 1413500 1937 – 1999 63 163

Summer Flow = f(tree rings) + error

#* #*

#* #*

#*^

^ ^

^ ^

Ulster

Delaware

Otsego

Greene

Sullivan

Schoharie

Albany

Wayne

Montgomery

Dutchess

Schenectady

Herkimer

Chenango

Madison

OneidaSaratoga

Columbia

Broome

Batavia Kill

Wes

t Bran

ch D

elaware River

Schoharie Creek

Little Delaware River

Ron

dout Creek

East Bra

nch

Del

awar

e Riv

er

East Branch Neversink RiverW

est Branch

Neversink River

Neversink R

iver

y5y4

y3y2

y1

Roundout

Pepacton

Schoharie

Neversink

Canonsville

MHH

MSB

MRH

MLQ

MBO

MiCO

MoTPMoRO

MoCO

New York

Pennsylvania

Preliminary Data Analysis – Bayesian Hypothesis(correlation – tree chronology Vs average summer seasonal flow)

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

MHH MLQ MRH MSB MPP MoCO MoTP MiCO

Cor

rela

tion

Coe

ffic

ien

t

Tree Chronology

Schoharie (ρ* = 0.20)

Neversink (ρ* = 0.25)

Roundout (ρ* = 0.25)

Canonsville (ρ* = 0.28)

Pepacton (ρ* = 0.25)

(a)

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

MHH MLQ MRH MSB MPP MoCO MoTP MiCO

Cor

rela

tion

Coe

ffic

ien

t

Tree Chronology

Schoharie (ρ* = 0.20)

Neversink (ρ* = 0.25)

Roundout (ρ* = 0.25)

Canonsville (ρ* = 0.28)

Pepacton (ρ* = 0.25)

Station-tree correlations similar!- pooling?

Bayesian Hierarchical Models

Partial Pooling – Hierarchical Model

Shrinkage on the coefficients to incorporate the predictive ability of each tree chronology on multiple

stations

)100,0(~1

covariance~

)0001.0,0(~

)0001.0,0(~

),(~

),(~)log(

2

0

2

0

unif

N

N

MVN

x

Ny

ii

i

j

jj

ij

trees

j

jt

ij

iit

iit

it

StreamflowLog Normal Distribution Regression Coefficients (β)

of the hierarchical model - multivariate normal distribution

),(~

0

jj

ij

trees

j

jt

ij

iit

MVN

x

Year t

Tree stand j

i2

it

)log( itQ

i0 j

tx

ij

Site i

j

j

T)(

)/()()/(

datap

datappdatap

Key ideas:1.Streamflow at each site comes from a pdf2.Parameters of each pdf informed by each tree3.Common multivariate distribution of parameters across trees4.Noniformative prior for parameters of multivariate distribution5.MCMC for parameter estimation

Delaware River Reconstruction and Performance

Models Developed•Hierarchical Bayesian Regression (Partial Pooling)

•Linear Regression (No Pooling)

Model Simulations•WinBUGS : Bayesian Inference Using Gibbs Sampler

•7500 simulations with 3 chains and convergence tests.

Cross Validated Performance Metrics •Reduction of Error (RE), Coefficient of Efficiency (CE)

Delaware River Reconstruction and PerformancePosterior PDF (Model Level 1)

Delaware River Reconstruction and Performance

No Pooling

Partial Pooling

Regression CoefficientsModel Level 2

Delaware River Reconstruction Cross-Validated Performance

Canonsville Pepacton

Paleo Hydrology Reconstruction

Traditional MethodsLinear/Nonlinear Regression

PCA of Tree RingsRegression on leading PCs

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Objective 1: Tree-ring ReconstructionsLCBR

Naturalize streamflow9 nodes in CRSS5 are well correlated

with precipitation (>0.5)

Referred to as “good nodes” (blue)

4 are not correlated (<0.1)

Referred to as “noise nodes” (yellow)

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Tree-Ring Reconstruction ApproachesMultiple Linear Regression

Individual chronologies are added in a stepwise fashion

Principle Component Linear RegressionEliminates multicollinearityParsimonious model since the majority of the

variance is represented in fewer variables.K-nearest neighbor nonparametric approach

No assumption of distributionCaptures nonlinearitiesRemoves undue influence of outliers

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New ApproachCluster analysis on the tree-ring chronologies

to find distinct, coherent climate signals.K-means clustering approachIncreases the amount of climate signal that can

be extractedPerform PCA on each cluster, provide the

leading PCs from each cluster as potential predictorsSignal that may have been washed out during

PCA on the entire pool of predictors is preserved

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Regression MethodsPresent two regression methods to add to the

tree-ring reconstruction repertoire Local Polynomial regression.Extreme Value Analysis (EVA)

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Method 1: Local Polynomial RegressionFind the K-nearest neighbors, fit a polynomial to the

neighborhoodPolynomials are fitted in the GLM framework, where Y

can be of any distribution in the exponential family (normal, gamma, binomial, etc)G(E(Y))=f(Y)+G(.) = link function, X = set of predictors/independent variables E(Y) is the expected value of the

response/dependent variable is the error, assumed to be normally distributed

Improvement over K-NN resampling Values beyond those found in the historical record can be

generated

Summary

• Bayesian Hierarchical Modeling •Powerful tool for all functional (regression) estimation problems(which is most of forecasting/simulation)

•Provides model and parameter uncertainties•Obviates the need for discarding covariates•Enables incorporation of expert opinions•Enables modeling a rich variety of variable types

•Continuous, skewed, bounded, categorical, discrete etc.•And distributions (Binomial, Poisson, Gammma, GEV)

•Generalized Framework •Traditional linear models are a subset

bayesian hierarchical modeling of hydroclimate problems

Documents

variable of y linear

model parameters

model structure

glmlinear model

near bayesian models

dependent variable y

normaltransformations

kleiber university of