Download Linear Regression Models - Civil, Environmental and Architectural

Bayesian Hierarchical Modeling of
Hydroclimate Problems
Balaji Rajagopalan
Department of Civil, Environmental and Architectural
Engineering
And
Cooperative Institute for Research in Environmental Sciences
(CIRES)
University of Colorado
Boulder, CO, USA
Bayes by the Bay Conference, Pondicherry
January 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
Linear Regression Models
where
Linear Regression Models
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
Generalized Linear Model (GLM)
Bayesian Perspective
Exponential
family PDF,
parameters
All distributions
Arise from this
Normal,
Exponential,
Gamma
Binomial,
Poisson, etc
• Noninformative prior on β
• Assuming Normal distribution for Y, g (.) is identity  Linear Regression
Generalized Linear Model (GLM)
Bayesian Perspective
• Log and logit – Canonical Link Functions
Generalized Linear Model (GLM)
Bayesian Perspective
Generalized Linear Model (GLM)
Bayesian Perspective
Generalized Linear Model (GLM)
Bayesian Perspective
Inverse
Chi-Square
Generalized Linear Model (GLM)
Bayesian Perspective
Summary
• 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.
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)
• Users most interested in sectoral/process
outcomes (streamflows, crop yields, risk of
disease X, etc.)
• Need for a robust spatial weather generator
Historical Synthetic series – Conditional on
Climate Information
Data
28.5
23.1
29.1
25.8
…
…
…
…
…
…
…
…
…
…
…
…
12.4
10.2
11.4
9.7
…
Process model
Frequency
distribution of
outcomes
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
1 T
As   a st
T t 1
1 T
Bs   bst
T t 1
1 T
C s   c st
T t 1
Ps (n)  log it 1 ( As  Bs sin( n)  C s cos(n))
Clusters on average day of max probability
Psmas  logit
1
A 
s
Bs2  Cs2

Day of Max Probability of Rainfall
• Max Probability of rainfall correlated
With climate variables – ENSO, etc.
• Characterize rainfall ‘onset’
• Prediction of ‘onset’
• Lima and Lall (2009, WRR)
Max Probability of Rainfall
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

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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
Occurrence Model
Latent Gaussian Process
Latent Gaussian Process

Parameter Estimation  MLE, two step
GLM +
Latent Gaussian
Process
Kleiber et al. (2012)
For Max and Min Temperature Models
Conditioned 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
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
Yichang
hydrological
station (YHS)
Streamflow Forecasting at Three Gorges Dam
a) SST Vs Mean JJA Flow(1970-2001)
c) SST Vs Peak Flow(1970-2001)
b) Snow Vs Mean JJA Flow(1970-2001)
d) Snow Vs Peak Flow(1970-2001)
Zone selected
Climate predictors
SST1
-10°N~10°N
150°E~180°E
SST2
-20°N~0°
75°E~110°E
SST3
10°N~30° N
130°E~150°E
Snow
-10°N~0°N
200°E~230°E
†: Significant at 95% confidence; ‡: Significant at 90% confidence
JJA Seasonal Flow
-0.27 ‡
0.51 †
0.38 †
0.42 †
Annual Peak Flow
-0.28 ‡
0.20 †
0.45 †
0.42 †
BHM for Seasonoal Streamflow

Model
Data showed mild nonlinearity  Quadratic
terms in the model
is distributed as half-Cauchy with parameter 25  “mildly informative”
Gelman (2006, Bayesian Analysis)
MCMC is used to obtain the posterior distributions
400
400
400
Streamflow Forecasting at Three Gorges Dam
Histogram of tau
Histogram of Beta1
300
400
Histogram of Beta2
300
400
300
400
Histogram of tau
Histogram of Beta1
Histogram of Beta2
200
300
200
300
200
300
100
200
100
200
100
200
10000
0
0
400
10
20
10
30
20
30
40
40
10001.8
0
1.8
400
2
2.2
2
2.4
2.2
Histogram of Beta3
2.6
2.4
2.6
0
100
-0.4
0
-0.4
400
Histogram of Beta4
200
300
200
300
100
200
100
200
100
200
Description
0
Interceptor
-0.4 -0.2
SST1
SST12
SST2
0
Beta1
0.2
Beta2
Beta3
0
100
-0.4
0.4
0.6
0.4
Mean
0
2.273
0.6
-0.4
-0.111
0.130
Node
0.2
0.4
0.4
-0.2
0
0.2
Standard Dev.
-0.2
0.074
0
0.050
0.048
0.2
0.4
0.6
0
100
-0.4
2.50%
0.4 2.129
0.6
-0.209
0.035
-0.2
0
0.2
Median
0
-0.4
2.273 0
-0.2
-0.111
0.130
0.4
0.6
0.6
97.50%
0.2
2.420 0.6
0.4
-0.011
0.224
Beta4
0.276
0.051
0.176
0.276
0.377
Beta5
Snow
Performance Measure
Seasonal (JJA)
0.083
R
0.802
0.025
CoE
0.643
0.034
IoA
0.886
0.083
Bias
0.001
0.132
RMSE
0.231
2
0.6
Histogram of Beta5
200
300
0.2
0
0.2
300
400
Histogram of Beta3
0
-0.2
0
Histogram of Beta5
300
400
-0.2
-0.2
Histogram of Beta4
300
400
0
100
-0.4
Predictors
2, 3, 4 and 5
Show tighter
Bounds
Uncertainty
in predictors
(i.e. model) is
obtained and
propogated in
the forecacsts
You can use
PCA or stepwis
etc. to reduce
the number of
predictors
(this can be
crude)
Streamflow Forecasting at Three Gorges Dam
Description
Node
Mean
Standard Dev.
2.50%
Median
97.50%
Interceptor
SST1
Beta1
Beta2
Beta3
2.273
-0.111
0.130
0.074
0.050
0.048
2.129
-0.209
0.035
2.273
-0.111
0.130
2.420
-0.011
0.224
Beta4
0.276
0.051
0.176
0.276
0.377
Beta5
Snow
Performance Measure
Seasonal (JJA)
0.083
R
0.802
0.025
CoE
0.643
0.034
IoA
0.886
0.083
Bias
0.001
0.132
RMSE
0.231
SST12
SST2
2
Maximum Seasonal Streamflow
Extreme Value Analysis – Floods
(Kwon et al.,2010, Hydrologic
Sciences)
Ann Max Flow
American River at Fair Oaks - Ann. Max.
Flood
180,000
160,000
140,000
120,000
100,000
80,000
60,000
40,000
20,000
0
1900
1920
1940
1960
Year
100 yr flood estimated from
21 & 51 yr moving windows
1980
2000
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 Variance given DJF
NINO3 and PDO
Flood mean given DJF
NINO3 and PDO
NINO3
NINO3
PDO
PDO
Derived using weighted local
regression with 30 neighbors
Correlations:
Log(Q) vs DJF NINO3 -0.34
Jain & Lall, 2000
vs DJF PDO
-0.32
Atmospheric River
generates flooding
CZD
Russian River, CA Flood Event
of 18-Feb-04
Slide from Paul Neiman’s
talk
Russian River flooding in Monte Rio, California
18 February 2004
IWV
(cm)
GPS IWV data from near CZD: 14-20 Feb 2004
Atmospheric
river
Cloverdale
10” rain at CZD
in ~48 hours
IWV (inches)
IWV (cm)
Bodega
Bay
photo courtesy of David Kingsmill
Flood Estimation Under Nonstationarity

Significant interannual/interdecadal variability of
floods



Stationarity assumptions (i.i.d) are invalid
Large scale climate features in the OceanAtmosphere-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)
1 / 


 

 z   
G( z )  exp  1   










3 Model parameters
Location parameter:  (where distribution is centered)
Scale parameter:
  0 (spread of the distribution)

Shape parameter:
(behavior of distribution tail)
Gumbell, Frischet, Weibull
“Unconditional” GEV
(Coles 2001)
Incorporate covariates into GEV parameters to
account for nonstationarity
Could apply to any parameter, but location is most intuitive:
   0  1 x
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)
Streamflow Forecasting at Three Gorges Dam
a) SST Vs Mean JJA Flow(1970-2001)
c) SST Vs Peak Flow(1970-2001)
b) Snow Vs Mean JJA Flow(1970-2001)
d) Snow Vs Peak Flow(1970-2001)
Zone selected
Climate predictors
SST1
-10°N~10°N
150°E~180°E
SST2
-20°N~0°
75°E~110°E
SST3
10°N~30° N
130°E~150°E
Snow
-10°N~0°N
200°E~230°E
†: Significant at 95% confidence; ‡: Significant at 90% confidence
JJA Seasonal Flow
-0.27 ‡
0.51 †
0.38 †
0.42 †
Annual Peak Flow
-0.28 ‡
0.20 †
0.45 †
0.42 †
BHM for Seasonal Maximum Flow

Model
Data showed mild nonlinearity  Quadratic
terms in the model
is distributed as half-Cauchy with parameter 25  “mildly informative”
Gelman (2006, Bayesian Analysis)
MCMC is used to obtain the posterior distributions
of Beta2
Streamflow Forecasting at ThreeHistogram
Gorges
Dam
400
400
400
Histogram of tau
Histogram of Beta1
300
400
300
400
300
400
Histogram of tau
Histogram of Beta1
Histogram of Beta2
200
300
200
300
200
300
100
200
100
200
100
200
0
1000
0.5
1
1.5
0
1003
3.5
4
4.5
5
5.5
0
-0.5
100
0
0.5
1
1.5
0
0
400
0.5
1
1.5
0
3
400
3.5
4
4.5
5
5.5
0
-0.5
0
0.5
1
1.5
Histogram of Beta3
Histogram of Beta4
300
400
300
400
Histogram of Beta3
Histogram of Beta4
200
300
200
300
100
200
100
200
0
-0.5
100
Predictors
3 and 5
Show tighter
Bounds
0
0.5
1
1.5
0
-0.5
100
0
0.5
1
1.5
Zone selected
Climate predictors
0
0
-10°N~10°N
-0.5
0 SST1
0.5
1
1.5
-0.5
0
0.5150°E~180°E
1
1.5
SST2
-20°N~0°
75°E~110°E
SST3
10°N~30° N
130°E~150°E
Snow
-10°N~0°N
200°E~230°E
†: Significant at 95% confidence; ‡: Significant at 90% confidence
JJA Seasonal Flow
-0.27 ‡
0.51 †
0.38 †
0.42 †
Annual Peak Flow
-0.28 ‡
0.20 †
0.45 †
0.42 †
Streamflow Forecasting at Three Gorges Dam
Description
Interceptor
Node
Beta1
Mean
4.174
Standard Dev.
0.195
2.50%
3.791
Median
4.171
97.50%
4.548
SST12
SST3
Beta2
Beta3
Beta4
0.198
0.699
-0.089
0.119
0.148
0.079
-0.055
0.410
-0.264
0.203
0.706
-0.085
0.423
0.986
0.053
Beta5
Snow2
Performance Measure
Annual Peak Flow
0.302
0.098
0.091
0.310
0.473
R
0.729
CoE
0.531
IoA
0.828
Bias
-0.001
RMSE
0.602
SST32
Nonstationary Flood Risk at Three Gorges Dam
Ann Max Flow
Dynamic 50-yea
flood from BHM
and Stationary
50-year flood
180,000
160,000
140,000
120,000
100,000
80,000
60,000
40,000
20,000
0
1900
1920
1940
1960
Year
1980
2000
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)
-Distribution
-No physical filtration (“unfiltered”)
Case study location: PWB
Precipitation
events
High Flows
Exceedances
(SWTR
criterion:
turbidity < 5
NTU)
Back-up
groundwater
source
(Pumping $$)
GEV Model
Uncond
CondT
CondR
CondRT
CondR+T
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
M 0*
-
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
Variable
* 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
0
0
Streamflow (cfs)
Maximum
Maximum Streamflow (cfs)
8000
6000
4000
2000
2000
4000
6000
8000
Conditional quantiles correspond well to observed record
1970
1970
1980
1990
1980
1990
Year
Year
2000
2000
Uses
concurrent
climate, but
could also be
used with
seasonal
forecast
2e-04
1e-04
0e+00
PDF
3e-04
4e-04
GEV distribution can be compared for specific historic
times
0
2000
4000
6000
Maximum Streamflow (cfs)
8000
P and T climate change projections from IPCC AR4
are readily available
12 km2 resolution
(1/8 of a grid cell)
http://gdo-dcp.ucllnl.org/downscaled_cmip3_projections/#Welcome
Bias correct P & T to historic data for PWB watershed area
Results indicate increasing maximum streamflow
anomalies
Observed
16 GCM models
Maximum Streamflow Anomaly (%)
75
50
25
0
-25
GCM model average
1950
2000
Year
2050
2100
Streamflow quantiles shift higher under CC
projections
Observed
16 GCM models
Likelihood of Turbidity Spike

P( E )   P( E | S ) P( S )
Conditional P(E)
0
Probability of a
turbidity spike given a
certain maximum flow
Maximum Flow (CFS)
(Ang and Tang 2007)
Likelihood of a turbidity spike increases under
CC projections
Observed
16 GCM models
Likelihood of a turbidity spike increases
1950-2007
2070-2099
95th (top whisker)
13
28
75th (box top)
6.3
11
50th (box middle)
4.2
5.9
P(E)
Percentile
Small shifts in risk can result in high
expected loss
Percent Increase in Expected Loss Relative to 1950-2007
Period
140
50th
75th
120
95th
115
100
80
75
60
62
40
41
40
20
23
24
16
10
0
2010-2039
2040-2069
2070-2099
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
Motivation
Paleo Hydrology
Colorado River Example
UC CRSS stream
gauges
LC CRSS stream gauges
20
Total Colorado River Use 9-year moving average.
18
NF Lees Ferry 9-year moving average
16
Annual Flow (MAF)
14
12
10
8
6
4
2
Calnder Year
19
98
20
02
20
06
19
94
19
90
19
86
19
82
19
78
19
74
19
70
19
66
19
62
19
58
19
54
19
50
19
46
19
42
19
38
19
34
19
30
19
26
19
18
19
22
19
14
Colorado River Demand - Supply
0
Streamflow and Tree Ring Data
Fulton
Oneida
Herkimer
Saratoga
Onondaga
Madison
Montgomery
MiCO
Schenectady
R
Otsego
Cortland
Albany
Schoharie
Chenango
^
Broome
are
Ri v
er
*# Schoharie
^
y1
ela
w
ra
st B
We Delaware
er
tB
ra
nc
h
Tioga
v
re Ri
D
New York
hD
nc
a
e la w
Ea
y4
* Pepacton
* Canonsville #
#
s
Sc
Batavia Kill
ho
ha r
ie Cre
^
Greene
MoCO
MoTP
Columbia
MHH
y5
nk
rsi
e
v
Ne
ek
er
Ri v
nd
Ro
o
C
ut
k
ree
MSBMPP
MRH
MLQ
^ ^
y2
* y3 #
#
* Roundout
Ulster
Susquehanna
Bradford
Neversink
Dutchess
Sullivan
Wayne
Wyoming
Lackawanna
Putnam
Orange
Streamflow and Tree Ring Data
Average Summer (JJA) Flows as Predictand
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
Annual Tree Ring Growth Index (Chronology) as Predictor – 246 years common data
Abbreviation
MHH
MLQ
MRH
MSB
MPP
MoCO
MoTP
MiCO
Site
Mohonk, NY
Mohonk, NY
Mohonk, NY
Mohonk, NY
Mohonk, NY
Montplace, NY
Montplace, NY
Middleburgh, NY
Species
Humpty Dumpty Helmlock
Long, QUSP
Rock Rift Hemlock
Sweet Birch, BELE
Pitch Pine
Chestnut Oak, QUPR
Tulip Popular, LITU
Chestnut Oak, QUPR
1754
Number of Trees
43
20
18
17
23
21
20
23
1999
1903
1937 1950
Summer Flow = f(tree rings) + error
Number of Series
25
34
25
27
45
34
32
42
Data Record
1754 - 1999
1754 - 1999
1754 - 1999
1754 - 1999
1754 - 1999
1754 - 1999
1754 - 1999
1754 - 1999
246 years chronology (Xt)
(8 tree ring chronologies)
1999
variable length streamflow record (Yt)
(5 sites)
Preliminary Data Analysis – Bayesian Hypothesis
(correlation – tree chronology Vs average summer seasonal flow)
Station-tree
correlations similar!
- pooling?
Oneida
Hypothesis
Herkimer
Saratoga
Montgomery
No Shrinkage of
Regression MiCO
Coefficients
(no pooling)
traditional regression
Madison
Shrinkage of
Regression Coefficients
across sites
(partial pooling)
hierarchical model
Schenectady
Otsego
Chenango
Albany
Schoharie
(a)
*# Schoharie
r
elawa
Little D elaware
^
r
R i ve
S choh
C r ee k
MoRO
MoTP
MoCO
Eas t
^
Columbia
y5
* Pepacton
#
MBO
r
Rive
sink
ever n k R ive r
N
h
c
si
Bran Never
East nch
a
Br
t
s
e
k
r ee
ou t C
^
y2
r
Nev e
Sullivan
Ro
R
sink
nd
iv e W
r
Broome
Wayne
Greene
arie
Br
^
y4
* Canonsville
#
iv
an c
hD
Delaware
Batavia Ki ll
y1
er
B
hD
eR
st
nc
ra
el a
wa r
We
New York
ve r
e Ri
^
y3
* Neversink #
#
* Roundout
MLQ
MSB
MHH
MRH
Ulster
Dutchess
Bayesian Hierarchical Models
Streamflow
Log Normal Distribution
Regression Coefficients (β)
of the hierarchical model - multivariate normal distribution
Partial Pooling – Hierarchical
Model
Shrinkage on the coefficients to
incorporate the predictive ability of
each tree chronology on multiple
stations
trees
ti   i    ij xtj
0
j
 ~ MVN (   j ,  j )
i
j
p( / data) 
p( )  p(data /  )
p(data)
Key ideas:
1. Streamflow at each site comes
from a pdf
2. Parameters of each pdf informed
by each tree
3. Common multivariate distribution
of parameters across trees
4. Noniformative prior for parameters
of multivariate distribution
5. MCMC for parameter estimation
T
log( yti ) ~ N ( ti ,  2 )
i

trees
ti   i    ij xtj
0
j
 ~ MVN (   j ,  j )
i
j
  j ~ N (0,0.0001)
 ~ N (0,0.0001)
i
0
Site i
 0i
 ti
 2i
log( Qti )
i
j
j

xt j
 ~ covariance
 1 
i  ~ unif (0, 100)
 2 
i
Year t
j
Tree stand j
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 Performance
Posterior PDF (Model Level 1)
Delaware River Reconstruction and Performance
Regression Coefficients
Model Level 2
No
Pooling
Partial
Pooling
Delaware River Reconstruction
Cross-Validated Performance
Canonsville
Pepacton
Paleo Hydrology Reconstruction
Traditional Methods
Linear/Nonlinear Regression
PCA of Tree Rings
Regression on leading PCs
Slide 88 of 49
Objective 1: Tree-ring
Reconstructions
 LCBR
 Naturalize streamflow
 9 nodes in CRSS
 5 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)
Slide 89 of 49
Tree-Ring Reconstruction
Approaches
 Multiple Linear Regression
 Individual chronologies are added in a stepwise fashion
 Principle Component Linear Regression
 Eliminates multicollinearity
 Parsimonious model since the majority of the variance is
represented in fewer variables.
 K-nearest neighbor nonparametric approach
 No assumption of distribution
 Captures nonlinearities
 Removes undue influence of outliers
Slide 90 of 49
New Approach
 Cluster analysis on the tree-ring chronologies to find
distinct, coherent climate signals.
 K-means clustering approach
 Increases the amount of climate signal that can be
extracted
 Perform PCA on each cluster, provide the leading PCs
from each cluster as potential predictors
 Signal that may have been washed out during PCA on
the entire pool of predictors is preserved
Slide 91 of 49
Slide 92 of 49
Regression Methods
 Present two regression methods to add to the tree-ring
reconstruction repertoire
 Local Polynomial regression.
 Extreme Value Analysis (EVA)
Slide 93 of 49
Method 1:
Local Polynomial Regression
 Find the K-nearest neighbors, fit a polynomial to the
neighborhood
 Polynomials 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
Slide 94 of 49
Slide 95 of 49
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