Post on 18-Dec-2015
Comparison of natural streamflows generated from a parametric and nonparametric stochastic model
James Prairie(1,2), Balaji Rajagopalan(1) and Terry Fulp(2)
1. University of Colorado at Boulder, CADSWES
2. U.S Bureau of Reclamation
Options
Motivation
• Generate future inflow scenarios for decision making models– reservoir operating rules, salinity control
• Estimate uncertainty in model output
• Parametric Techniques– AR, ARMA, PAR, PARMA
• Nonparametric Techniques– K-NN, density estimator, bootstrap
Objective of Study
• Compare nonparametric and parametric techniques for simulation of streamflows – at USGS stream gauge 09180500: Colorado
River near Cisco, UT
Outline of Talk
• Overview of parametric technique
• Explain nonparametric technique
• Compare various distribution attributes– mean
– standard deviation
– lag(1) correlation
– skewness
– marginal probability density function
– bivariate probability
• Conclusions
Parametric
• Periodic Auto Regressive model (PAR)– developed a lag(1) model
– Stochastic Analysis, Modeling, and Simulation (SAMS)
• Data must fit a Gaussian distribution– log and power transformation
– not guaranteed to preserve statistics after back transformation
• Expected to preserve– mean, standard deviation, lag(1) correlation
– skew dependant on transformation
– gaussian probability density function
( )å=
-- S+-F+=p
jjjj yy
1,,,, tnttntttn mm Salas (1992)
season
year
==
tn
Nonparametric
• K- Nearest Neighbor model (K-NN)– lag(1) model
• No prior assumption of data’s distribution– no transformations needed
• Resamples the original data with replacement using locally weighted bootstrapping technique– only recreates values in the original data
• augment using noise function
• alternate nonparametric method
• Expected to preserve– all distributional properties
• (mean, standard deviation, lag(1) correlation and skewness)
– any arbitrary probability density function
Nonparametric (cont’d)
• Markov process for resampling
Lall and Sharma (1996)
Nearest Neighbor Resampling
1. Dt (x t-1) d =1 (feature vector)
2. determine k nearest neighbors among Dt using Euclidean distance
3. define a discrete kernel K(j(i)) for resampling one of the xj(i) as follows
4. using the discrete probability mass function K(j(i)), resample xj(i) and update the feature vector then return to step 2 as needed
5. Various means to obtain k– GCV– Heuristic scheme
( ) ÷÷ø
öççè
æ-= å
=
d
jtjijjit vvwr
1
2/1
Where v tj is the jith component of Dt, and w j are scaling weights.
( )( )å
=
=k
j
j
jijK
1
1
1
Lall and Sharma (1996)Nk =
Annual Water Year Natural FlowUSGS stream gauge 09180500 (Colorado River near Cisco, UT)
0
2000
4000
6000
8000
10000
12000
14000
1906
1909
1912
1915
1918
1921
1924
1927
1930
1933
1936
1939
1942
1945
1948
1951
1954
1957
1960
1963
1966
1969
1972
1975
1978
1981
1984
1987
1990
flo
w (
1000
acr
e-fe
et/y
ear)
Monthly Natural FlowUSGS stream gauge 09180500 (Colorado River near Cisco, UT)
0
500
1000
1500
2000
2500
3000
3500
4000
Oct-05
Oct-08
Oct-11
Oct-14
Oct-17
Oct-20
Oct-23
Oct-26
Oct-29
Oct-32
Oct-35
Oct-38
Oct-41
Oct-44
Oct-47
Oct-50
Oct-53
Oct-56
Oct-59
Oct-62
Oct-65
Oct-68
Oct-71
Oct-74
Oct-77
Oct-80
Oct-83
Oct-86
Oct-89
flo
w (
1000
acr
e-fe
et/m
on
th)
Bivariate Probability Density Function
Conclusions
• Basic statistics are preserved– both models reproduce mean, standard deviation,
lag(1) correlation, skew
• Reproduction of original probability density function– PAR(1) (parametric method) unable to reproduce
non gaussian PDF
– K-NN (nonparametric method) does reproduce PDF
• Reproduction of bivariate probability density function– month to month PDF
– PAR(1) gaussian assumption smoothes the original function
– K-NN recreate the original function well
• Additional research• nonparametric technique allow easy incorporation
of additional influences to flow (i.e., climate)
Exploratory Data
Analysis
Stochastic Flow Model
Salinity Model in
RiverWare
Policy Analysis
GPAT
Comparison of
parametric and
nonparametric model
Natural Salt
Model
Climate Indicators
K-NN and AR(1)
Nonparametric
regression (lowess)
replacement for
CRSM
compare salt mass
Research Topics Interconnection
Investigation of CRSS
model and data