Comparison of natural streamflows generated from a parametric and nonparametric stochastic model

1. 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

2. Motivation • Generate future inflow scenarios for decision making models • reservoir operating rules, salinity control • Estimate uncertainty in model output Options • Parametric Techniques • AR, ARMA, PAR, PARMA • Nonparametric Techniques • K-NN, density estimator, bootstrap

3. Objective of Study • Compare nonparametric and parametric techniques for simulation of streamflows • at USGS stream gauge 09180500: Colorado River near Cisco, UT

4. 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

5. 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 Salas (1992)

6. 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

7. Nonparametric (cont’d) • Markov process for resampling Lall and Sharma (1996)

8. 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 Where v tj is the jith component of Dt, and w j are scaling weights. Lall and Sharma (1996)

17. Bivariate Probability Density Function

18. 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)