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. 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

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. D t (x t-1 ) d =1 (feature vector) 2. determine k nearest neighbors among D t using Euclidean distance 3. define a discrete kernel K(j(i)) for resampling one of the x j(i) as follows 4. using the discrete probability mass function K(j(i)), resample x j(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 D t, 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)