1. STOCHASTIC HYDROLOGY Stochastic Simulation (I) Univariate simulation Professor Ke-Sheng Cheng Department of Bioenvironmental Systems Engineering National Taiwan University

2. Stochastic Hydrology Hydrological processes exhibit variations in both space and time. As hydrological models are simplified versions of reality, they produce predictions or estimates of hydrological variables (e.g. runoff, hydraulic head, concentration) that are inherently erroneous. Stochastic hydrology is mainly concerned with the assessment of uncertainty in hydrological analysis, modeling and forecasting. 11/13/2015 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 2

3. In stochastic hydrology, the assessment of uncertainty is an integral part of hydrological analysis and modeling, being as important as the predictions themselves. Assessment of uncertainties is achieved by using “stochastic models”, which are models consisting of random components. These random components characterize the part of reality that is not explained by the deterministic components in our model. 11/13/2015 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 3

4. Errors in hydrological model prediction can occur because – the model concept is wrong, or – due to errors in parameters, boundary and initial conditions. We may choose to ignore these errors and accept our model predictions at face value. 11/13/2015 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 4 Model uncertainty Parameter uncertainty

5. However, in operational hydrology, when actual decisions have to be made based on hydrological model predictions (sometimes involving human life such as in flood control), it is imperative that uncertainty is taken into account in the decision-making process. – Government policy decision-making is a complicated process. (Holistic decision making) – An inappropriate decision may result in significant losses of life or over/under- investments in public infrastructures. Unintended consequences of government policy 11/13/2015 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 5

6. What does a prediction really represent? Example of a linear regression model 11/13/2015 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 6

7. What does a prediction really represent? Example of a linear regression model. 11/13/2015 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 7

8. Predictions of data-driven models. Predictions of deterministic models. Properties of the predictand variable – Unbiased ? – Confidence interval ? – Efficient ? 11/13/2015 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 8 For parameter estimation, we are concerned about the above properties of our estimators, but why not the predictions?

9. The objectives of this course are – To demonstrate the stochastic nature of many hydrological processes, – To facilitate students with a stochastic perspective of hydrological modeling and forecasting, and – To introduce techniques of stochastic simulation which will enable students to explore a wide range of applications. 11/13/2015 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 9

10. Univariate Simulation Pseudo random number generation Probability integral transformation Rejection method Frequency-factor based generation Random number generation using R 11/13/2015 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU 10

11. Pseudo Random Number Generator (PRNG) Computer simulation of random variables is the task of using computers to generate many random numbers that are independent and identically distributed (IID). It is also known as random number generation (RNG). In fact, these computer-generated random numbers form a deterministic sequence, and the same list of numbers will be cycled over and over again. This cycle can be made to be so long that the lack of true independence is unimportant. 11/13/2015 11 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

12. Therefore, such computer codes are often termed pseudo-random number generators (PRNG). There exist mathematical transformation methods to obtain other distributions from uniform variates. For this reason, most PRNGs found in software libraries produce uniform random numbers in the unit interval (0, 1). 11/13/2015 12 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

13. Linear Congruential Generator Generation of random samples of various probability densities is based random samples of the uniform density U[0,1). Therefore, the algorithm of generating random numbers of U[0,1) is essential. This can be achieved by the Linear Congruential Generator (LCG) described below. 11/13/2015 13 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

14. Let a sequence of numbers x n be defined by modulo m where a, c, and m are given positive integers. The above equation means that is divided by m and the remainder is taken as the value of. The quantity is then taken as an approximation to the value of a U[0,1) random variable. When c = 0, the algorithm is also called a pure multiplicative generator. 11/13/2015 14 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

15. A guideline for selection of a and m (c =0) is that m be chosen to be a large prime number that can be fitted to the computer word size. For a 32-bit word computer, m = and a = result in desired properties. For small computers without a random number generator, the following a, c, and m are found to be satisfactory when the LCG algorithm is used: and 11/13/2015 15 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

16. PROBABILITY INTEGRAL TRANSFORMATION The PIT method is based on the property that a random variable X with CDF can be transformed into a random variable U with uniform distribution over the interval (0,1) by defining Conversely, if U is uniformly distributed over the interval (0,1), then has cumulative distribution function. 11/13/2015 16 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

17. 11/13/2015 17 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

18. 11/13/2015 18 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

19. For random variables whose cumulative distribution function cannot be expressed by a close form the probability integral transformation technique cannot be used for generating random numbers of these random variables. The normal distribution is one such random variable. 11/13/2015 19 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

20. The Acceptance/Rejection Method This method uses an auxiliary density for generation of random quantities from another distribution. This method is particularly useful for generating random numbers of random variables whose cumulative distribution functions cannot be expressed in closed form. 11/13/2015 20 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

21. Suppose that we want to generate random numbers of a random variable X with density f(X). An auxiliary density g(X) which we know how to generate random samples is identified and cg(X) is everywhere no less than f(X) for some constant c, i.e., 11/13/2015 21 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

22. cg(X) f(X)f(X) X 11/13/2015 22 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

23. Generate a random number x of density g(X), Generate a random number u from the density U[0,cg(x)), Reject x if u > f(x); otherwise, x is accepted as a random number form f(X), Repeat the above steps until the desired number of random numbers are obtained. 11/13/2015 23 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

24. Frequency-factor-based generation An advantage of the method is that it does not require CDF inversion and frequency factors of the five commonly used distributions involves only the standard normal and the uniform deviates. 11/13/2015 24 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

25. GENERAL EQUATION FOR HYDROLOGICAL FREQUENCY ANALYSIS A random variable X has cumulative distribution function with mean and standard deviation. The magnitude of X corresponding to return period T, denoted by, is defined as Chow (1951) proposed the following general equation for hydrologic frequency analysis: where, the frequency factor, is a function of T and is distribution-specific. 11/13/2015 25 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

26. Apparently, if X is normally distributed, the frequency factor corresponds to the standard normal deviate with exceedence probability 1/T. Frequency factors of distributions commonly used in hydrologic frequency analysis have been developed (Kite, 1988). 11/13/2015 26 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

27. 11/13/2015 27 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

28. 11/13/2015 28 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

29. 11/13/2015 29 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

30. 11/13/2015 30 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

31. TEST AND VALIDATION In order to demonstrate the applicability of the FQFT approach, random numbers of normal, log-normal, extreme value type I (EV1), Pearson type III (PT3) and Log- Pearson type III (LPT3) distributions were generated and tested. 11/13/2015 31 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

32. For each type of distribution N random samples, each of size n, were generated and used in subsequent analysis. In this study the sample size n was set to vary from 50 to 500 at increment of 50 and number of random samples N was set to 1,000 and 10,000. 11/13/2015 32 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

33. Three means were adopted to test the validity of the generated random numbers: – Graphical comparison of CDF and empirical CDF (ECDF) derived from generated data, – Properties of estimated parameters, and – Type-I-error of goodness-of-fit (GOF) test. 11/13/2015 33 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

34. Graphical comparison of CDF and ECDF Figure 2 graphically illustrates the closeness of CDF and ECDF with regard to sample size of 50 and 500. Each ECDF in Figure 2 is based on one single random sample of size 50 or 500 and it may change when another random sample is used. It can be seen that even at sample size of 50 the ECDF is fairly close to CDF of the designated distribution. At sample size of 500, all ECDFs become almost indistinguishable from their corresponding CDFs. 11/13/2015 34 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

35. 11/13/2015 35 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

36. 11/13/2015 36 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

37. Properties of parameter estimators From each of the N generated random samples, distribution parameters including mean, standard deviation and coefficient of skewness can be estimated. Furthermore, from a total of N random samples, the sample mean and standard deviation of the above estimated parameters were calculated, with respect to sample size n ranging from 50 to 500. 11/13/2015 37 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

38. Uncertainty in estimation of mean reduces as sample size increases. 11/13/2015 38 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

39. Uncertainty in estimation of standard deviation reduces as sample size n increases. 11/13/2015 39 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

40. Uncertainty in estimation of skewness coefficient reduces as sample size n increases. 11/13/2015 40 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

41. With 1,000 and 10,000 random samples (N=1000 or 10,000), sample means (the center line) of the estimated parameters (including mean, standard deviation and coefficient of skewness) are very close to the theoretical values designated for random number generation. 11/13/2015 41 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

42. It is also seen clearly that standard deviations of all parameter estimators decrease with increase of the sample size n, indicating the unbiasedness of the estimator and reduction of uncertainty in parameter estimation. Such characteristics of parameter estimators suggest the generated random samples are indeed from the desired distributions. 11/13/2015 42 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

43. Type-I-Error of goodness-of-fit test Each random sample of size n is generated from a theoretical distribution with designated parameters and GOF test can be applied to test whether the random sample is drawn from the theoretical distribution. The widely applied Chi-square GOF test is adopted in this study. 11/13/2015 43 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

44. 11/13/2015 44 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

45. 11/13/2015 45 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

46. 11/13/2015 46 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

47. 11/13/2015 47 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

48. 11/13/2015 48 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

49. 11/13/2015 49 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

50. 11/13/2015 50 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

51. 11/13/2015 51 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

52. Generating random samples of normal distribution The Box-Muller method 11/13/2015 52 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

53. The Central Limit Theorem 11/13/2015 53 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

54. Acceptance/Rejection Method – The cumulative distribution function of the exponential density and its inverse function can be easily derived, and therefore, random samples of the exponential distribution can be generated with the probability integral transformation method. 11/13/2015 54 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

55. 11/13/2015 55 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

56. 11/13/2015 56 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

57. 11/13/2015 57 Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

58. Random number generation in R R commands for stochastic simulation (for normal distribution – pnorm – cumulative probability – qnorm – quantile function – rnorm – generating a random sample of a specific sample size – dnorm – probability density function For other distributions, simply change the distribution names. For examples, (punif, qunif, runif, and dunif) for uniform distribution and (ppois, qpois, rpois, and dpois) for Poisson distribution. 11/13/2015 58 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

59. Generating random numbers of discrete distribution using R Discrete uniform distribution – R does not provide default functions for random number generation for the discrete uniform distribution. – However, the following functions can be used for discrete uniform distribution between 1 and k. rdu<-function(n,k) sample(1:k,n,replace=T) # random number ddu =1 & x<=k & round(x)==x,1/k,0) # density pdu<-function(x,k) ifelse(x<1,0,ifelse(x<=k,floor(x)/k,1)) # CDF qdu 1, return("undefined"), ceiling(p*k)) # quantile 11/13/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 59

60. – Similar, yet more flexible, functions are defined as follows dunifdisc =min & x<=max & round(x)==x, 1/(max-min+1), 0) >dunifdisc(23,21,40) >dunifdisc(c(0,1)) punifdisc max, 1, floor(q-min+1)/(max-min+1))) >punifdisc(0.2) >punifdisc(5,2,19) qunifdisc<-function(p, min=0, max=1) floor(p*(max-min+1))+min >qunifdisc(0.2222222,2,19) >qunifdisc(0.2) runifdisc<-function(n, min=0, max=1) sample(min:max, n, replace=T) >runifdisc(30,2,19) >runifdisc(30) 11/13/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 60

61. Binomial distribution 11/13/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 61

62. Poisson distribution 11/13/2015 Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ. 62