STOCHASTIC HYDROLOGY Stochastic Simulation (I) Univariate simulation Professor Ke-Sheng Cheng...

STOCHASTIC HYDROLOGY Stochastic Simulation (I)

Univariate simulation

Professor Ke-Sheng ChengDepartment of Bioenvironmental Systems Engineering

National Taiwan University

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.

04/21/23Lab for Remote Sensing Hydrology and Spatial Modeling

Dept of Bioenvironmental Systems Engineering, NTU2

• 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.



• 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.



Model uncertainty

Parameter uncertainty

• 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



What does a prediction really represent?

• Example of a linear regression model



What does a prediction really represent?

• Example of a linear regression model.



• Predictions of data-driven models.

• Predictions of deterministic models.

• Properties of the predictand variable– Unbiased ?

– Confidence interval ?

– Efficient ?



For parameter estimation, we are concerned about the above properties of our estimators, but why not the predictions?

• 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.



Univariate Simulation

• Pseudo random number generation

• Probability integral transformation

• Rejection method

• Frequency-factor based generation

• Random number generation using R



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.

04/21/23 11Lab for Remote Sensing Hydrology and Spatial Modeling Dept of Bioenvironmental Systems Engineering, NTU

• 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).


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.


• Let a sequence of numbers xn 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.

][ 1 caxx nn

caxn 1

nx mxn /


• 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

1231 57

13849,25173 ca .65536m


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 .

)(XF

)(XFU X

)(1 UFX X)(XF




• 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.


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.


• 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.,

xxcgxf )()(


cg(X)

f(X)

X


• 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.


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.


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.

)(XF

Tx TxXP T

1

TT Kx

TK


• 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).

TK






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.


• 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.


• 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.


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.




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.


Uncertainty in estimation of mean reduces as sample size increases.


Uncertainty in estimation of standard deviation reduces as sample size n increases.


Uncertainty in estimation of skewness coefficient reduces as sample size n increases.


• 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.


• 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.


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.










Generating random samples of normal distribution • The Box-Muller method


• The Central Limit Theorem


• 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.





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.

04/21/23 58Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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<-function(x,k) ifelse(x>=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 <- function(p, k) ifelse(p <= 0 | p > 1, return("undefined"),

ceiling(p*k)) # quantile

04/21/23Laboratory for Remote Sensing Hydrology and Spatial Modeling, Dept of Bioenvironmental Systems Engineering, National Taiwan Univ.

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– Similar, yet more flexible, functions are defined as follows• dunifdisc<-function(x, min=0, max=1) ifelse(x>=min & x<=max &

round(x)==x, 1/(max-min+1), 0)>dunifdisc(23,21,40)>dunifdisc(c(0,1))

• punifdisc<-function(q, min=0, max=1) ifelse(q<min, 0, ifelse(q>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)


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• Binomial distribution


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• Poisson distribution


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STOCHASTIC HYDROLOGY Stochastic Simulation (I) Univariate simulation Professor Ke-Sheng Cheng...

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