CVEN 5393: Water Resources Systems and Management Prof. Joseph Kasprzyk Topic 6

Transcription

1 Statistical Approaches for Simulation Modeling CVEN 5393: Water Resources Systems and Management Prof. Joseph Kasprzyk Topic 6

2 Probability distributions are used to describe random variables. [courtesy T. Wagener]

3 PDF and CDF [courtesy T. Wagener]

4 A histogram is often used to show the empirical distribution of a set of observations. Range Frequency Divide the data into groups according to their magnitudes. The magnitudes are shown on the horizontal axes. Blocks are drawn to represent the groups. The height of a block is proportional to the number of occurrences in the group. The variability of the data is shown by the horizontal spread of the blocks, and the most common values are found in the highest blocks.

5 Plan for Today Look at a stochastic traffic modeling example to learn about stochastic processes Discuss mechanics of Monte Carlo Simulation for doing stochastic simulation List the different sources of input streamflow information for water resources management models

6 What causes a traffic jam? The longest traffic jam in history was 12 days and 62 miles long.

7 We can use a simulation model to create traffic jams, even with the following simple rules: 1. First, increase your speed by 1 unit. You are in a hurry to get where you are going! [Technical Resources/MATLAB/Monte Carlo/] 2. Check the distance between you and the next car (think bumper cars!). If your speed means you would hit that car, slow down. 3. With some probability, put your foot on the brake and slow down by one unit. Maybe you re checking text messages?

8 This is called the Nagel-Schreckenberg traffic model. [Technical Resources/MATLAB/Monte Carlo/] This example is from lecture notes by Art Owen ( Code in MATLAB by J. Kasprzyk, who takes responsibility for all errors

9 At each simulation timestep, the following updates occur for each vehicle: [Technical Resources/MATLAB/Monte Carlo/]

10 Results - The simulation starts with 100 cars as black dots on the top row - Vehicles move from left to right, as time increases from top to bottom. - Traffic jams (bunches of black dots) occur spontaneously - The traffic jams move backward as traffic moves forward. [Technical Resources/MATLAB/Monte Carlo/]

11 Monte Carlo simulation allows us to simulate processes that are random, due to uncertainties. Each driver brakes with a probability p. Therefore, braking is a random variable. The cars are initialized in random starting locations But, a burn-in or spin-up period can alleviate this, where you only plot or analyze simulations long after the initial timestep. Plots show emergent behavior based on these simple traffic rules! [Technical Resources/MATLAB/Monte Carlo/]

12 Let s look at how p is simulated within our model: - At every time step, there is a probability p that a driver will slow down. - So in the simulation, draw a random number. - Use that random number to decide whether or not, at this simulation, the driver will brake. - Think of a flip of the coin Lines of the model on D2L: [Technical Resources/MATLAB/Monte Carlo/]

13 Monte Carlo Implementation Sampling from a list of data

14 Approach Input data is gathered from observations, or possibly other simulations The data are provided in list or array form. Assumptions: Data are representative of a given process There is enough data, such that the data cover the desired probability distribution for your model In today s example, we will assume that two variables are independent, which is not always true.

15 Example See MonteCarloFromList.mon D2L Water supply and demand are given in list form see the description of the water supply example in Watkins and Mckinney (1997) Assume that cost of buying water is normally distributed with mean $200k and standard deviation $50k WARNING: This is not a realistic example! Each supply and demand is considered equally likely Supply and demand events are considered independent For a given shortage, it is assumed that the city will be able to buy all the water it needs, and the distribution of water pricing is known. This simple code is designed to show you the basics of some calculations in MATLAB [Technical Resources/MATLAB/Monte Carlo/]

16 Illustrative Results 300 Simulated Shortage 300 Cost Volume (mill. cubic m) Cost (mill. USD) Histograms of sample results. Notice how the distribution of cost is different than the shortage distribution. Why could this be?

17 Monte Carlo Implementation 3A+Why+median+and+IQR+are+often+better+than+mean+and+standard+deviation Sampling from a given probability distribution

18 Monte Carlo sampling uses a trick : the CDF always ranges from 0 to 1! A cumulative distribution function (CDF) of your distribution is known or assumed. Generate a uniformly distributed random number, x, in the range from 0 to 1 Then, x corresponds to a given position in the probability distribution!

19 Implementation You may be required to implement the approach on the previous slide if you are using an empirical distribution of a particular dataset These are also called non-parametric approaches, because your distribution does not have a classical form For parametric sampling of classical distributions (normal, uniform, chi-squared), you can often use built-in functions within programming languages But under the hood, those algorithms are following the approach on the previous slide!

20 MATLAB Statistics Toolbox: random

21 R random number generation

22 Python random number generator

23 Time-Series Analysis and Streamflow Inputs to Water Management Models This material is only a preview. In our program, the best source of information on this is CVEN 6833, Advanced Data Analysis taught by Prof. R. Balaji

24 Motivation Monte Carlo simulation techniques discussed previously are most appropriate for the following situations: Data are not changing in time or space, (no need to preserve correlation within the data) Data have known, or easily assumed probability distributions Different variables within a model exhibit different distributions, and a modeler wants to combine those distributions together to determine output distributions To effectively model data that are changing in time, we need to preserve temporal properties of those data!

25 Motivation: Financial Data One stock may show volatility Accessed 3/10/2015

26 Motivation: Financial Data so often, investors trade on index funds that combine multiple stocks together. Note the patterns in time (stock market crash in )

27 Motivation: Colorado River Source: US Bureau of Reclamation [courtesy K. Nowak, R. Balaji] 7 States, 2 Nations Upper Basin: CO, UT, WY, NM Lower Basin: AZ, CA, NV Irrigates 3.5 million acres Serves 30 million people 60 MAF of total storage 4x Annual Flow 50 MAF in Lakes Powell + Mead

28 Colorado River regulation is critically dependent on its streamflow Colorado River Compact Divides Basin at Lee Ferry, AZ Apportions 7.5 MaFto Upper and Lower Basins 1944 Treaty with Mexico 1.5 MaF to be delivered at international border Total of 16.5 MaF/yr allocated 30,000,000 25,000,000 20,000,000 15,000,000 10,000,000 5,000, [courtesy K. Nowak, R. Balaji] Colorado River natural flow at Imperial Dam

29 Approaches for Streamflow Input: (1) Historical Record Also called period of record analysis Often used, especially for firm and safe yield calculations This is limited, since the observational record is short. Excellent indicator of statistical properties such as autocorrelation, the tendency for a time-series to be correlated with itself. Example: historical floods But: stationarity may be dead press_release/2004/ /index.html

30 Approaches for Streamflow Input: (1) Historical Record Multiple authors have asked in recent years: Is it appropriate to use the historical record for simulation? [Milly et al. 2008]

31 Approaches for Streamflow Input: (2) Parametric Time-Series Models See discussion in Nowak et al. (2010) for limitations of parametric models

32 Approaches for Streamflow Input: (3) Paleo Reconstruction

33 Reconstructions of paleo hydrology paint a dire picture Reconstruction of Colorado River at Lees Ferry streamflow, , with 10-year running mean Six 10-year periods before 1900 with reconstructed mean flow lower than 12 MaF (lowest: ) one of the three wettest ~25-year periods in 1200 years Mid-1100s: 57-year period with mean flow of ~13 MaF Century-scale non-stationarity: 100-year mean varies from 13.9 to 15.4 MaF [courtesy K. Nowak, R. Balaji] * Slide courtesy of Jeff Lukas, NOAA/WWA

34 Approaches for Streamflow Input: (4) Non-Parametric Techniques Basic idea: a synthetic approach to resampling the historical record Does not require estimation of parameters The approaches can be made to fit the statistical patterns of data very well May be more computationally difficult to use for some applications See example next time For more info, see above! books/ /7783

35 Acknowledgments Traffic Engineering example from lecture notes by Art Owen ( Code in MATLAB by J. Kasprzyk, who takes responsibility for all errors Nowak et al. slides provided by Prof. R. Balaji, from Ken Nowak s defense