SIMULATION SEMINAR SERIES INPUT PROBABILITY DISTRIBUTIONS

Transcription

1 SIMULATION SEMINAR SERIES INPUT PROBABILITY DISTRIBUTIONS Zeynep F. EREN DOGU

2 PURPOSE & OVERVIEW Stochastic simulations involve random inputs, so produce random outputs too. The quality of the output is no better than the quality of inputs. Approaches for effective representation and generation of uncertain inputs, when building and exercising computersimulation models. 2

3 DETERMINISTIC VS. RANDOM INPUTS Most simulation models, and indeed most operationsresearch models, might be viewed as having two aspects: structural and quantitative. The structural components include the logical elements and relationships among them. Optimization model the form of the objective function, the number and form of the constraints whether we seek to minimize or maximize the objective function. In a dynamic simulation model the flowchart topology of entity movements and how they interact. 3

4 DETERMINISTIC VS. RANDOM INPUTS The quantitative components of a model are the values of numerical inputs, or ranges or probability distributions that describe what values these inputs might assume. In an optimization model these could be the coefficients and other parameters 4

5 DETERMINISTIC VS. RANDOM INPUTS Quantitative inputs to simulation models can be classified as deterministic or random. Deterministic inputs: have known and unchanging values, like the number of servers at a workstation if they never change. Random inputs: deemed to vary in an uncontrolled way (though not completely unpredictably), like part-processing times or arrival processes. 5

6 INPUT DATA In order to create a computer model, we need to... Measure the data in the real system Approximate the data by a distribution function Use this function in the simulation program This procedure is called input data analysis 6

7 WE LL DISCUSS The 4 steps of input model development: 1. Collect data from the real system (There are many difficulties in data collection, involves lots of work, difficult, expensive and error-prone) 2. Identify a probability distribution to represent the input process 3. Choose parameters for the distribution 4. Evaluate the chosen distribution and parameters for goodness of fit. 7

8 DATA COLLECTION Suggestions that may enhance and facilitate data collection: Plan ahead: begin by a practice or pre-observing session, watch for unusual circumstances Combine homogeneous data sets, e.g. successive time periods, during the same time period on successive days Be aware of data censoring: danger of leaving out long process times Check for relationship between variables, e.g. build scatter diagram Check for autocorrelation 8

9 IDENTIFYING THE DISTRIBUTION Histograms (determining the shape of a distribution & make an educated guess at the distribution) Selecting families of distribution Parameter estimation Goodness-of-fit tests 9

10 HISTOGRAMS Steps in generating a histogram: 1. Divide the n items of data into k intervals (must be chosen carefully). k depends on: The number of observations The dispersion of the data Suggested ~ the square root of the sample size 2. Plot the number of items of data n k in each interval 3. Compare the shape of the histogram to known probability density functions (pdfs) 10

11 HISTOGRAM OF A VARIABLE: BODY MASS INDEX k=12 k=20 11

12 SELECTING THE FAMILY OF DISTRIBUTIONS A family of distributions is selected based on: The context of the input variable Shape of the histogram Remember the physical characteristics of the process Is the process naturally discrete or continuous valued? Is it bounded? Frequently encountered distributions: Easier to analyze: exponential, normal and Poisson Harder to analyze: beta, gamma and Weibull 12

13 SELECTING THE FAMILY OF DISTRIBUTIONS Use the physical basis of the distribution as a guide, for example: Binomial: # of successes in n trials Poisson: # of independent events that occur in a fixed amount of time or space Normal: dist n of a process that is the sum of a number of component processes Exponential: time between independent events, or a process time that is memoryless Weibull: time to failure for components Discrete or continuous uniform: models complete uncertainty Triangular: a process for which only the minimum, most likely, and maximum values are known Empirical: resamples from the actual data collected 13

14 QUANTILE-QUANTILE PLOTS Q-Q plots test whether the type of distribution is correct. Useful when n < 30, when histograms are messy. If X is a random variable with cumulative density functions (cdf) F, then the q-quantile of X is the such that: When F has an inverse, 14

15 QUANTILE-QUANTILE PLOTS Steps in generating a Q-Q plot: 1. Let {x i, i = 1,2,., n} be a sample of data from X. 2. Sort x i to give y j : y 1 y 2... y n 3. Plot y j against If the graph yields a straight line, then X is of type F. The parameters are correct, if the slope of the line is 1, and the line passes through the origin 15 Note: Q-Q plot can also be used to check homogeneity (Check whether a single distribution can represent both sample sets)

16 Q-Q PLOTS FOR NORMAL AND LOGNORMAL DIST 16

17 PARAMETER ESTIMATION Estimate the parameters of the distribution Given samples X i of the random variable X the estimators of sample mean and sample variance: Mean: sd: 6.83 n=532 17

18 GOODNESS-OF-FIT TESTS Conduct hypothesis testing on input data distribution using: Kolmogorov-Smirnov test Chi-square test No single correct distribution in a real application exists. They must be used with caution: They will accept almost anything for small n They will reject almost everything for large n 18

19 P-VALUES AND BEST FITS p-value is: The significance level at which one would just reject H 0 for the given test statistic value. A measure of fit, the larger the better Large p-value: good fit Small p-value: poor fit 19

20 Kolmogorov-Smirnov test Normal: p-value = Lognormal: p-value = 0.85 Poor Fit Good Fit Shapiro-Wilk test Normal: p-value = 2.170e-12 Lognormal: p-value = Poor Fit Good Fit 20

21 SOME NOTES 21

22 NOTE 1. INDEPENDENCE ACROSS AND WITHIN RANDOM INPUTS In stochastic simulations it s common to assume that the various random inputs are mutually independent of each other. There might be some situations where this would be invalid: 1. A patient arriving to an urgent-care facility might require four processing times positively correlated with each other 2. If you want to determine the tempo of a music, musical beats are repeating events that are autocorrelated. (like signal processing) 3. A call center has two peak periods each work day, and flattening them to a constant average rate would clearly result in misrepresentation. 22

23 NOTE 2. NORMAL DISTRIBUTION The most comfortable of all distributions, the normal. (provided in most simulation software) However, many dynamic simulations have as their random input time durations (processing times, times between successive arrivals, etc.). The normal distribution has infinite tails both ways, i.e., is unbounded in both directions, and it s the left side that s troublesome, producing negative values at least once in a while, which make no physical sense for time durations. 23

24 NOTE 3. OTHER WAYS OF USING DATA We have discussed: To use statistical methods to fit one of a variety of standard distributions to the input data, and then use standard generation methods (the standard way) Other ways might be: To feed the observed data into the simulation where they belong in the model. (problematic) To use the data to specify some sort of empirical, or histogram-based input distribution, with no (or, at least, very little) fitting involved. There are several methods for this. 24

25 NOTE 4. ADVANCED TOPICS Selecting Model without Data Multivariate Input Models Time-Series Input Models Covariance and Correlation 25

26 SUMMARY We described the 4 steps in developing input data models: Collecting the raw data Identifying the underlying statistical distribution Estimating the parameters Testing for goodness of fit 26

27 A REAL LIFE EXAMPLE Centers for Disease Control and Prevention: National Health and Nutrition Examination Survey 27

28 R CODES nhanes <- sqlquery(channel = 2, select * from [18 +$]) Hist(nhanes$BMXBMI, scale="frequency", breaks="sturges", col="darkgray") Hist(nhanes$BMXBMI, scale="frequency", breaks=20, col="darkgray") qq.plot(nhanes$bmxbmi, dist= "norm", labels=false) qq.plot(log(nhanes$bmxbmi), dist= "norm", labels=false) numsummary(nhanes[,"bmxbmi"], statistics=c("quantiles"), quantiles=c(0,.25,.5,.75,1)) numsummary(nhanes[,"bmxbmi"], statistics=c("mean", "sd"), quantiles=c(0,.25,.5,.75,1)) ks.test(nhanes[,"bmxbmi"], "pnorm", mean=26.3, sd=6.83) lognhanes<-log(nhanes[,"bmxbmi"]) mean(lognhanes) sd(lognhanes) ks.test(log(nhanes[,"bmxbmi"]), "pnorm", mean=3.24, sd=0.25) shapiro.test(nhanes[,"bmxbmi"]) shapiro.test(log(nhanes[,"bmxbmi"])) 28