Chapter 11 Sections 121 and 122 of Applied Hydrology 04042006 2 Probability A measure of how likely an event will occur A number expressing the ratio of favorable outcome to the all possible outcomes ID: 1022405
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1. Hydrologic StatisticsReading: Chapter 11, Sections 12-1 and 12-2 of Applied Hydrology04/04/2006
2. 2ProbabilityA measure of how likely an event will occurA number expressing the ratio of favorable outcome to the all possible outcomes Probability is usually represented as P(.)P (getting a club from a deck of playing cards) = 13/52 = 0.25 = 25 %P (getting a 3 after rolling a dice) = 1/6
3. 3Random VariableRandom variable: a quantity used to represent probabilistic uncertaintyIncremental precipitation Instantaneous streamflowWind velocityRandom variable (X) is described by a probability distributionProbability distribution is a set of probabilities associated with the values in a random variable’s sample space
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5. 5Sampling terminologySample: a finite set of observations x1, x2,….., xn of the random variableA sample comes from a hypothetical infinite population possessing constant statistical propertiesSample space: set of possible samples that can be drawn from a populationEvent: subset of a sample spaceExamplePopulation: streamflowSample space: instantaneous streamflow, annual maximum streamflow, daily average streamflow Sample: 100 observations of annual max. streamflowEvent: daily average streamflow > 100 cfs
6. 6Hydrologic extremes Extreme eventsFloods DroughtsMagnitude of extreme events is related to their frequency of occurrenceThe objective of frequency analysis is to relate the magnitude of events to their frequency of occurrence through probability distributionIt is assumed the events (data) are independent and come from identical distribution
7. 7Return PeriodRandom variable:Threshold level:Extreme event occurs if: Recurrence interval: Return Period:Average recurrence interval between events equalling or exceeding a thresholdIf p is the probability of occurrence of an extreme event, then or
8. 8More on return periodIf p is probability of success, then (1-p) is the probability of failureFind probability that (X ≥ xT) at least once in N years.
9. 9Hydrologic data seriesComplete duration seriesAll the data availablePartial duration seriesMagnitude greater than base valueAnnual exceedance seriesPartial duration series with # of values = # yearsExtreme value seriesIncludes largest or smallest values in equal intervalsAnnual series: interval = 1 yearAnnual maximum series: largest valuesAnnual minimum series : smallest values
10. 10Return period exampleDataset – annual maximum discharge for 106 years on Colorado River near AustinxT = 200,000 cfsNo. of occurrences = 32 recurrence intervals in 106 yearsT = 106/2 = 53 yearsIf xT = 100, 000 cfs 7 recurrence intervalsT = 106/7 = 15.2 yrsP( X ≥ 100,000 cfs at least once in the next 5 years) = 1- (1-1/15.2)5 = 0.29
11. 11Summary statisticsAlso called descriptive statisticsIf x1, x2, …xn is a sample thenMean, Variance, Standard deviation, Coeff. of variation, m for continuous data s2 for continuous data s for continuous data Also included in summary statistics are median, skewness, correlation coefficient,
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13. 13Time series plotPlot of variable versus time (bar/line/points)Example. Annual maximum flow series Colorado River near Austin
14. 14HistogramPlots of bars whose height is the number ni, or fraction (ni/N), of data falling into one of several intervals of equal widthInterval = 50,000 cfsInterval = 25,000 cfsInterval = 10,000 cfsDividing the number of occurrences with the total number of points will give Probability Mass Function
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16. 16Probability density functionContinuous form of probability mass function is probability density functionpdf is the first derivative of a cumulative distribution function
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18. 18Cumulative distribution functionCumulate the pdf to produce a cdfCdf describes the probability that a random variable is less than or equal to specified value of xP (Q ≤ 50000) = 0.8P (Q ≤ 25000) = 0.4
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22. 22Probability distributions Normal familyNormal, lognormal, lognormal-IIIGeneralized extreme value familyEV1 (Gumbel), GEV, and EVIII (Weibull) Exponential/Pearson type familyExponential, Pearson type III, Log-Pearson type III
23. 23Normal distributionCentral limit theorem – if X is the sum of n independent and identically distributed random variables with finite variance, then with increasing n the distribution of X becomes normal regardless of the distribution of random variablespdf for normal distributionm is the mean and s is the standard deviationHydrologic variables such as annual precipitation, annual average streamflow, or annual average pollutant loadings follow normal distribution
24. 24Standard Normal distributionA standard normal distribution is a normal distribution with mean (m) = 0 and standard deviation (s) = 1Normal distribution is transformed to standard normal distribution by using the following formula:z is called the standard normal variable
25. 25Lognormal distributionIf the pdf of X is skewed, it’s not normally distributedIf the pdf of Y = log (X) is normally distributed, then X is said to be lognormally distributed.Hydraulic conductivity, distribution of raindrop sizes in storm follow lognormal distribution.
26. 26Extreme value (EV) distributionsExtreme values – maximum or minimum values of sets of dataAnnual maximum discharge, annual minimum dischargeWhen the number of selected extreme values is large, the distribution converges to one of the three forms of EV distributions called Type I, II and III
27. 27EV type I distributionIf M1, M2…, Mn be a set of daily rainfall or streamflow, and let X = max(Mi) be the maximum for the year. If Mi are independent and identically distributed, then for large n, X has an extreme value type I or Gumbel distribution.Distribution of annual maximum streamflow follows an EV1 distribution
28. 28EV type III distributionIf Wi are the minimum streamflows in different days of the year, let X = min(Wi) be the smallest. X can be described by the EV type III or Weibull distribution.Distribution of low flows (eg. 7-day min flow) follows EV3 distribution.
29. 29Exponential distributionPoisson process – a stochastic process in which the number of events occurring in two disjoint subintervals are independent random variables. In hydrology, the interarrival time (time between stochastic hydrologic events) is described by exponential distribution Interarrival times of polluted runoffs, rainfall intensities, etc are described by exponential distribution.
30. 30Gamma DistributionThe time taken for a number of events (b) in a Poisson process is described by the gamma distributionGamma distribution – a distribution of sum of b independent and identical exponentially distributed random variables. Skewed distributions (eg. hydraulic conductivity) can be represented using gamma without log transformation.
31. 31Pearson Type III Named after the statistician Pearson, it is also called three-parameter gamma distribution. A lower bound is introduced through the third parameter (e) It is also a skewed distribution first applied in hydrology for describing the pdf of annual maximum flows.
32. 32Log-Pearson Type IIIIf log X follows a Person Type III distribution, then X is said to have a log-Pearson Type III distribution