7. Concepts in Probability, Statistics and Stochastic Modelling


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1 7. Cocepts i Probability, Statistics ad Stochastic Modellig 1. Itroductio 169. Probability Cocepts ad Methods Radom Variables ad Distributios Expectatio Quatiles, Momets ad Their Estimators LMomets ad Their Estimators Distributios of Radom Evets Parameter Estimatio Model Adequacy Normal ad Logormal Distributios Gamma Distributios LogPearso Type 3 Distributio Gumbel ad GEV Distributios LMomet Diagrams Aalysis of Cesored Data Regioalizatio ad IdexFlood Method Partial Duratio Series Stochastic Processes ad Time Series Describig Stochastic Processes Markov Processes ad Markov Chais Properties of TimeSeries Statistics Sythetic Streamflow Geeratio Itroductio Streamflow Geeratio Models A Simple Autoregressive Model Reproducig the Margial Distributio Multivariate Models MultiSeaso, MultiSite Models Disaggregatio Models Aggregatio Models Stochastic Simulatio Geeratig Radom Variables River Basi Simulatio The Simulatio Model Simulatio of the Basi Iterpretig Simulatio Output Coclusios Refereces 3
2 169 7 Cocepts i Probability, Statistics ad Stochastic Modellig Evets that caot be predicted precisely are ofte called radom. May if ot most of the iputs to, ad processes that occur i, water resources systems are to some extet radom. Hece, so too are the outputs or predicted impacts, ad eve people s reactios to those outputs or impacts. To igore this radomess or ucertaity is to igore reality. This chapter itroduces some of the commoly used tools for dealig with ucertaity i water resources plaig ad maagemet. Subsequet chapters illustrate how these tools are used i various types of optimizatio, simulatio ad statistical models for impact predictio ad evaluatio. 1. Itroductio Ucertaity is always preset whe plaig, developig, maagig ad operatig water resources systems. It arises because may factors that affect the performace of water resources systems are ot ad caot be kow with certaity whe a system is plaed, desiged, built, maaged ad operated. The success ad performace of each compoet of a system ofte depeds o future meteorological, demographic, ecoomic, social, techical, ad political coditios, all of which may ifluece future beefits, costs, evirometal impacts, ad social acceptability. Ucertaity also arises due to the stochastic ature of meteorological processes such as evaporatio, raifall ad temperature. Similarly, future populatios of tows ad cities, per capita waterusage rates, irrigatio patters ad priorities for water uses, all of which affect water demad, are ever kow with certaity. There are may ways to deal with ucertaity. Oe, ad perhaps the simplest, approach is to replace each ucertai quatity either by its average (i.e., its mea or expected value), its media, or by some critical (e.g., worstcase ) value, ad the proceed with a determiistic approach. Use of expected or media values of ucertai quatities may be adequate if the ucertaity or variatio i a quatity is reasoably small ad does ot critically affect the performace of the system. If expected or media values of ucertai parameters or variables are used i a determiistic model, the plaer ca the assess the importace of ucertaity by meas of sesitivity aalysis, as is discussed later i this ad the two subsequet chapters. Replacemet of ucertai quatities by either expected, media or worstcase values ca grossly affect the evaluatio of project performace whe importat parameters are highly variable. To illustrate these issues, cosider the evaluatio of the recreatio potetial of a reservoir. Table 7.1 shows that the elevatio of the water surface varies over time depedig o the iflow ad demad for water. The table idicates the pool levels ad their associated probabilities as well as the expected use of the recreatio facility with differet pool levels. The average pool level L is simply the sum of each possible pool level times its probability, or L 10(0.10) 0(0.5) 30(0.30) 40(0.5) 50(0.10) 30 (7.1) This pool level correspods to 100 visitordays per day: VD(L ) 100 visitordays per day (7.) A worstcase aalysis might select a pool level of te as a critical value, yieldig a estimate of system performace equal to 100 visitordays per day: VD(L low ) VD(10) 5 visitordays per day (7.3)
3 170 Water Resources Systems Plaig ad Maagemet possible pool levels probability of each level recreatio potetial i visitordays per day for reservoir with differet pool levels Table 7.1. Data for determiig reservoir recreatio potetial. Neither of these values is a good approximatio of the average visitatio rate, that is VD 0.10 VD(10) 0.5 VD(0) 0.30 VD(30) 0.5 VD(40) 0.10 VD(50) 0.10(5) 0.5(75) 0.30(100) 0.5(80) 0.10(70) (7.4) 78.5 visitordays per day Clearly, the average visitatio rate, VD 78.5, the visitatio rate correspodig to the average pool level VD(L ) 100, ad the worstcase assessmet VD(L low ) 5, are very differet. Usig oly average values i a complex model ca produce a poor represetatio of both the average performace ad the possible performace rage. Whe importat quatities are ucertai, a comprehesive aalysis requires a evaluatio of both the expected performace of a project ad the risk ad possible magitude of project failures i a physical, ecoomic, ecological ad/or social sese. This chapter reviews may of the methods of probability ad statistics that are useful i water resources plaig ad maagemet. Sectio is a codesed summary of the importat cocepts ad methods of probability ad statistics. These cocepts are applied i this ad subsequet chapters of this book. Sectio 3 presets several probability distributios that are ofte used to model or describe the distributio of ucertai quatities. The sectio also discusses methods for fittig these distributios usig historical iformatio, ad methods of assessig whether the distributios are E01101a adequate represetatios of the data. Sectios 4, 5 ad 6 expad upo the use of these mathematical models, ad discuss alterative parameter estimatio methods. Sectio 7 presets the basic ideas ad cocepts of the stochastic processes or time series. These are used to model streamflows, raifall, temperature or other pheomea whose values chage with time. The sectio cotais a descriptio of Markov chais, a special type of stochastic process used i may stochastic optimizatio ad simulatio models. Sectio 8 illustrates how sythetic flows ad other timeseries iputs ca be geerated for stochastic simulatios. Stochastic simulatio is itroduced with a example i Sectio 9. May topics receive oly brief treatmet i this itroductory chapter. Additioal iformatio ca be foud i applied statistical texts or book chapters such as Bejami ad Corell (1970), Haa (1977), Kite (1988), Stediger et al. (1993), Kottegoda ad Rosso (1997), ad Ayyub ad McCue (00).. Probability Cocepts ad Methods This sectio itroduces the basic cocepts ad defiitios used i aalyses ivolvig probability ad statistics. These cocepts are used throughout this chapter ad later chapters i the book..1. Radom Variables ad Distributios The basic cocept i probability theory is that of the radom variable. By defiitio, the value of a radom variable caot be predicted with certaity. It depeds, at least i part, o the outcome of a chace evet. Examples are: (1) the umber of years util the flood stage of a river washes away a small bridge; () the umber of times durig a reservoir s life that the level of the pool will drop below a specified level; (3) the raifall depth ext moth; ad (4) ext year s maximum flow at a gauge site o a uregulated stream. The values of all of these radom evets or variables are ot kowable before the evet has occurred. Probability ca be used to describe the likelihood that these radom variables will equal specific values or be withi a give rage of specific values.
4 Cocepts i Probability, Statistics ad Stochastic Modellig 171 The first two examples illustrate discrete radom variables, radom variables that take o values that are discrete (such as positive itegers). The secod two examples illustrate cotiuous radom variables. Cotiuous radom variables take o ay values withi a specified rage of values. A property of all cotiuous radom variables is that the probability that the value of ay of those radom variables will equal some specific umber ay specific umber is always zero. For example, the probability that the total raifall depth i a moth will be exactly 5.0 cm is zero, while the probability that the total raifall will lie betwee 4 ad 6 cm could be ozero. Some radom variables are combiatios of cotiuous ad discrete radom variables. Let X deote a radom variable ad x a possible value of that radom variable X. Radom variables are geerally deoted by capital letters, ad particular values they take o by lowercase letters. For ay realvalued radom variable X, its cumulative distributio fuctio F X (x), ofte deoted as just the cdf, equals the probability that the value of X is less tha or equal to a specific value or threshold x: F X (x) Pr[X x] (7.5) This cumulative distributio fuctio F X (x) is a odecreasig fuctio of x because Pr[X x] Pr[X x δ] for δ 0 (7.6) I additio, lim F ( x) 1 X x ad lim F ( x) 0 X x (7.7) (7.8) The first limit equals 1 because the probability that X takes o some value less tha ifiity must be uity; the secod limit is zero because the probability that X takes o o value must be zero. The left half of Figure 7.1 illustrates the cumulative distributio fuctio (upper) ad its derivative, the probability desity fuctio, f X (x), (lower) of a cotiuous radom variable X. If X is a realvalued discrete radom variable that takes o specific values x 1, x,, the the probability mass fuctio p X (x i ) is the probability X takes o the value x i. p X (x i ) Pr[X x i ] (7.9) The value of the cumulative distributio fuctio F X (x) for a discrete radom variable is the sum of the probabilities of all x i that are less tha or equal to x. FX( x) px( xi) (7.10) The right half of Figure 7.1 illustrates the cumulative distributio fuctio (upper) ad the probability mass fuctio p X (x i ) (lower) of a discrete radom variable. The probability desity fuctio f X (x) (lower left plot i Figure 7.1) for a cotiuous radom variable X is the aalogue of the probability mass fuctio (lower right plot i Figure 7.1) of a discrete radom variable X. The probability desity fuctio, ofte called the pdf, is the derivative of the cumulative distributio fuctio so that: dfx ( x) fx ( x) 0 (7.11) dx It is ecessary to have (7.1) Equatio 7.1 idicates that the area uder the probability desity fuctio is 1. If a ad b are ay two costats, the cumulative distributio fuctio or the desity fuctio may be used to determie the probability that X is greater tha a ad less tha or equal to b where Pr[ a X b] F ( b) F ( a) f ( x) dx (7.13) The area uder a probability desity fuctio specifies the relative frequecy with which the value of a cotiuous radom variable falls withi ay specified rage of values, that is, ay iterval alog the horizotal axis. Life is seldomly so simple that oly a sigle quatity is ucertai. Thus, the joit probability distributio of two or more radom variables ca also be defied. If X ad Y are two cotiuous realvalued radom variables, their joit cumulative distributio fuctio is: F ( x, y) Pr[ X x ad Y y] XY x x fx ( x) 1 i x If two radom variables are discrete, the y X X X a f (, u v) dudv XY F ( x, y) p ( x, y ) XY XY i i xi x yi y b (7.14) (7.15)
5 17 Water Resources Systems Plaig ad Maagemet 1.0 a 1.0 b E0057a Figure 7.1. Cumulative distributio ad probability desity or mass fuctios of radom variables: (a) cotiuous distributios; (b) discrete distributios. F X (x) possible values of a radom variable X x F X (x) possible values of a radom variable X x F X (x) F X (x) possible values of a radom variable X x possible values of a radom variable X x where the joit probability mass fuctio is: p XY (x i, y i ) Pr[X x i ad Y y i ] (7.16) If X ad Y are two radom variables, ad the distributio of X is ot iflueced by the value take by Y, ad vice versa, the the two radom variables are said to be idepedet. For two idepedet radom variables X ad Y, the joit probability that the radom variable X will be betwee values a ad b ad that the radom variable Y will be betwee values c ad d is simply the product of those separate probabilities. Pr[a X b ad c Y d] Pr[a X b] Pr[c Y d] (7.17) This applies for ay values a, b, c, ad d. As a result, F XY (x, y) F X (x)f Y (y) (7.18) which implies for cotiuous radom variables that f XY (x, y) f X (x)f Y (y) (7.19) ad for discrete radom variables that p XY (x, y) p X (x)p Y (y) (7.0) Other useful cocepts are those of the margial ad coditioal distributios. If X ad Y are two radom variables whose joit cumulative distributio fuctio F XY (x, y) has bee specified, the F X (x), the margial cumulative distributio of X, is just the cumulative distributio of X igorig Y. The margial cumulative distributio fuctio of X equals F X (x) Pr[X x] lim F ( x, y) (7.1) where the limit is equivalet to lettig Y take o ay value. If X ad Y are cotiuous radom variables, the margial desity of X ca be computed from f ( x) f ( x, y) dy X XY y XY (7.) The coditioal cumulative distributio fuctio is the cumulative distributio fuctio for X give that Y has take a particular value y. Thus the value of Y may have bee observed ad oe is iterested i the resultig coditioal distributio for the so far uobserved value of X. The coditioal cumulative distributio fuctio for cotiuous radom variables is give by
6 Cocepts i Probability, Statistics ad Stochastic Modellig 173 fxy (, s y) ds FX Y( x y) Pr[ X x Y y] (7.3) fy ( y) where the coditioal desity fuctio is fxy ( x, y) fx Y( x y) (7.4) fy ( y) For discrete radom variables, the probability of observig X x, give that Y y equals pxy ( x, y) px Y( x y) (7.5) py ( y) These results ca be exteded to more tha two radom variables. Kottegoda ad Rosso (1997) provide more detail... Expectatio Kowledge of the probability desity fuctio of a cotiuous radom variable, or of the probability mass fuctio of a discrete radom variable, allows oe to calculate the expected value of ay fuctio of the radom variable. Such a expectatio may represet the average raifall depth, average temperature, average demad shortfall or expected ecoomic beefits from system operatio. If g is a realvalued fuctio of a cotiuous radom variable X, the expected value of g(x) is: E[ gx ( )] gxf ( ) ( x) dx whereas for a discrete radom variable E[ gx ( )] gx ( ) p ( x) (7.6) (7.7) The expectatio operator,e[ ], has several importat properties. I particular, the expectatio of a liear fuctio of X is a liear fuctio of the expectatio of X. Thus, if a ad b are two oradom costats, E[a bx] a be[x] (7.8) The expectatio of a fuctio of two radom variables is give by E[( gxy, )] gxyf (, ) (, xy) dxdy or i XY E[( gxy, )] gx (, y) p ( x, y ) i i X i j X i i x XY i i (7.9) If X ad Y are idepedet, the expectatio of the product of a fuctio g( ) of X ad a fuctio h( ) of Y is the product of the expectatios: E[g(X) h(y)] E[g(X)] E[h(Y)] (7.30) This follows from substitutio of Equatios 7.19 ad 7.0 ito Equatio Quatiles, Momets ad Their Estimators While the cumulative distributio fuctio provides a complete specificatio of the properties of a radom variable, it is useful to use simpler ad more easily uderstood measures of the cetral tedecy ad rage of values that a radom variable may assume. Perhaps the simplest approach to describig the distributio of a radom variable is to report the value of several quatiles. The pth quatile of a radom variable X is the smallest value x p such that X has a probability p of assumig a value equal to or less tha x p : Pr[X x p ] p Pr[X x p ] (7.31) Equatio 7.31 is writte to isist if at some value x p, the cumulative probability fuctio jumps from less tha p to more tha p, the that value x p will be defied as the pth quatile eve though F X (x p ) p. If X is a cotiuous radom variable, the i the regio where f X (x) 0, the quatiles are uiquely defied ad are obtaied by solutio of F X (x p ) p (7.3) Frequetly reported quatiles are the media x 0.50 ad the lower ad upper quartiles x 0.5 ad x The media describes the locatio or cetral tedecy of the distributio of X because the radom variable is, i the cotiuous case, equally likely to be above as below that value. The iterquartile rage [x 0.5, x 0.75 ] provides a easily uderstood descriptio of the rage of values that the radom variable might assume. The pth quatile is also the 100 p percetile. I a give applicatio particularly whe safety is of cocer it may be appropriate to use other quatiles. I floodplai maagemet ad the desig of flood cotrol structures, the 100year flood x 0.99 is a commoly selected desig value. I water quality maagemet, a river s miimum sevedayaverage low flow expected oce i te years is commoly used i the Uited States as the
7 174 Water Resources Systems Plaig ad Maagemet critical plaig value: Here the oeite year value is the 10 th percetile of the distributio of the aual miima of the seveday average flows. The atural sample estimate of the media x 0.50 is the media of the sample. I a sample of size where x (1) x () x () are the observatios ordered by magitude, ad for a oegative iteger k such that k (eve) or k 1 (odd), the sample estimate of the media is x for k 1 ( k 1 ) xˆ x( k) x( k 1 ) for k (7.33) Sample estimates of other quatiles may be obtaied by usig x (i) as a estimate of x q for q i/( 1) ad the iterpolatig betwee observatios to obtai xˆp for the desired p. This oly works for 1/( 1) p /( 1) ad ca yield rather poor estimates of x p whe ( 1)p is ear either 1 or. A alterative approach is to fit a reasoable distributio fuctio to the observatios, as discussed i Sectio 3, ad the estimate x p usig Equatio 7.3, where F X (x) is the fitted distributio. Aother simple ad commo approach to describig a distributio s cetre, spread ad shape is by reportig the momets of a distributio. The first momet about the origi is the mea of X ad is give by µ X E X [ X] xf ( x) dx (7.34) Momets other tha the first are ormally measured about the mea. The secod momet measured about the mea is the variace, deoted Var(X) or σ X, where: σ X Var( X) E[( X µ X) ] (7.35) The stadard deviatio σ X is the square root of the variace. While the mea µ X is a measure of the cetral value of X, the stadard deviatio σ X is a measure of the spread of the distributio of X about µ X. Aother measure of the variability i X is the coefficiet of variatio, X CV X σ (7.36) µ X The coefficiet of variatio expresses the stadard deviatio as a proportio of the mea. It is useful for comparig the relative variability of the flow i rivers of differet sizes, or of raifall variability i differet regios whe the radom variable is strictly positive. The third momet about the mea, deoted λ X, measures the asymmetry, or skewess, of the distributio: λ X E[(X µ X ) 3 ] (7.37) Typically, the dimesioless coefficiet of skewess γ X is reported rather tha the third momet λ X. The coefficiet of skewess is the third momet rescaled by the cube of the stadard deviatio so as to be dimesioless ad hece uaffected by the scale of the radom variable: γ (7.38) Streamflows ad other atural pheomea that are ecessarily oegative ofte have distributios with positive skew coefficiets, reflectig the asymmetric shape of their distributios. Whe the distributio of a radom variable is ot kow, but a set of observatios {x 1,,x } is available, the momets of the ukow distributio of X ca be estimated based o the sample values usig the followig equatios. The sample estimate of the mea: X The sample estimate of the variace: σˆx λ σ X X 3 X i 1 X / 1 SX Xi X ( ) ( 1) The sample estimate of skewess: λˆ ( X X i X) 3 ( 1)( ) (7.39a) (7.39b) (7.39c) The sample estimate of the coefficiet of variatio: CV ˆ X SX/ X (7.39d) The sample estimate of the coefficiet of skewess: γˆx λˆx /S X 3 i i 1 i 1 (7.39e) The sample estimate of the mea ad variace are ofte deoted as x _ ad s x where the lower case letters are used whe referrig to a specific sample. All of these
8 Cocepts i Probability, Statistics ad Stochastic Modellig 175 sample estimators provide oly estimates of actual or true values. Uless the sample size is very large, the differece betwee the estimators ad the true values of µ X, σ X, λx, CVX, ad γx may be large. I may ways, the field of statistics is about the precisio of estimators of differet quatities. Oe wats to kow how well the mea of twety aual raifall depths describes the true expected aual raifall depth, or how large the differece betwee the estimated 100year flood ad the true 100year flood is likely to be. As a example of the calculatio of momets, cosider the flood data i Table 7.. These data have the followig sample momets: _ x s X CV X 0.55 γˆx 0.71 As oe ca see, the data are positively skewed ad have a relatively large coefficiet of variace. Whe discussig the accuracy of sample estimates, two quatities are ofte cosidered, bias ad variace. A estimator θˆ of a kow or ukow quatity θ is a fuctio of the observed values of the radom variable X, say i differet time periods, X 1,,X, that will be available to estimate the value of θ; θˆ may be writte θˆ [X 1, X,, X ] to emphasize that θˆ itself is a radom variable. Its value depeds o the sample values of the radom variable that will be observed. A estimator θˆ of a quatity θ is biased if E[θˆ] θ ad ubiased if E[θˆ] θ. The quatity {E[θˆ] θ} is geerally called the bias of the estimator. A ubiased estimator has the property that its expected value equals the value of the quatity to be estimated. The sample mea is a ubiased estimate of the populatio mea µ X because 1 1 E[ X] E Xi E[ Xi] µ X (7.40) i 1 i 1 The estimator S X of the variace of X is a ubiased estimator of the true variace σ X for idepedet observatios (Bejami ad Corell, 1970): E S X σ X (7.41) However, the correspodig estimator of the stadard deviatio, S X, is i geeral a biased estimator of σ X because E[ S X ] date σ X (7.4) The secod importat statistic ofte used to assess the accuracy of a estimator θˆ is the variace of the estimator Var θˆ, which equals E{(θˆ E[θˆ]) }. For the mea of a set of idepedet observatios, the variace of the sample mea is: X Var(X) σ discharge m 3/s * Value for 1945 is missig. date discharge m 3/s Table 7.. Aual maximum discharges o Magra River, Italy, at Calamazza, *. (7.43) It is commo to call σ x / the stadard error of xˆ rather tha its stadard deviatio. The stadard error of a average is the most commoly reported measure of its precisio. The bias measures the differece betwee the average value of a estimator ad the quatity to be estimated. E01101b
9 176 Water Resources Systems Plaig ad Maagemet The variace measures the spread or width of the estimator s distributio. Both cotribute to the amout by which a estimator deviates from the quatity to be estimated. These two errors are ofte combied ito the mea square error. Uderstadig that θ is fixed ad the estimator θˆ is a radom variable, the mea squared error is the expected value of the squared distace (error) betwee θ ad its estimator θˆ: MSE(θˆ) E[(θˆ θ) ] E{[θˆ E(θˆ)] [E(θˆ) θ]} [Bias] Var(θˆ) (7.44) where [Bias] is E(θˆ) θ. Equatio 7.44 shows that the MSE, equal to the expected average squared deviatio of the estimator θˆ from the true value of the parameter θ, ca be computed as the bias squared plus the variace of the estimator. MSE is a coveiet measure of how closely θˆ approximates θ because it combies both bias ad variace i a logical way. Estimatio of the coefficiet of skewess γ X provides a good example of the use of the MSE for evaluatig the total deviatio of a estimate from the true populatio value. The sample estimate γˆx of γ X is ofte biased, has a large variace, ad its absolute value was show by Kirby (1974) to be bouded by the square root of the sample size : γˆx (7.45) The bouds do ot deped o the true skew, γ X. However, the bias ad variace of γˆx do deped o the sample size ad the actual distributio of X. Table 7.3 cotais the expected value ad stadard deviatio of the estimated coefficiet of skewess γˆx whe X has either a ormal distributio, for which γ X 0, or a gamma distributio with γ X 0.5, 0.50, 1.00,.00, or These values are adapted from Wallis et al. (1974 a,b) who employed momet estimators slightly differet tha those i Equatio For the ormal distributio, E[γˆ] 0 ad Var [γˆx] 5/. I this case, the skewess estimator is ubiased but highly variable. I all the other cases i Table 7.3, the skewess estimator is biased. To illustrate the magitude of these errors, cosider the mea square error of the skew estimator γˆx calculated from a sample of size 50 whe X has a gamma distributio with γ X 0.50, a reasoable value for aual streamflows. The expected value of γˆx is 0.45; its variace equals (0.37), its stadard deviatio is squared. Usig Equatio 7.44, the mea square error of γˆx is: MSE(γˆX) ( ) ( 037. ) (7.46) A ubiased estimate of γ X is simply (0.50/0.45) γˆx. Here the estimator provided by Equatio 7.39e has bee scaled to elimiate bias. This ubiased estimator has a mea squared error of: MSE 050. ˆ 048. γ X 050. ( ) ( 037. ) (7.47) The mea square error of the ubiased estimator of γˆx is larger tha the mea square error of the biased estimate. Ubiasig γˆx results i a larger mea square error for all the cases listed i Table 7.3 except for the ormal distributio for which γ X 0, ad the gamma distributio with γ X As show here for the skew coefficiet, biased estimators ofte have smaller mea square errors tha ubiased estimators. Because the mea square error measures the total average deviatio of a estimator from the quatity beig estimated, this result demostrates that the strict or uquestioig use of ubiased estimators is ot advisable. Additioal iformatio o the samplig distributio of quatiles ad momets is cotaied i Stediger et al. (1993)..4. LMomets ad Their Estimators Lmomets are aother way to summarize the statistical properties of hydrological data based o liear combiatios of the origial observatios (Hoskig, 1990). Recetly, hydrologists have foud that regioalizatio methods (to be discussed i Sectio 5) usig Lmomets are superior to methods usig traditioal momets (Hoskig ad Wallis, 1997; Stediger ad Lu, 1995). Lmomets have also proved useful for costructio of goodessoffit tests (Hoskig et al., 1985; Chowdhury et al., 1991; Fill ad Stediger, 1995), measures of regioal homogeeity ad distributio selectio methods (Vogel ad Feessey, 1993; Hoskig ad Wallis, 1997).
10 Cocepts i Probability, Statistics ad Stochastic Modellig 177 distributio of X expected value of γ X sample size E01101c Table 7.3. Samplig properties of coefficiet of skewess estimator. Source: Wallis et al. (1974b) who oly divided by i the estimators of the momets, whereas i Equatios 7.39b ad 7.39c, we use the geerallyadopted coefficiets of 1/( 1) ad /( 1)( ) for the variace ad skew. ormal gamma γ X γ X γ X γ X γ X γ X = = = = = = upper boud o skew ^ stadard deviatio of γ X distributio of X sample size ormal gamma γ X γ X γ X γ X γ X γ X = = = = = = The first Lmomet desigated as λ 1 is simply the arithmetic mea: λ 1 E[X] (7.48) Now let X (i ) be the i th largest observatio i a sample of size (i correspods to the largest). The, for ay distributio, the secod Lmomet, λ, is a descriptio of scale based upo the expected differece betwee two radomly selected observatios: λ (1/) E[X ( 1) X (1 ) ] (7.49) Similarly, Lmomet measures of skewess ad kurtosis use three ad four radomly selected observatios, respectively. λ 3 (1/3) E[X (3 3) X ( 3) X (1 3) ] (7.50) λ 4 (1/4) E[X (4 4) 3X (3 4) 3X ( 4) X (1 4) ] (7.51) Sample Lmomet estimates are ofte computed usig itermediate statistics called probability weighted momets (PWMs). The r th probability weighted momet is defied as: β r E{X[F(X)] r } (7.5) where F(X) is the cumulative distributio fuctio of X. Recommeded (Ladwehr et al., 1979; Hoskig ad Wallis, 1995) ubiased PWM estimators, b r, of β r are computed as: b0 X 1 b1 ( j 1) X ( j ) ( 1) j 1 b ( j 1)( j ) X ( j ) ( 1)( ) j 3 (7.53)
11 178 Water Resources Systems Plaig ad Maagemet These are examples of the geeral formula for computig estimators b r of β r. 1 j 1 1 br X j r r 1 j 1 r 1 r X( j) r 1 (7.54) for r 1,, 1. Lmomets are easily calculated i terms of PWMs usig: λ 1 β 0 j r 1 j r 1 λ β 1 β 0 ( ) λ 3 6β 6β 1 β 0 λ 4 0β 3 30β 1β 1 β 0 (7.55) Wag (1997) provides formulas for directly calculatig Lmomet estimators of λ r. Measures of the coefficiet of variatio, skewess ad kurtosis of a distributio ca be computed with Lmomets, as they ca with traditioal product momets. Where skew primarily measures the asymmetry of a distributio, the kurtosis is a additioal measure of the thickess of the extreme tails. Kurtosis is particularly useful for comparig symmetric distributios that have a skewess coefficiet of zero. Table 7.4 provides defiitios of the traditioal coefficiet of variatio, coefficiet of skewess ad coefficiet of kurtosis, as well as the Lmomet, Lcoefficiet of variatio, Lcoefficiet of skewess ad Lcoefficiet of kurtosis. The flood data i Table 7. ca be used to provide a example of Lmomets. Equatio 7.53 yields estimates of the first three Probability Weighted Momets: b 0 1,549.0 b b (7.56) Recall that b 0 is just the sample average x _. The sample Lmomets are easily calculated usig the probability weighted momets. Oe obtais: λˆ1 b 0 1,549 λˆ b 1 b λˆ3 6b 6b 1 b 0 80 (7.55) Thus, the sample estimates of the Lcoefficiet of variatio, t, ad Lcoefficiet of skewess, t 3, are: t 0.95 t (7.58) Table 7.4. Defiitios of dimesioless productmomet ad Lmomet ratios. ame commo symbol defiitio E01101d productmomet ratios coefficiet of variatio skewess kurtosis CVX γx κ X σx / µ X E [ (X µ X ) 3 ] / σ X E [ (X µ X ) 4 ] / σ X 3 4 Lmomet ratios * Lcoefficiet of variatio * LCV, τ skewess Lskewess, τ kurtosis Lkurtosis, τ 3 4 λ / λ / λ λ / λ Hoskig ad Wallis (1997) use τ istead of τ to represet the LCV ratio 3 4 λ 1
12 Cocepts i Probability, Statistics ad Stochastic Modellig Distributios of Radom Evets A frequet task i water resources plaig is the developmet of a model of some probabilistic or stochastic pheomea such as streamflows, flood flows, raifall, temperatures, evaporatio, sedimet or utriet loads, itrate or orgaic compoud cocetratios, or water demads. This ofte requires oe to fit a probability distributio fuctio to a set of observed values of the radom variable. Sometimes, oe s immediate objective is to estimate a particular quatile of the distributio, such as the 100year flood, 50year sixhourraifall depth, or the miimum sevedayaverage expected oceiteyear flow. The the fitted distributio ca supply a estimate of that quatity. I a stochastic simulatio, fitted distributios are used to geerate possible values of the radom variable i questio. Rather tha fittig a reasoable ad smooth mathematical distributio, oe could use the empirical distributio represeted by the data to describe the possible values that a radom variable may assume i the future ad their frequecy. I practice, the true mathematical form for the distributio that describes the evets is ot kow. Moreover, eve if it was, its fuctioal form may have too may parameters to be of much practical use. Thus, usig the empirical distributio represeted by the data itself has substatial appeal. Geerally, the free parameters of the theoretical distributio are selected (estimated) so as to make the fitted distributio cosistet with the available data. The goal is to select a physically reasoable ad simple distributio to describe the frequecy of the evets of iterest, to estimate that distributio s parameters, ad ultimately to obtai quatiles, performace idices ad risk estimates of satisfactory accuracy for the problem at had. Use of a theoretical distributio has several advatages over use of the empirical distributio: It presets a smooth iterpretatio of the empirical distributio. As a result quatiles, performace idices ad other statistics computed usig the fitted distributio should be more accurate tha those computed with the empirical distributio. It provides a compact ad easytouse represetatio of the data. It is likely to provide a more realistic descriptio of the rage of values that the radom variable may assume ad their likelihood. For example, by usig the empirical distributio, oe implicitly assumes that o values larger or smaller tha the sample maximum or miimum ca occur. For may situatios, this is ureasoable. Ofte oe eeds to estimate the likelihood of extreme evets that lie outside the rage of the sample (either i terms of x values or i terms of frequecy). Such extrapolatio makes little sese with the empirical distributio. I may cases, oe is ot iterested i the values of a radom variable X, but istead i derived values of variables Y that are fuctios of X. This could be a performace fuctio for some system. If Y is the performace fuctio, iterest might be primarily i its mea value E[Y], or the probability some stadard is exceeded, Pr{Y stadard}. For some theoretical Xdistributios, the resultig Ydistributio may be available i closed form, thus makig the aalysis rather simple. (The ormal distributio works with liear models, the logormal distributio with product models, ad the gamma distributio with queuig systems.) This sectio provides a brief itroductio to some useful techiques for estimatig the parameters of probability distributio fuctios ad for determiig if a fitted distributio provides a reasoable or acceptable model of the data. Subsectios are also icluded o families of distributios based o the ormal, gamma ad geeralizedextremevalue distributios. These three families have foud frequet use i water resources plaig (Kottegoda ad Rosso, 1997) Parameter Estimatio Give a set of observatios to which a distributio is to be fit, oe first selects a distributio fuctio to serve as a model of the distributio of the data. The choice of a distributio may be based o experiece with data of that type, some uderstadig of the mechaisms givig rise to the data, ad/or examiatio of the observatios themselves. Oe ca the estimate the parameters of the chose distributio ad determie if the fitted distributio provides a acceptable model of the data. A model is geerally judged to be uacceptable if it is ulikely that
13 180 Water Resources Systems Plaig ad Maagemet oe could have observed the available data were they actually draw from the fitted distributio. I may cases, good estimates of a distributio s parameters are obtaied by the maximumlikelihoodestimatio procedure. Give a set of idepedet observatios {x 1,, x } of a cotiuous radom variable X, the joit probability desity fuctio for the observatios is: fx x x 1, X, X3,, X( 1,, θ ) = f ( x θ) f ( x θ) f ( x θ) X 1 X X (7.59) where θ is the vector of the distributio s parameters. The maximum likelihood estimator of θ is that vector θ which maximizes Equatio 7.59 ad thereby makes it as likely as possible to have observed the values {x 1,, x }. Cosiderable work has goe ito studyig the properties of maximum likelihood parameter estimates. Uder rather geeral coditios, asymptotically the estimated parameters are ormally distributed, ubiased ad have the smallest possible variace of ay asymptotically ubiased estimator (Bickel ad Doksum, 1977). These, of course, are asymptotic properties, valid for large sample sizes. Better estimatio procedures, perhaps yieldig biased parameter estimates, may exist for small sample sizes. Stediger (1980) provides such a example. Still, maximum likelihood procedures are recommeded with moderate ad large samples, eve though the iterative solutio of oliear equatios is ofte required. A example of the maximum likelihood procedure for which closedform expressios for the parameter estimates are obtaied is provided by the logormal distributio. The probability desity fuctio of a logormally distributed radom variable X is: 1 1 fx( x) exp [l( x) µ ] x πσ σ (7.60) Here, the parameters µ ad σ are the mea ad variace of the logarithm of X, ad ot of X itself. Maximizig the logarithm of the joit desity for {x 1,,x } is more coveiet tha maximizig the joit probability desity itself. Hece, the problem ca be expressed as the maximizatio of the loglikelihood fuctio L l f[( x µσ, )] l( xi π ) l( σ ) (7.61) The maximum ca be obtaied by equatig to zero the partial derivatives L/ µ ad L/ σ whereby oe obtais: L 1 0 [l( xi) µ ] µ σ i 1 L [l( xi) µ ] σ σ σ These equatios yield the estimators 1 µˆ l( x i ) σˆ (7.6) (7.63) The secodorder coditios for a maximum are met ad these values maximize Equatio It is useful to ote that if oe defies a ew radom variable Y l(x), the the maximum likelihood estimators of the parameters µ ad σ, which are the mea ad variace of the Y distributio, are the sample estimators of the mea ad variace of Y: µˆ y _ i 1 i 1 l f( x µσ, ) i i 1 i 1 1 [l( x i ) µ ˆ ] i 1 i i 1 1 l( ) σ [ x i µ ] i 1 σˆ [( 1)/]S Y (7.64) The correctio [( 1)/] i this last equatio is ofte eglected. The secod commoly used parameter estimatio procedure is the method of momets. The method of momets is ofte a quick ad simple method for obtaiig parameter estimates for may distributios. For a distributio with m 1, or 3 parameters, the first m momets of the postulated distributio i Equatios 7.34, 7.35 ad 7.37 are equated to the estimates of those momets calculated usig Equatios The resultig oliear equatios are solved for the ukow parameters.
14 Cocepts i Probability, Statistics ad Stochastic Modellig 181 For the logormal distributio, the mea ad variace of X as a fuctio of the parameters µ ad σ are give by 1 µ X exp µ σ σ exp( µ σ)[exp ( σ) 1] X (7.65) Substitutig x _ for µ X ad s X for σ X ad solvig for µ ad σ oe obtais σˆ l s / x x 1 µˆ l l x σˆ (7.66) 1 s / x The data i Table 7. provide a illustratio of both fittig methods. Oe ca easily compute the sample mea ad variace of the logarithms of the flows to obtai µˆ 7.0 ( 1 ) X X σˆ (0.565) (7.67) Alteratively, the sample mea ad variace of the flows themselves are x _ s X 661,800 (813.5) (7.68) Substitutig those two values i Equatio 7.66 yields µˆ 7.4 σ X (0.4935) (7.69) Method of momets ad maximum likelihood are just two of may possible estimatio methods. Just as method of momets equates sample estimators of momets to populatio values ad solves for a distributio s parameters, oe ca simply equate Lmomet estimators to populatio values ad solve for the parameters of a distributio. The resultig method of Lmomets has received cosiderable attetio i the hydrological literature (Ladwehr et al., 1978; Hoskig et al., 1985; Hoskig ad Wallis, 1987; Hoskig, 1990; Wag, 1997). It has bee show to have sigificat advatages whe used as a basis for regioalizatio procedures that will be discussed i Sectio 5 (Lettemaier et al., 1987; Stediger ad Lu, 1995; Hoskig ad Wallis, 1997). Bayesia procedures provide aother approach that is related to maximum likelihood estimatio. Bayesia iferece employs the likelihood fuctio to represet the iformatio i the data. That iformatio is augmeted with a prior distributio that describes what is kow about costraits o the parameters ad their likely values beyod the iformatio provided by the recorded data available at a site. The likelihood fuctio ad the prior probability desity fuctio are combied to obtai the probability desity fuctio that describes the posterior distributio of the parameters: f θ (θ x 1, x,, x ) f X (x 1, x,, x θ)ξ(θ) (7.70) The symbol meas proportioal to ad ξ(θ) is the probability desity fuctio for the prior distributio for θ (Kottegoda ad Rosso, 1997). Thus, except for a costat of proportioality, the probability desity fuctio describig the posterior distributio of the parameter vector θ is equal to the product of the likelihood fuctio f X (x 1, x,, x θ) ad the probability desity fuctio for the prior distributio ξ(θ) for θ. Advatages of the Bayesia approach are that it allows the explicit modellig of ucertaity i parameters (Stediger, 1997; Kuczera, 1999) ad provides a theoretically cosistet framework for itegratig systematic flow records with regioal ad other hydrological iformatio (Vices et al., 1975; Stediger, 1983; Kuczera, 1983). Martis ad Stediger (000) illustrate how a prior distributio ca be used to eforce realistic costraits upo a parameter as well as providig a descriptio of its likely values. I their case, use of a prior of the shape parameter κ of a geeralized extreme value (GEV) distributio (discussed i Sectio 3.6) allowed defiitio of geeralized maximum likelihood estimators that, over the κrage of iterest, performed substatially better tha maximum likelihood, momet, ad Lmomet estimators. While Bayesia methods have bee available for decades, the computatioal challege posed by the solutio of Equatio 7.70 has bee a obstacle to their use. Solutios to Equatio 7.70 have bee available for special cases such as ormal data, ad biomial ad Poisso samples (Raiffa ad Schlaifer, 1961; Bejami ad Corell, 1970; Zeller, 1971). However, a ew ad very geeral set of Markov Chai Mote Carlo (MCMC) procedures (discussed i Sectio 7.) allows umerical computatio of the posterior distributios of parameters
15 18 Water Resources Systems Plaig ad Maagemet for a very broad class of models (Gilks et al., 1996). As a result, Bayesia methods are ow becomig much more popular ad are the stadard approach for may difficult problems that are ot easily addressed by traditioal methods (Gelma et al., 1995; Carli ad Louis, 000). The use of Mote Carlo Bayesia methods i flood frequecy aalysis, raifall ruoff modellig, ad evaluatio of evirometal pathoge cocetratios are illustrated by Wag (001), Bates ad Campbell (001) ad Craiiceau et al. (00), respectively. Fially, a simple method of fittig flood frequecy curves is to plot the ordered flood values o special probability paper ad the to draw a lie through the data (Gumbel, 1958). Eve today, that simple method is still attractive whe some of the smallest values are zero or uusually small, or have bee cesored as will be discussed i Sectio 4 (Kroll ad Stediger, 1996). Plottig the raked aual maximum series agaist a probability scale is always a excellet ad recommeded way to see what the data look like ad for determiig whether or ot a fitted curve is cosistet with the data (Stediger et al., 1993). Statisticias ad hydrologists have ivestigated which of these methods most accurately estimates the parameters themselves or the quatiles of the distributio. Oe also eeds to determie how accuracy should be measured. Some studies have used average squared deviatios, some have used average absolute weighted deviatios with differet weights o uder ad overestimatio, ad some have used the squared deviatios of the logquatile estimator (Slack et al., 1975; Kroll ad Stediger, 1996). I almost all cases, oe is also iterested i the bias of a estimator, which is the average value of the estimator mius the true value of the parameter or quatile beig estimated. Special estimators have bee developed to compute desig evets that o average are exceeded with the specified probability ad have the aticipated risk of beig exceeded (Beard, 1960, 1997; Rasmusse ad Rosbjerg, 1989, 1991a,b; Stediger, 1997; Rosbjerg ad Madse, 1998). 3.. Model Adequacy After estimatig the parameters of a distributio, some check of model adequacy should be made. Such checks vary from simple comparisos of the observatios with the fitted model (usig graphs or tables) to rigorous statistical tests. Some of the early ad simplest methods of parameter estimatio were graphical techiques. Although quatitative techiques are geerally more accurate ad precise for parameter estimatio, graphical presetatios are ivaluable for comparig the fitted distributio with the observatios for the detectio of systematic or uexplaied deviatios betwee the two. The observed data will plot as a straight lie o probability graph paper if the postulated distributio is the true distributio of the observatio. If probability graph paper does ot exist for the particular distributio of iterest, more geeral techiques ca be used. Let x (i) be the ith largest value i a set of observed values {x i } so that x (1) x () x (). The radom variable X (i) provides a reasoable estimate of the pth quatile x p of the true distributio of X for p i/( 1). I fact, whe oe cosiders the cumulative probability U i associated with the radom variable X (i), U i F X (X (i) ), ad if the observatios X (i) are idepedet, the the U i have a beta distributio (Gumbel, 1958) with probability desity fuctio: fu ( u )! ui 1 ( 1 u) i 0 u 1 i ( i 1)!( 1)! (7.71) This beta distributio has mea i E[ Ui] 1 ad variace i ( i 1) Var( Ui ) ( 1) ( ) (7.7a) (7.7b) A good graphical check of the adequacy of a fitted distributio G(x) is obtaied by plottig the observatios x (i) versus G 1 [i/( 1)] (Wilk ad Gaadesika, 1968). Eve if G(x) equalled to a exact degree the true Xdistributio F X [x], the plotted poits would ot fall exactly o a 45 lie through the origi of the graph. This would oly occur if F X [x (i) ] exactly equalled i/( 1), ad therefore each x (i) exactly equalled F X 1 [i/( 1)]. A appreciatio for how far a idividual observatio x (i) ca be expected to deviate from G 1 [i/( 1)] ca be obtaied by plottig G 1 [u i (0.75) ] ad G 1 [u i (0.5) ], where u i (0.75) ad u i (0.5) are the upper ad lower quartiles of the distributio of U i obtaied from itegratig the probability
16 Cocepts i Probability, Statistics ad Stochastic Modellig 183 desity fuctio i Equatio The required icomplete beta fuctio is also available i may software packages, icludig Microsoft Excel. Stediger et al. (1993) show that u (1) ad (1 u () ) fall betwee 5/ ad 3( 1) with a probability of 90%, thus illustratig the great ucertaity associated with the cumulative probability of the smallest value ad the exceedace probability of the largest value i a sample. Figures 7.a ad 7.b illustrate the use of this quatile quatile plottig techique by displayig the results of fittig a ormal ad a logormal distributio to the aual maximum flows i Table 7. for the Magra River, Italy, at Calamazza for the years The observatios of X (i), give i Table 7., are plotted o the vertical axis agaist the quatiles G 1 [i/( 1)] o the horizotal axis. A probability plot is essetially a scatter plot of the sorted observatios X (i) versus some approximatio of their expected or aticipated value, represeted by G 1 (p i ), where, as suggested, p i i/( 1). The p i values are called plottig positios. A commo alterative to i/( 1) is (i 0.5)/, which results from a probabilistic iterpretatio of the empirical distributio of the data. May reasoable plottig positio formulas have bee proposed based upo the sese i which G 1 (p i ) should approximate X (i). The Weibull formula i/( 1) ad the Haze formula (i 0.5)/ bracket most of the reasoable choices. Popular formulas are summarized by Stediger et al. (1993), who also discuss the geeratio of probability plots for may distributios commoly employed i hydrology. Rigorous statistical tests are available for tryig to determie whether or ot it is reasoable to assume that a give set of observatios could have bee draw from a particular family of distributios. Although ot the most powerful of such tests, the Kolmogorov Smirov test provides bouds withi which every observatio should lie if the sample is actually draw from the assumed distributio. I particular, for G F X, the test specifies that E0057c E0057d observed values X observed values X (i) ad KolmogorovSmirov bouds (m 3 (i) ad KolmogorovSmirov bouds (m 3 /sec) /sec) upper 90% cofidece iterval for all poits lower 90% cofidece iterval for all poits quatiles of fitted ormal distributio G 1 [ i /(+1)] m 3 /sec) upper 90% cofidece iterval for all poits lower 90% cofidece iterval for all poits quatiles of fitted logormal distributio G 1 [i/(+1)] (m 3 /sec) 1 i 1 1 Pr G Cα X() i G Cα i 1 α i (7.73) Figure 7.. Plots of aual maximum discharges of Magra River, Italy, versus quatiles of fitted (a) ormal ad (b) logormal distributios. where C α is the critical value of the test at sigificace level α. Formulas for C α as a fuctio of are cotaied i Table 7.5 for three cases: (1) whe G is completely
17 184 Water Resources Systems Plaig ad Maagemet specified idepedet of the sample s values; () whe G is the ormal distributio ad the mea ad variace are estimated from the sample with x _ ad s X ; ad (3) whe G is the expoetial distributio ad the scale parameter is estimated as 1/(x _ ). Chowdhury et al. (1991) provide critical values for the Gumbel ad geeralized extreme value (GEV) distributios (Sectio 3.6) with kow shape parameter κ. For other distributios, the values obtaied from Table 7.5 may be used to costruct approximate simultaeous cofidece itervals for every X (i). Figures 7.a ad b cotai 90% cofidece itervals for the plotted poits costructed i this maer. For the ormal distributio, the critical value of C α equals /( / ), where correspods to α For 40, oe computes C α As ca be see i Figure 7.a, the aual maximum flows are ot cosistet with the hypothesis that they were draw from a ormal distributio; three of the observatios lie outside the simultaeous 90% cofidece itervals for all the poits. This demostrates a statistically sigificat lack of fit. The fitted ormal distributio uderestimates the quatiles correspodig to small ad large probabilities while overestimatig the quatiles i a itermediate rage. I Figure 7.b, deviatios betwee the fitted logormal distributio ad the observatios ca be attributed to the differeces betwee F X (x (i) ) ad i/( 1). Geerally, the poits are all ear the 45 lie through the origi, ad o major systematic deviatios are apparet. The Kolmogorov Smirov test coveietly provides bouds withi which every observatio o a probability plot should lie if the sample is actually draw from the assumed distributio, ad thus is useful for visually evaluatig the adequacy of a fitted distributio. However, it is ot the most powerful test available for estimatig which distributio a set of observatios is likely to have bee draw from. For that purpose, several other more aalytical tests are available (Fillibe, 1975; Hoskig, 1990; Chowdhury et al., 1991; Kottegoda ad Rosso, 1997). The Probability Plot Correlatio test is a popular ad powerful test of whether a sample has bee draw from a postulated distributio, though it is ofte weaker tha alterative tests at rejectig thitailed alteratives (Fillibe, 1975; Fill ad Stediger, 1995). A test with greater power has a greater probability of correctly determiig that a sample is ot from the postulated distributio. The Probability Plot Correlatio Coefficiet test employs the correlatio r betwee the ordered observatios x (i) ad the correspodig fitted quatiles w i G 1 (p i ), determied by plottig positios p i for each x (i). Values of r ear 1.0 suggest that the observatios could have bee draw from the fitted distributio: r measures the liearity of the probability plot providig a quatitative assessmet of fit. If x _ deotes the average value of the observatios ad w _ deotes the average value of the fitted quatiles, the ( x() i x)( wi w) r (7.74) ( x() i x) ( wi w) 0. 5 ( ) Table 7.5. Critical values of Kolmogorov Smirov statistic as a fuctio of sample size (after Stephes, 1974). sigificace level α E01101e F x completely specified: C α ( / ) F x ormal with mea ad variace estimated as x ad s x Cα ( / ) F x expoetial with scale parameter b estimated as 1 / (x) ( Cα + 0. / ) ( / ) values of C α are calculated as follows: for case with α = 0.10, C α = / ( / )
18 Cocepts i Probability, Statistics ad Stochastic Modellig 185 Table 7.6 provides critical values for r for the ormal distributio, or the logarithms of logormal variates, based upo the Blom plottig positio that has p i (i 3/8)/( 1/4). Values for the Gumbel distributio are reproduced i Table 7.7 for use with the Grigorte plottig positio p i (i 0.44)/( 0.1). The table also applies to logarithms of Weibull variates (Stediger et al., 1993). Other tables are available for the GEV (Chowdhury et al., 1991), the Pearso type 3 (Vogel ad McMarti, 1991), ad expoetial ad other distributios (D Agostio ad Stephes, 1986). Recetly developed Lmomet ratios appear to provide goodessoffit tests that are superior to both the Kolmogorov Smirov ad the Probability Plot Correlatio test (Hoskig, 1990; Chowdhury et al., 1991; Fill ad Stediger, 1995). For ormal data, the Lskewess estimator τˆ3 (or t 3 ) would have mea zero ad Var τˆ3 ( /)/, allowig costructio of a powerful test of ormality agaist skewed alteratives usig the ormally distributed statistic Z t3 / ( / )/ (7.75) , sigificace level Table 7.6. Lower critical values of the probability plot correlatio test statistic for the ormal distributio usig p i (i 3/8)/( 1/4) (Vogel, 1987). E01101f with a reject regio Z z α/. Chowdhury et al. (1991) derive the samplig variace of the LCV ad Lskewess estimators τˆ ad τˆ3 as a fuctio of κ for the GEV distributio. These allow costructio of a test of whether a particular data set is cosistet with a GEV distributio with a regioally estimated value of κ, or a regioal κ ad a regioal coefficiet of variatio, CV. Fill ad Stediger (1995) show that the τˆ3 Lskewess estimator provides a test for the Gumbel versus a geeral GEV distributio usig the ormally distributed statistic sigificace level E01101g Z (τˆ3 0.17)/ ( / )/ (7.76) with a reject regio Z z α/. The literature is full of goodessoffit tests. Experiece idicates that amog the better tests there is ofte ot a great deal of differece (D Agostio ad Stephes, 1986). Geeratio of a probability plot is most ofte a good idea because it allows the modeller to see what the data look like ad where problems occur. The Kolmogorov Smirov test helps the eye , Table 7.7. Lower critical values of the probability plot correlatio test statistic for the Gumbel distributio usig p i (i 0.44)/( 0.1) (Vogel, 1987).
19 186 Water Resources Systems Plaig ad Maagemet iterpret a probability plot by addig bouds to a graph, illustratig the magitude of deviatios from a straight lie that are cosistet with expected variability. Oe ca also use quatiles of a beta distributio to illustrate the possible error i idividual plottig positios, particularly at the extremes where that ucertaity is largest. The probability plot correlatio test is a popular ad powerful goodessoffit statistic. Goodessoffit tests based upo sample estimators of the Lskewess τˆ3 for the ormal ad Gumbel distributio provide simple ad useful tests that are ot based o a probability plot Normal ad Logormal Distributios The ormal distributio ad its logarithmic trasformatio, the logormal distributio, are arguably the most widely used distributios i sciece ad egieerig. The probability desity fuctio of a ormal radom variable is fx( x) 1 1 exp ( x µ ) πσ σ for X (7.77) where µ ad σ are equivalet to µ X ad σ X, the mea ad variace of X. Iterestigly, the maximum likelihood estimators of µ ad σ are almost idetical to the momet estimates x _ ad s X. The ormal distributio is symmetric about its mea µ X ad admits values from to. Thus, it is ot always satisfactory for modellig physical pheomea such as streamflows or pollutat cocetratios, which are ecessarily oegative ad have skewed distributios. A frequetly used model for skewed distributios is the logormal distributio. A radom variable X has a logormal distributio if the atural logarithm of X, l(x), has a ormal distributio. If X is logormally distributed, the by defiitio l(x) is ormally distributed, so that the desity fuctio of X is for x 0 ad µ l(η). Here η is the media of the Xdistributio. A logormal radom variable takes o values i the rage [0, ]. The parameter µ determies the scale of the Xdistributio whereas σ determies the shape of the distributio. The mea ad variace of the logormal distributio are give i Equatio Figure 7.3 illustrates the various shapes that the logormal probability desity fuctio ca assume. It is highly skewed with a thick right had tail for σ 1, ad approaches a symmetric ormal distributio as σ 0. The desity fuctio always has a value of zero at x 0. The coefficiet of variatio ad skew are: CV X [exp(σ ) 1] 1/ (7.79) γ X 3CV X CV X 3 (7.80) The maximum likelihood estimates of µ ad σ are give i Equatio 7.63 ad the momet estimates i Equatio For reasoablesized samples, the maximum likelihood estimates geerally perform as well or better tha the momet estimates (Stediger, 1980). The data i Table 7. were used to calculate the parameters of the logormal distributio that would describe these flood flows. The results are reported i Equatio The twoparameter maximum likelihood ad method of momets estimators idetify parameter estimates for which the distributio skewess coefficiets f(x) σ = 0. σ = 0.5 σ = dl ( x) fx( x) exp [ 1( x) µ ] πσ σ dx 1 1 exp [ l( x / η)] x πσ σ (7.78) E0057e x Figure 7.3. Logormal probability desity fuctios with various stadard deviatios σ.
20 Cocepts i Probability, Statistics ad Stochastic Modellig 187 are.06 ad 1.7, which is substatially greater tha the sample skew of A useful geeralizatio of the twoparameter logormal distributio is the shifted logormal or threeparameter logormal distributio obtaied whe l(x τ) is described by a ormal distributio, ad X τ. Theoretically, τ should be positive if, for physical reasos, X must be positive; practically, egative values of τ ca be allowed whe the resultig probability of egative values of X is sufficietly small. Ufortuately, maximum likelihood estimates of the parameters µ, σ, ad τ are poorly behaved because of irregularities i the likelihood fuctio (Giesbrecht ad Kempthore, 1976). The method of momets does fairly well whe the skew of the fitted distributio is reasoably small. A method that does almost as well as the momet method for lowskew distributios, ad much better for highly skewed distributios, estimates τ by: x τˆ () x( ) xˆ. x x xˆ (7.81) provided that x (1) x () xˆ0.50 0, where x (1) ad x () are the smallest ad largest observatios ad xˆ0.50 is the sample media (Stediger, 1980; Hoshi et al., 1984). If x (1) x () xˆ0.50 0, the the sample teds to be egatively skewed ad a threeparameter logormal distributio with a lower boud caot be fit with this method. Good estimates of µ ad σ to go with τˆ i Equatio 7.81 are (Stediger, 1980): µˆ l () 1 ( ) x τˆ s /( x τˆ ) 1 X σˆ sx l 1 ( x τˆ ) (7.8) For the data i Table 7., Equatios 7.81 ad 7.8 yield the hybrid mometofmomets estimates of µˆ 7.606, σˆ (0.3659) ad τˆ for the threeparameter logormal distributio. This distributio has a coefficiet of skewess of 1.19, which is more cosistet with the sample skewess estimator tha were the values obtaied whe a twoparameter logoral distributio was fit to the data. Alteratively, oe ca estimate µ ad σ by the sample mea ad variace of l(x τˆ) which yields the hybrid maximum likelihood estimates µˆ 7.605, σˆ (0.3751) ad agai τˆ The two sets of estimates are surprisigly close i this istace. I this secod case, the fitted distributio has a coefficiet of skewess of 1.. Natural logarithms have bee used here. Oe could have just as well use base 10 commo logarithms to estimate the parameters; however, i that case the relatioships betwee the logspace parameters ad the realspace momets chage slightly (Stediger et al., 1993, Equatio ) Gamma Distributios The gamma distributio has log bee used to model may atural pheomea, icludig daily, mothly ad aual streamflows as well as flood flows (Bobée ad Ashkar, 1991). For a gamma radom variable X, x fx( x) β x x ( ) ( ) α β α 1 β e β 0 Γ α µ X β σ α X β γ X CVX for β 0 α (7.83) The gamma fuctio, Γ(α), for iteger α is (α 1)!. The parameter α 0 determies the shape of the distributio; β is the scale parameter. Figure 7.4 illustrates the differet shapes that the probability desity fuctio for a gamma variable ca assume. As α, the gamma distributio approaches the symmetric ormal distributio, whereas for 0 α 1, the distributio has a highly asymmetric Jshaped probability desity fuctio whose value goes to ifiity as x approaches zero. The gamma distributio arises aturally i may problems i statistics ad hydrology. It also has a very reasoable shape for such oegative radom variables as raifall ad streamflow. Ufortuately, its cumulative distributio fuctio is ot available i closed form, except for iteger α, though it is available i may software packages icludig Microsoft Excel. The gamma
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