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Parameters of distribution Location Parameter Scale Parameter Shape Parameter

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Plotting position of xi means, the probability assigned to each data point to be plotted on probability paper. The plotting of ordered data on extreme probability paper is done according to a general plotting position function: P = (m-a) / (N+1-2a). Constant 'a' is an input variable and is default set to 0.3. Many different plotting functions are used, some of them can be reproduced by changing the constant 'a'. Gringorton P = (m-0.44)/(N+0.12) a = 0.44 Weibull P = m/(N+1) a = 0 Chegadayev P = (m-0.3)/(N+0.4) a = 0.3 Blom P = (m-0.375)/(N+0.25) a = 0.375 Plotting position

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Curve Fitting Methods The method is based on the assumption that the observed data follow the theoretical distribution to be fitted and will exhibit a straight line on probability paper. Graphical Curve fitting Method Mathematical Curve fitting Method.- Method of Moments- Method of Least squares Method of Maximum Likelihood

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Estimation of statistical parameters To apply theoretical distribution functions following steps are required: –investigate homogeneity of series –estimate the parameters of postulated distribution –test goodness of fit of theoretical distribution to observed frequency distribution Estimation of parameters: –graphical method –analytical methods: *method of moments *maximum likelihood method *method of least squares *mixed moment-maximum likelihood method

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Estimation of statistical parameters (2) Estimation procedures differ Comparison of quality by: –mean square error or its root –error variance and standard error –bias –efficiency –consistency Mean square error in of :

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Estimation of statistical parameters (3) Consequently: –First part is the variance of = average of squared differences about expected mean, it gives the random portion of the error –Second part is square of bias, bias = systematic difference between expected and true mean, it gives the systematic portion of the error Root mean square error: Standard error Consistency: Mind effective number of data

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Graphical estimation Variable is function of reduced variate: –e.g. for Gumbel: Reduced variate function of non-exceedance prob.: Determine non-exceedance prob. from rank number of data in ordered set, e.g. for Gumbel: Unbiased plotting position depends on distribution

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Graphical estimation (2) Procedure: –rank observations in ascending order –compute non-exceedance frequency F i –transform F i into reduced variate z i –plot x i versus z i –draw straight line through points by eye-fitting –estimate slope of line and intercept at z = 0 to find the parameters

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Graphical estimation: example Annual maximum river flow at Chooz on Meuse

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Graphical estimation

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Graphical estimation: example (2) Gumbel parameters: –graphical estimation: x 0 = 590, = 247 –MLM-method: x 0 = 591, = 238 100-year flood: –T = 100 FX(x) = 1-1/100 = 0.99 –z = -ln(-ln(0.99)) = 4.6 –graphical method: x = x 0 + z = 590 + 247x4.6 = 1726 m 3 /s –MLM method: x = x 0 + z = 591 + 238x4.6 = 1686 m 3 /s Graphical method: pro’s and con’s –easily made –visual inspection of series –strong subjective element in method: not preferred for design; only useful for first rough estimate –confidence limits will be lacking

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Plotting positions Plotting positions should be: –unbiased –minimum variance General:

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Product moment Standard procedure: –estimate mean and variance for 2-par distributions –estimate mean, variance and skewness for 3-par distr. In HYMOS method available for: –normal –LN-2 & LN-3 –G-2 & P-3 –EV-1 Pro’s and con’s of Method: –simple procedure –for small sample sizes ( N < 30) sample moments may substantially differ from population values, due to too much weight to outliers –3-parameters from small samples provides poor quality estimates

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PWM and L-moments Definition of Probability weighted moments (PWM): Choose p=1 and s=0: It follows that in PWM the non-exceedance probability is raised to a power, rather than the variable itself Since F X (x) < 1 PWM’s are much less sensitive for outliers But: higher weight is put on higher ranked values

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L-moments Order statistics (data in ascending order): then X i:N is i th order statistic first four L-moments:

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L-moments (2) L-skewness and L-kurtosis: Relation of L-moments with distribution parameters: –Normal Gumbel

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L-moments and PWM Determination of L-moments requires numerous combination = cumbersome Relation between L-moments and PWM:

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Sample estimates of PWM’s Sample estimates:

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Example L-moments Higher weight on higher ranked values

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Example L-moments (2)

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L-moment diagram

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Maximum likelihood method Principle: –Given r.v. X with pdf f X (x) and sample of X of size N –pdf f X (x) has parameters 1, 2,…, k –sample values independent and identically distributed –then with parameter set the prob. that r.v. will fall in interval including x i is given by: pdf f X (x i | )dx –joint prob. of occurrence of sample set x i, i=1 to N is –procedure involves maximisation of:

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Maximum likelihood method (2) Likelihood function: Best set of parameters follow from: Log-likelihood maximisation often easier, since products are replaced by sums:

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Example MLM LN-2 distribution: Likelihood for sample of N is: Log-likelihood: Parameters follow from:

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Example MLM Parameters for LN-2: Parameters are seen to be first and second moments about origin and mean of ln(x) Similarly the parameters for LN-3 can be derived Equations become more complicated For small samples convergence is poor then mixed-moment MLM method is preferred

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Parameter estimation by Least Squares Method is extension of graphical method Instead of fitting line by eye, the Least Squares method is used Quality of method strongly dependent on probability assigned to ranked sample values (plotting position) Example of annual flow extremes of Meuse at Chooz carried out using Gringorten and Weibull plotting position Difference for 100 year flood is resp. 3% and 9% with MLM estimate Weibull plotting position gives low T to highest sample value, hence in extrapolated part higher values are found than e.g. with Gringorten

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Method of Least Squares

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Mixed-moment maximum likelihood method Method applies generally MLM to 2 parameters and the product moment method to the location parameter Example is for LN-3: –use MLM-estimates with X replaced by X - x 0 : –The location parameter is taken from the first moment:

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Mixed-moment max.likelihood meth. (2) The location parameter is solved iteratively –the following modified form is used: –For each value of x 0 : Y and Y 2 are estimated using Newton-Raphson method: –the denominator is estimated by its expected value: –The new x 0 is determined (until improvement is small) from:

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Censoring of data Right censoring: eliminating data from analysis at the high side of the data set Left censoring: eliminating data from analysis at the low side of the data set Relative frequencies of remaining data is left unchanged. Right censoring may be required because: –extremes in data set have higher T than follows from series –extremes may not be very accurate Left censoring may be required because: –physics of lower part is not representative for higher values

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Quantile uncertainty and confidence limits Estimation errors in parameters lead to estimation errors in quantiles Procedure demonstrated for quantile of N( X, X 2 ) –quantile estimated by: –estimation variance of quantile: –Hence: Var m X = s X 2 /N Var (s X ) = s X 2 /2N Cov(m X,s X ) =0 Var m X = s X 2 /N Var (s X ) = s X 2 /2N Cov(m X,s X ) =0

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Quantile uncertainty and conf. limits (2) Confidence limits become: –CL diverge away from the mean –Number of data N also determine width of CL Uncertainty in non-exceedance probability for a fixed x p : –standard error of reduced variate It follows with z p approx N(z p, zp ): hence:

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Confidence limits for frequency distribution

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Example rainfall Vagharoli

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Example rainfall Vagharoli (2) Ranked observations F i =(i-3/8)/(N+1/4) Normal distribution F X (z) for z=(x-877)/357 T=1/(1-F X (z))

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Example rainfall Vagharoli (3)

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Example Vagharoli (4) T F X (x) = 1 - 1/T

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Investigating homogeneity Prior to fitting, tests required on: –1. stationarity (properties do not vary with time) –2. homogeneity (all element are from the same population) –3. randomness (all series elements are independent) –First two conditions transparent and obvious. Violating last condition means that effective number of data reduces when data are correlated –lack of randomness may have several causes; in case of a trend there will be serial correlation HYMOS includes numerous statistical test : –parametric (sample taken from appr. Normal distribution) –non-parametric or distribution free tests (no conditions on distribution, which may negatively affect power of test

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Summary of tests On randomness: –median run test –turning point test –difference sign test On correlation: –Spearman rank correlation test –Spearman rank trend test –Arithmetic serial correlation coefficient –Linear trend test On homogeneity: –Wilcoxon-Mann-Whitney U-test –Student t-test –Wilcoxon W-test –Rescaled adjusted range test

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Chi-square goodness of fit test Hypothesis –F(x) is the distribution function of a population from which sample x i, i =1,…,N is taken –Actual to theoretical number of occurrences within given classes is compared Procedure: –data set is divided in k class intervals containing at least each 5 values –Class limits from all classes have equal probability p j = 1/k = F(z j ) - F(z j -1) e.g. for 5 classes this is p = 0.20, 0.40, 0.60, 0.80 and 1.00 –the interval j contains all x i with: U C (j-1)

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Chi-squared goodness of fit test

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Chi-square goodness of fit test (2) –Consider following test statistic: –under H0 test statistic has 2 distr, with df = k-1-m –k= number classes, m = number of parameters –simplified test statistic: –H0 not rejected at significance level if:

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Number of classes in Chi-squared goodness of fit test

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Example –Annual rainfall Vagharoli (see parameter estimation) –test on applicability of normal distribution –4 class intervals were assumed (20 data) –upper class levels are at p=0.25, 0.50, 0.75 and 1.00 –the reduced variates are at -0.674, 0.00, 0.674 and –hence with mean = 877, and stdv = 357 the class limits become: 877 - 0.674x357 = 636 877 = 877 877 + 0.674x357 = 1118

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Example continued (2) From the table it follows for the test statistic: At significance level = 5%, according to Chi-squared distribution for = 4-1-2 df the critical value is at 3.84, hence c 2 < critical value, so H0 is not rejected

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