Parameters of distribution
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Parameters of distribution. Location Parameter Scale Parameter Shape Parameter. Plotting position. Plotting position of xi means, the probability assigned to each data point to be plotted on probability paper.

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Parameters of distribution

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Parameters of distribution

Parameters of distribution

  • Location Parameter

  • Scale Parameter

  • Shape Parameter


Parameters of distribution

Plotting position

  • 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


Parameters of distribution

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


Estimation of statistical parameters 2

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 :


Estimation of statistical parameters 3

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


Graphical estimation

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


Graphical estimation 2

Graphical estimation (2)

  • Procedure:

    • rank observations in ascending order

    • compute non-exceedance frequency Fi

    • transform Fi into reduced variate zi

    • plot xi versus zi

    • draw straight line through points by eye-fitting

    • estimate slope of line and intercept at z = 0 to find the parameters


Graphical estimation example

Graphical estimation: example

  • Annual maximum river flow at Chooz on Meuse


Parameters of distribution

Graphical estimation


Graphical estimation example 2

Graphical estimation: example (2)

  • Gumbel parameters:

    • graphical estimation: x0 = 590,  = 247

    • MLM-method: x0 = 591,  = 238

  • 100-year flood:

    • T = 100  FX(x) = 1-1/100 = 0.99

    • z = -ln(-ln(0.99)) = 4.6

    • graphical method: x = x0 + z = 590 + 247x4.6 = 1726 m3/s

    • MLM method: x = x0 + z = 591 + 238x4.6 = 1686 m3/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


Plotting positions

Plotting positions

  • Plotting positions should be:

    • unbiased

    • minimum variance

  • General:


Censoring of data

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


Quantile uncertainty and conf limits 2

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 xp:

    • standard error of reduced variate

  • It follows with zp approx N(zp,zp):

hence:


Parameters of distribution

Confidence limits for frequency distribution


Example rainfall vagharoli

Example rainfall Vagharoli


Example rainfall vagharoli 2

Example rainfall Vagharoli (2)

Normal distribution

FX(z) for

z=(x-877)/357

Ranked observations

T=1/(1-FX(z))

Fi=(i-3/8)/(N+1/4)


Example rainfall vagharoli 3

Example rainfall Vagharoli (3)


Example vagharoli 4

Example Vagharoli (4)

T

FX(x) = 1 - 1/T


Investigating homogeneity

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


Summary of tests

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


Chi square goodness of fit test

Chi-square goodness of fit test

  • Hypothesis

    • F(x) is the distribution function of a population from which sample xi, 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

      pj = 1/k = F(zj) - F(zj-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 xi with: UC(j-1)<xi UC(j)

    • the number of samples falling in class j = bj is computed

    • the number of values expected in class j = ej according to the theoretical distribution is computed

    • the theoretical number of values in any class = N/k because of the equal probability in each class


Parameters of distribution

Chi-squared goodness of fit test


Chi square goodness of fit test 2

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:


Number of classes in chi squared goodness of fit test

Number of classes in Chi-squared goodness of fit test


Example

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


Example continued 2

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 c2 < critical value, so H0 is not rejected


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