1. Stochastic Hydrology Hydrological Frequency Analysis (II) LMRD-based GOF tests
Prof. Ke-Sheng Cheng
Department of Bioenvironmental Systems Engineering
National Taiwan University

2. More advanced topics LMRD-based GOF tests Regional frequency analysis
Reference:
Liou, J.J., Wu, Y.C., Cheng, K.S., Establishing acceptance regions for L-moments-based goodness-of-fit test by stochastic simulation. Journal of Hydrology, Vol. 355, No.1-4,
Wu, Y.C., Liou, J.J., Cheng, K.S., Establishing acceptance regions for L-moments based goodness-of-fit tests for the Pearson type III distribution. Stochastic Environmental Research and Risk Assessment, DOI /s z.
Regional frequency analysis
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

3. GOF test using L-moment-ratios diagram (LMRD)
Concept of identifying appropriate distributions using moment-ratio diagrams (MRD).
Product-moment-ratio diagram (PMRD)
L-moment-ratio diagram (LMRD)
Two-parameter distributions
Normal, Gumbel (EV-1), etc.
Three-parameter distributions
Log-normal, Pearson type III, GEV, etc.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

4. Moment ratios are unique properties of probability distributions and sample moment ratios of ordinary skewness and kurtosis have been used for selection of probability distribution.
The L-moments uniquely define the distribution if the mean of the distribution exists, and the L-skewness and L-kurtosis are much less biased than the ordinary skewness and kurtosis.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

5. A two-parameter distribution with a location and a scale parameter plots as a single point on the LMRD, whereas a three-parameter distribution with location, scale and shape parameters plots as a curve on the LMRD, and distributions with more than one shape parameter generally are associated with regions on the diagram.
However, theoretical points or curves of various probability distributions on the LMRD cannot accommodate for uncertainties induced by parameter estimation using random samples.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

6. Ordinary (or product) moment-ratios diagram (PMRD)
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

7. The ordinary (or product) moment ratios diagram
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

8. 11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

9. Sample estimates of product moment ratios
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

10. 11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

11. (D'Agostino and Stephens, 1986)
95％
90％
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

12. Even though joint distribution of the ordinary sample skewness and sample kurtosis is asymptotically normal, such asymptotic property is a poor approximation in small and moderately samples, particularly when the underlying distribution is even moderately skew.
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

13. 11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

14. Scattering of sample moment ratios of the normal distribution
(100,000 random samples)
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

15. L-moments and the L-moment ratios diagram
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

19. L-moment-ratio diagram of various distributions
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

20. Sample estimates of L-moment ratios (probability weighted moment estimators)
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

21. 11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

22. Sample estimates of L-moment ratios (plotting-position estimators)
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

23. Hosking and Wallis (1997) indicated that is not an unbiased estimator of , but its bias tends to zero in large samples.
and are respectively referred to as the probability-weighted-moment estimator and the plotting-position estimator of the L-moment ratio .
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

24. Packages for sample L-moments calculation and parameter estimation in R
lmomco
Lmoments
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

25. Establishing acceptance region for L-moment ratios
The standard normal and standard Gumbel distributions (zero mean and unit standard deviation) are used to exemplify the approach for construction of acceptance regions for L-moment ratio diagram.
L-moment-ratios ( , ) of the normal and Gumbel distributions are respectively (0, ) and (0.1699, ).
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

26. Stochastic simulation of the normal and Gumbel distributions
For either of the standard normal and standard Gumbel distribution, a total of 100,000 random samples were generated with respect to the specified sample size20, 30, 40, 50, 60, 75, 100, 150, 250, 500, and 1,000.
For each of the 100,000 samples, sample L-skewness and L-kurtosis were calculated using the probability-weighted-moment estimator and the plotting-position estimator.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

27. Scattering of sample L-moment ratios Normal distribution
(100,000 random samples)
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

28. (100,000 random samples) 11/18/2018 Normal distribution ?
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

29. Non-normal distribution !
95% acceptance region
99% acceptance region
Non-normal distribution !
(100,000 random samples)
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

30. Scattering of sample L-moment ratios Gumbel distribution
(100,000 random samples)
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

31. (100,000 random samples)
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

32. (100,000 random samples)
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

33. For both distribution types, the joint distribution of sample L-skewness and L-kurtosis seem to resemble a bivariate normal distribution for a larger sample size (n = 100).
However, for sample size n = 20, the joint distribution of sample L-skewness and L-kurtosis seems to differ from the bivariate normal. Particularly for Gumbel distribution, sample L-moments of both estimators are positively skewed.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

34. For smaller sample sizes (n = 20 and 50), the distribution cloud of sample L-moment-ratios estimated by the plotting-position method appears to have its center located away from ( , ), an indication of biased estimation.
However, for sample size n = 100, the bias is almost unnoticeable, suggesting that the bias in L-moment-ratio estimation using the plotting-position estimator is negligible for larger sample sizes.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

35. By contrast, the distribution cloud of the sample L-moment-ratios estimated by the probability-weighted-moment method appears to have its center almost coincide with ( , ).
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

36. Bias of sample L-skewness and L-kurtosis - Normal distribution
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

37. 11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

38. Bias of sample L-skewness and L-kurtosis - Gumbel distribution
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

39. 11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

40. Mardia test for bivariate normality of sample L-skewness and L-kurtosis
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

44. Mardia test for bivariate normality of sample L-skewness and L-kurtosis
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

45. Mardia test for bivariate normality of sample L-skewness and L-kurtosis
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

46. It appears that the assumption of bivariate normal distribution for sample L-skewness and L-kurtosis of both distributions is valid for moderate to large sample sizes. However, for random samples of normal distribution with sample size , the bivariate normal assumption may not be adequate. Similarly, the bivariate normal assumption for sample L-skewness and L-kurtosis of the Gumbel distribution may not be adequate for sample size
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

47. Establishing acceptance regions for LMRD-based GOF tests
For moderate to large sample sizes, the sample L-skewness and L-kurtosis of both the normal and Gumbel distributions have asymptotic bivariate normal distributions.
Using this property, the acceptance region of a GOF test based on sample L-skewness and L-kurtosis can be determined by the equiprobable density contour of the bivariate normal distribution with its encompassing area equivalent to
Difference between the acceptance region and confidence region
Confidence region (interval) of a parameter, for example, population mean
Acceptance region of a test statistic, for example, sample mean
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

48. The probability density function of a multivariate normal distribution is generally expressed by
The probability density function depends on the random vector X only through the quadratic form which has a chi-square distribution with p degrees of freedom.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

49. Therefore, probability density contours of a multivariate normal distribution can be expressed by
for any constant
For a bivariate normal distribution (p=2) the above equation represents an equiprobable ellipse, and a set of equiprobable ellipses can be constructed by assigning to c for various values of .
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

50. Consequently, the acceptance region of a GOF test based on the sample L-skewness and L-kurtosis is expressed by
where is the upper quantile of the distribution at significance level .
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

51. For bivariate normal random vector , the density contour of can also be expressed as
However, the expected values and covariance matrix of sample L-skewness and L-kurtosis are unknown and can only be estimated from random samples generated by stochastic simulation.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

52. The Hotelling’s T2 statistic
Thus, in construction of the equiprobable ellipses, population parameters must be respectively replaced by their sample estimates
The Hotelling’s T2 statistic
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

53. The Hotelling’s T2 is distributed as a multiple of an F-distribution, i.e.,
For large N,
Therefore, the distribution of the Hotelling’s T2 can be well approximated by the chi-square distribution with degree of freedom 2.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

54. Thus, if the sample L-moments of a random sample of size n falls outside of the corresponding ellipse, i.e.
the null hypothesis that the random sample is originated from a normal or Gumbel distribution is rejected.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

55. Scattering of sample L-moment ratios Normal distribution
(100,000 random samples)
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

56. (100,000 random samples) 11/18/2018 Normal distribution ?
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

57. Variation of 95% acceptance regions with respect to sample size n
Non-normal distribution !
What if n=36?
(100,000 random samples)
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

58. Empirical relationships between parameters of acceptance regions and sample size
Since the 95% acceptance regions of the proposed GOF tests are dependent on the sample size n, it is therefore worthy to investigate the feasibility of establishing empirical relationships between the 95% acceptance region and the sample size. Such empirical relationships can be established using the following regression model
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

59. Empirical relationships between the sample size and parameters of the bivariate distribution of sample L-skewness and L-kurtosis
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

60. Empirical relationships between the sample size and parameters of the bivariate distribution of sample L-skewness and L-kurtosis
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

61. Example
Suppose that a random sample of size n = 44 is available, and the plotting-position sample L-skewness and L-kurtosis are calculated as ( , ) = (0.214, 0.116). We want to test whether the sample is originated from the Gumbel distribution.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

62. From the regression models for plotting-position estimators, we find
to be respectively , , , , and The Hotelling’s T2 is then calculated as
The value of T2 is much smaller than the threshold value
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

63. The null hypothesis that the random sample is originated from the Gumbel distribution is not rejected.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

64. 95% acceptance regions of L-moments-based GOF test for the normal distribution
Acceptance ellipses corresponding to various sample sizes (n = 20, 30, 40, 50, 60, 75, 100, 150, 250, 500, and 1,000).
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

65. Acceptance ellipses corresponding to various sample sizes (n = 20, 30, 40, 50, 60, 75, 100, 150, 250, 500, and 1,000).
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

66. 95% acceptance regions of L-moments-based GOF test for the Gumbel distribution
Acceptance ellipses corresponding to various sample sizes (n = 20, 30, 40, 50, 60, 75, 100, 150, 250, 500, and 1,000).
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

67. Acceptance ellipses corresponding to various sample sizes (n = 20, 30, 40, 50, 60, 75, 100, 150, 250, 500, and 1,000).
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

68. Validity check of the LMRD acceptance regions
The sample-size-dependent confidence intervals established using empirical relationships described in the last section are further checked for their validity. This is done by stochastically generating 10,000 random samples for both the standard normal and Gumbel distributions, with sample size20, 30, 40, 50, 60, 75, 100, 150, 250, 500, and 1,000.
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

69. 11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

70. For validity of the sample-size-dependent 95% acceptance regions, the rejection rate should be very close to the level of significance ( ) or the acceptance rate be very close to 0.95.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

71. Acceptance rate of the validity check for sample-size-dependent 95% acceptance regions of sample L-skewness and L-kurtosis pairs.
Based on 10,000 random samples for any given sample size n.
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

83. Establishing acceptance regions for L-moments based goodness-of-fit tests for the Pearson type III distribution
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

84. Establishing LMRD acceptance regions for three-parameter distributions is more complicated than that for normal and Gumbel distributions. For a two-parameter distribution with a location and a scale parameter, there exists a unique point characterizing its LMRD. Whereas for a three-parameter distribution with location, scale and shape parameters, the LMRD plots as a curve and each point on the curve represents a valid parameter vector.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

85. Following the same concept of establishing acceptance region for the normal distribution, it appears that the acceptance region for L-moments based GOF test of a three-parameter distribution will vary with sample size and the L-skewness.
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

86. L-moments and the L-moment ratio diagram of the PE3 distribution
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

87. 11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

88. Stochastic simulation of the Pearson type III distribution
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

95. Establishing acceptance regions for GOF test of the PE3 distribution
Rationale for establishment of acceptance regions
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

100. Conditional density of 3 given t3
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

105. Determining acceptance regions
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

112. Validation of the LMRD acceptance regions
Acceptance rates of t3-specific validity check
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Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU

113. Acceptance rates of 3-specific validity check
11/18/2018
Lab for Remote Sensing Hydrology and Spatial Modeling, Dept. of Bioenvironmental Systems Eng., NTU