Inference of the optimal probability distribution model for geotechnical parameters by using Weibull and NID distributions
Fengqiang Gong^{1} , Tiancheng Wang^{2} , Shanyong Wang^{3}
^{1, 2}School of Resources and Safety Engineering, Central South University, Changsha, Hunan, 410083, P. R. China
^{3}ARC Centre of Excellence for Geotechnical Science and Engineering, Faculty of Engineering and Built Environment, The University of Newcastle, Callaghan, New South Wales, 2308, Australia
^{1}Corresponding author
Journal of Vibroengineering, Vol. 21, Issue 4, 2019, p. 876887.
https://doi.org/10.21595/jve.2018.19758
Received 21 February 2018; received in revised form 26 April 2018; accepted 9 May 2018; published 30 June 2019
JVE Conferences
To obtain the optimal probability distribution models of geotechnical parameters, the goodness of fit of the normal information diffusion (NID) distribution and Weibull distribution were investigated and compared for actual engineering samples and Monte Carlo (MC) simulated samples. Two datasets from actual engineering parameters (the strength of a rock mass and the average wind speed) were used to test the fitting abilities of these two distributions. The results show that the parameters of the NID distribution are easily estimated, the KolmogorovSmirnov (KS) test results of the NID distribution are smaller than those of the Weibull distribution, and the NID distribution curves can perfectly reflect the stochastic volatility of geotechnical parameters with small sample sizes. The sample size effects on the fitting accuracy of the NID distribution and Weibull distribution were also investigated in this paper. Eight simulated samples with different sample sizes, namely, 15, 20, 30, 50, 100, 200, 500, and 1000, were produced by using the MC method based on two known Weibull distributions. The results show that with an increase in the sample size, the KS test results of the NID distribution gradually decrease and tend to converge, while the chisquare test results of the NID distribution remain low and are always lower than those of the Weibull distribution. The cumulative probability values of the NID distribution are larger than those of the Weibull distribution and are always equal to 1.0000. Finally, the comparison of the fitting accuracy between the NID distribution and normalized Weibull distribution was also analyzed.
Keywords: reliability analysis, geotechnical parameters, the optimal probability distribution, probability density function (PDF), normal information diffusion, Weibull distribution.
1. Introduction
Due to the natural attributes of rock materials, the complexity of the geological environment and the randomness of external loading (such as impact loads, seismic response, vibratory action, etc.), uncertainty is inevitable in geotechnical engineering [13]. To quantitatively evaluate the influence of this uncertainty, reliability analysis has been widely used in many fields of geotechnical engineering [4, 5], such as slope reliability [612], tunnel and underground cavity reliability [13, 14], etc. In the reliability analysis of geotechnical engineering under quasistatic loads or vibrations loads, the inference of optimal probability density function (PDF) or cumulative distribution function (CDF) of a geotechnical parameter is one of the most essential tasks; this is the first step in a reliability analysis and plays a central role in ensuring the precision and accuracy of the geotechnical reliability analysis [15, 16]. Through the comparison and selection of the classical distributions (normal distribution, lognormal distribution, Weibull distribution, gamma distribution, etc.), some previous studies have shown that many geotechnical parameters will accept a Weibull distribution as the optimal PDFs [1720]. However, there are some unsolved issues in the application process of the Weibull distribution. The specific PDF forms of the Weibull distribution are not uniform (including the twoparameter, threeparameter and mixed Weibull distributions), and the parameters of the Weibull distribution, such as the shape parameter $m$, the scale parameter $\sigma $ or the position parameter $\mu $, are sometimes difficult to estimate. In addition, the total cumulative probability value of the Weibull distribution is generally less than 1.0000 because its defined interval does not match the actual finite interval of geotechnical parameters. It is necessary to study the inference method, which more accurately represents the actual distribution.
In recent years, the normal information diffusion (NID) theory has been the focus of the attention of many scholars and has been further developed by C. F. Huang et al. [21, 22]. NID theory provides a new way to study function approximation based on the information assignment method of a fuzzy set. In NID theory, the original information is directly transferred to the fuzzy relation in a way that avoids calculation of the membership function and preserves the original information contained in the original data as much as possible. Due to the advantages of the information diffusion principle, NID theory has been successfully applied to some fields of study, especially to natural disaster and risk assessment [2325].
In this paper, NID theory was introduced to fit the optimal PDFs or CDFs of geotechnical parameters. Two geotechnical parameters, the strength of a rock mass affected by acid [26] and the average wind speed [27], were used as examples to investigate the goodness of fit in a comparative analysis of the NID distribution and Weibull distribution. In addition, the effect of the sample size on the fitting accuracy of these two distributions was also illustrated with MC simulation samples. The results show that NID distribution can make full use of the sample information to deduce the PDFs of the geotechnical parameters and that its fit is more accurate than that of the Weibull distribution.
2. Weibull distribution
In mathematical statistics, the Weibull distribution has a range of function forms, including the twoparameter, threeparameter and mixed Weibull distributions, which are widely used in various fields of research. The specific Weibull distribution function is determined by the shape parameter $m$, the scale parameter $\sigma $ and the position parameter $\mu $. Among these parameters, the most important parameter is the shape parameter, which determines the basic shape of the PDF curve. In addition, the scale parameter effects the scaling of the PDF curve. In the geotechnical engineering reliability, the twoparameter Weibull distribution is one of the most commonly used models [18, 19]. The PDF of the twoparameter Weibull distribution can be written as Eq. (1):
where $F(\cdot )$ denotes the cumulative distribution function. $m$ and $\sigma $ is the shape parameter and the scale parameter, respectively.
3. NID distribution
The basic principle of the NID distribution was developed by C. F. Huang [21], and a brief introduction is as follows.
Suppose that the PDF of a random variable $x$ is $f\left(x\right)$; then, $\mu \left(x\right)$ is defined as a Borel measurable function in $(\mathrm{\infty},+\mathrm{\infty})$. The diffusion estimation of $f\left(x\right)$ can be expressed as Eq. (2):
where ${\mathrm{\Delta}}_{n}>$ 0 is defined as the window width and $\mu \left(x\right)$ is defined as a diffusion function $f\left(x\right)$. According to the information diffusion process, $\mu \left(x\right)$ can be written as Eq. (3):
Substituting Eq. (3) into Eq. (2), the normal information diffusion function can be written as follows Eq. (4):
where $h$ denotes the window width of the standard normal diffusion function $\mu \left(x\right)$, $n$ denotes the sample size of a random variable, ${x}_{i}$ ($i=$ 1, 2, 3, …) is the observed values of the random variable, and ${x}_{max}$ and ${x}_{min}$ are the maximum value and minimum value of ${x}_{i}$, respectively. According to the principle of choosing the nearest normal information diffusion, $h=\gamma \left({x}_{max}{x}_{min}\right)/(n1)$, in which $\gamma $ is related to the sample size (Table 4). When the sample size is greater than or equal to 17, $\gamma $ is always equal to 1.420693101. The details of the information diffusion process are discussed in Huang’s study [22].
4. Fitting comparison of the NID distribution and Weibull distribution
4.1. Data of actual samples
In this paper, two datasets from actual engineering parameters (the strength of a rock mass and the average wind speed) were used as the examples, which accepted the Weibull distribution as the optimal PDF in previous studies [26, 27]. The specific data are given in Tables 1 and 2.
Table 1. Sample 1# data
92

107

113

114

119

120

122

127

128

130

134

141

142

146

147

148

153

156

167


Note: The data of the strength of a rock mass affected by acid (unit: MPa) [26]

Table 2. Sample 2# data
4.6

5.0

5.3

5.5

5.6

5.6

5.7

5.7

6.0

6.0

6.3

6.4

6.5

6.5

6.6

7.0

7.1

7.6

7.8

7.8

7.9

8.1

8.2

8.9

8.9

9.0

9.0

9.7

9.9

10.2

Note: The data of the average wind speed (unit: mph) [27]

4.2. Distribution interval determination for the actual samples
Normally, the actual distribution interval of geotechnical parameters is limited. The sample values of the geotechnical parameters are no less than zero and cannot approach positive infinity; truncated processing is necessary to determine the distribution interval of geotechnical parameters. Here, we provide a new integral interval standard combining a 3$\sigma $ statistical principle and the effect of skewness $c$: the value of the left end of the interval should not be less than zero. When $c>$ 0, [$\mu 3\sigma $, $\mu +(3+c)\sigma $], and when $c<$ 0, [$\mu (3c)\sigma $, $\mu +3\sigma $], where $\mu $ and $\sigma $ are the mean and standard deviation of the sample parameter, respectively. The truncated intervals for the two actual samples are given in Table 3.
4.3. Distribution parameters of the actual samples
The parameters of the NID distribution and Weibull distribution are given in Tables 4 and 5. The window width $hs$ of the NID distribution for the samples 1# and 2# are 5.9196 and 0.2743, respectively. The distribution parameters of sample 1# are obtained from [26] and 1# belongs to the twoparameter Weibull distribution because its position parameter $\mu $ is equal to zero. For determining the distribution parameters of sample 2#, compared with the fitting goodness of the threeparameter Weibull distribution obtained from [27] and twoparameter Weibull distribution obtained from the maximum likelihood estimation (MLE) method, as shown in Fig. 1(b), it was found that the twoparameter Weibull distribution can be accepted as the probability distribution more accurately than the threeparameter Weibull distribution.
Table 3. The interval values of the actual samples
Sample

Size

Mean

Standard deviation

Skewness

Truncated interval


Left

Right


1#

19

131.8947

18.9616

–0.1464

72.2338

188.7797

2#

30

7.1467

1.5684

0.3385

2.4415

12.3827

Table 4. The parameters of the NID distributions
Sample

$n$

${x}_{max}$

${x}_{min}$

$\gamma $

$h$

1#

19

167

92

1.420693101

5.9196

2#

30

10.2

4.6

1.420693101

0.2743

Table 5. The results of the KS test values and CDF values
Sample

distribution parameters

Comparison of the KS test results

CDF values


$m$

$\sigma $

$\mu $

Weibull

NID

${D}_{n,0.05}$

Weibull

NID


1#

7.2500

140.3000

0.0000

0.0969

0.0683

0.3100

0.9917

1.0000

2# (2P)

5.0339

7.7777

0.0000

0.1495

0.0576

0.2420

0.9966

1.0000

2# (3P)

2.1754

3.4344

3.4395

0.2141

0.0576

0.2420

0.9996

1.0000

Note: 2P and 3P denote the twoparameter and threeparameter Weibull distributions, respectively

Fig. 1. Comparative KS test results of the actual sample data
a) Sample 1#
b) Sample 2#
4.4. Comparison of goodness of fit
The KS test is one of the most widely used goodnessoffit tests [28]. In this paper, the KS test was used to discriminate the relative superiority of the NID distribution and Weibull distribution, and the differences between the empirical cumulative frequencies versus theoretical CDF values at every sample point are shown in Fig. 1. The maximum discrepancy of the KS test results ${D}_{n}s$, critical values and cumulative probability values of the NID distribution and Weibull distribution are given in Table 5. The critical values of samples 1# and 2# are 0.3100 and 0.2420 under 95 % confidence level, respectively. The ${D}_{n}s$ results of the NID distribution are 0.0683 and 0.0576, and those of the Weibull distribution are 0.0969 and 0.1495, respectively. Clearly, both of the Weibulltype distributions pass the KS testing, while the ${D}_{n}s$ of the NID distribution are much less than those of the Weibull distribution. In particular, the ${D}_{n}$ of the Weibull distribution is 2.6 times that of the NID distribution for sample 2#. In addition, both the cumulative probability values of the NID distribution are 1.0000. However, the cumulative probability values of the Weibull distribution are 0.9917 and 0.9966, respectively. It can be concluded that the fitting accuracy of the NID distribution is higher than that of the Weibull distribution.
4.5. Comparison of the fitting probability distribution curves
The empirical cumulative frequency curves and theoretical CDF curves for the two actual sample datasets are given Fig. 2. Within the truncated interval, the goodness of fit of the NID distribution is much more accurate than that of the Weibull distribution.
Fig. 2. Comparative CDF curves of the actual sample datasets
a) Sample 1#
b) Sample 2#
Fig. 3. Comparative PDF curves of the actual sample datasets
a) Sample 1#
b) Sample 2#
The PDF curves and histograms for the two actual samples are also given in Fig. 3. Due to the uncertainty in and complexity of the geotechnical parameters, the distributions of the actual samples often present a certain fluctuation. As one of the singlepeak distributions, the Weibull distribution cannot be used to describe the characteristics of the fluctuation in the actual distribution. However, the NID distribution is very flexible and can be used to describe this fluctuation (Fig. 3).
To summarize, whether for CDF curves or PDF curves, the NID distribution will approximate the actual distribution more accurately than the Weibull distribution will. The superiority of the NID distribution can be further confirmed by describing the actual distributions of the geotechnical parameters.
5. Effect of sample size on fitting accuracy
Considering that the sample sizes obtained in actual geotechnical engineering are generally small, to study the effect of the sample size on the fitting accuracy with the NID distribution and Weibull distribution, eight simulated samples of different sizes were produced by using the MC method in this paper. Two known Weibull distributions estimated by samples 1# and 2#, WBL1# (7.2500, 140.3000) and WBL2# (5.0339, 7.7777), were used as the generating functions in the MC method, and the simulated sample sizes are 15, 20, 30, 50, 100, 200, 500, and 1000 (partial sample datasets are shown in Table 6).
The KS test was first used to test the validity of the NID distribution and Weibull distribution. The KS test results and critical values under different sizes are given in Table 7 and Table 8. The effect of the sample size on the KS test results is shown in Fig. 4. With an increase in the sample size, the KS test results of the two fitting methods gradually decrease and tend to converge to the horizontal axis. However, compared with the KS test results of the Weibull distribution, those of the NID distribution are much lower. The convergence speed and stability and the KS test results of the NID distribution are all superior to those of the Weibull distribution.
In addition, the chisquare test was used to investigate the fitting ability for all the samples with a sample size larger than 50. The chisquare test results for a 95 % confidence level are shown in Table 9.
Table 6. Partial simulated samples with the MC method
Size

Simulated data


15

1#

122.1827, 111.8577, 111.4772, 144.3043, 143.4258, 132.1535, 142.5631, 111.0938, 113.1547, 130.3050, 146.0154, 122.9854, 147.7328, 135.6769, 139.4246

2#

5.6766, 4.9123, 8.9816, 4.8271, 6.6610, 9.1989, 8.1665, 7.0353, 4.1709, 4.0127, 8.7865, 3.8716, 4.1776, 7.2920, 5.7718


20

1#

131.3119, 72.4864, 117.7505, 81.6449, 147.6651, 131.8557, 163.0097, 117.6235, 127.7709, 108.4023, 76.0449, 97.8086, 138.1223, 186.8482, 131.1880, 149.3262, 148.6061, 142.5358, 157.7919, 118.3333

2#

9.1276, 4.0796, 10.8625, 5.9287, 5.6591, 5.2686, 9.3094, 7.6447, 8.2525, 5.7731, 7.5141, 4.8584, 8.6470, 8.2342, 8.8605, 8.9211, 5.2635, 6.8948, 7.0227, 8.8642


30

1#

118.2600, 131.0561, 141.8753, 111.0446, 130.5486, 91.0131, 103.9049, 140.8998, 130.8867, 141.4201, 126.5545, 114.3751, 118.4575, 155.1633, 112.0200, 167.9393, 137.8663, 119.5188, 115.6696, 140.3312, 118.5326, 103.9849, 147.2000, 154.8328, 148.2517, 141.2433, 144.6531, 98.2004, 163.0277, 128.3052

2#

5.3974, 6.7077, 7.8491, 7.1765, 7.6363, 9.3872, 8.3475, 9.0068, 8.6356, 8.3473, 7.5724, 9.6763, 4.9454, 4.3994, 7.2694, 7.2760, 7.9056, 4.9734, 7.7720, 9.0935, 5.8966, 7.6864, 8.3389, 7.6276, 9.2076, 8.9481, 4.4431, 4.1979, 6.9143, 9.5544


$\vdots $

$\vdots $

$\vdots $

Table 7. The KS test results and CDF values of sample 1#
Size

Truncated interval

${D}_{n,0.05}$

KS test results

CDF values


Left

Right

Weibull

NID

Weibull

NID


15

85.4387

171.6572

0.3380

0.2607

0.1113

0.9596

1.0000

20

31.5233

216.2854

0.2940

0.1476

0.0799

1.0000

1.0000

30

71.3670

187.9536

0.2420

0.1388

0.0334

0.9923

1.0000

50

38.9533

196.7983

0.1923

0.1127

0.0200

0.9999

1.0000

100

46.1519

195.3722

0.1360

0.0894

0.0100

0.9997

1.0000

200

64.3473

193.7082

0.0962

0.0417

0.0050

0.9965

1.0000

500

59.6480

196.2190

0.0608

0.0487

0.0020

0.9980

1.0000

1000

56.5405

196.9676

0.0430

0.0301

0.0010

0.9986

1.0000

Table 8. The KS test results and CDF values of sample 2#
Size

Truncated interval

${D}_{n,0.05}$

KS test results

CDF values


Left

Right

Weibull

NID

Weibull

NID


15

0.4664

12.5077

0.3380

0.3336

0.0730

1.0000

1.0000

20

1.7339

12.8139

0.2940

0.1857

0.0500

0.9995

1.0000

30

1.5277

12.2911

0.2420

0.1865

0.0344

0.9997

1.0000

50

3.3735

11.1177

0.1923

0.1238

0.0200

0.9828

1.0000

100

1.9711

11.8769

0.1360

0.0552

0.0137

0.9988

1.0000

200

2.2851

11.8036

0.0962

0.0310

0.0050

0.9976

1.0000

500

1.9713

11.8592

0.0608

0.0475

0.0020

0.9988

1.0000

1000

2.2985

11.9048

0.0430

0.0209

0.0010

0.9976

1.0000

Fig. 4. KS test results of the simulated data with the sample size
a) Sample 1#
b) Sample 2#
Table 9. The results of the chisquare tests of samples 1# and 2#
The sizes of MC data

The number of intervals

The chisquare test results


Critical value for Weibull

Weibull

Critical value for NID

NID


1#

50

7

9.4877

7.5426

12.5916

0.5208

100

10

14.0671

8.8863

16.9190

1.2689


200

14

19.6751

11.1422

22.3621

0.1825


500

22

30.1435

15.0571

32.6705

0.5096


1000

31

41.3372

30.2494

43.7729

0.6558


2#

50

7

9.4877

4.7083

12.5916

1.0037

100

10

14.0671

4.0272

16.9190

0.6340


200

14

19.6751

5.1070

22.3621

0.4919


500

22

30.1435

9.6412

32.6705

0.3132


1000

31

41.3372

28.9021

43.7729

0.5684

The change in the chisquare test results with an increase in the sample size are shown in Fig. 5 for the simulated samples. It can be seen that both the NID distribution and Weibull distribution have passed the chisquare test. However, the test results of the NID distribution are considerably lower than those of the Weibull distribution; the test results of the Weibull distribution are one to two orders of magnitude greater than those of the NID distribution. Thus, the goodness of fit of the NID distribution is superior to that of the Weibull distribution. Moreover, the test results of the NID distribution are much more stable than those of the Weibull distribution.
Fig. 5. The chisquare test results of the simulated data with the sample size
a) Sample 1#
b) Sample 2#
The CDF curves of the NID distribution and Weibull distribution for the simulated data of sample 1# are shown in Fig. 6. It is easy to see that, with an increase in the sample size, the CDF curves of the NID distribution are always closer to the empirical cumulative distribution function (EDF) curves than those of the Weibull distribution. Clearly, when the sample size is equal to 1000, the curves of the NID, Weibull and empirical distributions are nearly coincident.
Fig. 6. Comparative CDF curves of the simulated data of sample 1# (sample size $n$)
a)
b)
c)
d)
e)
f)
g)
h)
The CDF values for simulated samples 1# and 2# with different sizes are shown in Tables 7 and 8, and the trends of the CDF values with sample size are shown in Fig. 7. Clearly, with an increase in the sample size, the cumulative probability values of the NID distribution are always equal to 1.0000 and are completely unaffected by the sample size. However, the cumulative probability values of the Weibull distribution are generally less than 1.0000, and there is a considerable amount of volatility when the sample size increases. It is evident from the above analysis that the NID distribution has a higher fitting precision and wider applicability.
Fig. 7. Cumulative probability values of the simulated data with the sample size
a) Sample 1#
b) Sample 2#
6. Discussion
In the truncated interval, the cumulative probability values of classical distributions are usually less than 1.0000. To solve this problem, the normalization of the truncated classical distribution was introduced. The basic principle of normalized distribution is introduced as follows:
where $\stackrel{~}{f}\left(x\right)$ is the normalized PDF, $F\left(x\right)$ is the cumulative PDF, $f\left(x\right)$ is the classical PDF, $x$ is the value of the sample, and $R$ and $L$ are the maximum and minimum values of the sample, respectively.
Table 10. The results of KS test values of the normalized Weibull distributions
Sample

${D}_{n,0.05}$

$k$

Normalized Weibull

NID

Weibull


Actual

1#

0.3100

1.0047

0.1046

0.0683

0.0969

2#

0.2420

1.0030

0.1475

0.0576

0.1495


MC 1#

15

0.3380

1.0038

0.1670

0.1113

0.2607

20

0.2940

1.0006

0.1031

0.0799

0.1476


30

0.2420

1.0065

0.1225

0.0334

0.1388


50

0.1923

1.0002

0.0736

0.0200

0.1127


100

0.1360

1.0005

0.0769

0.0100

0.0894


200

0.0962

1.0031

0.0428

0.0050

0.0417


500

0.0608

1.0027

0.0334

0.0020

0.0487


1000

0.0430

1.0018

0.0287

0.0010

0.0301


MC 2#

15

0.3380

1.0002

0.1579

0.0730

0.3336

20

0.2940

1.0008

0.1409

0.0500

0.1857


30

0.2420

1.0001

0.1091

0.0344

0.1865


50

0.1923

1.0051

0.0722

0.0200

0.1238


100

0.1360

1.0008

0.0494

0.0137

0.0552


200

0.0962

1.0014

0.0432

0.0050

0.0310


500

0.0608

1.0011

0.0292

0.0020

0.0475


1000

0.0430

1.0020

0.0207

0.0010

0.0209

The KS test values of the normalized Weibull distribution for a 95 % confidence level are shown in Table 10. The sequence of the KS test value of actual sample 1# is 0.0683 (NID) < 0.0969 (Weibull) < 0.1046 (normalized Weibull) < 0.3100 (Critical value). The sequence of the KS test value of actual sample 2# is 0.0576 (NID) < 0.1475 (normalized Weibull) < 0.1495 (Weibull) < 0.2420 (Critical value). It can be found that all of the KS test values pass the testing. However, all of the KS test values of the normalized Weibull distribution are much more than those of the NID distribution, which indicates that the fitting ability of NID distribution is better than that of normalized Weibull distribution.
7. Conclusions
To accurately approximate the PDFs for geotechnical parameters, the NID method was introduced; several conclusions of this study are given below.
1) The PDFs of two sets of geotechnical samples were fitted with the NID distribution and Weibull distribution. The results show that, for the KS test results, the chisquare test results and the cumulative probability values, the NID distribution is more accurate than the Weibull distribution. In addition, compared with the PDF curves of the Weibull distribution, those of the NID distribution can overcome the singlepeak feature of the classical distributions and agree more closely with those of the actual distribution.
2) The effect of the sample size on the fitting accuracy for the NID distribution and Weibull distribution was investigated with the MC method, and eight simulated samples were produced. It can be found that with an increase in the sample size, the KS test results of the NID distribution are all lower than those of the Weibull distribution. In addition, its convergence speed and stability are superior to those of the Weibull distribution. The cumulative probability values of the NID distribution are always equal to 1.0000 in the truncated interval and are unaffected by the sample size. However, the cumulative probability values of the Weibull distribution are generally less than 1.0000 and are unstable.
3) The comparison of the fitting accuracy between the NID distribution and the normalized Weibull distribution was also discussed, and the results show that, even if the cumulative probability values are equal to 1 for those two distributions, the fitting accuracy of the NID distribution is still higher than that of normalized Weibull distribution.
Acknowledgements
This research was financially supported by the Natural Science Foundation of China (Grant No. 41102170) and the Fundamental Research Funds for the Central Universities of Central South University (Grant No. 2017zzts536).
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