Abstract
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.
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 1Sample 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 2Sample 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 3The 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 4The 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 5The 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. 1Comparative 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. 2Comparative CDF curves of the actual sample datasets
a) Sample 1#
b) Sample 2#
Fig. 3Comparative 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 6Partial 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 7The 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 8The 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. 4KS test results of the simulated data with the sample size
a) Sample 1#
b) Sample 2#
Table 9The 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. 5The 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. 6Comparative 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. 7Cumulative 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 10The 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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