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Prediction of Coefficient of Lateral Earth Pressure Using Artificial Neural Networks
Research Scholar, Department of Civil Engineering and
Professor, Department of Civil Engineering |
ABSTRACT
This paper describes the use of artificial neural network (ANN) for prediction of lateral earth pressure ratio (K0) based on dilatometer test results. Feed forward back propagation neural networks have been used and the best model is chosen based on different statistical parameters. The best fit line for predicted K0 (K0p) and observed K0 (K0obs), correlation coefficient, coefficient of determination, and the mean and standard deviation of the ratio K0p/K0obs are used to compare different ANN models . The importance of using different statistical criteria for the evaluations of the ANN models is discussed. Using sensitivity analysis, the parameters influencing the value of K0 are identified.
Keywords: Earth pressure coefficient, K0, ANN
INRODUCTION
Correlations have been a significant part of soil mechanics from its earliest days. In some cases it is essential as it is difficult to measure the amount directly and in other cases it is desirable, to ascertain the results with other tests through correlations. The correlations are generally, semi-empirical based on some mechanics or purely empirical based on statistical analysis. The determination of in situ horizontal stress (or the lateral earth pressure ratio (K0) is an important research in geotechnical engineering. The in situ horizontal stress is required in finding, the skin friction of pile, pressure on walls, fracturing of dams, evaluating borehole stability and interpretation of results of other in situ tests. There have been several empirical correlations to find the K0 based on laboratory and in situ tests (Bowles, 1988). However, the uncertainty and complex nature of soil necessitates the use of in-situ tests, for reliable prediction of K0 value. The most widely used in-situ test for field investigation test is standard penetration test (SPT), followed by cone penetration test (CPT). Though CPT may be used to find out the K0 for sandy soil, it is difficult to draw the horizontal stress information for clay. The dilatometer test (DMT) is more reliable and mostly used for determination of in-situ horizontal stress. The K0 value from DMT is determined using some statistical empirical correlations with its primary readings (information from the DMT).
Soft computing technique, artificial neural network (ANN) is now being used as alternate statistical tool. Studies dealing with various engineering applications indicate that the ANN models are not significantly different from a number of statistical models. The statistician’s primary objective is to develop universal methodology under some strict statistical rules and guide lines. In contrast, neural network practitioners are primarily concerned about prediction accuracy and finding methods that work. In general, the problems in real field dealt by engineers are more complex, the dimensionality of the models tends to be much higher. Due to its convenience in use and versatile nature in approximately complex relationship, the use of ANN is phenomenal for different geotechnical problems. Toll (1996) and Shahin et al. (2001) have presented the state-of the art report on the different applications (liquefaction prediction, soil classification, compaction, pile capacity, settlement analysis etc.) of ANN in geotechnical engineering. ANNs have been also used for site characterization, based on SPT (Itani and Najjar, 2000; Das and Basudhar, 2004) and CPT ( Juang et al., 2001) results.
With the above in view, feed forward back propagation neural network (BPNN) has been used here to predict the K0 based on DMT results and other index properties of soil. This in turn, to explore the potential of ANNs in predicting another important property of soil and compare with the results obtained form available statistical methods.
Artificial Neural Networks
ANNs developed by engineers and computer scientists, have their roots in artificial intelligence (AI). As per the architectural difference, the ANNs can be classified as back propagation neural network (BPNN), categorical learning (self-organizing map (SOM)) networks and probabilistic neural networks (PBNN) (Hagan et al., 2002). The BPNN is most widely used as a universal approximator. The network geometry determines the number of connection weights and how these are arranged. This is generally done by fixing the number of hidden layers and choosing the number of nodes in each of these layers. Fig. 1 shows the typical architecture of a single neuron, in which the Oi refers to the input to the neurons and WiB refers to the weight of the corresponding neuron. The details of mathematically modeling of ANN are available in Hagan et al. (2002).
Figure 1. Typical architecture of a back propagation neural network
The statistical approaches are model driven, where the data points are used to find out the model parameters only. In contrast, the ANNs belong to data driven approach i.e. the input and output data decided the type of model and the model parameters suitable for that particular problem. The data driven approaches have the ability to determine which model inputs are critical. Thus, in ANNs the inputs are provided in different combination to find out the best ANN model for prediction. It has been found in statistics that the input in isolation may not contribute to the model but may be an important parameter in combination with other parameter (Guyon and Elisseeff 2003). So in the present study the inputs are provided in different combinations and accordingly the ANN models are designated. Some of the models used in the present study are shown in Table 1.
To study the generalization in applicability of the neural network models, it is common practice to divide the data into two sub-sets: a training set and an independent testing set. However, depending upon the number of data points a set of data may be used as validation set to avoid overfitting (Shahin et al., 2002). In the present study with limited data, it is divided into training and testing set. In geotechnical engineering mostly, the data have been randomly divided into two subsets. However, Shahin et al.(2002) have divided the data points such that the statistical parameters like mean, standard deviation, maximum and minimum value of the input parameters are consistent for the three subsets. Shahin et al. (2004) found that division of data based on SOM clustering and fuzzy clustering have advantages over the random division of data points. In the present study the data is divided into training and testing subsets using fuzzy clustering following Shahin et al. (2004).
The preprocessing/normalization helps in avoiding the dimensional dissimilarities of different input parameters. The variables have to be scaled in such a way as to be commensurate with the limits of the transfer function used in the output layer. In the present study data need to scale in the range [-1, 1] as hyperbolic tangent sigmoid function is used as transfer function.
RESULTS AND DISCUSSION
In the present study data base available in Lunne et al. (1990) have been considered. The data base consists of index properties of soil (plastic limit (PL or wp), plasticity index (PI or Ip), clay content, natural moisture content (w), and unit weight (g)), strength data along with the data from DMT (KD, ID, ED). Based on the above data Lunne et al. (1990) have proposed the correlation of K0 with KD as shown Eq. 1. They observed that the value of the coefficient “a” depends upon the geological age of clays “young” or “old” which was reflected in the ratio su/sv0. The value of K0 may also depend upon ID and IP. The value of “a” for “young” clays is 0.34 and that for “old” clays it is 0.68.
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(1) |
In the present study, out of 36 data points, 25 were used for training and 11 data points were used for testing. So keeping in mind the findings of Lunne et al. (1990) different models are tried and the models with better performances are presented here. In the present study, MATLAB subroutines are used for the implementation of ANN models. The different statistical performances criteria considered for evaluation of the models are correlation coefficient (R) between predicted and observed, coefficient of determination (R2), and the mean (m) and standard deviation (s) of the ratio, predicted K0 to the observed K0 (K0p/K0obs).
Table 1 shows the models with different input parameters and their statistical performances. Except Model B, the close value of R during training and testing show the generalization of the models. It can be also pointed out that the ANN models are valid for both “young” and “old” clays. Figs. 2-5 show the correlation between the predicted and observed values of the K0 for Model A, B, C and D respectively.
The performances of the ANN models were found to depend upon the input parameters. Comparing different models in terms of coefficient of determination (R2), it can be observed that, Model D which has very high value of R during training and testing shows poor value of R2 during testing. This may be due to the fact that correlation coefficient may be biased towards high observation values. So the models should be evaluated using other statistical performances along with correlation coefficient. Based on the performance during testing, Model A found to be the better models. The poor performance of Model D (with six input parameters) during testing may be due to overfitting of the model as the number of training data is limited.
Table 1. Different models and the corresponding coefficient of correlation (R)
and coefficient of determination (R2)
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| Model | Input Parameters | R (Training) |
R (Testing) |
R2 (Training) |
R2 (Testing) |
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| Model A | ID, IP, KD and su/sv0 | 0.986 | 0.979 | 0.971 | 0.919 |
| Model B | IP, KD and su/sv0 | 0.969 | 0.812 | 0.938 | 0.472 |
| Model C | ID, KD and su/sv0 | 0.973 | 0.919 | 0.946 | 0.795 |
| Model D | PL, w, ID, IP, KD and su/sv0 | 0.982 | 0.975 | 0.998 | 0.815 |
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(a)
(b)
Figure 2. The correlation between predicted and the observed values of K0
during (a) training and (b) testing for Model A
(a)
(b)
Figure 3. The correlation between predicted and the observed values of K0
during (a) training and (b) testing for Model B
(a)
(b)
Figure 4. The correlation between predicted and the observed values of K0
during (a) training and (b) testing for Model C
(a)
(b)
Figure 5. The correlation between predicted and the observed values of K0
during (a) training and (b) testing for Model D
The R and R2 value of the models reflect the overall error performances of the model. It is also necessary to evaluate the models in terms of its predictability capability, in terms of under prediction or over prediction. Such a study also presented here in terms of finding out the mean (m) and standard deviation (s) of the ratio K0p/K0obs. The m value greater than 1.0 indicates over prediction and less than 1.0 indicate under prediction of the model. The best model is represented by K0 value close to 1.0 and close to 0. Table 2 presents the K0obs and K0p values for the above ANN models. The Model D can be considered as the best model with training data set, but poor prediction for testing data set. Considering Model C it can be seen that the performances of this based on s is the worst, though it has good R2 value. This shows the importance of comparing the models based on their predictability capabilities.
Table 2. Predictability performances of different models
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| Model | Input Parameter | m (Training) | m (Testing) | s (Training) | s (Testing) |
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| Model A | ID, IP, KD and su/sv0 | 0.989 | 1.144 | 0.17 | 0.159 |
| Model B | IP, KD and su/sv0 | 1.056 | 1.2 | 0.198 | 0.505 |
| Model C | ID,KD and su/sv0 | 1.013 | 1.46 | 0.143 | 0.700 |
| Model D | PL, w, ID, IP, KD and su/sv0 | 1.006 | 0.931 | 0.064 | 0.252 |
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Sensitivity analysis
The sensitivity analysis for the model studied was performed as per Garson’s method (Garson, 1991) to find out important input parameters (Table 3). It can be seen that starting with different initial weights the parameters gets changed in the order of importance. Such, observation has been observed by Shahin et al. (2002). But in both the case KD found to be the most important parameter, so most of the statistical correlation is based on KD values. The other parameters arranged in order of importance are ID, su/sv0 and Ip.
Table 3 Relative Importance of different inputs
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| Result | ID | IP | KD | su/sv0 |
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| Relative importance1 | 26.3% | 21.8% | 30.7% | 21.2% |
| Relative importance2 | 24.31% | 22.3% | 28.5% | 24.85% |
| Av. Relative importance | 25.3% | 22.05% | 29.6% | 23.02% |
1, 2 for different initial weights | ||||
CONCLUSIONS
The prediction of coefficient of lateral earth pressure based on DMT results, using ANN models was presented with high value of correlation coefficient for both training and testing data set. The performances of the models were governed by the input parameters. It was observed that models should be evaluated based on different statistical parameters. A high predictability performance of the chosen ANN model was also observed. Based on sensitivity analysis, KD found to be the most important parameter for determination of K0 value.
REFERENCES
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