Dynamic analysis of a modified algae and fish model with aggregation and Allee effect

1.   Introduction

Slope collapses are complicated natural disasters with devastating consequences ^[1]. Every year, such hazards cause significant damage to public and private property, traffic disruptions, and lives lost ^[2,3,4]. As a result, slope stability analyses are essential to prevent and mitigate damages, and better tools for slope assessment are desperately needed in the field of civil engineering. The results of the analysis can be used to identify collapse-prone areas. Based on this data, government agencies can gain a better understanding of slope failure occurrences, and the task of providing financial resources to build retaining structures and developing evacuation plans can be completed more efficiently ^[2].

The accurate prediction of a rock or soil slope's stability is a difficult problem, owing to the slope's dependency on multiple parameters and the difficulty in determining these values ^[5]. The term factor of safety (FoS), is commonly used to describe the stability of slopes. FoS is calculated by dividing the resisting forces to the driving forces. The FoS is more than one when the resisting forces of a slope are greater than the driving forces; when the resisting forces are less than the driving forces, the FoS is less than one and the slope is unstable.

Slope stability analysis and prediction approaches have been the focus of many researchers. These efforts have led to development of a number of different and sophisticated formulations for determining FoS and also slope design approaches such as limit equilibrium methods (LEM) ^[6,7,8,9] continuum mechanics-based numerical techniques ^[10,11,12], methods based on probabilistic methodologies, such as variational and combination methods ^[13] and numerical approaches have been widely used as traditional methods for studying slope stability in geotechnical problems. Due of its computational time, LEM simulations may become inadequate. In recent years, however, a number of studies have been conducted to create a number of computational intelligence systems for slope stability analysis.

Data mining approaches have recently proved successful in paving the way for many promising opportunities in slope stability ^{[14,15,16,17,18]} and other fields of civil engineering ^{[19,20,21,22,23,24,25,26,27,28,29]}. Table 1 lists some representative references for data mining applications for slope stability prediction. The majority of these studies investigated slopes subjected to circular-type failure and stability of these slopes based on geotechnical, geometrical and pore-water pressure parameters. In these studies, data mining approaches based on historical data have been used for two purposes: 1) prediction of slope FoS: the output of these models is the FoS, and 2) prediction of SS status: the output of proposed models shows the slope's stability or instability. However soft computing techniques have proved successful in predicting SS; the fact that most of these techniques are black boxes. The novelty of this article is the development of a transparent and understandable model for predicting SS in slopes that have experienced circular mode failure. The TAN-based model overcomes the shortcomings of other soft computing techniques by producing transparent and structural model showing the relationship between input and output parameters.

A critical review of existing literature suggests that TAN algorithm implementation in the analysis of geotechnical engineering is scarcely explored. Unlike other soft computing technologies, the TAN algorithm can produce a model that is simple to understand and interpret. The main contributions of this paper are as follows: 1) a new TAN model is developed to predict the slope stability subjected to circular slope failures; 2) most probable explanation slope sites of unstable is presented; 3) to test the performance of the models proposed, it is applied to field data given in open source literatures; 4) sensitivity analysis is presented owing to know the most sensitive factor; 5) data discretization was conducted to reduce and elucidate the data set, develop the model quickly and easily, and acquire easily interpretable outputs in this study; 6) The difference in the class ratio between the sample and the population i.e., sampling bias in the training and test datasets is almost negligible.

The structure of the paper is as follows: in Section 2, the materials and methods are presented. The construction process of the proposed prediction model is described in Section 3. Section 4 presents results and discussion. Finally, the concluding remarks are presented.

2.   Methodology

2.1.   Tree Augmented Naive-Bayes (TAN)

The TAN classifier was presented as an extension of the Naive Bayes classifier. TAN allows the independence assumption by permitting arcs between variables. The impact of variable Xi on the class variable is likewise determined by the value of Xj, as indicated by an arc from variable Xi to variable Xj. An example of a TAN is shown in Figure 1. This approach is based on a Chow and Liu algorithm ^[50] that was proposed earlier. The method is divided into the following five steps.

1) Given the class variable C, compute the conditional mutual information. $I\left.\left({X}_{i}; {X}_{j}\left|C\right.\right.\right)$, between each pair of variables, i ≠ j. $I\left.\left({X}_{i}; {X}_{j}\left|C\right.\right.\right)$ is defined as follows:

2) When the value of C is known, this function approximates the information provided by Xj about Xi (and vice versa).

3) Use the variables as nodes in a complete undirected graph. Assign a weight $I\left.\left({X}_{i}; {X}_{j}\left|C\right.\right.\right)$ to each arc connecting X_i and X_j.

4) Create the maximum weighted spanning tree.

5) To convert an undirected tree into a directed tree, select a root variable and determine the direction of all arcs to be outward from it.

6) Connect the classification node C to each Xi using an arc.

The aforementioned technique, produces TANs that optimize the network's log likelihood given the training data and has a time complexity of O(n².N), where n is the number of variables and N is the number of data points ^[51]. With the same computational complexity and robustness, experimental data demonstrated that TANs outperformed Naive Bayes ^[51].

A sufficient threshold probability for classification must be chosen. Because there are only two classes in this study, a 0.5 threshold is typically used ^[52]. For example, "Stable" slopes, are defined as P(Stable|X) > 0.5.

2.2.   Database description

Slope stability study results suggest that six parameters, {γ, c, ϕ, β, H and ru}, influence circular failure of a slope. They are in line with the parameters that are usually found in the literature ^{[18,40,41,46,49]}. Other indicators are theoretically possible, but collecting these would be a significant challenge before they could be used in practice. Slope height (H) and slope angle (β) are geometric properties of a slope that are frequently used to determine slope failure conditions. The slope stability rapidly reduces as the slope height rises. The slope stability decreases as the slope angle increases. Water infiltration reduces the shear strength of the rock and soil owing to softening. Slope stability suffers as a result of all of these changes. The investigations in this research were performed on a dataset encompassing 87 case studies that were investigated for circular critical failure mechanisms and were acquired from various literatures ^[5,31,39,53]. The input parameters are H, γ, c, ϕ, β and ru, whereas the output parameter is SS status. SS in the database was classified as stable or unstable based on whether or not considerable soil movement was observed on the slope surface. If there is no considerable movement of the soil in the slope surface that affects safety, the slope status is considered stable. Otherwise, a slope's status is unstable. 42 of the 87 database cases are stable, while the rest are unstable. The SS is coded as 0 for unstable slopes and 1 for stable slopes in this study.

ANN, ELM, and SVM algorithms were used in much previous research to predict specific FoS values ^[5,41,54,55], the FoS cannot always reflect the actual condition of the slopes, i.e., when the FoS exceeds one, the actual condition of the slopes is sometimes considered "unstable" (see for example Case Nos. 7, 17, 19 in Table A1 shown in Appendix). Therefore, rather than predicting specific FoS values, in the present study, the dataset was used to establish a relation among these six factors as input and the actual slope condition as output.

Figure 2 depicts the cumulative percentage and frequency distributions for all of the input and output parameters of the mentioned database utilized in the modeling of SS of circular mode failure. The data points of every input parameter are distributed over its range. Table 2 shows minimum (Min) and maximum (Max) values, mean, and standard deviation (SD) of all six input parameters. It's worth noting that each parameter's min and max values establish the ranges in which predictions can be made.

2.3.   Correlation analysis

To check the suitability of the TANs applied in this study, correlation coefficients (ρ) are determined to verify the strength of the relationship between the various factors (see Table 3). Given a pair of random variables (m, n) the formula for ρ is:

where cov is the covariance, σm is the standard deviation of m and σn is the standard deviation of n. Values of |ρ| > 0.8 testify a strong correlation between m and n, values between 0.3 and 0.8 a moderate correlation, whereas values of |ρ| < 0.30 testify a weak correlation ^[56]. According to Song et al. ^[57], a correlation is considered "strong" if |ρ| > 0.8. According to Table 3, H, α, c, ϕ, β, γ, and ru are correlated in order of moderate to weakest. So, none of the parameters was deleted for developing the slope stability predicting model. From Table 3, the maximum absolute value of correlation coefficient is found to be 0.639 and no "strong" correlation exists between the different pairs of factors.

2.4.   Data discretization

Data discretization is a method of converting continuous data into discrete data with a set of intervals. Several reasons are there to discretize data, the most significant of which are as follows ^[19]: i) simplifying the dataset; ii) make modeling simple and quick; iii) getting outputs that are simple to understand and iv) only discrete data can be employed in the statistical method. When no prior knowledge from domain experts is available, discretization methods are employed to discretize continuous variables. In the literature, there are various discretization methods. Among them are: equal frequency discretization ^[58], information-preserving discretization ^[59], error-based discretization ^[60], entropy-based discretization ^[61], and the one-rule discretization ^[62]. The six parameters employed are continuous. All six parameters in this study are continuous, and the data was partitioned into intervals with nearly the same number of cases using the equal frequency binning algorithm from the Waikato Environment for Knowledge Analysis (WEKA) software package. Table 4 shows the state intervals as well as their definitions.

2.5.   Model evaluation criteria

To evaluate and compare the proposed model's performance to that of existing models in the literature, a number of measures were used: accuracy (Acc), precision (Prec), recall (Rec), F-score, and Matthews correlation coefficient (Mcc). The percentage of successfully identified samples to the total number of samples is called Acc. The Prec assesses the accuracy of predictions for a particular class (stable or unstable), whereas the Rec measures the accuracy of predictions only taking into account predicted values. The correlation coefficient between predicted and actual is measured by Mcc. The weighted harmonic mean of precision and recall is the F-score. All of the measures used are based on the confusion matrix.

The confusion matrix is shown in Table 5, where true positive (TP) refers the number of correctly predicted stable slopes and true negative (TN) defines the number of correctly predicted unstable slopes. False positive (FP) reflects the number of unstable slopes that were wrongly predicted, whereas false negative (FN) represents the number of stable slopes that were incorrectly predicted. The mathematical equations of the performance metrics are given below respectively.

Figure 3 depicts the overall flowchart of the TAN prediction model development procedure based on the preceding description.

3.   Development of proposed model

The TAN algorithm was employed in this paper to develop a Bayesian belief network as presented in Figure 4. The TAN algorithm is used to create a network with 7 nodes and numerous lines. The lines linking the nodes represent the relationships between the variables, and the nodes represent the variables. Figure 4 depicts the hierarchical interactions of influencing and being affected by others among various slope stability parameters. The interaction of variables like "slope geometry", "geomaterial shear strength", and "water condition" results in slope stability, all of which are fully captured by the Bayesian belief network structure. In a Bayesian belief network, the interactional hierarchical relationship of variables can fully encompass the actual situation of slope stability.

Once a Bayesian belief network topology has been established, parameter learning is carried out to obtain the conditional probability distribution of nodes in Netica. Finally, as illustrated in Figure 5, the Bayesian belief network model for the SS causation analysis can be established.

4.   Results and discussion

4.1.   TAN model's performance

The TAN model in Section 2.1 was built using historical data from 74 slope stability cases (unstable slope instances (38) and stable slope instances (36)). The sampling bias in the training dataset due to the class ratio of 38:36 for the 74 dataset is 1.05. Table 6 illustrates the training performance results, such as Acc, Prec, Rec, F-score, and Mcc. The accuracy of the 36 stable slope instances is 0.889 (called Rec, which is TP / (TP + FN)). The accuracy is 0.816 for the 38 unstable slope instances (called Rec, which is TN / (TN + FP)). The overall Acc is around 85.1 percent, and the Mcc is 0.705, both of which are excellent for practical engineering.

4.2.   Validation of model

In this section, the model's performances are validated using testing dataset that have not been used during the process of model's construction. The significance of validation is to find the capabilities of developed model to be generalized for the conditions that have not been attended during training phase. As mentioned before, the testing dataset consist of 13 slope cases, which are shown in Table 7. It is worthwhile to mention here that the sampling bias in the testing dataset due to the class ratio (i.e., unstable: stable) of 7:6 for the 13 dataset is 1.16. For these 13 cases, the input parameters were fed into developed TAN-based models and the predicted values for SS were obtained. The method of SS prediction for the first case of testing dataset has been schematically indicated in Figure 6. A comparison of predicted and real values of SS for the testing dataset has been given in Table 7. This model has only one unsuccessful prediction case and its overall accuracy is equal to 92.3%. From practical point of view, these results show that the developed TAN classification model is useful and efficient. Finally, to assess the accuracy of developed TAN classification model, it was compared with recently developed soft computing/data mining models in the literature. Table 8 shows the results of this comparison. The confusion matrices were developed using Table 5, and the Prec, Rec, F-score, Mcc, and Acc were determined using Eqs (3)-(7) in Table 8. As can be seen, the prediction accuracy of proposed TAN classification model is as good as those of other reported techniques. The fundamental advantage of the proposed model is that it may be considered as a "white box" that clearly demonstrates the link between input and output parameters. As a result, users (geotechnical engineers) may use these models to analyze and predict slope stability quickly.

4.3.   Causal inference

System fault diagnostics is also another useful application of the Bayesian belief network. The bidirectional reasoning technology of the Bayesian belief network can quantify not only the probability of a system failure under combined fault conditions, as well as the posterior probabilities of different components under the system fault condition, allowing users to quickly ascertain the most likely combination that caused system failure. Computational analysis becomes more intuitive and adaptable as a result of this. Consider the "unstable" state in "slope stability" as an example of causal inference. Because the evidence variable is "unstable" in this example, the status probability is 100%. As seen in Figure 7, using Netica's automated updating feature, the probability of "cohesion" state "low" increases significantly from 33.8 to 40.6% after inputting the data. In addition, the probability of "low" in "internal friction angle" increases from 35.9 to 47.2%, reaching the maximum probability. In the absence of additional evidence, this shows that "low" grades of cohesion and internal friction angle are the most likely cause of "unstable" state in slope stability.

4.4.   Most probable explanation

The TAN model can be used to find the most probable explanations from sets of multiple causes (node states) that are likely to lead to a conclusion; Netica can be used to find the set that is most likely to lead to the result, and the set with the maximum likelihood will be the most probable explanation. Figure 8 depicts the most probable explanation cause (node state) set of "unstable" slope is {pore pressure ratio (ru): wet, slope height (H): low, internal friction angle (ϕ): low, slope angle (β): medium, cohesion (c): medium}.

4.5.   Sensitivity analysis

To examine the impact of each factor on the slope stability, a sensitivity analysis was performed on six input factors. Mutual information between nodes can reveal whether or not they are interconnected and, if so, how close they are ^[64]. According to the sensitivity analysis, a basic event with a reasonably large contribution to the probability of a resulting event makes it easier to reduce the probability of these basic events by taking into account effective measures, thereby lowering the probability of a resulting event. For sensitivity analysis, the target node "slope stability" is selected, and the results are displayed in Table 9. Table 9 shows that node "slope height" has the highest mutual info ( = 0.10294), that implying the greatest impact on "slope stability", followed by "cohesion" and "unit weight", which have mutual info = 0.08706 and 0.06945, respectively.

5.   Conclusions

In this study, TAN model was trained and tested using a circular mode failure slope stability database acquired from the literature to predict slope stability based on the input variables such as γ, c, ϕ, β, H and ru. The following are the major findings of this study:

1) The results obtained from TAN modeling suggest that the TAN model has an appropriate capability to accurate prediction of the SS for circular slip failure. The TAN-based model also gives improved performance than other models (i.e., SVM, RF, NB) proposed in literature.

2) Results of sensitivity conclude that the slope height (mutual info = 0.10294) is the main important parameter when the TAN-based model is selected for prediction of SS for circular mode failure for this dataset.

3) The "most probable explanation" set of "unstable" slope is {unit weight (γ): medium, pore pressure ratio (r_u): wet, slope height (H): low, internal friction angle (ϕ): low, slope angle (β): medium, cohesion (c): medium}. This is quite compatible with engineering judgment and well matched.

Follow-up research will look at the rationality of the TAN model as well as other parameters such as rainwater infiltration that could lead to slope instability, in order to develop a more accurate and comprehensive model. Since the TAN model is a probabilistic model, it requires more detailed and extensive basic data to improve its reliability. Furthermore, because the influencing slope stability parameters in reality are greater than that considered in this study, and as the TAN model is also appropriate for the development of a larger and more complex slope stability analysis model, the model can be expanded to a more sophisticated model that takes into account more parameters such as applied seismic acceleration, depth of rock, soil type, and rainfall characteristics.

Acknowledgments

The work was supported by the National Key Research and Development Plan of China under Grant No. 2021YFB2600703.

Conflict of interests

The authors declare no conflict of interest.

Appendix