Saturated coral sand damping ratio prediction method based on interpretable artificial intelligence

Abstract

The invention discloses a saturated coral sand damping ratio prediction method based on interpretable artificial intelligence. The method comprises the following steps: preparing a saturated coral sand sample; a saturated coral sand sample is used for testing, and a data set is generated; establishing a saturated coral sand damping ratio prediction model by using a CNN-BiLSTM combined neural network; inputting the data set into a saturated coral sand damping ratio prediction model for training; and performing explanatory analysis on the saturated coral sand damping ratio prediction model by using an SHAP method. According to the method, saturated coral sand is used for testing, the change of the damping ratio of the saturated coral sand under different material parameters is recorded, a preliminary conclusion among all influence factors is obtained, then a CNN-BiLSTM mixed deep learning model is established, feature information is extracted and predicted, and the accuracy of the prediction result is improved. Finally, the SHAP is used for analyzing the relation between the input parameters and the output values of all the characteristics, and the application of artificial intelligence in the coral sand dynamic deformation characteristic field is expanded.

Description

Technical Field

[0001] The invention relates to the field of coral sand performance research and digital modeling, and in particular to a saturated coral sand damping ratio prediction method based on explainable artificial intelligence. Background Art

[0002] Coral sand is a carbonate sediment that has undergone long-term biological accumulation and geological action. Its main components are the skeletal remains of marine organisms such as corals, minerals such as high-magnesium calcite and aragonite, and contain more than 90% calcium carbonate. The special formation conditions of coral sand make it have the characteristics of irregular particles, a large number of pores within the particles, a rough surface and easy to break. The particle characteristics of coral sand make it easy to liquefy when subjected to cyclic loads related to earthquakes, causing engineering buildings to tilt and collapse. So far, in the evaluation of earthquake impact and the seismic fortification of engineering structures, the engineering site conditions are usually regarded as the most important factors affecting the seismic effects, and the dynamic characteristics of the site soil are in the primary position among the site conditions. Therefore, the study of the dynamic characteristics of coral sand has become the focus of scientific researchers.

[0003] The dynamic characteristics of soil are mainly reflected in nonlinearity, hysteresis and strain accumulation. In order to more accurately reflect the energy dissipation and viscosity performance indicators of soil under dynamic loads, the damping ratio λ is considered to be the most basic dynamic parameter to characterize the nonlinearity, hysteresis and other viscoelastic-plastic characteristics of soil in various dynamic strain ranges. In order to study the influencing factors of the damping ratio, scholars have carried out a large number of experimental studies. Sands of different types and regions have different dynamic characteristics. Most of the current experimental studies are aimed at quartz sand, and there are relatively few studies on coral sand. The empirical relationship of quartz sand is not applicable to coral sand. Therefore, in order to ensure the safety of island and reef buildings, it is necessary to discuss and study the dynamic characteristics of coral sand by distinguishing and comparing it with quartz sand.

[0004] The resonant column test and the dynamic triaxial test are currently the main methods for measuring the damping ratio. However, the two tests have different measurement methods and the magnitude of the measured damping ratio is different. The damping ratio of a single test cannot fully reflect the mechanical properties of the soil. The commonly used method is to complement the results of the resonant column and dynamic triaxial tests. However, a combination of multiple tests will generate a large amount of data, and the test data will also differ due to different test methods, which makes scientific research complicated and cumbersome. At the same time, the dynamic characteristics of the soil are not only affected by the shear strain factor, but also closely related to other factors such as the initial effective confining pressure of the soil sample.

[0005] At present, a large number of studies are applicable to specific objects, and the application conditions of the damping ratio prediction mathematical formula are single and multi-parameter dependent. There is no ideal mathematical formula that can comprehensively reflect the nonlinearity of the original soil and the variability of the sedimentation depth. Therefore, how to establish an optimization model that can quickly and effectively analyze a large amount of data and predict model data is a technical problem that needs to be solved urgently. Summary of the invention

[0006] In order to address the shortcomings of the prior art, the present invention provides a method for predicting the damping ratio of saturated coral sand based on explainable artificial intelligence, proposes an explainable deep learning model for accurate prediction of the damping ratio of saturated coral sand, and verifies the reliability and accuracy of the model through additional experiments.

[0007] The embodiment of the present invention provides the following solution:

[0008] The embodiment of the present invention provides a method for predicting the damping ratio of saturated coral sand based on explainable artificial intelligence, the method comprising:

[0009] Step 1, preparing saturated coral sand sample;

[0010] Step 2: Use saturated coral sand samples to conduct experiments and generate a data set. The influencing factors of each sample in the data set include relative density, confining pressure, fine particle content and strain, and the influencing target is damping ratio;

[0011] Step 3: Use CNN-BiLSTM (Convolutional Neural Networks-Bidirectional Long Short Term Memory) combined neural network to establish a saturated coral sand damping ratio prediction model;

[0012] Step 4: using the data set to input the saturated coral sand damping ratio prediction model for training, optimizing the parameters of the saturated coral sand damping ratio prediction model, and obtaining an optimized saturated coral sand damping ratio prediction model;

[0013] Step 5: Use the SHAP (SHapley Additive exPlanation, global interpretative analysis) method to conduct an interpretative analysis on the optimized saturated coral sand damping ratio prediction model.

[0014] In an optional embodiment, the preparation of a saturated coral sand sample in step one includes loading the coral sand, saturating the loaded sample by a combination of CO2 replacement, introduction of air-free water and graded back-pressure saturation, and when the pore pressure coefficient B ≥ 0.97, subjecting the sample to stepwise isobaric consolidation treatment to obtain a saturated coral sand sample.

[0015] In an optional embodiment, the test using the saturated coral sand sample described in step 2 includes the following process: using an excitation device to perform a resonant column test on the saturated coral sand sample to obtain damping ratios corresponding to different relative densities, different confining pressures, different fine particle contents and different strains.

[0016] In an optional embodiment, the saturated coral sand damping ratio prediction model includes, from input to output, an input layer, a CNN (Convolutional Neural Networks) layer, a BiLSTM (Bidirectional Long Short Term Memory) layer, a fully connected layer and an output layer.

[0017] In an optional embodiment, the CNN layer includes 1 convolution layer and 1 pooling layer; the BiLSTM layer includes 1 bidirectional LSTM layer.

[0018] In an optional embodiment, the processing flow of the saturated coral sand damping ratio prediction model is as follows:

[0019] S3.1, taking the influencing factors in the data set as input variables, performing convolution and pooling processing in sequence through the CNN layer to extract feature values;

[0020] S3.2, adjust the dimension of the extracted feature values ​​and input them into the BiLSTM layer to capture the dependency between different input variables and extract deep features;

[0021] S3.3. Map the extracted deep features to the output space through the fully connected layer to generate prediction results.

[0022] In an optional embodiment, in the data set, 80% of the samples are used as a training data set, and the remaining 20% ​​of the samples are used as a testing data set.

[0023] In an optional embodiment, the explanatory analysis described in step five includes global explanatory analysis, feature interaction analysis, and interaction analysis between input variables.

[0024] The beneficial effects of the present invention based on its technical solution are:

[0025] (1) The present invention firstly uses saturated coral sand to conduct experiments, records the changes in the damping ratio of saturated coral sand under different material parameters, and obtains preliminary conclusions between various influencing factors. Then, a CNN-BiLSTM hybrid deep learning model is established to extract feature information and make predictions. Finally, SHAP is used to analyze the relationship between the input parameters and output values ​​of each feature, thus expanding the application of artificial intelligence in the field of dynamic deformation characteristics of coral sand.

[0026] (2) The CNN-BiLSTM model established in this paper can accurately predict the damping ratio of saturated coral sand. The R 2 The accuracy of the model is 0.975 and 0.959 respectively, which is higher than that of single CNN (Convolutional Neural Networks), LSTM (Long Short Term Memory), BiLSTM (Bidirectional Long Short Term Memory) and CNN-LSTM (Convolutional Neural Networks-Long Short Term Memory). The model combines the feature extraction capability of the CNN model with the precise prediction capability and strong robustness of the prediction results of the BiLSTM model, and is superior in predicting the damping ratio of saturated coral sand.

[0027] (3) The present invention uses SHAP analysis to solve the black box problem of deep learning models and derive the influence of each feature on the output variable. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] In order to more clearly illustrate the embodiments of this specification or the technical solutions in the prior art, the drawings required for use in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of this specification. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0029] Figure 1 Schematic diagram of the process of predicting the damping ratio of saturated coral sand based on explainable artificial intelligence.

[0030] Figure 2 Schematic diagram of the Pearson correlation matrix of sample space distribution.

[0031] Figure 3 Schematic diagram of the damping ratio prediction model for saturated coral sand.

[0032] Figure 4 is the prediction result of the training sample.

[0033] Figure 5 is the prediction result of the test sample.

[0034] Figure 6 It is the regression distribution diagram of predicted value and true value.

[0035] Figure 7 It is a global SHAP value swarm and feature importance graph.

[0036] Figure 8 SHAP interaction value graph and heat map.

[0037] Fig. 9 Schematic diagram of the mutual influence relationship between different characteristic parameters.

[0038] Fig.10 Schematic diagram for comparing different prediction models. DETAILED DESCRIPTION

[0039] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by ordinary technicians in this field based on the embodiments of the present invention belong to the scope of protection of the embodiments of the present invention.

[0040] This embodiment provides a method for predicting the damping ratio of saturated coral sand based on explainable artificial intelligence. Figure 1 , the method comprising:

[0041] Step 1. Prepare a saturated coral sand sample through the following process: sample the coral sand, and saturate the sample by combining CO2 replacement, introduction of air-free water and graded back pressure saturation. When the pore pressure coefficient B≥0.97, perform step-by-step isobaric consolidation on the sample to obtain a saturated coral sand sample.

[0042] Step 2: Use saturated coral sand samples to conduct experiments, that is, use a vibration device to conduct a resonant column test on the saturated coral sand samples to obtain the damping ratios corresponding to different relative densities, different confining pressures, different fine particle contents, and different strains. Generate a data set, in which the influencing factors of each sample include relative density, confining pressure, fine particle content, and strain, and the influencing target is the damping ratio.

[0043] In this embodiment, a total of 659 sets of data were collected, and there were 4 influencing factors, i.e., input variables, which were relative density (kg / m 3 ), confining pressure (kPa), fine particle content (%) and strain are shown in the following table:

[0044]

[0045] The Pearson correlation coefficient matrix of the sample space distribution used for CNN-BiLSTM model training is as follows Figure 2As shown in the figure, the highest correlation coefficient of the input variables is only -0.13, indicating that the correlation between any two variables is very low, thus verifying the high independence of each variable. This result means that the constructed database of saturated coral sand design parameters has good feature independence, providing a reasonable and reliable data basis for the development and training of artificial intelligence models.

[0046] Step 3: Use the CNN-BiLSTM combined neural network to establish a saturated coral sand damping ratio prediction model.

[0047] Reference Figure 3 The saturated coral sand damping ratio prediction model includes an input layer, a CNN layer, a BiLSTM layer, a fully connected layer, and an output layer from input to output. The convolution layer contains 64 filters, the size of the convolution kernel is 3, the pooling size of the pooling layer is 2, and the bidirectional LSTM layer is set to 256 neurons. The number of training rounds of the training model is set to 170 rounds. The structure of the CNN-BiLSTM prediction model is as follows Fig.10 As shown in the figure. t is the input value, h t and h t is the output vector of the bidirectional LSTM, σ is the vector obtained by concatenating the output vectors, and y t It is the value formed by mapping the concatenated vector through the Softmax Layer.

[0048] The processing flow of the saturated coral sand damping ratio prediction model is as follows:

[0049] S3.1, taking the influencing factors in the data set as input variables, performing convolution and pooling processing in sequence through the CNN layer to extract feature values;

[0050] S3.2, adjust the dimension of the extracted feature values ​​and input them into the BiLSTM layer to capture the dependency between different input variables and extract deep features;

[0051] S3.3. Map the extracted deep features to the output space through the fully connected layer to generate prediction results.

[0052] Step 4: Use the data set to input the saturated coral sand damping ratio prediction model for training. During the model training process, 80% of the samples are set as the training set, and the remaining 20% ​​of the samples are used as the test set. The parameters of the saturated coral sand damping ratio prediction model are optimized to obtain the optimized saturated coral sand damping ratio prediction model. The prediction results of the training samples and the test samples are shown in Figure 2. Figure 4 and Figure 5 The results are shown in the following table:

[0053]

[0054] The regression distribution diagram of the predicted value and the true value is as follows Figure 6 As shown in the figure, most of the data points in the training set and the test set fall within the 95% confidence interval, which further proves that the prediction results are accurate and reliable.

[0055] Step 5: Use the SHAP method to conduct an explanatory analysis on the optimized saturated coral sand damping ratio prediction model, including global explanatory analysis, feature interaction analysis, and interaction analysis between input variables.

[0056] (1) Global explanatory analysis

[0057] Figure 7 SHAP summary plot of all features related to the damping ratio of saturated coral sand and feature importance bar chart to visualize the global feature importance. Figure 7 Each point in the figure represents a data sample. The redder the color, the higher the value of the characteristic parameter, and the bluer the color, the lower the value. The positive and negative signs of the SHAP value on the horizontal axis respectively indicate the positive or negative impact of the feature on the output result, while the absolute value of the SHAP value reflects the strength of its impact. The larger the absolute value, the more significant the impact of the feature on the model output. The bar chart represents the importance of the characteristic parameter's contribution to the damping ratio of saturated coral sand, and the importance is arranged in descending order from top to bottom.

[0058] Depend on Figure 7 It can be seen that strain is the most important parameter affecting the damping ratio. As strain increases, its corresponding positive SHAP value also increases accordingly, indicating that strain has a significant positive contribution to the damping ratio. The SHAP value gradually turns to a positive value with the increase of relative density, indicating that relative density also contributes positively to the damping ratio. The SHAP value of fine particle content tends to be more negative as the parameter value increases, indicating that there is a negative correlation between fine particle content and damping ratio. The SHAP value first turns to a negative value and then turns to a positive value as the confining pressure increases.

[0059] (2) Feature interaction analysis

[0060] Feature interaction analysis can consider the interaction between features. Figure 8 As shown in the SHAP interaction value diagram on the left, the diagonal line is the main effect of the feature, which shows the independent contribution of the feature to the model prediction value; the off-diagonal line is the pairwise interaction between the feature and other features, which shows the contribution excluding the independent effect, and purely reflects the gain or loss of synergy. In each independent small figure, each point represents a data sample, the horizontal axis is the SHAP value, and the color of the point represents the value of the feature value parameter on the vertical line. The redder the color, the higher the value, and the bluer the color, the lower the value. In order to evaluate the impact of each feature on the model prediction of a single sample, and at the same time view the model prediction response to a single feature, we further draw Figure 8The heat map on the right is used to visualize these relationships. The horizontal axis of the heat map is each instance, and the left y-axis is the feature importance ranking, which decreases from top to bottom, corresponding to the rectangular graph on the right. The longer the rectangular graph, the higher the importance. The right y-axis is the visualization of the SHAP value, red represents positive values, blue represents negative values, and the darker the color, the larger the absolute value of SHAP. The top curve is the prediction result of the model under the corresponding feature parameters. The heat map can intuitively show the impact of features on model predictions and enhance the interpretability of the prediction model.

[0061] Depend on Figure 8 It can be seen that the main effects of strain and relative density on the damping ratio are positive. Under the individual effect, the strain and relative density increase, and the damping ratio increases. Under the interaction, the strain and relative density increase, the SHAP value becomes positive, and the damping ratio of the saturated coral sand increases accordingly. Through the experimental data of step 2, it is known that there is a negative correlation between the fine particle content and the damping ratio. The SHAP value first becomes negative and then becomes positive as the confining pressure increases. Figure 8 The sub-graph on the left does not show obvious changes in the SHAP values ​​of these two features, and further analysis is needed. Figure 8 The heatmap on the right shows that high predictions (high values ​​in f(x) on the left) are associated with high relative density and high strain (red). Low predictions (low values ​​in f(x) on the left) are associated with low strain and low relative density (blue). The SHAP interaction value plot is consistent with the results shown in the SHAP heatmap.

[0062] (3) Interaction analysis between input variables

[0063] In order to understand the influence of characteristic parameters on the damping ratio of saturated coral sand in more detail, the dependence diagram between each characteristic parameter should be drawn, such as Fig. 9 As shown. The SHAP dependency graph can show the effect of the interaction between features on the SHAP value in more detail, and can be used as a supplement to the SHAP interaction value graph. The x-axis of the dependency graph represents the feature parameter, the y-axis on the left represents the SHAP value corresponding to the feature on the x-axis, and the y-axis on the right is the parameter of another feature. The redder the color, the higher the value, and the bluer the color, the lower the value. Fig. 9 The mutual influence relationship between different features can be further analyzed. Fig. 9 (a) and Fig. 9 (d) shows that with the increase of relative density and strain, the corresponding SHAP value also increases. When the relative density is 70kg / m3, that is, under high relative density conditions, the increase of strain will increase the SHAP value of relative density. From 9(b), when the confining pressure is 300KPa, with the increase of fine particle content, the SHAP value of confining pressure increases, indicating that when the confining pressure is relatively high, the increase of fine particle content parameters will increase the damping ratio of saturated coral sand. Fig. 9(c) It can be seen that when the fine particle content is high, the increase in strain will make the SHAP value smaller and reduce the damping ratio of the saturated coral sand.

[0064] It is known from the experimental data of step 2 that strain and relative density have the greatest influence on the damping ratio of saturated coral sand, which is significantly greater than the influence of confining pressure and relative density on confining pressure, and there is a negative correlation between fine particle content and damping ratio. Therefore, combined with the analysis of the dependency graph, it can be summarized as follows: setting the confining pressure to 100KP, appropriately increasing the relative density and strain values, and reducing the fine particle content can effectively improve the damping ratio of saturated coral sand.

[0065] In order to further verify the effectiveness of the CNN-BiLSTM prediction model used in the invention, the CNN, LSTM, BiLSTM, and CNN-LSTM models were used to predict the damping ratio, and the prediction results of these models were compared with the prediction results of the CNN-BiLSTM model. The following table summarizes the prediction performance of each model on the training set and test set.

[0066]

[0067] In addition, in order to further evaluate the prediction performance of the CNN-BiLSTM model, the determination coefficient (R 2 ), root mean square error (RMSE) and mean absolute error (MAE), and their distribution is shown in the Taylor diagram, as shown in Fig.10 shown.

[0068] From the above comparison, we can see that when testing the CNN-BiLSTM prediction model, the R2 value of the training set reached 0.975, and the R2 value of the test set 2 The value reaches 0.959, and the prediction accuracy is higher than that of the other four models, with the highest accuracy. The accuracy of CNN-LSTM is higher than that of CNN and LSTM because the ability of CNN to extract features is combined with the special recursive neural network LSTM, which complements each other and improves the model's ability to predict features. The accuracy of CNN-BiLSTM is higher than that of CNN-LSTM because BiLSTM is composed of a bidirectional LSTM layer. Compared with LSTM, BiLSTM can use contextual information to improve the robustness and generalization ability of the model. In short, the CNN-BiLSTM model adopted by the present invention has better performance in data prediction tasks, and the accuracy is significantly improved compared to traditional neural network models.

[0069] Those skilled in the art will appreciate that embodiments of the present invention may be provided as methods, systems, or computer program products. Therefore, the present invention may take the form of a complete hardware embodiment, a complete software embodiment, or an embodiment combining software and hardware. Moreover, the present invention may take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0070] The present invention is described with reference to flowcharts and / or block diagrams of methods, devices (modules, systems), and computer program products according to embodiments of the present invention. It should be understood that each process and / or block in the flowchart and / or block diagram, as well as the combination of processes and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, an embedded computer, or other programmable data processing device to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing device generate instructions for implementing the processes in the flowchart and / or block diagram. Figure 1 A process or multiple processes and / or boxes Figure 1 A device that provides the functions specified in a block or multiple blocks.

[0071] These computer program instructions may also be stored in a computer-readable memory capable of directing a computer or other programmable data processing device to operate in a specific manner, so that the instructions stored in the computer-readable memory produce an article of manufacture comprising an instruction device, which implements the process Figure 1 A process or multiple processes and / or boxes Figure 1 A function specified in one or more boxes.

[0072] These computer program instructions can also be loaded onto a computer or other programmable data processing device so that a series of operating steps are executed on the computer or other programmable device to produce a computer-implemented process, thereby providing instructions for implementing the process. Figure 1 A process or multiple processes and / or boxes Figure 1 The steps for the functions specified in one or more boxes.

[0073] Although the preferred embodiments of the present invention have been described, those skilled in the art may make other changes and modifications to these embodiments once they have learned the basic creative concept. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present invention.

[0074] Obviously, those skilled in the art can make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include these modifications and variations.

Claims

1. A method for predicting damping ratio of saturated coral sand based on explainable artificial intelligence, characterized in that: The method comprises: Step 1, preparing saturated coral sand sample; Step 2: Use saturated coral sand samples to conduct experiments and generate a data set. The influencing factors of each sample in the data set include relative density, confining pressure, fine particle content and strain, and the influencing target is damping ratio; Step 3: Use the CNN-BiLSTM combined neural network to establish a saturated coral sand damping ratio prediction model; Step 4: using the data set to input the saturated coral sand damping ratio prediction model for training, optimizing the parameters of the saturated coral sand damping ratio prediction model, and obtaining an optimized saturated coral sand damping ratio prediction model; Step 5: Use the SHAP method to conduct an explanatory analysis on the optimized saturated coral sand damping ratio prediction model.

2. The method for predicting damping ratio of saturated coral sand based on explainable artificial intelligence according to claim 1, characterized in that: The preparation of the saturated coral sand sample described in step 1 includes loading the coral sand, saturating the loaded sample by a method of combined CO2 replacement, introduction of air-free water and graded back pressure saturation, and when the pore pressure coefficient B≥0.97, subjecting the sample to stepwise isobaric consolidation treatment to obtain a saturated coral sand sample.

3. The method for predicting damping ratio of saturated coral sand based on explainable artificial intelligence according to claim 1, characterized in that: The test using the saturated coral sand sample described in step 2 includes the following process: using a vibration device to perform a resonant column test on the saturated coral sand sample to obtain damping ratios corresponding to different relative densities, different confining pressures, different fine particle contents and different strains.

4. The method for predicting damping ratio of saturated coral sand based on explainable artificial intelligence according to claim 1, characterized in that: The saturated coral sand damping ratio prediction model includes an input layer, a CNN layer, a BiLSTM layer, a fully connected layer and an output layer from input to output.

5. The method for predicting damping ratio of saturated coral sand based on explainable artificial intelligence according to claim 4, characterized in that: The CNN layer includes 1 convolution layer and 1 pooling layer; the BiLSTM layer includes 1 bidirectional LSTM layer.

6. The method for predicting damping ratio of saturated coral sand based on explainable artificial intelligence according to claim 4, characterized in that: The processing flow of the saturated coral sand damping ratio prediction model is as follows: S3.1, taking the influencing factors in the data set as input variables, performing convolution and pooling processing in sequence through the CNN layer to extract feature values; S3.2, adjust the dimension of the extracted feature values ​​and input them into the BiLSTM layer to capture the dependency between different input variables and extract deep features; S3.

3. Map the extracted deep features to the output space through the fully connected layer to generate prediction results.

7. The method for predicting damping ratio of saturated coral sand based on explainable artificial intelligence according to claim 1, characterized in that: In the data set, 80% of the samples are used as training data sets, and the remaining 20% ​​of the samples are used as testing data sets.

8. The method for predicting damping ratio of saturated coral sand based on explainable artificial intelligence according to claim 1, characterized in that: The explanatory analysis described in step five includes global explanatory analysis, feature interaction analysis, and interaction analysis between input variables.

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