A method and system for constructing a knowledge network based on the precision matrix of Gaussian distribution

Abstract

The present invention provides a method for constructing a knowledge network based on the precision matrix of Gaussian distribution, comprising the following steps: Step 1: Collect numerical behavior data, perform data preprocessing on the collected data, and divide it into a training set, a validation set, and a test set; wherein, the data preprocessing includes standardization; Step 2: Use the training set, validation set, and test set processed in Step 1 to train the precision matrix model, and save the trained precision matrix result; Step 3: Use the precision matrix trained in Step 2 to construct a knowledge network, correspond the elements in the precision matrix to the edges in the knowledge network one by one, and use the networkx drawing tool to draw a knowledge association network diagram; Step 4: Use the information of the knowledge network in Step 3 to construct a regression predictor, take the behavior data of the students on the previous part of the knowledge points as input, and output the predicted behavior data of the students on the next part of the knowledge points.

Description

Technical Field

[0001] The present invention belongs to the technical field of data mining, and particularly relates to a method and system for constructing a knowledge network based on a precision matrix. Background Art

[0002] With the progress of modern society, the combination of pedagogy and artificial intelligence has become increasingly close. However, existing curriculum education has various defects, such as neglecting the overall structural characteristics and lacking scientific quantitative analysis methods. As a visualization method, the knowledge network solves many existing problems in curriculum education. It is a network structure formed by curriculum knowledge nodes and forms a knowledge system through internal associations. From the relevant research on the curriculum knowledge network, most application scenarios are in the teaching field, mainly presenting the curriculum knowledge of a certain subject in the form of a network diagram, so as to facilitate teachers to understand the overall structure of the curriculum knowledge and improve students' cognition of the subject at the same time. However, there is little research content on the junior high school mathematics knowledge network, only a few papers, which are constructed by using the method of knowledge graph.

[0003] Therefore, combining the curriculum knowledge network with the content of junior high school mathematics is of great significance to the curriculum teaching of junior high school mathematics, which not only improves the teaching efficiency of junior high school mathematics discipline but also promotes the development and application of the mathematics discipline field. However, from the current research status of the curriculum knowledge network, there are many deficiencies in the research of the curriculum knowledge network.

[0004] 1) The text analysis method neglects the logical relationship between contents

[0005] At present, most of the construction of the curriculum knowledge network is based on the text analysis method. A typical method is the knowledge graph method. It is mainly based on the co-occurrence of knowledge concepts or keywords. If two concepts or keywords appear in the same paragraph or the same sentence at the same time, it is considered that these two concepts have an association relationship. The curriculum knowledge network established based on this concept or keyword constructs the knowledge network in a shallow and abstract way, neglecting the various logical relationships between knowledge contents, and does not conform to the characteristics of knowledge contents in the actual curriculum.

[0006] 2) Not applicable to numerical behavior data

[0007] The text analysis method depends on text data, and the problems it can solve are limited. It is not applicable to numerical behavior data such as the number of questions asked or grades, while numerical behavior data is common data in teaching.

[0008] 3) The relevance of the reflected knowledge points is not conditional relevance

[0009] For a network established based on traditional methods, the relevance of knowledge points it reflects is overall. The relevance between two knowledge points is interfered by other knowledge points, and it is impossible to determine whether the existing relevance is affected by other knowledge points or excludes the influence of other knowledge points and is only caused by the internal connection between these two knowledge points.

[0010] In summary, a method for analyzing the conditional relevance of knowledge points based on numerical behavior data and constructing a knowledge network diagram has not yet emerged, and the present invention can just solve the above problems. Therefore, it is very necessary to introduce the method of the present invention. Summary of the Invention

[0011] To solve the deficiencies of the prior art, the present invention proposes a method for constructing a knowledge network based on the precision matrix of Gaussian distribution. Its feature is to use the precision matrix method in multivariate analysis to analyze the relevance between knowledge points, visualize the precision matrix trained by the model as a knowledge network relationship diagram, the elements of the precision matrix correspond to the edges in the knowledge network diagram, the variables of the precision matrix correspond to the nodes in the knowledge network diagram. Finally, fuse the information of the knowledge network diagram to construct a regression predictor, enabling the model to consider the conditional relevance between knowledge points when predicting the behavior performance of students in the future learning process, and improving the accuracy of model prediction to a certain extent.

[0012] The method for constructing a knowledge network based on the precision matrix proposed by the present invention includes the following steps:

[0013] Step 1: Data preprocessing

[0014] Collect numerical behavior data, perform data preprocessing on the collected data, and divide it into a training set, a validation set, and a test set; wherein, the data preprocessing includes standardization.

[0015] Step 2: Precision matrix model training

[0016] Use all the data processed in Step 1 to train the precision matrix model, and save the trained precision matrix results.

[0017] Step 3: Construct a knowledge network based on the precision matrix

[0018] Use the precision matrix trained in Step 2 to construct a knowledge network, make the elements in the precision matrix correspond one by one to the edges in the knowledge network, and use the networkx drawing tool to draw the knowledge association network diagram.

[0019] Step 4: Construct a regression predictor based on the knowledge network

[0020] Use the information of the knowledge network in Step 3 to construct a regression predictor, take the behavior data of students on the previous part of knowledge points as input, and output the predicted behavior data of students on the subsequent part of knowledge points.

[0021] For step 1, the data collection is to collect the numerical behavior data of students on major educational platforms, including the number of questions asked, the number of absences, exam scores, etc.; the standardization method uses z-score standardization; the default division of the data set is 70% for the training set, 10% for the validation set, and 20% for the test set.

[0022] For step 2, the training of the precision matrix model uses the glasso method and the clime method. The glasso method is as follows, and the objective function is:

[0023] min Θ {log detΘ - tr(SΘ) - ρ||Θ|| 1}

[0024] where Θ ∈ R p×p is the target precision matrix, p is the number of knowledge points, ρ is the regularization parameter, and the formula for S is as follows:

[0025]

[0026]

[0027] where X i is the behavior data of the i-th student sample on p knowledge points, is the mean vector of the samples, n is the total number of samples, and T represents the transpose operation of the matrix;

[0028] Let W be the estimate of ∑. W and S are partitioned as follows:

[0029]

[0030]

[0031] Substituting W and S into the original objective function becomes the following lasso problem:

[0032]

[0033] where, and w 12 = W 11 β. When β is obtained, w 12 is also obtained; repeat the above process for each column of W, move the target column to be solved to the last column, solve the lasso problem after partitioning, and update the target column with the latest solution. After updating all rows and columns of W, finally, the result of taking the inverse of W is the final Θ;

[0034] The clime method is as follows, and the objective function is:

[0035] minimize ||Θ|| 1

[0036] subject to |SΘ - I| ∞ ≤ λ

[0037] where Θ ∈ R p×p is the target precision matrix, λ is the regularization parameter that controls the sparsity of the precision matrix; the objective function is decomposed into p problems as follows:

[0038] minimize |α| 1

[0039] subject to |Sα - e i | ∞ ≤ λ

[0040] where e i is the standard p-dimensional unit vector with the i-th element being 1 and other elements being 0, and α is the unknown parameter of the above p optimization problems. Solve the above optimization problem for each column of Θ, and merge the solutions of the p sub-problems according to the corresponding positions to obtain a preliminary version of Θ, and then perform the following symmetrization processing:

[0041]

[0042] where I{x} is the indicator function, which is 1 when X is true and 0 when X is false. represents the trained precision matrix, represents the element in the i-th row and j-th column of the trained precision matrix;

[0043] Both the glasso and clime methods select the optimal parameters through the seven-fold cross-validation method, and the loss function uses the likelihood function, and the formula is as follows:

[0044] loss = tr(SΘ) - log det Θ

[0045] where loss represents the objective loss function, Θ represents the precision matrix output by the model, and S represents the empirical covariance matrix of the training set samples.

[0046] For step 3, the specific construction process is as follows:

[0047] Step A1: Initialize the knowledge network graph and add nodes in the graph with a number matching the number of knowledge points;

[0048] Step A2: Traverse the precision matrix and decide whether to add an edge between the corresponding nodes in the graph according to whether the elements in the precision matrix are 0;

[0049] Step A3: Use the Python tool networkx for drawing network diagrams to draw a knowledge network relationship diagram;

[0050] For Step 4, the specific construction process is as follows:

[0051] The prediction module includes a point prediction module and an interval prediction module;

[0052] Let the total number of knowledge points be p, the behavior data of the first m knowledge points be used as features, and the behavior data of the (m + 1)-th to p-th knowledge points be used as the target data to be predicted; here, use to represent the behavior data of the i-th student sample on the first m knowledge points, to represent the behavior data of the i-th student sample on the (m + 1)-th to p-th knowledge points;

[0053] The point prediction module: Let the input be the precision matrix Θ trained in Step 2, and the output be represented by The formula is as follows:

[0054] ∑ = Θ -1

[0055]

[0056] is the covariance matrix of the knowledge points, which is the inverse matrix of the precision matrix, and p is the total number of knowledge points; μ represents the mean vector of the behavior data of all student samples in the training set on all knowledge points, with a dimension of p; μ 1 , μ 2 , ∑ 11 , ∑ 12 , ∑ 22 are the parts after partitioning μ and ∑ according to the following structure:

[0057]

[0058] μ 1 represents the mean vector of the behavior data of all student samples on the first m knowledge points in the training set, with a dimension of m, and μ 2 represents the mean vector of the behavior data of all student samples on the (m + 1)-th to p-th knowledge points in the training set, with a dimension of p - m; at the same time, use RMSE as the evaluation index, and the calculation formula is as follows:

[0059]

[0060] Among them, z i represents the true value, represents the value predicted by the model, and k represents the number of samples participating in the prediction;

[0061] The interval prediction module: Let the input be The output is the range interval of the behavior data of the predicted i-th student sample on the j-th (j = m + 1, …, p) knowledge point; D represents the behavior data of n student samples on the first m knowledge points. Add a column of all 1s to the matrix D to form a new matrix X, as follows:

[0062]

[0063] X = (1, D)

[0064]

[0065] where y represents the behavior data of all student samples in the training set on the j-th knowledge point, is the behavior data of the predicted i-th student sample in the test set on the j-th knowledge point.

[0066] The interval finally output by the interval prediction module is:

[0067]

[0068] where t 1-α / 2 (n - p - 1) is the t-quantile with a confidence level of 1 - α / 2 and a degree of freedom of n - p - 1.

[0069] The present invention also provides a knowledge network construction system based on the precision matrix of Gaussian distribution, including: a memory and a processor; a computer program is stored on the memory, and when the computer program is executed by the processor, the method of the present invention described above is implemented.

[0070] The present invention also provides a computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the method of the present invention described above is implemented.

[0071] The knowledge network construction method disclosed by the present invention uses the precision matrix model in multivariate analysis to analyze the correlation between knowledge points, visualizes the precision matrix trained by the model as a knowledge network relationship diagram, the elements of the precision matrix correspond to the edges in the knowledge network diagram, and the variables of the precision matrix correspond to the nodes in the knowledge network diagram. Finally, the information of the knowledge network diagram is fused to construct a regression predictor to predict the behavior performance of students in the future learning process.

[0072] The present invention solves the problem that the existing knowledge network construction methods cannot be applied to numerical behavior data. By introducing the method of precision matrix to construct a knowledge network diagram, and at the same time, a regression predictor is constructed based on the knowledge network diagram to predict which knowledge points students need help in the future learning process, thereby assisting teaching.

[0073] By adopting the method of the present invention, the problem that the existing methods for constructing knowledge networks are not applicable to numerical behavioral data is solved. A knowledge network graph is constructed by means of the precision matrix. At the same time, a predictor is constructed based on the knowledge network graph, improving the accuracy of model prediction. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the drawings in the following description are only some embodiments of the present invention. For those skilled in the art, without creative efforts, other drawings can be obtained based on these drawings.

[0075] Figure 1 It is a schematic flowchart of the present invention.

[0076] Figure 2 It is a schematic diagram of constructing a knowledge network based on a trained precision matrix.

[0077] Figure 3 It is a schematic diagram of the prediction result of a specific example interval. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0078] Combined with the following specific embodiments and drawings, the invention will be further described in detail. The processes, conditions, experimental methods, etc. for implementing the present invention, except for the specifically mentioned content below, are all common knowledge and well-known common sense in the art, and the present invention has no special restrictive content.

[0079] The present invention first performs standardized preprocessing on the numerical behavioral data of students, trains a precision matrix model using the preprocessed data, then constructs a knowledge network graph based on the trained precision matrix, and visualizes it through the networkx tool of python. At the same time, the trained precision matrix and the preprocessed student behavioral data are input into a regression point predictor and a regression interval predictor to obtain the behavioral data of students on different knowledge points in the future learning process.

[0080] Embodiment

[0081] As Figure 1 shown in the schematic flowchart, construct a knowledge network based on the precision matrix:

[0082] First step, collect the dataset and preprocess the dataset. The datasets used in this invention are MathMale and MathFemale datasets. These are two datasets from an educational platform, which are the number of questions asked by 1000 male students and 1000 female students for 83 junior high school math chapters respectively. The dataset also includes the names and classifications of each chapter. In this embodiment, the chapters are regarded as knowledge points, and the number of questions asked is regarded as a kind of numerical behavior data of students. Preprocess the two datasets and divide them into training set, validation set and test set. Among them, the preprocessing includes standardization. This invention additionally adds BiasCorrectionUcl and Wind datasets. These data are all numerical data, and through these data, the generality of this invention is verified. It is not limited to educational data and still has good performance on other types of datasets.

[0083] Second step, train the models clime and glasso of the precision matrix. The training data includes the number of questions asked by students in all chapters in the training set. Use the maximum likelihood function as the loss function and select the optimal parameters through seven-fold cross-validation method. Then use the selected optimal parameters to iterate 100 times to obtain the final version of the precision matrix.

[0084] Third step, construct a knowledge network based on the precision matrix trained in the second step. Traverse the precision matrix. For the elements in the precision matrix that are 0, there is no edge at the corresponding position in the knowledge network diagram. For the elements in the precision matrix that are not 0, there is an edge at the corresponding position in the knowledge network diagram. Refer to Figure 2 , Figure 2 The figure shows a schematic diagram of constructing a knowledge network based on the precision matrix. The element at the 1st row and 2nd column in the left matrix is 0.15, which is not 0. So there is an edge between node 1 and 2 in the right network diagram; the element at the 1st row and 3rd column in the left matrix is 0. So there is no edge between node 1 and 3 in the right network diagram; the element at the 2nd row and 3rd column in the left matrix is 0.3, which is not 0. So there is an edge between node 2 and 3 in the right network diagram.

[0085] Fourth step, construct a regression predictor based on the precision matrix used to construct the knowledge network in the third step. The training data includes 83 knowledge points from the first grade to the third grade of junior high school. Select the chapters from the first grade to the second grade, that is, the first 59 chapters as features, and the last 24 chapters as the targets to be predicted. Let the input be representing the number of questions asked by the i-th student in the first 59 chapters, and the trained precision matrix is Θ;

[0086] The point predictor is as follows:

[0087] The point prediction output result is represented by and Denote the number of questions asked by the \(i\)-th predicted student in the last 24 chapters, and the formula is as follows:

[0088] \(\sum=\Theta\) -1

[0089]

[0090] is the covariance matrix; \(\mu\) represents the mean vector of the samples, with a dimension of 83; \(\mu\) 1 and \(\mu\) 2 , \(\sum\) 11 , \(\sum\) 12 , \(\sum\) 22 are the structures after partitioning \(\mu\) and \(\sum\) as follows:

[0091]

[0092] \(\mu\) 1 represents the mean vector of the behavior data of all student samples in the first 59 knowledge points in the training set, with a dimension of 59, \(\mu\) 2 represents the mean vector of the behavior data of all student samples from the 60th knowledge point to the 83rd knowledge point in the training set, with a dimension of 24;

[0093] The interval predictor is as follows:

[0094] Let \(D\) denote the data of the number of questions asked by 1000 samples in the first 59 knowledge points. Add a column of all 1s to the matrix \(D\) to form a new matrix \(X\), as shown below:

[0095]

[0096] \(X=(1, D)\)

[0097]

[0098] where \(y\) represents the behavior data of all student samples in the training set at the \(j\)-th knowledge point, is the behavior data of the \(i\)-th predicted student sample at the \(j\)-th knowledge point in the test set.

[0099] The interval finally output by the interval prediction module is:

[0100]

[0101] where \(t\) 1-α / 2 (n - p - 1) is the \(t\)-quantile with a confidence level of \(1-\alpha / 2\) and a degree of freedom of \(n - p - 1\).

[0102] The present invention conducted experiments on the MathMale and MathFemale datasets, and compared the point prediction model with linear regression LR, support vector regression SVR, random forest regression RFR and XGBoost regression models. The point prediction experimental results are shown in Table 1:

[0103] Table 1

[0104]

[0105] The method of the present invention is better than all the comparative prediction algorithms in terms of point regression prediction performance, which also indirectly illustrates the better performance of the method of constructing a knowledge network relationship diagram based on the precision matrix. Figure 3 As shown in Figure 2, a student, a boy and a girl, were randomly selected from the MathMale dataset and the MathFemale dataset, respectively. The confidence level was set to 0.95, and the range of the number of questions they asked on different chapters of the third-year mathematics course was predicted. The prediction results are shown in Figure 2. Figure 3 As shown in the figure. The lower line represents the lower limit of the prediction interval, the upper line represents the upper limit of the prediction interval, and the real line represents the actual number of questions asked by students. It can be clearly seen from the figure that the interval predicted by the method of the present invention completely covers the actual number of questions asked by students in all chapters. This result once again verifies that the method of the present invention has good performance.

[0106] The present invention adopts the precision matrix method to construct a knowledge network based on numerical behavior data, and fuses the information of the knowledge network to construct a regression point predictor and a regression interval predictor. Compared with the existing regression prediction model, the present invention has higher prediction accuracy in the regression prediction task after fusing the information of the knowledge network. At the same time, the knowledge network constructed by the present invention reflects the conditional correlation between knowledge points, which has better interpretability than the traditional method of constructing a knowledge network.

[0107] The protection content of the present invention is not limited to the above embodiments. Without departing from the spirit and scope of the inventive concept, changes and advantages that can be thought of by those skilled in the art are included in the present invention and are protected by the attached claims.

Claims

1. A method for constructing a knowledge network based on the precision matrix of Gaussian distribution, characterized in that, it includes the following steps: Step 1: Collect numerical behavior data, preprocess the collected data, and divide it into a training set, a validation set, and a test set; wherein, the data preprocessing includes standardization; Step 2: Use the training set, validation set, and test set processed in Step 1 to train the precision matrix model, and save the results of the trained precision matrix model; Step 3: Use the precision matrix model trained in Step 2 to construct a knowledge network, map the elements in the precision matrix model to the edges in the knowledge network one by one, and use the networkx drawing tool to draw a knowledge association network diagram; Step 4: Use the information of the knowledge network in Step 3 to construct a regression predictor, take the behavior data of the students on the previous part of the knowledge points as input, and output the predicted behavior data of the students on the next part of the knowledge points; The construction process of the regression predictor in Step 4 is as follows: The prediction module includes a point prediction module and an interval prediction module; Let the total number of knowledge points be p, the behavior data of the first m knowledge points be used as features, and the behavior data of the (m + 1)-th to p-th knowledge points be used as the target data to be predicted; The point prediction module: Given the input as the precision matrix Θ trained in step 2, and the output is denoted by as follows: E = Θ -1 Denote the behavior data of the $i$-th student sample on the first $m$ knowledge points, Denote the behavior data of the $i$-th student sample on the knowledge points from the $(m + 1)$-th to the $p$-th; is the covariance matrix of knowledge points, which is the inverse matrix of the precision matrix, and $p$ is the total number of knowledge points; $\mu$ represents the mean vector of the behavior data of all student samples in the training set on all knowledge points, with dimension $p$; $\mu$ 1 , $\mu$ 2 , $E$ 11 , $E$ 12 , $E$ 22 are the parts after partitioning $\mu$ and $E$ according to the following structure respectively: μ 1 represents the mean vector of the first m knowledge point behavior data of all student samples in the training set, with a dimension of m, μ 2 represents the mean vector of the behavior data of the (m + 1)-th to p-th knowledge points of all student samples in the training set, with a dimension of p - m; at the same time, RMSE is used as the evaluation index, and the calculation formula is as follows: Among them, z i represents the true value, represents the value predicted by the model, and k represents the number of samples participating in the prediction; The interval prediction module: Let the input be The output is the range interval of the behavior data of the predicted i-th student sample on the j-th (j = m + 1, …, p) knowledge point; D represents the behavior data of n student samples on the first m knowledge points. Add a column of all 1s to matrix D to synthesize a new matrix X as follows: X=(1,D) Among them, y represents the behavioral data of all student samples in the training set on the k-th knowledge point. It is the behavioral data of the i-th student sample predicted in the test set on the j-th knowledge point. The interval finally output by the interval prediction module is: where t 1-α / 2 (n - p - 1) is the t-quantile with a confidence level of 1 - α / 2 and degrees of freedom of n - p - 1.

2. The knowledge network construction method according to claim 1, characterized in that, in Step 1, the collection of numerical behavior data is to collect the behavior data of students on the education platform, including: the number of questions asked, the number of absences, and the test scores; the standardization method uses z-score standardization; by default, 70% of the dataset is the training set, 10% is the validation set, and 20% is the test set.

3. The knowledge network construction method according to claim 1, characterized in that, in Step 2, the training of the precision matrix model uses the glasso method and the clime method for training; The glasso method is as follows, and the objective function is: min Θ {log det Θ - tr(SΘ) - ρ||Θ|| 1} Among them, is the target precision matrix, p is the number of knowledge points, ρ is the regularization parameter, and the S formula is as follows: Among them, X i is the behavioral data of the i-th student sample on p knowledge points, is the mean vector of the samples, n is the total number of samples, and T represents the transpose operation of the matrix; Let W be the estimate of E, and partition W and S as follows: Substituting W and S into the original objective function becomes the following lasso problem: Among them, while w 12 = W 11 β, when β is obtained, w is also obtained 12 ; Repeat the above process for each column of W, move the target column to be required to the last column, solve the lasso problem after partitioning, and update the target column with the latest solution; After updating all rows and columns of W, finally, the result of taking the inverse of W is the final Θ; The clime method is as follows, and the objective function is: min||Θ|| 1 s.t. |SΘ - I| ∞ ≤ λ Among them, is the target precision matrix, and λ is the regularization parameter that controls the sparsity of the precision matrix; the objective function is decomposed into p optimization problems as follows: min|α| 1 such that |Sα - e i | ∞ ≤ λ where e i is the standard p-dimensional unit vector, with the i-th element being 1 and other elements being 0, and α is the unknown parameter of the above p optimization problems; the solutions of the above optimization problems are obtained for each column of Θ, and the solutions of the p sub-problems are merged according to the corresponding positions to obtain the preliminary version of Θ, and then the following symmetrization process is carried out: Where, I{x} is an indicator function, which is 1 when x is true and 0 when x is false; represents the trained precision matrix, represents the element in the i-th row and j-th column of the trained precision matrix.

4. The knowledge network construction method according to claim 3, characterized in that, the glasso and clime methods select the optimal parameters through the seven-fold cross-validation method, and the loss function uses the maximum likelihood function, and the formula is as follows: loss = tr(SΘ)-log detΘ wherein, loss represents the objective loss function, Θ represents the precision matrix output by the model, and S represents the empirical covariance matrix of the training set samples.

5. The knowledge network construction method according to claim 1, characterized in that, the specific construction process in Step 3 is as follows: Step A1: Initialize the knowledge network diagram, and add nodes with a number matching the number of knowledge points in the diagram; Step A2: Traverse the precision matrix, and decide whether to add an edge between the corresponding nodes in the diagram according to whether the elements in the precision matrix are 0; Step A3: Use the python network drawing tool networkx to draw the knowledge network relationship diagram.

6. A knowledge network construction system based on the precision matrix of Gaussian distribution, characterized in that, comprising: a memory and a processor; a computer program is stored on the memory, and when the computer program is executed by the processor, the method described in any one of claims 1-5 is implemented.

7. A computer-readable storage medium, on which a computer program is stored, characterized in that, when the computer program is executed by the processor, the method described in any one of claims 1-5 is implemented.

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