Research on identifying psychological health problems of college students by logistic regression model based on data mining

Data mining is a complicated process. Through a series of calculations in a large amount of data, it is necessary to find the potential and hidden internal relations in the data and extract valuable information. The specific process is shown in Figure 1:

Selection and preparation of data set. According to the actual demand, select the initial data set and collect as much relevant data as possible. The more relevant the data, the higher the accuracy, but the amount of calculation will also increase [11].

Fig. 1

Logistic regression is a kind of data mining technology. For a given data set, it can get a nonlinear model through linear model transformation to predict the actual test value as much as possible and output it [13, 14]. Actually, it is a classification model. Compared with other algorithms, logical regression has a simple form, strong interpretability and fast training speed, whose main idea is to fit the decision boundary as much as possible and output the predicted value. On the basis of linear regression, that is, for multivariate input X = (x₁, x₂, …, x_n)^T, the linear expression of predicted output y is as follows: (1) f(x)=w1x1+w2x2+…+wnxn+b f(x) = {w_1}{x_1} + {w_2}{x_2} + \ldots + {w_n}{x_n} + b

Sigmoid function is usually selected as the mapping function, and its expression is: (2) g(z)=11+e−z g(z) = {1 \over {1 + {e^{ - z}}}}

Introduce Formula (1) so that: (3) g(x)=11+e−(wTx+b) g(x) = {1 \over {1 + {e^{ - \left( {{w^T}x + b} \right)}}}}

The value of g(x) is: (4) g(x)={1,wTx+b>00.5,wTx+b=00,wTx+b<0 g(x) = \left\{ {\matrix{ {1,} \hfill & {{w^T}x + b > 0} \hfill\cr {0.5,} \hfill& {{w^T}x + b = 0} \hfill\cr {0,} \hfill& {{w^T}x + b < 0} \hfill\cr } } \right.

It can be seen that through sigmoid the function can define the domain as R and any real number is mapped between [0, 1], and it can be judged whether the sample belongs to the positive or negative class by judging whether the value of g(x) is larger than the threshold value of 0.5. Different thresholds can be adjusted according to the actual situation, and logarithmic conversion is usually obtained by: (5) g(x)lng(x)1−g(x)=wTx+b g(x)\ln {{g(x)} \over {1 - g(x)}} = {w^T}x + b

Regarding g(x) as the probability that x is a positive example, then 1 − g(x) represents the probability of being a counterexample. Therefore, the classification problem using logistic regression algorithm is finally transformed into finding the best solution for w. And through the loss function of the model, that is, the gap between the predicted value and the actual value, the classification result of the model is optimised. When solving the parameters in the model, the gradient descent method is used to solve the minimum value of loss function, which is simple to implement.

Due to the serious data imbalance between normal and abnormal samples, classification without other operations will affect the results of the model [15]. Therefore, before the dichotomy, it is necessary to balance the data, and combine their features with the differences.

The samples of normal students are marked as negative examples, whose code is 0, and the samples of abnormal students are marked as positive examples, whose code is 1. Before training the model, in order to avoid the influence of dimensional units between feature samples, the data should first be standardised. Data standardisation refers to scaling all feature data to a fixed range to avoid the influence of different units and different numerical values when training the model. Similarly, because there is a large numerical span in the features selected in this experiment, in order to eliminate the influence of numerical value on the classification model, the data should be normalised at first. That is, the result is mapped to [0–1] by linear transformation.

For sample X₁, X₂, X_n, the maximum value is max (X_i) and the minimum value is min (X_i), whose formula is: (6) x*=Xi−min(Xi)max(Xi)−min(Xi) {x^*} = {{{X_i} - \min ({X_i})} \over {\max ({X_i}) - min({X_i})}}

According to this formula, the range of sample data can be compressed to [0,1], so as to eliminate the differences among different feature samples and avoid the influence of numerical dimension among feature samples.

The logistic regression model has an important influence on the results of sample classification, so the parameters of the logistic regression model are optimised. According to the transformation of sigmoid function, the assumed function is: (7) x*=Xi−min(Xi)max(Xi)−min(Xi) {x^*} = {{{X_i} - \min ({X_i})} \over {\max ({X_i}) - min({X_i})}}

Among them, θ is the parameter variable, x is the eigenvector. In logistic regression, loss function is generally used to measure the accuracy of classification results. That is, according to a given sample {(x₁,y₁) … (x_i,y_i)}, the degree of deviation between the value of predicted samples ŷ_i and the value of the actual sample y_i are obtained by the loss function, that is, the error between the measured actual value and the predicted value. The logistic regression loss function here is defined as: (8) J=1m∑i=1mL(y^i,yi)=−1m∑i=1m[yilog(y^i)+(1−yi)log(1−y^i)] J = {1 \over m}\sum\limits_{i = 1}^m L({\hat y_i},{y_i}) = - {1 \over m}\sum\limits_{i = 1}^m \left[ {{y_i}\log ({{\hat y}_i}) + (1 - {y_i})\log (1 - {{\hat y}_i})} \right]

To prevent the model from over-fitting, a regularisation term is added to the loss function and is represented as: (9) x*=J=−1m∑i=1m[yilog(y^i)+(1−yi)log(1−y^i)]+λ2m∑j=1λθj2 {x^*} = J = - {1 \over m}\sum\limits_{i = 1}^m \left[ {{y_i}\log ({{\hat y}_i}) + (1 - {y_i})\log (1 - {{\hat y}_i})} \right] + {\lambda \over {2m}}\sum\limits_{j = 1}^\lambda \theta _j^2 where λ is the regularisation parameter, the gradient descent method is used to solve the parameter θ, and the learning rate α is used to control the step size of gradient descent. The iterative update formula is: (10) θ0=θ0−αm∑i=1m(h0(xi)−yi)x0(i) {\theta _0} = {\theta _0} - {\alpha \over m}\sum\limits_{i = 1}^m \left( {{h_0}({x_i}) - {y_i}} \right){x_{{0_{(i)}}}} (11) θj=θj(1−λam)−αm∑i=1m(hθ(xi)−yi)xi(j), j=1,2,…n {\theta _j} = {\theta _j}\left( {1 - {{\lambda a} \over m}} \right) - {\alpha \over m}\sum\limits_{i = 1}^m \left( {{h_\theta }({x_i}) - {y_i}} \right)x_i^{(j)},\quad j = 1,2, \ldots n

Introducing features into the model: (12) hθ(x)=11+e−(θ0+θ1x1+θ2x2+θ3x3+θ4x4) {h_\theta }(x) = {1 \over {1 + {e^{ - \left( {{\theta _0} + {\theta _1}{x_1} + {\theta _2}{x_2} + {\theta _3}{x_3} + {\theta _4}{x_4}} \right)}}}} the gradient value is obtained by gradient descent method, and the solution is θ = [− 3.294, 5.145, − 0.042, 2.238, 3.479]^T. That is, according to the optimised parameters, the prediction model for identifying students with abnormal psychology is constructed as follows: (13) hθ(x)=11+e−(−3.294+5.145x1−0.042x2+2.238x3+3.479x4) {h_\theta }(x) = {1 \over {1 + {e^{ - \left( { - 3.294 + 5.145{x_1} - 0.042{x_2} + 2.238{x_3} + 3.479{x_4}} \right)}}}}

In this paper, the model updated with the above parameters is used to predict students’ psychological problems, in which x₁, x₂, x₃, x₄, and y, respectively, represent the number of friends, sleep condition, economic condition, academic achievement and sample category.

Most of the students’ studies in colleges and universities are closed or semi-closed, so their health problems can be reflected through their classmates. The co-occurrence times of two students are directly proportional to the probability of becoming friends, therefore, association rules can be used to determine the classmate relationship.

In order to obtain students’ friends at college, it is necessary to calculate the co-occurrence data set between pairs of students, which is represented as: (14) U={(i,j,τ(t∪j)|i,j∈u1,2,…N)} U = \{ (i,j,\tau (t \cup j)|i,j \in {u_{1,2, \ldots N}})\} where i and j represent students, τ(i ∪ j) and refers to the number of simultaneous occurrences of students i and j.

In order to eliminate contingency, if the co-occurrence times of student A and student B are greater than T, the two students are considered as friends. Considering that the threshold value of each location is related to the total number of times, it is defined as: (15) Til=sumilαl T_i^l = {{{\rm{sum}}_i^l} \over {{\alpha ^l}}}

Among them, for students i in position l the threshold of co-occurrence friends at, sumil {\rm{sum}}_i^l refers to students i in position l the total number of occurrences, α^l. For location l the ratio coefficient, remember students. i and j the co-occurrence of times are τ(i ∪ j), when τ(i ∪ j) > T_i. When, think students j are i in position l friends, recorded as G(i → j). On the basis of co-occurrence data pairs, use association rules to determine the scale coefficient. α^l, which is used here with i → j.

Form to explore the relationship, that is i → j, if there is the association rules, the students i and j have good friends. Support is defined here. S(i → j) and confidence. C(i → j), the support degree is:

where Til T_i^l is the threshold of student's i the co-occurrence friends at position l, sumil {\rm{sum}}_i^l is the total number of times student i appears at position l, α^l is the ratio coefficient at position l, and the co-occurrence times of student i and j are τ(i ∪ j). When τ(i ∪ j) > T_i, student J is considered to be a friend of i at position l and is denoted as G(i → j). On the basis of co-occurrence data pairs, association rules are used to determine the proportional coefficient α^l. Here, the relationship is mined by using the form i → j, that is, if there is an association rule i → j, then students i and j have a friend relationship. The support degree S(i → j) and confidence degree C(i → j) are defined here, and the support degree is: (16) S(i→j)=τ(i∪j)D, i≠j S(i \to j) = {{\tau (i \cup j)} \over D},\quad i \ne j

Confidence is defined as: (17) C(i→j)=τ(i∪j)τ(i), i≠j C(i \to j) = {{\tau (i \cup j)} \over {\tau (i)}},\quad i \ne j

Among them D, indicates the total number of credit card records in the whole data set, τ(i ∪ j) for students i and j. The number of co-occurrences of, τ(i) are for students i. All credit card record entries of support degree S(i → j) represent students i and j Frequency and confidence of simultaneous occurrence in all data sets C(i → j) can be used to indicate students i and j the degree of relevance. Support and confidence are important indicators in social relationship association analysis. Association rules with low support are mostly random co-occurrence and generally meaningless.

Where D represents the total number of credit card records in the entire data set, τ(i ∪ j) is the number of co-occurrences of students i and j, and τ(i) is all credit card records of student i. Support degree S(i → j) represents the frequency of simultaneous occurrence of students i and j in all data sets, and confidence degree C(i → j) can be used to represent the degree of association between students i and j. Support degree and confidence degree are important indicators in social relationship association analysis. Association rules with low support degree are mostly from random co-occurrence and are generally meaningless.

The regularity of students’ behaviour reflects their self-discipline and orderliness in college. Students with strong regularity have better binding force, and the regularity of students’ behaviour is closely related to their academic performance [19,20,21]. Therefore, it can be considered that students with strong behavioural regularity can arrange their own plans in life. In this paper, the regularity of students’ behaviour is mainly quantified, such as eating and bathing, and whether there are significant differences between students with abnormal psychology and normal students in the regularity of students’ campus life behaviours, are explored. Shannon entropy is used to calculate and measure, and its definition is as follows: (18) H=−∑ip(i)logp(i) H = - \sum\limits_i p(i)\log p(i)

Here p(i) is said to be the first in the specified unit time i. In this experiment, 24 h a day is defined as the total time period, and students’ daily behaviour data is divided into 48 time series (30 minutes is a time series). For example, the behaviour data of a student in a certain period of time is {2021-12-13, 10:05; 2021-12-14, 12:23; 2021-12-15, 11:32; 2021-12-16, 11:23} and its chronological order is in the information entropy of {22, 24, 23, 22}.

where p(i) represents the probability of doing an activity in the ith time period within a specified unit time. In this experiment, 24 h a day is defined as the total time period, and students’ daily behaviour data is divided into 48 time series (30 minutes is a time series). For example, the behavioural data of a student in a certain period is {2021-12-13, 10:05; 2021-12-14, 12:23; 2021-12-15, 11:32; 2021-12-16, 11:23}, and its time sequence is listed as the information entropy of {22, 24, 23, 22,}. The result calculated by the formula is 1.58 Bit, in statistical informatics. Shannon entropy is used to measure the uncertainty of information [22], so it is used to measure the regularity of students’ behaviour. In this model, entropy is used to measure the regularity of three meals and fetching water. In the data set, all students have restaurant consumption records; about 96% of the students have a record of drawing water from the dormitory. The behaviour of fetching water usually occurs in the evening, and the time of washing face and rinsing mouth can indirectly reflect the bedtime of students, so these two high-frequency behaviours can objectively measure the regular habits of students. The greater the entropy of a student in a time period, the higher is the uncertainty and the lower the regularity.

Studies [23] have shown that the psychological status of college students is influenced by the family economic situation, and many students from poor families will suffer from long-term inferiority and depression; also they seem to be unwilling to communicate with others and other behaviours. However, due to the limitation of conditions, it is impossible to know the family status of students’ original families. In order to explore whether there are significant differences in the college performance between psychologically normal and abnormal students, in this paper, students’ financial situation is measured from the students’ financial aid and their consumption at college.

Given the time and amount of students’ consumption each time, first of all, the annual consumption data of 2,000 students (436,044 items in total) were processed and analysed, and the statistics of the average consumption of two kinds of samples between one semester and 1 year showed that there was no obvious difference in data distribution. Through Wilcoxon S hypothesis test, the zero hypothesis is as follows: There is no significant difference in consumption level between normal students and psychologically abnormal students. The alternative hypothesis is that there are significant differences in consumption level between normal students and students with psychological disorders. It is found that under the condition of 0.05 confidence, the consumption level of the two groups is P = 0.07>0.05, which means that there is no significant difference between them. In addition, a comparative analysis of the students’ financial aid in college has been made, where no difference is found.

According to the above characteristics, the quantitatively extracted sample data is introduced into the improved logistic regression model obtained before, which can effectively reflect the psychological health problems of college students. The evaluation index of this model is defined as:

Accuracy, that is, the proportion of correctly predicted samples to all samples. In the confusion matrix, T P and T N represent positive and negative samples, respectively, the formula for calculating the accuracy rate is: (19) Accuracy=TP+TNTP+FP+TN+FN {\rm{Accuracy}} = {{TP + TN} \over {TP + FP + TN + FN}}

Precision, which refers to the proportion of correctly predicted positive samples (T P) to the positive samples (T P and FP), the calculation formula is: (20) Precision=TPTP+FP {\rm{Precision}} = {{TP} \over {TP + FP}}

Recall represents the proportion of correctly predicted positive samples (T P) to the total of positive samples (T P and FP), which also use T P_rate to indicate the formula as: (21) Recall=TPrate=TPTP+FN {\rm{Recall}} = T{P_{rate}} = {{TP} \over {TP + FN}}

In this model, for the positive sample in the original data tag, if it is predicted as a positive example, the prediction result of this sample is a True Positive example, or TP in short. On the contrary, if it is predicted as a negative example, the result is a False Negative example. Similarly, a False Positive example (FP) means that a negative example sample in the original label is predicted, and a True Negative example (T N) is to predict a negative example sample in the original label.

The original sample ratio of this model is about 10:1, as shown in Figures 2–4; if it is directly used to train the model, the logistic regression effect of the sample data is poor, but with the approaching of positive and negative proportion, the accuracy, precision and recall of the sample are better. When the positive-negative ratio of the sample is 1:2, the model regression effect is the best, with the accuracy, precision and recall reaching 76.2% and 83.4%, respectively. At the same time, when the positive–negative ratio of the sample is less than 1:2, the regression effect of the model is somewhat reduced.

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In addition, it is necessary to identify students with psychological disorders in the shortest time. For this reason, the data sets of one semester, 1 year, are selected for comparison. The data set used in this model is for students in the second semester of sophomore year (one semester), sophomore year (1 year) and college stage (freshman to junior year; there is no data for senior students here, because there are few exam classes for senior students in our college, and most of the students are in internship outside during their senior years). Feature training model as previously described is used to classify these data.

The recall rate indicates the proportion of the number of samples correctly predicted as positive cases in the sum of all samples divided into positive cases. The purpose of the experiment in this paper is to identify students with abnormal psychology among all students, that is, to identify a relatively large number of positive examples, so recall rate is an important evaluation index for this model. The higher is the recall rate, the more students with abnormal psychology are identified by the model. The results show that the improved logistic regression model performs well when adopting all 3-year data. In addition, the recall rate obtained by using 1 year's data is slightly lower than that obtained by using all the data. Combined with the classification accuracy, it can be considered that it takes at least 1 year's student behaviour data to identify students with psychological disorders, and the more data used, the better the robustness of the logistic regression model.