A time-series analysis study on the correlation between college students’ mental health status and the ideology courses in colleges and universities

A collection of historical points in time is called a time series. Time series are arranged in a certain time interval, and based on the existing historical time to predict future trends, from predicting the sales of a product to estimate the number of users of a product, commonly used time series forecasting model there are a lot of models, including the ARIMA model is the most commonly used in the forecasting of time series of practical cases of the model, this model is mainly for the smooth non-white noise series of data.

The theoretical components of the ARIMA model consist of three parts: an autoregressive model (AR), a moving average model (MA), and a difference model (Integration). The AR term is only used to predict the past value of the next value, which is defined by the p in the time series parameter, and the p can be identified by a pacf plot. The MA term defines the past prediction error in predicting the future value of a number, which is represented by parameter q for the MA value, and q can be identified by the pacf plot. For the difference d, i.e., the order of the difference, the number of difference operations to be performed on the sequence is specified. The purpose of performing differencing operations on the data is to keep the time series smooth, and the adf test can usually be used to determine whether the series is smooth or not.

The ARIMA (p,d,q) model is developed from three special models, the AR autoregressive model, the MA sliding average model, and the I difference model. Where p is the number of autoregressive terms, d is the difference coefficient term, also known as the number of differences or terms, while q is the number of sliding average terms [26].

Assuming that the time series data set is represented by x₁, x₂, x₃, …, x_i, AR(p) the autoregressive model can only be used to de-predict phenomena related to points in time prior to that time series, and is a model that describes the relationship between current and historical values, which requires that the predicted series satisfy smoothness. Expressed as a mathematical model, the equation is shown in (1): (1)xt=μ+∑i=1pφixt−i+εt

Where, x_i represents the current value, μ represents the constant term, p represents the order, φ_i represents the autocorrelation coefficient, and ε_i represents the error, which is also white noise. The formula expansion is shown in (2). (2)Xt=φ1Xt−1+φ2Xt−2+...+φpXt−p+μt

When μ = 0 implies μ_t = ε_t, i.e., the random perturbation term is a white noise. Then the AR model is called a pure AR process. It follows that historical values can predict current values, and p is the order in the autoregressive model, which is the number of p observations involving the past and represents how many historical values are used to predict current values. The autoregressive model, while simple to understand, has significant limitations. Firstly, it can only use its own data to make predictions, which means that it is only suitable for making predictions about phenomena related to its own prior period. Secondly, the own time series data must have a smooth nature, otherwise there will be a large error. Finally, the series must be correlated and if the autocorrelation coefficient is less than 0.5 then it is not suitable for autoregressive modeling, otherwise it will also have a large error [27].

Another component of ARIMA- MA(q) the moving average model focuses on focusing on the cumulative sum of the error terms in the model, which implies that the current observation is a linear combination of the variable’s previous q random errors in the model. Expressed in terms of a mathematical model, the equation is shown in (3), the (3)xt=μ+∑i=1qθiεt−i+εt

Where, x_i represents the current value, μ represents the constant term, q represents the order, θ_i represents the autocorrelation coefficient and ε_i represents the error, which is also white noise. It has the advantage of being very effective in eliminating random fluctuations in the predicted series, and thus it is complementary to the autoregressive process.

Combining the above two models gives the autoregressive moving average ARMA(p, q) model. A random time series that can be represented by the ARMA(p, q) model must be predictable and interpretable by its own lagged values and random perturbation terms. Expanding the above two equations together is shown in (4): (4)Xt=φ1Xt−1+φ2Xt−2+...+φpXt−p+εt+θ1εt−1+...+θqεt−q

where X_t represents the current value, which is also a normal and smooth time series, q and p represent the autoregressive and moving average orders, θ_i and φ_i represent the non-zero coefficients to be determined, and ε_i represents the error term. Note here that for the ARMA(p, q) model to be meaningful, the original stochastic time series must be smooth and unchanging over time, and it must be invertible.

The ADF test is an augmented form of Dickey-Fuller, which is based on the principle of identifying the presence or absence of a unit root in a time series, if the time series is smooth, then there is no unit root. On the contrary, there is. ADF assumes the existence of unit root and if it finally presents significance, i.e., the value of p is less than 5% or less than 1%, it means that the original hypothesis is rejected and the time series is a smooth time series.

According to CRAMER decomposition theorem (5): (5)s∇d∑dβ=c

c is some constant, and the observed time series {x_i} is expanded for first-order differencing, which leads to Eq. (6): (6)∇xt=xt−xt−1

The above equation is equivalent to equation (7): (7)xt=xt−1+∇xt

It follows that the nature of the first-order difference is a process of self-national normalization, which leads to any d-order difference with Eq. (8): (8)∇dxt=(1−B)dxt(−1)iCidxt−i

The essence of the above equation is also a dst order autoregressive process as shown in equation (9): (9)xt=∇dxt+∑d(−1)iCdixt−i

The autocorrelation function, ACF, whose coefficients express the linear relationship between the observed values of a time series and its values in past time periods. The autocorrelation coefficient takes values from -1 to 1. The closer its absolute value is to 1, the more autocorrelated the series is. For time series with significant cyclical or seasonal variations and those with a strong upward and downward trend, the absolute value of the autocorrelation coefficient will be closer to 1. The value of its coefficient ACF can be calculated according to the following formula (10): (10)ϕ^kk=D^kD^,∀0≺k≺n

Where the above equation is expanded in terms to obtain equations (11) and (12): (11)D^=| 1 ρ^1 ... ρ^k−1 ρ^1 1 ... ρ^k−2 ⋮ ⋮ ⋮ ρ^k−1 ρ^k−2 ... 1| (12)D˜k=| 1 ρ˜1 ... ρ˜1 ρ^1 1 ... ρ^2 ⋮ ⋮ ⋮ ρ˜k−1 ρ˜k−2 ... ρ^k|

ϕ^kk is the partial autocorrelation coefficient and k denotes the number of lag periods.

Data integration is the process of combining data from multiple data sources into one consistent data store, thus providing a complete data source for data mining. The data used in this case involves a table of basic information about individual students and a table of individual students’ psychological problems. The table of mental health issues analyzed for each student is generated by joining through the common primary key XH (student number). Part of the table after joining through the school number is shown in Table 1. From the table, it can be seen that the 10 randomly selected students have the highest mean score of 1.787 for obsessive-compulsive symptoms.

Some of the students’ psychological problems are shown in Table 2. Data mining using the matrix-based Apriori algorithm starts with transforming the related transaction database into a Boolean matrix. After data preprocessing, each student’s mental health problem analysis table is a transaction Ti (TID) with transaction set T={T1,T2,T3,......,Ti}. Each psychological dimension factor is an item set. The corresponding 9 psychological factors in each transaction correspond to 1 if they exhibit symptoms and 0 if they are asymptomatic. Assuming minimum support minsup = 20%, the minimum support count is sup_count = minsup × |T|.

A Boolean matrix that is transformed based on the contents of the table. [ 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1]

By utilizing matrix-based Apriori algorithm for data mining, some of the association rules between college students’ mental health status and college Civics courses are shown in Table 3. The above rules are converted to values by code table for codes and stored in the association rule base. By analyzing the obtained rules, it is found that the support between stress adjustment and anxiety is 0.92 and the confidence level is 0.4. These data indicate that there are indeed some potential relationships between college students’ mental health status and college Civics courses, and that the college Civics courses have a certain guiding role in intervening and guiding college students’ mental health status.

The time series of the depression severity index of the two students is shown in Figure 3. From the ARIMA model, it can be seen that the data series of student A shows more fluctuation, and when the time series is 16, the depression severity index reaches 0.7, and the degree of mood depression is more serious overall.

Figure 3.

The time series of the two studies of the severity of depression

Several possible models are screened by smoothing the non-smooth series, based on the initial determination of the model and order of the series from the ACF and PACF plots of the smooth difference series. Based on the AIC and SC criteria, the p and q that minimize the AIC and SC values are selected. secondly, other statistical parameters such as the adjusted R² and the inverse root of the lag polynomial are also referred to select the most appropriate model.

For the sequence of student B, the partial autocorrelation coefficient of the sequence after first-order differencing is significantly non-zero at k=1, and quickly converges to 0 after k=2. The autocorrelation coefficient is significantly non-zero at k=1. k=2 is on the edge of the confidence interval, which can be considered to be taken as q=1 or q=2. The initial model chosen is ARMA(2,1) or ARMA(2,2), and after attempting to build the model, we found that the value of its adjusted R² is too small. So the second-order difference series was obtained by differencing again for parameter estimation and model testing.

The test results of each model are shown in Table 4. The adjusted R² value of ARIMA(3,2,0) model is 0.6431, which is larger than the values of the other three models, while the AIC and SC values are -6.0231 and -5.9031, which are smaller than those of the other models, so it can be determined that ARIMA(3,2,0) is the optimal model.

The estimation results of the parameters of each model are shown in Table 5. The corresponding autoregressive parameters of the ARIMA(3,2,0) model were -1.0631, -0.7531, and -0.3812, respectively. From this model, it can be seen that the change of the three-difference series of the severity of depression of Student B was governed by the level of depression in the previous three days, and also had a relationship with the random error of the day.

After ordering, estimating and testing to get a more satisfactory model it is time for forecasting. Forecasting is the estimation of the values taken at future moments of the series based on the past and present sample values. The forecasting results of the ARIMA model are shown in Table 6. The results of the prediction made by the model can be seen that the predicted values are closer to the real values, and the minimum error value in the relative error is -0.004%. Therefore, it can be considered that the ARIMA (3,2,0) model finally established by the A student sequence has a better prediction effect and can reflect the change rule of the A student sequence. And the results of the prediction by the model established by the B student sample can be seen, the relative error between the predicted value and the real value is within the acceptable range, the model ARIMA (3,2,2) has a better prediction effect, basically portraying the change rule of the sequence of B students.