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Optimization Simulation System of University Science Education Based on Finite Differential Equations

Published Online: 15 Jul 2022
Volume & Issue: AHEAD OF PRINT
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Received: 24 Apr 2022
Accepted: 28 Jun 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

Driven by the macro-social and economic development and the reform of the educational, financial investment system, and other dynamic mechanisms, Chinese financial education expenditure has shown a rapid growth trend [1]. As shown in Table 1, in 1980, Chinese fiscal education funding was 13.489 billion yuan. After more than 30 years of growth, in 2020, Chinese fiscal education expenditure exceeded 3 trillion yuan for the first time. The total amount is as high as 3137.3 billion yuan. In this paper, the price in 1980 is used as the constant price after deducting the factors of price changes. The actual value of Chinese fiscal education expenditure in 2020 is 550.519 billion yuan. This value increased by 537.03 billion yuan compared with 1980. The introductory average annual growth rate remained above 10%. For a long time, Chinese education financial investment has been dramatically affected by the executive order of 4%. Under the circumstance that the education financial investment system is not yet perfect, this single expansion or replication will inevitably be limited by “scarcity of educational resources” and “exhaustion of executive order dividends.” This will lead to the lack of long-term effect of the growth of financial education funding [2]. We can draw the following conclusions by analyzing the actual development of fiscal education funding since Chinese reform and opening up. In 1981 and 1987, Chinese fiscal education expenditures showed an apparent negative increase after deducting price changes. There are many uncertain factors in the development of education financial investment under various current mechanisms.

In the process of studying the autocorrelation and memory of price or funding fluctuations in academia, many scholars use the improved fractional Brownian motion model as an analytical tool [3]. However, this analyzing memory can be divided into two categories: Gaussian and non-Gaussian processes. Fractional Brownian motion is a Gaussian process. It has the characteristics of additive invariance, long-term correlation, thick tail, and discontinuity. This makes fractional Brownian motion an excellent tool for studying price or funding fluctuations. This paper addresses Chinese educational and financial investment research issues. We incorporate the analytical framework of the fractional Brown model to make a comprehensive judgment on the memory of the current financial education funding growth in China.

A review of research on fractional Brownian motion models
Overview of the Fractional Brownian Motion Model

In 1941, the former Soviet mathematician Kolmogorov further analyzed and defined fractional Brownian motion based on the Hilbert space theory. At the same time, they gave the stochastic integral expression based on standard Brownian movement [4]. After that, British hydrologist Hirst found that using the fractional Brownian motion method can more appropriately describe the actual situation when studying the long-term storage capacity of the Nile Reservoir. At the same time, he set up the Hurst index and used the index as an essential indicator to judge whether the time series data obey the fractional Brownian motion. In general, the Hurst exponent H takes a value between 0 and 1. Different values express different actual conditions [5].

Suppose (Ω, F, P) is a probability space. There is a particle moving randomly on the X-axis. In each time interval, the particle will move to the right or left of the cloth of length ξ. Then the ξ distribution density function is: P(ξ,α)=14πDαexp(ξ24Dα) P\left({\xi,\alpha} \right) = {1 \over {\sqrt {4\pi D\alpha}}}\exp \left({{{{\xi ^2}} \over {4D\alpha}}} \right)

Where α is the time interval. D is the diffusion coefficient. It is assumed that the particle is an IID time series with step ξ1, ξ2, ⋯⋯, ξt. After n random motions, the position of the particle on the X axis is expressed as follows: B(t=nt)=t=1nξt {B_{\left({t = nt} \right)}} = \sum\nolimits_{t = 1}^n {{\xi _t}}

Thus we derive a function Bt concerning time. This is the Brownian motion function, and the distribution density function of S, BtBs at a fixed time point is: P(BtBs)=14pD|ts|exp((BtBs)24d|ts|) P\left({{B_t} - {B_s}} \right) = {1 \over {\sqrt {4pD\left| {t - s} \right|}}}\exp \left({{{{{\left({{B_t} - {B_s}} \right)}^2}} \over {4d\left| {t - s} \right|}}} \right)

0<H<1 and any, s, tR, then the random processes BtH B_t^H and Bt+sHBtH B_{t + s}^H - B_t^H are both fractional Brownian motions with the Hurst exponent H. At the same time BtH B_t^H is a continuous Gaussian process with zero means. Its covariance function is expressed as: E(BtH,BsH)=12(|t|2H+|s|2H|ts|2H) E\left({B_t^H,B_s^H} \right) = {1 \over 2}\left({{{\left| t \right|}^{2H}} + {{\left| s \right|}^{2H}} - {{\left| {t - s} \right|}^{2H}}} \right)

Any t ≥ 0 has EBtH=0 EB_t^H = 0 , E(BtH)2=t2H E{\left({B_t^H} \right)^2} = {t^{2H}} .

BtH B_t^H has smooth increments. For any s>0 random process Bt+sHBtH B_{t + s}^H - B_t^H and random process BtH B_t^H have the same finite-dimensional distribution.

When μ>0, BμtH B_{\mu t}^H and μHBμtH {\mu ^H}B_{\mu t}^H also have the same finite-dimensional distribution.

The correlation coefficient of BtH B_t^H and BsH B_s^H calculated by formula (4) is: C(t)=E(BsHBstH)(BtHBsH)E(BsHBstH)2E(BtHBsH)2 C\left(t \right) = {{E\left({B_s^H - B_{s - t}^H} \right)\left({B_t^H - B_s^H} \right)} \over {\sqrt {E{{\left({B_s^H - B_{s - t}^H} \right)}^2}} \sqrt {E{{\left({B_t^H - B_s^H} \right)}^2}}}}

Assuming BsH=0 B_s^H = 0 , adjust the length of step ξ so that α=1, 2Dα=1, we can get: C(t)=E[(BstH)(BtH)]E(BtH)2=22Ht1 C\left(t \right) = {{E\left[{\left({B_{s - t}^H} \right)\left({B_t^H} \right)} \right]} \over {E{{\left({B_t^H} \right)}^2}}} = {2^{2H - t}} - 1

It can be known from the above formula that when H=1/2, C(t)=0, which is the standard Brownian motion. This means that there is no correlation between past and future increments. This is a particular case of fractional Brownian motion [6]. And when H≠1/2, there is a specific correlation between history and future increments. The fractional Brownian motion at this time has a particular memory. The value of the H index can show the positive and negative and the length of this memory effect. The process of growth shows long-term effects. Past growth (or decline) trends can continue in the future.

Measure the memory of financial investment in education

We can draw the following conclusions by analyzing the growth of Chinese fiscal education funding from 1980 to 2020. At present, Chinese education financial investment is also a random movement that keeps wandering under the impact of various dynamic mechanisms such as macro-social economic development and educational, financial investment system reform [7]. This is consistent with the assumption of fractional Brownian motion.

Model setting and sample processing

The fluctuation of financial education funding in China is assumed to be a continuous Gaussian process with zero mean and obeying fractional Brownian motion [8]. Suppose BtH(T=1,2,,n) B_t^H\left({T = 1,\,2, \cdots,n} \right) is the observed financial education expenditure in China over the years. BtH B_t^H obeys fractional Brownian sign and the Hurst exponent is H. Also assume ξ1=B1H,ξ2=B2HB1H,,ξt=BtHBN1H {\xi _1} = B_1^H,\,{\xi _2} = B_2^H - B_1^H, \cdots,{\xi _t} = B_t^H - B_{N - 1}^H . X is the integral of ξ and E(k) represents their covariance.

This article uses Chinese fiscal education funding from 1980 to 2020 as an indicator. This dataset meets and exceeds the minimum number of sample observations for the R/S and Whittle methods. The financial education expenditure data are compiled from the “China Education Fund Statistical Yearbook” over the years. The selection of the research data set in this paper has deducted the impact of price changes on the analysis results. This article still uses the price in 1980 as the constant price to recalculate the data indicators.

selection of model calculation method

The Hurst exponent H estimate calculation methods in academia mainly include the R/S process was first proposed by Hirst himself in 1951 when he calculated the long-term storage capacity of the Nile Reservoir. Its central role is to analyze the rescaled cumulative mean deviation [9]. Then the R/S method was introduced into economics and finance by Mandelbrot B. This method becomes an effective research method for fractal features of time series. Greene and FielitzBD use the R/S method to calculate the Hurst index value of daily returns of more than 200 stocks in the New York Stock Exchange. From this, it is judged that they have significant long memory and correlation. Then you systematically revised the R/S method based on the memory problem of daily stock returns. They established the MR/S process. This overcomes the short-term correlation and long-term correlation indistinguishable in the R/S process to a certain extent. Domestic scholars have used R/S and MR/S methods to analyze China's stock market's memory and nonlinear characteristics.

However, traditional estimation methods such as R/S modified R/S (MR/S) and V/S can only judge whether the time series has long memory [10]. They have a significant deviation in estimating the memory strength Hurst exponent H. Professor O. Rose of the University of Wuerzburg, Germany, systematically compared the advantages and disadvantages of the R/S method and the Whittle method in estimating the Hurst exponent H value. They verified that Whittle's practice has higher efficiency and robustness in estimating parameters through specific empirical cases. Whittle's approach can effectively avoid the defects of other ways in terms of stability and calculation accuracy. This method is more suitable for statistical inference. Therefore, this paper uses both Whittle and R/S methods to estimate the Hurst index H of the fluctuation of financial education funding in China over the years. We use this to judge whether the current growth of Chinese fiscal education funding has long-term effects.

Whittle method estimation and calculation

First, we use Whittle's method to estimate the Hurst exponent H. From the above assumptions, the spectral density function of Chinese fiscal education funding over the years can be obtained: f(x,H)=k=E(k)etkx=(1cosx)k=|x+2kπ|12H f\left({x,H} \right) = \sum\nolimits_{k = - \infty}^\infty {E\left(k \right){e^{tkx}} = \left({1 - \cos x} \right)} \sum\nolimits_{k = - \infty}^\infty {{{\left| {x + 2k\,\pi} \right|}^{- 1 - 2H}}}

Then we make the following assumptions In(x)=12πn|j=1Nεjejx|2 {I_n}\left(x \right) = {1 \over {2\pi n}}{\left| {\sum\nolimits_{j = 1}^N {{\varepsilon _j}{e^{jx}}}} \right|^2} f˜(x,H)=exp(12πππlogf(x,H)dx)f(x,H) \tilde f\left({x,\,H} \right) = \exp \left({{1 \over {2\pi}}\int_{- \pi}^\pi {\log \,f\,\left({x,\,H} \right)dx}} \right)f\left({x,\,H} \right)

Where In(x) denotes the periodogram of a given time series ξ of length n. Thus, we can further obtain the following relation: Lnω(H)=ππIn(x)f˜(x,H)dx+ππlog[f˜(x,H)]dx L_n^\omega \left(H \right) = \int_{- \pi}^\pi {{{{I_n}\left(x \right)} \over {\tilde f\left({x,\,H} \right)}}dx +} \int_{- \pi}^\pi {\log \left[{\tilde f\left({x,H} \right)} \right]dx}

Then Whittle estimates that H \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over H} satisfies H=argmaxLnω(H) \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over H} = \arg \,\max L_n^\omega \left(H \right) . And the confidence interval of the Hurst index at the 95% confidence level. H±1.962D1(H)n \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}}\over H} \pm 1.96\sqrt {{{2{D^{- 1}}\left(H \right)} \over n}}

In D(H)=12πππ[Hlogf(x,H)]2dx D\left(H \right) = {1 \over {2\pi}}\int_{- \pi}^\pi {{{\left[{{\partial \over {\partial H}}\log f\left({x,H} \right)} \right]}^2}dx}

We write sample data into C++ software. We can estimate the Hurst exponent H=0.4357<0 for the entire sample period by writing a program. This shows that under the impact of various dynamic mechanisms, there is a certain negative correlation between Chinese current fiscal education expenditure and its increment [11]. Its growth process has a short memory. The current education financial investment system has not shown a good long-term effect in promoting the growth of financial education funds. At the same time, we use the software R-3.1.1 to write a program that can also estimate the Hurst exponent H=0.4736728<0.5 for the entire sample period. The confidence interval for the Hurst index at 95% confidence is [0.35738, 0.58996]. This again supports the above conclusion.

R/S method estimation and calculation

We refer to the relevant assumptions of domestic and foreign scholars on the R/S method in the fractional Brownian motion model. We use the R/S method to estimate the Hurst exponent H of the growth of fiscal education funding in China, and we can obtain the functional equation: R(α)S(α)=(λα)H {{R\left(\alpha \right)} \over {S\left(\alpha \right)}} = {\left({\lambda \alpha} \right)^H} ξt=BtBt1 {\xi _t} = {B_t} - {B_{t - 1}}

α is the time interval. λ is a constant. {ξt} (t = 1, 2, ⋯, n) represents the incremental sequence of fiscal education funding starting in the second year. n is the number of samples observed. There are 37 in total. The mean A(α), standard deviation S(α), cumulative deviation S(t, α), and range R(α) in the time interval α are expressed as follows A(α)=1αt=1αξt A\left(\alpha \right) = {1 \over \alpha}\sum\nolimits_{t = 1}^\alpha {{\xi _t}} S(α)=[1αt=1α(ξtA(α))2]12 S\left(\alpha \right) = {\left[{{1 \over \alpha}\sum\nolimits_{t = 1}^\alpha {{{\left({{\xi _t} - A\left(\alpha \right)} \right)}^2}}} \right]^{{1 \over 2}}} C(t,α)=t=1α[ξtA(α)] C\left({t,\alpha} \right) = \sum\nolimits_{t = 1}^\alpha {\left[{{\xi _t} - A\left(\alpha \right)} \right]} R(α)=maxC(t,α)minC(t,α)(1tn) R\left(\alpha \right) = \max C\left({t,\alpha} \right) - \min \,C\left({t,\alpha} \right)\left({1 \le t \le n} \right)

We now estimate the Hurst exponent H. We now take the natural logarithm of both sides of functional equation (13) to obtain the following formula ln[R(a)S(a)]=Hln(λ)+Hln(α) \ln \left[{{{{R_{\left(a \right)}}} \over {{S_{\left(a \right)}}}}} \right] = H\,\ln \left(\lambda \right) + H\ln \left(\alpha \right)

H is a constant. Then H ln(λ) is also a constant. So we can assume λ = H ln(λ), then functional equation (19) can be rewritten as: ln[R(a)S(a)]=λ+Hln(α) \ln \left[{{{{R_{\left(a \right)}}} \over {{S_{\left(a \right)}}}}} \right] = \lambda + H\ln \left(\alpha \right)

Combining the above functional equations, the mean A(α), the standard deviation S(α), the cumulative deviation C(t, α), and the range R(α) can be calculated from the incremental sequence {ξt} (t = 1, 2, ⋯, n) of fiscal education funding. This way we get R(a)S(a)(a=1,2,35) {{{R_{\left(a \right)}}} \over {{S_{\left(a \right)}}}}\left({a = 1,\,2,\, \cdots 35} \right) .

Then we estimate the parameter λ and the Hurst exponent H in the equation respectively according to the functional equation (20) and using the ordinary least squares method OLS. Now we write the sample data into the analysis software Matlab. 2012. We will program the above calculation process [12]. Finally, we use the ordinary least squares method OLS to calculate the Hurst exponent H, and the results are shown in Table 1:

Estimation results of Hurst exponent H based on R/S method

Estimated value t-statistic P-value
a=1–11
H 0.6371 111.7922 0
a=1–18
H 0.5334 120.8717 0
a=1–24
H 0.6762 137.558 0
a=1–35
H 0.4707 247.1779 0
Analysis of empirical conclusions
Memory analysis in the stage of “dividing revenue and expenditure, and grading and contracting.”

The government avoids the drawbacks of excessive and strict control of the central government's local fiscal revenue and expenditure under the traditional planned economic system. It fully mobilized the enthusiasm of local governments to increase revenue and reduce costs. They decided to implement a new financial management system of “dividing revenue and expenditure, and grading responsibility” from that year [13]. This gives the local government the motivation and ability to develop local education and actively promotes the reform and change of Chinese education financial investment system.

On May 27, 1985, the Central Committee of the Communist Party of China issued the “Decision on the Reform of the Education System.” In the “Decision,” they linked the financial investment in education with fiscal revenue growth for the first time. At the same time, they put forward the strategic goal of “two growth.” As shown in Table 1, the time interval of this stage is 1980–1992. When α=1–11, the estimated value of the Hurst exponent H is 0.6371>0.5. And the P-value is 0. This shows that the growth of fiscal education funding has a long memory in the stage of “dividing revenue and expenditure, grading and contracting.”

Memory analysis of the tax-sharing system stage

The growth mechanism has not fully exerted its due effect in the early stage of the reform, resulting in a small estimated value of the Hurst index H and a downward trend at this stage [14]. As shown in Table 2, the time interval at this stage is extended to 1980–1999. When α=1–18, the estimated value of the Hurst index H is 0.5334>0.5, and the past growth (or decline) of fiscal education funding is It can continue in the future. Still, after comparing with the first stage, it can be found that the estimated value of the Hurst exponent H in the second stage has dropped significantly. In 1993, the proportion of Chinese fiscal education funding to GDP was 2.46%, and then it kept declining until it recovered to 2.41% in 1998.

Memory analysis of public finance stage

The tax-sharing fiscal management system reform focuses on adjusting the distribution of budgetary and tax revenue between the central and local governments. To a certain extent, the upward shift of the focus of fiscal revenue has resulted in the mismatch between local government financial power and administrative power. This has resulted in the lack of clarity and funding for many public projects. The time interval of this stage is extended to 1980–2005. When α=1–24, the estimated value of Hurst exponent H is 0.6762>0.5. In the initial phase of implementing the public financial system, the growth of fiscal education funding still maintains an excellent long-term memory.

Memory analysis in the post “4%” stage

The time interval at this stage is extended to 1980–2020—the Hurst index H=0.4707<0.5 for the whole sample period. The estimated value of the Hurst exponent H has dropped significantly. For the first time in more than 30 years of reform and opening up, it fell below 0.5. At present, there is a specific negative correlation between Chinese fiscal education expenditure and its increment. Its growth process has a short memory. Financial education funding cannot continue the growth trend of the previous stage in the future. The “policy dividend” of 4% of educational financial investment has gradually disappeared. The current “index-linked” education financial investment system has not shown a good long-term effect in the process of promoting the growth of financial education funds.

Conclusion

This paper incorporates the current memory analysis of Chinese educational, financial investment into the research framework of fractional Brownian motion. We select Chinese fiscal education expenditure from 1980 to 2020 as the accounting indicator. This satisfies the model's minimum requirements for the number of sample observations. According to the development process of Chinese education financial investment system, we divide the calculation of Hurst index H into four different stages. The calculation results do not consider the practical significance of the model research while meeting the statistical requirements. Finally, this paper uses R/S and Whittle methods to calculate the Hurst index H of fiscal education funding. At the same time, we compare and analyze the calculation results to ensure the accuracy of the calculation results.

Estimation results of Hurst exponent H based on R/S method

Estimated value t-statistic P-value
a=1–11
H 0.6371 111.7922 0
a=1–18
H 0.5334 120.8717 0
a=1–24
H 0.6762 137.558 0
a=1–35
H 0.4707 247.1779 0

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