Study of Change-Point Detection and Applications Based on Several Statistical Methods

Abstract

In the current global context of economic integration, unexpected events have an important influence in the financial field. In 2020, the “COVID-19” outbreak triggered financial turmoil throughout the whole country and even in the global market. In the wake of this era, how to sum up past developments and predict future development through change-point detection is particularly important. In this paper, four methods for detecting change-points are presented: the likelihood ratio method, least squares method, CUSUM method, and local comparison method. Considering that Bernstein polynomials have worked well in density function approximation, the multi-dimensional Bernstein polynomials are presented. The study applies multiple change-point detection methods to determine the most suitable degree of freedom $m_j$ for multi-dimensional Bernstein models, after which various rewriting expressions can be obtained. Next, “COVID-19” data and money supply data are used for change-point detection with good results. Then, we focus on conducting change-point testing on the S&P 500 index and SSE 50 index, indicating strong symmetry when major crisis events occur. All analyses indicate that change-point detection plays an important role in identifying major crisis events and financial shocks.

1. Introduction

In recent years, with the continuous development of statistics, the change-point problem has become a hot research topic. From a historical perspective, it is generally believed that research into the problem can be traced back to the first article on change-point statistical analysis published by Page in the internationally renowned statistical journal Biometrika in 1954 [1]. The article used continuous sampling theory and the cumulative sum method to detect whether there were significant fluctuations in product quality on a production line, and to prevent changes as well as provide warnings of them.

The following problems of the change-point problem have been mainly solved: (1) Detection of unknown change moments and determining when the parameters of a random process change to correct the process and change the control parameters. (2) Mechanism analysis of change-points. By analyzing the stability of the system structure, external control variables are identified to describe and predict the occurrence of morphological changes. (3) Practical application of the change-point model [2].

Currently, the research on change-point detection is widely used in many fields such as finance, medicine, industry, economics, and earthquake forecasting. For example, in epidemiology, one of the most important issues is the magnitude and moment of change in the rate of transmission of an infectious disease over the course of its transmission, which is particularly critical in determining effective means of treatment and control [3]. In seismology, qualitative changes in seismic signal sequences are very likely to indicate the presence of more prominent faults in the subsurface rock layers [4]. In the fields of economics and finance, the occurrence of major political events and health and safety issues often leads to the occurrence of stock price surges or crashes. Other specific applications of the change-point problem include pattern recognition for computers and rhythmic analysis of electrocardiograms.

Especially in the context of the big data era, unexpected events in the financial sector cannot be ignored in the context of global economic integration. In 1997, the sudden attack on Thailand’s foreign exchange market by the U.S. Quantum Fund led to a financial crisis in Southeast Asia; in 1998, the Russian government’s announcement of a delay in the repayment of short-term bonds led to the Long-Term Capital Management Fund episode in the U.S.; and in 2007, the crisis in the U.S. subprime mortgage market triggered a global upheaval in financial markets. How to determine or predict the occurrence or location of a change-point based on existing (historical, all observable) information is a highly important task in various fields of society, including industry, economics, finance, and in the statistical community [5].

Since the issue of change-points was proposed, a large number of scholars have begun to participate in this research, which has led to rapid development in the theory and application of change-point problems. The change-point problem plays a very important role in practical applications, and most international scholarly research is based on specific distribution functions and models. In their monograph, Csörgö and Horváth [6] provide an in-depth study of the limit theory of change-point statistical analysis and summarize the last thirty years of research on change-point theory.

There are many methods for change-point detection, and currently, commonly used methods include the likelihood ratio method, least squares method, cumulative sum method (CUSUM method), and local average method. For classical change-point detection methods, many scholars have accomplished many complete theoretical achievements. Brown and Durbin et al. [7] proposed the cumulative sum of squares method to discuss the problem of variance change-points in time series and studied the asymptotic distribution of change-points. Kokoszka and Leipus [8] proved the consistency of the CUSUM statistic and used the CUSUM method to complete mean change-point detection in the ARCH model, achieving good results. Kim and Cho et al. [9] used the CUSUM statistic to detect changes in financial time series data. Lee and Maekawa et al. [10] analyzed the parameter changes of the GARCH (1,1) model based on the residual cumulative sum of squares hypothesis test, provided the limit distribution of the CUSUM test statistic, and applied it to the detection of change-points in dollar exchange rate data. James and Siegmund et al. [11] used the likelihood ratio test method to study the multivariate normal distribution mean vector change-point model. Bai and Perron [12] proposed a multi variable likelihood ratio test method in regression models to determine the position and quantity of variables. Gurevich and Vexler [13] used the generalized likelihood ratio method to detect parameter change-points in Logistic regression models and analyzed the asymptotic nature of the change-point likelihood ratio estimates.

Given the excellent approximation properties of Bernstein polynomials, many scholars have conducted extensive research on their estimation of continuous functions. Cheng [14] studied the Bernstein polynomial estimation of smooth quantile functions. Guan et al. [15] studied non-parametric estimation of error detection rate based on Bernstein polynomials. Igarashi et al. [16] studied improving the convergence rate of Bernstein polynomial density estimation. Guan [17,18] used a modified Bernstein polynomial approximation of density functions and used the likelihood ratio test for variable points in finding degrees of freedom, then an approximate model for estimating univariate density function based on grouped data was proposed, which was called a Bernstein polynomial model. It has been proven that the convergence speed of its density estimation is almost parameterized, and this method has been compared with some existing methods through simulation, and applied to chicken embryo data. Belalia et al. [19] provided a non-parametric test for the independence of continuous random variables based on the Bernstein distribution function and Bernstein Copula density function. In terms of multi-dimensional Bernstein polynomial density estimation, Axel Tenbusch [20] studied two-dimensional Bernstein polynomial density estimation. Wang and Guan [21] used the Bernstein polynomial model to estimate non-parametric multivariate density functions.

Based on the status of such research, this study will be carried out along the following lines. First, in Section 2, an introduction to four methods for detecting change-points is presented: the likelihood ratio method, CUSUM method, local comparison method and least squares method. Considering that Bernstein polynomials have worked well in density function approximation, the multi-dimensional Bernstein polynomials are presented. Then, in Section 3, the multi-dimensional Bernstein polynomial in the new form is uniquely determined only by the coefficient $p_m$, and its convergent iterative form can be derived using the EM algorithm. Next, Section 4 applies the four change-point detection methods from Section 2 to find the most suitable degree of freedom $m_j$ for the multi-dimensional Bernstein model. Combining the multi-dimensional Bernstein log likelihood function $l\left(p_m\right)$, a representation of the change-point ${\tau }_j$ is given, and numerical simulations of the binary normal distribution mean single-change-point model are conducted for each change-point detection method. Finally, in Section 5, an empirical analysis based on the change-point detection method mentioned in Section 4 is carried out.

The innovation of this study lies in applying multiple change-point detection methods to find the most suitable degree of freedom $m_j$ for the multi-dimensional Bernstein model, obtaining various rewritten expressions for change-point detection quantities. On the basis of this innovation, through numerical simulation and empirical analysis of the single-change-point model for the mean of a bivariate normal distribution, the point detection method is applied to financial data, and practical conclusions are drawn.

2. Introduction to Change-Point Detection Methods

At present, the research methods for the analysis of change-points mainly include the maximum likelihood method, the least squares method, local comparison method, Bayes method, etc. The main idea of the Bayes method is to treat the parameters and variables in a model with change-points as random variables, and then use the prior distribution and sample distribution to gradually derive the posterior distribution of the change-points in the model. Based on the posterior distribution of the change-point position, Bayes’ inference of the remaining parameters is discussed.

2.1. Likelihood Ratio Test

The earliest method of studying change-point detection in change-point theory was to employ a likelihood ratio test to detect the existence of change-points. This method is commonly used to distinguish the distribution of samples. The idea of the likelihood ratio method is based on the idea of maximum likelihood estimation, which uses likelihood ratios to identify and test certain parameters in the model, including change-points, by calculating their maximum likelihood estimation. The time when the change-point occurs is also considered as a parameter. At present, most of the research work is focused on detecting changes in the mean of normal distributions.

Here, we consider a simple form of the normal distribution change-point model: Assume that the random variables $X_1,X_2,L,$ and $X_n$ are independent, and

(1)$\begin{array}{c}{X}_i\sim N\left({\mu }_1,{\sigma }_1^2\right),i=1,2,\Lambda ,k.\end{array}$

(2)$\begin{array}{c}{X}_i\sim N\left({\mu }_2,{\sigma }_2^2\right),i=k+1,\Lambda ,k.\end{array}$

where k, ${\mu }_1$, ${\mu }_2$, ${\sigma }_1^2$, and ${\sigma }_2^2$ are unknown. The existence of a mean change-point is considered when ${\mu }_1\ne {\mu }_2$. The following test problem is considered:

(3)$H_0:{\mu }_1={\mu }_2\leftrightarrow H_1:{\mu }_1\ne {\mu }_2$

This means the null hypothesis is that there is no mean change-point, and the alternative hypothesis is that there is a mean change-point. The estimation steps for the mean change-point in this model are as follows:

The first step is to use the likelihood ratio test idea for a fixed $k\left(2\le k\le n\right)$. Let

(4)$\begin{array}{c}{\lambda }_k={\displaystyle \frac{{\displaystyle \underset{(\mu ,{\sigma }^2)\in {\Theta }_0}{sup}}{\displaystyle \prod _{i=1}^n}f\left(x_i;\mu ,{\sigma }^2\right)}{{\displaystyle \underset{({\mu }_1,{\mu }_2,{\sigma }^2)\in \Theta }{sup}}{\displaystyle \prod _{i=1}^k}f\left(x_i;{\mu }_1,{\sigma }^2\right){\displaystyle \prod _{i=k+1}^n}f\left(x_i;{\mu }_2,{\sigma }^2\right)}},{\Lambda }_k=-2ln{\lambda }_k.\end{array}$

For the second step, the test statistic can be taken as ${\Lambda }_k={\displaystyle \underset{2\le k\le n}{max}}\left(-2ln{\lambda }_k\right)$. If ${\Lambda }_k$ is large, reject the original hypothesis. This indicates the existence of a mean change-point and the estimated change-point is $\widehat{k}=arg\left\{k:{\Lambda }_n={\displaystyle \underset{2\le k\le n}{max}}{\Lambda }_k\right\}$. Otherwise, accepting the original hypothesis means that there is no change-point.

2.2. Cumulative Sum Method

The Cumulative Sum (CUSUM) method is a commonly used change-point estimation method both domestically and internationally. The simple form of this method is as follows: Consider the model $X_t=\mu \left(t\right)+{\epsilon }_t$, $t=1,2,\Lambda ,n$. Here, $\mu \left(t\right)$ is a non-random function and ${\epsilon }_t$ is a linear process. Consider the simplest scenario where $\mu \left(t\right)$ only takes two values:

(5)$\mu \left(t\right)=\left\{\begin{array}{c}{\mu }_1,\begin{array}{cc}& t\le k^{*}\end{array}\\ {\mu }_2,\begin{array}{cc}& t>k^{*}\end{array}\end{array}\right.$

where ${\mu }_1$, ${\mu }_2$ and $k^{*}$ are unknown. Then, the CUSUM estimate for point $k^{*}$ is defined as

(6)${\widehat{k}}^{*}=min\left\{k:\left|U_k\right|={\displaystyle \underset{1\le j\le n}{max}}\left|U_k\right|\right\}$

where $U_k={\left\{{\displaystyle \frac{k(n-k)}{n}}\right\}}^{1-\gamma }\left\{{\displaystyle \frac{1}{k}}{\displaystyle \sum _{i=1}^k}{X}_j-{\displaystyle \frac{1}{n-k}}{\displaystyle \sum _{i=k+1}^n}{X}_j\right\}$, $0\le \gamma \le 1$.

2.3. Local Comparison Method

The local comparison method is also called the sliding comparison method or sliding window method; the main idea is to examine the change of a certain statistic in each location, in the vicinity of the non-change-point of the location. The estimation will remain stable, and take as significant the location of the change-point in the estimation. Generally speaking, this method has two advantages: (1) The test statistic derived from the hypothesis testing problem has a relatively simple asymptotic distribution. (2) This provides a convenient approach for estimating the efficiency of testing and for the interval estimation of change-points. The following first describes the steps for estimating the change-points using the method. The model is as follows:

(7)$X_i=\left\{\begin{array}{c}\begin{array}{cc}{a}_1+e_i,& 1\le i\le k;\end{array}\\ \begin{array}{cc}{a}_2+e_i,& k\le i\le n.\end{array}\end{array}\right.$

The random error $e_i(1\le i\le n)$ in the equation is independent and identically distributed, so $E\left(e_i\right)=0$, $\mathrm{Var}\left(e_i\right)={\sigma }^2<+\infty$. Meanwhile, n, $a_1$, $a_2$, and ${\sigma }^2$ in the equation are all unknown. The estimation steps for the mean change-point $k^{*}$ of the model are as follows:

For the first step, since changes in means are involved here, it is natural to look at changes in sample means or sample sums within neighboring localities. Therefore, take $i\in \left[1,n\right]$ and an appropriate natural number d to sum and subtract the d observed values on each side of i (including i on the right side) in order to obtain $Y_i=(X_i+\Lambda +X_{i+d-1})-(X_{i-d}+\Lambda +X_{i-1})$, $i=d+1,d+2,\Lambda ,n-d+1$.

For step two, if i is not a change-point and its distance from the change-point m is not less than 0, then $Y_i$ tends to be closer to 0. Otherwise, if $i=m$ or is close to m, which means it has a change-point, then if the difference between the values of the two terms on the right of $Y_i$ is large, there is a change-point. Thus, a large difference between the two values to the right of $Y_i$ means that there is a change-point. Therefore, set $\widehat{m}$ as the index that maximizes $\left|Y_i\right|$, where $\left|y_{\widehat{m}}\right|=max\left(\left|y_{d+1}\right|,\left|y_{d+1}\right|,L,\left|y_{n-d+1}\right|\right)$, then take $\widehat{m}$ to be the estimate of the variable point m.

For the final step, when $\left|y_{\widehat{m}}\right|$ exceeds a certain critical value C, denying $H_0$ means there is a change-point; otherwise, $H_0$ is considered to have no change-point.

2.4. Least Squares Method

The basic idea of the least squares method is to use the sum of the squares of the differences between observed values and theoretical values as the objective function to estimate the relevant parameters at the minimum point. Due to the fact that this method does not require knowledge of the distribution of model errors and the model is simple and easy to design, it has been widely used in dealing with change-point problems.

To construct the discrete model for the mean change-point problem, set $X_i=a_i+e_i$, $i=1,2,\Lambda ,n$. Define $a_1={\Lambda }_1=a_{m_1-1}=b_1$, $a_{m_1}=\Lambda =a_{m_2-1}=b_2$, L, and $a_{m_q}=\Lambda =a_n=b_{q+1}$, where $1\le m_1<m_2<\Lambda <m_q\le n$; if $b_{j-1}\ne b_j$, then $m_j$ is a change-point. We assume that the random errors $e_1$, $e_2$, $\Lambda$, and $e_n$ are independent and identically distributed, and $E\left(e_i\right)=0$, $\mathrm{Var}\left(e_i\right)={\sigma }^2<+\infty \left(i=1,2,L,n\right)$. To determine the presence or absence of change points, the following hypothesis test can be obtained:

(8)$H_0:b_1=b_2=\Lambda =b_{q+1}\leftrightarrow H_1:b_1=\Lambda =b_j\ne b_{j+1}=\Lambda =b_{q+1}$

where $H_0$ represents the absence of change-points. The steps of this test are as follows:

For the first step, calculate the statistical measures as follows:

(9)$S={\displaystyle \sum _{i=1}^n}\left(X_i-\overline{X}\right)$

where $\overline{X}={\displaystyle \sum _{i=1}^n}{\displaystyle \frac{X_i}{n}}$.

In step two, divide the samples $X_1$, $X_2$, $\Lambda$, and $X_n$ into two parts: $X_1$, $X_2$, $\Lambda$, and $X_{i-1}$ and $X_i$, $X_{i+1}$, $\Lambda$, and $X_n$. Next, calculate their sample variances separately and add them up to obtain

(10)$S_i={\displaystyle \sum _{j=1}^{i-1}}{\left(X_j-{\overline{X}}_{j_1}\right)}^2+{\displaystyle \sum _{j=1}^n}{\left(X_j-{\overline{X}}_{j_2}\right)}^2,\left(2\le i\le n\right)$

where ${\overline{X}}_{j_1}={\displaystyle \frac{1}{i-1}}{\displaystyle \sum _{j=1}^{i-1}}{X}_j$, and ${\overline{X}}_{j_2}={\displaystyle \frac{1}{n-i+1}}{\displaystyle \sum _{j=1}^n}{X}_j$.

For step three, according to the identity equations $S=S_i+n^{-1}(i-1)(n-i+1){({\overline{X}}_{j_1}-{\overline{X}}_{j_2})}^2,\left(2\le i\le n\right)$ and $E\left(n^{-1}(i-1)(n-i+1){({\overline{X}}_{j_1}-{\overline{X}}_{j_2})}^2\right)={\sigma }^2+n^{-1}(i-1)(n-i+1){(\mathrm{E}{\overline{X}}_{j_1}-E{\overline{X}}_{j_2})}^2$, if there is a change-point, the above equation is generally not 0. Therefore, it is known that the existence of the change-point increases the difference between $S_1$ and $S_2$, denoted as $S^{*}=min\left\{S_2,\Lambda ,S_n\right\}$.

In step four, take the test significance level $\alpha$ and calculate the c value:

(11)$C={\sigma }^2\left(2lnlnn+lnlnlnn-ln\pi +x_{\alpha }\right)$

Here, $x_{\alpha }=-xln\left(-{\displaystyle \frac{1}{2}}ln(1-\alpha )\right)$. If ${\sigma }^2$ is unknown in the general equation, the following estimate can be used instead:

(12)${\widehat{\sigma }}^2={\displaystyle \frac{S^{*}}{(n-2)(lnlnn-lnlnlnn-2.4)}}.$

For the final step, if $S-S^{*}>C$, then negate $H_0$ and consider there to be a change-point—otherwise, accept $H_0$ (assuming the test has asymptotic level $\alpha$). The idea behind this type of testing method is very intuitive, but it should be noted that the least squares method is most effective for testing with exactly one change-point. The difference between S and $S_i$ may be offset by changes in the mean when multiple change-points exist.

3. Bernstein Polynomial Form and Coefficient Determination

The use of polynomial consistent approximation for continuous functions is an important result in function approximation theory, which has wide applications in finance and the engineering sciences. Among them, the Bernstein polynomial has a good fitting effect, as shown in the example in Figure 1.

However, estimating its degree of freedom m has become a thought-provoking issue, as has applying change-point detection methods to find the most suitable degree of freedom m. In this section, we first introduce Bernstein polynomials.

3.1. One-Dimensional Bernstein Polynomial

Any continuous (density) function defined on $\left[0,1\right]$ can be approximated by a one-dimensional Bernstein polynomial with m degrees of freedom, which takes the form

(13)$f_m\left(t\right)=B_{m}f\left(t\right)={\displaystyle \sum _{i=0}^m}f\left({\displaystyle \frac{i}{m}}\right)b_{mi}\left(t\right),0\le t\le 1$

where $b_{mi}\left(t\right)=\left(\begin{array}{c}m\\ i\end{array}\right)t^i{(1-t)}^{m-i}$, $i=0,\Lambda ,m$. Here, call $B_m$ the m-th Bernstein operator.

It is not difficult to see that $B_m$ can be seen as a linear combination of the beta distribution $Be(i+1,m+1-i)$,$i=0,\Lambda ,m$, which can be rewritten as follows:

(14)$f_B(t;p)={\displaystyle \sum _{i=0}^m}{p}_{mi}{\beta }_{mi}\left(t\right)$

where $p_m={(p_{m0},p_{m1},L,p_{mm})}^T$, $p_{mi}\ge 0$, ${\beta }_{mi}\left(t\right)=(m+1)\left(\begin{array}{c}m\\ i\end{array}\right)t^i{(1-t)}^{m-i}$, $i=0,L,m$.

At this point, if f has continuous r-order derivatives ($r\ge 2$) on $\left[0,1\right]$, the convergence rate of $f\left(t\right)$ can reach at least $O\left(m^{-r/2}\right)$ [22]. In multiple studies, it is referred to as a polynomial with positive coefficients.

Theorem 1 

([23]). Set f as a continuous function or $C^1$ function on $\left[0,1\right]$. Then, if $\forall \epsilon >0$, $\exists N\in N^{*}$, so that when $m>N$, $\forall x\in \left[0,1\right]$, the following holds:

(15)$\left|B_{m}f\left(x\right)-f\left(x\right)\right|<\epsilon .$

Therefore, for $\forall x\in \left[0,1\right]$, it satisfies that in the condition of $\forall \epsilon >0$, we have the following equation:

(16) $\underset{m\to \infty }{lim}p\left\{\left|B_{m}f\left(x\right)-f\left(x\right)\right|>\epsilon \right\}=0.$

$B_{m}f\left(x\right)$ is the consistent estimator of $f\left(x\right)$, denoted as $B_{m}f\left(x\right)\stackrel{p}{\to }f\left(x\right)$.

3.2. Multi-Dimensional Bernstein Polynomial

3.2.1. Multi-Dimensional Bernstein Polynomial Forms and Properties

After approximating the one-dimensional density function with a Bernstein polynomial and extending it to the joint distribution, its d-dimensional Bernstein polynomial form is as follows:

(17)$P_m\left(t\right)={\displaystyle \sum _{i=0}^m}p\left(i\right)·{\beta }_{mi}\left(t\right),p\left(i\right)\ge 0$

where $t=(t_1,L,t_d)$, $m=(m_1,L,m_d)$, $i=(i_1,L,i_d)$, the operator symbol ${\sum }_{i=0}^m={\sum }_{i_{1=0}}^{m_1}L{\sum }_{i_{d=0}}^{m_d}$, ${\beta }_{mi}\left(t\right)={\prod }_{j=1}^d{\beta }_{m_j{i}_j}\left(t_j\right)$, and the maximum number of non-zero coefficients is $K={\prod }_{j=1}^d(m_j+1)$.

It is not difficult to observe that $P_m\left(t\right)$ can be seen as a linear combination of K-fold beta distributions. Therefore, assuming $X={(X_1,L,X_d)}^T$ is a d-dimensional random variable with a continuous distribution function F, the multivariate continuous density function f can be approximated in the following form:

(18)$f_m(t;p)={\displaystyle \sum _{i=0}^m}p\left(i\right)·{\beta }_{mi}\left(t\right)$

where $p=p_m=\left\{p\left(i\right):0\le i\le m\right\}\in S_m\equiv \left\{p\left(i\right):0\le i\le m,p\left(i\right)\ge 0,{\sum }_{i=0}^{m}p\left(i\right)=1\right\}$. Meanwhile, the $X_j$ edge density function is also a $(m_j+1)$-fold beta distribution.

The multi-dimensional Bernstein polynomial approximation of multivariate density functions also exhibits good asymptotic results. When m is large enough, there exists $p_0\in S_m$, $k>0$, and for all $t\in {(0,1)}^d$, the following is satisfied:

(19)${\displaystyle \frac{\left|f_m(t;p_0)-f\left(t\right)\right|}{f\left(t\right)}}\le C(d,f)m_0^{-k/2},$

where $m_0={min}_{1\le j\le d}{m}_j$ and $C(d,f)$ depends on d and f, but is not related to the degree of freedom m.

3.2.2. Multi-Dimensional Bernstein Log Likelihood Function

For convenience when it comes to subsequent theoretical research, the definition of multi-dimensional Bernstein log likelihood functions is given below. Let $x_i={(x_{1i},L,x_{di})}^T$, $i=1,L,n$, with n samples taken from the distribution function F. The multi-dimensional Bernstein log likelihood function takes the following form:

(20)$l\left(p_m\right)={\displaystyle \sum _{k=1}^n}log{f}_m(x_k;p)={\displaystyle \sum _{k=1}^n}log\left\{{\displaystyle \sum _{i=0}^m}p\left(i\right){\beta }_{mi}\left(x_k\right)\right\}.$

Here, the maximum ${\widehat{p}}_m$ value obtained by taking $l\left(p_m\right)$ is called the maximum Bernstein likelihood estimate of $p_m$.

3.2.3. EM Algorithm Determines Coefficients

Using the EM algorithm, given the initial value ${p_m}^{\left(0\right)}$, one can iteratively determine the maximum likelihood estimate of using the following equation:

(21)${\widehat{p}}_{ml}^{(s+1)}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{j=1}^n}{\displaystyle \frac{{\widehat{p}}_{ml}^{\left(s\right)}{\beta }_{ml}\left(x_j\right)}{{\displaystyle \sum _{i=0}^m}{\widehat{p}}_{mi}^{\left(s\right)}{\beta }_{mi}\left(x_j\right)}},l=0,L,m;s=0,1,L.$

Here, one obtains

(22)${\widehat{p}}^{(s+1)}(l_1,L,l_d)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{j=1}^n}{\displaystyle \frac{{\widehat{p}}^{\left(s\right)}(l_1,L,l_d){\prod }_{v=1}^d{\beta }_{m_v{l}_v}\left(x_{vj}\right)}{{\displaystyle \sum _{i=0}^m}{\widehat{p}}^{\left(s\right)}(i_1,L,i_d){\prod }_{v=1}^d{\beta }_{m_v{i}_v}\left(x_{vj}\right)}}.$

The definition of the initial value $p_m^{\left(0\right)}$ can be determined by the actual situation. Redner and Walker (1984) proved that $p_m^{(s+1)}$ converges to the maximum likelihood estimate ${\widehat{p}}_m$ of $p_m$ [24].

In summary, the multi-dimensional Bernstein polynomial in the new form is uniquely determined only by the coefficient $p_m$, and its convergent iterative form can be derived using the EM algorithm.

4. Change-Point Detection Methods with Degrees of Freedom and Numerical Simulations

In the previous section, the EM algorithm was used to obtain the maximum likelihood estimates ${\widehat{p}}^{(s+1)}(l_1,L,l_d)$ of the coefficients of the Bernstein model (Equation (5)), which were then combined with a finite number of K known beta distributions ${\beta }_{mi}\left(t\right)$ to obtain an estimate of $f_m(t;p)$. However, this estimation presupposes first determining the most appropriate degree of freedom $m_j$.

Obviously, as the degree of freedom $m_j$ grows, the Bernstein log likelihood function $l\left(m\right)=l_B\left({\widehat{p}}_m\right)$ grows as well, but choosing too large a degree of freedom $m_j$ complicates the problem under study, which is similar to over-fitting in statistics and loses the significance of the Bernstein polynomial fitting. Therefore, the degree of freedom $m_j$ should be selected within an interval, and the change-point detection method can be applied to find the most suitable degree of freedom $m_j$.

For each component j, let $M_j=\left\{m_{j0},L,m_{j{k}_j}\right\}$, $m_{ji}=m_{j0}+i$, $i=0,1,L,$ and $k_j$. We now substitute $x_{j1},L,x_{jn}$ to obtain the Bernstein log likelihood function $l_{ji}=l_j\left(m_{ji}\right)$ with a degree of freedom of $m_{ji}\in M_j$, whose variation can be represented by $y_{ji}=l_{ji}-l_{j,i-1}$, $i=1,L,k_j$. If the appropriate degree of freedom $m_j$ is in $M_j$, when the selected degree of freedom is greater than this appropriate degree of freedom $m_j$, the result is over-fitting. At this point, the corresponding mean and variance of $y_{ji}$ will be very small. So, a change-point ${\tau }_j$ is also generated, where $m_{j{\tau }_j}$ is the appropriate degree of freedom at the same time. There is a lower bound $m_{bj}=max\left\{1,\left[\mu \left(1-\mu \right)/{\sigma }^2-3\right]\right\}$ for $m_{j0}$ in $M_j$, where $\left[x\right]$ represents the maximum integer that does not exceed x.

Integrating multi-dimensional Bernstein log likelihood functions into different change-point detection methods can provide different representations of change-point $\tau$. Thus, numerical simulations of a normally distributed mean single-change-point model are proposed to analyze and compare the advantages and disadvantages of several methods. We take 300 random variables $x_j(1\le j\le 300)$, where every 100 of them follow a binary normal distribution with the same mean and covariance, while the mean decreases. It may be useful to set the covariance matrix of the random variable to be $\left(\begin{array}{cc}0.5& 0.3\\ 0.3& 0.2\end{array}\right)$ and the mean to be

(23)$\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)=\left\{\begin{array}{ccc}\left(\begin{array}{c}50\\ 100\end{array}\right),& & 1\le j\le 100,\\ \left(\begin{array}{c}49\\ 100\end{array}\right),& & 101\le j\le 200,\\ \left(\begin{array}{c}49\\ 99\end{array}\right),& & 201\le j\le 300.\end{array}\right.$

Consider again the case where the difference in means is even smaller, with a mean of

(24)$\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)=\left\{\begin{array}{ccc}\left(\begin{array}{c}50\\ 100\end{array}\right),& & 1\le j\le 100,\\ \left(\begin{array}{c}49.7\\ 100\end{array}\right),& & 101\le j\le 200,\\ \left(\begin{array}{c}49.7\\ 99.7\end{array}\right),& & 201\le j\le 300.\end{array}\right.$

This study was performed using the RStudio software (1.4.1717), which requires input commands and allows you to write functions and scripts for batch operations with a simple and flexible syntax. There are many program packages available on the R website, which are widely used in basic statistics, ecology, geography, sociology, economics, medical statistics, bioinformatics, and many other areas.

4.1. Likelihood Ratio Method for Determining Degrees of Freedom

4.1.1. Change-Point Representation

Csörgö and Horváth used the likelihood ratio method to detect the change-point $\widehat{\tau }$ in 1997, and then the appropriate degree of freedom $m_j$ can be estimated using ${\widehat{m}}_j=m_{j{\widehat{\tau }}_j}$. The study considered $y_{j1}, \Lambda ,$ and $y_{j{\tau }_j}$ as distributions with a mean of ${\mu }_{j1}$, and $y_{j,\tau +1},\Lambda ,$ and $y_{jk}$ as distributions with a mean of ${\mu }_{j0}$, with ${\mu }_{j1}>{\mu }_{j0}$. The form of the change-point is as follows:

(25)${\widehat{\tau }}_j=arg{max}_{1\le \tau \le k_j}\left\{R_j\left(\tau \right)\right\}.$

The likelihood ratio of $\tau$ is

(26)$R_j\left(\tau \right)=-\tau log\left({\displaystyle \frac{l_{j\tau }-l_{j0}}{\tau }}\right)-(k_j-\tau )log\left({\displaystyle \frac{l_{jk}-l_{j\tau }}{k_j-\tau }}\right)+k_{j}log\left({\displaystyle \frac{l_{jk}-l_{j0}}{k_j}}\right),\tau =1,L,k_j.$

If there are many ${\tau }_j$ values that make $R_j\left(\tau \right)$ reach its maximum value, we choose the smallest one as the inflection point ${\widehat{\tau }}_j$.

4.1.2. Numerical Simulation

Using numerical simulations combined with reality, this study considers the case where the change-point does not take the first and last points. The case of Equation (23) is considered first to obtain the curve graph shown in Figure 2. The $j=1,2$ in the equation refers to two components, and the equation can be used to determine the maximum points of the likelihood ratio test function values for both components. If a change-point exists in the first component, then, due to consistency, a change-point must also exist in the second component. These two components can be effectively visualized in a three-dimensional graph. The rest of the figures are the same as this one. Next, we find the point that makes $R_j\left(\tau \right)$ reach its maximum value, as shown in Figure 3.

It can be clearly seen from the graph that there are change points at $\tau =99$ and $\tau =199$, which are the 100th and 200th data points, which is completely consistent with the mean of the given data undergoing a sudden change.

Subsequently, we consider the case where the mean difference is smaller, and adopt Equation (24) to obtain the curve shown in Figure 4. We find the point that makes $R_j\left(\tau \right)$ reach its maximum value, as shown in Figure 5.

From the figure, it can be seen that when the variance remains constant and the difference in mean is small, change points occur at ${\tau }_1=99$ and ${\tau }_2=193$, which are the 100th and 194th data points, with small errors compared to the given data.

4.2. Least Squares Method for Determining Degrees of Freedom

4.2.1. Change-Point Representation

For the first step, calculate the statistical measure

(27)$S_j={\displaystyle \sum _{i=1}^k}{(y_{ji}-{\overline{y}}_j)}^2$

where ${\overline{y}}_j={\displaystyle \sum _{i=1}^k}{\displaystyle \frac{y_{ji}}{k}}$.

For step two, divide the samples $y_{j1},L,$ and $y_{jk}$ into $y_{j1},L,$ and $y_{j,i-1}$ and $y_{ji},L,$ and $y_{jk}$, calculate their sample variances separately, and add them to obtain

(28)$S_{ji}={\displaystyle \sum _{p=1}^{i-1}}{(y_{jp}-{\overline{y}}_{jp1})}^2+{\displaystyle \sum _{p=i}^k}{(y_{jp}-{\overline{y}}_{jp2})}^2,2\le i\le k,$

where ${\overline{y}}_{jp1}={\displaystyle \frac{1}{i-1}}{\displaystyle \sum _{p=1}^{i-1}}{y}_{jp}={\displaystyle \frac{1}{i-1}}\left(l_{j,i-1}-l_{j0}\right)$ and ${\overline{y}}_{jp2}={\displaystyle \frac{1}{k-i+1}}{\displaystyle \sum _{p=1}^k}{y}_{jp}={\displaystyle \frac{1}{k-i+1}}\left(l_{j,k}-l_{j,i-1}\right)$.

In step three, according to the identity equations

(29)$S_j=S_{ji}+k^{-1}(i-1)(k-i+1){({\overline{y}}_{jp1}-{\overline{y}}_{jp2})}^2,2\le i\le k$

and

(30)$E\left(k^{-1}(i-1)(k-i+1){({\overline{y}}_{jp1}-{\overline{y}}_{jp2})}^2\right)={\sigma }^2+k^{-1}(i-1)(k-i+1){(E{\overline{y}}_{jp1}-E{\overline{y}}_{jp2})}^2,$

if there is a change-point, the above equation is generally not 0.

From these equations, it can be inferred that the existence of a change-point increases the gap between $S_j$ and $S_{ji}$. Therefore, the change-point is

(31)${\overline{\tau }}_j=argmax\left\{k^{-1}(i-1)(k-i+1){({\overline{y}}_{jp1}-{\overline{y}}_{jp2})}^2\right\},$

where ${\overline{y}}_{jp1}={\displaystyle \frac{1}{i-1}}{\displaystyle \sum _{p=1}^{i-1}}{y}_{jp}={\displaystyle \frac{1}{i-1}}\left(l_{j,i-1}-l_{j0}\right)$ and ${\overline{y}}_{jp2}={\displaystyle \frac{1}{k-i+1}}{\displaystyle \sum _{p=1}^k}{y}_{jp}={\displaystyle \frac{1}{k-i+1}}\left(l_{j,k}-l_{j,i-1}\right)$.

4.2.2. Numerical Simulation

For the numerical simulation combined with reality, this study considers the case where the change-point does not take the first and last points. The case of Equation (23) is considered first to obtain the curve graph shown in Figure 6, and the point that maximizes $t_j$ is found, as shown in Figure 7.

It can be clearly seen from the graph that there are change points at $\tau =100$ and $\tau =200$, which are the 101st and 201st data points, completely consistent with the point at which the given data mean undergoes a sudden change.

Subsequently, we consider the case where the mean difference is smaller, and adopt Equation (24) to obtain the curve shown in Figure 8, and find the point that makes $t_j$ reach its maximum value, as shown in Figure 9.

From the figure, it can be clearly seen that when the variance remains constant and the difference in mean is small, change points occur at ${\tau }_1=100$ and ${\tau }_2=194$, which are the 101st and 195th data points, with small errors compared to the given data. In addition, it can be observed that the least squares method obtained images and results very similar to the likelihood ratio method.

4.3. CUSUM Method for Determining Degrees of Freedom

4.3.1. Change-Point Representation

For the model, let $y_{jt}={\mu }_j\left(t\right)+{\epsilon }_{jt}$, $t=1,2,\Lambda ,k$. Now, consider the simplest scenario where ${\mu }_{jt}$ only takes two values:

(32)${\mu }_j\left(t\right)=\left\{\begin{array}{cc}{\mu }_{j1},& t\le {\tau }_j,\\ {\mu }_{j2},& t\le {\tau }_j.\end{array}\right.$

where ${\mu }_{j1}$, ${\mu }_{j2}$, and ${\tau }_j$ are unknown, the CUSUM estimate for the change-point ${\tau }_j$ is defined as

(33)${\widehat{\tau }}_j=min\left\{t:\left|U_t\right|=\underset{1\le p\le k}{max}\left|U_p\right|\right\}$

with $U_t={\left\{{\displaystyle \frac{t(k-t)}{k}}\right\}}^{1-\gamma }\left\{{\displaystyle \frac{1}{t}}{\displaystyle \sum _{p=1}^t}{y}_{jp}-{\displaystyle \frac{1}{k-t}}{\displaystyle \sum _{p=t+1}^k}{y}_{jp}\right\}={\left\{{\displaystyle \frac{t(k-t)}{k}}\right\}}^{1-\gamma }\left\{{\displaystyle \frac{1}{t}}\left(l_{jt}-l_{j0}\right)-{\displaystyle \frac{1}{k-t}}\left(l_{jk}-l_{jt}\right)\right\},0\le \gamma \le 1$.

4.3.2. Numerical Simulation

For the numerical simulation combined with reality, this study considers the case where the change-point does not take the first and last points and takes $\gamma =0.5$. The case of Equation (23) is considered first to obtain the curve graph shown in Figure 10, and the point that maximizes $U_t$ is found, as shown in Figure 11.

It can be clearly seen from the graph that there are change points at $\tau =99$ and $\tau =199$, which are the 100th and 200th data points, completely consistent with the point where the given data mean undergoes a sudden change. Subsequently, we consider the case where the mean difference is smaller, and adopt Equation (24) to obtain the curve shown in Figure 12, and find the point that makes $R_j\left(\tau \right)$ reach its maximum value, as shown in Figure 13.

From the figure, it can be seen that when the variance remains constant and the difference in mean is small, change points occur at ${\tau }_1=99$ and ${\tau }_2=193$, which are the 100th and 194th data points, with small errors compared to the given data.

4.4. Local Comparison Method for Determining Degrees of Freedom

4.4.1. Change-Point Representation

Consider the following model:

(34)$y_{ji}=\left\{\begin{array}{cc}{a}_{j1}+e_{ji},& 1\le i\le {\tau }_j,\\ a_{j2}+e_{j2},& {\tau }_j\le i\le k.\end{array}\right.$

Here, the random error $e_{ji}(1\le i\le k)$ is independent and identically distributed and ${\tau }_j,a_{j1},a_{j2},$ and ${\sigma }_j^2$ in the equation $E\left(e_{ji}\right)=0$, $\mathrm{Var}({\mathrm{e}}_{ji})={\sigma }^2<+\infty$, are all unknown. The estimation steps for the mean change-point ${\tau }_j$ of the model are as follows.

For step one, take $i\in \left[1,k\right]$. Given an appropriate natural number c, summing and subtracting the c observations from the left and right of i (with i included on the right), the result is

(35)$\begin{array}{cc}\hfill D_{ji}& =\left(y_{ji}+L+y_{j,i+c-1}\right)-\left(y_{j,i-c}+L+y_{j,i-1}\right)\hfill \\ \hfill & =\left(l_{j,i+c-1}-l_{j,i-1}\right)-\left(l_{j,i-1}-l_{j,i-c-1}\right)\hfill \\ \hfill & =l_{j,i+c-1}-2l_{j,i-1}+l_{j,i-c-1}\hfill \end{array}$

with $i=c+1,c+2,L,k-c+1$.

For step two, when $i={\tau }_j$ or is close to ${\tau }_j$, the difference between the two values on the right side of the $D_{ji}$ expression is significant. Therefore, ${\widehat{\tau }}_j$ can be set as the index that maximizes $\left|D_{ji}\right|$. Due to the monotonic decrease in $y_{ji}=l_j\left(m_{ji}\right)-l_j\left(m_{j,i-1}\right)$, $i=1,L,k$, where the value of $D_{ji}$ is negative, $\left|D_{ji}\right|=-D_{ji}=-l_{j,i=c-1}+2l_{j,i-1}-l_{j,i-c-1}$. So, the estimated change-point is

(36)${\widehat{\tau }}_j=arg{max}_{c+1\le i\le k-c+1}\left\{-D_{ji}\right\}.$

The appropriate local length can be selected based on the data length, and then adjusted appropriately in practical operations.

4.4.2. Numerical Simulation

For the numerical simulation, the case of Equation (23) is considered first; we take the length of the experimental interval as 5. The study considers the case where the change-point does not take the first six points and the last six points. We obtain the curve graph shown in Figure 14, and the point that maximizes $D_{ji}$ is found, as shown in Figure 15.

It can be clearly seen from the graph that there are change points at $\tau =96$ and $\tau =195$, which are the 102nd and 201st data points, completely consistent with the point where the given data mean undergoes a sudden change. Subsequently, we consider the case where the mean difference is smaller, and adopt Equation (24) to obtain the curve shown in Figure 16, and find the point that makes $D_{ji}$ reach its maximum value, as shown in Figure 17.

From the figure, it can be seen that when the variance remains constant and the difference in mean is small, change points occur at ${\tau }_1=96$ and ${\tau }_2=254$, which are the 102nd and 260th data points, with significant errors compared to the given data.

According to all the data simulations, it can be seen that under several different methods, the detection of the binary normal distribution mean single-change-point model reaches its maximum value around $t_1=100$ and $t_2=200$, where the change-points are ${\widehat{\tau }}_1=100$ and ${\widehat{\tau }}_2=200$. The results are significant when the mean values before and after the change-point change dramatically. The larger the difference in the mean values of the change-points, the more accurate the test results at the same time.

In conclusion, the likelihood ratio method, least squares method, and CUSUM method have more significant maximum values compared to the local comparison method, and it is not necessary to consider the length of the local interval taken, making them more practical. The local comparison method is not able to accurately detect the change-points when the difference in the mean value of normal distribution is small, and it is necessary for the interval length to be reasonably selected according to the background of different problems.

5. Several Applications of Change-Point Detection

In the context of the widespread application of big data technology, change-point detection has an undeniable impact on global economic integration and unexpected events in the financial sector.

For example, in the 2007 subprime mortgage crisis in the United States, the economy suffered a heavy blow, with many once-dominant Wall Street investment banks either collapsing or being acquired, and a group of commercial banks going bankrupt. For another example, the sudden outbreak of a new coronavirus pneumonia epidemic (COVID-19) at the end of 2019 triggered global financial market turbulence, with a rare four meltdowns in U.S. stocks in April 2020 alone. In the above two examples, both the economy and financial markets suffered a great crisis. How to determine or predict whether or where a change-point occurs based on available (historical, observable) information is thus a very important task in many social fields such as economics, finance, and industrial manufacturing.

Thus, for applications of change-point detection, this study considers detecting the existence of change-points from a complete sample observation value sequence, and summarizes the observations through post-change-point testing.

5.1. Research on Epidemic Data of COVID-19 in China

In December 2019, China began to deal with many cases of viral pneumonia, all of which were diagnosed as viral pneumonia or lung infections. After laboratory research and detection, COVID-19 was discovered. On 30 January 2020, the WHO announced that COVID-19 was a public health emergency of international concern. Up to 26 May 2020, more than 5.4 million cases of COVID-19 had been confirmed worldwide and more than 340,000 people had died, with more than 200 countries and regions with COVID-19 cases. Against this background, the International Monetary Fund (IMF) lowered the global GDP growth forecast for 2020 by 6.3%, from 3.3% in January to −3%, representing the worst economic recession since the “Great Depression” in the 1930s. The epidemic also affected the expectations of global investors. In March, the US stock market fell sharply four times, triggering the circuit breaker mechanism. On April 20th, the international crude oil futures settlement price fell to a negative value for the first time in history, and financial risks resonated within the sluggish real economy.

The National Health Commission of the People’s Republic of China began reporting the COVID-19 infection situation in the country on 11 January 2020. This article analyzes the data from 23 January 2020 to 15 May 2020, as shown in Table 1 (only the top 10 groups of data are listed in the table).

Our empirical analysis of China’s COVID-19 data based on actual conditions considers the situation where the first and last points are not taken as the change-points. Due to the inability to visualize five-dimensional data using images, this study utilizes the likelihood ratio method to visualize and analyze each component, resulting in the curve shown in Figure 18. We identified the point at which $R_j\left(\tau \right)$ reached its maximum value, as shown in Figure 19.

From these figures, it can be seen that the number of newly confirmed cases has a change-point at $\tau =38$, which is the 39th data point, corresponding to 1 March 2020. The number of newly suspected cases has a change-point at $\tau =43$, which is the 44th data point, corresponding to 6 March 2020. They are close in time. In making sense of this model, it is believed that the new data change-point of the COVID-19 occurred in early March, and the virus transmission was effectively controlled.

The number of newly cured cases showed a change-point at $\tau =67$, which is the 68th data point, corresponding to 30 March 2020. This reflects that the pressure on patients admitted to major hospitals significantly decreased since the end of March. In addition, there is another small spike at about the 10th data point, reflecting a significant increase in hospital capacity at the beginning of February, which is attributed to the completion of two new square-cabin hospitals to relieve pressure on patient beds. The rapid response to COVID-19 also led to a significant increase in cured cases in China in early stages and the change-point also appeared earlier.

5.2. Research on S&P 500 and SSE 50 Index

This study selects the S&P 500 and SSE 50 indices and compares their index changes. The S&P 500 Index is given by the U.S. Standard & Poor’s Company (New York, NY, USA) as a record of 500 listed companies in the U.S. It is a stock index that is associated with the largest U.S. turnover and is one of the most significant indicators of index futures when it comes to reflecting broader market changes. The SSE 50 Index is based on a scientific and objective methodology for selecting the 50 most representative stocks at a large scale and with good liquidity in the Shanghai stock market to form a sample stock, so as to comprehensively reflect the overall situation of the most influential enterprises in the Shanghai stock market.

We selected historical data concerning the S&P 500 Index and the SSE 50 Index from 22 May 2017 to 15 May 2020 and searched for change points; we now compare and analyze the results. The S&P 500 data consist of 752 sets, sourced from yahoo!finance (https://finance.yahoo.com/quote/%5EGSPC/ (accessed on 1 July 2020)), The Shanghai Stock Exchange 50 Index consists of 727 groups, sourced from Investing.com (https://www.investing.com/indices/shanghai-se-50-historical-data (accessed on 1 July 2020)).

This study first conducts an empirical analysis of the opening and closing price data of the S&P 500 index. Based on the actual situation, considering the case where the change-point does not take the first and last points, the curve graph shown in Figure 20 is obtained, and the point that maximizes $R_j\left(\tau \right)$ is found, as shown in Figure 21.

From these figures, it can be seen that the opening price has a change-point at $\tau =457$, which is the 458th data point, corresponding to 16 March 2019. The closing price has a change-point at $\tau =456$, which is the 457th data point, corresponding to 15 March 2019, occurring at an adjacent closing and opening time.

Next, this study conducts an empirical analysis of the opening and closing price data of the SSE 50 Index. Based on the actual situation, considering the case where the change-point does not take the first and last points, the curve graph shown in Figure 22 is obtained, and the point that maximizes $R_j\left(\tau \right)$ is found, as shown in Figure 23.

From these figures, it can be seen that the opening price has a change-point at $\tau =432$, which is the 433rd data point, corresponding to 28 February 2019. The closing price has a change-point at $\tau =429$, which is the 430th data point, corresponding to 25 February 2019, occurring at an adjacent closing and opening time.

Comparing the change-point results of the S&P 500 index and the SSE 50 index, it is not difficult to see that the US economic turmoil at the end of 2018 had a significant impact on both the US stock market and the stocks of leading Chinese enterprises, indicating strong symmetry between the two stock indices when major crisis events occur. However, the change-point of the SSE 50 index data appeared earlier, reflecting that the Chinese stock market was affected for a shorter period of time. In addition, it is not difficult to see from the figure that the change-point detection image of the S&P 500 data has many fluctuations. Based on the judgment criterion that the highest point in the figure is the position of the change-point, it can be concluded that the distribution characteristics of the US stock market change rapidly and the stock market is relatively volatile.

Finally, this study continues to perform change-point testing on the opening and closing price data of the S&P 500 index from 2 January 2020 to 15 May 2020. The curve graph shown in Figure 24 is obtained, and the point that maximizes $R_j\left(\tau \right)$ is found, as shown in Figure 25.

From these figures, it can be seen that the position of the change-point is very obvious; the opening price has a change-point at $\tau =43$, which is the 44th data point, corresponding to 5 March 2020. The closing price has a change-point at $\tau =44$, which is the 45th data point, corresponding to 6 March 2020. The statistical characteristics of US stock data around March 6th underwent significant changes. Based on the four circuit breakers that occurred on March 9th, 12th, 16th, and 18th, it is not difficult to see that circuit breakers have a huge impact on the statistical characteristics of US stock prices.

This result to some extent indicates that global financial regulatory agencies need to analyze and predict financial data in a timely manner. When a qualitative change in its structure is detected, which means a change-point is created, there should be increased vigilance to prevent major financial shocks from having a disruptive and significant impact on financial markets.

5.3. Research on Money Supply

Money supply is an important indicator for measuring economic activity in a country or region, which includes different levels of currency such as M0, M1, and M2. Specifically, M0 only contains cash in circulation, M1 includes not only cash but also corporate current deposits, and M2 adds fixed deposits for enterprises and personal deposits on the basis of M1, which can still be relatively easily converted into cash currency. So, it is generally believed that M1 can reflect current actual purchasing power, while M2 can only reflect potential purchasing power. Central banks and various commercial banks can use this to determine monetary policy. Such as when M1 is too high and M2 is too low, indicating strong demand, insufficient investment, and the risk of inflation; when M2 is too high and M1 is too low, indicating that investment is overheated, demand is not strong, and there is an asset foam risk. Below, this study uses money supply data from June 2015 to March 2020, as shown in Table 2 (only the top 10 sets of data are listed in the table).

We conducted numerical simulations on the data in Table 2 by using the local comparison method, taking the length of the experimental interval as 5. We consider the case where the change-point does not take the first six points and the last six points, then obtain the curve graph shown in Figure 26. Then, the M1 and M2 components have been plotted to obtain the curve graph shown in Figure 27.

From these figures, it can be clearly seen that the peak of the M2 supply end-of-term value detection shows a lag compared to the M1 supply, which means that the change-point of M2 supply occurs about 6 months after the change-point of M1 supply. This confirms that consumption and the terminal market will have an impact on investment and monetary reduction in the intermediate market.

This paper mainly uses the change-point detection method based on multi-dimensional Bernstein polynomials mentioned to empirically analyze the epidemic data of COVID-19 in China, stock data of the S&P 500 Index and the SSE 50 Index, and money supply data. We have summarized the development and changes of the epidemic in China, the difference in stock price changes between the U.S. and China, and the impact of the market on monetary tightening.

6. Conclusions

This article mainly presented four methods for detecting change-points. The best estimation of the multi-dimensional Bernstein polynomial degree of freedom $m_j$ was achieved by combining the change-point detection method, and numerical simulations of the binary normal mean single-change-point model were carried out. Finally, an application of change-point detection in finance was studied and conclusions were obtained:

In terms of theory:

(1) The Bernstein model can be represented as a finite linear combination of K-fold beta distributions, and the EM algorithm can be used to obtain a convergent iterative form of its coefficient $p_{ml}$. The definition of its initial value $p_m^{\left(0\right)}$ can be determined by practical situations.

(2) In numerical simulations, under several different change-point detection methods, the detection of the binary normal distribution mean single-change-point model reaches its maximum value near the change-point, and the results are significant. The larger the mean difference of the change-points, the clearer the peak values of the image, and the more accurate the results.

(3) Combined with multi-dimensional Bernstein log likelihood functions, the likelihood ratio method, least squares method, and CUSUM method have more significant maximum values compared to local comparison methods. Also, it is not necessary to consider the length of the local interval taken, making them more practical. The local comparison method cannot accurately detect change points when the mean difference is small, and the interval length needs to be reasonably selected according to different problem backgrounds.

In terms of application:

(4) In combination with the epidemic notification data associated with the novel coronavirus pneumonia in China, the data change-point of newly confirmed and suspected cases appeared in early March, which reflects that the virus transmission was effectively controlled. The data change-point of new death-resulting cases appeared on March 10th, 9 days behind the data of new confirmed cases, and the feedback showed that the novel coronavirus had a certain incubation period. The data change-point for the number of newly cured cases occurred on March 29th, which reflects that since the end of March, the pressure on patients admitted to major hospitals significantly decreased. In addition, the peak in the images in early February reflects a significant enhancement in hospitals’ abilities to admit patients.

(5) By comparing the daily opening and closing price changes of the S&P 500 Index and the Shanghai Stock Exchange 50 Index over three years, it is concluded that the US economic turmoil at the end of 2018 had a greater impact on both the US stock market and the stock of leading Chinese enterprises, but the data changes of the Shanghai Stock Exchange 50 Index appeared earlier, reflecting that the Chinese stock market was relatively affected for a short time. In addition, the change-point detection images of the S&P 500 data show significant fluctuations, reflecting the rapid changes in the distribution characteristics of the US stock market, which is relatively volatile.

(6) In the money supply data from June 2015 to March 2020, the peak of the M2 supply final value detection showed a lag compared to the M1 supply, which means that the change-point of M2 supply occurred about 6 months after the change-point of M1 supply. This confirms that consumption and the terminal market will have an impact on reductions in money in investments and the intermediate market.

The research work in this study has achieved certain results in the theory and application of change-point detection, but there are still many issues worth further studying in the future:

(1) In various fields of practical life, if the time series is long, there is often more than one moment of qualitative change, so the detection of multiple changes has a wider range of applications and is worth further studying.

(2) At present, most research on change-points has been conducted on small data volumes and low dimensions, and the data structures of large data volumes and high dimensions have not been thoroughly analyzed.

(3) Most of the existing research work has been based on post-change-point detection; thus, mid-term change-point detection can be considered, which will have a good effect on predicting change-points.

Author Contributions

Conceptualization, F.T. and B.T.; methodology, F.T. and Y.Q.; software, Y.Q.; validation, Y.W. and B.T.; formal analysis, F.T., Y.Q. and B.T.; data curation, Y.Q.; writing—original draft preparation, F.T.; writing—review and editing, Y.Q., B.T. and Y.W.; supervision, B.T. and Y.W.; funding acquisition, B.T. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The data used in this study can be found at the National Bureau of Statistics website (http://data.stats.gov.cn/easyquery.htm?cn=A01 (accessed on 1 July 2020)), yahoo!finance (https://finance.yahoo.com/quote/%5EGSPC/ (accessed on 1 July 2020)), and Investing.com (https://www.investing.com/indices/shanghai-se-50-historical-data (accessed on 1 July 2020)).

Acknowledgments

The authors are grateful to the editor and referees for their valuable time and effort put into this manuscript.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Histogram and density function of duration of eruption of Old Faithful. Note: [Forumla omitted. See PDF.] represents log-normal parameter density estimation, [Forumla omitted. See PDF.] represents Bernstein density estimation, and [Forumla omitted. See PDF.] represents kernel density estimation.

Figure 2. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 3. Subscript result taken from maximum detection value point in Figure 2.

Figure 4. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 5. Subscript result taken from maximum detection value point in Figure 4.

Figure 6. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 7. Subscript result taken from maximum detection value point in Figure 6.

Figure 8. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 9. Subscript result taken from maximum detection value point in Figure 8.

Figure 10. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 11. Subscript result taken from maximum detection value point in Figure 10.

Figure 12. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 13. Subscript result taken from maximum detection value point in Figure 12.

Figure 14. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 15. Subscript result taken from maximum detection value point in Figure 14.

Figure 16. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 17. Subscript result taken from maximum detection value point in Figure 16.

Figure 18. COVID−19 data concerning number of newly confirmed, suspected, death-resulting, and cured cases detected [Forumla omitted. See PDF.].

Figure 19. Index result obtained from point with maximum detection value.

Figure 20. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 21. Subscript result taken from maximum detection value point in Figure 20.

Figure 22. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 23. Subscript result taken from maximum detection value point in Figure 22.

Figure 24. Model detection quantity [Forumla omitted. See PDF.], [Forumla omitted. See PDF.].

Figure 25. Subscript result taken from maximum detection value point in Figure 24.

Figure 26. End of term measurement of M1 and M2 supply quantities.

Figure 27. End of term measurement of M1 and M2 supply quantities.

Table 1

Reported data of novel coronavirus pneumonia from 23 January, 2020 to 15 May 2020 (top 10 data groups).

Date	No.	New Confirmed	New Suspected	New Deaths	New Cured
23/01	1	259	680	8	6
24/01	2	444	1118	16	3
25/01	3	688	1309	15	11
26/01	4	769	3806	24	2
27/01	5	1771	2077	26	9
28/01	6	1459	3248	26	43
29/01	7	1737	4148	38	21
30/01	8	1982	4812	43	47
31/01	9	2102	5019	46	72
01/02	10	2590	4562	45	85

Data sourced from National Health Commission of People’s Republic of China: http://www.nhc.gov.cn/xcs/yqtb/list_gzbd.shtml (accessed on 1 May 2020).

Table 2

Closing value of money supply from June 2015 to March 2020 (first 10 datasets).

Date	No.	Closing Value of M2 Supply (in Billions of CNY)	End of Term Value of M1 Supply (in Billions of CNY)
June 2015	1	1,333,375.36	356,082.86
July 2015	2	1,353,210.92	353,122.19
August 2015	3	1,356,907.98	362,793.73
September 2015	4	1,359,824.06	364,416.90
October 2015	5	1,361,020.70	375,806.45
November 2015	6	1,373,956.01	387,618.32
December 2015	7	1,392,278.11	400,953.44
January 2016	8	1,416,319.55	412,685.64
February 2016	9	1,424,618.68	392,504.70
March 2016	10	1,446,198.03	411,581.31

The data is sourced from the website of the National Bureau of Statistics (http://data.stats.gov.cn/easyquery.htm?cn=A01 (accessed on 1 July 2020)).

References

1. Page, E.S. Continuous Inspection Schemes. Biometrika; 1954; 41, pp. 100-115. [DOI: https://dx.doi.org/10.1093/biomet/41.1-2.100]

2. Zhang, X. Overview of the latest progress in change point detection problem. J. Jianghan Univ. (Nat. Sci. Ed.); 2012; 40, pp. 18-23.

3. Tang, C. Statistical Inference for Change-Point Problems and Its Applications in Finance. Doctoral Dissertation; University of Science and Technology of China: Hefei, China, 2007.

4. Li, Y. Multiple Change Points Detection Based on Binary Segementation and Its Applications in Finance. Master’s Thesis; Harbin Institute of Technology: Harbin, China, 2018.

5. Liu, X.; Fang, L.; Zhang, Y. Change Point Detection of Stock Market Liquidity in the Context of European Sovereign Debt Crisis. J. Manag. Sci. Eng.; 2014; 7, pp. 82-94.

6. Csörgö, M.; Horváth, L. Limit Theorems in Change-Point Analysis; John Wiley and Sons: Hoboken, NJ, USA, 1997; pp. 128-154.

7. Brown, R.L.; Durbin, J.; Evans, J.M. Techniques for Testing the Constancy of Regression Relationships over Time. J. R. Stat. Soc.; 1975; 37, pp. 149-163. [DOI: https://dx.doi.org/10.1111/j.2517-6161.1975.tb01532.x]

8. Kokoszka, P.; Leipus, R. Change-point in the Mean of Dependent Observations. Stat. Probab. Lett.; 1998; 40, pp. 385-393. [DOI: https://dx.doi.org/10.1016/S0167-7152(98)00145-X]

9. Kim, S.; Cho, S.; Lee, S. On the CUSUM Testing for Parameter Changes in GARCH(1,1) Models. Commun. Stat.-Theory Methods; 2000; 29, pp. 445-462. [DOI: https://dx.doi.org/10.1080/03610920008832494]

10. Lee, S.; Tokutsu, Y.; Maekawa, K. The CUSUM Test for Parameter Change inRegression Models with ARCH Errors. J. Jpn. Stat. Soc.; 2004; 34, pp. 173-188. [DOI: https://dx.doi.org/10.14490/jjss.34.173]

11. James, B.; James, K.L.; Siegmund, D. Asymptotic Approximations for Likeli-hood Ratio Tests and Confidence Regions for a Change-Point in the Mean of a Multivariate Normal Distribution. Stat. Sin.; 1992; 2, pp. 69-90.

12. Bai, J.; Perron, P. Critical Values for Multiple Structural Change Tests. Econom. J.; 2003; 6, pp. 72-78. [DOI: https://dx.doi.org/10.1111/1368-423X.00102]

13. Gurevich, G.; Vexler, A. Change Point Problems in the Model of Logistic Regression. J. Stat. Plan. Inference; 2005; 131, pp. 313-331. [DOI: https://dx.doi.org/10.1016/j.jspi.2004.03.004]

14. Cheng, C. The Bernstein Polynomial Estimator of a Smooth Quantile Function. Stat. Probab. Lett.; 1995; 24, pp. 321-330. [DOI: https://dx.doi.org/10.1016/0167-7152(94)00190-J]

15. Guan, Z.; Wu, B.; Zhao, H. Nonparametric Estimator Of False Discovery Rate Based On Bernstein Polynomials. Stat. Sin.; 2008; 18, pp. 905-923.

16. Igarashi, G.; Kakizawa, Y. On Improving Convergence Rate of Bernstein Polynomial Density Estimator. J. Nonparametric Stat.; 2014; 26, pp. 61-84. [DOI: https://dx.doi.org/10.1080/10485252.2013.827195]

17. Guan, Z. Efficient and Robust Density Estimation Using Bernstein Type Polynomials. J. Nonparametric Stat.; 2016; 28, pp. 250-271. [DOI: https://dx.doi.org/10.1080/10485252.2016.1163349]

18. Guan, Z. Bernstein Polynomial Model for Grouped Continuous Data. J. Nonparametric Stat.; 2017; 29, pp. 831-848. [DOI: https://dx.doi.org/10.1080/10485252.2017.1374384]

19. Belalia, M.; Bouezmarni, T.; Lemyre, F.C.; Taamouti, A. Testing Independence Based on Bernstein Empirical Copula and Copula Density. J. Nonparametric Stat.; 2017; 29, pp. 346-380. [DOI: https://dx.doi.org/10.1080/10485252.2017.1303063]

20. Tenbusch, A. Two-Dimensional Bernstein Polynomial Density Estimators. Metrika; 1994; 41, pp. 233-253. [DOI: https://dx.doi.org/10.1007/BF01895321]

21. Wang, T.; Guan, Z. Bernstein Polynomial Model for Nonparametric Multivariate Density. Statistics; 2019; 53, pp. 321-338. [DOI: https://dx.doi.org/10.1080/02331888.2019.1574299]

22. Lorentz, G.G. The Degree of Approximation by Polynomials with Positive Coefficients. Math. Ann.; 1963; 151, pp. 239-251. [DOI: https://dx.doi.org/10.1007/BF01398235]

23. Cheney, E.W. Introduction to Approximation Theory; 2nd ed. AMS Chelsea Publishing: New York, NY, USA, 1982; pp. 2-5.

24. Redner, R.A.; Walker, H.F. Mixture Densities, Maximum Likelihood and the EM Algorithm. SIAM Rev.; 1984; 26, pp. 195-239. [DOI: https://dx.doi.org/10.1137/1026034]