Abstract
Random matrix theory is applied in the financial field to study the correlation of the financial correlation coefficient matrix, which is a key factor in network construction. In this paper, the random matrix theory is combined with network construction to study the art financial risk prediction algorithm based on the random matrix. Based on the stochastic matrix theory and the key nodes of the network, the financial network and the “noise” network before and after the “denoising” are analyzed and compared. It is found that the key and important information of the original network is still retained after the “denoising” of the network, and the noise information corresponds to the information represented by the smaller nodes in the original network. Based on the stochastic matrix of artwork financial risk prediction, the topological properties of the financial network before and after denoising are analyzed and compared from the perspectives of minimum spanning tree, model, and community structure. Based on the random matrix theory, this paper discusses the financial correlation coefficient matrix and the statistical properties of the eigenvalues of the random matrix, and on this basis, the existing denoising methods are improved, the correlation coefficient matrix more suitable for constructing the financial network is established, and the art financial risk prediction algorithm is constructed. Then, based on the stochastic matrix theory and the key nodes of the network, the financial network and the noise network before and after denoising are analyzed and compared. It is found that the key and important information of the original network is still retained after denoising the network, and the noise information corresponds to the information represented by the relatively small nodes in the original network. Finally, based on the stochastic matrix of art financial risk prediction, analysis of the financial network topology, such as minimum spanning tree, model, and community structure, found that the improved financial network topology is more obvious and the structure is closer.
1. Introduction
Art financial market conditions are constantly changing with time, so the correlation between two stocks is not fixed, and the correlation between limited time series estimation will be disturbed by some noise information. The empirical correlation coefficient matrix contains a lot of uncertainty and a lot of noise information, which will have a certain impact on the topology of the research network. Many scholars are also studying the dynamic changes in financial markets. A multidimensional subdiffusion model is introduced to describe the stagnation period of each stock and the behavior of the dependence relationship between multiple stocks in the financial market [1], which indicates that the relationship between stocks is not invariable, and analyzes the relevant characteristics of the financial market. This paper studies the systemic risk of the financial market [2, 3], analyzes the dynamic behavior of the financial risk system, and achieves the purpose of controlling system chaos through a series of controls. The noise in lung cancer gene expression data was removed by the random matrix theory method and the conclusion was drawn [4] that the random matrix theory hierarchical clustering method is an effective new method to identify the gene network.
Random matrix theory is more and more closely related to the correlation coefficient matrix. Using the method of random matrix theory [5], by comparing the difference between mobile crowd data and random data in the spectrum distribution of the cross-correlation matrix, it is pointed out that the space-time pattern of weight of eigenvector members is very consistent with the fluctuation process of the overall behavioral characteristics of the urban mobile crowd. In recent years, with the application of economic physics in financial time series analysis, stochastic matrix theory is more and more applied in the financial field, which provides an effective method for analyzing correlation noise. Random matrix theory originated from nuclear physics and is currently used in quantum field theory, multivariate statistics, wireless communication, and finance. In the financial field, the stochastic matrix theory has been applied more and more. How to remove the noise of the financial correlation coefficient matrix is studied [6]. The eigenvalues and distribution of eigenvectors of the correlation coefficient matrix, improvement measures, and applications of the correlation coefficient matrix are studied in detail [7]. The random matrix theory is used to study the asset portfolio [8]. The random matrix theory and correlation coefficient distribution row analysis are used to compare Teheran Stock Exchange and Dow Jones Industrial Average under local disturbance and global disturbance [9]. In recent years, random matrix theory has been applied to complex networks. The random matrix theory was used to study the traffic behavior of the interconnected backbone router and virtual LAN of the University of Louisville [10, 11], to analyze the change of real and virtual traffic rates with different traffic time series and to apply the eigenvalue and inverse participation ratio of the random matrix prediction results to the traffic congestion mechanism of the network. The gene correlation network was constructed by selecting correlation values, and the random matrix theory was used to detect the network community. Compared with other hierarchical clustering methods, it was found that the community structure delineated by this method reflected all the characteristics of other methods [12]. Random matrix theory was used to analyze the maximum eigenvalue of the correlation coefficient matrix of the market mode of the stock network. In order to better analyze risk management, the market mode was eliminated and the financial network was constructed [13]. Topology structure of the financial network under different values was analyzed [14], and it was found that the network was scale-free. The random matrix theory and the network method were applied to investigate the correlation of 20 financial indicators and the nature of the network. The results before and during the 2008 financial crisis were compared and analyzed to analyze the topological properties of the cluster network under different values.
Firstly, RMT (records management taskforce) was introduced into the analysis of the high-dimensional cross-correlation matrix of the financial time series, thus creating a new method to study market correlation from the perspective of statistical physics [15]. The distribution of eigenvalues and eigenvectors of the cross-correlation matrix of the 406 stock market returns in the S&P500 stock market was investigated and compared with the random cross-correlation matrix calculated with the same number and length of the random time series [16]. The results show that 94% of the eigenvalues of the actual cross-correlation matrix fall within the RMT range, and the eigenvectors corresponding to the remaining 6% of the eigenvalues are related to 26% of the fluctuations, which indicates that the deviated eigenvalues hide a lot of characteristic information about the market. The distribution of the nearest neighbor spacing of the eigenvalues of the cross incidence matrix is investigated, which is no different from that of the real symmetric random matrix [17]. It is proved that the cross-correlation matrix satisfies the global property of Gaussian vertical set of real symmetric matrices. He further studied the antiparticipation rate intellectual property of feature vectors and found that the intellectual property corresponding to feature vectors within RMT is about the reciprocal of the number of investigated stocks [18], indicating that all company stocks contribute to these feature vectors. However, the eigenvectors corresponding to the maximum and minimum eigenvalues deviating from the RMT range have higher IPR, indicating that only a few companies' stocks contribute to these eigenvectors [19]. This result implies that it is possible to classify stocks through in-depth study of eigenvalues that deviate from the RMT range. Further, the contribution of stocks divided according to the standard market segment to the eigenvalue corresponding to the eigenvector deviating from the RMT range is calculated [20, 21], and it is found that in addition to the approximate equal value of each component of the eigenvector corresponding to the maximum eigenvalue (also known as the overall movement of the market). The rest of the top 10 sublarge eigenvalues can be found corresponding to the corresponding sector (this is also known as the local movement of the market), thus confirming the view that large eigenvalues can realize stock classification.
At present, many scholars use the random matrix theory to analyze the correlation coefficient matrix and use the random matrix theory to analyze the related properties of the network, mainly studying portfolio optimization and risk management in the financial market. However, based on the random matrix theory and the correlation coefficient matrix, there are few research studies on dealing with the noise problem in the artwork financial network and analyzing the topological properties of the network. Since the financial system is a highly complex dynamic system, there is a lot of noise in the financial network, which is not conducive to the study of the topological properties of the network and the analysis of the dynamic behavior of the network. Therefore, we still need to discuss the method of removing noise information in the network based on the random matrix theory. Random matrix theory is used to provide a theoretical model for studying the characteristics of large dimensional ECM noise. The analytical framework of random matrix theory is described in detail, including the limit distribution of empirical spectrum distribution function of the random matrix, the eigenroots extreme value of the random matrix, and the convergence rate of empirical spectrum distribution function of the random matrix. The theoretical distribution of the random matrix introduces the characteristic spectrum and Wigner distribution of the random matrix. In addition, the common random matrix in practical application is introduced, and the principle and steps of random matrix analysis are described according to the actual research situation of this paper.
2. Research on the Stochastic Matrix of Art Financial Risk Prediction
2.1. Comparative Analysis of the Return Correlation Matrix and the Random Correlation Matrix of the Art Stock
From the knowledge module, the technical route of research can be summarized for the study of the theoretical framework and empirical study, model research, applied research, and summary analysis of five steps: theory research mainly related to the knowledge framework, including the limiting distribution of the experience of the random matrix spectral distribution function, the characteristics of the random matrix root extreme value. The convergence speed of the experience of the random matrix spectral distribution function, the characteristic spectrum of the random matrix, Wigner distribution of the random matrix, and large dimensional random matrix ECM are shown in Figure 1.
Random matrix (RMT) is a theory about the distribution characteristics of eigenvalues and eigenvectors of the ideal random time series cross-correlation matrix. Random matrix theory was first developed by statistical physicists dealing with the interactions of particles in complex quantum systems. The research object of random matrix theory is the cross-correlation matrix composed of multiple independent random variables. The cross-correlation matrix of stocks can be tested by the random matrix method. If the fluctuations between stocks are independent, the cross-correlation matrix of stocks should be consistent with the prediction of random matrix theory. If there is deviation in the conclusion, the deviation part indicates that the stock correlation matrix contains unique information of the market. Therefore, the purpose of random matrix research on stock cross-correlation is twofold: (a) to find the characteristics of the cross-correlation matrix deviating from the characteristics of the random matrix and (b) to try to isolate deviating information. The cross-correlation matrix is defined in terms of the matrix. Suppose G is an N × L dimension matrix composed of L consecutive price and return rates of N stocks, that is, the elements are the N × L dimension matrix, then the cross-correlation matrix can be expressed as
The eigenvalue distribution of the random cross-correlation matrix R, which is fixed in the limit case, can be solved analytically. Specifically, the probability density of the eigenvalue brother x of R can be obtained by the following formula:
Finally, we calculate the contribution F of each group to the feature vector:
The calculation of this contribution degree is similar to the wave function analysis in the chaotic system, so it has certain physical significance. As shown in Figure 2, through the calculation of the contribution of the corresponding eigenvectors of abnormal eigenvalues to various industries, it can be found that, except for the largest eigenvalues covering the whole city, the contribution of other abnormal eigenvalues to a few one or two industries is far greater than that of other industries, indicating that abnormal eigenvalues are closely related to the classification of stock industry.
2.2. Establishment of the Index Return Correlation Coefficient Matrix of the Artwork Financial Risk
By comparing the properties of the correlation coefficient matrix C and the random matrix, the random matrix theory divides C into two parts: one part conforms to the properties of the random matrix (noise), and the other part is the difference part, so as to improve the correlation coefficient matrix and remove noise information in the network. Through the summary of denoising methods, it is found that the purpose of denoising is to distinguish the noise and potential real information in the correlation coefficient matrix, so as to further extract real information. In a financial network, we need to compare the property difference between the correlation coefficient matrix C and the random correlation matrix, find out the difference between them, and then extract noise information and real information. The eigenvalues are calculated according to the random matrix theory (see Table 1).
To further verify the relationship between eigenvalue and the correlation coefficient, we observed the coefficient distribution histogram under two conditions: first, correlation coefficient distribution after removing the maximum eigenvalue; the other is correlation coefficient distribution after removing the eigenvalues beyond the prediction range. The specific calculation steps are as follows:(1)Work out the eigenvalues of the empirical correlation matrix.(2)The eigenvalue to be removed is set to zero and other values remain unchanged.(3)The new correlation coefficient matrix is obtained, the correlation coefficient is extracted, and the filtered correlation coefficient distribution histogram can be obtained after calculation. Figure 3 is the histogram of correlation coefficient distribution after removing the maximum eigenvalue.
Edge correlation coefficient threshold is a common method for network construction. Pearson correlation coefficient was calculated by using stock data, and the correlation coefficient matrix C was calculated. The stock was taken as the node, and the correlation between stocks was the linked edge. The appropriate minimum value was determined according to the distribution of the correlation coefficient matrix. Conversely, there is no edge between the two nodes. According to this, a financial network can be constructed. The edge correlation coefficient threshold method is adopted to construct the financial network model, and the specific steps of constructing the financial network are as follows:
Step 1. Calculate the original financial correlation coefficient matrix.
Step 2. Improve the correlation coefficient matrix based on the improved “denoising” method.
Step 3. Use the edge correlation coefficient threshold method to determine the threshold value.
Step 4. Establish a financial network model.
Since the calculation of the correlation coefficient depends on the time series examined, the conclusion of cross correlation matrix analysis is also affected by the different time series. In order to get more reliable conclusions as far as possible, first of all, in the selection of data, we select the data of the two art markets at the same time to exclude the influence of time, and then we make innovations in the processing of data. In order to eliminate the influence of time series selection on the conclusion, the static cross-correlation matrix and the dynamic cross-correlation matrix are constructed, respectively, according to the different scales and ways of time series selection. Static and dynamic cross-correlation matrices are defined as follows.
Static cross-correlation matrix, data to calculate the cross-correlation coefficient, namely,The statistical properties of the dynamic cross-correlation matrix are further analyzed. Since the purpose of the analysis is to investigate the characteristics of the market structure through the dynamic evolution of statistical indicators, some common statistical indicators are calculated. First, calculate the mean value of the first-order statistical index as
2.3. Improvement of the Stochastic Matrix Method for Art Amount Risk Prediction and Construction of the Network Model
By using random matrix theory, the maximum and minimum eigenvalues of artwork financial risk prediction of the random matrix are calculated as follows:
However, the dynamic evolution of the rate of the return correlation coefficient reflects the dynamic game process of risk contagion. The occurrence of art financial events will have an impact on financial subjects, and the change of financial subjects will affect the correlation between financial subjects. Therefore, based on the proposed time window, this paper established a model to analyze the dynamic evolution of the correlation coefficient and obtained the corresponding return rate series and constructed the correlation coefficient of index i and j according to the different ∆T step size:
T represents the year, DT represents the step length, which is 3 months, establish the corresponding average correlation coefficient dynamic correlation coefficient evolution model, and get Figure 4.
As shown in Figure 5, in order to timely analyze the risk contagion effect, this paper adopts relevant data to construct the artwork financial network. To sum up, the 5-day smoothed average closing price of STOCK indexes of 36 major global art finance STOCK markets (as shown in Table 2) released by the database and the STOCK Q database is selected as the research object in this paper.
Through the statistics of the eigenvector components in each range, it is found that the mean value and standard deviation of the eigenvector components in the left deviation and the main spectrum are about 0 and 0.1, respectively. The right deviation is different, and the mean of the eigenvector components of their eigenvalues is not zero.
According to random matrix theory, a random matrix does not have typicality. Therefore, according to the RMT method, the correlation coefficient matrix extreme eigenvalue, the random matrix R extreme eigenvalue, and the RMT prediction eigenvalue range are compared, and the results are shown in Table 3. It can be seen that, first, the eigenvalues of the random matrix almost fall within the range of the maximum and minimum predicted values of RMT, which is more consistent with the average of the possible eigenvalues of all the random matrices within the range predicted by the random matrix. Second, the maximum and minimum eigenvalues of the correlation coefficient matrix greatly deviate from the RMT prediction range, indicating that there are special nonrandom attributes in the correlation coefficient matrix.
As shown in Table 4, the eigenvalue of the correlation coefficient of the random matrix is close to 1, indicating that the information contained in it lacks economic meaning 29. For the correlation coefficient matrix, the eigenvalues of its full eigenvalues are smaller, indicating that the correlation matrix contains more information. Removing the maximum 3 eigenvalues for the correlation coefficient matrix means removing the eigenvalues in the correlation coefficient matrix that are greater than the maximum predicted eigenvalues of RMT. As shown in Table 4, after removal, the eigenvalue lineal value increases significantly, indicating that the effective information in the retention matrix decreases significantly. For the correlation coefficient, removing the minimum 8 eigenvalues means removing the eigenvalues in the correlation coefficient matrix that are smaller than the predicted minimum eigenvalues of RMT. As shown in Table 4, the eigenvalue lineal value increases after removal, indicating that the effective information in the retention matrix is also reduced. In other words, in the art financial risk prediction index correlation coefficient matrix studied in this paper, the part whose eigenvalue is less than the minimum predicted eigenvalue of RMT contains valid information and cannot be ignored.
Inverse reference ratio (IRP) can reflect the contribution of vector elements to the vector, and the larger its value is, the less contributing vector elements are. IRP of each feature vector is obtained according to equal calculation, as shown in Figure 6.
Figure 6 shows the empirical correlation matrix arranged in the ascending order of eigenvalues, and the IRP line of the corresponding eigenvector is the theoretical predicted value. As can be seen from the figure, the IRP values of the vectors corresponding to several small eigenvalues and those close to the upper limit of the predicted values are relatively large; the IRP values of most of the intermediate eigenvectors are near the predicted values; the IRP values of the vectors corresponding to the eigenvalues beyond the predicted range are smaller. This may be because the feature vectors at both ends of the prediction range can represent fewer income sequences. In the middle part, because most of the information is noise, the IRP value fluctuates around the predicted value of 0.015. Large eigenvalues also have a great impact on most stocks in the market, so the IRP value is relatively small.
2.4. Experimental Design
In a scale-free network, there exist a certain number of nodes with an extremely high degree value (connection number), that is, those nodes with the largest degree are usually called hub nodes, also known as the key nodes of the network. A few hub nodes have extremely many connections, while most nodes have only a few. Minority hub point plays a dominant role on the operation of the network. As a research network changes before and after denoising, select the first Shanghai all of the data, the art of financial stocks, the original data to construct the random matrix of art financial risk prediction, as well as the noise network, and find the key network nodes, respectively, and comparison analysis (see Table 5).
In the financial network, the structure of the model can reflect a certain connection between stocks to a certain extent, which is of great significance to the study of the interaction between different stocks in the art financial market and the buying behavior of investors. At present, software used to detect the modules in the network mainly includes the random matrix algorithm, which is relatively fast and can detect the modules from three nodes to eight nodes in the network and can detect more types of modules. We choose this algorithm to detect the modules in the financial network. Based on the Shanghai financial network constructed by all the art stock data in Shanghai, this paper examines the three nodes and four nodes of the original network and the random matrix for art financial risk prediction and makes comparative analysis (see Table 6).
3. Analysis of Experimental Results
Figure 7 shows the corresponding experimental results. In order to investigate the influencing factors of network parameter changes, the trend of the return rate of the art financial risk prediction index of CSM (computer system manual) in corresponding periods is also given. The analysis shows that there is a strong correlation with the change of time t, and the mean value of the eigenvalue of the cross-correlation matrix changes consistently, which further shows a positive correlation with the change of the mean value of the correlation coefficient. In addition, the fluctuation range is large, the fluctuation coefficient is 0.815; the two indexes of NYSE are almost 0 and the fluctuation coefficient is 0.051. This shows that CSM’s network community division results are good and bad, and the network structure fluctuates violently, while NYSE’s network community division results are consistent, and the network structure fluctuation is weak.
In order to observe whether the distribution of eigenvalues is related to the distribution of price and return, we adopt two additional methods to construct the simulated time series. The first method is to construct a random time series of the same length and the same number with 0 as the mean value and 1 as the standard deviation, so that the time series obtained will obey normal distribution. The second method makes the order of the real financial time series suffer for 1000 times and gets time series with the same length and number. The distribution of the new time series is consistent with the actual time series, but the randomization order will lose most information of the real time series except distribution. The eigenvalue distribution of the cross correlation matrix obtained by these four types of methods is reflected in the same figure, as shown in Figure 8. It can be found from the distribution, the CSM market or the NYSE market will be randomized and the random time sequence generated by the normal distribution of the distribution of time series with the same, and the distribution of the theoretical calculation is relatively close, and the distribution of the real price time series obvious deviation from the theoretical prediction and simulation results. This result indicates that two conclusions are valid: first, the distribution of the time series has no effect on the distribution of eigenvalues of the cross-correlation matrix, and the distribution of eigenvalues of the correlation matrix is also random as long as the calculated correlation coefficients are random. The cross-correlation matrix C of the time series of real artwork finance is not a pure random matrix, which hides more information than the cross-correlation matrix obtained by the random time series.
Antiparticipation rate (IPR) is an important index to determine the number of important components of a feature vector. We calculate the antiparticipation rate of each eigenvalue of CSM and NYSE. Figure 9 shows the relationship between the antiparticipation rate (IPR) of the two market feature vectors and their corresponding feature vectors in logarithmic coordinates. The mean of IPR corresponding to the eigenvalues in the MRT prediction shows that most components are well balanced. The maximum eigenvalue IPR is even lower than 1, indicating that almost all components contribute to it. The IPR of the sublarge eigenvalue is significantly higher than that of 1, indicating that only a few components contribute to it. In addition, it is not difficult to see that the characteristics of IPR in the two markets are basically the same, indicating that this feature has universality in different markets.
4. Conclusion
Based on the random matrix theory, this paper puts forward a new risk prediction method by analyzing the art financial correlation coefficient matrix and the statistical properties of the eigenvalues of the random matrix. Using the key nodes of the network, the minimum spanning tree to the original network, random matrix art financial risk prediction, and compares random matrix analysis, found that the random matrix of the original artwork financial risk prediction of key important information, which kept the art of important information in financial markets, on the contrary, and noise filtering original network important information. Through the analysis and comparison of the model structure and the community structure of the original network and the random matrix, the result shows that the model structure of the random matrix is more important in the network, which is more conducive to the study of the interaction between different works of art in the financial market and the buying behavior of investors. In addition, the community distribution of the random matrix is more uniform, showing the trend of centralized development of the art financial market, which is in line with the development of the actual market. Based on the improved denoising method, the stochastic matrix of artwork financial risk prediction shows better topological properties. The closer topology of the constructed network can better reflect the inherent nature of the financial network. There are still many rules of the evolution of financial markets, and it remains to be explored and studied to apply new denoising methods to other financial markets. Considering that the financial correlation coefficient matrix plays a very important role in portfolio and risk control, and the empirical correlation coefficient matrix used in it has a lot of noise information based on the new denoising method, the research on the financial market portfolio and risk control is our next work. Considering that the financial correlation coefficient matrix plays a very important role in portfolio and risk control, and the empirical correlation coefficient matrix used in it has a lot of noise information based on the new denoising method, it is our next step to study the portfolio and risk control of different financial markets.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the Department of Education of Jiangsu Province “Research on the rural settlement transformation under the background of new urbanization” (2021SJA1805).