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In order to what is the best definition of identification describe the risk dependence structure and correlation between financial variables, carry out scientific financial risk assessment, and provide the basis for accurate financial decision-making, first the basic theory of Copula function is established and the mixed Copula model is constructed. Then combined standard deviation formula class 11 hybrid Copula model is nested in a hidden Markov model HMM claes, the risk dependences among banking, insurance, securities and trust industries are analysed, and the Copula—Garch model is constructed for empirical analysis of investment portfolio.
Finally, the deep learning Markov model is adopted to predict the financial index. The forjula show that the mixed Copula model based on HMM is more effective than the single Copula and the mixed Copula models. The empirical structure shows that among the four major financial industries in China, the banking and insurance industries have strong interdependence and high probability of risk contagion.
With the deepening of global economic integration, the financial markets of various countries and regions have become more closely related, and their correlation has become more obvious [ 1 ]. However, the economic globalisation has made the financial crisis more widespread, and financial crises in the local regions or certain countries often trigger global financial crises, such as the subprime crisis in the United States [ 23 ].
Therefore, understanding and seizing the correlation between financial markets is of great significance for effectively avoiding the spread of financial crisis. In traditional correlation studies of financial variables, methods such as Pearson correlation coefficient, Spearman correlation and Granger causality test are often used, which are with great limitations [ 456 ]. Therefore, the Copula function has gradually become one of the main methods to explore the correlation between financial variables.
Yet the situation is complex and changeable in the actual financial market, and there deivation close correlations among various enterprises or fields. The single Copula function adopted to explore correlations in financial markets can combined standard deviation formula class 11 to large errors [ 78 ]. Moreover, existing scholars can accurately describe the tail correlation between financial variables by capturing the linear combination of Copula functions with tail characteristics to form a mixed Copula function [ 910 ].
Since there are many hidden states behind the yield rate in financial variables, special methods are required to speculate the hidden state. Hidden Markov model HMM is first applied in speech recognition and other fields, and then applied in finance and other fields [ 1112 ]. However, there are few studies on how to measure the portfolio risk by combining the Copula function with HMM.
To fill in the gap, the basic concept of the hybrid Copula model is analysed first, the hybrid Copula model is then nested in the HMM framework and applied to portfolio risk dependency and metric analysis, and deviatiin deep Markov model is adopted to predict the stock risk. To sum up, the results of this study aim to provide a theoretical basis for exploring the risk-dependent structure among financial variables in China combined standard deviation formula class 11 as to conduct accurate risk assessment.
The model based on Copula function can be adopted to study the correlation between variables and their characteristics. The binary Copula function is defined as follows. The Archimedes Copula function contains multiple Copula functions, which can be expressed by their generative meta-functions [ 13 ]. The mathematical expressions of these three functions are as follows, respectively.
Then the Gumbel Copula, Clayton Copula and Frank Copula functions are combined to build a mixed Copula function, and the weight values of these three functions are w 1w 2w 3. The mixed Copula function clasz be expressed as follows. Then the density function of the mixed Copula function is as follows. The state in HMM is hidden, which can be obtained by the observation sequence and transition probability matrix between states.
The four parameters wtandard HMM can be described as the number of states, distribution of initial probability, transition probability matrix and density function of observation sequence [ 16 ]. Therefore, the unconditional probability vector of the initial probability distribution is as follows. Based on the HMM model, the Copula model is nested in the framework to obtain the dynamic hybrid Copula model. The construction process is as follows. According to Sklar theorem [ 17 ], the mathematical expression of the joint density function f i of d dimension in the i state at time h is as follows.
Therefore, it needs to iterate to step K to maximise the Combined standard deviation formula class 11 function, which needs K steps to maximise each part. If the weights of the initial distribution, transition matrix and Copula functions are conditional, the Lagrange multiplication can be what is the equation of a linear relationship called to maximise the parameters and then the maximum likelihood function combined standard deviation formula class 11 adopted to estimate the edge distribution.
In Eq. Taking the optimisation of Autoencoder model in deep learning as an example [ 18 ], the L-BFGS can solve similar problems with gradient descent function, SGD and other functions, as shown in Figure 1. But in most cases, the L-BFGS function converges faster and has less memory overhead than other algorithms [ 19 ]. To construct the portfolio model with Copula function, it is necessary to determine the edge distribution of Copula and determine the edge distribution function.
The appropriate Copula function is selected to describe the dependence structure relation of multi-variable edge distribution. The generalised autoregressive conditional heteroskedast GARCH model is adopted to generalise the marginal distribution of variables [ 20 ]. Copula function is adopted to combine edge distribution functions of different variables, and the resulting joint distribution function is the portfolio function of financial assets.
It contains binary normal Copula and T-Copula models, whose mathematical expressions are as follows. And the logarithmic yield on the portfolio is as follows. VaR satisfies the following equation. In practice, the solution of Eq. Recurrent neural network RNN is used for prediction; RNN is also known as a deep learning Markov model, as its properties conform to the specific generalised Markov model [ 2223 ]. The basic structure of RNN model is shown in Figure 3.
The neurons in the hidden layer in the RNN are affected by the neurons in the upper layer. Different from other models, the parameters in the RNN model are shared with each other, which makes the RNN model carry out the same processing at different moments, which reduces the workload of parameter learning for what is direct relationship model [ 24 ].
At time t why are predator prey relationships important to every ecosystem, the gradient of the error function E in the cyclic layer weight matrix W with respect to RNN is as follows. After the parameter gradient of weight matrix in RNN is determined, the stochastic gradient descent algorithm is adopted to train RNN.
Then the RNN prediction results are evaluated by calculating the error rate and the average error rate. The calculation equations of error rate e and average error rate at time t are what is minimum orbital velocity follows. The closing prices of solstice from 2 January to 31 December are collected as the research object.
As all of the data are time series, data preprocessing is required to ensure the stability of data before building the model by these data [ 25 ]. In this study, the logarithmic rate of yield r of various industries is selected for empirical analysis. The descriptive statistics of data of various industries are conducted, and the results are shown in Figure 4. Compared with the standard normal distribution, the distribution of exponential logarithmic yields of various industries has a more obvious kurtosis and skewness.
Descriptive statistical results of the exponential logarithmic yield of each industry. A banking industry; B insurance industry; C securities industry; D trust industry. Through descriptive statistics, the kurtosis values of logarithmic yields of these entity relational data model industries series index are, respectively, 9. As they have high peak value, they do not deviatuon to the normal distribution, and the yield rates have the characteristics of peak distribution.
Tormula to the analysis of skewness value, the skewness value of the exponential logarithmic yield of these four industries is, respectively, 0. The final constructed model is the edge distribution function. After multiple verifications, it is found that GARCH 1,1 is the most effective method to verify all the yield rate sequence models. Therefore, normal distribution and T distribution are used in this study to simulate the residuals of yield series.
The parameters of each model are estimated respectively. The final estimation results of edge distribution are shown in Table 2. Akaike information criterion AIC value can reflect the optimal benign of statistical model fitting. The estimation results of edge distribution parameters obtained from Table 1 show that the AIC value of normal distribution is greater than that of T -distribution, which indicates the fitting effect of edge distribution of exponential logarithmic yield rates of A, B, C and D industry sequences based on T distribution is obviously better than that of normal distribution.
As A, B, C, and D exponential logarithmic yield sequence have the feature of thick tail, it indicates that the T distribution has an obvious advantage in describing the yield sequence with peak and thick tail distribution feature. Estimation of edge distribution parameters of sequential exponential logarithmic yield sequence based on GARCH model. In order to eliminate autocorrelation, autoregressive moving average ARMA model [ 26 ] deviatoin used for data processing in this study.
Compared with the GARCH model, the generalised autoregressive score GAS comined can make full use of the density function, so it is adopted to overcome the deviatlon in the data [ 27 ]. The claxs function of y is p y fFwhere F is the information set. Then the mathematical expression of GAS 1,1 model is as follows. The GAS model is adopted to determine the sequence edge distribution of each industry, and the parameter estimation value of the GAS model is finally obtained, as shown in Table 3.
As shown in Table 3 and Figure 5A5B and 5C all obey the Laplace standqrd, which confirms that the sequences of the three industries have certain skewness, and it is the same as the analysis results of the GARCH model. The distribution of D industry is different from that of other industries, and it belongs to partial t -distribution. Estimation of edge distribution parameters standrad sequence exponential logarithmic rate of yield based on GAS model.
PIT statistical results. PIT, probability integral transform. The uniformly distributed variables obtained from the above PIT are calculated in the mathematical expressions of three single Gumbel Copula, Clayton Copula, Frank Copula and mixed Copula functions, coass are Eqs 3—6. The parameters of the Gumbel Copula, Clayton Copula, Frank Copula, combined standard deviation formula class 11 mixed Copula functions are estimated, respectively, and the iteration results of parameters in different functions are shown in Figure 7.
The expectation-maximisation EM algorithm is adopted for parameter iteration, standaard the parameters of different functions gradually become stable after 10 times of iteration. Therefore, the parameter estimates of the three single Copula functions are taken as the initial values of the parameters of the mixed Copula function, and the obtained parameter estimates of the single Copula and mixed Copula functions are shown in Table 4.
According to the principle of maximum log-likelihood function and minimum AIC and Combined standard deviation formula class 11 information criterion BIC [ 28 ], the log-likelihood function of the mixed Copula function is the largest and its AIC and BIC values are the combined standard deviation formula class 11, which indicates the mixed Copula model constructed in this study has veviation effects compared sfandard the single Copula model.
State 1 is a low dependent state, while state 2 is a high dependent state. After analysing the probability of Copula example of symbiotic plants class 7 based on HMM in these two states, the probability of state 2 is the highest As shown in Table 4the mixed Commbined model based on HMM proposed in this study has the largest logarithmic likelihood function and the lowest AIC and BIC values, which indicates that the model effect of nesting mixed Copula functions into HMM is optimal, which is consistent with the research results of Chun et al.
Subsequently, the dynamic transition diagrams of states 1 and 2 of the sample from to are obtained, as shown in Figure atandard. These two dependent states will change at a formkla frequency, yet the evaluation of this transformation is not regular and mainly depends on the degree of interdependence between industries. Studies show that the increase in interdependence will lead to the occurrence of risk combned, and the probability of risk infection is very high during this period.
As combined standard deviation formula class 11 in the low dependence probability graph Figure 9B and the high dependence probability graph Figure 9Cthe probability graph is basically consistent with the trend of the dynamic transition graph of states. Therefore, when an industry is in state 2 high dependence statethere is a very strong dependence relationship between industries, which may cause macro or systemic risks from the perspective of dependency and risk contagion [ 30 ].
Transition and state conditional probability of mixed Copula model based on HMM. A the dynamic transition graph of the state; B the probability stqndard of low dependence; and C the probability graph of high dependence. The interdependence of industries under these two states is calculated, and the results are shown in Table 5. Under different states, the tail dependence between A banking and B insurance is the strongest, which indicates that they have the most sensitivity to the impact of risks, and are more susceptible to be impacted by the own risks of extreme events.
Assuming that the investor invests according to the weight of [0.
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