# A Dynamic Prediction Model of Financial Distress in the Financial Sharing Environment

Authors: Jie Zhu; Hong Zhu; Nan Lin
Journal: Discrete Dynamics in Nature and Society
 (2023)

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## Abstract

The dynamic prediction of financial distress can monitor the financial status of an enterprise in real time and provide evidence for financial analysts. However, currently, there are few studies concerning the dynamic prediction of financial distress in the financial sharing environment, so in order to fill this research gap, this study established a dynamic prediction model of financial distress in the financial sharing environment. Firstly, this study employed the analytic hierarchy process (AHP) and entropy weight theory to determine an index system for the dynamic evaluation of financial distress in the financial sharing environment and gave the weight assignment method of the evaluation indexes. Then, based on the probabilistic neural network (PNN), this study constructed a dynamic prediction model of financial distress and used experimental results to verify the effectiveness and feasibility of the constructed model.


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## Body

## 1. Introduction

At the present time, the rapid development of information technologies such as cloud computing, big data, and mobile Internet has promoted the application of financial informatization [1–8], providing conditions for the realization of financial sharing services (FSS) [9–16]. The FSS mode not only realizes the information management and control of functional departments inside the companies but also greatly improves the efficiency and quality of financial information transmission, financial data processing, and analysis; however, at the same time, it also brings unprecedented challenges to the integrated financial information systems and reform of paperless audit environment [17–22].In the emerging credit market, the demand for risk management increases day by day. Meanwhile, many new algorithms are emerging. As a result, the dynamic prediction of financial stress rises to prominence again. El Bannany et al. [23] studied the performance of various data mining models applied to the prediction of financial distress of companies in the Middle East and North Africa and optimized the multilayer perceptron model by adjusting hyperparameters such as network depth and width; at last, they used a real-time sample data set of companies in North Africa to evaluate the performance of the proposed prediction model. Sun et al. [24] divided the company’s financial status into four types: sound status, false-sound status, moderate distressed, and severe distressed; then, the support vector machine (SVM) was combined with three kinds of decomposition and fusion methods of one-to-one, one-to-static, and error-correcting output codes to establish three multitype financial distress early-warning models. The prediction of corporate financial distress is very important to companies, investors, and regulators. However, most financial distress prediction models are established based on a single time dimension, and they generally ignore two key characteristics of the financial distress data: data set imbalance and data flow concept drift. Currently, financial stress prediction is being applied and studied in many financial fields as an important risk management tool. Shen et al. [25] proposed a new dynamic financial distress prediction method: the adaptive neighbor SMOTE recursive integrated approach, which allows multiple prediction results to be obtained from unbalanced data streams; through overall average AUC, it is found that the random forest classifier outperforms other commonly used classifiers in terms of data classification. Christopoulos et al. [26] assumed that liquidity and profitability constitute the key criteria for the status configuration of corporate financial distress status; then, with companies listed in the New York Stock Exchange as samples, they adopted a survival model based on dynamic logit to study the predictability of financial distress. In order to identify all the factors that have a significant impact on the company’s financial status and rank them according to their relative importance, Mahtani et al. [27] adopted a multi-criteria decision-making method based on fuzzy AHP and further verified the robustness of the method through sensitivity analysis. Li et al. [28] extended the cross-sectional DEA model to the time-varying Malmquist DEA, and this decision support system can intelligently adjust the efficiency boundary over time and make robust predictions. The experimental results showed that, besides accurately predicting financial distress based on DEA efficiency measurement, the Malmquist DEA also provides insights into the company’s competitive position.After comprehensively reviewing the existing literature, it is found that although the research on financial distress prediction has achieved certain results, there are still problems that cannot be ignored, such as the dimensionality reduction of financial index data, the low prediction accuracy of integrated algorithms, and the negligence of the dynamic and unbalanced nature of sample data. In addition, currently, there are few studies concerning the dynamic prediction of financial distress in the financial sharing environment, so in order to fill in this research gap, this study establishes an index system for the dynamic prediction of financial stress in the financial sharing environment and further explores the integrated model for the dynamic forecast of financial stress. The main contents fall into two aspects: (1) Section2 employs the analytic hierarchy process (AHP) and entropy weight theory to determine an index system for the dynamic evaluation of financial distress in the financial sharing environment and scientifically weighs the evaluation indexes. (2) Section 3 details the idea of building a dynamic prediction model for financial stress based on probabilistic neural network (PNN), clarifies the flow of input data processing and the construction process of PNN model and optimal combination prediction model. Finally, the model prediction was carried out in the light of timeliness, and the proposed model was proved feasible and effective through comparative experiments.


## 2. Determination of the Weights of Evaluation Indexes of the Prediction Model

After combing through the domestic and foreign literature on financial stress prediction in the financial sharing environment, it can be learned that the idea of sharing service has been well applied in some Chinese and foreign enterprises. However, this idea is still being explored for most enterprises. In addition, the relevant literature is fragmented. Most focus on theories and cases, while few talk about empirical research.Based on the existing results, this study employs AHP and the entropy weight theory to construct a dynamic prediction model of financial distress in the financial sharing environment. In 1865, R. Clausius proposed the entropy weight theory, which illustrates the uniformity of spatial distribution of energy. Shannon applied this theory to information theory and put forward the information entropy weight. Referring to the analysis of the impact of specific financial sharing risks on the identification link of the dynamic significance of financial distress and the possible consequences, the analysis results were taken as the basis for the evaluation index selection of the proposed financial distress prediction model in this study.Financial shared service (FSS) is the improvement direction of the current financial operation mode. FSS operation modes could be divided into four types of basic mode, market mode, advanced market mode, and independent business mode, which respectively aim at the functional departments inside the company, the responsible entities that operate independently, and the profit organizations. Figure1 gives a schematic diagram of the FSS operation modes. As shown in the figure, according to the development trend of FSS, the four operation modes develop progressively over time. Figure 2 shows the process framework of FSS. Only by fully integrating the existing financial systems and realizing the standard FSS process, specific applications of the FSS technology can be guaranteed. According to Figure 2, the process of FSS contains 11 first-level procedures including tax accounting, FSS data maintenance, financial report analysis, and general ledger accounting of the financial system, etc., and under this framework, there are more-detailed second-level procedures and specific businesses. The advancement progress of FSS mode is greatly affected by the framework of the sharing process.Figure 1
Operation modes of FSS.Figure 2
Overall process framework of FSS.Following principles of pertinence, comparability, and operability, this study analyzes actual FSS cases and selects 6 major financial distress types to construct the first-level evaluation indexes for judging whether there is financial distress in the financial system in the financial sharing environment, and then, the 6 first-level evaluation indexes were further divided into 28 second-level evaluation indexes, representing subtypes of the financial distress; together these indexes constituted a relatively complete evaluation index system of financial distress in the financial sharing environment. Each type of financial distress involves different financial risk items, and all these evaluation indexes are closely related to the financial distress and its dynamic changes during the FSS implementation process.The evaluation target of the evaluation index system is the dynamic evaluation result of financial distress based on the application of the FSS system (FD). The 6 first-level evaluation indexes include strategy planning FD1, organization management FD2, financial analysts FD3, financial risk control process FD4, financial system construction FD5, and compliance operation management FD6. Strategy planning FD1 contains 6 second-level indexes: strategy formulation conflict FD11, insufficient preparation for system implementation FD12, unreasonable center location FD13, unreasonable financial business scope definition FD14, management’s emphasis on risks FD15, excessive initial investment and operation management costs FD16. Organization management FD2 contains 4 second-level indexes: internal conflicts of financial organization FD21, unintegrated financial and business departments FD22, unreasonable organization structure FD23, and unreasonable system formulation FD24. Financial analysts FD3 contains 3 second-level indexes: difficult internal communication of financial analysts FD31, financial analysts’ insensitivity to data FD32, and nonstandard operation of financial analysts FD33. Financial risk control process FD4 contains 5 second-level indexes: non-standard instruction for initial process operations FD41, unreasonably designed standard process FD42, difficult integration of financial business processes FD43, lack of balance control FD44, and poor adaptability of new process FD45. Financial system construction FD5 contains 5 second-level indexes: system integration inability FD51, unreasonable system development and design FD52, weak system support FD53, insufficient system stability FD54, and insufficient system electronic data security FD55. Compliance operation management FD6 contains 5 second-level indexes: noncompliant organizational form of the system FD61, noncompliant system operation mode FD62, noncompliant system authority setting FD63, corporate and financial operation risks FD64, and incomplete internal audit system FD65.Figure3 gives a scree plot of the evaluation indexes. According to the figure, when the abscissa is greater than 6, the eigenvalue of the evaluation indexes is less than 2, and the curve tends to become flattered, indicating that an inflection point appeared after the 6-th evaluation index, and this has also verified that it is reasonable to extract 6 types of financial distress.Figure 3
The screen plot of evaluation indexes.If a financial system hasm different dynamic distress statuses, which are represented by R1, R2, R3, …, Rm, and the probability values corresponding to each status are represented by QA1, QA2, QA3, …, QAm, then the dynamic information volume of the uncertain financial distress of the system can be described by the following formula:(1)D−∑i=1mQAilnQAi,satisfies0≤QAi≤1i=1,2,3,…,m,∑i=1mQAi=1.The entropy weight method was used in the dynamic prediction model of financial distress in the financial sharing environment and taken as the method for determining the weight values of the model indexes. If the dynamic evaluation of financial distress needs to measurem types of factors and ωij represents the evaluation index of factor FD, then it satisfies the following formula:(2)∑j=1nωij=1i=1,…,m;j=1,…,n.The relative importance of FDi can be measured by entropy value:(3)Di=−∑j=1nωijlnωij.The relative importance of FDi was normalized, and the entropy value of the relative importance of FDi was calculated by the following formula:(4)si=−1lnn∑j=1nωijlnωij.The weight of FDi can be determined based on 1 − si using the following formula:(5)ψi=1m−P1−si0≤ψi≤1,P=∑i=1msi,∑i=1mψi=1.


## 3. Research Methods of the Prediction Model

### 3.1. Model Analysis Steps

The specific analysis steps of the proposed dynamic prediction model of financial distress constructed based on AHP and entropy weight theory are described in detail as follows:Step 1. Modeling foundation Basic works of constructing the proposed model are as follows: (1) correctly selecting dynamic evaluation indexes of financial distress; (2) reasonably setting the scope of the conformance weight of the prediction model; (3) scientifically defining the level of significance of financial distress in the financial sharing environment.Step 2. Assign values to the conformance weightβijl of the second-level evaluation indexes in the third layer of the prediction model.Step 3: Suppose that the number of inducing factors corresponding to thei-th type of financial distress is represented by q, and the number of control measures corresponding to the j-th inducing factor of the i-th type of financial distress is represented by e, then, based on the following formula, the entropy value αij and weight ψij of the second-level indexes, namely, the second layer of the model can be measured as follows:(6)αij=−1lne∑l=1eβijllnβijl;ϕij=1−αijq−∑j=1qαij.In order to make the data of the evaluation indexes meet the calculation requirements of the entropy weight method, it is necessary to normalizeβijl before the calculation of αij, and then use the formula to calculate weight value ψij corresponding to αij.Step 4: Based on the following formula, the entropy valueγi and the corresponding weight ψi of the evaluation indexes in the first layer of the model could be calculated:(7)γi=−1lnq∑j=1qϕijlnϕij;ϕi=1−γiq−∑j=1qγi.Step 5: Suppose the number of evaluation indexes under the overall significance level of the target layer (namely, the financial distress in the financial sharing environment) is represented byg, then, based on the following formula, the entropy value d and weight value ψ of the overall significance of the target layer (the financial distress in the financial sharing environment) could be calculated as follows:(8)d=−1lng∑j=1gϕilnϕi;ϕ=1−d.The PNN, proposed by Dr. Specht in 1988, is a multilayer hierarchical neural network. This study adopts the model to dynamically evaluate and predict the financial stress in the financial sharing environment. The input data of the network were a grayscale image composed of the data of indexes from yearτ − 1 to year τ − w, and the indexes were divided into two types: quantitative indexes and qualitative indexes. In this study, the data of these two types of indexes were processed respectively, and the specific steps are introduced as follows:


### 3.2. Input Data Processing

After normalization, the value range of all elements in the matrix formed by the evaluation index data of the corporate financial system to be evaluated satisfied [0, 1]. The matrix was converted into a grayscale image that uses pixel value to represent the shade of the color; the pixel value range was [0, 255], and 0 and 255, respectively, represent absolute black and absolute white. The value range of the index data was transformed from [0, 1] to [0, 255], and the obtained integer value was represented byUAτ−wij:(9)UAijτ−w=Aijτ−w∗255.Among them, <x> means to take the integer part of x. Based on the idea of energy minimization, the excepted grayscale image could be generated, and the specific steps are as follows:Step 1: Initialize settings.TheM evaluation indexes of the i-th corporate financial system to be evaluated in the τ-th year were arranged in a M1/2 × M1/2 matrix from top to bottom and from left to right, and the set of the corresponding serial numbers of the evaluation indexes was represented by O = {1,2, …, M}. Then, the initial grayscale image was generated based on the matrix and represented by GPi,0τ. Suppose, NSτ −w represents the energy of a company in the τ − w-th year, qτ −w(V(j1), V(j2)) represents the correlation coefficient between V(j1) and V(j2) in the τ − w-th year, and V(j) represents the value corresponding to the j-th index, then the following formula gives the expression of NSτ −w, the energy of the grayscale image:(10)NSτ−w=∑j1,j2qτ−wVj1,Vj2ξτ−wj1,j2.Suppose,ξτ −w(j1, j2) represents the distance between the j1-th index and the j2-th index in the τ − w-th year, aτ −w (j1) and bτ −w (j1) respectively represent the abscissa and ordinate of the pixel corresponding to the j1-th index in the τ − w-th year, then there is(11)ξτ−wj1,j2=aτ−wj1−aτ−wj22+bτ−wj1−bτ−wj22.The data of the evaluation indexes of thei-th company in the τ − w-th year were converted to generate an initial grayscale image GPi,0τ−w with an initial energy NS0τ−w.Step 2. Randomly select two pixels inGPi,0τ−w and exchange their positions. If the energy value of the grayscale image can be reduced, then the positions are exchanged; otherwise, they are not exchanged.Step 3: Repeat the previous step. When the number of repetitions reaches 3 times the number of evaluation indexes and there is still no pixel position swap, terminate the repetition operation and output the exchanged imageGPiτ−w and the corresponding energy NSτ −w.


### 3.3. Modeling of the PNN

After grayscale image TWτ −wi (w = 1, 2, 3, 4, 5; i = 1, 2, …, N) of the τ − w-th year of the i-th corporate financial system to be evaluated was input into the PNN, the evaluation prediction result (DRiτ−w, 1 − DRiτ−w) could be obtained. Wherein, DRiτ−w represents the probability that the financial system of the i-th company to be evaluated is evaluated as in financial distress in the time window of the τ − w-th year; 1 − DRiτ−w represents the probability that the financial system of the i-th company to be evaluated is evaluated as not in financial distress in the time window of the τ − w-th year.Taking the grayscale images of theτ − w-th year corresponding to N companies to be evaluated as the input variables of the PNN, then, based on the activation function, the following equation could be constructed:(12)CZτ−wτ=GAτ−w,ωτ−w.The output of formula (12) is CZτ−wτ(w = 1, 2, 3, 4, 5), namely, the financial status of the financial system of the company to be evaluated of the τ-th year in the time window of the τ − w-th year. Suppose, TPiτ represents the real status of the financial system of the i-th company to be evaluated in the τ-th year, then, in order to describe the output variable more clearly, in this study, the data of the financial system of the i-th company to be evaluated in the τ-w-th year are written in the form of W arrays of TWiτ−1,TPiτ, TWiτ−2,TPiτ, TWiτ−3,TPiτ, TWiτ−4,TPiτ, and TWiτ−5,TPiτ.PNN is a neural network model containing multiple layers of input layer, pattern layer, summation layer, and output layer. Figure4 shows the structure of the constructed PNN model. Since the evaluation of financial distress in the financial sharing environment is a kind of two-category problem, the output of the model is the final category of the input grayscale image.Figure 4
Structure of the constructed PNN.Bayesian discriminant analysis and probability density function together constitute the basic theories of PNN. Assuming in a financial distress classification problem in the financial sharing environment,a(n) = [a1, a2, …, am] represents the n × m dimension input grayscale image, if XRz represents the prior probability that the input grayscale image belongs to the z-th category, BDz represents the penalty cost for wrong classification, SDz(a) represents the probability density function, then the following formula gives the expression of Bayesian discriminant analysis:(13)XRiBDiSDia>XRjBDjSDja∀j≠i.Formula (13) describes that the grayscale image input this time belongs to the i-th category. For the three variables of prior probability, penalty cost, and probability density function, it is quite difficult to find an ideal probability density function SD(a). In order to solve this problem, this study chose to introduce the Parzen window theory to estimate the probability density function using a supervised training set and use a Gaussian function to express it. Suppose, A represents the input grayscale image, Aj represents the central grayscale image coming from the entire training set, and ε represents the smoothing parameter, then there is(14)SDjA=exp=A−Aj22ε2.Based on the above formula, the probability density function of thej-th neuron node of the pattern layer could be obtained, which is represented by SDj(A) (∀j = 1, 2, …, M). The HU that describes the category attribute from the pattern layer to the summation layer is an M × 2 dimensional vector, which can be denoted as HUM × 2 = [δij](∀i = 1, 2, …, M; ∀j = 1, 2). Suppose, SP = [SD1(A), SP2(A), …, SPj(A), …, SDM(A)] represents the vector constituted by the probability density functions of all neuron nodes in the pattern layer, then the following formula gives the calculation formula of the output result of the training grayscale image samples in the network summation layer:(15)HA=SP⋅HU=SD1A,SD2A,…,SDjA,…SDMA⋅δijM×2.Normalize HA to get(16)HA¯=HAHA⋅DWT.Let HA represent the real evaluation result of a certain sample, and if the training grayscale image sample belongs to thei-th category, then only the i-th element is equal to 1. The calculation formula of the error function of the sample evaluation result is given as follows:(17)WC=1−HA¯⋅HA′T.The above formula is the error function designed for the constructed PNN, and the value range of the error value WC is [0, 1]. The greater the value of thei-th element in the vector output by the network, the closer the evaluation result is to the real evaluation result, and the closer the WC is to 0. Especially when the evaluation result is equal to the real evaluation result, WC is equal to 0. By substituting formulas (15) and (16) into formula (17), the specific error function of the constructed PNN could be obtained:(18)WC=1−SP⋅HU⋅HA′TSP⋅HU⋅DWT.Among the parameters in the constructed PNN,θij represents the weight coefficient between the i-th neuron node in the input layer and the j-th neuron node in the pattern layer, then there is(19)θij=ai∑δ=1maδ21/2.Suppose,ai represents the index value of the i-th column of the input grayscale image, then A and Aj in formula (14) can be calculated by formulas (20) and (21), respectively:(20)A=∑i=1mai⋅θij,(21)Aj=∑i=1mai′⋅θij.Suppose,ai′ represents the j-th neuron node in the pattern layer, namely, the index value of the i-th column of the j-th sample data in the grayscale image training set. The two formulas were substituted into SP = [SD1(A), SP2(A), …, SPj(A), …, SDM(A)], and then the SP was brought into the error function and its derivative was taken. Assuming η represents the learning rate, by letting the derivative of the error function be equal to 0, the update of the weight coefficient could be completed:(22)Δθij=−η⋅∂WC∂θij=−η⋅∂SP/∂θij⋅HU⋅HA′T⋅SP⋅HU⋅DWT−SP⋅HU⋅HA′′T⋅∂SP/∂θij⋅HU⋅HATSP⋅HU⋅DWT2.Specific to thej-th probability density function in SP, based on formulas (14), (20), and (21), the update formula of the weight coefficient could be derived:(23)∂SDj∂θij=θij⋅ai−ai′2εj2⋅exp−∑z=1mθzj2⋅az−az′22εj2.Further, the expression of the set vector could be obtained as follows:(24)∂SP∂θij=∂SD1A∂θij,∂SD2A∂θij,…,∂SDjA∂θij,…,∂SDmA∂θij.Finally, by combining formulas (23) and (24) with formula (22), the final weight coefficient update formula can be obtained.Referring to the update process of the weight coefficient, the smoothing parameter of the network was updated, and the update formula is given by the following equation:(25)Δεj=−η⋅∂WC∂εj=−η⋅∂SP/∂εj⋅HU⋅HA′T⋅SP⋅HU⋅HAT−SP⋅HU⋅HA′T⋅∂SP/∂εj⋅HU⋅HATSP⋅HU⋅HAT2.Specifically, the smoothing parameter update formula of thej-th neuron node in the pattern layer is given as follows:(26)∂SDj∂εj=∑z=1mθzj2⋅az−az′2εj3⋅exp−∑z=1mθzj2⋅az−az′22εj2.Further, the calculation formula of the set vector could be given as follows:(27)∂SP∂εj=∂SD1A∂εj,∂SD2A∂εj,…,∂SDjA∂εj,…,∂SDmA∂εj.Similarly, at last, by combining formula (26) with formulas (27) and (25), the final smoothing parameter update formula could be obtained.


### 3.4. Construction of the Optimal Combination Prediction Model

In a financial sharing environment, the dynamic evaluation of financial distress needs to consider the information content volume and timeliness of the evaluation index data. The information content volume of the evaluation index data of financial distress decreases with the passage of time, that is, the shorter the time from the occurrence of the financial distress, the greater the information content volume of the evaluation index data, and the higher the accuracy of the evaluation result. Conversely, the longer the time from the occurrence of the financial distress, the smaller the information content volume of the evaluation index data, and the lower the accuracy of the evaluation results. In order to find the optimal ratio of different time windows, this study comprehensively considered the impact of different time periods on the financial distress evaluation of the financial system of each company to be evaluated.Suppose, BR represents the fold number of the cross-validation; DRτ −wi, v represents the probability that the v− th fold (rv = 1, 2, …, BR) of the financial system of the i-th company to be evaluated is evaluated as in financial distress in the τ-th year predicted in the time window of the τ − w-th year; 1 − DRτwi,v represents the probability that the v-th fold of the financial system of the i-th company to be evaluated is evaluated as not in financial distress in the τ-th year predicted in the time window of the τ − w-th year; TP wi,v (i = 1, 2, …, N) represents the real probability that the v-th fold of the financial system of the i-th company to be evaluated is evaluated as in financial distress; 1 − TP wi,v represents the real probability that the v-th fold of the financial system of the i-th company to be evaluated is evaluated as not in financial distress; Ψτ −w represents the parameter that describes the heterogeneity information of different years to be estimated, and then, based on the grayscale image verification sample set and the optimization model, the time window could be weighted as follows:(28)minΨτ−1,Ψτ−2,Ψτ−3,Ψτ−4,Ψτ−51BR∑v=1BR∑i=1N∑w=15Ψτ−σθi,vτ−w−TPi,vτ2+∑w=15Ψτ−w1−θi,vτ−w−1−TPi,vτ−w−1−TPi,vτ2⇔minΨτ−1,Ψτ−2,Ψτ−3,Ψτ−4,Ψτ−51BR∑v=1BR∑i=1N∑w=15Ψτ−wθi,vτ−w−TPi,vτ2,(29)s.t.0<Ψτ−w<1;w=1,2,3,4,5,(30)Ψτ−5<Ψτ−4<Ψτ−3<Ψτ−2<Ψτ−1,(31)∑w=15Ψτ−w=1;w=1,2,3,4,5.


## 3.1. Model Analysis Steps

The specific analysis steps of the proposed dynamic prediction model of financial distress constructed based on AHP and entropy weight theory are described in detail as follows:Step 1. Modeling foundation Basic works of constructing the proposed model are as follows: (1) correctly selecting dynamic evaluation indexes of financial distress; (2) reasonably setting the scope of the conformance weight of the prediction model; (3) scientifically defining the level of significance of financial distress in the financial sharing environment.Step 2. Assign values to the conformance weightβijl of the second-level evaluation indexes in the third layer of the prediction model.Step 3: Suppose that the number of inducing factors corresponding to thei-th type of financial distress is represented by q, and the number of control measures corresponding to the j-th inducing factor of the i-th type of financial distress is represented by e, then, based on the following formula, the entropy value αij and weight ψij of the second-level indexes, namely, the second layer of the model can be measured as follows:(6)αij=−1lne∑l=1eβijllnβijl;ϕij=1−αijq−∑j=1qαij.In order to make the data of the evaluation indexes meet the calculation requirements of the entropy weight method, it is necessary to normalizeβijl before the calculation of αij, and then use the formula to calculate weight value ψij corresponding to αij.Step 4: Based on the following formula, the entropy valueγi and the corresponding weight ψi of the evaluation indexes in the first layer of the model could be calculated:(7)γi=−1lnq∑j=1qϕijlnϕij;ϕi=1−γiq−∑j=1qγi.Step 5: Suppose the number of evaluation indexes under the overall significance level of the target layer (namely, the financial distress in the financial sharing environment) is represented byg, then, based on the following formula, the entropy value d and weight value ψ of the overall significance of the target layer (the financial distress in the financial sharing environment) could be calculated as follows:(8)d=−1lng∑j=1gϕilnϕi;ϕ=1−d.The PNN, proposed by Dr. Specht in 1988, is a multilayer hierarchical neural network. This study adopts the model to dynamically evaluate and predict the financial stress in the financial sharing environment. The input data of the network were a grayscale image composed of the data of indexes from yearτ − 1 to year τ − w, and the indexes were divided into two types: quantitative indexes and qualitative indexes. In this study, the data of these two types of indexes were processed respectively, and the specific steps are introduced as follows:


## 3.2. Input Data Processing

After normalization, the value range of all elements in the matrix formed by the evaluation index data of the corporate financial system to be evaluated satisfied [0, 1]. The matrix was converted into a grayscale image that uses pixel value to represent the shade of the color; the pixel value range was [0, 255], and 0 and 255, respectively, represent absolute black and absolute white. The value range of the index data was transformed from [0, 1] to [0, 255], and the obtained integer value was represented byUAτ−wij:(9)UAijτ−w=Aijτ−w∗255.Among them, <x> means to take the integer part of x. Based on the idea of energy minimization, the excepted grayscale image could be generated, and the specific steps are as follows:Step 1: Initialize settings.TheM evaluation indexes of the i-th corporate financial system to be evaluated in the τ-th year were arranged in a M1/2 × M1/2 matrix from top to bottom and from left to right, and the set of the corresponding serial numbers of the evaluation indexes was represented by O = {1,2, …, M}. Then, the initial grayscale image was generated based on the matrix and represented by GPi,0τ. Suppose, NSτ −w represents the energy of a company in the τ − w-th year, qτ −w(V(j1), V(j2)) represents the correlation coefficient between V(j1) and V(j2) in the τ − w-th year, and V(j) represents the value corresponding to the j-th index, then the following formula gives the expression of NSτ −w, the energy of the grayscale image:(10)NSτ−w=∑j1,j2qτ−wVj1,Vj2ξτ−wj1,j2.Suppose,ξτ −w(j1, j2) represents the distance between the j1-th index and the j2-th index in the τ − w-th year, aτ −w (j1) and bτ −w (j1) respectively represent the abscissa and ordinate of the pixel corresponding to the j1-th index in the τ − w-th year, then there is(11)ξτ−wj1,j2=aτ−wj1−aτ−wj22+bτ−wj1−bτ−wj22.The data of the evaluation indexes of thei-th company in the τ − w-th year were converted to generate an initial grayscale image GPi,0τ−w with an initial energy NS0τ−w.Step 2. Randomly select two pixels inGPi,0τ−w and exchange their positions. If the energy value of the grayscale image can be reduced, then the positions are exchanged; otherwise, they are not exchanged.Step 3: Repeat the previous step. When the number of repetitions reaches 3 times the number of evaluation indexes and there is still no pixel position swap, terminate the repetition operation and output the exchanged imageGPiτ−w and the corresponding energy NSτ −w.


## 3.3. Modeling of the PNN

After grayscale image TWτ −wi (w = 1, 2, 3, 4, 5; i = 1, 2, …, N) of the τ − w-th year of the i-th corporate financial system to be evaluated was input into the PNN, the evaluation prediction result (DRiτ−w, 1 − DRiτ−w) could be obtained. Wherein, DRiτ−w represents the probability that the financial system of the i-th company to be evaluated is evaluated as in financial distress in the time window of the τ − w-th year; 1 − DRiτ−w represents the probability that the financial system of the i-th company to be evaluated is evaluated as not in financial distress in the time window of the τ − w-th year.Taking the grayscale images of theτ − w-th year corresponding to N companies to be evaluated as the input variables of the PNN, then, based on the activation function, the following equation could be constructed:(12)CZτ−wτ=GAτ−w,ωτ−w.The output of formula (12) is CZτ−wτ(w = 1, 2, 3, 4, 5), namely, the financial status of the financial system of the company to be evaluated of the τ-th year in the time window of the τ − w-th year. Suppose, TPiτ represents the real status of the financial system of the i-th company to be evaluated in the τ-th year, then, in order to describe the output variable more clearly, in this study, the data of the financial system of the i-th company to be evaluated in the τ-w-th year are written in the form of W arrays of TWiτ−1,TPiτ, TWiτ−2,TPiτ, TWiτ−3,TPiτ, TWiτ−4,TPiτ, and TWiτ−5,TPiτ.PNN is a neural network model containing multiple layers of input layer, pattern layer, summation layer, and output layer. Figure4 shows the structure of the constructed PNN model. Since the evaluation of financial distress in the financial sharing environment is a kind of two-category problem, the output of the model is the final category of the input grayscale image.Figure 4
Structure of the constructed PNN.Bayesian discriminant analysis and probability density function together constitute the basic theories of PNN. Assuming in a financial distress classification problem in the financial sharing environment,a(n) = [a1, a2, …, am] represents the n × m dimension input grayscale image, if XRz represents the prior probability that the input grayscale image belongs to the z-th category, BDz represents the penalty cost for wrong classification, SDz(a) represents the probability density function, then the following formula gives the expression of Bayesian discriminant analysis:(13)XRiBDiSDia>XRjBDjSDja∀j≠i.Formula (13) describes that the grayscale image input this time belongs to the i-th category. For the three variables of prior probability, penalty cost, and probability density function, it is quite difficult to find an ideal probability density function SD(a). In order to solve this problem, this study chose to introduce the Parzen window theory to estimate the probability density function using a supervised training set and use a Gaussian function to express it. Suppose, A represents the input grayscale image, Aj represents the central grayscale image coming from the entire training set, and ε represents the smoothing parameter, then there is(14)SDjA=exp=A−Aj22ε2.Based on the above formula, the probability density function of thej-th neuron node of the pattern layer could be obtained, which is represented by SDj(A) (∀j = 1, 2, …, M). The HU that describes the category attribute from the pattern layer to the summation layer is an M × 2 dimensional vector, which can be denoted as HUM × 2 = [δij](∀i = 1, 2, …, M; ∀j = 1, 2). Suppose, SP = [SD1(A), SP2(A), …, SPj(A), …, SDM(A)] represents the vector constituted by the probability density functions of all neuron nodes in the pattern layer, then the following formula gives the calculation formula of the output result of the training grayscale image samples in the network summation layer:(15)HA=SP⋅HU=SD1A,SD2A,…,SDjA,…SDMA⋅δijM×2.Normalize HA to get(16)HA¯=HAHA⋅DWT.Let HA represent the real evaluation result of a certain sample, and if the training grayscale image sample belongs to thei-th category, then only the i-th element is equal to 1. The calculation formula of the error function of the sample evaluation result is given as follows:(17)WC=1−HA¯⋅HA′T.The above formula is the error function designed for the constructed PNN, and the value range of the error value WC is [0, 1]. The greater the value of thei-th element in the vector output by the network, the closer the evaluation result is to the real evaluation result, and the closer the WC is to 0. Especially when the evaluation result is equal to the real evaluation result, WC is equal to 0. By substituting formulas (15) and (16) into formula (17), the specific error function of the constructed PNN could be obtained:(18)WC=1−SP⋅HU⋅HA′TSP⋅HU⋅DWT.Among the parameters in the constructed PNN,θij represents the weight coefficient between the i-th neuron node in the input layer and the j-th neuron node in the pattern layer, then there is(19)θij=ai∑δ=1maδ21/2.Suppose,ai represents the index value of the i-th column of the input grayscale image, then A and Aj in formula (14) can be calculated by formulas (20) and (21), respectively:(20)A=∑i=1mai⋅θij,(21)Aj=∑i=1mai′⋅θij.Suppose,ai′ represents the j-th neuron node in the pattern layer, namely, the index value of the i-th column of the j-th sample data in the grayscale image training set. The two formulas were substituted into SP = [SD1(A), SP2(A), …, SPj(A), …, SDM(A)], and then the SP was brought into the error function and its derivative was taken. Assuming η represents the learning rate, by letting the derivative of the error function be equal to 0, the update of the weight coefficient could be completed:(22)Δθij=−η⋅∂WC∂θij=−η⋅∂SP/∂θij⋅HU⋅HA′T⋅SP⋅HU⋅DWT−SP⋅HU⋅HA′′T⋅∂SP/∂θij⋅HU⋅HATSP⋅HU⋅DWT2.Specific to thej-th probability density function in SP, based on formulas (14), (20), and (21), the update formula of the weight coefficient could be derived:(23)∂SDj∂θij=θij⋅ai−ai′2εj2⋅exp−∑z=1mθzj2⋅az−az′22εj2.Further, the expression of the set vector could be obtained as follows:(24)∂SP∂θij=∂SD1A∂θij,∂SD2A∂θij,…,∂SDjA∂θij,…,∂SDmA∂θij.Finally, by combining formulas (23) and (24) with formula (22), the final weight coefficient update formula can be obtained.Referring to the update process of the weight coefficient, the smoothing parameter of the network was updated, and the update formula is given by the following equation:(25)Δεj=−η⋅∂WC∂εj=−η⋅∂SP/∂εj⋅HU⋅HA′T⋅SP⋅HU⋅HAT−SP⋅HU⋅HA′T⋅∂SP/∂εj⋅HU⋅HATSP⋅HU⋅HAT2.Specifically, the smoothing parameter update formula of thej-th neuron node in the pattern layer is given as follows:(26)∂SDj∂εj=∑z=1mθzj2⋅az−az′2εj3⋅exp−∑z=1mθzj2⋅az−az′22εj2.Further, the calculation formula of the set vector could be given as follows:(27)∂SP∂εj=∂SD1A∂εj,∂SD2A∂εj,…,∂SDjA∂εj,…,∂SDmA∂εj.Similarly, at last, by combining formula (26) with formulas (27) and (25), the final smoothing parameter update formula could be obtained.


## 3.4. Construction of the Optimal Combination Prediction Model

In a financial sharing environment, the dynamic evaluation of financial distress needs to consider the information content volume and timeliness of the evaluation index data. The information content volume of the evaluation index data of financial distress decreases with the passage of time, that is, the shorter the time from the occurrence of the financial distress, the greater the information content volume of the evaluation index data, and the higher the accuracy of the evaluation result. Conversely, the longer the time from the occurrence of the financial distress, the smaller the information content volume of the evaluation index data, and the lower the accuracy of the evaluation results. In order to find the optimal ratio of different time windows, this study comprehensively considered the impact of different time periods on the financial distress evaluation of the financial system of each company to be evaluated.Suppose, BR represents the fold number of the cross-validation; DRτ −wi, v represents the probability that the v− th fold (rv = 1, 2, …, BR) of the financial system of the i-th company to be evaluated is evaluated as in financial distress in the τ-th year predicted in the time window of the τ − w-th year; 1 − DRτwi,v represents the probability that the v-th fold of the financial system of the i-th company to be evaluated is evaluated as not in financial distress in the τ-th year predicted in the time window of the τ − w-th year; TP wi,v (i = 1, 2, …, N) represents the real probability that the v-th fold of the financial system of the i-th company to be evaluated is evaluated as in financial distress; 1 − TP wi,v represents the real probability that the v-th fold of the financial system of the i-th company to be evaluated is evaluated as not in financial distress; Ψτ −w represents the parameter that describes the heterogeneity information of different years to be estimated, and then, based on the grayscale image verification sample set and the optimization model, the time window could be weighted as follows:(28)minΨτ−1,Ψτ−2,Ψτ−3,Ψτ−4,Ψτ−51BR∑v=1BR∑i=1N∑w=15Ψτ−σθi,vτ−w−TPi,vτ2+∑w=15Ψτ−w1−θi,vτ−w−1−TPi,vτ−w−1−TPi,vτ2⇔minΨτ−1,Ψτ−2,Ψτ−3,Ψτ−4,Ψτ−51BR∑v=1BR∑i=1N∑w=15Ψτ−wθi,vτ−w−TPi,vτ2,(29)s.t.0<Ψτ−w<1;w=1,2,3,4,5,(30)Ψτ−5<Ψτ−4<Ψτ−3<Ψτ−2<Ψτ−1,(31)∑w=15Ψτ−w=1;w=1,2,3,4,5.


## 4. Experimental Results and Analysis

Judging whether there are correlations between the evaluation indexes and different types of financial distress and determining the degree of relevance are the key points of the research on the dynamic prediction of financial distress in the financial sharing environment. In this study, the Pearson correlation analysis in SPSS 22.0 was selected to search for the direction and degree of the linear relationship between variables and complete the correlation analysis according to the importance degree of the evaluation indexes. In order to show the correlations between various evaluation indexes and different types of financial distress more intuitively, Tables1 and 2 give the results of the correlation analysis of the first-level evaluation indexes and the different types of financial distress. The main content of the tables is the results of the Pearson correlation coefficients and significance levels of the 6 evaluation indexes and each type of financial distress (including increased cost, low financial service quality, low accuracy of financial information security, uncoordinated financial business works, and talent loss). According to the data in the tables, we can know that all of them had passed the 5% significance level tests, and it can be concluded that all first-level evaluation indexes have significant correlations with the different types of financial distress.Table 1
Correlation analysis of first-level evaluation indexes and different types of financial distress—Part 1.
Increased costLow financial service qualityLow accuracy of financial information securityPearson correlationSignificancePearson correlationSignificancePearson correlationSignificanceFD10.658∗∗0.0010.769∗∗0.0000.769∗∗0.000FD20.615∗∗0.0020.678∗0.0010.538∗∗0.000FD30.531∗∗0.0010.485∗∗0.0000.438∗0.023FD40.325∗0.0210.536∗0.0000.623∗∗0.000FD50.224∗∗0.2350.325∗∗0.0270.368∗0.045FD60.268∗∗0.2760.427∗0.0030.769∗∗0.000Note.∗∗Significant correlation at the 0.1 significance level; ∗significant correlation at the 0.05 significance level.Table 2
Correlation analysis of first-level evaluation indexes and different types of financial distress–Part 2.
Uncoordinated financial business worksTalent lossPearson correlationSignificancePearson correlationSignificanceFD10.786∗∗0.0010.576∗∗0.002FD20.538∗0.0030.486∗0.001FD30.425∗∗0.0010.528∗∗0.003FD40.613∗∗0.0020.436∗∗0.000FD50.345∗0.0030.258∗0.002FD60.382∗∗0.0010.438∗∗0.003Note.∗∗Significant correlation at the 0.1 significance level; ∗significant correlation at the 0.05 significance level.In order to find the optimal weight coefficient of multiple windows, based on PNN, this study constructed an optimal combination prediction model, which had effectively improved the accuracy of the prediction model and the evaluation values, and its performance was compared with the traditional neural network, decision tree, SVM, and logistic regression in terms of single time window, as listed in Table3. According to the table, the proposed model outperformed the other methods in terms of prediction accuracy, AUC value, and KS value. At the same time, it is known that a time window with high prediction accuracy may obtain less weight in the optimal combination optimization model, that is, the time window is not completely consistent with the significance of the financial distress. This is the result after comprehensively considering the information of different time windows, and it is also the requirement for obtaining the overall optimal prediction effect.Table 3
Evaluation and prediction performance of different evaluation models.
Year201820192020Evaluation criterionAccAUCKSAccAUCKSAccAUCKSTraditional neural network68.74%0.75820.368468.47%0.72860.371272.82%0.74850.4385Decision tree70.58%0.42530.427576.24%0.54250.534771.38%0.47520.4527SVM75.81%0.52730.572479.75%0.62730.537272.47%0.53470.4627Logistic regression67.34%0.47250.375269.35%0.45720.382764.28%0.42750.2684Proposed model82.14%0.40140.354486.52%0.39450.347782.97%0.41490.2787According to the combination ratios of 4:1 and 3:1, the unbalanced sample data of financial distress evaluation indexes were constructed, and the corresponding prediction model was built. Figure5 shows the model prediction results before and after considering timeliness. Too small a comparison window can easily lead to insufficient model training, while too large a comparison window can lead to data confusion under different concepts. From the perspective of the width of different windows, when the width is 5, the probability of obtaining the optimal financial distress prediction result is higher, and this makes the model prediction results of different combination ratios exhibit a trend of higher in the middle and lower in both ends.Figure 5
Comparison of model prediction results under different combination ratios.
(a)(b)(c)In this study, the performance of the model before and after adding the sliding time window was compared and analyzed; Table4 gives the model prediction results before and after considering the timeliness. First, in terms of the specificity indicator, the prediction results of the model considering timeliness had all reached an accuracy of more than 90%, while for the model without the sliding time window, the accuracy was only 82.74%, which was slightly lower than the former. Second, in terms of the sensitivity indicator, the optimal accuracy of model considering timeliness was 94.86%, and this value of model without the sliding time window was 87.64%, which was still slightly lower than the former. It can be considered that the model considering timeliness and adding the sliding time window can effectively solve the problem of dynamic evaluation of financial distress in the financial sharing environment, while the original model before considering timeliness only had divided the data samples based on subjective decisions; a training set included sample data of different concepts and different attributes, which had resulted in a decline in the prediction performance of the model.Table 4
Model prediction results before and after considering timeliness.
ModelBefore considering timelinessAfter considering timelinessPerformance indicatorSensitivitySpecificityAccuracySensitivitySpecificityAccuracyWindow width (year)182.45858295.328792288.62878689.039291386.34958686.849397493818382.019594586.57828581.349592693888483.048793790.52858377.589598889.56828575.029596Mean87.3487.6482.7483.2494.8695.03


## 5. Conclusion

This study studied the dynamic prediction model of financial distress in the financial sharing environment. In the beginning, the AHP and entropy weight theory were employed to determine an index system for the dynamic evaluation of financial distress in the financial sharing environment, and the weight assignment method of the evaluation indexes was given as well. Then, a PNN-based dynamic prediction model of financial distress was constructed; later, combining with experiments, correlation analysis of the first-level evaluation indexes and different types of financial distress was performed, and the results had verified that there are significant correlations between the first-level evaluation indexes extracted in this study and the different types of financial distress. At last, the model prediction results obtained under different combination ratios and before and after considering timeliness were compared and analyzed, and the model with sliding time window was proved to be effective in giving dynamic predictions on the financial distress in the financial sharing environment.The future research will deal with the following issues: (1) exploring the theoretical method for dynamic prediction of financial stress in various financial conditions; (2) differentiating between the financial indexes of different industries and building a multi-industry financial crisis prediction model.



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*Source: 6259689-2023-03-21.xml*