This article first uses a new method of nonlinear combination forecasting based on neural networks to construct a financial crisis early warning model and conduct an empirical study. The drafting article uses Fisher's second-class linear discriminant analysis and binary logistic regression to establish a three-year early warning model for listed companies before the financial crisis. Empirical research shows that this early warning model applies to various industries. It can play a certain role in predicting and preventing the financial crisis of Chinese companies.

#### Keywords

- Fisher linear discriminant
- Listed company
- Financial crisis
- Early warning

#### MSC 2010

- 62H86

The multivariate model uses multiple financial indicators as the sample variables for the study. Combine multiple variables (indicators) to build a multivariate predictive analysis model. The statistical methods used in financial crisis early warning mainly include: multiple linear discrimination, multiple logistic regression, multiple probability ratio regression methods, etc. The multiple linear discriminant model (MDA) is mostly used from the research literature at home and abroad. Because its model is relatively simple, it is widely used, and the multivariate models established from that place mainly include Fisher's linear judgment model.

The actual situation and foreign research show different industries make the financial indicators and parameters included in the early warning financial crisis model different, so different models should be used for research and analysis [1]. Therefore, the empirical analysis of the multivariate model is carried out for the listed companies in the home appliance industry in China using multivariate statistical analysis methods. This can improve the pertinence, practicability, and operability of the model. First, according to the 28 financial indicators we selected and using cluster analysis, we will make a scientific, statistical classification of listed companies in the home appliance industry in China. Finally, the principal component analysis method is used to extract the principal components to calculate each component's scores and total scores. Sort the samples according to the total scores of the principal components to discover their financial crisis.

The article takes 30 home appliance companies in China's listed companies as the research objects. At the same time, we use its financial data in 2021, 2020, and 2019 as the basis for analysis. This financial information all comes from its periodic public reports [2]. According to the characteristics of listed companies in China, we select net sales margin, main business profit margin, return on net assets, net asset interest rate, earnings per share (EPS), accounts receivable turnover rate, inventory turnover rate, total asset turnover rate, Working capital total assets ratio, debt-to-asset ratio, current ratio, quick ratio, equity ratio (debt-to-equity ratio), multiple of interest earned, the growth rate of main business income, the growth rate of net profit, the growth rate of net assets (increased equity Ratio), total asset growth rate, capital preservation and appreciation ratio, total cash-to-debt ratio (debt protection ratio), cash flow-to-debt ratio, cash inflow and outflow ratio, cash-to-sales ratio, profit quality index (operating index), net operating cash per share Flow, the cash recovery rate of all assets, retained earnings total assets ratio, net assets per share to establish an index system as analysis variables. These 28 indicators comprehensively reflect the company's profitability, solvency, asset management ability, growth ability, ability to obtain cash flow, and capital strength. The above indicators can more comprehensively reflect the enterprise's financial status in terms of short-term and long-term factors [3].

It can be seen from the cluster analysis that this article divides the research samples into three categories, so it uses a multi-category linear decision model.

The Fisher linear discriminant function is most commonly used in discriminant analysis. Its general form is: _{i}_{1}_{1} + _{2} _{2} +…_{n} x_{n}_{1}, _{2},…, _{n}_{1}, _{2},…, _{n}j

Suppose there is a set _{1},…, _{n}_{1} samples belong to _{1} class and denoted as
_{2} samples belonging to a class _{2} are denoted as
^{*}.

_{i}

Therefore, maximizing the essence

For most of the actual data, the method of linear discriminant analysis is too simple. Nuclear Fisher discriminant analysis uses “nuclear techniques” similar to SVM and nuclear PCA methods. First, the data is non-linearly mapped to a certain feature space F. Then perform Fisher linear discrimination in this feature space [5]. This implicitly realizes the nonlinear discrimination of the original input space. To find the linear discriminant in F, you need to maximize

Here

In

To first need to obtain the expression (3) only in the dot product form of the input samples. Then use a kernel function to replace the dot product operation. Any

Using the expansions (7) and (6) and replacing the dot product with the kernel function, we get

Define

It is necessary to classify the research object first [6]. We use the cluster analysis method in multivariate statistical analysis to scientifically divide listed companies in the home appliance industry into three categories according to their financial status.

Since there are as many as 28 sample variables for establishing the discriminant model in this paper, it is obvious that the full model method cannot be used to incorporate all of them into the discriminant function. Therefore, we need to use stepwise analysis to eliminate variables that are not significant. This will also simplify the model and improve the efficiency of discrimination. Specific discriminant analysis methods: 1) Use the system default Wilks’lambda. Each step is the entry discriminant function with the smallest λ statistic of Wilk. 2) The criterion of stepwise discrimination stops adopting the F value. When a variable is added, the variance analysis of the variables in the discriminant function is performed. 3) The unstandardized Fisher coefficient is used to discriminate and classify new samples directly. 4) Use the correlation matrix within the group when selecting the required independent variable coefficient matrix. Before calculating the correlation matrix, calculate the intra-class correlation matrix after averaging the covariance matrix of each group (class). 5) The prior probabilities in the classification parameters are equal to all prior probabilities. 6) The covariance matrix used for classification uses within-groups

We use statistical test quantity to perform a statistical test-significance test on the model to judge whether the discriminant function can separate the three categories well.

1) Tests for the equality of the means of variables in different classes. When the value of λ is 1, the mean of each group is equal [7]. The lambda values of the 6 variables in the table are all less than 1, indicating that the mean values of each group are not equal. The significance level of the six variables except for the account receivable turnover rate is less than 0.05. Therefore, the null hypothesis that the mean of each group is equal can be rejected. At the 0.05 level of significance, the mean values of 5 of the 6 variables are significantly different, and we can perform discriminant analysis.

Variable Mean Test.

Financial indicator | Wilks ‘λ value | F | Degree of freedom 1 | Degrees of freedom 2 | Significance level |
---|---|---|---|---|---|

Roe | 0.471 | 10.152 | 2 | 27 | 0.0005 |

Net asset interest rate | 0.522 | 8.221 | 2 | 27 | 0.0016 |

Accounts Receivable Turnover Rate | 0.887 | 0.179 | 2 | 27 | 0.2372 |

Turnover rate of total assets | 0.684 | 3.722 | 2 | 27 | 0.0374 |

Main business income growth rate | 0.418 | 12.56 | 2 | 27 | 0.0001 |

Net operating cash flow per share | 0.509 | 8.677 | 2 | 27 | 0.0012 |

2) Test for equality of variance-covariance matrix. The test statistic is Box'sM test results shown in Table 2. The null hypothesis that the overall covariance matrices are equal is rejected at a significance level of 0.000. The variance and covariance of each category are not equal.

Box'sM test table for variance.

Box′sM | 211.218 | |

F | Approx. (Asymptotically) | 7.417 |

df1 (degree of freedom 1) | 21 | |

df2 (degrees of freedom 2) | 2062.047 | |

Sig. (Significance level) | 0 |

3) Goodness of fit test. We use Wilks’λ statistic to test whether the mean values of the discriminant functions of each group are equal. A test of the validity of the function. Table 3 is a test of the two canonical discriminant functions constructed. The null hypothesis that the mean values of the discriminant functions of each group are equal is rejected at the significance level of 0.000. We believe that the discriminant function can distinguish the three categories well. Both canonical discriminant functions are statistically significant. It can be considered that the established discriminant function is statistically effective through the above statistical test established on the model. On this basis, the model results of the discriminant analysis can be written specifically. Otherwise, the established model is invalid.

The goodness of fit test.

Test function | Wilks’ λ value | Chi-square | Degree of freedom | Significance level |
---|---|---|---|---|

1 to 2 | 0.002 | 150.019 | 12 | 0 |

2 | 0.36 | 25.003 | 5 | 0 |

The canonical decision function can be constructed by calculating the coefficient value of the canonical decision function. Replacing the original multiple variables with a small number of canonical variables can conveniently describe the relationship between various types [8]. The original variables can directly discriminate the Fisher discriminant function constructed by the Bayes criterion. Table 4 is the result of stepwise discriminant analysis of the obtained Fisher linear discriminant function. The Fisher linear discriminant function model containing 6 variables is obtained by removing the less important indicators from the 28 indicators (variables). The expressions of these three functions are as follows:

_{1} = 2.273_{1} + 3.735_{2} − 0.174_{3} + 7.026_{4} − 0.276_{5} + 2.374_{6} − 4.350

_{2} = 12.182_{1} − 45.9_{2} + 0.010 29_{3} + 5.25_{4} − 0.878_{5} − 2.088_{6} − 2.897

_{3} = − 774.553_{1} +1722.38_{2} + 4.485_{3} − 146.593_{4} + 67.021_{5} − 50.065_{6} − 1161.22

Fisher's linear discriminant function coefficient table.

Financial indicator | Series | ||
---|---|---|---|

Non-financial crisis | Intermediate state | Financial Crisis | |

Roe | 2.273 | 12.182 | −774.553 |

Net asset interest rate | 3.735 | −45.9 | 1722.38 |

Accounts Receivable Turnover Rate | −0.174 | 0.01 | 4.485 |

The turnover rate of total assets | 7.026 | 5.25 | −146.593 |

Main business income growth rate | −0.276 | −0.878 | 67.021 |

Net operating cash flow per share | 2.374 | −2.088 | 50.065 |

(constant) | −4.35 | −2.897 | −1161.22 |

Among them _{1}, _{2}, _{3} represents the function discriminant value of the non-financial crisis group, the intermediate state group, and the financial crisis group. _{1}, _{2}, _{3}, _{4}, _{5}, _{6} respectively represents the return on net assets, net asset interest rate, accounts receivable turnover rate, total assets turnover rate, main business income growth rate, and net operating cash flow per share.

The main indicators that affect the financial status of China's home appliance industry are the return on net assets and the net interest rate that reflect profitability. The account receivable turnover rate and total asset turnover rate reflect the asset management capability. The main business income growth rate reflects the growth ability. The net operating cash flow per share reflects the ability to obtain cash. Among them, two indicators reflect profitability, which accounts for one-third of the model variables. This shows that the profitability of listed companies in China's home appliance industry is particularly important. From the financial sensitivity analysis, we know that the most sensitive factor to corporate profits is the price [9]. The unprecedented fierce price wars of China's home appliance enterprises will inevitably seriously affect the enterprises’ profitability and then affect the cash capacity, debt solvency, and asset turnover ability of the enterprises. The two indicators that reflect the asset management capability in the model reflect the liquidity of corporate assets and the liquidity of corporate assets. The poor liquidity and liquidity of assets indicate that the company’s sales or cash recovery capabilities are not strong, and the efficiency of asset utilization is not high. The main business income growth rate is also a very important indicator. If an enterprise wants to seize market share to survive and develop, it must do everything possible to increase sales revenue. But on the other hand, blindly seeking to increase sales revenue, the vicious price war between companies in the industry at all costs will inevitably seriously affect the company's profitability.

After the discriminant model is established, the discriminant rules must be determined. This article is a Fisher discriminant function constructed by the Bayes criterion. The category with the highest score is the corresponding category of the enterprise. For example, the three function values of Sichuan Changhong in 2020 are _{1} =1.91,_{2} = − 1.91,_{3} = − 1172.19 respectively. The maximum value is _{1}, then Sichuan Changhong belongs to a non-financial crisis company. The verification methods of the discriminant function effect mainly include self-verification, external data verification, sample dichotomy, interactive verification, Bootstrap method, etc.

Self-verification is to substitute the samples used by the constructor into the discriminant function in turn. In this way, the effect of the model can be judged. The results of self-verification are shown in Table 5.

Self-verification result table.

category | Non-financial crisis | Intermediate state | Financial Crisis | |||
---|---|---|---|---|---|---|

Original value | Count | Non-financial crisis | 12 | 0 | 0 | 12 |

Intermediate state | 1 | 15 | 0 | 16 | ||

Financial Crisis | 0 | 0 | 2 | 2 | ||

Verification percentage | Non-financial crisis | 100 | 0 | 0 | 100 | |

Intermediate state | 6.3 | 93.8 | 0 | 100 | ||

Financial Crisis | 0 | 0 | 100 | 100 | ||

The overall classification accuracy rate is 96.7% |

It can be seen from the table that the result of discriminant analysis is that the classification of 12 non-financial crisis companies is all correct. The classification accuracy rate is 100%. Only one of the 16 intermediate state companies was mistakenly classified as a non-financial crisis company. The classification accuracy rate is 93.75%. The classification accuracy of the two financial crisis companies was 100%. The overall classification accuracy rate was 96.7%. This shows that the discriminative ability of the model is better.

Interactive verification is an important discriminative effect verification technology that has gradually developed in recent years [10]. The specific method is to remove one case from each category when establishing the discriminant function. Then use the established discriminant function to discriminate the case. This method can effectively avoid the interference of strong influence points.

Interactive verification result table.

Cross-authentication | Category | Prediction group | Overall | |||

Non-financial crisis | Intermediate state | Financial Crisis | ||||

Count | Non-financial crisis | 11 | 1 | 0 | 12 | |

Intermediate state | 1 | 15 | 0 | 16 | ||

Financial Crisis | 1 | 1 | 0 | 2 | ||

Verification percentage | Non-financial crisis | 91.7 | 8.3 | 0 | 100 | |

Intermediate state | 6.3 | 93.8 | 0 | 100 | ||

Financial Crisis | 50 | 50 | 0 | 100 | ||

The overall classification accuracy rate is 86.7% |

The prediction accuracy rates of the non-financial crisis group, intermediate state, and financial crisis groups were 91.7%, 93.8%, and 0%, respectively. The correct overall rate of judgment is 86.7%. The reason why all the predictions of the financial crisis group are not correct is that the sample size of this group is small (only two companies). This affects the prediction effect of the group and the whole.

After the discriminant function is established, a part of the sample data is collected again. We use the discriminant function to discriminate and test the discriminant effect. External data back-generation verification is due to the strong homogeneity between the 2020 and 2019 data and the 2021 data sample.

The experiment correctly predicted all 12 non-financial crisis companies. The correct rate is 100%. 15 of the 16 intermediate state companies were correctly predicted. Only one company was predicted to be a non-financial crisis company, with a correct rate of 93.8%. For 2 financial crisis companies, one was correctly predicted, and the other was predicted to be an intermediate company. The correct rate is 50%. For all 30 companies, 28 were correctly predicted, with a correct rate of 93.3%.

Judging from the three-year forecasting effect (Table 7), the forecasting accuracy of the model showed a downward trend in the three years. The closer it is to the model year, the higher the accuracy of the forecast. This is also in line with the objective law that the closer the forecast period, the better the forecast effect.

Summary of prediction model test results.

2021 | 2020 | 2019 | |
---|---|---|---|

Number of test samples | 30 | 30 | 28 |

Check the correct number | 29 | 28 | 20 |

Detection accuracy | 96.70% | 93.30% | 71.40% |

Misjudgment rate | 3.30% | 6.70% | 28.60% |

Even though the correct rate in the first year of the financial crisis was 96.7%, there was a misjudgment rate of 3.3%. And the farther the forecast year, the worse the accuracy. Only the size of a discriminant value cannot completely determine the financial status. The factors that affect the accuracy of discrimination include the size of the sample, the rationality of the original sample classification (grouping), the similarity (or degree of dispersion) of the sample indicators, the degree of change in financial status in different years, and so on.

The analysis method of this article can be used as a reference for listed companies in the home appliance industry in China and enterprises in other industries. This method can also play a certain role in listed companies’ credit rating, business performance evaluation, financial risk control, and other aspects.

#### Self-verification result table.

category | Non-financial crisis | Intermediate state | Financial Crisis | |||
---|---|---|---|---|---|---|

Original value | Count | Non-financial crisis | 12 | 0 | 0 | 12 |

Intermediate state | 1 | 15 | 0 | 16 | ||

Financial Crisis | 0 | 0 | 2 | 2 | ||

Verification percentage | Non-financial crisis | 100 | 0 | 0 | 100 | |

Intermediate state | 6.3 | 93.8 | 0 | 100 | ||

Financial Crisis | 0 | 0 | 100 | 100 | ||

The overall classification accuracy rate is 96.7% |

#### Box'sM test table for variance.

Box′sM | 211.218 | |

F | Approx. (Asymptotically) | 7.417 |

df1 (degree of freedom 1) | 21 | |

df2 (degrees of freedom 2) | 2062.047 | |

Sig. (Significance level) | 0 |

#### Summary of prediction model test results.

2021 | 2020 | 2019 | |
---|---|---|---|

Number of test samples | 30 | 30 | 28 |

Check the correct number | 29 | 28 | 20 |

Detection accuracy | 96.70% | 93.30% | 71.40% |

Misjudgment rate | 3.30% | 6.70% | 28.60% |

#### The goodness of fit test.

Test function | Wilks’ λ value | Chi-square | Degree of freedom | Significance level |
---|---|---|---|---|

1 to 2 | 0.002 | 150.019 | 12 | 0 |

2 | 0.36 | 25.003 | 5 | 0 |

#### Fisher's linear discriminant function coefficient table.

Financial indicator | Series | ||
---|---|---|---|

Non-financial crisis | Intermediate state | Financial Crisis | |

Roe | 2.273 | 12.182 | −774.553 |

Net asset interest rate | 3.735 | −45.9 | 1722.38 |

Accounts Receivable Turnover Rate | −0.174 | 0.01 | 4.485 |

The turnover rate of total assets | 7.026 | 5.25 | −146.593 |

Main business income growth rate | −0.276 | −0.878 | 67.021 |

Net operating cash flow per share | 2.374 | −2.088 | 50.065 |

(constant) | −4.35 | −2.897 | −1161.22 |

#### Interactive verification result table.

Cross-authentication | Category | Prediction group | Overall | |||

Non-financial crisis | Intermediate state | Financial Crisis | ||||

Count | Non-financial crisis | 11 | 1 | 0 | 12 | |

Intermediate state | 1 | 15 | 0 | 16 | ||

Financial Crisis | 1 | 1 | 0 | 2 | ||

Verification percentage | Non-financial crisis | 91.7 | 8.3 | 0 | 100 | |

Intermediate state | 6.3 | 93.8 | 0 | 100 | ||

Financial Crisis | 50 | 50 | 0 | 100 | ||

The overall classification accuracy rate is 86.7% |

#### Variable Mean Test.

Financial indicator | Wilks ‘λ value | F | Degree of freedom 1 | Degrees of freedom 2 | Significance level |
---|---|---|---|---|---|

Roe | 0.471 | 10.152 | 2 | 27 | 0.0005 |

Net asset interest rate | 0.522 | 8.221 | 2 | 27 | 0.0016 |

Accounts Receivable Turnover Rate | 0.887 | 0.179 | 2 | 27 | 0.2372 |

Turnover rate of total assets | 0.684 | 3.722 | 2 | 27 | 0.0374 |

Main business income growth rate | 0.418 | 12.56 | 2 | 27 | 0.0001 |

Net operating cash flow per share | 0.509 | 8.677 | 2 | 27 | 0.0012 |

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Tool Based on Mathematical Modeling Computer Vision Communication Technology in Mathematical Modeling Mathematical Statistics Technology in the Educational Grading System of Preschool Students Music Recommendation Index Evaluation Based on Logistic Distribution Fitting Transition Probability Function Children's Educational Curriculum Evaluation Management System in Mathematical Equation Model The Effect of Children’s Innovative Education Courses Based on Fractional Differential Equations Fractional Differential Equations in the Standard Construction Model of the Educational Application of the Internet of Things Optimization in Mathematics Modeling and Processing of New Type Silicate Glass Ceramics MCM of Student’s Physical Health Based on Mathematical Cone Sports health quantification method and system implementation based on multiple thermal physiology simulation Research on visual optimization design of machine–machine interface for mechanical industrial equipment based on nonlinear partial equations Research on identifying psychological health problems of college students by logistic regression model based on data mining Abnormal Behavior of Fractional Differential Equations in Processing Computer Big Data Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching A mathematical model of PCNN for image fusion with non-sampled contourlet transform Nonlinear Differential Equations in Computer-Aided Modeling of Big Data Technology The Uniqueness of Solutions of Fractional Differential Equations in University Mathematics Teaching Based on the Principle of Compression Mapping Influence of displacement ventilation on the distribution of pollutant concentrations in livestock housing Recognition of Electrical Control System of Flexible Manipulator Based on Transfer Function Estimation Method Automatic Knowledge Integration Method of English Translation Corpus Based on Kmeans Algorithm Real Estate Economic Development Based on 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Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression Optimisation of Modelling of Finite Element Differential Equations with Modern Art Design Theory Mathematical function data model analysis and synthesis system based on short-term human movement Human gait modelling and tracking based on motion functionalisation The Control Relationship Between the Enterprise's Electrical Equipment and Mechanical Equipment Based on Graph Theory Financial Accounting Measurement Model Based on Numerical Analysis of Rigid Normal Differential Equation and Rigid Functional Equation Mathematical Modeling and Forecasting of Economic Variables Based on Linear Regression Statistics Nonlinear Differential Equations in Cross-border E-commerce Controlling Return Rate Differential equation model of financial market stability based on Internet big data 3D Mathematical Modeling Technology in Visualized Aerobics Dance Rehearsal System Children’s cognitive function and mental health based on finite element nonlinear mathematical model Fractional Differential Equations in Electronic Information Models BIM Engineering Management Oriented to Curve Equation Model Mathematical Method to Construct the Linear Programming of Football Training The Size of Children's Strollers of Different Ages Based on Ergonomic Mathematics Design Stiffness Calculation of Gear Hydraulic System Based on the Modeling of Nonlinear Dynamics Differential Equations in the Progressive Method Relationship Between Enterprise Talent Management and Performance Based on the Structural Equation Model Method Professional English Translation Corpus Under the Binomial Theorem Coefficient Geometric Tolerance Control Method for Precision Machinery Based on Image Modeling and Novel Saturation Function Retrieval and Characteristic Analysis of Multimedia Tester Based on Bragg Equation Evaluation of ecosystem health in Futian mangrove wetland based on the PSR-AHP model Towards more efficient control of the ironmaking blast furnace: modelling gaseous reduction of iron ores in H _{2}-N_{2}atmosphereAlgorithm of overfitting avoidance in CNN based on maximum pooled and weight decay Mathematical Calculus Modeling in Improving the Teaching Performance of Shot Put Application of Nonlinear Differential Equation in Electric Automation Control System Higher Mathematics Teaching Curriculum Model Based on Lagrangian Mathematical Model Optimization of Color Matching Technology in Cultural Industry by Fractional Differential Equations The Marketing of Cross-border E-commerce Enterprises in Foreign Trade Based on the Statistics of Mathematical Probability Theory The Evolution Model of Regional Tourism Economic Development Difference Based on Spatial Variation Function The Inner Relationship between Students' Psychological Factors and Physical Exercise Based on Structural Equation Model (SEM) Fractional Differential Equations in Sports Training in Universities Higher Education Agglomeration Promoting 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