Author Affiliations
1School of Finance and Economics, Xizang Minzu University, Xianyang, 712082, China2Research Center for Quality Development of Xizang Special Industries, Xianyang, 712082, China3School of Computer Science and Technology, Xinjiang University, Urumqi, 830017, Chinashow less
Fig. 1. Comparison of the hinge loss with the huberized hinge loss.
Fig. 2. Flowchart of the algorithm.
| Algorithm | | 1. Initialize $ {\tilde \beta _0} $ and $ \tilde {\Cambriabfont\text{β}} $. | | 2. Iterate 1) and 2) until convergence: | | 1) Cyclic coordinate descent: j = 1, 2, ···, p. | | i) Calculate $ {r_i} = {y_i}({\tilde \beta _0} + {\mathbf{x}}_i^{\text{T}}\tilde {\Cambriabfont\text{β}}) $; | | ii) Calculate $ \hat \beta _j^{{\text{new}}} = \left( {{\delta \mathord{\left/ {\vphantom {\delta 2}} \right. } 2}} \right)S\left( {z{\rm{,}}{\text{ }}{P_{\lambda _1}'}\left( {\left| {{{\tilde \beta }_j}} \right|} \right)} \right) $; | | iii) Let $ {\tilde \beta _j} = \hat \beta _j^{{\text{new}}}{\rm{,}}{\text{ }}j = j + 1 $. | | 2) Update the intercept term: | | i) Calculate $ {r_i} = {y_i}\left( {{{\tilde \beta }_0} + {\mathbf{x}}_i^{\text{T}}\tilde {\Cambriabfont\text{β}}} \right) $; | | ii) Calculate $ \hat \beta _0^{{\text{new}}} = {\tilde \beta _0} - \left( {{\delta \mathord{\left/ {\vphantom {\delta {2n}}} \right. } {2n}}} \right)\sum\limits_{i = 1}^n {{l_\delta' }\left( {{r_i}} \right){y_i}} $; | | iii) Let $ {\tilde \beta _j} = \hat \beta _j^{{\text{new}}}{\rm{,}}{\text{ }}j = j + 1 $. | | 3. If $\mathop {{\rm{max}} }\limits_{0 \leq j \leq p} \left| {\hat \beta _j^{{\text{new}}} - {{\tilde \beta }_j}} \right| < \varepsilon $, stop the iteration; otherwise, return to step 2. |
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Table 1. Desciption of the resulted algorithm for solving the model (1).
| Indicators | Methods | Min | 1st Qu | Median | Mean | 3rd Qu | Max | | ACC | SVM | 0.7016 | 0.7435 | 0.8010 | 0.8000 | 0.8534 | 0.8952 | | LASSO-SVM | 0.7612 | 0.8114 | 0.8680 | 0.8653 | 0.9025 | 0.9589 | | MCP-HSVM | 0.7773 | 0.8260 | 0.8721 | 0.8749 | 0.9180 | 0.9708 | | SCAD-HSVM | 0.7842 | 0.8396 | 0.8862 | 0.8884 | 0.9372 | 0.9804 | | AUC | SVM | 0.7840 | 0.8170 | 0.8627 | 0.8608 | 0.9021 | 0.9335 | | LASSO-SVM | 0.8436 | 0.8916 | 0.9378 | 0.9290 | 0.9657 | 0.9902 | | MCP-HSVM | 0.8489 | 0.8801 | 0.9239 | 0.9239 | 0.9625 | 0.9969 | | SCAD-HSVM | 0.8502 | 0.8867 | 0.9328 | 0.9256 | 0.9626 | 0.9974 | | TPR | SVM | 0.6691 | 0.7075 | 0.7546 | 0.7602 | 0.8055 | 0.8645 | | LASSO-SVM | 0.7553 | 0.7988 | 0.8370 | 0.8463 | 0.8809 | 0.9500 | | MCP-HSVM | 0.7602 | 0.8068 | 0.8434 | 0.8512 | 0.8978 | 0.9581 | | SCAD-HSVM | 0.7595 | 0.7925 | 0.8525 | 0.8542 | 0.9102 | 0.9555 | | FPR | SVM | 0.2101 | 0.2342 | 0.2535 | 0.2571 | 0.2807 | 0.3097 | | LASSO-SVM | 0.1470 | 0.1710 | 0.1974 | 0.1960 | 0.2195 | 0.2456 | | MCP-HSVM | 0.1643 | 0.1868 | 0.2100 | 0.2121 | 0.2381 | 0.2634 | | SCAD-HSVM | 0.1683 | 0.1982 | 0.2239 | 0.2209 | 0.2457 | 0.2670 |
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Table 2. Results of evaluation indicators for the classification prediction of each model.
| Indicators | Methods | Min | 1st Qu | Median | Mean | 3rd Qu | Max | | NV | LASSO-SVM | 5.0000 | 10.0000 | 17.0000 | 16.6400 | 24.2500 | 38.0000 | | MCP-HSVM | 6.0000 | 9.0000 | 15.0000 | 19.6000 | 25.7500 | 47.0000 | | SCAD-HSVM | 8.0000 | 9.0000 | 10.0000 | 10.2600 | 11.0000 | 13.0000 | | FNR | LASSO-SVM | 0.0176 | 0.1029 | 0.2294 | 0.2512 | 0.3247 | 0.5112 | | MCP-HSVM | 0.0765 | 0.1252 | 0.2000 | 0.2182 | 0.3059 | 0.4824 | | SCAD-HSVM | 0.1029 | 0.2294 | 0.2512 | 0.2824 | 0.3471 | 0.5647 | | FDR | LASSO-SVM | 0.0000 | 0.0764 | 0.2248 | 0.2679 | 0.3564 | 0.4578 | | MCP-HSVM | 0.0000 | 0.1193 | 0.2556 | 0.2880 | 0.3810 | 0.4610 | | SCAD-HSVM | 0.0000 | 0.1000 | 0.1577 | 0.1735 | 0.2662 | 0.3500 |
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Table 3. Results of evaluation indicators for variable selection of LASSO-SVM, MCP-HSVM, and SCAD-HSVM.
| ρ | Indicators | SVM | LASSO-SVM | MCP-HSVM | SCAD-HSVM | | ρ = 0.3 | ACC | 0.7758 | 0.8339 | 0.8436 | 0.8508 | | AUC | 0.8351 | 0.8802 | 0.9050 | 0.8956 | | TPR | 0.7290 | 0.7996 | 0.8306 | 0.8260 | | FPR | 0.2827 | 0.2103 | 0.2206 | 0.2230 | | ρ = 0.5 | ACC | 0.7496 | 0.8188 | 0.8348 | 0.8310 | | AUC | 0.7828 | 0.8650 | 0.8861 | 0.8853 | | TPR | 0.6939 | 0.7647 | 0.8245 | 0.8130 | | FPR | 0.3238 | 0.2357 | 0.2394 | 0.2345 | | ρ = 0.8 | ACC | 0.6760 | 0.7786 | 0.8200 | 0.8196 | | AUC | 0.7209 | 0.8212 | 0.8701 | 0.8635 | | TPR | 0.6331 | 0.7211 | 0.8135 | 0.7992 | | FPR | 0.3301 | 0.2567 | 0.2412 | 0.2451 |
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Table 4. Results of evaluation indicators for the classification prediction of each model under different correlations.
| ρ | Indicators | LASSO-SVM | MCP-HSVM | SCAD-HSVM | | ρ = 0.3 | NV | 15.8200 | 18.4800 | 14.2600 | | FNR | 0.2424 | 0.1612 | 0.2118 | | FDR | 0.2360 | 0.2057 | 0.1885 | | ρ = 0.5 | NV | 12.9000 | 16.0800 | 13.7600 | | FNR | 0.3117 | 0.2294 | 0.2941 | | FDR | 0.3298 | 0.2325 | 0.2916 | | ρ = 0.8 | NV | 10.2400 | 15.7600 | 12.9800 | | FNR | 0.3729 | 0.2800 | 0.3329 | | FDR | 0.1655 | 0.2397 | 0.3244 |
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Table 5. Means of evaluation indicators for variable selection of LASSO-SVM, MCP-HSVM, and SCAD-HSVM.
| Level 1 indicators | Level 2 indicators | | Solvency | Current ratio: X1Quick ratio: X2Equity ratio: X3Total equity/liabilities attributable to shareholders of the parent company: X4Earnings before interest, taxes, depreciation, and amortization/total liabilities: X5Net cash flow from operating activities/total liabilities: X6Interest coverage ratio (earnings before interest and taxes/interest expense): X7 | | Indicators per share | Earnings per share (RMB): X8Operating cash flow per share (RMB): X9Net assets per share (net of minority interests) (RMB): X10 | | Revenue quality | Net income from operating activities/total profit (%): X11Net gain from changes in value/total profit (%): X12Net non-operating income and expenses/total profit (%): X13Net income after deducting non-recurring gain and losses/net income (%): X14 | | Profitability | Gross profit margin on sales (%): X15Net sales margin (%): X16Net return on assets (%): X17Net margin on total assets (%): X18 | | Operating capacity | Inventory turnover ratio (times): X19Accounts receivable turnover ratio (times): X20Current assets turnover ratio (times): X21Fixed assets turnover ratio (times): X22Total assets turnover ratio (times): X23 | | Capital structure | Asset-liability ratio (%): X24Equity multiplier: X25Current assets/total assets (%): X26Non-current assets/total assets (%): X27Current liabilities/total liabilities (%): X28Non-current liabilities/total liabilities (%): X29 |
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Table 6. Financial indicators used in this paper.
| Methods | ACC | AUC | TPR | NV | | SVM | 0.8776 | 0.7676 | 0.7057 | / | | Lasso-SVM | 0.8632 | 0.7967 | 0.7301 | 17.8 | | MCP-HSVM | 0.8652 | 0.8187 | 0.7682 | 23.6 | | SCAD-HSVM | 0.8754 | 0.8385 | 0.7494 | 22.1 |
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Table 7. Prediction results of each method (average of 100 predictions).
| Variables | LASSO-SVM | MCP-HSVM | SCAD-HSVM | Variables | LASSO-SVM | MCP-HSVM | SCAD-HSVM | | X1 | 0 | 15 | 11 | X16 | 100 | 100 | 100 | | X2 | 100 | 100 | 100 | X17 | 0 | 93 | 100 | | X3 | 40 | 50 | 45 | X18 | 100 | 100 | 100 | | X4 | 0 | 8 | 0 | X19 | 0 | 0 | 55 | | X5 | 100 | 100 | 100 | X20 | 100 | 100 | 100 | | X6 | 50 | 100 | 58 | X21 | 100 | 100 | 100 | | X7 | 0 | 0 | 5 | X22 | 100 | 100 | 100 | | X8 | 100 | 100 | 100 | X23 | 95 | 100 | 55 | | X9 | 100 | 100 | 100 | X24 | 91 | 95 | 100 | | X10 | 100 | 100 | 100 | X25 | 0 | 40 | 0 | | X11 | 92 | 95 | 100 | X26 | 100 | 100 | 100 | | X12 | 50 | 95 | 100 | X27 | 100 | 100 | 100 | | X13 | 0 | 0 | 3 | X28 | 85 | 95 | 100 | | X14 | 100 | 100 | 100 | X29 | 0 | 55 | 76 | | X15 | 0 | 51 | 0 | | | | |
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Table 8. Selection frequency of each financial indicator.
| Selection frequency | Level 2 indicators | | 100 | X2 (quick ratio), X5 (earnings before interest, taxes, depreciation, and amortization/total liabilities), X6 (net cash flow from operating activities/total liabilities), X8 (earnings per share), X9 (operating cash flow per share), X10 (net assets per share), X14 (net income after deducting non-recurring gain and losses/net income), X16 (net sales margin), X18 (net margin on total assets), X20 (accounts receivable turnover ratio), X21 (current assets turnover ratio), X22 (fixed assets turnover ratio), X23 (total assets turnover ratio), X26 (current assets/total assets), and X27 (non-current assets/total assets) | | 90−100 | X11 (net income from operating activities/total profit), X12 (net gain from changes in value/total profit), X17 (return on net assets), X24 (asset-liability ratio), and X28 (current liabilities/total liabilities) | | 60−90 | None | | 40−60 | X3 (equity ratio), X15 (gross profit margin on sales), X25 (equity multiplier), and X29 (non-current liabilities/total liabilities) | | 20−40 | None | | 1−20 | X1 (current ratio) and X4 (total equity/liabilities attributable to shareholders of the parent company) | | 0 | X7 (interest coverage ratio (earnings before interest and taxes/interest expense)), X13 (net non-operating income and expenses/total profit), and X19 (inventory turnover ratio) |
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Table 9. Classification of variables by the selection frequency of MCP-HSVM.
