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- 1. Outline Motivation Dimensionality Reduction Proposed approach Experimental setup Conclusions and Future Work Bankrupcy Analysis for Credit Risk using Manifold LearningB Ribeiro, A Vieira, J Duarte, C Silva, J Carvalho das Neves, University of Coimbra, ISEP and ISEG, Portugal and Q Liu, A H Sung New Mexico Tech, USA November, 2008 ICONIP 2008
- 2. Outline Motivation Dimensionality Reduction Proposed approach Experimental setup Conclusions and Future Work1 Motivation2 Dimensionality Reduction Manifold Learning Isomap Supervised Isomap3 Proposed approach Overview Operation4 Experimental setup Data set Evaluation metrics Results5 Conclusions and Future Work ICONIP 2008
- 3. Outline Motivation Dimensionality Reduction Proposed approach Experimental setup Conclusions and Future WorkCredit Risk Analysis Predicting bankruptcy has been a very important topic in accounting and ﬁnance attracting considerable research both from academic and business areas The question of how to determine the credit-worthiness of a customer or how safe is to grant credit remains a main concern for banks and investors, particularly, with the recent ﬁnancial crisis ICONIP 2008
- 4. Outline Motivation Dimensionality Reduction Proposed approach Experimental setup Conclusions and Future WorkImportance of Risk (1) ICONIP 2008
- 5. Outline Motivation Dimensionality Reduction Proposed approach Experimental setup Conclusions and Future WorkImportance of Risk (2) ICONIP 2008
- 6. Outline Motivation Dimensionality Reduction Proposed approach Experimental setup Conclusions and Future WorkProblem deﬁnition The problem of bankruptcy prediction can be addressed as follows: Given a set of ﬁnancial ratios describing the situation of a company over a given period, predict the probability that this company may become bankrupted in a near future, normally during the following year ICONIP 2008
- 7. Outline Motivation Manifold Learning Dimensionality Reduction Isomap Proposed approach Supervised Isomap Experimental setup Conclusions and Future WorkObjectives of dimensionality reduction Nonlinear dimensionality reduction permits severe reduction on the feature space A direct consequence of nonlinear dimension reduction is the visualization of data which can help to reveal the data structures Aims at choosing from the available set of features, a smaller set that more eﬃciently represents the data Supervised or unsupervised Supervised methods use the label of the training examples in the reduction step and usually perform better ICONIP 2008
- 8. Outline Motivation Manifold Learning Dimensionality Reduction Isomap Proposed approach Supervised Isomap Experimental setup Conclusions and Future WorkIntroduction Emerging technique that estimates a low-dimensional structure, embedded in high-dimensional data The underpinning idea is to invert a generative model for a given set of observations Manifold learning can be used as a pre-processing technique to tackle the curse of dimensionality ICONIP 2008
- 9. Outline Motivation Manifold Learning Dimensionality Reduction Isomap Proposed approach Supervised Isomap Experimental setup Conclusions and Future WorkFormulation Given data points x1 , x2 , · · · , xn ∈ IRD , we assume that the data lies on a d-dimensional M manifold embedded into IRD , where d < D A manifold M can be described by a single coordinate chart f : M −→ IRd . The manifold learning consists of ﬁnding y1 , · · · yn ∈ IRd , where yi = f (xi ). ICONIP 2008
- 10. Outline Motivation Manifold Learning Dimensionality Reduction Isomap Proposed approach Supervised Isomap Experimental setup Conclusions and Future WorkIsomap Algorithm 1 Estimates which points are neighbors on the manifold M, based on the distances dX (i, j) between pairs of points i, j in the input space X by computing the weighted graph G of neighborhood relations given by the edges of weight dX (i, j). 2 Estimates the geodesic distances between all pairs of data points in the manifold M by computing the shortest path distance on the k’s nearest neighbor graph built on the data set. 3 Applies classical MDS to the matrix of graph distances DG = {dG (i, j)}, constructing an embedding of the data in a d-dimensional Euclidean space Y that best preserves the manifolds estimated intrinsic geometry ICONIP 2008
- 11. Outline Motivation Manifold Learning Dimensionality Reduction Isomap Proposed approach Supervised Isomap Experimental setup Conclusions and Future WorkAnalysis Isomap assumes that there is an isometric chart that preserves distances between points. If xi and xj are two points in the manifold M embedded into IRD and the geodesic distance between them is dG (xi , xj ) , then there is a chart f : M −→ IRd such that ||f (xi ) − f (xj )|| = dG (xi , xj ) For nearby points in the high-dimensional space the Euclidean distance is a good approximation of the geodesic distance whereas for distant points this is not true ICONIP 2008
- 12. Outline Motivation Manifold Learning Dimensionality Reduction Isomap Proposed approach Supervised Isomap Experimental setup Conclusions and Future WorkImage Processing Example [J. Tenenbaum, de Silva, & Langford, 2000] ICONIP 2008
- 13. Outline Motivation Manifold Learning Dimensionality Reduction Isomap Proposed approach Supervised Isomap Experimental setup Conclusions and Future WorkAnalysis A weighted graph with k’s nearest neighbors is built where its edges are weighted by the Euclidean distances between nearby data points Then a shortest path computation algorithm such as, Dijkstra’s or Floyd’s, will complete the calculus of the remainder geodesic distances. MDS is then used to estimate the points whose Euclidean distance equal the geodesic distances. Given a matrix D ∈ IRn×n of dissimilarities, MDS constructs a set of points whose interpoint Euclidean distances match those in D closely. ICONIP 2008
- 14. Outline Motivation Manifold Learning Dimensionality Reduction Isomap Proposed approach Supervised Isomap Experimental setup Conclusions and Future WorkSupervised version The training labels are used to reﬁne the distances between inputs, since both classiﬁcation and visualization can beneﬁt when the inter-class dissimilarity is larger than the intra-class dissimilarity The mapping function given by Isomap is only implicitly deﬁned and nonlinear interpolation techniques, such as GRNN have to be used to learn it This can also make the algorithm overﬁt the training set and can often make the neighborhood graph of the input data disconnected ICONIP 2008
- 15. Outline Motivation Manifold Learning Dimensionality Reduction Isomap Proposed approach Supervised Isomap Experimental setup Conclusions and Future WorkDetermining distances The Euclidean distance dij = d(xi , xj ) between two given observations xi and xj , labeled ci and cj respectively, is replaced by a dissimilarity measure: ((a − 1)/a)1/2 if ci = cj D(xi , xj ) = (1) a1/2 − d0 if ci = cj 2 where a = 1/e −dij /σ with dij set to one of the distance measures described above, σ is a smoothing parameter (set according to the data ’density’), do is a constant (0 ≤ d0 ≤ 1) and ci , cj are the data class labels. ICONIP 2008
- 16. Outline Motivation Dimensionality Reduction Overview Proposed approach Overview Experimental setup Conclusions and Future WorkS-Isomap Semi Supervised Approach ICONIP 2008
- 17. Outline Motivation Dimensionality Reduction Overview Proposed approach Overview Experimental setup Conclusions and Future WorkTesting instances When a reduced space is reached, our aim is to learn a kernel-based model that can be applied for testing new cases of failed and non-failed ﬁrms For testing, however, Isomap does not provide an explicit mapping in the embedded mapping. Therefore we can not generate the test set directly, since we would need to use the labels We use a generalized regression neural network (GRNN) to learn the mapping, before the SVM prediction phase takes place ICONIP 2008
- 18. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkDiane database Financial statements of French companies, initially of 60,000 industrial French companies, for the years of 2002 to 2006, with at least 10 employees 3,000 were declared bankrupted in 2007 or presented a restructuring plan 30 ﬁnancial ratios which allow the description of ﬁrms in terms of the ﬁnancial strength, liquidity, solvability, productivity of labor and capital, margins, net proﬁtability and return on investment ICONIP 2008
- 19. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkFinancial ratios 1. Number of employees 2. Financial Debt/Capital Employed % 3. Capital Employed/Fixed Assets 4. Depreciation of Tangible Assets 5. Working capital/current assets 6. Current ratio 7. Liquidity ratio 8. Stock Turnover days 9. Collection period 10. Credit Period 11. Turnover per Employee 12. Interest/Turnover 13. Debt Period days 14. Financial Debt/Equity 15. Financial Debt/Cashﬂow 16. Cashﬂow/Turnover 17. Working Capital/Turnover (days) 18. Net Current Assets/Turnover (days) 19. Working Capital Needs/Turnover 20. Export 21. Value added per employee 22. Total Assets/Turnover 23. Operating Proﬁt Margin 24. Net Proﬁt Margin 25. Added Value Margin 26. Part of Employees 27. Return on Capital Employed 28. Return on Total Assets 29. EBIT Margin 30. EBITDA Margin ICONIP 2008
- 20. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkPreprocessing Many cases with missing values, especially for defaults companies Default cases sorted out by the number of missing values. Examples with 10 missing values at most were considered 600 default examples was obtained To balance the dataset we selected randomly 600 non-default examples ICONIP 2008
- 21. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkPreprocessing For the ratios of the years 2003 and 2006, each missing value was replaced by the closest available year value For 2004 and 2005, if values of the next and previous years were available, each missing value was replaced by their mean, otherwise it was replaced by the remaining value In some cases there was no data available for a ratio in any of the years. In this very few cases the missing data was replaced by the median value of the ratio in each year All ratios were logarithmized and then standardized to zero mean and unity variance ICONIP 2008
- 22. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkHistorical data Companies are often subjected to ﬂuctuation of the market, economy cycles and unavoidable contingencies related to its business activity Yearly variations of important ﬁnancial ratios reﬂecting the balance sheet, sometimes quite relevant, are common particularly for small companies We included information from the past 3 years preceding the default. The number of inputs is therefore increased from 30 to 90 ratios More relevant than the ratios themselves, are the variations that occur over the period range of the analysis. ICONIP 2008
- 23. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkContingency table and error measures Class Positive Class Negative Assigned Positive tp fp (True Positives) (False Positives) Assigned Negative fn tn (False Negatives) (True Negatives) tp tp Recall ( tp+fn ) and Precision ( tp+fp ) fp Error type I ( fp+tn ) - % of companies classiﬁed as bankrupt when in reality they are healthy fn Error type II ( fn+tp ) - % number of samples classiﬁed as healthy when they are observed to be bankrupt fp+fn Error Rate - tp+fp+fn+tn ) ICONIP 2008
- 24. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkTrustworthiness A projection is trustworthy if the set of the k nearest neighbors of each data point in the low-dimensional space are also close-by in the original space: N 2 M(k) = 1 − (r (i, j) − k), (2) Nk(2N − 3k − 1) i=1 j∈Uk (i) where r (i, j) is the rank of the data point j in the ordering according to the distance from i in the original data space, and Uk (i) denotes the set of those data points that are among the k-nearest neighbors of the data point i in the low-dimensional space but not in the original space. ICONIP 2008
- 25. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkVisualization Trustworthiness with S-Isomap 0.95 nldr=3 nldr=5 nldr=10 0.9 Trustworthiness 0.85 0.8 0.75 0.7 3 4 5 7 10 15 20 40 60 80 100 150 200 K Nearest Neighbors ICONIP 2008
- 26. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkS-ISOMAP with k-Nearest Neighbors in Historical2006-2005 Data Set k KNN SVM Test Acc Error TypeI Error TypeII Test Acc Error TypeI Error TypeII 3 89.20±1.35 9.05±2.60 12.56±1.60 89.55±1.01 10.31±2.30 10.62±1.98 4 88.13±1.23 9.52±1.71 14.24±1.68 88.78±1.25 9.84±1.27 12.59±1.66 5 88.35±2.06 10.21±1.85 12.97±2.93 88.68±1.94 10.51±1.86 12.07±2.51 7 89.33±1.71 8.35±2.49 13.05±2.24 89.93±1.41 8.92±2.23 11.25±1.73 10 89.30±0.89 8.86±1.74 12.50±2.18 89.90±1.61 9.01±2.10 11.13±2.52 15 88.35±1.70 8.78±2.21 14.48±3.63 89.30±1.49 8.74±1.79 12.65±2.44 20 87.90±0.98 8.66±2.04 15.74±2.84 88.95±1.44 9.13±1.82 13.05±2.79 40 88.33±0.97 9.59±1.15 13.76±1.86 89.20±1.22 9.57±1.40 12.00±1.47 60 88.75±0.93 8.02±1.89 14.52±2.38 89.13±0.68 9.02±1.55 12.77±2.17 80 89.15±0.78 8.57±1.63 13.05±2.55 89.93±1.05 9.06±1.22 11.02±2.30 100 89.10±1.04 8.80±2.87 12.96±1.98 89.40±1.23 9.15±2.89 12.02±1.56 150 88.23±1.39 9.42±1.86 14.04±1.63 88.50±1.38 10.32±2.38 12.61±1.53 200 89.13±1.71 8.29±1.11 13.12±2.77 89.33±1.85 9.36±1.05 11.99±2.99 ICONIP 2008
- 27. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkPerformance Measures on Diane Financial Data Sets S-Isomap Train Test Recall Precision ErrorTypeI ErrorTypeII 2006 91.85 ± 0.54 87.73 ± 1.54 86.79 ± 2.62 87.94 ± 1.96 11.30 ± 1.91 13.21 ± 2.62 2005 78.70 ± 0.91 77.08 ± 2.02 77.13 ± 2.66 76.64 ± 3.62 22.87 ± 3.37 22.87 ± 2.66 2006-2005 94.26 ± 0.41 89.55 ± 1.01 89.38 ± 1.98 89.72 ± 1.94 10.31 ± 2.30 10.62 ± 1.98 2005-2004 96.74 ± 0.27 79.65 ± 1.42 77.61 ± 2.71 80.61 ± 2.12 18.38 ± 2.79 22.39 ± 2.71 KNN Train Test recall precision errorTypeI errorTypeII 2006 90.92 ± 0.76 85.77 ± 1.68 77.95 ± 3.29 92.51 ± 2.00 6.32 ± 1.69 22.05 ± 3.29 2005 84.78 ± 0.76 76.86 ± 1.71 73.22 ± 3.33 79.02 ± 1.98 19.46 ± 1.66 26.78 ± 3.33 2006-2005 91.18 ± 1.00 86.09 ± 1.88 76.99 ± 3.87 94.22 ± 3.03 4.74 ± 2.81 23.01 ± 3.87 2005-2004 84.39 ± 0.81 75.60 ± 1.79 64.80 ± 3.50 82.72 ± 1.65 13.58 ± 1.38 35.20 ± 3.50 SVM Train Test recall precision errorTypeI errorTypeII 2006 95.09 ± 0.42 90.54 ± 1.28 89.33 ± 2.24 91.73 ± 1.76 8.19 ± 1.90 10.67 ± 2.24 2005 86.06 ± 0.76 81.63 ± 1.76 81.01 ± 3.81 82.42 ± 2.84 17.64 ± 2.92 18.99 ± 3.81 2006-2005 95.85 ± 0.55 91.18 ± 1.28 92.10 ± 1.93 90.56 ± 1.69 9.74 ± 1.72 7.90 ± 1.93 2005-2004 89.93 ± 0.66 80.29 ± 1.54 81.04 ± 2.34 79.81 ± 2.58 20.42 ± 2.53 18.96 ± 2.34 RVM Train Test recall precision errorTypeI errorTypeII 2006 97.88 ± 0.63 81.25 ± 1.78 67.35 ± 2.98 92.31 ± 1.98 5.39 ± 2.01 32.65 ± 1.45 2005 93.25 ± 0.54 76.75 ± 1.25 72.64 ± 2.19 79.35 ± 2.34 19.09 ± 1.78 27.36 ± 2.03 2006-2005 99.68 ± 0.35 80.71 ± 2.11 72.47 ± 6.08 89.47 ± 2.55 8.71 ± 2.56 27.53 ± 6.08 2005-2004 100.00 ± 0.0 70.75 ± 1.74 65.36 ± 2.29 73.68 ± 1.53 23.46 ± 1.03 34.64 ± 2.29 ICONIP 2008
- 28. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkS-Isomap with Euclidean distance - KNN and SVM SVM Testing Accuracy with S-Isomap nldr=3 94 nldr=5 nldr=10 92 Accuracy in % 90 88 86 3 4 5 7 10 15 20 40 60 80 100 150 200 K Nearest Neighbors ICONIP 2008
- 29. Outline Motivation Data set Dimensionality Reduction Evaluation metrics Proposed approach Results Experimental setup Conclusions and Future WorkDiscussion of Results S-Isomap presents better results in testing accuracy than single KNN and RVM by 2% and 10% S-isomaps presents comparable results with SVM, however, with much reduced embedded space (nldr=3) whereas SVM algorithm is used with all ﬁnancial ratios The error of type II, corresponding to a failure of the correct prediction of bankruptcy is lower for the SVM. The same happens with a false alarm, i.e., indicating a bankruptcy for a healthy ﬁrm, which corresponds to the error of type I. The fact that ﬁrms clump nicely in the reduced space not only enhances ﬁnancial data visualization but also improves prediction results as compared with the kernel machines. ICONIP 2008
- 30. Outline Motivation Dimensionality Reduction Proposed approach Experimental setup Conclusions and Future WorkConclusions and Future Work We proposed an approach for bankruptcy analysis and prediction based on a supervised Isomap algorithm where class label information is incorporated Assuming that corporate ﬁnancial statuses lie in a manifold we attempt to uncover this embedded structure using manifold learning Isomap acts as a preprocessing stage allowing ﬁnancial data visualization Results have shown that comparable testing accuracy can be obtained even using a 3-dimensional reduced space Although the results in the ﬁnance setting seem promising, further work is necessary to design a method for avoiding the interpolation error resulting from the mapping learning stage. ICONIP 2008
- 31. Outline Motivation Dimensionality Reduction Proposed approach Experimental setup Conclusions and Future Work Bankrupcy Analysis for Credit Risk using Manifold LearningB Ribeiro, A Vieira, J Duarte, C Silva, J Carvalho das Neves, University of Coimbra, ISEP and ISEG, Portugal and Q Liu, A H Sung New Mexico Tech, USA November, 2008 ICONIP 2008

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