1 / 17

Linear Programming

Linear Programming. As Used for Discriminant Analysis. Objectives. Maximize minimum distance from critical value Minimize sum of deviations from critical value Simple Direct Free of statistical assumptions Flexible. Requirements to use LP. LP modeling skills Commercial software.

dillon
Download Presentation

Linear Programming

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Linear Programming As Used for Discriminant Analysis

  2. Objectives • Maximize minimum distance from critical value • Minimize sum of deviations from critical value • Simple • Direct • Free of statistical assumptions • Flexible

  3. Requirements to use LP • LP modeling skills • Commercial software

  4. Linear Discriminant Analysis • Separate data into groups such that • Minimize distance within group • Maximize distance to other groups • Can have: • Binary (2 groups) • Multiple categories (more than 2 groups)

  5. Minimize Sum of Deviations (MSD) Minimize 1 + … + r Subject to: A11 x1 + … + A1r xr b + 1for A1 in B, ………… An1x1+ … + Anr xr b - rfor An in G, , … , r 0,

  6. Maximize Minimum Distance(MMD) Maximize 1 + … + r Subject to: A11 x1+ … + A1r xr b - 1forA1 in B, ………… An1 x1+ … + Anr xrb + rfor An in G, 1, …, r 0

  7. Example Minimize 1 + 2 Subject to: 6 x1+ 8 x2 b + 1for A1 in B, 15 x1+ 31 x2 b - 2for A2 in G, 1,2 0 Use b = 9 Optimal solution: x1* = 0, x2* = 0.290323 A1= 2.35 < 9 so BAD; A2 = 9.00001 > 9 so GOOD

  8. i i Bad Good 2.345840 9.000013 Perfect SeparationAX* = 9

  9. Overlapping Data

  10. bL1 bU1 bL2 C1 bU2 bL3 C2 bU3 X C3 a1 Three-Class Linear Discriminant Analysis

  11. MCLP Classification • Two or more criteria • Create deviational variables for each Functiona + da- - da+ = Targeta Objective: Min weighted sum of deviations IDEAL POINT: all desired deviations = 0

  12. Fuzzy LP Classification • Not all data precise • Fuzzy concept: • Membership function 0 ≤ MF ≤ 1 • Can have MF for any number of states • 50 degrees • Cold MF might be 0.7 • Warm MF might be 0.4 • Hot MF might be 0

  13. Fuzzy MOLP • Discriminate to various classes available X-axis is alpha; Y-axis is beta

  14. Real Application: Credit Card • Outcomes • Bankruptcy • Good • Scoring techniques • Behavior Score • Credit Bureau Scores • Proprietary Bankruptcy Score • Set Enumeration Decision Tree

  15. Real Application – Credit Card • LP an alternative to these scoring methods • Classify cardholders in terms of payment • Common variables: • Balance • Purchase • Payment • Cash advance • State of residence • Job security

  16. Real Application – Credit Card • FDR model • 38 original variables over 7 months • 65 derived variables generated • Separation criteria: • Information value – mean difference/STD • Concordance • Kolmogorov-Smirnov (best)

  17. Real Application – Credit Cards • Sampled 6,000 records • 2-class output • 65 attributes • 50 LP solutions computed • Varied fuzzy parameters, setoff limits • Used 1000, 3000, 6000 records • Compared with decision tree, neural network model • MCLP best at not calling actual bad cases good • But this was on a small test set • Fuzzy LP best on large test set

More Related