## Relaxed Maximum a Posteriori Fault Identification (2007)

Citations: | 12 - 9 self |

### BibTeX

@MISC{Zymnis07relaxedmaximum,

author = {A. Zymnis and S. Boyd and D. Gorinevsky},

title = {Relaxed Maximum a Posteriori Fault Identification},

year = {2007}

}

### OpenURL

### Abstract

We consider the problem of estimating a pattern of faults, represented as a binary vector, from a set of measurements. The measurements can be noise corrupted real values, or quantized versions of noise corrupted signals, including even 1-bit (sign) measurements. Maximum a posteriori probability (MAP) estimation of the fault pattern leads to a difficult combinatorial optimization problem, so we propose a variation in which an approximate maximum a posteriori probability estimate is found instead, by solving a convex relaxation of the original problem, followed by rounding and simple local optimization. Our method is extremely efficient, and scales to very large problems, involving thousands (or more) possible faults and measurements. Using synthetic examples, we show that the method performs extremely well, both in identifying the true fault pattern, and in identifying an ambiguity group, i.e., a set of alternate fault patterns that explain the observed measurements almost as well as our estimate. 1

### Citations

3916 |
Convex Optimization
- Boyd, Vandenberge
- 2004
(Show Context)
Citation Context ...ve occurred. We can derive simple conditions under which z = 0 solves the relaxed MAP (and hence, MAP) problem. This occurs if and only if ∇ly(0) ≥ 0, where the inequality is elementwise. (See, e.g., =-=[BV04]-=-.) For the case of linear measurements, this corresponds to λ ≥ 1 σ 2AT y. If this inequality is satisfied, we can immediately conclude that the null hypothesis x = 0 maximizes the posterior probabili... |

2031 | Numerical Optimization - Nocedal, Wright - 1999 |

1964 | Regression shrinkage and selection via the lasso
- Tibshirani
- 1996
(Show Context)
Citation Context ... The idea of using convex relaxation as the basis for a heuristic for solving a combinatorial problem is quite old. Some recent examples include compressed sensing [Don06], sparse regressor selection =-=[Tib96]-=-, sparse signal detection [Tro06], and sparse decoding [FKW03]. Other applications that use convex relaxations include portfolio optimization with transaction costs [LFB07], controller design [HHB98],... |

1850 | Compressed sensing
- Donoho
- 2006
(Show Context)
Citation Context ...thod based on Lagrangian relaxation. The idea of using convex relaxation as the basis for a heuristic for solving a combinatorial problem is quite old. Some recent examples include compressed sensing =-=[Don06]-=-, sparse regressor selection [Tib96], sparse signal detection [Tro06], and sparse decoding [FKW03]. Other applications that use convex relaxations include portfolio optimization with transaction costs... |

1251 | Factor graphs and the sum-product algorithm - Kschischang, Frey, et al. |

905 | A theory of diagnosis from first principles - Reiter - 1987 |

651 | Diagnosing Multiple Faults - Kleer, Williams - 1987 |

437 | Constructing free energy approximations and generalized belief propagation algorithms - Yedidia, Freeman, et al. - 2005 |

317 | Just Relax: Convex programming methods for identifying sparse signals
- Tropp
- 2006
(Show Context)
Citation Context ...ion as the basis for a heuristic for solving a combinatorial problem is quite old. Some recent examples include compressed sensing [Don06], sparse regressor selection [Tib96], sparse signal detection =-=[Tro06]-=-, and sparse decoding [FKW03]. Other applications that use convex relaxations include portfolio optimization with transaction costs [LFB07], controller design [HHB98], circuit design [VBG97], and sens... |

207 | Fault diagnosis in dynamic systems using analytical and knowledge base redundancy - A survey and some new results - Frank - 1990 |

192 | Fault detection and diagnosis in engineering systems, Ed - Gertler - 1998 |

150 | Branch-and-bound methods: a survey - Lawler, Wood - 1966 |

145 | Process fault detection based on modelling and estimation methods-a survey - Isermann - 1984 |

84 | Bayesian fault detection and diagnosis in dynamic systems - Lerner, Parr, et al. - 2000 |

83 | A fault detection service for wide area distributed computations - Stelling, Foster, et al. - 1998 |

64 | Proactive Network Fault Detection - Hood, Ji - 1997 |

49 |
Portfolio optimization with linear and fixed transaction costs
- Lobo, Fazel, et al.
- 2007
(Show Context)
Citation Context ... sparse regressor selection [Tib96], sparse signal detection [Tro06], and sparse decoding [FKW03]. Other applications that use convex relaxations include portfolio optimization with transaction costs =-=[LFB07]-=-, controller design [HHB98], circuit design [VBG97], and sensor selection [JB]. The fault estimation problem is closely related to the problem of multi-user detection in CDMA systems. In this setting,... |

45 | Fault detection in an ethernet network using anomaly signature matching - Feather, Siewiorek, et al. - 1993 |

44 |
Fault diagnosis of analog circuits
- Bandler, Salama
- 1985
(Show Context)
Citation Context ...th monitoring [VAB + 07, GDG04]), industrial process control [HKH91], automotive [MCF + 93], telecommunications and networking [HJ97, FSM93], computer systems [SDF + 99], circuit fault identification =-=[BS85]-=-, and many others. In most papers in the literature it is assumed that faults affect measurements in an additive way. In fact there is a whole body of research devoted to optimal signature matrix desi... |

31 | The application of semidefinite programming for detection in CDMA - Tan, Rasmussen - 2001 |

30 | Low-authority controller design via convex optimization, AIAA
- Hassibi, How, et al.
- 1999
(Show Context)
Citation Context ... [Tib96], sparse signal detection [Tro06], and sparse decoding [FKW03]. Other applications that use convex relaxations include portfolio optimization with transaction costs [LFB07], controller design =-=[HHB98]-=-, circuit design [VBG97], and sensor selection [JB]. The fault estimation problem is closely related to the problem of multi-user detection in CDMA systems. In this setting, the received signal plays ... |

28 | Gamal, Optimal wire and transistor sizing for circuits with non-tree topology
- Vandenberghe, Boyd, et al.
- 1997
(Show Context)
Citation Context ...detection [Tro06], and sparse decoding [FKW03]. Other applications that use convex relaxations include portfolio optimization with transaction costs [LFB07], controller design [HHB98], circuit design =-=[VBG97]-=-, and sensor selection [JB]. The fault estimation problem is closely related to the problem of multi-user detection in CDMA systems. In this setting, the received signal plays the role of the measurem... |

23 |
Analysis of belief propagation for non-linear problems: The example of CDMA (or: How to prove Tanaka’s formula
- Montanari, Tse
(Show Context)
Citation Context ...blem: belief propagation. This is an iterative message passing algorithm, that has been proven to work very well in the context of multi-user detection in CDMA (code division multiple access) systems =-=[MT06]-=-. We have not found any literature on belief propagation applied to fault identification; but we can refer the reader to the tutorial papers [KFL01, YFW05] for a general discussion of belief propagati... |

21 | LP decoding
- Feldman, Karger, et al.
- 2003
(Show Context)
Citation Context ...tic for solving a combinatorial problem is quite old. Some recent examples include compressed sensing [Don06], sparse regressor selection [Tib96], sparse signal detection [Tro06], and sparse decoding =-=[FKW03]-=-. Other applications that use convex relaxations include portfolio optimization with transaction costs [LFB07], controller design [HHB98], circuit design [VBG97], and sensor selection [JB]. The fault ... |

20 | Constrained Control and Estimation: An optimisation approach - Goodwin, Seron, et al. - 2005 |

16 | Global optimization to prescribed accuracy - Moore - 1991 |

13 | Fundamental problems in fault detection and identification - Saberi, Stoorvogel, et al. - 2000 |

11 |
Fault diagnosis in complex chemical plants using artificial neural networks
- Hoskins, Kaliyur, et al.
- 1991
(Show Context)
Citation Context ...d related work Fault detection problems arise in most computer based engineering systems. Examples include aerospace (e.g., jet engine health monitoring [VAB + 07, GDG04]), industrial process control =-=[HKH91]-=-, automotive [MCF + 93], telecommunications and networking [HJ97, FSM93], computer systems [SDF + 99], circuit fault identification [BS85], and many others. In most papers in the literature it is assu... |

10 | Computationally efficient algorithms for multiple fault diagnosis in large graph-based systems - Tu, Pattipati, et al. |

9 | Semidefinite positive relaxation of the maximum likelihood criterion applied to multiuser detection in CDMA context - Abdi, Nahas, et al. - 2002 |

7 | Parametric fault modeling and diagnostics of a turbofan engine
- Ganguli, Deo, et al.
- 2004
(Show Context)
Citation Context ... of research devoted to optimal signature matrix design [Fra90, SSS00]. A number of publications are concerned with the problem of fault identification in linear dynamical systems for both parametric =-=[GDG04]-=- and binary faults [LPKB00, MBM99]. A number of heuristics have been proposed to tackle this problem, including variations of least-squares [GDG04] and methods based on Kalman filtering [LPKB00, VAB +... |

7 | Model-based on-board fault detection and diagnosis for automotive engines - Gertler, Costin, et al. - 1993 |

5 | A framework for control, state estimation, fault detection, and verification of hybrid systems - Morari, Bemporad, et al. - 1999 |

4 | Advanced estimation for aircraft engines - Viassolo, Adibhatla, et al. |

3 | Multiple fault diagnosis in graph-based systems - Tu, Pattipati, et al. |

3 | SDPT3 version 4.0 – a MATLAB software for semidefinite-quadratic-linear programming - Toh, Todd, et al. - 2009 |

2 | Available from www.stanford.edu/ ∼ boyd/cvx - Grant, Boyd, et al. - 2007 |

2 |
Sensor selection via convex optimization. Available from www.stanford.edu/ ∼ boyd/sensor selection.html
- Joshi, Boyd
(Show Context)
Citation Context ...decoding [FKW03]. Other applications that use convex relaxations include portfolio optimization with transaction costs [LFB07], controller design [HHB98], circuit design [VBG97], and sensor selection =-=[JB]-=-. The fault estimation problem is closely related to the problem of multi-user detection in CDMA systems. In this setting, the received signal plays the role of the measurements, and the transmitted b... |