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## Abstract Suboptimal LULU-estimators in Measurements Containing Outliers (2013)

### Citations

3274 |
An introduction to probability theory and its applications vol I
- Feller
- 1968
(Show Context)
Citation Context ... interval a < θ ≤ b is determined uniquely by the sum rule which, since A and C are mutually exclusive, reduces to 3The popular statistical texts of the latter half of the twentieth century by Feller =-=[9, 10]-=- and volumes 1-2A of Kendall and Stuart [15] concentrate solely on calculating these. Stellenbosch Universityshttp://scholar.sun.ac.za CHAPTER 3. BAYESIAN INFERENCE 42 P (B | I) = P (A | I) + P (C | I... |

3080 |
An introduction to probability theory and its applications
- Feller
- 1971
(Show Context)
Citation Context ... interval a < θ ≤ b is determined uniquely by the sum rule which, since A and C are mutually exclusive, reduces to 3The popular statistical texts of the latter half of the twentieth century by Feller =-=[9, 10]-=- and volumes 1-2A of Kendall and Stuart [15] concentrate solely on calculating these. Stellenbosch Universityshttp://scholar.sun.ac.za CHAPTER 3. BAYESIAN INFERENCE 42 P (B | I) = P (A | I) + P (C | I... |

920 | Probability Theory – The Logic of Science
- Jaynes
(Show Context)
Citation Context ... [5]. These rules, central to probability theory as extended logic, turn out to be nothing more than the standard product and sum rules of probability theory. Their derivation (see for example [5] or =-=[13]-=-), using Boolean algebra and calculus, remarkably follows from only a few verbal statements2 (qualitative rules, not axioms) describing the desired attributes of a 1Representation by real numbers auto... |

881 | Theory of Probability - Jeffreys - 1983 |

688 |
An essay towards solving a problem in the doctrine of chances
- Bayes
- 1763
(Show Context)
Citation Context ...reverse of this problem: Given the data, what were the probable causes of the data, or contents of the urn? Bernoulli himself pondered this problem [3], but it was Bayes’ posthumously published paper =-=[2]-=- that appears to articulate a solution which resembles what we today call Bayes Theorem. Laplace [16], seemingly unaware of Bayes earlier claims, was responsible for writing it down in its general and... |

342 |
Data analysis: A Bayesian tutorial
- Sivia
- 1996
(Show Context)
Citation Context ...s Jaynes, de Finetti and others. Aided by Cox’s theorems [5] they advocated a new way of interpreting probability theory, in contrast with the predominant classical statistical thinking of their time =-=[30]-=-. 3.1 Probability theory as extended logic Cox showed that if degrees of belief of various propositions are represented by real numbers1 (the larger implying a greater degree of belief), there do exis... |

326 |
La prevision: ses lois logiques, ses sources subjectives," Annales de VInstitute Henri Poincare
- Finetti
- 1937
(Show Context)
Citation Context ...rials at which they occur. The distribution (5.6) is then invariant under permutations of the zi, and is thus what is referred to as an exchangeable prior. There is an important theorem by de Finetti =-=[6]-=- that now becomes relevant to the problem. The de Finetti theorem asserts that any exchangeable probability function is determined by a single generating function f(α). Thus there is a function f(α) s... |

228 |
Probability, frequency and reasonable expectation.
- Cox
- 1946
(Show Context)
Citation Context ...lities on a principle of ‘insufficient reason’ [11]. The potential of their work went undiscovered until Jeffreys [14] inspired thinkers such as Jaynes, de Finetti and others. Aided by Cox’s theorems =-=[5]-=- they advocated a new way of interpreting probability theory, in contrast with the predominant classical statistical thinking of their time [30]. 3.1 Probability theory as extended logic Cox showed th... |

106 |
Théorie Analytique des Probabilités
- Laplace
(Show Context)
Citation Context ...he urn? Bernoulli himself pondered this problem [3], but it was Bayes’ posthumously published paper [2] that appears to articulate a solution which resembles what we today call Bayes Theorem. Laplace =-=[16]-=-, seemingly unaware of Bayes earlier claims, was responsible for writing it down in its general and continuous parameter form as we know it today and, unlike Bayes he motivated the assignment of the p... |

40 |
Ars Conjectandi
- Bernoulli
- 1713
(Show Context)
Citation Context ...d. The real scientist however is concerned with the reverse of this problem: Given the data, what were the probable causes of the data, or contents of the urn? Bernoulli himself pondered this problem =-=[3]-=-, but it was Bayes’ posthumously published paper [2] that appears to articulate a solution which resembles what we today call Bayes Theorem. Laplace [16], seemingly unaware of Bayes earlier claims, wa... |

25 |
Definition and comparison of robust nonlinear data smoothing algorithms.
- Velleman
- 1980
(Show Context)
Citation Context ...be appropriate for removing outliers, a supporting theory is ‘extremely difficult’ even in the simplest cases [18]. Nonlinear analysis had thus previously been approached by largely heuristic methods =-=[31, 32]-=-. The accompanying theory enabled by the LULU operators promises a deeper understanding. LULU-theory, is so named due to its constituent nonlinear operators Ln and Un that are usually applied in compo... |

24 |
The Advanced -, Theory of Statistics", vol 1
- -Kendall-, Stuart
- 1958
(Show Context)
Citation Context ...he sum rule which, since A and C are mutually exclusive, reduces to 3The popular statistical texts of the latter half of the twentieth century by Feller [9, 10] and volumes 1-2A of Kendall and Stuart =-=[15]-=- concentrate solely on calculating these. Stellenbosch Universityshttp://scholar.sun.ac.za CHAPTER 3. BAYESIAN INFERENCE 42 P (B | I) = P (A | I) + P (C | I) . (3.14) Thus P (a < θ ≤ b | I) = G(b)−G(a... |

21 |
Some theory of nonlinear smoothers.
- MALLOWS
- 1980
(Show Context)
Citation Context ... LULU smoothers of Stellenbosch mathematician Carl Rohwer. While the median smoothers can be appropriate for removing outliers, a supporting theory is ‘extremely difficult’ even in the simplest cases =-=[18]-=-. Nonlinear analysis had thus previously been approached by largely heuristic methods [31, 32]. The accompanying theory enabled by the LULU operators promises a deeper understanding. LULU-theory, is s... |

10 |
Robust nonlinear' data smoothers: definitions and recommendations.
- Velleman
- 1977
(Show Context)
Citation Context ...be appropriate for removing outliers, a supporting theory is ‘extremely difficult’ even in the simplest cases [18]. Nonlinear analysis had thus previously been approached by largely heuristic methods =-=[31, 32]-=-. The accompanying theory enabled by the LULU operators promises a deeper understanding. LULU-theory, is so named due to its constituent nonlinear operators Ln and Un that are usually applied in compo... |

8 |
Idempotent One-Sided Approximation of Median Smoothers
- Rohwer
- 1989
(Show Context)
Citation Context ...oduction It was in 1989 when Carl Rohwer, working on practical problems for the Maritime institute in Simonstown, that the founding ideas of LULU-theory and the Discrete Pulse Transform (DPT) emerged =-=[24, 19]-=-. LULU-theory is so named due to its constituent non-linear operators, or smoothers Ln and Un that are usually applied in composition LnUn (or UnLn). They were initially developed to be applied to one... |

6 |
Locally Monotone Robust Approximation Sequences
- Rohwer, Toerien
- 1991
(Show Context)
Citation Context ...y developed to be applied to one-dimensional sequences in order to remove impulsive noise [24], but since their appearance a series of articles followed by Rohwer [20, 21, 22, 23], Rohwer and Toerien =-=[27]-=- and Rohwer and Wild [28], detailing their alluring mathematical properties which include for example idempotency, co-idempotency, stability, trend preserving and variation decomposing [12]. The work ... |

5 |
Natural alternatives for one dimensional median filtering
- Rohwer, Wild
- 2002
(Show Context)
Citation Context ... to one-dimensional sequences in order to remove impulsive noise [24], but since their appearance a series of articles followed by Rohwer [20, 21, 22, 23], Rohwer and Toerien [27] and Rohwer and Wild =-=[28]-=-, detailing their alluring mathematical properties which include for example idempotency, co-idempotency, stability, trend preserving and variation decomposing [12]. The work culminated in a self cont... |

4 |
The Discrete Pulse Transform
- Laurie, Rohwer
(Show Context)
Citation Context ...uence. Thus the probability for an upward pulse appearing in U1x−U1L1x is 24 120 = 1 5 . Because the number of pulses in a decomposition is always smaller than or equal to the number of data points N =-=[26]-=-, we see that more than half (13 + 1 5 = 8 15 ) of the pulses are expected to be in the first resolution level [25] if the sequence is random (i.i.d.). An operator S is defined to be idempotent if S2 ... |

3 |
Exact and asymptotic distributions of LULU smoothers
- Conradie, Wet, et al.
- 2006
(Show Context)
Citation Context ...since continued with numerous ventures. For example, the theory was later extended to higher dimensional arrays [1], knowledge concerning some statistical properties of LULU smoothers has been gained =-=[4]-=- [12] and fast implementation algorithms have been presented [8] [17]. A Discrete Pulse Transform follows naturally from the theory [24] [26], and is a multiresolutional decomposition of a sequence x ... |

3 |
The Roadmaker’s Algorithm and the Discrete Pulse Transform, private communication with the author on a paper submitted for publication
- Laurie
- 2009
(Show Context)
Citation Context ...later extended to higher dimensional arrays [1], knowledge concerning some statistical properties of LULU smoothers has been gained [4] [12] and fast implementation algorithms have been presented [8] =-=[17]-=-. A Discrete Pulse Transform follows naturally from the theory [24] [26], and is a multiresolutional decomposition of a sequence x into a sum of positive and negative pulses. Each resolution level w i... |

3 |
Multiresolution Analysis with Pulses
- Rohwer
- 2002
(Show Context)
Citation Context ...on LnUn (or UnLn). They were initially developed to be applied to one-dimensional sequences in order to remove impulsive noise [24], but since their appearance a series of articles followed by Rohwer =-=[20, 21, 22, 23]-=-, Rohwer and Toerien [27] and Rohwer and Wild [28], detailing their alluring mathematical properties which include for example idempotency, co-idempotency, stability, trend preserving and variation de... |

3 |
Variation reduction and LULU—Smoothing
- Rohwer
- 2002
(Show Context)
Citation Context ...on LnUn (or UnLn). They were initially developed to be applied to one-dimensional sequences in order to remove impulsive noise [24], but since their appearance a series of articles followed by Rohwer =-=[20, 21, 22, 23]-=-, Rohwer and Toerien [27] and Rohwer and Wild [28], detailing their alluring mathematical properties which include for example idempotency, co-idempotency, stability, trend preserving and variation de... |

3 |
Fully trend preserving operators
- Rohwer
- 2004
(Show Context)
Citation Context ...on LnUn (or UnLn). They were initially developed to be applied to one-dimensional sequences in order to remove impulsive noise [24], but since their appearance a series of articles followed by Rohwer =-=[20, 21, 22, 23]-=-, Rohwer and Toerien [27] and Rohwer and Wild [28], detailing their alluring mathematical properties which include for example idempotency, co-idempotency, stability, trend preserving and variation de... |

3 |
Bayes estimators for the continuous uniform distribution
- Rossman, Short, et al.
- 1998
(Show Context)
Citation Context ...ure 4.9: Mean estimator and standard deviation (σ = √ 〈L2〉 − 〈L〉) of L with Jeffreys prior (λ = 1). Parameter X = 0 is assumed known. The results for two different data sets are shown. As is noted in =-=[29]-=-, some of the many classical estimators proposed for this problem can be attained by a certain choice of improper prior. For example, when we set X = 0 and λ = 2, then 〈L〉 corresponds with the minimum... |

2 |
The discrete pulse transform in two dimensions
- Fabris-Rotelli, Walt
(Show Context)
Citation Context ...was later extended to higher dimensional arrays [1], knowledge concerning some statistical properties of LULU smoothers has been gained [4] [12] and fast implementation algorithms have been presented =-=[8]-=- [17]. A Discrete Pulse Transform follows naturally from the theory [24] [26], and is a multiresolutional decomposition of a sequence x into a sum of positive and negative pulses. Each resolution leve... |

2 |
Some Statistical Aspects of LULU smoothers
- Jankowitz
- 2007
(Show Context)
Citation Context ...nd Toerien [27] and Rohwer and Wild [28], detailing their alluring mathematical properties which include for example idempotency, co-idempotency, stability, trend preserving and variation decomposing =-=[12]-=-. The work culminated in a self contained monograph on the subject in 2005 [24], and has since continued with numerous ventures. For example, the theory was later extended to higher dimensional arrays... |

2 |
Projections and separators
- Rohwer
- 1999
(Show Context)
Citation Context |

2 |
Nonlinear Smoothing and Multiresolution Analysis, volume 150
- Rohwer
- 2005
(Show Context)
Citation Context ...o remove impulsive noise, but since their appearance a series of articles followed detailing their alluring mathematical properties. A Discrete Pulse Transform (DPT) follows naturally from the theory =-=[24]-=- [26], and is a multiresolutional decomposition of a sequence x into a sum of positive and negative pulses. Each resolution level w is then a sequence which contains essentially zeros except for w con... |

2 |
The Estimation of Moments of an Unknown Error Distribution
- Rohwer
(Show Context)
Citation Context ...into a sum of positive and negative pulses. Each resolution level w is then a sequence which contains essentially zeros except for w consecutive entries of the same constant value (pulses of width w) =-=[25]-=-. The DPT is viewed as a competitor to median based decompositions and, although not well-known in the physics community, promises major advantages over currently popular wavelet decompositions in cer... |

1 | LULU operators and discrete pulse transform for multidimensional arrays
- Anguelov, Fabris-Rotelli
- 2010
(Show Context)
Citation Context ... The work culminated in a self contained monograph on the subject in 2005 [24], and has since continued with numerous ventures. For example, the theory was later extended to higher dimensional arrays =-=[1]-=-, knowledge concerning some statistical properties of LULU smoothers has been gained [4] [12] and fast implementation algorithms have been presented [8] [17]. A Discrete Pulse Transform follows natura... |

1 |
Applications of the Discrete Pulse Transform in Image Analysis
- Fabris-Rotelli
- 2012
(Show Context)
Citation Context ... of width w) [25]. The successes of the DPT are heralded in the field of image processing. Here the acquisition of data can come from digital cameras, long-wave (infrared) and laser capturing devices =-=[7]-=-. The DPT is viewed as a competitor to median based decompositions, and is superior to wavelet decomposition in some applications. The interest in research and applications continues to grow with new ... |

1 |
Idempotent one-sided approximation of median smoothers
- Fienberg
- 1989
(Show Context)
Citation Context ...e for writing it down in its general and continuous parameter form as we know it today and, unlike Bayes he motivated the assignment of the prior probabilities on a principle of ‘insufficient reason’ =-=[11]-=-. The potential of their work went undiscovered until Jeffreys [14] inspired thinkers such as Jaynes, de Finetti and others. Aided by Cox’s theorems [5] they advocated a new way of interpreting probab... |