Multiple Models — Fixed, Switching, Interacting

Gundlich, Brigitte; Teunissen, Peter

doi:10.1007/978-3-662-10735-5_18

Cited by 10 publications

(7 citation statements)

References 8 publications

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“…As was pointed out in Teunissen (2003), the expression for theâ BIEE is identical to its Bayesian counterpart as given in Betti et al (1993) and Gundlich and Koch (2002); also see Teunissen (2001) and Gundlich and Teunissen (2004). This equivalence nicely bridges the gap between the present nonBayesian approach and the Bayesian approach.…”

Section: Corollary 7 (Bmep In Gaussian Case) If Y 0 and Y Have A Joinsupporting

confidence: 74%

Best prediction in linear models with mixed integer/real unknowns: theory and application

Teunissen

2007

J Geod

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In this contribution, we extend the existing theory of minimum mean squared error prediction (best prediction). This extention is motivated by the desire to be able to deal with models in which the parameter vectors have real-valued and/or integer-valued entries. New classes of predictors are introduced, based on the principle of equivariance. Equivariant prediction is developed for the real-parameter case, the integer-parameter case, and for the mixed integer/real case. The best predictors within these classes are identified, and they are shown to have a better performance than best linear (unbiased) prediction. This holds true for the mean squared error performance, as well as for the error variance performance. We show that, in the context of linear model prediction, best predictors and best estimators come in pairs. We take advantage of this property by also identifying the corresponding best estimators. All of the best equivariant estimators are shown to have a better precision than the best linear unbiased estimator. Although no restrictions are placed on the probability distributions of the random vectors, the Gaussian case is derived separately. The best predictors are also compared with least-squares predictors, in particular with the integer-based least-squares predictor introduced in Teunissen (J Geodesy, in press, 2006).

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Section: Corollary 7 (Bmep In Gaussian Case) If Y 0 and Y Have A Joinsupporting

confidence: 74%

Best prediction in linear models with mixed integer/real unknowns: theory and application

Teunissen

2007

J Geod

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“…a BIE ¼ a a ILS Proof of Lemma 3 is given in the appendix. It is interesting to observe that the above expressions given forâ a BIE andb b BIE are identical to their Bayesian counterparts as given in Betti et al (1993) and Gundlich and Koch (2002); see also Teunissen (2001) and Gundlich and Teunissen (2003). This is not quite true for the general case.…”

Section: The Gaussian Casementioning

confidence: 54%

Theory of integer equivariant estimation with application to GNSS

Teunissen

2003

Journal of Geodesy

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Abstract. Carrier phase ambiguity resolution is the key to high-precision global navigation satellite system (GNSS) positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo. The so-called 'fixed' baseline estimator is known to be superior to its 'float' counterpart in the sense that its probability of being close to the unknown but true baseline is larger than that of the 'float' baseline, provided that the ambiguity success rate is sufficiently close to its maximum value of one. Although this is a strong result, the necessary condition on the success rate does not make it hold for all measurement scenarios. It is discussed whether or not it is possible to take advantage of the integer nature of the ambiguities so as to come up with a baseline estimator that is always superior to both its 'float' and its 'fixed' counterparts. It is shown that this is indeed possible, be it that the result comes at the price of having to use a weaker performance criterion. The main result of this work is a GaussMarkov-like theorem which introduces a new minimum variance unbiased estimator that is always superior to the well-known best linear unbiased (BLU) estimator of the Gauss-Markov theorem. This result is made possible by introducing a new class of estimators. This class of integer equivariant estimators obeys the integer remove-restore principle and is shown to be larger than the class of integer estimators as well as larger than the class of linear unbiased estimators. The minimum variance unbiased estimator within this larger class is referred to as the best integer equivariant (BIE) estimator. The theory presented applies to any model of observation equations having both integer and real-valued parameters, as well as for any probability density function the data might have.

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“…The Bayesian approach was already used in [13]- [15], our technique is substantially different in several ways such as the optimality criterion, the hypotheses used for the integer parameters and the use of a sequential implementation. [17] applies several Bayesian models to the ambiguity resolution problem but it remains in a general context, somewhat far from practice. Although our technique is primarily devoted to the estimation of static parameters, it is extended to time-varying real parameters (e.g., as in the case of kinematic baselines in GNSS differential positioning) following a linear dynamical model.…”

Section: E E E P R O O F W E B V E R S I O Nmentioning

confidence: 99%

“…Based on the IA theory, the known ratio test used for validating the ambiguities solution was revisited in [12]. Finally, the Bayesian approach was employed in the context of GNSS ambiguity resolution in [13]- [17]. [13]- [16] yield a posterior distribution for all variables, discrete and continuous, conditional to the observed quantities; [16] determines confidence regions for the baseline estimation based on GPS measurements; and [17] presents several well known Bayesian estimation schemes and applies them to the ambiguity resolution problem.…”

mentioning

confidence: 99%

“…Finally, the Bayesian approach was employed in the context of GNSS ambiguity resolution in [13]- [17]. [13]- [16] yield a posterior distribution for all variables, discrete and continuous, conditional to the observed quantities; [16] determines confidence regions for the baseline estimation based on GPS measurements; and [17] presents several well known Bayesian estimation schemes and applies them to the ambiguity resolution problem. Very often, the proposed ambiguity resolution methods deal with the problem of differential positioning, where the difference of the position of two antennas (with its associated receivers) is estimated.…”

mentioning

confidence: 99%

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