Enhancements to GPS operations and clock evaluations using a "total" Hadamard deviation

Howe, David A.; Beard, Ronald; Greenhall, C. A.; Vernotte, F.; Riley, William J.; Peppler, T.K.

doi:10.1109/tuffc.2005.1509784

Cited by 26 publications

(22 citation statements)

References 11 publications

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“…Several techniques have been developed for the analysis of the noise altering the output signal of sensors, such as the AV, the HV or the TV [16,17,18,19]. The AV was invented in 1966 by David Allan when he criticized the use of the sample variance estimator in the context of non iid time series.…”

Section: Allan Hadamard and Total Variance Techniquesmentioning

confidence: 99%

Constrained expectation-maximization algorithm for stochastic inertial error modeling: study of feasibility

Stebler¹,

Guerrier²,

Skaloud³

et al. 2011

Meas. Sci. Technol.

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Abstract. Stochastic modeling is a challenging task for low-cost sensors which errors can have complex spectral structures. This makes the tuning process of the INS/GNSS Kalman filter often sensitive and difficult. For example, first-order Gauss-Markov processes are very often used in inertial sensor models. But the estimation of their parameters is a non-trivial task if the error structure is mixed with other types of noises. Such estimation is often attempted by computing and analyzing Allan variance plots. This contribution demonstrates solving situations when the estimation of error parameters by graphical interpretation is rather difficult. The novel strategy performs direct estimation of these parameters by means of the Expectation-Maximization (EM) algorithm. The algorithm results are first analyzed with a critical and practical point of view using simulations with typically encountered error signals. These simulations show that the EM algorithm seems to perform better than the Allan variance and offers a procedure to estimate first-order Gauss-Markov processes mixed with other types of noises. At the same time, the conducted tests revealed limits of this approach that are related to the convergence and stability issues. Suggestions are given to circumvent or mitigate these problems when complexity of error structure is "reasonable". This work also highlights the fact that the suggested approach via EM algorithm and the Allan variance may not be able to estimate reasonably well parameters of complex error models, and shows the need for new estimation procedures to be developed in this context. Finally, an empirical scenario is presented to support the former findings. There, the positive effect of using the more sophisticated EM-based error modeling on a filtered trajectory is highlighted.

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Section: Allan Hadamard and Total Variance Techniquesmentioning

confidence: 99%

Constrained expectation-maximization algorithm for stochastic inertial error modeling: study of feasibility

Stebler¹,

Guerrier²,

Skaloud³

et al. 2011

Meas. Sci. Technol.

View full text Add to dashboard Cite

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“…Specifically, we have just shown that such observable statistics will underestimate 2 r σ by removing portions of L p (f) below f T for x poly,M (t) fitting. For Δ-variance statistics of random error, one can show that such underestimation will peak as τ approaches T. This can explain the well-known tendency of non-total [23,3] statistics of Δ-variances with x poly,M (t) removed to generate downwardly biased results for neg-p noise when τ approaches T. For total statistics [23,3], which double the data to remove such downward biases, it is conjectured that the interaction between the doubling and fitting process causes the appearance bias removal without truly increasing the statistical confidence of the estimate. This, however, needs further investigation.…”

Section: Spectral Representation Of Accuracy and Precisionmentioning

confidence: 85%

The profound impact of negative power law noise on the estimation of causal behavior

Reinhardt

2009

2009 IEEE International Frequency Control Symposium Joint With the 22nd European Frequency and Time Forum

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This paper demonstrates that highly correlated negative power law (neg-p) noise-noise with a PSD ∝ |f| p when p<0-and causal behavior contained in data cannot be properly separated from each other by any causal fitting or estimation technique. It then shows: (a) that this leads such techniques to generate anomalous estimates of the true causal behavior and to underreport the true fit error, and (b) that these anomalies cannot be corrected by adding modeling or increasing the data collection interval T. Further consequences of the above are also discussed. These include anomalous behavior in noise whitening, M-corner hat, and cross-correlation techniques and the nonobservability of unbiased measures of pure random error when neg-p noise is present. The paper also investigates the relationship between ergodic-like behavior-the approximate equality of time averages over a finite T and ensemble averages-and such anomalous behavior in general statistical processing techniques. It is shown that such ergodic-like behavior is a necessary condition for practical implementations of these techniques to behave like their theoretical counterparts and, furthermore, a necessary condition for ergodic-like behavior is T>>τ c , where τ c is the correlation time of the noise process involved. Finally, it is shown that this is unachievable for neg-p noise, because τ c = ∞ for such noise (unless system highpass filtering makes τ c finite for the system filtered noise variable).

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“…These constants are summarized for each of the five integer-slope noise types in Table I. A method for determining other noise types is to estimate Avar's own bias, or B 1 , function relative to the standard variance [19], [20]. Unfortunately, Avar bias cannot be defined beyond T 2 , or half the data run, since Avar is undefined, which leads us to a difficulty in using this method for τ > T 2 .…”

Section: Methods Of Removing Bias Relative To Avarmentioning

confidence: 99%

“…1. The upper plot shows Thêo1 and Adev operating on WHPM noise type and the lower plot shows Thêo1 with bias removed relative to Adev using (2) and Allan dead-time bias, or B 1 , function [19], [20].…”

Section: Hybrid Statisticsmentioning

confidence: 99%

ThêoH Bias-removal Method

Howe

McGee-Taylor

Tassett

2006

2006 IEEE International Frequency Control Symposium and Exposition

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A version of Thêo1 variance, called ThêoBR variance (BR for "bias-removed" relative to the Allan variance), is constructed. This relative bias correction is applied over a range of the longest recommended Allan τ values for a given data run. ThêoH deviation ('H' to indicate a hybrid combination of ThêoBR and Allan deviations) is the Allan deviation in short term and switches to the ThêoBR deviation in long term. In the presence of non-integer-power-law and mixed noise types, the approach is as effective and less cumbersome than past approaches, and requires little, if any, a priori knowledge or human judgment of data being analyzed. The substantially gained properties of Thêo1 at large τ , to 50 % beyond the longest possible τ using the Allan deviation alone, can be obtained without Allan bias. Long-term frequency stability can be obtained in essentially onethird less time. For example, a two-month stability can be obtained with three months of data, rather than the four months of data that are usually required for such a point.

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Enhancements to GPS operations and clock evaluations using a "total" Hadamard deviation

Cited by 26 publications

References 11 publications

Constrained expectation-maximization algorithm for stochastic inertial error modeling: study of feasibility

Constrained expectation-maximization algorithm for stochastic inertial error modeling: study of feasibility

The profound impact of negative power law noise on the estimation of causal behavior

ThêoH Bias-removal Method

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