Automatic detection of AutoPEEP during controlled mechanical ventilation
- Quang-Thang Nguyen^{1}Email author,
- Dominique Pastor^{1}Email author and
- Erwan L’Her^{2}
https://doi.org/10.1186/1475-925X-11-32
© Nguyen et al.; licensee BioMed Central Ltd. 2012
Received: 24 March 2012
Accepted: 5 June 2012
Published: 20 June 2012
Abstract
Background
Dynamic hyperinflation, hereafter called AutoPEEP (auto-positive end expiratory pressure) with some slight language abuse, is a frequent deleterious phenomenon in patients undergoing mechanical ventilation. Although not readily quantifiable, AutoPEEP can be recognized on the expiratory portion of the flow waveform. If expiratory flow does not return to zero before the next inspiration, AutoPEEP is present. This simple detection however requires the eye of an expert clinician at the patient’s bedside. An automatic detection of AutoPEEP should be helpful to optimize care.
Methods
In this paper, a platform for automatic detection of AutoPEEP based on the flow signal available on most of recent mechanical ventilators is introduced. The detection algorithms are developed on the basis of robust non-parametric hypothesis testings that require no prior information on the signal distribution. In particular, two detectors are proposed: one is based on SNT (Signal Norm Testing) and the other is an extension of SNT in the sequential framework. The performance assessment was carried out on a respiratory system analog and ex-vivo on various retrospectively acquired patient curves.
Results
The experiment results have shown that the proposed algorithm provides relevant AutoPEEP detection on both simulated and real data. The analysis of clinical data has shown that the proposed detectors can be used to automatically detect AutoPEEP with an accuracy of 93% and a recall (sensitivity) of 90%.
Conclusions
The proposed platform provides an automatic early detection of AutoPEEP. Such functionality can be integrated in the currently used mechanical ventilator for continuous monitoring of the patient-ventilator interface and, therefore, alleviate the clinician task.
Keywords
Introduction
Mechanical ventilation is routinely used in the clinical ward and/or in nursing/rehabilitation institutions. Unfortunately, imperfect interaction between patient and ventilator is frequently exhibited in intubated patients [1] and those undergoing non-invasive ventilation [2].
It has been demonstrated that the graphical curves (flow, airway pressure and air volume) available on most recent mechanical ventilators provide much information to analyze the patient-ventilator interface [3]. By visually monitoring these curves, patient-ventilator mismatching can be observed and detected by the clinician. Various automatic detection algorithms either embedded in a ventilatory system to detect ineffective triggering and double triggering [4], or recently in a computerized monitoring system (BetterCare) to determine ineffective respiratory efforts during expiration [5] have been reported with positive results. However, to the best of our knowledge, the automatic detection of other types of ventilatory abnormalities, including AutoPEEP, has not yet been adequately considered.
This paper addresses automatic detection of AutoPEEP, a common ventilatory abnormality that usually occurs in patients with acute severe asthma or chronic obstructive pulmonary disease. The presence of AutoPEEP basically indicates an insufficient expiratory time. The amount of time given over to expiration therefore needs to be lengthened, either by reducing the respiration rate or by decreasing the inspiratory time, or both. AutoPEEP can be measured at the patient’s bedside by using the pressure transducer of the ventilator. However, this quantification requires intervention from the therapist, who must perform an expiratory pause, in order to monitor tele-expiratory pressure [6]. On the contrary, although not readily quantifiable, AutoPEEP can easily be recognized on the expiratory portion of the flow waveform. If expiratory flow does not return to zero before the next inspiration, AutoPEEP is present. This detection however requires the eye of an expert clinician at the patient’s bedside. Using flow signal as the input, an automatic detection of AutoPEEP (dynamic hyperinflation) due to either expiratory flow limitation and/or inappropriate ventilatory cycling should be helpful to optimize care. Our focus is thus early detection of AutoPEEP for continuous monitoring of the patient-ventilator interface. In what follows, AutoPEEP detection is performed by Signal Norm Testing (SNT) on the flow signal captured from the patient-ventilator interface. SNT involves testing the norm of a signal observed in noisy condition with respect to a certain tolerance fixed by users on the basis of their know-how and/or experience of the domain [7]. An extension of SNT in a sequential framework is also investigated. Other practical aspects, including phase change detection and parameter estimation are considered as well. The performance assessment is provided in three levels. First, the detection performance of the proposed detectors will be illustrated with data synthesized on computer. Then, further evaluation is performed on data derived from a respiratory system analog. Finally, an ex-vivo performance assessment on retrospective data acquired from patients is carried out.
Methods
Automatic detection of AutoPEEP and System overview
Data acquisition and Serial/Parallel conversion
This very-first module acquires the discrete flow signal y _{ n } provided by the ventilator or by an independent flow sensor installed inside the air-tube during the mechanical ventilation. Although every flow datum is acquired, only end-expiration flow data of each breath is useful for the detection of AutoPEEP. When the end-expiration instant t _{ k } of the k-th breath is provided by the phase change detector, the Data Acquisition and Serial/Parallel conversion module will log L samples at the end of the expiratory phase to form the observation vector ${\mathbf{Y}}_{k}={[{y}_{{t}_{k}-L+1},{y}_{{t}_{k}-L+2},\dots ,{y}_{{t}_{k}}]}^{T}$ for the k-th breath. This output observation vector Y _{ k } is finally injected into the AutoPEEP detector module.
Respiration phase change detection
The main role of this module is to detect the end-expiration of each breath and provide this instant to trigger the data logging process and the Serial/Parallel conversion described above. This can also be regarded as a breath detector, which separates the continuous flow signal into different breaths.
Estimator
This module consists of two estimators, which estimate necessary parameters for the AutoPEEP detection algorithms. These parameters are the so-called waveform vector (p _{ k }for the k-th breath) and the noise standard deviation estimate ( $\widehat{\sigma}$). The waveform vector will be used to aggregate multi-samples at the end of the expiratory phase of a breath into a decision (cf. Section Single-breath detector), while the noise standard deviation estimate will be provided to adjust the AutoPEEP detector.
AutoPEEP detector
The AutoPEEP detector is the main core of the whole platform. Given a specified tolerance τ and the desired maximum false-alarm rate (level) γ, the AutoPEEP detector will decide whether an AutoPEEP is present or not for a given breath, on the basis of its observation Y _{ k } and estimated parameters ${\mathbf{p}}_{k},\widehat{\sigma}$.
AutoPEEP detectors
Given tolerance τ and observation y _{ n } of the noisy flow signal, the AutoPEEP detection is the testing of the null hypothesis $\left|{f}_{{t}_{k}}\right|\le \tau $ against the alternative one $\left|{f}_{{t}_{k}}\right|>\tau $. The SNT (Signal Norm Testing) problem introduced in [7] provides such a test. In this section, two AutoPEEP detectors are proposed. One is based directly on SNT and takes each of the breaths into account independently. The other one is an extension of SNT in a sequential framework. The latter detector is developed under the assumption that the state (AutoPEEP/NON-AutoPEEP) of the patient-ventilator interface is regular and remains the same within a certain number of breaths. This assumption usually holds in practice.
Single-breath detector
Signal Norm Testing
where θ is some unknown clean deterministic signal and z is its observation in noise. The additive noise x is assumed to be centered and gaussian with variance ${\sigma}_{\phantom{\rule{0.3em}{0ex}}x}^{2}$, i.e. $x\sim \mathcal{N}(0,{\sigma}_{\phantom{\rule{0.3em}{0ex}}x}^{2})$. Given observation z, SNT is the problem of testing the composite hypothesis h _{0}:|θ|≤τ versus its alternative h _{1}:|θ|>τ.
in which λ _{ γ }(ρ) is the unique solution in η to the equation 1−Φ(η−ρ)−Φ(−η−ρ)]=γ, where Φ(.) is the cumulative distribution of any standard normal random variable. Additionally, the test is UMPU (UMP unbiased) [7]. This thresholding test will be used for the detection of AutoPEEP, one of the most frequent abnormalities exhibited during mechanical ventilation.
Single-breath SNT-based AutoPEEP detector
where ${\mathbf{p}}_{k}={\left[{p}_{1}^{\left(k\right)}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{p}_{2}^{\left(k\right)}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\dots \phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}\phantom{\rule{2.77626pt}{0ex}}{p}_{L}^{\left(k\right)}\right]}^{T}$ is the waveform vector. It should be noted that ${p}_{L}^{\left(k\right)}=1$. This vector p _{ k } corresponds to the local form of the flow signal near the end of the expiratory phase. It is also worth mentioning that this local waveform vector p _{ k } depends mainly on the configuration of the interface, including the patient condition and the ventilator settings. As long as the interface stays unchanged, the waveform vector remains almost the same regardless whether or not an AutoPEEP might occur. In practice, either p _{ k } is known prior to the detection or it can be estimated from the observation using one of the methods proposed in Section Waveform regression to compute p _{ k }.
where ${u}_{k}={\mathbf{p}}_{k}^{T}{\mathbf{Y}}_{k}/\parallel {\mathbf{p}}_{k}{\parallel}^{2},\phantom{\rule{2.77626pt}{0ex}}{w}_{k}={\mathbf{p}}_{k}^{T}{\mathbf{X}}_{k}/\parallel {\mathbf{p}}_{k}{\parallel}^{2}$ and $\parallel {\mathbf{p}}_{k}{\parallel}^{2}={\mathbf{p}}_{k}^{T}{\mathbf{p}}_{k}$ is the L _{2} norm of waveform vector p _{ k }. By such proceeding, noise w _{ k } follows normal distribution with zero mean and variance ${\sigma}_{w}^{2}={\sigma}^{2}/\parallel {\mathbf{p}}_{k}{\parallel}^{2}$. According to [9, Theorem 27.4, p. 362] and equation (14), it can be proved that, even when the original noise is not gaussian, the resulting noise w _{ k } tends to a normally distributed random variable, as long as L is large enough and the original noise samples are i.i.d (independent and identically distributed). In practice, the i.i.d condition can be significantly relaxed. The problem in (4) is the same as that in previous section, except that the noise level is reduced (σ _{ w }≤ σ). Moreover, no information on the correlation among samples of noise vector X _{ k } is required. The two hypotheses are unchanged: ${h}_{0}:\left|{f}_{{t}_{k}}\right|\le \tau $ and ${h}_{1}:\left|{f}_{{t}_{k}}\right|>\tau $. The detection is thus carried out as follows. We decide that there is an AutoPEEP if $\left|{u}_{k}\right|>{\sigma}_{w}{\lambda}_{\gamma}\left(\frac{\tau}{{\sigma}_{w}}\right)$, where λ _{ γ }(.) is calculated as in (3). Otherwise, the considered breath is labeled with Non-AutoPEEP.
It should be noted that ∥p _{ k }∥ increases with respect to the number L of samples. The noise standard deviation σ _{ w } will thus decreases when more samples are taken into account. By reducing the noise standard deviation, the detection probability is improved while the false-alarm rate is still limited to the specified level γ. Theoretically, L is only limited by the length of expiratory phase. However, L must not be too long so that the local waveform vector can be considered stable and stays almost unchanged for a large number of breaths.
Sequential detector
SNT extension in sequential decision framework
for testing the hypothesis h _{0}:|θ| ≤ τ against the alternative one h _{1}:|θ| > τ. It is inherited from (7) and (8) that, once the decision has been made by (9), the probability of false-alarm (P _{FA}) is limited to the specified value γ (i.e. P _{FA} < γ) and the detection probability (P _{D}) is guaranteed to be higher than 1−γ(i.e. P _{D} > 1 −γ).
In a sequential framework, the test ${\mathcal{T}}^{\ast}\left(z\right)$ is firstly carried out to attempt a decision based on current observation z. If ${\mathcal{T}}^{\ast}\left(z\right)$ returns 1 or 0 then the decision is made. The value returned by ${\mathcal{T}}^{\ast}\left(z\right)$ is the index of the accepted hypothesis. Otherwise, the decision cannot be made yet since current observation z does not provide enough evidence to either accept or reject any of the two hypotheses. More data are required. The decision is then postponed until enough evidence has been collected. For AutoPEEP detection, the process is detailed below.
Sequential SNT-based AutoPEEP detector
It is worth mentioning that ${w}_{1:K}\sim \mathcal{N}(0,{\sigma}_{w,K}^{2})$ and that ${\sigma}_{w,K}=\frac{{\sigma}_{w}}{\sqrt{K}}$ is strictly decreasing with the number K of breaths used.
where ${\lambda}_{1:K}^{\left(h\right)}>{\lambda}_{1:K}^{\left(\ell \right)}$ for any 0<γ<0.5.
If the decision cannot be made yet (i.e. ${\lambda}_{1:1}^{\left(\ell \right)}\le \left|{u}_{1:1}\right|\le {\lambda}_{1:1}^{\left(h\right)}$), it will be delayed until the next observation (i.e. u _{2}) is obtained and the test is performed based on u _{1:2} using ${\mathcal{T}}^{\ast}\left({u}_{1:2}\right)$. If the decision still cannot be performed, it will be delayed again until the next observation, where the test ${\mathcal{T}}^{\ast}\left({u}_{1:3}\right)$ is used. The process is iterated until the decision is made. Then the process is restarted for a new sequence of observations.
Phase change detection
where 2h + 1 is the length of the moving window.
with threshold height ${\lambda}_{\text{SNT}}={\sigma}_{D}{\lambda}_{\gamma}\left({\lambda}_{u}\left(N\right)/{\sigma}_{D}\right)$ where λ _{ γ }(ρ) is defined as in Section Single-breath detector. Level γ is set to be very small, for example γ = 10^{−4},10^{−5},10^{−7}, etc. Since a peak is only one point, the results of the thresholding test should be post-processed in such a way that consecutive 1s are removed. In particular, in case of consecutive decisions equal to 1, only the first one will be kept. End-expirations are negative peaks.
Estimation
As aforementioned, some estimations have to be made prior to the AutoPEEP detection, including: the waveform vector (p _{ k }) and the standard deviation of the unknown noise. In the following, these two estimations will be addressed.
Waveform regression to compute p _{ k }
According to Section Single-breath detector, waveform vector p _{ k } concerns the current (k-th) breath. However, as aforementioned, this waveform vector does not vary much. It is then sensible to use estimates from previous breaths so as to improve the estimation of p _{ k }. In this respect, the following strategies can be considered to compute the waveform vector estimate to be used in AutoPEEP detectors:
This waveform vector will be updated each time the machine is tuned or after a verification session by the clinician. One may also want to update the estimation on a regular time basis.
Estimation of the noise standard deviation
Noise is unknown in practice. As long as the noise standard deviation in concerned, it must be estimated from the observation. In this work, we propose two solutions: one based directly on the result obtained by waveform regression, whereas the other is based on an estimation from the wavelet coefficients of the flow signal.
Estimate from regression
To aggregate $\widehat{\sigma}$ from ${\widehat{\sigma}}_{k}$, the same strategies as those proposed for the waveform vector can be considered.
Estimation from wavelet coefficients
knowing that med_{ i } c _{ i }= 0.
It has been shown in [17] that this estimator outperforms the MAD when the number of outliers increases. The DATE can thus be employed as an alternative to the MAD mentioned above in such situation. For the cases considered in this work, because the number of large wavelet coefficients pertaining to signal remains small, the two estimators yield similar performance. The MAD estimator is thus adopted for its lower complexity and higher rapidity.
Results and discussion
Simulations
Emulations with a respiratory system analog
AutoPEEP detection results provided by the proposed detectors on emulated flow data
Parameters | True | N. of | Det. by SNT ^{c} | Det. by Sequential SNT ^{c} | ||||||
---|---|---|---|---|---|---|---|---|---|---|
Id | Ventilator ^{a} | Lung model ^{b} | Label | breaths | P | N | Label | P | N | Label |
1 | PEP=0, Vt=500, f=15, P=0, I:E=1:2 | C=80, R=5 | N | 21 | 0 | 21 | N | 0 | 21 | N |
2 | PEP=0, Vt=500, f=15, P=0, I:E=1:2 | C=30, R=5 | N | 20 | 0 | 20 | N | 0 | 20 | N |
3 | PEP=0, Vt=500, f=25, P=0, I:E=1:2 | C=80, R=5 | P | 33 | 33 | 0 | P | 33 | 0 | P |
4 | PEP=0, Vt=500, f=25, P=0, I:E=1:1 | C=80, R=5 | P | 34 | 34 | 0 | P | 34 | 0 | P |
5 | PEP=0, Vt=300, f=20, P=0, I:E=1:2 | C=80, R=5 | N | 27 | 0 | 27 | N | 0 | 27 | N |
6 | PEP=0, Vt=500, f=12, P=0, I:E=1:2 | C=80, R=5 | N | 16 | 0 | 16 | N | 0 | 16 | N |
7 | PEP=0, Vt=500, f=20, P=15, I:E=1:3 | C=80, R=5 | N | 27 | 0 | 27 | N | 0 | 27 | N |
8 | PEP=5, Vt=500, f=20, P=0, I:E=1:3 | C=80, R=5 | N | 27 | 0 | 27 | N | 0 | 27 | N |
9 | PEP=5, Vt=500, f=20, P=0, I:E=1:2 | C=120, R=10 | P | 27 | 27 | 0 | P | 27 | 0 | P |
10 | PEP=0, Vt=700, f=20, P=0, I:E=1:2 | C=120, R=10 | P | 27 | 27 | 0 | P | 27 | 0 | P |
11 | PEP=0, Vt=700, f=20, P=0, I:E=1:6 | C=120, R=10 | P | 24 | 24 | 0 | P | 24 | 0 | P |
12 | PEP=0, Vt=700, f=20, P=0, I:E=1:1 | C=120, R=10 | P | 27 | 27 | 0 | P | 27 | 0 | P |
13 | PEP=0, Vt=700, f=20, P=0, I:E=1:2 | C=140, R=25 | P | 13 | 13 | 0 | P | 13 | 0 | P |
Analysis of clinical data
For further evaluation, the AutoPEEP detectors were tested ex-vivo on various patient curves. These curves were retrospectively extracted from data files issued from the Medical Intensive Care Unit of Brest University Hospital, France and from the Institut Universitaire de Cardiologie et de Pneumologie de Québec, Canada. For each patient undergoing mechanical ventilation, the flow signal was recorded. All these data were then mixed up to form a unique dataset. In total, the final dataset contains 1998 breaths from 15 patients with different health conditions and different treatments. The parameters of the ventilator also varied depending on the situation. According to the retrospective aspect of the study and to the fact that the files were anonymized, the study was considered to be in accordance with French legislation by our local ethics committee.
The analysis was performed both manually by a set of experts and automatically by the proposed methods. On the one hand, each breath was carefully screened by two experts of the domain. They performed a dual analysis, separately, before confronting their points of view and delivering a final assessment of the data. For each breath of the dataset, their decision was then regarded as the ground-truth label (AutoPEEP/NON-AutoPEEP). On the other hand, the proposed detectors were used to predict the label of every breath of the dataset. The two analyses were carried out independently and anonymously. The results were then compared together to evaluate the detection performance of the proposed methods.
In these experiments, the tolerance was set to τ = 2 [l/min] as before. In this respect, the dataset includes 1383 breaths with AutoPEEP and 615 breaths with NON-AutoPEEP. The dataset is somehow unbalanced with the presence of AutoPEEP in 69% of the cases. For the proposed detectors, level γ was set to 0.01 as usual. Figure 7 presents a typical case with the regression at end-expiration and the corresponding detection. It can be seen that the detection algorithm can precisely reveal the true label for all the breaths.
Detection performance with flow data from patients
Measure | Single-breath SNT-based detector | Sequential SNT-based detector |
---|---|---|
Accuracy | 93.09% | 93.09% |
Precision | 99.44% | 99.37% |
Recall | 90.53% | 90.60% |
Specificity | 98.86% | 98.70% |
Conclusion
To the best of our knowledge, this is the first work on the automatic detection of AutoPEEP for continuous monitoring of the patient-ventilator interface during controlled mechanical ventilation. With the introduction of the waveform vector to aggregate multiple samples into a unique decision, the SNT has been successfully applied to provide a good AutoPEEP detector. Finally, we have extended SNT in a sequential framework, namely Sequential SNT. The resulting sequential AutoPEEP detector has been shown to yield high detection performance. Besides, the proposed algorithms have very low complexity and require very little computational power. The platform can then be deployed as a real-time functional block.
Although the algorithm is proposed for the detection of AutoPEEP during controlled mechanical ventilation, it could be extended to assisted mechanical ventilation and pressure support ventilation since the algorithm investigates the expiratory part of the flow curve, which mainly depends on characteristics of the patient rather than on the ventilatory settings and mode of ventilation. The platform may also be extended to the detection of other types of ventilatory abnormalities that are deviations of the observed signal from some reference. In this respect, other signals such as pressure and volume curves could also be taken into account.
For the present work, by using the retrospective data files with a double-blinded and dual expert analysis, we were able to assess whether the system automatic analysis was concordant with that of the experts. In the next validation step, continuous and prospective recordings of the curves will be carried out to get better insight into cases where any disagreement between the proposed system and the therapist might occur. Furthermore, it is also worth performing a semi-closed-loop analysis, in which the therapist supervises, validates the decisions yielded by the proposed platform and adjusts the ventilatory parameters to correct any possible abnormality.
The deviation detection approach proposed in this paper is very general and could be used in many other applications, including fault detection and structural health monitoring. A theoretical general approach in Sequential SNT should also be investigated.
Declarations
Acknowledgements
The authors are very grateful to François Lellouche, Département de médecine - Institut universitaire de cardiologie et de pneumologie de Québec - Université Laval, Québec, Canada for his help while collecting data and for his valuable remarks without which this work might have never been done.
Authors’ Affiliations
References
- Thille A, Rodriguez P, Cabello B, Lellouche F, Brochard L: Patient-ventilator asynchrony during assisted mechanical ventilation. Intensive Care Medicine 2006, 32: 1515–1522. http://dx.doi.org/10.1007/s00134–006–0301–8 10.1007/s00134-006-0301-8View ArticleGoogle Scholar
- Vignaux L, Vargas F, Roeseler J, Tassaux D, Thille A, Kossowsky M, Brochard L, Jolliet P: Patient-ventilator asynchrony during non-invasive ventilation for acute respiratory failure: a multicenter study. Intensive Care Medicine 2009, 35: 840–846. http://dx.doi.org/10.1007/s00134–009–1416–5 10.1007/s00134-009-1416-5View ArticleGoogle Scholar
- Roeseler J, Michotte JB, Sottiaux T: Patient-ventilator asynchrony during pressure support: Usefulness of the curves displayed by the ventilator. Réanimation 2010, 19: 62–65.View ArticleGoogle Scholar
- Mulqueeny Q, Ceriana P, Carlucci A, Fanfulla F, Delmastro M, Nava S: Automatic detection of ineffective triggering and double triggering during mechanical ventilation. Intensive Care Med 2007, 33: 2014–2018. 10.1007/s00134-007-0767-zView ArticleGoogle Scholar
- Blanch L, Sales B, Montanya J, Lucangelo U, Garcia-Esquirol O, Villagra A, Chacon E, Estruga A, Borelli M, Burgueño MJ, Oliva JC, Fernandez R, Villar J, Kacmarek R, Murias G: Validation of the Better Care(Ⓡ) system to detect ineffective efforts during expiration in mechanically ventilated patients: a pilot study. Intensive Care Med 2012, 38: 772–780. 10.1007/s00134-012-2493-4View ArticleGoogle Scholar
- Blanch L, Bernabé F, Lucangelo U: Measurement of air trapping, intrinsic positive end-expiratory pressure, and dynamic hyperinflation in mechanically ventilated patients. Respiratory Care 2005, 50: 110–123.Google Scholar
- Pastor D: Signal norm testing in additive standard Gaussian noise. Technical Report RR-2011001-SC, Institut Télécom, Télécom Bretagne 2011.Google Scholar
- Lehmann E, Romano J: Testing Statistical Hypotheses. 3rd edition. New York: Springer; 2005.MATHGoogle Scholar
- Billingsley P: Probability and Measure. 3rd edition. New York: John Wiley & Sons; 1995.MATHGoogle Scholar
- Serfling R: Approximation theorems of mathematical statistics. New York: John Wiley & Sons; 1980.View ArticleMATHGoogle Scholar
- Berman S: Sojourns and extremes of stochastic processes. California: Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove; 1992.MATHGoogle Scholar
- Donoho DL, Johnstone IM: Ideal Spatial Adaptation by Wavelet Shrinkage. Biometrika 1994, 81(3):425–455. 10.1093/biomet/81.3.425MathSciNetView ArticleMATHGoogle Scholar
- Pastor D, Atto A: Wavelet shrinkage: from sparsity and robust testing to smooth adaptation. In Recent Developments in Fractals and Related Fields. Edited by: Barral J, Seuret S. Boston: Birkhäuser; 2010.Google Scholar
- Seber G, Wild C: Nonlinear Regression. Hoboken, NJ: Wiley-Interscience; 2003.Google Scholar
- Hampel FR: The Influence Curve and Its Role in Robust Estimation. Journal of the American Statistical Association 1974, 69(346):383–393. http://www.jstor.org/stable/2285666 10.1080/01621459.1974.10482962MathSciNetView ArticleMATHGoogle Scholar
- Rousseeuw PJ, Croux C: Alternatives to the Median Absolute Deviation. Journal of the American Statistical Association 1993, 88(424):1273–1283. http://www.jstor.org/stable/2291267 10.1080/01621459.1993.10476408MathSciNetView ArticleMATHGoogle Scholar
- Pastor D, Socheleau FX: Robust Estimation of Noise Standard Deviation in Presence of Signals With Unknown Distributions and Occurrences. Signal Processing, IEEE Transactions on 2012, 60(4):1545–1555.MathSciNetView ArticleGoogle Scholar
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