# Random Distortion Testing (RDT)

The Random Distortion Testing (RDT) approach was introduced in [RDT] for the fixed sample size (FSS) case, that is, when the testing must be achieved once for all on the basis of a given multi-dimensional observation. It was extended in [Subspace], [BlockRDT] and [RDTlm], yet for observations with FSS before addressing sequential testing in [SeqRDT] [TSeqRDT]. This theoretical corpus provides an alternative framework to the standard likelihood theory for statistical hypothesis testing problems, with FSS or sequential, where the lack of prior knowledge on the observations does not make it possible to calculate likelihood ratios.

[RDT] Dominique Pastor, Quang-Thang Nguyen (2013). Random Distortion testing and optimality of thresholding tests. IEEE Transactions on Signal Processing, 61(16):4161-4171

[Subspace] François-Xavier Socheleau, Dominique Pastor (2014). Testing the Energy of Random Signals in a Known Subspace: An Optimal Invariant Approach. IEEE Signal Processing Letters, 21(10):1182-1186

[BlockRDT] D. Pastor and Q. T. Nguyen, “Robust statistical process control in Block-RDT framework,” in 2015 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), April 2015, pp. 3896–3900

[RDTlm] Dominique Pastor, François-Xavier Socheleau (2018). Random distortion testing with linear measurements. Signal processing, 145:116-126

[SeqRDT] Khanduri P., Pastor D., Sharma V. K., Varshney P. (2019). Sequential Random Distortion Testing of Non-Stationary Processes. IEEE Transactions on Signal Processing, 67 (21): 5450-5462, https://doi.org/10.1109/TSP.2019.2940124

[TSeqRDT] Khanduri P., Pastor D., Sharma V. K., Varshney P. (2019). Truncated Sequential Non-Parametric Hypothesis Testing Based on Random Distortion Testing. IEEE Transactions on Signal Processing, 67(15):4027-4042. DOI: https://doi.org/10.1109/TSP.2019.2923140

In a nutshell, we can say that [RDT] is an extension of the notion of test with Uniformly Best Constant Power (UBCP) proposed by Abraham Wald in 'Tests of statistical hypotheses concerning several parameters when the number of observations is large' (1943) to the case of random signals with unknown distributions and observed in independent Gaussian noise.

In turn, [Subspace] extends the RDT approach to the case when a projection of the signal in a certain subspace is observed in presence of independent Gaussiaon noise. Scharf and Friedlander’s matched subspace detector in 'Matched Subspace Detectors' (1994) is a particular case of the results established in [Subspace].

For observations with FSS, [RDTlm] encompasses all the foregoing results in one single framework that makes also possible to prove the optimality of the Generalized Lilekihood Ratio Test for testing the amplitude level of a waveform in noise.

The RDT formulation makes it possible to address the change-in-mean detection problem in [BlockRDT]. [BlockRDT] requires no iid assumption and no prior knowledge on the distributions of the observations before and after change. It's an alternative to Shewhart charts and outperforms the latter when possible model mismatches may cause likelihood theory to fail.

In [SeqRDT], we extend [RDT] to non-parametric sequential testing. As above, the purpose is to test whether or not a random signal observed in independent and identically distributed (i.i.d) additive noise deviates by more than a speciﬁed tolerance from a ﬁxed value. A preliminary version of this algorithm with its application in biomedical engineering can be found in 'Contributions to statistical signal processing with applications in biomedical engineering' (2012) by Q.T. Nguyen, whose PhD I supervised, and in [MV1-MV5] cited below. However, this first version doesn't guarantee upper bounds on the probabilities of false alarm (PFA) and missed detection (PMD). In contrast, our algorithm [SeqRDT] can achieve arbitrarily small upper bounds for PFA and PMD. In comparison to [RDT], [SeqRDT] doesn't require noise to be Gaussian. Simulations show that [SeqRDT] leads to faster decision-making on an average compared to its FSS counterpart [BlockRDT] and that [SeqRDT] is robust to model mismatches compared to the Wald's Sequential Probability Ratio Test (SPRT).

[TSeqRDT] is a truncated sequential algorithm based on [RDT]. By 'truncated', we mean that the new algorithm, [TSeqRDT], is asked to achieve its decision within a certain time interval. If no decision is attained at the end of this interval, the decision is forced by using the optimal FSS counterpart, [BlockRDT]. [TSeqRDT] requires fewer assumptions on the signal model than [SeqRDT], while guaranteeing the error probabilities to be below pre-speciﬁed levels. At the same time, [TSeqRDT] makes a decision faster than [BlockRDT]. However, unlike [SeqRDT], noise is required to be Gaussian in [TSeqRDT]. [TSeqRDT] is experimentally shown to be robust in comparison to standard likelihood ratio based procedures, in particular SPRT.

# The RDT formulation vs. standard ones

The brief outline above roughly positions the RDT theoretical corpus with respect to standard statistical hypothesis testing. However, the RDT approach, by essence, differs significantly from the usual way to state statisticsl hypothesis testing problems. This discrepancy has not been enhanced clearly in our former papers, simply because we didn't notice it. Hence, I pinpoint these differences below with some formal arguments. More details can be found by the interested reader in the Matlab notebook Introduction to RDT_v4 that you can dowload from the tab Shared files. The mathematical details about the optimality of the approach are not provided yet in this notebook or even below, since they wouldn't bring much added-value with respect to the contents of our papers.

# Some applications

The RDT approach originates from two different applications: radar processing and physiological signal monitoring (anomaly detection) in intensive care units (ICUs). These two applications, and more particularly the detection of anomalies in physiological signals, strongly motivated my research and the development of the RDT theory.

1) In radar processing, I had the opportunity to work on the detection of radar targets in presence of sea clutter. The interested reader can refer to Laurent Dejean's PhD dissertation or my HDR (Habilitation à Diriger des Recherches/Accreditation to supervise research) dissertation, both downlable from the 'Shared files' page. In short, the problem is that, in presence of sea clutter, you always an interference (the sea clutter) even in absence of any radar target. In other words, the model summarized by Eq. (1) above does not apply because, under the null hypothesis, you don't have noise alone but noise and clutter. And under the alternative hypothesis, you don't have signal + noise only but signal + noise + ... clutter! One way to cope with this situation is to introduce a statistical model for the clutter (that's the approach followed in Dejean's PhD) and to resort to likelihood ratio tests. When I was with Signaal in the Netherlands, I already worked on the same topic. One of my conclusions was that it was rather intricate to come up with a statistical model whose parameters are sufficiently well known to adjust a dedicated detector of the target in sea clutter: “even if the model is reasonably good, our knowledge of the parameters in it, [...] may not be enough to justify a direct numerical evaluation of formulas derived from the model.” [Kailath, 1999]. The RDT approach is thus an alternative to likelihood ratio tests because it does not require prior knowledge on the clutter distribution. Instead, it will assume that the clutter amplitude is upper-bounded via the tolerance. This application of RDT is treated in [RDT, Section VI].

2) In mechanical ventilation, you actively insufflate air into the patient's lungs. The monitoring system then lets the lungs empty in a passive way and during an expiratory time specified by the clinician. At the end of this expiratory time, the expiratory flow is expected to return to zero and a new cycle of mechanical ventilation starts. Unfortunately, if the time given to the expiratory is insufficient, the system will prematurely re-insufflate air into the not yet empty patient's lungs, which may induce dynamic hyperinflation.

This real expiratory flow is far from 0 at the end of the expiratory time and this may entail dynamic hyperinflation. Excerpt from 'Contributions to statistical signal processing with applications in biomedical engineering', Q.T. Nguyen, PhD dissertation, 2012

The early detection of a non-zero expiratory air flow at the end of the expiratory phase and before the beginning of new cycle is thus helpful to avoid dynamic hyperinflation and thus, to optimize care. But the problem does not boil down to deciding whether the expiratory flow is zero or not. The problem is more difficult than that. Indeed, the measure of the expiratory flow is performed in presence of noise (electronic noise, noise in the pipework). In addition, and that's crucial, can we really expect a null expiratory flow? No! The expiratory flow will often be non-null in practice at the end of the cycle, even when the expiratory time is well-tuned. Therefore, a way to proceed is to introduce an interval [\tau,0], within which the expiratory flow should remain in absence of hyperinflation. An alarm should therefore be triggered if the expiratory flow is outside this interval at the end of the cycle. It behooves to the clinician to fix the tolerance \tau on the basis of his/her experience. The detection of dynamic hyperinflation is an RDT problem (see Eq. (2)). Indeed, this practical problem motivated strongly all the development of the RDT approach. In this respect, the interested reader can refer to the following papers dedicated to the monitoring of physiological signals.

[MV1] Nguyen Q.-T., Pastor D., L'her E. (2012). Automatic detection of AutoPEEP during controlled mechanical ventilation. BioMedical Engineering OnLine, 11(1):32. DOI: https://doi.org/10.1186/1475-925X-11-32

[MV2] Nguyen Q. T., Pastor D., L'her E. (2012). Patient-ventilator asynchrony: automatic detection of autopeep, ICCASSP 2012: 7th International Conference on Acoustics, Speech, and Signal Processing, 25-30 mars 2012, Kyoto (Japon). Proceedings ICCASSP 2012: 7th International Conference on Acoustics, Speech, and Signal Processing.

[MV3] L'her E., Nguyen Q. T., Lellouche F., Pastor D. (2012). Automatic flow Curves analysis during mechanical ventilation (CURVEX): application to intrinsic PEEP Detection, ATS 2012, 18-23 mai 2012, San Francisco (États-Unis). Proceedings ATS 2012.

[MV4] L'her E., Lellouche F., Nguyen Q. T., Pastor D. (2013). Automatic dynamic hyperinflation and asynchrony detection during mechanical ventilation using the random-distorsion test, ESICM LIVES 2013: 26th annual congress of the European Society of Intensive Care Medicine, 5-9 octobre 2013, Paris (France).

[MV5] Nguyen Q. T., Pastor D., Lellouche F., L'her E. (2013). Mechanical ventilation system monitoring: Automatic detection of dynamic hyperinlation and asynchrony, EMBC 2013: 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 3-7 juillet 2013, Osaka (Japon). Proceedings EMBC 2013: 35th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 5207 - 5210. DOI: https://doi.org/10.1109/EMBC.2013.6610722

[PPG] Cherif S., Pastor D., Nguyen Q. T., L'her E. (2016). Detection of Artifacts on Photoplethysmography Signals Using Random Distortion Testing, EMBC 2016: 38th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 16 août-20 septembre 2016, Orlando (États-Unis). Proceedings EMBC 2016: 38th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 6214 - 6217. DOI: https://doi.org/10.1109/EMBC.2016.7592148

[MV1-MV5] are dedicated to the detection of dynamic hyperinflations and other anomalies in expiratory flow signals during mechanical ventilation. In [PPG], we address the detection of artifacts in PPG signals by RDT. In all these papers, the detections are performed by RDT. An issue in these applications of the RDT approach is noise estimation and the choice of the model from which deviations must be detected. In [MV1-MV5], the choice of the model derives from a theoretical model of the expiratory flow signal. The noise standard deviation is obtained by robust estimation via the MAD estimator on the detail coefficients of the wavelet decomposition. In [PPG], the same issues arise. In contrast to [MV1-MV5], the model is estimated as the mean of real signals and the noise standard deviation is approximated by the empirical standard deviation. The tolerance is used to make up for the coarseness of these estimations.

In [Weak sparseness in speech denoising], we apply a shrinkage function to the time-frequency coefficients of some noisy speech signal to denoise it. This shrinkage function depends on a threshold aimed at discriminating components with large amplitude from components with small amplitude. The former are likely to pertain to the speech signal, whereas the latter can be expected to derive from noise alone or speech of poor interest. The shrinkage functions tends to attenuate coefficients with small amplitude and to keep unaltered the components exceeding the threshold (the larger the amplitude, the lesser the attenuation and the smaller the amplitude, the stronger the attenuation). Separating components with large amplitudes from components with small amplitudes can be regarded as detection problem and the RDT framework can thus be employed to adjust this threshold. The RDT framework avoids modeling the distributions of the speech signal time-frequency components and the tolerance plays the role of the least amplitude expected for the speech signal time-frequency components.

[Weak sparseness in speech denoising] V. K. Mai, D. Pastor, A. Aissa El Bey, R. Le Bidan (2018). Semi-Parametric Joint Detection and Estimation for Speech Enhancement based on Minimum Mean Square Error. Speech Communication, 102:27-38. https://doi.org/10.1016/j.specom.2018.05.005