# Random Distortion Testing (RDT)

The Random Distortion Testing (RDT) provides an alternative theoretical framework to the standard likelihood theory for statistical hypothesis testing problems, either in a Fixed Sample Size (FSS) setting or in a sequential setting, when the lack of prior knowledge on the observations does not make it possible to calculate likelihood ratios.

The RDT approach was introduced in Random Distortion testing and optimality of thresholding tests (2013), authored byDominique Pastor & Quang-Thang Nguyen and published in IEEE Transactions on Signal Processing, 61(16):4161-4171. It was then extended through several papers, of which a brief description is given below. Asymptotic RDT is the latest extension of the RDT approach and was established in Guillaume Ansel's PhD. During his defence, which held on Wednesday the 17th of February 2021, Guillaume made a very nice presentation of the RDT approach and the extension he has provided. So, have a look at this defence (sorry, in french) whose scope was initially dedicated to cyber-security. You'll see how the RDT approach and its extensions can be employed in this active and crucial application field. A short video in english presenting the main principles of the RDT approaches and its extensions should follow.

Meanwhile, here a few words about the approach. 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.

​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 RDT 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].

References:

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

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

[Subspace] F.-X. Socheleau, D. 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] D. Pastor, F.-X. Socheleau (2018). Random distortion testing with linear measurements. Signal processing, 145:116-126

[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.

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. 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.