Optimal discretization of inverse problems in Hilbert scales. Regularization and self-regularization of projection methods

We study the efficiency of the approximate solution of ill-posed problems, based on discretized observations, which we assume to be given afore-hand. We restrict ourselves to problems which can be formulated in Hilbert scales. Within this framework we shall quantify the degree of ill-posedness, provide general conditions on projection schemes to achieve the best possible order of accuracy. We pay particular attention on the problem of self-regularization vs. Tikhonov regularization. Moreover, we study the information complexity. Asymptotically, any method, which achieves the best possible order of accuracy must use at least such amount of noisy observations. We accomplish our study with two specific problems, Abel's integral equation and the recovery of continuous functions from noisy coefficients with respect to a given orthonormal system, both classical ill-posed problems

The discretized discrepancy principle under general source conditions

AbstractWe discuss adaptive strategies for choosing regularization parameters in Tikhonov–Phillips regularization of discretized linear operator equations. Two rules turn out to be based entirely on data from the underlying regularization scheme. Among them, only the discrepancy principle allows us to search for the optimal regularization parameter from the easiest problem. This potential advantage cannot be achieved by the standard projection scheme. We present a modified scheme, in which the discretization level varies with the successive regularization parameters, which has the advantage, mentioned before

Optimal discretization and Degrees of ill-posedness for inverse estimation in Hilbert scales in the presence of random noise

The problem of minimizing the difficulty of the inverse estimation of some unknown element x0 from noisy observations yδ = Ax0 + δξ in dependence of the nature of the random noise ξ is considered. It is shown that a combination of a Tikhonov regularization estimator with a certain projection scheme is order optimal in the sense of difficulty for a wide class of operators A acting along Hilbert scales

Regularization of statistical inverse problems and the Bakushinskii veto

In the deterministic context Bakushinskii's theorem excludes the existence of purely data driven convergent regularization for ill-posed problems. We will prove in the present work that in the statistical setting we can either construct a counter example or develop an equivalent formulation depending on the considered class of probability distributions. Hence, Bakushinskii's theorem does not generalize to the statistical context, although this has often been assumed in the past. To arrive at this conclusion, we will deduce from the classic theory new concepts for a general study of statistical inverse problems and perform a systematic clarification of the key ideas of statistical regularization.Comment: 20 page

Patients in a permanent vegetative state or minimally conscious state in the Maine-et-Loire county of France: A cross-sectional, descriptive study

PurposesTo determine how many patients in a permanent vegetative state or a minimally conscious state are living in healthcare institutions in the Maine-et-Loire county of western France. To evaluate patient management, physical complications, problems encountered by nursing staff and the patient care teams’ wishes. Patients and methods We performed a cross-sectional, descriptive study in physical medicine and rehabilitation departments, nursing homes, geriatric units and local hospitals. All patients and their medical records were examined by the same investigator. A questionnaire for carers was used to evaluate nursing tasks and a second questionnaire for head nurses served to assess staff needs and the patient care teams’ wishes. Results Thirteen patients were identified. Four were in a permanent vegetative state and nine were in a minimally conscious state. Ten patients were cared for in geriatric units, one in a physical medicine and rehabilitation department and two in local hospitals. All patients displayed limited joint angle ranges. All the patient care teams reported practical difficulties and ethical issues. Discussion Our survey highlighted the variety of care scenarios for patients in a permanent vegetative state or a minimally conscious state. It revealed practical difficulties and, above all, ethical questions. The present work could serve as a basis for implementation of a recently issued French government circular on defining specific wards for these patients

General regularization schemes for signal detection in inverse problems

The authors discuss how general regularization schemes, in particular linear regularization schemes and projection schemes, can be used to design tests for signal detection in statistical inverse problems. It is shown that such tests can attain the minimax separation rates when the regularization parameter is chosen appropriately. It is also shown how to modify these tests in order to obtain (up to a $\log\log$ factor) a test which adapts to the unknown smoothness in the alternative. Moreover, the authors discuss how the so-called \emph{direct} and \emph{indirect} tests are related via interpolation properties

Duration learning for analysis of nanopore ionic current blockades

<p>Abstract</p> <p>Background</p> <p>Ionic current blockade signal processing, for use in nanopore detection, offers a promising new way to analyze single molecule properties, with potential implications for DNA sequencing. The alpha-Hemolysin transmembrane channel interacts with a translocating molecule in a nontrivial way, frequently evidenced by a complex ionic flow blockade pattern. Typically, recorded current blockade signals have several levels of blockade, with various durations, all obeying a fixed statistical profile for a given molecule. Hidden Markov Model (HMM) based duration learning experiments on artificial two-level Gaussian blockade signals helped us to identify proper modeling framework. We then apply our framework to the real multi-level DNA hairpin blockade signal.</p> <p>Results</p> <p>The identified upper level blockade state is observed with durations that are geometrically distributed (consistent with an a physical decay process for remaining in any given state). We show that mixture of convolution chains of geometrically distributed states is better for presenting multimodal long-tailed duration phenomena. Based on learned HMM profiles we are able to classify 9 base-pair DNA hairpins with accuracy up to 99.5% on signals from same-day experiments.</p> <p>Conclusion</p> <p>We have demonstrated several implementations for <it>de novo </it>estimation of duration distribution probability density function with HMM framework and applied our model topology to the real data. The proposed design could be handy in molecular analysis based on nanopore current blockade signal.</p

Additional predictors of stroke and transient ischaemic attack in BEFAST positive patients in out-of-hours emergency primary care

INTRODUCTION: In patients suspected of stroke or transient ischemic attack (TIA), rapid triaging is imperative to improve clinical outcomes. For this purpose, balance-eye-face-arm-speech-time (BEFAST) items are used in out-of-hours primary care (OHS-PC). We explored the risk of stroke and TIA among BEFAST positive patients calling to the OHS-PC, and assessed whether additional predictors could improve risk stratification. METHODS: This is a cross-sectional study of retrospectively gathered routine care data from telephone triage tape-recordings of patients calling the OHS-PC with neurological deficit symptoms, classified as BEFAST positive. Four models-with the predictors age, sex, a history of cardiovascular or cerebrovascular disease, and cardiovascular risk factors-were fitted using logistic regression to predict the outcome stroke or TIA. Likelihood ratio testing was used to select the best model, which was subsequently internally validated. RESULTS: The risk of stroke or TIA diagnosis was 52% among 1,289 BEFAST positive patients, median age 72 years, 56% female sex. Of patients with the outcome stroke/TIA, 24% received a low urgency allocation, while 92% had signs or symptoms when calling. Only the addition of age and sex improved predicting stroke or TIA (internally validated c-statistic 0.72, 95%CI 0.69-0.75). The predicted risk of stroke or TIA remained below 20% in those aged below 40. Females aged 70 or over and males aged 55 or over, had a predicted risk above 50%. DISCUSSION: Urgency allocation appears to be suboptimal in BEFAST positive patients calling the OHS-PC. Risk stratification could be improved in this setting by adding age and sex

Expression Profiling of a Genetic Animal Model of Depression Reveals Novel Molecular Pathways Underlying Depressive-Like Behaviours

The Flinders model is a validated genetic rat model of depression that exhibits a number of behavioural, neurochemical and pharmacological features consistent with those observed in human depression.In this study we have used genome-wide microarray expression profiling of the hippocampus and prefrontal/frontal cortex of Flinders Depression Sensitive (FSL) and control Flinders Depression Resistant (FRL) lines to understand molecular basis for the differences between the two lines. We profiled two independent cohorts of Flinders animals derived from the same colony six months apart, each cohort statistically powered to allow independent as well as combined analysis. Using this approach, we were able to validate using real-time-PCR a core set of gene expression differences that showed statistical significance in each of the temporally distinct cohorts, representing consistently maintained features of the model. Small but statistically significant increases were confirmed for cholinergic (chrm2, chrna7) and serotonergic receptors (Htr1a, Htr2a) in FSL rats consistent with known neurochemical changes in the model. Much larger gene changes were validated in a number of novel genes as exemplified by TMEM176A, which showed 35-fold enrichment in the cortex and 30-fold enrichment in hippocampus of FRL animals relative to FSL.These data provide significant insights into the molecular differences underlying the Flinders model, and have potential relevance to broader depression research

MetWAMer: eukaryotic translation initiation site prediction

<p>Abstract</p> <p>Background</p> <p>Translation initiation site (TIS) identification is an important aspect of the gene annotation process, requisite for the accurate delineation of protein sequences from transcript data. We have developed the MetWAMer package for TIS prediction in eukaryotic open reading frames of non-viral origin. MetWAMer can be used as a stand-alone, third-party tool for post-processing gene structure annotations generated by external computational programs and/or pipelines, or directly integrated into gene structure prediction software implementations.</p> <p>Results</p> <p>MetWAMer currently implements five distinct methods for TIS prediction, the most accurate of which is a routine that combines weighted, signal-based translation initiation site scores and the contrast in coding potential of sequences flanking TISs using a perceptron. Also, our program implements clustering capabilities through use of the <it>k</it>-medoids algorithm, thereby enabling cluster-specific TIS parameter utilization. In practice, our static weight array matrix-based indexing method for parameter set lookup can be used with good results in data sets exhibiting moderate levels of 5'-complete coverage.</p> <p>Conclusion</p> <p>We demonstrate that improvements in statistically-based models for TIS prediction can be achieved by taking the class of each potential start-methionine into account pending certain testing conditions, and that our perceptron-based model is suitable for the TIS identification task. MetWAMer represents a well-documented, extensible, and freely available software system that can be readily re-trained for differing target applications and/or extended with existing and novel TIS prediction methods, to support further research efforts in this area.</p