χ2\chi^2-confidence sets in high-dimensional regression

We study a high-dimensional regression model. Aim is to construct a confidence set for a given group of regression coefficients, treating all other regression coefficients as nuisance parameters. We apply a one-step procedure with the square-root Lasso as initial estimator and a multivariate square-root Lasso for constructing a surrogate Fisher information matrix. The multivariate square-root Lasso is based on nuclear norm loss with $\ell_1$-penalty. We show that this procedure leads to an asymptotically $\chi^2$-distributed pivot, with a remainder term depending only on the $\ell_1$-error of the initial estimator. We show that under $\ell_1$-sparsity conditions on the regression coefficients $\beta^0$ the square-root Lasso produces to a consistent estimator of the noise variance and we establish sharp oracle inequalities which show that the remainder term is small under further sparsity conditions on $\beta^0$ and compatibility conditions on the design.Comment: 22 page

Sharp Oracle Inequalities for Square Root Regularization

We study a set of regularization methods for high-dimensional linear regression models. These penalized estimators have the square root of the residual sum of squared errors as loss function, and any weakly decomposable norm as penalty function. This fit measure is chosen because of its property that the estimator does not depend on the unknown standard deviation of the noise. On the other hand, a generalized weakly decomposable norm penalty is very useful in being able to deal with different underlying sparsity structures. We can choose a different sparsity inducing norm depending on how we want to interpret the unknown parameter vector $\beta$. Structured sparsity norms, as defined in Micchelli et al. [18], are special cases of weakly decomposable norms, therefore we also include the square root LASSO (Belloni et al. [3]), the group square root LASSO (Bunea et al. [10]) and a new method called the square root SLOPE (in a similar fashion to the SLOPE from Bogdan et al. [6]). For this collection of estimators our results provide sharp oracle inequalities with the Karush-Kuhn-Tucker conditions. We discuss some examples of estimators. Based on a simulation we illustrate some advantages of the square root SLOPE

Oscillatory patterns in the electroencephalogram at sleep onset

Falling asleep is a gradually unfolding process. We investigated the role of various oscillatory activities including sleep spindles and alpha and delta oscillations at sleep onset (SO) by automatically detecting oscillatory events. We used two datasets of healthy young males, eight with four baseline recordings, and eight with a baseline and recovery sleep after 40 h of sustained wakefulness. We analyzed the 2-min interval before SO (stage 2) and the five consecutive 2-min intervals after SO. The incidence of delta/theta events reached its maximum in the first 2-min episode after SO, while the frequency of them was continuously decreasing from stage 1 onwards, continuing over SO and further into deeper sleep. Interestingly, this decrease of the frequencies of the oscillations were not affected by increased sleep pressure, in contrast to the incidence which increased. We observed an increasing number of alpha events after SO, predominantly frontally, with their prevalence varying strongly across individuals. Sleep spindles started to occur after SO, with first an increasing then a decreasing incidence and a continuous decrease in their frequency. Again, the frequency of the spindles was not altered after sleep deprivation. Oscillatory events revealed derivation dependent aspects. However, these regional aspects were not specific of the process of SO but rather reflect a general sleep related phenomenon. No individual traits of SO features (incidence and frequency of oscillations) and their dynamics were observed. Delta/theta events are important features for the analysis of SO in addition to slow waves

Diurnal variations in multi-sensor wearable-derived sleep characteristics in morning- and evening-type shift workers under naturalistic conditions

Consumer-grade, multi-sensor, rest-activity trackers may be powerful tools, to help optimize rest-activity management in shiftwork populations undergoing circadian misalignment. Nevertheless, performance testing of such devices under field conditions is scarce. We previously validated Fitbit Charge 2TM against home polysomnography and now evaluated the potential of this device to document differences in rest-activity behavior, including sleep macrostructure, in first-responder shift workers in an operational setting. We continuously monitored 89 individuals (54% females; mean age: 33.9 ± 7.7 years) for 32.5 ± 9.3 days and collected 2,974 individual sleep episodes scattered around the clock. We stratified the study participants according to their self-reported circadian preference on the reduced Horne-Östberg Morningness-Evening Questionnaire (rMEQ; the scores from 4 participants were missing). Fitbit estimates of sleep duration, wakefulness after sleep onset (WASO), REM sleep percentage in the first NREM-REM sleep cycle, and REM sleep latency formed approximately sinusoidal oscillations across 24 hours. Generalized additive mixed model analyses revealed that the phase position of sleep duration minimum was delayed by 2.8 h in evening types (ET; rMEQ ≤ 11; n = 20) and by 2.6 h in intermediate types (IT; 11 < rMEQ < 18; n = 45) when compared to morning types (MT; rMEQ ≥ 18; n = 20). Similarly, the phase position of WASO was delayed by 2.7 h in ET compared to MT. While nocturnal sleep duration did not differ among the three groups, sleep episodes during the biological day decreased in duration from ET to IT to MT. Together, the findings support the notion that a consumer-grade, rest-activity tracker allows estimation of behavioral sleep/wake cycles and sleep macrostructure in shift workers under naturalistic conditions that are consistent with their self-reported chronotype

Nocturnal sodium oxybate increases the anterior cingulate cortex magnetic resonance glutamate signal upon awakening

Clinical guidelines recommend sodium oxybate (SXB; the sodium salt of γ-hydroxybutyrate) for the treatment of disturbed sleep and excessive daytime sleepiness in narcolepsy, yet the underlying mode of action is elusive. In a randomised controlled trial in 20 healthy volunteers, we aimed at establishing neurochemical changes in the anterior cingulate cortex (ACC) following SXB-enhanced sleep. The ACC is a core neural hub regulating vigilance in humans. At 2:30 a.m., we administered in a double-blind cross-over manner an oral dose of 50 mg/kg SXB or placebo, to enhance electroencephalography-defined sleep intensity in the second half of nocturnal sleep (11:00 p.m. to 7:00 a.m.). Upon scheduled awakening, we assessed subjective sleepiness, tiredness and mood and measured two-dimensional, J-resolved, point-resolved magnetic resonance spectroscopy (PRESS) localisation at 3-Tesla field strength. Following brain scanning, we used validated tools to quantify psychomotor vigilance test (PVT) performance and executive functioning. We analysed the data with independent t tests, false discovery rate (FDR) corrected for multiple comparisons. The morning glutamate signal (at 8:30 a.m.) in the ACC was specifically increased after SXB-enhanced sleep in all participants in whom good-quality spectroscopy data were available (n = 16; pFDR < 0.002). Further, global vigilance (10th-90th inter-percentile range on the PVT) was improved (pFDR < 0.04) and median PVT response time was shorter (pFDR < 0.04) compared to placebo. The data indicate that elevated glutamate in the ACC could provide a neurochemical mechanism underlying SXB's pro-vigilant efficacy in disorders of hypersomnolence

Asymptotic Confidence Regions and Sharp Oracle Results under Structured Sparsity

To restrict ourselves to the regime of sparse solutions has become the new paradigm for modern statistics, machine learning and in particular for high dimensional linear regression models. Sparse solutions aim at representing the information by a small core of active explanatory parameters. As a pleasant side effect the resulting models bear plain and relatively easy interpretations. This indistinctive sparsity is commonly represented by the ℓ1-norm, which is a convex relaxation of the number of active variables. Reducing complexity in this way is well understood. Nevertheless in practical applications we commonly have more knowledge about the structure of possible arrangements. Therefore the topic of structured sparsity has recently emerged as a new promising way to represent the prior knowledge of the underlying sparsity structure. In this thesis we focus on embodying the prior knowledge of potential sparsity patterns through general norm penalties. Weak decomposability is a fundamental concept in understanding the sparsity structure a norm yields. The idea of weak decomposability is further generalized to LASSO type estimators with concave penalties. We also see that sharp oracle results can be obtained in the multivariate model. The square root LASSO is generalized to all weakly decomposable norm penalties, where sharp oracle results are given. The properties of the scaling of these square root estimators have nice applications for constructing χ2 confidence regions for the LASSO. Furthermore assigning uncertainty in high dimensionality for structured sparsity estimators is tackled by means of two related frameworks

Validation of Fitbit Charge 2 Sleep and Heart Rate Estimates Against Polysomnographic Measures in Shift Workers: Naturalistic Study

Background: Multisensor fitness trackers offer the ability to longitudinally estimate sleep quality in a home environment with the potential to outperform traditional actigraphy. To benefit from these new tools for objectively assessing sleep for clinical and research purposes, multisensor wearable devices require careful validation against the gold standard of sleep polysomnography (PSG). Naturalistic studies favor validation. Objective: This study aims to validate the Fitbit Charge 2 against portable home PSG in a shift-work population composed of 59 first responder police officers and paramedics undergoing shift work. Methods: A reliable comparison between the two measurements was ensured through the data-driven alignment of a PSG and Fitbit time series that was recorded at night. Epoch-by-epoch analyses and Bland-Altman plots were used to assess sensitivity, specificity, accuracy, the Matthews correlation coefficient, bias, and limits of agreement. Results: Sleep onset and offset, total sleep time, and the durations of rapid eye movement (REM) sleep and non-rapid-eye movement sleep stages N1+N2 and N3 displayed unbiased estimates with nonnegligible limits of agreement. In contrast, the proprietary Fitbit algorithm overestimated REM sleep latency by 29.4 minutes and wakefulness after sleep onset (WASO) by 37.1 minutes. Epoch-by-epoch analyses indicated better specificity than sensitivity, with higher accuracies for WASO (0.82) and REM sleep (0.86) than those for N1+N2 (0.55) and N3 (0.78) sleep. Fitbit heart rate (HR) displayed a small underestimation of 0.9 beats per minute (bpm) and a limited capability to capture sudden HR changes because of the lower time resolution compared to that of PSG. The underestimation was smaller in N2, N3, and REM sleep (0.6-0.7 bpm) than in N1 sleep (1.2 bpm) and wakefulness (1.9 bpm), indicating a state-specific bias. Finally, Fitbit suggested a distribution of all sleep episode durations that was different from that derived from PSG and showed nonbiological discontinuities, indicating the potential limitations of the staging algorithm. Conclusions: We conclude that by following careful data processing processes, the Fitbit Charge 2 can provide reasonably accurate mean values of sleep and HR estimates in shift workers under naturalistic conditions. Nevertheless, the generally wide limits of agreement hamper the precision of quantifying individual sleep episodes. The value of this consumer-grade multisensor wearable in terms of tackling clinical and research questions could be enhanced with open-source algorithms, raw data access, and the ability to blind participants to their own sleep data. Keywords: actigraphy; mobile phone; multisensory; polysomnography; validation; wearables