The mode of a distribution provides an important summary of data and is often estimated based on some non-parametric kernel density estimator. This article develops a new data analysis tool called modal linear regression in order to explore highdimensional data. Modal linear regression models the conditional mode of a response Y given a set of predictors x as a linear function of x. Modal linear regression differs from standard linear regression in that standard linear regression models the conditional mean (as opposed to mode) of Y as a linear function of x. We propose an Expectation-Maximization algorithm in order to estimate the regression coefficients of modal linear regression. We also provide asymptotic properties for the proposed estimator without the symmetric assumption of the error density. Our empirical studies with simulated data and real data demonstrate that the proposed modal regression gives shorter predictive intervals than mean linear regression, median linear regression, and MM-estimators.

The principal impetus for the fabrication of functional nanotube materials comes from the promise of discovering unique structure-dependant properties and superior performance that are derived from their intrinsic nanotubular architecture. [1][2][3][4] 1D TiO 2 nanotube arrays prepared by the electrochemical anodization of self-organized porous structures on Ti foil [5][6][7] have attracted great research interest in recent years owing to their peculiar architecture, remarkable properties, and potential for wide-ranging applications. Uniform TiO 2 nanotubes are quite remarkably different in structure from other forms of TiO 2 , and are highly ordered, high-aspect-ratio structures with nanocrystalline walls perpendicular to electrically conductive Ti substrates, thereby naturally forming a Schottky-type contact. Moreover, these structures can be directly used as electrodes for photoelectric applications since the size of the nanotubes is very precisely controllable. The technological applications of TiO 2 nanotube arrays are still at an early stage, but these remarkable structures have already been shown to be very promising for applications in sensing, [8] catalysis, [9] photovoltaics, [10] photoelectrolysis, [11] and nanotemplating. [12] The electrical resistance of the TiO 2 nanotubes changes by almost 7 orders of magnitude upon exposure to 1000 ppm H 2 , [13] the largest ever reported sensitivity of a material to a gas. Furthermore, the H 2 evolution rate of TiO 2 nanotube arrays has been reported to be 76 mL hw -1 , [11] which is the highest reported H 2 generation rate for any oxide system upon photoelectrolysis. TiO 2 nanotube arrays have also attracted great interest for enhancing the photocatalytic degradation of various organics, which makes them promising materials for the detection of pollutants. Given the increasing quantities of pollutants that are being dumped into water bodies, environmental monitoring and control have become issues of global concern. Chemical oxygen demand (COD) is one of the most widely used metrics in the field of water-quality analysis in many countries, and is frequently used as an important index for controlling the operation of wastewater treatment plants, wastewater effluent monitoring, and taxation of wastewater pollution. [14] or ultrasound-assisted oxidation.[15]Other alternative assays have also been developed such as electrocatalytic determination using PbO 2 or Cu sensors in thin-cell reactors, [16,17] and photocatalytic and photoelectrocatalytic methods based on TiO 2 nanomaterial sensors. [18,19] However, all these modified K 2 Cr 2 O 7 methods are still plagued by the secondary pollution caused by highly toxic Cr(VI) ions, and moreover, the PbO 2 sensors pose the risk of the potential release of hazardous Pb during the preparation and disposal of the active material of the sensors. As compared to traditional analytical methods, photoelectrocatalytic approaches are more promising because of the superior oxidative abilities of illuminated TiO 2 . Furthermore, TiO 2 ...

BackgroundFalls pose major health problems to the middle-aged and older adults and may potentially lead to various levels of injuries. Sleep duration and disturbances have been shown to be associated with falls in literature; however, studies of the joint and distinct effects of those sleep problems are still sparse. To fill this gap, we aimed to determine the association between sleep duration, sleep disturbances and falls among middle-aged and older adults in China controlling for psychosocial, lifestyle, socio-demographical factors and comorbidity.MethodsData were derived from the China Health and Retirement Longitudinal Study (CHARLS) based on multi-stage sampling designs, with respondents aged 50 and older. Associations were evaluated by using multiple logistic regression adjusting for confounders and complex survey design. To further determine if the association of sleep duration/disturbance and falls depends on age groups, the study data were divided into two samples (age 50–64 vs. age 65+) and comparison was made between the two age groups.ResultsOf the 12,759 respondents, 2172 (17%) had falls within the last 2 years. Our findings indicated that the participants who had nighttime sleep duration ≤5 were more likely to report falls than those who had nighttime sleep duration ≥6 h; whereas no association between nighttime sleep duration > 8 h and falls. Participants having sleep disturbances 1–2 days, or 3–4 days, and 5–7 days per week were also more likely to report falls than those who had no sleep disturbance. The nap sleep duration was not significantly associated with falls. Although the combined sample found both sleep duration and sleep disturbance to be strongly associated with falls after adjusting for various confounders, sleep disturbance was not significantly related to falls among participants aged 65 + .ConclusionsOur study suggested that there is an independent association between falls and short sleep duration and disturbed sleep among middle-aged and older adults in China. Findings underscore the need for evidence-based prevention and interventions targeting sleep duration and disturbance among this study population.

Looking at predictive accuracy is a traditional method for comparing models. A natural method for approximating out-of-sample predictive accuracy is leave-one-out crossvalidation (LOOCV) -we alternately hold out each case from a full data set and then train a Bayesian model using Markov chain Monte Carlo (MCMC) without the held-out; at last we evaluate the posterior predictive distribution of all cases with their actual observations. However, actual LOOCV is time-consuming. This paper introduces two methods, namely iIS and iWAIC, for approximating LOOCV with only Markov chain samples simulated from a posterior based on a full data set. iIS and iWAIC aim at improving the approximations given by importance sampling (IS) and WAIC in Bayesian models with possibly correlated latent variables. In iIS and iWAIC, we first integrate the predictive density over the distribution of the latent variables associated with the held-out without reference to its observation, then apply IS and WAIC approximations to the integrated predictive density. We compare iIS and iWAIC with other approximation methods in three kinds of models: finite mixture models, models with correlated spatial effects, and a random effect logistic regression model. Our empirical results show that iIS and iWAIC give substantially better approximates than non-integrated IS and WAIC and other methods.

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