Directional data consist of observations distributed on a (hyper)sphere, and appear in many applied fields, such as astronomy, ecology, and environmental science. This paper studies both statistical and computational problems of kernel smoothing for directional data. We generalize the classical mean shift algorithm to directional data, which allows us to identify local modes of the directional kernel density estimator (KDE). The statistical convergence rates of the directional KDE and its derivatives are derived, and the problem of mode estimation is examined. We also prove the ascending property of the directional mean shift algorithm and investigate a general problem of gradient ascent on the unit hypersphere. To demonstrate the applicability of the algorithm, we evaluate it as a mode clustering method on both simulated and real-world data sets.

A cluster tree provides an intuitive summary of a density function that reveals essential structure about the high-density clusters. The true cluster tree is estimated from a finite sample from an unknown true density. This paper addresses the basic question of quantifying our uncertainty by assessing the statistical significance of different features of an empirical cluster tree. We first study a variety of metrics that can be used to compare different trees, analyzing their properties and assessing their suitability for our inference task. We then propose methods to construct and summarize confidence sets for the unknown true cluster tree. We introduce a partial ordering on cluster trees which we use to prune some of the statistically insignificant features of the empirical tree, yielding interpretable and parsimonious cluster trees. Finally, we provide a variety of simulations to illustrate our proposed methods and furthermore demonstrate their utility in the analysis of a Graft-versus-Host Disease (GvHD) data set.

We introduce the concept of coverage risk as an error measure for density ridge estimation.The coverage risk generalizes the mean integrated square error to set estimation.We propose two risk estimators for the coverage risk and we show that we can select tuning parameters by minimizing the estimated risk.We study the rate of convergence for coverage risk and prove consistency of the risk estimators.We apply our method to three simulated datasets and to cosmology data.In all the examples, the proposed method successfully recover the underlying density structure.

This paper presents a 3.2 Gb/s low-power clock and data recovery (CDR). The improved architecture using two half-rate gated voltage-controlled oscillators (GVCOs) shared between frequency presetting and data recovery modes is presented to remove the LC-tank voltage-controlled oscillator in a cascaded CDR. Moreover, using the proposed active inductive loading technique instead of the on-chip inductor reduces the power consumption and area in high-speed operation. This CDR circuit has been designed in TSMC 0.18 μm CMOS technology. It consumes 22.5 mW from a 1.8-V supply and occupies an active area of 0.26 mm2. The peak-to-peak and rms jitter of the recovered clock are 95.6 ps and 12.1 ps for a 3.2 Gb/s 27-1 PRBS, respectively.

In this paper, we propose evaluation flow for estimating the performance degradation of motion estimation (ME) architectures when faults occur and decide their acceptability. Moreover, when a fault is evaluated as unacceptable, a swap-based error-tolerance (ET) technique is proposed to increase the acceptability of this fault. According to experimental results, the acceptability and effective yield can be improved significantly with negligible hardware overhead.