General Sampling Scheme Research Articles

Surprise sampling: Improving and extending the local case-control sampling

Fithian and Hastie (2014) proposed a new sampling scheme called local case-control (LCC) sampling that achieves stability and efficiency by utilizing a clever adjustment pertained to the logistic model. It is particularly useful for classification with large and imbalanced data. This paper proposes a more general sampling scheme based on a working principle that data points deserve higher sampling probability if they contain more information or appear "surprising" in the sense of, for example, a large error of pilot prediction or a large absolute score. Compared with the relevant existing sampling schemes, as reported in Fithian and Hastie (2014) and Ai, et al. (2018), the proposed one has several advantages. It adaptively gives out the optimal forms to a variety of objectives, including the LCC and Ai et al. (2018)'s sampling as special cases. Under same model specifications, the proposed estimator also performs no worse than those in the literature. The estimation procedure is valid even if the model is misspecified and/or the pilot estimator is inconsistent or dependent on full data. We present theoretical justifications of the claimed advantages and optimality of the estimation and the sampling design. Different from Ai, et al. (2018), our large sample theory are population-wise rather than data-wise. Moreover, the proposed approach can be applied to unsupervised learning studies, since it essentially only requires a specific loss function and no response-covariate structure of data is needed. Numerical studies are carried out and the evidence in support of the theory is shown.

Developing generalized sampling schemes with known error properties: the case of a moving observer

Pattern in space and time is central to ecology, and adequately designed ecological sampling is needed to resolve those patterns, pursue ecological questions and design conservation strategies. Recently, there has been an explosion of various ecological data due to the proliferation of online data‐sharing platforms, citizen science programs and new technology such as unmanned aerial vehicles (UAVs), but data reliability, consistency and the error properties of the sampling method are usually uncertain. While there are a number of standard survey protocols for different taxa, they often subjectively designed and standardization is meant to facilitate repeatability rather than produce a quantitative evaluation of the data (e.g. error properties). Here, we describe an ecological survey scheme consisting of an ‘algorithm' to be followed in the field that will result in a standard set of data as well as the error properties of the data. While many such sampling schemes could be developed that target different types of organisms, we focus on one case of a moving observer attempting to detect a species in the field (e.g. a birder, UAV, etc.) with the goal of producing a presence–absence map. The multiscale model developed is spatially explicit and accommodates inherent survey tradeoffs such as sampling speed, detectability and map resolution. Given a set of sampling parameters, the model provides estimates of the total sampling time and map accuracy translated into the probability of false negative. Additionally it also provides an actual and sampled occupancy–area curve across mapping resolutions that can be utilized to discuss sampling effects. While the proposed sampling framework is simple, the same general approach could be adapted for other conditions to meet the needs of a particular taxon. If a set of ‘canonical' sampling algorithms could be developed with known mathematical properties, it would enhance reliability and usage of ecological datasets.

Effects of mitigation of pixel cross-talk in the encoding of complex fields using the double-phase method

We report on unwanted effects of pixel cross-talk and its mitigation on the experimental realization of the double-phase method with phase-only spatial light modulators. We experimentally demonstrate that a generalized sampling scheme can reduce nonuniform phase modulation due to the pixel cross-talk phenomenon and, consequently, improve the quality of amplitude and phase images obtained with this encoding method. To corroborate our proposal, several experiments to reconstruct amplitude-only as well as fully independent amplitude and phase patterns under different spatial sampling schemes were carried out. We also show how a convenient implementation of the well-known polarization-based phase-shifting technique can be employed to measure the encoded complex field using only a conventional CMOS camera.

Leveraging mixed and incomplete outcomes via reduced-rank modeling

Multivariate outcomes with multivariate features of possibly high dimension are routinely produced in various fields. In many real-world problems, the collected outcomes are of mixed types, including continuous measurements, binary indicators and counts, and a substantial proportion of values may also be missing. Regardless of their types, these mixed outcomes are often interrelated, representing diverse reflections or views of the same underlying data generation mechanism. As such, an integrative multivariate model can be beneficial. We develop a mixed-outcome reduced-rank regression, which effectively enables information sharing among different prediction tasks. Our approach integrates mixed and partially observed outcomes belonging to the exponential dispersion family, by assuming that all the outcomes are associated through a shared low-dimensional subspace spanned by the features. A general singular value regularized criterion is proposed, and we establish a non-asymptotic performance bound for the proposed estimators in the context of supervised learning with mixed outcomes from an exponential family and under a general sampling scheme of missing data. An iterative singular value thresholding algorithm is developed for optimization with convergence guarantee. The effectiveness of our approach is demonstrated by simulation studies and an application on predicting health-related outcomes in longitudinal studies of aging.

Estimation and Inference of Quantile Regression for Survival Data Under Biased Sampling

ABSTRACTBiased sampling occurs frequently in economics, epidemiology, and medical studies either by design or due to data collecting mechanism. Failing to take into account the sampling bias usually leads to incorrect inference. We propose a unified estimation procedure and a computationally fast resampling method to make statistical inference for quantile regression with survival data under general biased sampling schemes, including but not limited to the length-biased sampling, the case-cohort design, and variants thereof. We establish the uniform consistency and weak convergence of the proposed estimator as a process of the quantile level. We also investigate more efficient estimation using the generalized method of moments and derive the asymptotic normality. We further propose a new resampling method for inference, which differs from alternative procedures in that it does not require to repeatedly solve estimating equations. It is proved that the resampling method consistently estimates the asymptotic covariance matrix. The unified framework proposed in this article provides researchers and practitioners a convenient tool for analyzing data collected from various designs. Simulation studies and applications to real datasets are presented for illustration. Supplementary materials for this article are available online.

A Relative Entropy Approach to Stochastic Dominance Analysis

This study formulates portfolio analysis in terms of Stochastic Dominance, Relative Entropy and Empirical Likelihood. We define a portfolio inefficiency measure based on the divergence between given probabilities and the nearest probabilities that rationalize a given portfolio for some admissible utility function. When applied to a sample of time-series observations in a blockwise fashion, the inefficiency measure becomes a Likelihood Ratio statistic for inequality moment conditions. The limiting distribution of the test statistic is bounded by a chi-squared distribution under general sampling schemes, allowing for conservative large-sample testing. We develop a tight numerical approximation for the test statistic based on a two-stage optimization procedure and piecewise linearization techniques. A Monte Carlo simulation study of the Empirical Likelihood Ratio test shows superior small-sample properties compared with various Generalized Method of Moments tests. An application analyzes the efficiency of a passive stock market index in data sets from the empirical asset pricing literature.

Portfolio Analysis Using Stochastic Dominance, Relative Entropy, and Empirical Likelihood

This study formulates portfolio analysis in terms of stochastic dominance, relative entropy, and empirical likelihood. We define a portfolio inefficiency measure based on the divergence between given probabilities and the nearest probabilities that rationalize a given portfolio for some admissible utility function. When applied to a sample of time-series observations in a blockwise fashion, the inefficiency measure becomes a likelihood ratio statistic for testing inequality moment conditions. The limiting distribution of the test statistic is bounded by a chi-squared distribution under general sampling schemes, allowing for conservative large-sample testing. We develop a tight numerical approximation for the test statistic based on a two-stage optimization procedure and piecewise linearization techniques. A Monte Carlo simulation study of the empirical likelihood ratio test shows superior small-sample properties compared with various generalized method of moments tests. An application analyzes the efficiency of a passive stock market index in data sets from the empirical asset pricing literature. This paper was accepted by Manel Baucells, decision analysis.

Generalized Coprime Sampling of Toeplitz Matrices for Spectrum Estimation

Increased demand on spectrum sensing over a broad frequency band requires a high sampling rate and thus leads to a prohibitive volume of data samples. In some applications, e.g., spectrum estimation, only the second-order statistics are required. In this case, we may use a reduced data-sampling rate by exploiting a low-dimensional representation of the original high-dimensional signals. In particular, the covariance matrix can be reconstructed from compressed data by utilizing its specific structure, e.g., the Toeplitz property. Among a number of techniques for compressive covariance sampler design, the coprime sampler is considered attractive because it enables a systematic design capability with a significantly reduced sampling rate. In this paper, we propose a general coprime sampling scheme that implements effective compression of Toeplitz covariance matrices. Given a fixed number of data samples, we examine different schemes on covariance matrix acquisition for performance evaluation, comparison, and optimal design, based on segmented data sequences.

Sampling and empirical risk minimization

ABSTRACTIn certain situations that shall be undoubtedly more and more common in the Big Data era, the datasets available are so massive that computing statistics over the full samples is hardly feasible, if not unfeasible. A natural approach in this context consists in using survey schemes and substituting the ‘full data’ statistics with their counterparts based on the resulting random samples, of manageable size. It is the main purpose of this paper to investigate the impact of survey sampling on statistical learning methods based on empirical risk minimization through the standard binary classification problem, considered here as a ‘case in point’. Precisely, we prove that, in presence of auxiliary information, appropriate use of optimally coupled Poisson survey plans may not affect much the learning rates, while possibly reducing significantly the number of terms that must be averaged to compute the empirical risk functional with overwhelming probability. These striking results are next shown to extend to more general sampling schemes by means of a coupling technique, originally introduced by Hajek [Asymptotic theory of rejective sampling with varying probabilities from a finite population. Ann Math Stat. 1964;35(4):1491–1523].

Local measurement and reconstruction for noisy bandlimited graph signals

Signals and information related to networks can be modeled and processed as graph signals. It has been shown that if a graph signal is smooth enough to satisfy certain conditions, it can be uniquely determined by its decimation on a subset of vertices. However, instead of the decimation, sometimes local combinations of signals on different sets of vertices are obtained in potential applications such as sensor networks with clustering structures. In this work, a generalized sampling scheme is proposed based on local measurement, which is a linear combination of signals associated with local vertices. It is proved that bandlimited graph signals can be perfectly reconstructed from the local measurements through a proposed iterative local measurement reconstruction (ILMR) algorithm. Some theoretical results related to ILMR including its convergence and denoising performance are given. Then the optimal partition of local sets and local weights are studied to minimize the error bound. It is shown that in noisy scenarios the proposed local measurement scheme is more robust than the traditional decimation scheme.

Accelerated failure time model under general biased sampling scheme.

Right-censored time-to-event data are sometimes observed from a (sub)cohort of patients whose survival times can be subject to outcome-dependent sampling schemes. In this paper, we propose a unified estimation method for semiparametric accelerated failure time models under general biased estimating schemes. The proposed estimator of the regression covariates is developed upon a bias-offsetting weighting scheme and is proved to be consistent and asymptotically normally distributed. Large sample properties for the estimator are also derived. Using rank-based monotone estimating functions for the regression parameters, we find that the estimating equations can be easily solved via convex optimization. The methods are confirmed through simulations and illustrated by application to real datasets on various sampling schemes including length-bias sampling, the case-cohort design and its variants.

Sampling of Graph Signals With Successive Local Aggregations

A new scheme to sample signals defined in the nodes of a graph is proposed. The underlying assumption is that such signals admit a sparse representation in a frequency domain related to the structure of the graph, which is captured by the so-called graph-shift operator. Most of the works that have looked at this problem have focused on using the value of the signal observed at a subset of nodes to recover the signal in the entire graph. Differently, the sampling scheme proposed here uses as input observations taken at a single node. The observations correspond to sequential applications of the graph-shift operator, which are linear combinations of the information gathered by the neighbors of the node. When the graph corresponds to a directed cycle (which is the support of time-varying signals), our method is equivalent to the classical sampling in the time domain. When the graph is more general, we show that the Vandermonde structure of the sampling matrix, which is critical to guarantee recovery when sampling time-varying signals, is preserved. Sampling and interpolation are analyzed first in the absence of noise and then noise is considered. We then study the recovery of the sampled signal when the specific set of frequencies that is active is not known. Moreover, we present a more general sampling scheme, under which, either our aggregation approach or the alternative approach of sampling a graph signal by observing the value of the signal at a subset of nodes can be both viewed as particular cases. The last part of the paper presents numerical experiments that illustrate the results developed through both synthetic graph signals and a real-world graph of the economy of the United States.

Fitting Accelerated Failure Time Models in Routine Survival Analysis withRPackageaftgee

Accelerated failure time (AFT) models are alternatives to relative risk models which are used extensively to examine the covariate effects on event times in censored data regression. Nevertheless, AFT models have been much less utilized in practice due to lack of reliable computing methods and software. This paper describes an R package aftgee that implements recently developed inference procedures for AFT models with both the rank-based approach and the least squares approach. For the rank-based approach, the package allows various weight choices and uses an induced smoothing procedure that leads to much more efficient computation than the linear programming method. With the rank-based estimator as an initial value, the generalized estimating equation approach is used as an extension of the least squares approach to the multivariate case. Additional sampling weights are incorporated to handle missing data needed as in case-cohort studies or general sampling schemes. A simulated dataset and two real life examples from biomedical research are employed to illustrate the usage of the package.

A Unified Approach to Semiparametric Transformation Models Under General Biased Sampling Schemes

We propose a unified estimation method for semiparametric linear transformation models under general biased sampling schemes. The new estimator is obtained from a set of counting process-based unbiased estimating equations, developed through introducing a general weighting scheme that offsets the sampling bias. The usual asymptotic properties, including consistency and asymptotic normality, are established under suitable regularity conditions. A closed-form formula is derived for the limiting variance and the plug-in estimator is shown to be consistent. We demonstrate the unified approach through the special cases of left truncation, length bias, the case-cohort design, and variants thereof. Simulation studies and applications to real datasets are presented.

Accelerated MRI by SPEED with generalized sampling schemes

To enhance the fast imaging technique of skipped phase encoding (PE) and edge deghosting (SPEED) for more general sampling options, and thus more flexibility in implementations and applications. SPEED uses skipped PE steps to accelerate MRI scan. Previously, the PE skip size was chosen from prime numbers only. This restriction has been relaxed in this study to allow choice of any integers rather than merely prime numbers. Various sampling patterns were studied under all possible combinations of PE skip size and PE shifts. A criterion based on the rank values of ghost phasor matrices was introduced to evaluate SPEED reconstruction. The reconstruction quality was found to correlate with the rank value of the ghost phasor matrix and the skipped PE size N. A low-rank value indicates a singular matrix that causes failure of the SPEED reconstruction. Composite numbers combined with appropriately chosen PE shifts yielded satisfactory reconstruction results. With properly chosen PE shifts, it was found that any integers, including both prime numbers and composite numbers, could be used as PE skip size for SPEED. This finding allows much more flexible data acquisition options that may lead to more freedom in practical implementations and applications.

MOMENTUM SHIFT IN NONADIABATIC DYNAMICS

We review the momentum-jump approximation in the quantum-classical Liouville formulation of nonadiabatic dynamics. We consider different momentum shift rules and study their numerical importance in computer simulations of the spin-boson model. Our findings might help the design of generalized sampling schemes of nonadiabatic dynamics at long time.

Editorial introduction

Recent years have witnessed great interest in financial models based on Lévy processes as possible alternatives to the traditional Black–Scholes model of financial markets, which is based on the geometric Brownian motion. The main criticism of this model concerns the fact that in the Black–Scholes world, the distribution of the log returns is normal irrespective of the spacing of the data. For widely spaced data, such as security prices sampled on a monthly time grid, the empirical distribution of the returns is close to normal. However, the daily return distributions typically are far from being normal, since they have more mass at the origin and in the tails. The discrepancy is even more visible if one goes down to an intra-day time grid. Models based on Lévy processes, however, can account for such departures from normality and they also have the ability to reproduce other important stylized features of financial time series, for instance, abrupt changes in asset prices because of financial cataclysms. As any stochastic model, a financial model based on a Lévy process depends on various parameters (finite or possibly infinite-dimensional). Estimation of these parameters, or, in financial terminology, calibration of the model to the available data, is of critical importance to successful application of these models in practice. This is a new, challenging and rapidly growing area of statistical research. To assess recent progress achieved in inference methods for Lévy processes, to identify problems of interest, to outline future research directions and to contribute to the development of this new area, the three of us came up with the idea of organizing a workshop that would emphasize more the statistical challenges of financial modelling with Lévy processes, rather than probabilistic aspects associated with the theory of Lévy processes (path properties, fluctuation theory). EURANDOM, the Eindhoven-based European Research Institute in Stochastics and Stochastic Operations Research, kindly agreed to host the workshop, which subsequently took place on 15–17 July, 2009, and attracted some 60 participants. At the same time, the editorial board of Statistica Neerlandica supported our idea to publish contributions by invited speakers as a special issue of Statistica Neerlandica. The six papers constituting this issue give a good impression of modern research in statistical inference for Lévy processes, as well as financial modelling with them. In particular, Antonis Papapantoleon reviews the construction and properties of some popular approaches using Lévy processes to modelling London Interbank Offered Rate (LIBOR) interest rates. He discusses different frameworks: classical LIBOR market models, forward price models and Markov-functional models, with finally some special attention to recently developed affine models. José Manuel Corcuera and Gergely Farkas consider the asymptotic behaviour of the power variation of processes that are integral transforms of α-stable processes with index of stability α between 0 and 2. They establish stable convergence of the corresponding fluctuations. These results provide tools for statistical inference from discrete observations. Their main result is a central limit theorem for the pth power variation. Non-parametric estimation of the Lévy density for pure jump Lévy processes is the topic of a paper by Fabienne Comte and Valentine Genon-Catalot. Under various sampling schemes, non-adaptive and adaptive estimators of the Lévy density are proposed and the corresponding risk bounds are derived. Johanna Kappus and Markus Reiß construct an estimator of the Lévy–Khinchine characteristic triplet corresponding to the given Lévy process and derive optimal rates of convergence under a general sampling scheme which encompasses the usual low- and high-frequency data assumptions and covers also the mid-frequency regime. Mark Podolskij and Mathias Vetter survey limit theorems for certain functionals of semi-martingales that are observed at high frequency. They explain the main ideas of the theory using the concept of stable convergence, and they demonstrate some laws of large numbers and stable central limit theorems. Starting from multivariate elliptic distributions, Nick Bingham, John Fry and Rüdiger Kiesel introduce and study multivariate elliptic processes. They discuss Lévy processes, diffusions, discrete versus continuous time, jumps versus diffusions and semi-martingales. They also analyze some data to illustrate the theory. We expect that the topics touched upon in this issue will continue to play an important role. Concluding this introduction, we would like to thank the authors who contributed to this issue, as well as Statistica Neerlandica for hosting the proceedings. It is our pleasant duty to acknowledge the contribution of EURANDOM to the organization of the workshop and to thank the sponsors: the European Science Foundation (through its programme AMaMeF), the Netherlands Organization for Scientific Research (NWO), the Mathematical Research Institute (MRI) and the Thomas Stieltjes Institute for Mathematics.

Realized volatility with stochastic sampling

A central limit theorem for the realized volatility of a one-dimensional continuous semimartingale based on a general stochastic sampling scheme is proved. The asymptotic distribution depends on the sampling scheme, which is written explicitly in terms of the asymptotic skewness and kurtosis of returns. Conditions for the central limit theorem to hold are examined for several concrete examples of schemes. Lower bounds for mean squared error and for asymptotic conditional variance are given, which are attained by using a specific sampling scheme.

Adaptive Interleaved Sampling for Interactive High‐Fidelity Rendering

Abstract Recent advances have made interactive ray tracing (IRT) possible on consumer desktop machines. These advances have brought about the potential for interactive global illumination (IGI) with enhanced realism through physically based lighting. IGI, unlike IRT, has a much higher computational complexity. Furthermore, since non‐primary rays constitute the majority of the computation, the rays are predominantly incoherent, making impractical many of the methods that have made IRT possible. Two methods that have already shown promise in decreasing the computational time of the GI solution are interleaved sampling and adaptive rendering. Interleaved sampling is a generalized sampling scheme that smoothly blends between regular and irregular sampling while maintaining coherence. Adaptive rendering algorithms adjust rendering quality, non‐uniformally, using a guidance scheme. While adaptive rendering has shown to provide speed‐up when used for off‐line rendering it has not been utilized in IRT due to its naturally incoherent nature. In this paper, we combine adaptive rendering and interleaved sampling within a component‐based solution into a new approach we term adaptive interleaved sampling. This allows us to tailor new adaptive heuristics for interleaved sampling of the individual components of the GI solution significantly improving overall performance. We present a novel component‐based IGI framework for which we achieve interactive frame rates for a range of effects such as indirect diffuse lighting, soft shadows and single scatter homogeneous participating media.

Central limit theorem for the realized volatility based on tick time sampling

A central limit theorem for the realized volatility estimator of the integrated volatility based on a specific random sampling scheme is proved, where prices are sampled with every ‘continued price change’ in bid or ask quotation data. The estimator is shown to be robust to market microstructure noise induced by price discreteness and bid–ask spreads. More general sampling schemes also are treated in case that the price process is a diffusion.