Asymptotic Precision Research Articles

Double machine learning and design in batch adaptive experiments

Abstract We consider an experiment with at least two stages or batches and O ( N ) O\left(N) subjects per batch. First, we propose a semiparametric treatment effect estimator that efficiently pools information across the batches, and we show that it asymptotically dominates alternatives that aggregate single batch estimates. Then, we consider the design problem of learning propensity scores for assigning treatment in the later batches of the experiment to maximize the asymptotic precision of this estimator. For two common causal estimands, we estimate this precision using observations from previous batches, and then solve a finite-dimensional concave maximization problem to adaptively learn flexible propensity scores that converge to suitably defined optima in each batch at rate O p ( N − 1 ⁄ 4 ) {O}_{p}\left({N}^{-1/4}) . By extending the framework of double machine learning, we show this rate suffices for our pooled estimator to attain the targeted precision after each batch, as long as nuisance function estimates converge at rate o p ( N − 1 ⁄ 4 ) {o}_{p}\left({N}^{-1/4}) . These relatively weak rate requirements enable the investigator to avoid the common practice of discretizing the covariate space for design and estimation in batch adaptive experiments while maintaining the advantages of pooling. Our numerical study shows that such discretization often leads to substantial asymptotic and finite sample precision losses outweighing any gains from design.

Precision motion control for electro-hydraulic axis systems under unknown time-variant parameters and disturbances

This article focuses on asymptotic precision motion control for electro-hydraulic axis systems under unknown time-variant parameters, mismatched and matched disturbances. Different from the traditional adaptive results that are applied to dispose of unknown constant parameters only, the unique feature is that an adaptive-gain nonlinear term is introduced into the control design to handle unknown time-variant parameters. Concurrently both mismatched and matched disturbances existing in electro-hydraulic axis systems can also be addressed in this way. With skillful integration of the backstepping technique and the adaptive control, a synthesized controller framework is successfully developed for electro-hydraulic axis systems, in which the coupled interaction between parameter estimation and disturbance estimation is avoided. Accordingly, this designed controller has the capacity of low-computation costs and simpler parameter tuning when compared to the other ones that integrate the adaptive control and observer/estimator-based technique to dividually handle parameter uncertainties and disturbances. Also, a nonlinear filter is designed to eliminate the “explosion of complexity” issue existing in the classical back-stepping technique. The stability analysis uncovers that all the closed-loop signals are bounded and the asymptotic tracking performance is also assured. Finally, contrastive experiment results validate the superiority of the developed method as well.

Quantum metrology with imperfect measurements

The impact of measurement imperfections on quantum metrology protocols has not been approached in a systematic manner so far. In this work, we tackle this issue by generalising firstly the notion of quantum Fisher information to account for noisy detection, and propose tractable methods allowing for its approximate evaluation. We then show that in canonical scenarios involving N probes with local measurements undergoing readout noise, the optimal sensitivity depends crucially on the control operations allowed to counterbalance the measurement imperfections—with global control operations, the ideal sensitivity (e.g., the Heisenberg scaling) can always be recovered in the asymptotic N limit, while with local control operations the quantum-enhancement of sensitivity is constrained to a constant factor. We illustrate our findings with an example of NV-centre magnetometry, as well as schemes involving spin-1/2 probes with bit-flip errors affecting their two-outcome measurements, for which we find the input states and control unitary operations sufficient to attain the ultimate asymptotic precision.

Non-perturbative effects in corrections to quantum master equations arising in Bogolubov–van Hove limit

We study the perturbative corrections to the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation which arises in the weak coupling limit. The spin-boson model in the rotating wave approximation at zero temperature is considered. We show that the perturbative part of the density matrix satisfies the time-independent GKSL equation for arbitrary order of the perturbation theory (if all the moments of the reservoir correlation function are finite). But to reproduce the right asymptotic precision at long times, one should use an initial condition different from the one for exact dynamics. Moreover, we show that the initial condition for this master equation even fails to be a density matrix under certain resonance conditions.

Localization Attack by Precoder Feedback Overhearing in 5G Networks and Countermeasures

In fifth-generation (5G) cellular networks, users feed back to the base station the index of the precoder (from a codebook) to be used for downlink transmission. The precoder is strongly related to the user channel and in turn to the user position within the cell. We propose a method by which an external attacker determines the user position by passively overhearing this unencrypted layer-2 feedback signal. The attacker first builds a map of fed back precoder indices in the cell. Then, by overhearing the precoder index fed back by the victim user, the attacker finds its position on the map. We focus on the type-I single-panel codebook, which today is the only mandatory solution in the 3GPP standard. We analyze the attack and assess the obtained localization accuracy against various parameters. We analyze the localization error of a simplified precoder feedback model and describe its asymptotic localization precision. We also propose a mitigation against our attack, wherein the user randomly selects the precoder among those providing the highest rate. Simulations confirm that the attack can achieve a high localization accuracy, which is significantly reduced when the mitigation solution is adopted, at the cost of a negligible rate degradation.

Tensor-network approach for quantum metrology in many-body quantum systems

Identification of the optimal quantum metrological protocols in realistic many particle quantum models is in general a challenge that cannot be efficiently addressed by the state-of-the-art numerical and analytical methods. Here we provide a comprehensive framework exploiting matrix product operators (MPO) type tensor networks for quantum metrological problems. The maximal achievable estimation precision as well as the optimal probe states in previously inaccessible regimes can be identified including models with short-range noise correlations. Moreover, the application of infinite MPO (iMPO) techniques allows for a direct and efficient determination of the asymptotic precision in the limit of infinite particle numbers. We illustrate the potential of our framework in terms of an atomic clock stabilization (temporal noise correlation) example as well as magnetic field sensing (spatial noise correlations). As a byproduct, the developed methods may be used to calculate the fidelity susceptibility—a parameter widely used to study phase transitions.

On the choice of high-dimensional regression parameters in Gaussian random tomography

The stochastic Radon transform was introduced by Panaretos in order to estimate a biophysical particle, modelled as a three-dimensional probability density subject to random and unknown rotations before being projected onto the imaging plane. Assuming that the particle has a Gaussian mixture distribution, the question arises as to how to recover the modes of the mixture, and the mixing weights. Panaretos and Konis proposed to do this in a sequential manner, using high-dimensional regression. Their approach has two drawbacks: first, the mixing weight estimates often have incorrect rank; second, much information inherent in the coefficients from the penalized regression approach (Lasso, or more generally elastic net) is lost. We propose a procedure, based on the asymptotic precision matrix, that exploits the information from the high-dimensional regression more efficiently, and yields in particular a method for choosing the parameter that balances the Lasso with ridge regression.

Quantum metrology in the presence of limited data

Quantum metrology protocols are typically designed around the assumption that we have an abundance of measurement data, but recent practical applications are increasingly driving interest in cases with very limited data. In this regime the best approach involves an interesting interplay between the amount of data and the prior information. Here we propose a new way of optimising these schemes based on the practically-motivated assumption that we have a sequence of identical and independent measurements. For a given probe state we take our measurement to be the best one for a single shot and we use this sequentially to study the performance of different practical states in a Mach–Zehnder interferometer when we have moderate prior knowledge of the underlying parameter. We find that we recover the quantum Cramér–Rao bound asymptotically, but for low data counts we find a completely different structure. Despite the fact that intra-mode correlations are known to be the key to increasing the asymptotic precision, we find evidence that these could be detrimental in the low data regime and that entanglement between the paths of the interferometer may play a more important role. Finally, we analyse how close realistic measurements can get to the bound and find that measuring quadratures can improve upon counting photons, though both strategies converge asymptotically. These results may prove to be important in the development of quantum enhanced metrology applications where practical considerations mean that we are limited to a small number of trials.

Improved precision in the analysis of randomized trials with survival outcomes, without assuming proportional hazards.

We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the Kaplan-Meier and inverse probability weighted estimators, under identifiability conditions; (iii) it remains consistent under violations of independent censoring (unlike the Kaplan-Meier estimator) when either the censoring or survival distributions, conditional on covariates, are estimated consistently; and (iv) it achieves the nonparametric efficiency bound when both of these distributions are consistently estimated. We illustrate the performance of our method using simulations based on resampling data from a completed, phase 3 randomized clinical trial of a new surgical treatment for stroke; the proposed estimator achieves a 12% gain in relative efficiency compared to the Kaplan-Meier estimator. The proposed estimator has potential advantages over existing approaches for randomized trials with time-to-event outcomes, since existing methods either rely on model assumptions that are untenable in many applications, or lack some of the efficiency and consistency properties (i)-(iv). We focus on estimation of the restricted mean survival time, but our methods may be adapted to estimate any treatment effect measure defined as a smooth contrast between the survival curves for each study arm. We provide R code to implement the estimator.

Convergence acceleration for observers by gain commutation

ABSTRACTIncreasing convergence rates of observers and differentiators for a class of nonlinear systems in the output canonical form (under presence of bounded matched disturbances, Lipschitz uncertainties and measurement noises) is investigated. A supervisory algorithm is designed that switches among different values of observer gains to accelerate the estimation. In the noise-free case, the presented switched-gain observer ensures global uniform time of convergence of the estimation error to the origin. In the presence of noise, the goals of overshoot reducing for the initial phase, acceleration of convergence and improvement of asymptotic precision of estimation are achieved. Efficiency of the proposed switching-gain observer is demonstrated by numerical comparison with a sliding mode and linear high-gain observers.

Detecting small-study effects and funnel plot asymmetry in meta-analysis of survival data: A comparison of new and existing tests.

Small‐study effects are a common threat in systematic reviews and may indicate publication bias. Their existence is often verified by visual inspection of the funnel plot. Formal tests to assess the presence of funnel plot asymmetry typically estimate the association between the reported effect size and their standard error, the total sample size, or the inverse of the total sample size. In this paper, we demonstrate that the application of these tests may be less appropriate in meta‐analysis of survival data, where censoring influences statistical significance of the hazard ratio. We subsequently propose 2 new tests that are based on the total number of observed events and adopt a multiplicative variance component. We compare the performance of the various funnel plot asymmetry tests in an extensive simulation study where we varied the true hazard ratio (0.5 to 1), the number of published trials (N=10 to 100), the degree of censoring within trials (0% to 90%), and the mechanism leading to participant dropout (noninformative versus informative). Results demonstrate that previous well‐known tests for detecting funnel plot asymmetry suffer from low power or excessive type‐I error rates in meta‐analysis of survival data, particularly when trials are affected by participant dropout. Because our novel test (adopting estimates of the asymptotic precision as study weights) yields reasonable power and maintains appropriate type‐I error rates, we recommend its use to evaluate funnel plot asymmetry in meta‐analysis of survival data. The use of funnel plot asymmetry tests should, however, be avoided when there are few trials available for any meta‐analysis.

Switched gain differentiator with fixed-time convergence

Acceleration of estimation for a class of nonlinear systems in the output canonical form is considered in this work. The acceleration is achieved by a supervisory algorithm design that switches among different values of observer gain. The presence of bounded matched disturbances, Lipschitz uncertainties and measurement noises is taken into account. The proposed switched-gain observer guarantees global uniform time of convergence of the estimation error to the origin in the noise-free case. In the presence of noise our commutation strategy pursuits the goals of overshoot reducing for the initial phase, acceleration of convergence and improvement of asymptotic precision of estimation. Efficacy of the proposed switching-gain observer is illustrated by numerical comparison with a sliding mode and linear high-gain observers.

Nonlinear Integral Type Observer Design for State Estimation and Unknown Input Reconstruction

This paper is concerned with model-based robust observer designs for state observation and its application for unknown input reconstruction. Firstly, a sliding mode observer (SMO), which provides exponential convergence of estimation error, is designed for a class of multivariable perturbed systems. Observer gain matrices subject to specific structures are going to be imposed such that the unknown perturbation will not affect estimate precision during the sliding modes; Secondly, to improve discontinuous control induced in the SMO as well as pursue asymptotic estimate precision, a proportional-integral type observer (PIO) is further developed. Both the design procedures of the SMO and PIO algorithms are characterized as feasibility issues of linear matrix inequality (LMI) and thus the computations of the control parameters can be efficiently solved. Compared with the SMO, it will be demonstrated that the PIO is capable of achieving better estimation precision as long as the unknown inputs are continuous. Finally, a servo-drive flexible robot arm is selected as an example to demonstrate the applications of the robust observer designs.

Enhanced Precision in the Analysis of Randomized Trials with Ordinal Outcomes

We present a general method for estimating the effect of a treatment on an ordinal outcome in randomized trials. The method is robust in that it does not rely on the proportional odds assumption. Our estimator leverages information in prognostic baseline variables, and has all of the following properties: (i) it is consistent; (ii) it is locally efficient; (iii) it is guaranteed to have equal or better asymptotic precision than both the inverse probability-weighted and the unadjusted estimators. To the best of our knowledge, this is the first estimator of the causal relation between a treatment and an ordinal outcome to satisfy these properties. We demonstrate the estimator in simulations based on resampling from a completed randomized clinical trial of a new treatment for stroke; we show potential gains of up to 39% in relative efficiency compared to the unadjusted estimator. The proposed estimator could be a useful tool for analyzing randomized trials with ordinal outcomes, since existing methods either rely on model assumptions that are untenable in many practical applications, or lack the efficiency properties of the proposed estimator. We provide R code implementing the estimator.

True precision limits in quantum metrology

We show that quantification of the performance of quantum-enhanced measurement schemes based on the concept of quantum Fisher information (QFI) yields results that are asymptotically equivalent to those from the rigorous Bayesian approach, provided generic uncorrelated noise is present in the setup. At the same time, we show that for the problem of decoherence-free phase estimation this equivalence breaks down, and the achievable estimation uncertainty calculated within the Bayesian approach is larger by a factor of π than that predicted from the QFI even in the large prior knowledge (small parameter fluctuation) regime, where the QFI is conventionally regarded as a reliable figure of merit. We conjecture that an analogous discrepancy is present in the arbitrary decoherence-free unitary parameter estimation scheme, and propose a general formula for the asymptotically achievable precision limit. We also discuss protocols utilizing states with an indefinite number of particles, and show that within the Bayesian approach it is legitimate to replace the number of particles with the mean number of particles in the formulas for the asymptotic precision, which as a consequence provides another argument for proposals based on the properties of the QFI of indefinite particle number states leading to sub-Heisenberg precisions not being practically feasible.

Comparison of tuning frequency estimation methods

In this paper, a comparison of different algorithms for concert pitch (i.e., tuning frequency or reference frequency) estimation is presented and discussed. The unavailability of ground-truth datasets makes this evaluation on real music recordings less trivial than it may initially appear. Hence, in this paper we use two datasets, one of real music (covers80 provided by LabROSA) and one of synthesized music (MS2012, constructed by the authors). The algorithms have been compared in terms of speed of convergence and stability of the estimated value over an increasing length of the analysed signal. A local tuning frequency estimation was also performed in order to compare the ability of the algorithms to follow the local variations of the reference frequency in a real-time environment. Moreover, an analysis of the execution time have been provided. While the various algorithms perform comparably in terms of asymptotic precision, they show a quite different behaviour in terms of speed of convergence and local tuning frequency estimation accuracy.

Visual Long-Term Memory Has the Same Limit on Fidelity as Visual Working Memory

Visual long-term memory can store thousands of objects with surprising visual detail, but just how detailed are these representations, and how can one quantify this fidelity? Using the property of color as a case study, we estimated the precision of visual information in long-term memory, and compared this with the precision of the same information in working memory. Observers were shown real-world objects in random colors and were asked to recall the colors after a delay. We quantified two parameters of performance: the variability of internal representations of color (fidelity) and the probability of forgetting an object’s color altogether. Surprisingly, the fidelity of color information in long-term memory was comparable to the asymptotic precision of working memory. These results suggest that long-term memory and working memory may be constrained by a common limit, such as a bound on the fidelity required to retrieve a memory representation.

Agnostic notes on regression adjustments to experimental data: Reexamining Freedman’s critique

Freedman [Adv. in Appl. Math. 40 (2008) 180–193; Ann. Appl. Stat. 2 (2008) 176–196] critiqued ordinary least squares regression adjustment of estimated treatment effects in randomized experiments, using Neyman’s model for randomization inference. Contrary to conventional wisdom, he argued that adjustment can lead to worsened asymptotic precision, invalid measures of precision, and small-sample bias. This paper shows that in sufficiently large samples, those problems are either minor or easily fixed. OLS adjustment cannot hurt asymptotic precision when a full set of treatment–covariate interactions is included. Asymptotically valid confidence intervals can be constructed with the Huber–White sandwich standard error estimator. Checks on the asymptotic approximations are illustrated with data from Angrist, Lang, and Oreopoulos’s [Am. Econ. J.: Appl. Econ. 1:1 (2009) 136–163] evaluation of strategies to improve college students’ achievement. The strongest reasons to support Freedman’s preference for unadjusted estimates are transparency and the dangers of specification search.

Direction information in multiple object tracking is limited by a graded resource

Is multiple object tracking (MOT) limited by a fixed set of structures (slots), a limited but divisible resource, or both? Here, we answer this question by measuring the precision of the direction representation for tracked targets. The signature of a limited resource is a decrease in precision as the square root of the tracking load. The signature of fixed slots is a fixed precision. Hybrid models predict a rapid decrease to asymptotic precision. In two experiments, observers tracked moving disks and reported target motion direction by adjusting a probe arrow. We derived the precision of representation of correctly tracked targets using a mixture distribution analysis. Precision declined with target load according to the square-root law up to six targets. This finding is inconsistent with both pure and hybrid slot models. Instead, directional information in MOT appears to be limited by a continuously divisible resource.

Artificial boundary conditions for viscoelastic flows

AbstractThe steady three‐dimensional exterior flow of a viscoelastic non‐Newtonian fluid is approximated by reducing the corresponding nonlinear elliptic–hyperbolic system to a bounded domain. On the truncation surface with a large radius R, nonlinear, local second‐order artificial boundary conditions are constructed and a new concept of an artificial transport equation is introduced. Although the asymptotic structure of solutions at infinity is known, certain attributes cannot be found explicitly so that the artificial boundary conditions must be constructed with incomplete information on asymptotics. To show the existence of a solution to the approximation problem and to estimate the asymptotic precision, a general abstract scheme, adapted to the analysis of coupled systems of elliptic–hyperbolic type, is proposed. The error estimates, obtained in weighted Sobolev norms with arbitrarily large smoothness indices, prove an approximation of order O(R−2+ε), with any ε>0. Our approach, in contrast to other papers on artificial boundary conditions, does not use the standard assumptions on compactly supported right‐hand side f, leads, in particular, to pointwise estimates and provides error bounds with constants independent of both R and f. Copyright © 2007 John Wiley & Sons, Ltd.