Finite Sample Effects Research Articles

Representing turbulent statistics with partitions of state space. Part 1. Theory and methodology

This is the first of a two-part paper. We formulate a data-driven method for constructing finite-volume discretizations of an arbitrary dynamical system's underlying Liouville/Fokker–Planck equation. A method is employed that allows for flexibility in partitioning state space, generalizes to function spaces, applies to arbitrarily long sequences of time-series data, is robust to noise and quantifies uncertainty with respect to finite sample effects. After applying the method, one is left with Markov states (cell centres) and a random matrix approximation to the generator. When used in tandem, they emulate the statistics of the underlying system. We illustrate the method on the Lorenz equations (a three-dimensional ordinary differential equation) saving a fluid dynamical application for Part 2 (Souza, J. Fluid Mech., vol. 997, 2024, A2).

Free-field method for inverse characterization of finite porous acoustic materials using feed forward neural networks.

This paper presents a free-field method for inverse estimation of acoustic porous material parameters from sound pressure measurements above small rectangular samples. The finite sample effect, the spherical propagation of the sound field, and a potential lateral material reaction are considered. Using an extensive series of systematically varied finite element simulations, neural network models are developed to replace computationally expensive simulations as a forward model for the calculation of the complex sound pressure above small samples in the inverse optimization. The method is experimentally validated using various porous material samples. The results show that the influence of the finite sample size is successfully removed and thus, the acoustic properties of the materials can be estimated from the determined porous parameters with high accuracy, even based on a single sound pressure measurement over small samples with pronounced edge diffraction. The poroacoustic parameters hence derived can be used directly, e.g., in simulation applications, or to calculate complex surface impedances or absorption coefficients.

Energy-based clustering: Fast and robust clustering of data with known likelihood functions.

Clustering has become an indispensable tool in the presence of increasingly large and complex datasets. Most clustering algorithms depend, either explicitly or implicitly, on the sampled density. However, estimated densities are fragile due to the curse of dimensionality and finite sampling effects, for instance, in molecular dynamics simulations. To avoid the dependence on estimated densities, an energy-based clustering (EBC) algorithm based on the Metropolis acceptance criterion is developed in this work. In the proposed formulation, EBC can be considered a generalization of spectral clustering in the limit of large temperatures. Taking the potential energy of a sample explicitly into account alleviates requirements regarding the distribution of the data. In addition, it permits the subsampling of densely sampled regions, which can result in significant speed-ups and sublinear scaling. The algorithm is validated on a range of test systems including molecular dynamics trajectories of alanine dipeptide and the Trp-cage miniprotein. Our results show that including information about the potential-energy surface can largely decouple clustering from the sampled density.

Effects of finite sampling on fatigue damage estimation of wind turbine components: A statistical study

The variability of the wind turbine loads complicates fatigue assessment in the design phase, as performing simulations covering the entire lifetime is computationally expensive. The current work provides important information for assessing the uncertainty in fatigue damage estimation due to finite data. We study the sample size effect on mean, variance, and skewness of damage in each wind bin, identify the important wind bins, and study the uncertainty propagation from each wind bin to the lifetime damage using 3600 aeroelastic simulations and bootstrapping. To achieve less than 1% error in the damage estimation across all load channels in the current case study, at least 100 turbulence seeds are needed. Damage in different wind bins follows a lognormal distribution when using the conventional approach of six seeds. The provided insights and information allow the designer to achieve a specific level of accuracy for a given computational cost using strategic bin sampling.

Observational Evidence for Large-scale Gas Heating in a Galaxy Protocluster at z = 2.30

We report a z = 2.30 galaxy protocluster (COSTCO-I) in the COSMOS field, where the Lyα forest as seen in the CLAMATO IGM tomography survey does not show significant absorption. This departs from the transmission–density relationship (often dubbed the fluctuating Gunn–Peterson approximation; FGPA) usually expected to hold at this epoch, which would lead one to predict strong Lyα absorption at the overdensity. For comparison, we generate mock Lyα forest maps by applying the FGPA to constrained simulations of the COSMOS density field and create mocks that incorporate the effects of finite sight-line sampling, pixel noise, and Wiener filtering. Averaged over r = 15 h −1 Mpc around the protocluster, the observed Lyα forest is consistently more transparent in the real data than in the mocks, indicating a rejection of the null hypothesis that the gas in COSTCO-I follows the FGPA (p = 0.0026, or 2.79σ significance). It suggests that the large-scale gas associated with COSTCO-I is being heated above the expectations of the FGPA, which might be due to either large-scale AGN jet feedback or early gravitational shock heating. COSTCO-I is the first known large-scale region of the IGM that is observed to be transitioning from the optically thin photoionized regime at cosmic noon to eventually coalesce into an intracluster medium (ICM) by z = 0. Future observations of similar structures will shed light on the growth of the ICM and allow constraints on AGN feedback mechanisms.

Estimating population size: The importance of model and estimator choice.

We consider estimator and model choice when estimating abundance from capture-recapture data. Our work is motivated by a mark-recapture distance sampling example, where model and estimator choice led to unexpectedly large disparities in the estimates. To understand these differences, we look at three estimation strategies (maximum likelihood estimation, conditional maximum likelihood estimation, and Bayesian estimation) for both binomial and Poisson models. We show that assuming the data have a binomial or multinomial distribution introduces implicit and unnoticed assumptions that are not addressed when fitting with maximum likelihood estimation. This can have an important effect in finite samples, particularly if our data arise from multiple populations. We relate these results to those of restricted maximum likelihood in linear mixed effectsmodels.

Uncertainty-aware data-driven predictive control in a stochastic setting

Data-Driven Predictive Control (DDPC) has been recently proposed as an effective alternative to traditional Model Predictive Control (MPC), in that the same constrained optimization problem can be addressed without the need to explicitly identify a full model of the plant. However, DDPC is built upon input/output trajectories. Therefore, the finite sample effect of stochastic data, due to, e.g., measurement noise, may have a detrimental impact on closed-loop performance. Exploiting a formal statistical analysis of the prediction error, in this paper we propose the first systematic approach to deal with uncertainty due to finite sample effects. To this end, we introduce two regularization strategies for which, differently from existing regularization-based DDPC techniques, we propose a tuning rationale allowing us to select the regularization hyper-parameters before closing the loop and without additional experiments. Simulation results confirm the potential of the proposed strategy when closing the loop.

Multi-axis control of a qubit in the presence of unknown non-Markovian quantum noise

In this paper, we consider the problem of open-loop control of a qubit that is coupled to an unknown fully quantum non-Markovian noise (either bosonic or fermionic). A graybox model that is empirically obtained from measurement data is employed to approximately represent the unknown quantum noise. The estimated model is then used to calculate the open-loop control pulses under constraints on the pulse amplitude and timing. For the control pulse optimization, we explore the use of gradient descent and genetic optimization methods. We consider the effect of finite sampling on estimating expectation values of observables and show results for single- and multi-axis control of a qubit.

Manifold-adaptive dimension estimation revisited.

Data dimensionality informs us about data complexity and sets limit on the structure of successful signal processing pipelines. In this work we revisit and improve the manifold adaptive Farahmand-Szepesvári-Audibert (FSA) dimension estimator, making it one of the best nearest neighbor-based dimension estimators available. We compute the probability density function of local FSA estimates, if the local manifold density is uniform. Based on the probability density function, we propose to use the median of local estimates as a basic global measure of intrinsic dimensionality, and we demonstrate the advantages of this asymptotically unbiased estimator over the previously proposed statistics: the mode and the mean. Additionally, from the probability density function, we derive the maximum likelihood formula for global intrinsic dimensionality, if i.i.d. holds. We tackle edge and finite-sample effects with an exponential correction formula, calibrated on hypercube datasets. We compare the performance of the corrected median-FSA estimator with kNN estimators: maximum likelihood (Levina-Bickel), the 2NN and two implementations of DANCo (R and MATLAB). We show that corrected median-FSA estimator beats the maximum likelihood estimator and it is on equal footing with DANCo for standard synthetic benchmarks according to mean percentage error and error rate metrics. With the median-FSA algorithm, we reveal diverse changes in the neural dynamics while resting state and during epileptic seizures. We identify brain areas with lower-dimensional dynamics that are possible causal sources and candidates for being seizure onset zones.

Geometrical meaning of statistical isotropy of smooth random fields in two dimensions

We revisit the geometrical meaning of statistical isotropy that is manifest in excursion sets of smooth random fields in two dimensions. Using the contour Minkowski tensor, ${\mathcal{W}}_{1}$, as our basic tool we first examine geometrical properties of single structures. For simple closed curves in two dimensions we show that ${\mathcal{W}}_{1}$ is proportional to the identity matrix if the curve has $m$-fold symmetry, with $m\ensuremath{\ge}3$. Then we elaborate on how ${\mathcal{W}}_{1}$ maps any arbitrary shaped simple closed curve to an ellipse that is unique up to translations of its centroid. We also carry out a comparison of the shape parameters, $\ensuremath{\alpha}$ and $\ensuremath{\beta}$, defined using ${\mathcal{W}}_{1}$, with the filamentarity parameter defined using two scalar Minkowski functionals---area and contour length. We show that they contain complementary shape information, with ${\mathcal{W}}_{1}$ containing additional information of orientation of structures. Next, we apply our method to boundaries of excursion sets of random fields and examine what statistical isotropy means for the geometry of the excursion sets. Focusing on Gaussian isotropic fields, and using a seminumerical approach we quantify the effect of finite sampling of the field on the geometry of the excursion sets. In doing so we obtain an analytic expression for $\ensuremath{\alpha}$ which takes into account the effect of finite sampling. Finally we derive an analytic expression for the ensemble expectation of ${\mathcal{W}}_{1}$ for Gaussian anisotropic random fields. Our results provide insights that are useful for designing tests of statistical isotropy using cosmological data.

Data-Driven Langevin Modeling of Nonequilibrium Processes.

Given nonstationary data from molecular dynamics simulations, a Markovian Langevin model is constructed that aims to reproduce the time evolution of the underlying process. While at equilibrium the free energy landscape is sampled, nonequilibrium processes can be associated with a biased energy landscape, which accounts for finite sampling effects and external driving. When the data-driven Langevin equation (dLE) approach [Phys. Rev. Lett. 2015, 115, 050602] is extended to the modeling of nonequilibrium processes, an efficient way to calculate multidimensional Langevin fields is outlined. The dLE is shown to correctly account for various nonequilibrium processes, including the enforced dissociation of sodium chloride in water, the pressure-jump induced nucleation of a liquid of hard spheres, and the conformational dynamics of a helical peptide sampled from nonstationary short trajectories.

Synchronized oscillations in growing cell populations are explained by demographic noise

Understanding synchrony in growing populations is important for applications as diverse as epidemiology and cancer treatment. Recent experiments employing fluorescent reporters in melanoma cell lines have uncovered growing subpopulations exhibiting sustained oscillations, with nearby cells appearing to synchronize their cycles. In this study, we demonstrate that the behavior observed is consistent with long-lasting transient phenomenon initiated and amplified by the finite-sample effects and demographic noise. We present a novel mathematical analysis of a multistage model of cell growth, which accurately reproduces the synchronized oscillations. As part of the analysis, we elucidate the transient and asymptotic phases of the dynamics and derive an analytical formula to quantify the effect of demographic noise in the appearance of the oscillations. The implications of these findings are broad, such as providing insight into experimental protocols that are used to study the growth of asynchronous populations and, in particular, those investigations relating to anticancer drug discovery.

Mi3-GPU: MCMC-based inverse Ising inference on GPUs for protein covariation analysis

Inverse Ising inference is a method for inferring the coupling parameters of a Potts/Ising model based on observed site-covariation, which has found important applications in protein physics for detecting interactions between residues in protein families. We introduce Mi3-GPU (“mee-three”, for MCMC Inverse Ising Inference) software for solving the inverse Ising problem for protein-sequence datasets with few analytic approximations, by parallel Markov-Chain Monte Carlo sampling on GPUs. We also provide tools for analysis and preparation of protein-family Multiple Sequence Alignments (MSAs) to account for finite-sampling issues, which are a major source of error or bias in inverse Ising inference. Our method is “generative” in the sense that the inferred model can be used to generate synthetic MSAs whose mutational statistics (marginals) can be verified to match the dataset MSA statistics up to the limits imposed by the effects of finite sampling. Our GPU implementation enables the construction of models which reproduce the covariation patterns of the observed MSA with a precision that is not possible with more approximate methods. The main components of our method are a GPU-optimized algorithm to greatly accelerate MCMC sampling, combined with a multi-step Quasi-Newton parameter-update scheme using a “Zwanzig reweighting” technique. We demonstrate the ability of this software to produce generative models on typical protein family datasets for sequence lengths L∼300 with 21 residue types with tens of millions of inferred parameters in short running times. Program summaryProgram Title: Mi3-GPUProgram Files doi:http://dx.doi.org/10.17632/ftbcfy2p35.1Licensing provisions: GPLv3Programming languages: Python3, OpenCL, CNature of problem: Mi3-GPU solves the inverse Ising problem for application in protein covariation analysis. The goal is to infer “coupling” parameters between positions in a Multiple Sequence Alignment of a protein family, with many applications including protein-contact prediction and fitness prediction.Solution method: Mi3-GPU solves the inverse Ising problem with few approximations using Markov-Chain Monte Carlo methods with Quasi-Newton optimization on GPUs. This problem previously has been approached by more approximate methods using analytic approximations including “message Passing”, “Susceptibility Propagation”, “mean-field” methods, pseudolikelihood approximations, and cluster expansion. The software leverages GPU to accelerate MCMC sampling and a histogram reweighting technique to accelerate parameter optimization.

Integrating Molecular Simulation and Experimental Data: A Bayesian/Maximum Entropy Reweighting Approach.

We describe a Bayesian/Maximum entropy (BME) procedure and software to construct a conformational ensemble of a biomolecular system by integrating molecular simulations and experimental data. First, an initial conformational ensemble is constructed using, for example, Molecular Dynamics or Monte Carlo simulations. Due to potential inaccuracies in the model and finite sampling effects, properties predicted from simulations may not agree with experimental data. In BME we use the experimental data to refine the simulation so that the new conformational ensemble has the following properties: (1) the calculated averages are close to the experimental values taking uncertainty into account and (2) it maximizes the relative Shannon entropy with respect to the original simulation ensemble. The output of this procedure is a set of optimized weights that can be used to calculate other properties and distributions of these. Here, we provide a practical guide on how to obtain and use such weights, how to choose adjustable parameters and discuss shortcomings of the method.

A New Variable Forgetting Factor-Based Bias-Compensated RLS Algorithm for Identification of FIR Systems With Input Noise and Its Hardware Implementation

This paper proposes a new variable forgetting factor QRD-based recursive least squares algorithm with bias compensation (VFF-QRRLS-BC) for system identification under input noise. A new variable forgetting factor scheme is proposed to improve its convergence speed and steady-state mean squares error. A new method for recursive estimation of the additive noise variance is also proposed for reliable bias compensation. The mean and mean-square asymptotic behaviors of the algorithm are analyzed and a self-calibration scheme is further proposed to improve the steady-state mean squares error (MSE) due to finite sample effect. Simulations show that the proposed VFF approach offers improved tracking and steady-state MSE performance over the conventional recursive least squares method and its fixed FF counterpart. A linear array architecture is proposed for the realization of this algorithm and several hardware efficient techniques are introduced to avoid the expensive cubic root and division operations required. The proposed algorithm is validated on Xilinx Zynq®-7000 AP SoC ZC702 Field Programmable Gate Array (FPGA). For a 10-tap finite impulse response (FIR) system, the implementation requires only about 11.5k slice look-up table (LUT)s, 4.5k slice registers and 50 DSP48s and it can work up to about 0.58 MHz sample rate with a 200 MHz system clock. The hardware resources are considerably lower than traditional techniques using divider and cubic root realization. The linear array architecture also serves as an attractive alternative to the systolic array in medium to low rate applications due to its reduced hardware usages.

Constant <sc>on/off</sc>-Time Hybrid Modulation in Digital Current-Mode Control Using Event-Based Sampling

Ripple-based constant on -time and off -time control techniques can achieve fast step-down and step-up transient performance along with an improved light-load efficiency. However, they suffer from varying switching frequency at the steady state and often require a phase-locked loop for frequency regulation. This paper proposes a constant on/off -time hybrid modulation technique in a digitally current-mode controlled synchronous buck converter. The proposed technique considers an event-based sampling mechanism using one sample per switching cycle, which makes it useful for high-frequency implementation. This requires a monoshot timer and a digital voltage controller $G_c(z)$ . The proposed technique can be configured to a constant on -time or off -time modulator with seamless transition because of sharing $G_c(z)$ ; thus, improved step-up/down transient performance can be retained. Furthermore, a discrete-time framework is proposed for fast-scale stability analysis, and the effects of finite sampling and practical parasitics are discussed. Discrete-time small-signal models are derived for the direct digital control design with enhanced stability and performance. A frequency adaptation method is discussed to customize the switching frequency in real time. A buck converter prototype is tested, and the usefulness of the proposed modulation technique is verified experimentally.

Influence of multiple-sequence-alignment depth on Potts statistical models of protein covariation.

Potts statistical models have become a popular and promising way to analyze mutational covariation in protein multiple sequence alignments (MSAs) in order to understand protein structure, function, and fitness. But the statistical limitations of these models, which can have millions of parameters and are fit to MSAs of only thousands or hundreds of effective sequences using a procedure known as inverse Ising inference, are incompletely understood. In this work we predict how model quality degrades as a function of the number of sequences N, sequence length L, amino-acid alphabet size q, and the degree of conservation of the MSA, in different applications of the Potts models: in "fitness" predictions of individual protein sequences, in predictions of the effects of single-point mutations, in "double mutant cycle" predictions of epistasis, and in 3D contact prediction in protein structure. We show how as MSA depth N decreases an "overfitting" effect occurs such that sequences in the training MSA have overestimated fitness, and we predict the magnitude of this effect and discuss how regularization can help correct for it, using a regularization procedure motivated by statistical analysis of the effects of finite sampling. We find that as N decreases the quality of point-mutation effect predictions degrade least, fitness and epistasis predictions degrade more rapidly, and contact predictions are most affected. However, overfitting becomes negligible for MSA depths of more than a few thousand effective sequences, as often used in practice, and regularization becomes less necessary. We discuss the implications of these results for users of Potts covariation analysis.

Tractable Bayesian Variable Selection: Beyond Normality

ABSTRACTBayesian variable selection often assumes normality, but the effects of model misspecification are not sufficiently understood. There are sound reasons behind this assumption, particularly for large p: ease of interpretation, analytical, and computational convenience. More flexible frameworks exist, including semi- or nonparametric models, often at the cost of some tractability. We propose a simple extension that allows for skewness and thicker-than-normal tails but preserves tractability. It leads to easy interpretation and a log-concave likelihood that facilitates optimization and integration. We characterize asymptotically parameter estimation and Bayes factor rates, under certain model misspecification. Under suitable conditions, misspecified Bayes factors induce sparsity at the same rates than under the correct model. However, the rates to detect signal change by an exponential factor, often reducing sensitivity. These deficiencies can be ameliorated by inferring the error distribution, a simple strategy that can improve inference substantially. Our work focuses on the likelihood and can be combined with any likelihood penalty or prior, but here we focus on nonlocal priors to induce extra sparsity and ameliorate finite-sample effects caused by misspecification. We show the importance of considering the likelihood rather than solely the prior, for Bayesian variable selection. The methodology is in R package ‘mombf.’ Supplementary materials for this article are available online.

Effects of power inversion spatial only adaptive array on GNSS receiver measurements

The Spatial Only Processing Power Inversion (SOP-PI) algorithm is frequently used in Global Navigation Satellite System (GNSS) adaptive array receivers for interference mitigation because of its simplicity of implementation. This study investigates the effects of SOP-PI on receiver measurements for high-precision applications. Mathematical deductions show that if an array with a centro-symmetrical geometry is used, ideally, SOP-PI is naturally bias-free; however, this no longer stands when non-ideal factors, including array perturbations and finite-sample effect, are added. Simulations are performed herein to investigate how exactly the array perturbations affect the carrier phase biases, while diagonal loading and forward-backward averaging are proposed to counter the finite-sample effect. In conclusion, whether SOP-PI with a centro-symmetrical array geometry will satisfy the high precision demands mainly depends on the array perturbation degree of the element amplitude and the phase center.

Optical sectioning enhancement using higher-order moment signals in random speckle-structured illumination microscopy.

The optical sectioning capability of structured illumination using random speckle patterns is shown by simulations to improve when cumulants beyond the traditional second order are used as the image-forming signals. The improvement scales with the cumulant order, asymptotically approaching confocal performance. As actual experiments operate with finite-size sample estimators instead of true cumulants, purely statistical (nonoptical) effects can result in nonideal behavior. We analyze the finite ensemble effects along with the experimental effects of detector dynamic range through Monte Carlo simulations. Despite these real-world factors, we show when the third-order-derived signal can demonstrate improved sectioning at good signal-to-noise levels set by finite-sample effects.