Optimal Bin Width Research Articles

Determining minimum sample size for the conditioned Latin hypercube sampling algorithm

In digital soil mapping (DSM), a fundamental assumption is that the spatial variability of the target variable can be explained by the predictors or environmental covariates. Strategies to adequately sample the predictors have been well documented, with the conditioned Latin hypercube sampling (cLHS) algorithm receiving the most attention in the DSM community. Despite advances in sampling design, a critical gap remains in determining the number of samples required for DSM projects. We propose a simple workflow and function coded in R language to determine the minimum sample size for the cLHS algorithm based on histograms of the predictor variables using the Freedman-Diaconis rule for determining optimal bin width. Data preprocessing was included to correct for multimodal and non-normally distributed data, as these can affect sample size determination from the histogram. Based on a user-selected quantile range (QR) for the sample plan, the densities of the histogram bins at the upper and lower bounds of the QR were used as a scaling factor to determine minimum sample size. This technique was applied to a field-scale set of environmental covariates for a well-sampled agricultural study site near Guelph, Ontario, Canada, and tested across a range of QRs. The results showed increasing minimum sample size with an increase in the QR selected. Minimum sample size increased from 44 to 83 when the QR increased from 50% to 95% and then increased exponentially to 194 for the 99% QR. This technique provides an estimate of minimum sample size that can be used as an input to the cLHS algorithm.

Optimization and inference of bin widths for histogramming inelastic neutron scattering spectra.

A data-driven bin-width optimization for the histograms of measured data sets based on inhomogeneous Poisson processes was developed in a neurophysiology study [Shimazaki & Shinomoto (2007). Neural Comput. 19, 1503-1527], and a subsequent study [Muto, Sakamoto, Matsuura, Arima & Okada (2019). J. Phys. Soc. Jpn, 88, 044002] proposed its application to inelastic neutron scattering (INS) data. In the present study, the results of the method on experimental INS time-of-flight data collected under different measurement conditions from a copper single crystal are validated. The extrapolation of the statistics on a given data set to other data sets with different total counts precisely infers the optimal bin widths on the latter. The histograms with the optimized bin widths statistically verify two fine-spectral-feature examples in the energy and momentum transfer cross sections: (i) the existence of phonon band gaps; and (ii) the number of plural phonon branches located close to each other. This indicates that the applied method helps in the efficient and rigorous observation of spectral structures important in physics and materials science like novel forms of magnetic excitation and phonon states correlated to thermal conductivities.

Extracellular voltage thresholds for maximizing information extraction in primate auditory cortex: implications for a brain computer interface

Objective. Research by Oby (2016 J. Neural. Eng. 13 036009) demonstrated that the optimal threshold for extracting information from visual and motor cortices may differ from the optimal threshold for identifying single neurons via spike sorting methods. The optimal threshold for extracting information from auditory cortex has yet to be identified, nor has the optimal temporal scale for representing auditory cortical activity. Here, we describe a procedure to jointly optimize the extracellular threshold and bin size with respect to the decoding accuracy achieved by a linear classifier for a diverse set of auditory stimuli. Approach. We used linear multichannel arrays to record extracellular neural activity from the auditory cortex of awake squirrel monkeys passively listening to both simple and complex sounds. We executed a grid search of the coordinate space defined by the voltage threshold (in units of standard deviation) and the bin size (in units of milliseconds), and computed decoding accuracy at each point. Main results. The optimal threshold for information extraction was consistently near two standard deviations below the voltage trace mean, which falls significantly below the range of three to five standard deviations typically used as inputs to spike sorting algorithms in basic research and in brain-computer interface (BCI) applications. The optimal binwidth was minimized at the optimal voltage threshold, particularly for acoustic stimuli dominated by temporally dynamic features, indicating that permissive thresholding permits readout of cortical responses with temporal precision on the order of a few milliseconds. Significance. The improvements in decoding accuracy we observed for optimal readout parameters suggest that standard thresholding methods substantially underestimate the information present in auditory cortical spiking patterns. The fact that optimal thresholds were relatively low indicates that local populations of cortical neurons exhibit high temporal coherence that could be leveraged in service of future auditory BCI applications.

Enhanced MDT-Based Performance Estimation for AI Driven Optimization in Future Cellular Networks

Minimization of drive test (MDT) allows coverage estimation at a base station by leveraging measurement reports gathered at the user equipment (UE) without the need for drive tests. Therefore, MDT is a key enabling feature for data and artificial intelligence driven autonomous operation and optimization in future cellular networks. However, to date, the utility of MDT feature remains thwarted by issues such as sparsity of user reports and user positioning inaccuracy. We characterize three key types of errors in MDT-based coverage estimation that stem from inaccurate user positioning, scarcity of user reports and quantization. For the first time, the presented analysis shows existence of joint interplay between these errors on coverage estimation that result from inter-dependency between positioning error and bin width. The analysis also shows that there exists an optimal bin width for given user positioning inaccuracy and user density that minimizes the overall error in MDT-based estimated coverage. Utility of our framework is presented by addressing two applications from network optimization perspective: determining optimal bin width to maximize accuracy of MDT-based coverage estimation and its calibration to further improve its accuracy.

Optimal Bin Width for Autonomous Coverage Estimation Using MDT Reports in the Presence of User Positioning Error

Minimization of drive test (MDT) allows coverage to be estimated at the base station using user equipment measurement reports with the objective of eliminating the need for drive tests. In this letter, we quantify various types of errors in MDT-based autonomous coverage estimation that stem from inaccurate user positioning, for example, as a result of GPS measurement uncertainties and quantization due to dividing the coverage area into bins. By investigating the interplay between quantization and positioning error to estimate coverage, we show that there exists an optimal bin width for coverage estimation and determine it as a function of positioning error and user density. This can enable network operators to configure the bin size for given positioning accuracy that results in the most accurate MDT-based coverage estimation.

Suppression of Multimodality in Inter-Spike Interval Distribution: Role of External Damped Oscillatory Input.

A theoretical model based on the generalized neuronal model with distributed delay (GNMDD) is proposed to generate multimodal/bimodal inter-spike interval (ISI) distribution. The distributed delay is assumed to be general in the sense that it is a linear combination of gamma distributed weak and strong memory kernels. It is found that the expected membrane potential in the subthreshold regime exhibits damped oscillatory behavior. This causes the ISI pattern to become bimodal/multimodal. Furthermore, the effect of external damped oscillatory current in the GNMDD model is investigated. Based on extensive simulation study of a set of coupled stochastic differential equations in the presence of a threshold, a vast amount of spiking data are generated. Using the non-parametric modeling approach, the optimal bin width of the histogram is obtained by using the Freedman-Diaconis rule. It is to be noted that, with the increasing amplitude of the damped oscillatory current, the multimodal ISI distribution changes to unimodal ISI distribution when the magnitude of external current reaches a critical value. It is found that the entropy also shows a sudden transition around the critical point. This phenomenon is akin to a phase transition.

The Rate of Asymptotic Normality of Frequency Polygon Density Estimation for Spatial Random Fields

This paper is to investigate the convergence rate of asymptotic normality of frequency polygon estimation for density function under mixing random fields, which include strongly mixing condition and some weaker mixing conditions. A Berry-Esseen bound of frequency polygon is established and the convergence rates of asymptotic normality are derived. In particularly, for the optimal bin width , it is showed that the convergence rate of asymptotic normality reaches to when mixing coefficient tends to zero exponentially fast.

Choosing noise dose bin widths when modeling community response to noise

While there are many studies on how communities respond to noise, there are few that explicitly define how the noise dose, used to fit community response models, are binned. When modeling the dose response relationship between noise and community annoyance, the choice of the noise bin width can drastically affect the community response or tolerance models fit to the data. To explore the optimal noise bin width, we fit various dose response models to bin widths between 1 and 10 dB and data that had been divided into bins of varying widths but an equal number of points in each bin. Additionally, we fit the models using a weighted regression procedure that weights each bin by its number of points, and find that this approach mitigates the modeling errors implicit to the (often arbitrary) noise bin width decision.

Biases in biomass estimates: The effect of bin width in size-structured stock assessment methods

Size-structured assessment models require size data to be binned, but it is unclear how a specified bin width can impact the ability of an assessment to estimate quantities important in management. Biases in estimated mature biomass are demonstrated here when increasing the bin width using a simulation framework based on the Pribilof Islands red king crab (PIRKC) assessment. Several potential sources of bias were explored through simulation. The bias was removed by applying an alternative assessment method which used a finer approximation of the integral representing the probability of a crab molting from one size bin. Trade-offs exist at both sides of the bin width spectrum. Precision and model stability may suffer when bin width is too large; slowed parameter estimation and problems fitting multinomial likelihoods to length frequency data may appear when bin widths are too small. Simulation studies (like the one presented here) are useful for identifying optimal bin width for management related tasks.

Statistical computation of Boltzmann entropy and estimation of the optimal probability density function from statistical sample

In this work, we investigate the statistical computation of the Boltzmann entropy of statistical samples. For this purpose, we use both histogram and kernel function to estimate the probability density function of statistical samples. We find that, due to coarse-graining, the entropy is a monotonic increasing function of the bin width for histogram or bandwidth for kernel estimation, which seems to be difficult to select an optimal bin width/bandwidth for computing the entropy. Fortunately, we notice that there exists a minimum of the first derivative of entropy for both histogram and kernel estimation, and this minimum point of the first derivative asymptotically points to the optimal bin width or bandwidth. We have verified these findings by large amounts of numerical experiments. Hence, we suggest that the minimum of the first derivative of entropy be used as a selector for the optimal bin width or bandwidth of density estimation. Moreover, the optimal bandwidth selected by the minimum of the first derivative of entropy is purely data-based, independent of the unknown underlying probability density distribution, which is obviously superior to the existing estimators. Our results are not restricted to one-dimensional, but can also be extended to multivariate cases. It should be emphasized, however, that we do not provide a robust mathematical proof of these findings, and we leave these issues with those who are interested in them.

Multifractal diffusion entropy analysis: Optimal bin width of probability histograms

In the framework of Multifractal Diffusion Entropy Analysis we propose a method for choosing an optimal bin-width in histograms generated from underlying probability distributions of interest. The method presented uses techniques of Rényi’s entropy and the mean squared error analysis to discuss the conditions under which the error in the multifractal spectrum estimation is minimal. We illustrate the utility of our approach by focusing on a scaling behavior of financial time series. In particular, we analyze the S&P500 stock index as sampled at a daily rate in the time period 1950–2013. In order to demonstrate a strength of the method proposed we compare the multifractal δ-spectrum for various bin-widths and show the robustness of the method, especially for large values of q. For such values, other methods in use, e.g., those based on moment estimation, tend to fail for heavy-tailed data or data with long correlations. Connection between the δ-spectrum and Rényi’s q parameter is also discussed and elucidated on a simple example of multiscale time series.

The effect of temporal separation of two Geiger mode avalanche photodiodes and variation of time bin widths on detection probabilities in LADAR system using two Geiger mode avalanche photodiodes

In this paper, the effect of temporal separation of two Geiger mode avalanche photodiodes and the variation of time bin widths on detection probabilities in LADAR system using two Geiger mode avalanche photodiodes (GmAPDs) was investigated. The system is implemented by using two GmAPDs with a beam splitter and applying comparative process to their ends. Then, the timing circuitry receives the electrical signals only if each GmAPD generates the electrical signals simultaneously. Although this system decreases the energy of a laser-return pulse scattered from the target, it is highly effective in reducing the false alarm probability because of the randomly distributed noise on the time domain. The temporal separation of two GmAPDs and the variation of the time bin widths result in the variation of detection probabilities. The experiments were performed to verify the effect of temporal separation of two GmAPDs and the variation of time bin widths on the detection probability in the LADAR system. As a result, the optimal temporal difference and the optimal time bin width were. Using these optimal values, a clear 3D image could be obtained by the simple acquisition of the raw time-of-flight data with high SNR.

Recovering 3D clustering information with angular correlations

We study how to recover the full 3D clustering information of P(\vec{k},z), including redshift space distortions (RSD), from 2D tomography using the angular auto and cross spectra of different redshift bins C_\ell(z,z'). We focus on quasilinear scales where the minimum scale \lambda_{min} or corresponding maximum wavenumber k_{max}= 2\pi/\lambda_{min} is targeted to be between k_{max}={0.05-0.2} h/Mpc. For spectroscopic surveys, we find that we can recover the full 3D clustering information when the redshift bin width \Delta z used in the 2D tomography is similar to the targeted minimum scale, i.e. \Delta z ~ {0.6-0.8} \lambda_{min} H(z)/c which corresponds to \Delta z ~ 0.01-0.05 for z<1. This value of \Delta z is optimal in the sense that larger values of \Delta z lose information, while smaller values violate our minimum scale requirement. For a narrow-band photometric survey, with photo-z error \sigma_z=0.004, we find almost identical results to the spectroscopic survey because the photo-z error is smaller than the optimal bin width \sigma_z<\Delta z. For a typical broad-band photometric survey with \sigma_z=0.1, we have that \sigma_z>\Delta z and most radial information is intrinsically lost. The remaining information can be recovered from the 2D tomography if we use \Delta z ~ 2\sigma_z. While 3D and 2D analysis are shown here to be equivalent, the advantage of using angular positions and redshifts is that we do not need a fiducial cosmology to convert to 3D coordinates. This avoids assumptions and marginalization over the fiducial model. In addition, it becomes straight forward to combine RSD, clustering and weak lensing in 2D space.

Polynomial Histograms for Multivariate Density and Mode Estimation

Abstract. We consider the problem of efficiently estimating multivariate densities and their modes for moderate dimensions and an abundance of data. We propose polynomial histograms to solve this estimation problem. We present first‐ and second‐order polynomial histogram estimators for a general d‐dimensional setting. Our theoretical results include pointwise bias and variance of these estimators, their asymptotic mean integrated square error (AMISE), and optimal binwidth. The asymptotic performance of the first‐order estimator matches that of the kernel density estimator, while the second order has the faster rate of O(n−6/(d+6)). For a bivariate normal setting, we present explicit expressions for the AMISE constants which show the much larger binwidths of the second order estimator and hence also more efficient computations of multivariate densities. We apply polynomial histogram estimators to real data from biotechnology and find the number and location of modes in such data.

A health indicator method for degradation detection of electronic products

A methodology for detecting the gradual health degradation of electronic products is presented by defining a health indicator to represent a product’s health state in a time interval. A health indicator is defined as the weighted sum of a histogram bin’s fractional contribution in a time interval that is based on the data collection rate, the likelihood of change in a product’s performance, and a user’s readiness to accept risk. The unified distance measure, Mahalanobis distance (MD), is used to create histograms where optimal bin-width is calculated using a kernel estimator. Summarization of a product’s performance in a time interval for health and degradation estimation together with the behavior of the MD values can reduce false alarms regarding the presence of fault and increase the capability of detecting intermittent and “no fault found” events. A case study performed on computers is presented. A healthy baseline for the computers was created and a threshold for both fault detection and degradation identification was defined. After fault injection into one of the computers, the increase in the MD values detected the initial change in performance parameters. After a short time period, the health indicator detected the degradation whereas the MD values returned to below threshold as the degradation in this case study was an aftereffect of the damaging event.

Histogram-kernel error and its application for bin width selection in histograms

Histogram and kernel estimators are usually regarded as the two main classical data-based non- parametric tools to estimate the underlying density functions for some given data sets. In this paper we will integrate them and define a histogram-kernel error based on the integrated square error between histogram and binned kernel density estimator, and then exploit its asymptotic properties. Just as indicated in this paper, the histogram-kernel error only depends on the choice of bin width and the data for the given prior kernel densities. The asymptotic optimal bin width is derived by minimizing the mean histogram-kernel error. By comparing with Scott's optimal bin width formula for a histogram, a new method is proposed to construct the data-based histogram without knowledge of the underlying density function. Monte Carlo study is used to verify the usefulness of our method for different kinds of density functions and sample sizes.

Histogram density estimators based upon a fuzzy partition

This paper presents a density estimator based upon a histogram computed on a fuzzy partition. We prove the consistency of this estimator in the Mean Squared Error ( MSE ). We give the optimal bin width of the estimator which minimizes the Asymptotic Integrated Mean Squared Error ( AIMSE ).

Frequency polygons for random fields (density estimation for random fields)

The purpose of this paper is to investigate the frequency polygon as a density estimator for stationary random fields indexed by multidimensional lattice points space. Optimal bin widths which asymptotically minimize integrated errors (IMSE) are derived. Under weak conditions, frequency polygons achieve the same rate of convergence to zero of the IMSE as kernel estimators. They can also attain the optimal uniform rate of convergence under general conditions. Frequency polygons thus appear to be very good density estimators with respect to both criteria of IMSE and uniform convergence.

Optimizing fluctuation analysis of GABA Aergic IPSCs for accurate unitary currents

Nonstationary fluctuation analysis of synaptic currents requires division of currents into bins of time, with little agreement on how to select an optimal bin width. We used simulated inhibitory postsynaptic currents (simIPSCs) in an empirical approach to establish the optimal bin width needed for estimation of the unitary current, i e. We found acceptable accuracy (≤5%) at bin widths shorter than the length of the stationary segment of simIPSCs that persisted when Gaussian noise was added to the simulated currents. We also studied evoked and spontaneous IPSCs mediated by receptors for γ-aminobutyrate (GABA) in thalamic neurons. Similar to simIPSCs, analysis of the IPSCs yielded saturating relationships between bin width and accuracy of unitary current estimate. Whereas standard error decreased, the accuracy of i e estimates increased with decreasing bin width, forming a plateau at bins below 2–3 ms in duration. Using this approach, one can reliably determine the optimal bin width for nonstationary noise analysis.

Polygone des fréquences pour des champs aléatoires

The purpose of this Note is to investigate the frequency polygon as a density estimator for stationary random fields indexed by multidimensional lattice points space. Optimal bin widths which asymptotically minimize integrated errors (IMSE) are derived. Under mild regularity assumptions, frequency polygons achieve the same rate of convergence to zero of the IMSE as kernel estimators. They can also attain the rate of uniform convergence under general conditions. Frequency polygons thus appear to be very good density estimators with respect to both criteria of IMSE and uniform convergence. To cite this article: M. Carbon, C. R. Acad. Sci. Paris, Ser. I 342 (2006).