Optimal Bin Width Research Articles

Statistical and Computational Methods for Comparative Proteomic Profiling Using Liquid Chromatography-Tandem Mass Spectrometry

The combined method of LC-MS/MS is increasingly being used to explore differences in the proteomic composition of complex biological systems. The reliability and utility of such comparative protein expression profiling studies is critically dependent on an accurate and rigorous assessment of quantitative changes in the relative abundance of the myriad of proteins typically present in a biological sample such as blood or tissue. In this review, we provide an overview of key statistical and computational issues relevant to bottom-up shotgun global proteomic analysis, with an emphasis on methods that can be applied to improve the dependability of biological inferences drawn from large proteomic datasets. Focusing on a start-to-finish approach, we address the following topics: 1) low-level data processing steps, such as formation of a data matrix, filtering, and baseline subtraction to minimize noise, 2) mid-level processing steps, such as data normalization, alignment in time, peak detection, peak quantification, peak matching, and error models, to facilitate profile comparisons; and, 3) high-level processing steps such as sample classification and biomarker discovery, and related topics such as significance testing, multiple testing, and choice of feature space. We report on approaches that have recently been developed for these steps, discussing their merits and limitations, and propose areas deserving of further research.

Histogram bin width selection for time-dependent Poisson processes

In constructing a time histogram of the event sequences derived from a nonstationary point process, we wish to determine the bin width such that the mean squared error of the histogram from the underlying rate of occurrence is minimized. We find that the optimal bin widths obtained for a doubly stochastic Poisson process and a sinusoidally regulated Poisson process exhibit different scaling relations with respect to the number of sequences, time scale and amplitude of rate modulation, but both diverge under similar parametric conditions. This implies that under these conditions, no determination of the time-dependent rate can be made. We also apply the kernel method to these point processes, and find that the optimal kernels do not exhibit any critical phenomena, unlike the time histogram method.

Frequency polygons for weakly dependent processes

The purpose of this paper is to investigate the frequency polygon as a density estimator for stationary strong mixing processes. Optimal bin widths which asymptotically minimize integrated mean square 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 ( (n −1 logn) 1 3 under general conditions. Frequency polygons thus appear to be very good density estimators with respect to both criteria of IMSE and uniform convergence.

Data-Based Choice of Histogram Bin Width

The most important parameter of a histogram is the bin width because it controls the tradeoff between presenting a picture with too much detail (“undersmoothing”) or too little detail (“oversmoothing”) with respect to the true distribution. Despite this importance there has been surprisingly little research into estimation of the “optimal” bin width. Default bin widths in most common statistical packages are, at least for large samples, quite far from the optimal bin width. Rules proposed by, for example, Scott lead to better large sample performance of the histogram, but are not consistent themselves. In this paper we extend the bin width rules of Scott to those that achieve root-n rates of convergence to the L 2-optimal bin width, thereby providing firm scientific justification for their use. Moreover, the proposed rules are simple, easy and fast to compute, and perform well in simulations.

Optimal bin widths for frequency histograms and ROC curves

Optimal bin widths for frequency histograms and ROC curves

Density estimation for time series by histograms

Histogram estimates of the density of time series are studied. optimal bin widths which asymptotically minimize integrated mean square errors are derived. Uniform strong consistency of the estimates and their rates of convergence are obtained. Histogram estimates can achieve the uniform optimal rate of convergence (n−1logn)13 for appropriate choices of the bin widths. The time series are assumed to satisfy some strong mixing or absolute regularity conditions.

On derivaton and application of aic as a data-based criterion for histograms

In this paper a derivation of the Akaike's Information Criterion (AIC) is presented to select the number of bins of a histogram given only the data, showing that AIC strikes a balance between the “bias” and “variance” of the histogram estimate. Consistency of the criterion is discussed, an asymptotically optimal histogram bin width for the criterion is derived and its relationship to penalized likelihood methods is shown. A formula relating the optimal number of bins for a sample and a sub-sample obtained from it is derived. A number of numerical examples are presented.

The construction of optimal histograms

In this note, a stopping rule is derived that gives the optimal histogram bin width to approximate a probability density function, The procedure minimizes the mean integrated square error between the density function and the histogram, and requires no prior knowledge of the density function.

On Optimal and Data-Based Histograms

In this paper the formula for the optimal histogram bin width is derived which asymptotically minimizes the integrated mean squared error. Monte Carlo methods are used to verify the usefulness of this formula for small samples. A data-based procedure for choosing the bin width parameter is proposed, which assumes a Gaussian reference standard and requires only the sample size and an estimate of the standard deviation. The sensitivity of the procedure is investigated using several probability models which violate the Gaussian assumption.

On optimal and data-based histograms

SUMMARY In this paper the formula for the optimal histogram bin width is derived which asymptotically minimizes the integrated mean squared error. Monte Carlo methods are used to verify the usefulness of this formula for small samples. A data-based procedure for choosing the bin width parameter is proposed, which assumes a Gaussian reference standard and requires only the sample size and an estimate of the standard deviation. The sensitivity of the procedure is investigated using several probability models which violate the Gaussian assumption.