Normalization in the fitting of data by iterative methods. Application to tracer kinetics and enzyme kinetics

1. The normalization of biochemical data to weight them appropriately for parameter estimation is considered, with reference particularly to data from tracer kinetics and enzyme kinetics. If the data are in replicate, it is recommended that the sum of squared deviations for each experimental variable at each time or concentration point is divided by the local variance at that point. 2. If there is only one observation for each variable at each sampling point, normalization may still be required if the observations cover more than one order of magnitude, but there is no absolute criterion for judging the effect of the weighting that is produced. The goodness of fit that is produced by minimizing the weighted sum of squares of deviations must be judged subjectively. It is suggested that the goodness of fit may be regarded as satisfactory if the data points are distributed uniformly on either side of the fitted curve. A chi-square test may be used to decide whether the distribution is abnormal. The proportion of the residual variance associated with points on one or other side of the fitted curve may also be taken into account, because this gives an indication of the sensitivity of the residual variance to movement of the curve away from particular data points. These criteria for judging the effect of weighting are only valid if the model equation may reasonably be expected to apply to all the data points. 3. On this basis, normalizing by dividing the deviation for each data point by the experimental observation or by the equivalent value calculated by the model equation may both be shown to produce a consistent bias for numerically small observations, the former biasing the curve towards the smallest observations, the latter tending to produce a curve that is above the numerically smaller data points. It was found that dividing each deviation by the mean of observed and calculated variable appropriate to it produces a weighting that is fairly free from bias as judged by the criteria mentioned above. This normalization factor was tested on published data from both tracer kinetics and enzyme kinetics.     

Stochastic Dynamic Model for Estimation of Rate Constants and Their Variances from Noisy and Heterogeneous PET Measurements

Tissue heterogeneity, radioactive decay and measurement noise are the main error sources in compartmental modeling used to estimate the physiologic rate constants of various radiopharmaceuticals from a dynamic PET study. We introduce a new approach to this problem by modeling the tissue heterogeneity with random rate constants in compartment models. In addition, the Poisson nature of the radioactive decay is included as a Poisson random variable in the measurement equations. The estimation problem will be carried out using the maximum likelihood estimation. With this approach, we do not only get accurate mean estimates for the rate constants, but also estimates for tissue heterogeneity within the region of interest and other possibly unknown model parameters, e.g. instrument noise variance, as well. We also avoid the problem of the optimal weighting of the data related to the conventionally used weighted least-squares method. The new approach was tested with simulated time–activity curves from the conventional three compartment – three rate constants model with normally distributed rate constants and with a noise mixture of Poisson and normally distributed random variables. Our simulation results showed that this new model gave accurate estimates for the mean of the rate constants, the measurement noise parameter and also for the tissue heterogeneity, i.e. for the variance of the rate constants within the region of interest.     

脑磁图神经活动源数目的估计

在脑磁图信号的分析中,正确估计出脑磁图神经活动源的数目是进一步分析脑磁图信号的前提。目前广泛采用的信息论方法和主成分分析方法都是根据特征值来确定源的数目,这两种方法在源数目较多、噪声较强的情况下,会导致误判。该文提出了一种噪声调节自动阈值的脑磁图源数目判断方法,利用基于噪声调节的主成分分析并结合聂曼- 皮尔逊准则对脑磁图源数目进行估计。同时,该方法采用了基于小波的噪声方差估计,实现了脑磁图信号中噪声方差的精确估计。通过对基于信息论方法、主成分分析方法以及该文所提议方法的实验结果的比较,表明该文所提议方法能更准确地估计脑磁图源数目,特别是在源数目较多、信噪比较小的情况下,仍能准确地估计脑磁图源数目,具有较大的实际意义。     

A quantitative method for the detection of edges in noisy time-series

A modification of the edge detector of Chung & Kennedy is proposed in which the output provides confidence limits for the presence or absence of sharp edges (steps) in the input waveform. Their switching method with forward and backward averaging windows is retained, but the output approximates an ideal output function equal to the difference in these averages divided by the standard deviation of the noise. Steps are associated with peak output above a pre-set threshold. Formulae for the efficiency and reliability of this ideal detector are derived for input waveforms with Gaussian white noise and sharp edges, and serve as benchmarks for the switching edge detector. Efficiency is kept high if the threshold is a fixed fraction of the step size of interest relative to noise, and reliability is improved by increasing the window width W to reduce false output. For different steps sizes D, the window width for fixed efficiency and reliability scales as 1/D2. Versions with weighted averaging (flat, ramp, triangular) or median averaging but the same window width perform similarly. Binned above-threshold output is used to predict the locations and signs of detected steps, and simulations show that efficiency and reliability are close to ideal. Location times are accurate to order square root of W. Short pulses generate reduced output if the number of data points in the pulse is less than W. They are optimally detected by choosing W as above and collecting data at a rate such that the pulse contains approximately W data points. A Fortran program is supplied.     

Quantifying reproducibility for differential proteomics: noise analysis for protein liquid chromatography-mass spectrometry of human serum

SUMMARY: Using replicated human serum samples, we applied an error model for proteomic differential expression profiling for a high-resolution liquid chromatography-mass spectrometry (LC-MS) platform. The detailed noise analysis presented here uses an experimental design that separates variance caused by sample preparation from variance due to analytical equipment. An analytic approach based on a two-component error model was applied, and in combination with an existing data driven technique that utilizes local sample averaging, we characterized and quantified the noise variance as a function of mean peak intensity. The results indicate that for processed LC-MS data a constant coefficient of variation is dominant for high intensities, whereas a model for low intensities explains Poisson-like variations. This result leads to a quadratic variance model which is used for the estimation of sample preparation noise present in LC-MS data.     

Few-sweep estimation of evoked potential based on a generalized subspace approach

A generalized subspace approach is proposed for single channel brain evoked potential (EP) extraction from background electroencephalogram (EEG) signal. The method realizes the optimum estimate of EP signal from the observable noisy signal. The underlying principle is to project the signal and noise into signal and noise coefficient subspace respectively by applying projection matrix at first. Secondly, coefficient weighting matrix is achieved based on the autocorrelation matrices of the noise and the noisy signal. With the coefficient weighting matrix, we can remove the noise projection coefficients and estimate the signal ones. EP signal is then obtained by averaging the signals estimated with the reconstruction matrix. Given different signal-to-noise ratio (SNR) conditions, the algorithm can estimate the EP signal with only two sweeps observable noisy signals. Our approach is shown to have excellent capability of estimating EP signal even in poor SNR conditions. The interference of spontaneous EEG has been eliminated with significantly improved SNR. The simulation results have demonstrated the effectiveness and superior performance of the proposed method.     

ESTIMATING PHENOTYPIC SELECTION IN AGE‐STRUCTURED POPULATIONS BY REMOVING TRANSIENT FLUCTUATIONS

An extension of the selection differential in the Robertson–Price equation for the mean phenotype in an age‐structured population is provided. Temporal changes in the mean phenotype caused by transient fluctuations in the age‐distribution and variation in mean phenotype among age classes, which can mistakenly be interpreted as selection, will disappear if reproductive value weighting is applied. Changes in any weighted mean phenotype in an age‐structured population may be decomposed into between‐ and within‐age class components. Using reproductive value weighting the between‐age class component becomes pure noise, generated by previous genetic drift or fluctuating selection. This component, which we call transient quasi‐selection, can therefore be omitted when estimating age‐specific selection on fecundity or viability within age classes. The final response can be computed at the time of selection, but can not be observed until lifetime reproduction is realized unless the heritability is one. The generality of these results is illustrated further by our derivation of the selection differential for the continuous time age‐structured model with general age‐dependent weights. A simple simulation example as well as estimation of selection components in a house sparrow population illustrates the applicability of the theory to analyze selection on the mean phenotype in fluctuating age‐structured populations.     

Adjustment uncertainty in effect estimation

Often there is substantial uncertainty in the selection of confounderswhen estimating the association between an exposure and health.We define this type of uncertainty as `adjustment uncertainty'.We propose a general statistical framework for handling adjustmentuncertainty in exposure effect estimation for a large numberof confounders, we describe a specific implementation, and wedevelop associated visualization tools. Theoretical resultsand simulation studies show that the proposed method providesconsistent estimators of the exposure effect and its variance.We also show that, when the goal is to estimate an exposureeffect accounting for adjustment uncertainty, Bayesian modelaveraging with posterior model probabilities approximated usinginformation criteria can fail to estimate the exposure effectand can over- or underestimate its variance. We compare ourapproach to Bayesian model averaging using time series dataon levels of fine particulate matter and mortality.     

A Fast Method that Uses Polygenic Scores to Estimate the Variance Explained by Genome-wide Marker Panels and the Proportion of Variants Affecting a Trait

Several methods have been proposed to estimate the variance in disease liability explained by large sets of genetic markers. However, current methods do not scale up well to large sample sizes. Linear mixed models require solving high-dimensional matrix equations, and methods that use polygenic scores are very computationally intensive. Here we propose a fast analytic method that uses polygenic scores, based on the formula for the non-centrality parameter of the association test of the score. We estimate model parameters from the results of multiple polygenic score tests based on markers with p values in different intervals. We estimate parameters by maximum likelihood and use profile likelihood to compute confidence intervals. We compare various options for constructing polygenic scores, based on nested or disjoint intervals of p values, weighted or unweighted effect sizes, and different numbers of intervals, in estimating the variance explained by a set of markers, the proportion of markers with effects, and the genetic covariance between a pair of traits. Our method provides nearly unbiased estimates and confidence intervals with good coverage, although estimation of the variance is less reliable when jointly estimated with the covariance. We find that disjoint p value intervals perform better than nested intervals, but the weighting did not affect our results. A particular advantage of our method is that it can be applied to summary statistics from single markers, and so can be quickly applied to large consortium datasets. Our method, named AVENGEME (Additive Variance Explained and Number of Genetic Effects Method of Estimation), is implemented in R software.     

Spectral processing methods for the removal of t1 noise and solvent artifacts from NMR spectra

Summary A data processing method is described which reduces the effects of t₁ noise artifacts and improves the presentation of 2D NMR spectral data. A t₁ noise profile is produced by measuring the average noise in each column. This profile is then used to determine weighting coefficients for a sliding weighted smoothing filter that is applied to each row, such that the amount of smoothing each point receives is proportional to both its estimated t₁ noise level and the level of t₁ noise of neighbouring points. Thus, points in the worst t₁ noise bands receive the greatest smoothing, whereas points in low-noise regions remain relatively unaffected. In addition, weighted smoothing allows points in low-noise regions to influence neighbouring points in noisy regions. This method is also effective in reducing the noise artifacts associated with the solvent resonance in spectra of biopolymers in aqueous solution. Although developed primarily to improve the quality of 2D NMR spectra of biopolymers prior to automated analysis, this approach should enhance processing of spectra of a wide range of compounds and can be used whenever noise occurs in discrete bands in one dimension of a multi-dimensional spectrum.     

Why weight?

Whether phylogenetic data should be differentially or equally weighted is currently debated. Further, if differential weighting is to be explored, there is no consensus among investigators as to which weighting scheme is most appropriate. Mitochondrial genome data offer a powerful tool in assessment of differential weighting schemes because taxa can be selected from which a highly corroborated phylogeny is available (so that accuracy can be assessed), and it can be assumed that different data partitions share the same history (so that gene-sorting issues are not so problematic). Using mitochondrial data from 17 mammalian genomes, we evaluated the most commonly used weighting schemes, such as successive weighting, transversion weighting, codon-based weighting, and amino acid coding, and compared them to more complex weighting schemes including a 6-parameter weighting, pseudoreplicate reweighting, and tri-level weighting. We found that the most commonly used weighting schemes perform the worst with these data. Some of the more complex schemes perform well, however, none of them is consistently superior. These results support ones biases; if one has a predilection to avoid differential weighting, these data support equally weighted parsimony and maximum likelihood. Others might be encouraged by these results to try weighting as a form of data exploration.     

Weighted rank regression for clustered data analysis

Summary .   We consider ranked-based regression models for clustered data analysis. A weighted Wilcoxon rank method is proposed to take account of within-cluster correlations and varying cluster sizes. The asymptotic normality of the resulting estimators is established. A method to estimate covariance of the estimators is also given, which can bypass estimation of the density function. Simulation studies are carried out to compare different estimators for a number of scenarios on the correlation structure, presence/absence of outliers and different correlation values. The proposed methods appear to perform well, in particular, the one incorporating the correlation in the weighting achieves the highest efficiency and robustness against misspecification of correlation structure and outliers. A real example is provided for illustration.     

Efficient weighting methods for genomic best linear-unbiased prediction (BLUP) adapted to the genetic architectures of quantitative traits

Genomic best linear-unbiased prediction (GBLUP) assumes equal variance for all marker effects, which is suitable for traits that conform to the infinitesimal model. For traits controlled by major genes, Bayesian methods with shrinkage priors or genome-wide association study (GWAS) methods can be used to identify causal variants effectively. The information from Bayesian/GWAS methods can be used to construct the weighted genomic relationship matrix (G). However, it remains unclear which methods perform best for traits varying in genetic architecture. Therefore, we developed several methods to optimize the performance of weighted GBLUP and compare them with other available methods using simulated and real data sets. First, two types of methods (marker effects with local shrinkage or normal prior) were used to obtain test statistics and estimates for each marker effect. Second, three weighted G matrices were constructed based on the marker information from the first step: (1) the genomic-feature-weighted G, (2) the estimated marker-variance-weighted G, and (3) the absolute value of the estimated marker-effect-weighted G. Following the above process, six different weighted GBLUP methods (local shrinkage/normal-prior GF/EV/AEWGBLUP) were proposed for genomic prediction. Analyses with both simulated and real data demonstrated that these options offer flexibility for optimizing the weighted GBLUP for traits with a broad spectrum of genetic architectures. The advantage of weighting methods over GBLUP in terms of accuracy was trait dependant, ranging from 14.8% to marginal for simulated traits and from 44% to marginal for real traits. Local-shrinkage prior EVWGBLUP is superior for traits mainly controlled by loci of a large effect. Normal-prior AEWGBLUP performs well for traits mainly controlled by loci of moderate effect. For traits controlled by some loci with large effects (explain 25–50% genetic variance) and a range of loci with small effects, GFWGBLUP has advantages. In conclusion, the optimal weighted GBLUP method for genomic selection should take both the genetic architecture and number of QTLs of traits into consideration carefully.Subject terms: Quantitative trait, Genome-wide association studies, Animal breeding, Quantitative trait, Genome-wide association studies     

Homogeneous amplified single-molecule detection: Characterization of key parameters

We recently presented a method that enables single-molecule enumeration by transforming specific molecular recognition events at nanometer dimensions to micrometer-sized DNA macromolecules. This transformation process is mediated by target-specific padlock probe ligation, followed by rolling circle amplification (RCA), resulting in the creation of one rolling circle product (RCP) for each recognized target. The transformation makes optical detection and quantification possible using standard fluorescence microscopy by counting the number of generated RCPs in a sample pumped through a microfluidic channel. In this study, we demonstrate that confocal volume definition is crucial to achieve high-precision measurements in the microfluidic quantification (coefficient of variance typically 3%). We further demonstrate that complementary sequence motifs between RCPs is only a weak inducer of aggregates and that all detection sites of the RCPs are occupied at detection oligonucleotide concentrations greater than 5 nM if hybridized in the proper buffer conditions. Therefore, the signal/noise ratio is limited by the number of detection sites. By increasing the density of detection sites in the RCP by a factor of 1.9, we show that the optical signal/noise level can be increased from 42 to 75.     

Regularized estimation in the accelerated failure time model with high-dimensional covariates

We consider two regularization approaches, the LASSO and the threshold-gradient-directed regularization, for estimation and variable selection in the accelerated failure time model with multiple covariates based on Stute's weighted least squares method. The Stute estimator uses Kaplan-Meier weights to account for censoring in the least squares criterion. The weighted least squares objective function makes the adaptation of this approach to multiple covariate settings computationally feasible. We use V-fold cross-validation and a modified Akaike's Information Criterion for tuning parameter selection, and a bootstrap approach for variance estimation. The proposed method is evaluated using simulations and demonstrated on a real data example.     

Phylogenetic Inference with Weighted Codon Evolutionary Distances

We develop a new approach to estimate a matrix of pairwise evolutionary distances from a codon-based alignment based on a codon evolutionary model. The method first computes a standard distance matrix for each of the three codon positions. Then these three distance matrices are weighted according to an estimate of the global evolutionary rate of each codon position and averaged into a unique distance matrix. Using a large set of both real and simulated codon-based alignments of nucleotide sequences, we show that this approach leads to distance matrices that have a significantly better treelikeness compared to those obtained by standard nucleotide evolutionary distances. We also propose an alternative weighting to eliminate the part of the noise often associated with some codon positions, particularly the third position, which is known to induce a fast evolutionary rate. Simulation results show that fast distance-based tree reconstruction algorithms on distance matrices based on this codon position weighting can lead to phylogenetic trees that are at least as accurate as, if not better, than those inferred by maximum likelihood. Finally, a well-known multigene dataset composed of eight yeast species and 106 codon-based alignments is reanalyzed and shows that our codon evolutionary distances allow building a phylogenetic tree which is similar to those obtained by non-distance-based methods (e.g., maximum parsimony and maximum likelihood) and also significantly improved compared to standard nucleotide evolutionary distance estimates.     

Learning a Weighted Meta-Sample Based Parameter Free Sparse Representation Classification for Microarray Data

Sparse representation classification (SRC) is one of the most promising classification methods for supervised learning. This method can effectively exploit discriminating information by introducing a regularization terms to the data. With the desirable property of sparisty, SRC is robust to both noise and outliers. In this study, we propose a weighted meta-sample based non-parametric sparse representation classification method for the accurate identification of tumor subtype. The proposed method includes three steps. First, we extract the weighted meta-samples for each sub class from raw data, and the rationality of the weighting strategy is proven mathematically. Second, sparse representation coefficients can be obtained by regularization of underdetermined linear equations. Thus, data dependent sparsity can be adaptively tuned. A simple characteristic function is eventually utilized to achieve classification. Asymptotic time complexity analysis is applied to our method. Compared with some state-of-the-art classifiers, the proposed method has lower time complexity and more flexibility. Experiments on eight samples of publicly available gene expression profile data show the effectiveness of the proposed method.     

How repeatable is the estimation of developmental stability by fluctuating asymmetry?

The estimation of individual fluctuating asymmetry (FA) is subject to large sampling variabilities. Heritability estimates, as well as correlations between developmental stability and any other individual character and/or between-trait correlations, are consequently biased downward if FA is used as an estimate of an individual''s ability to buffer its development against developmental noise. The estimation of the hypothetical repeatability, defined as the ratio of the between-individual component of variation in the unsigned FA divided by the total variance, allows correction for these biases such that patterns observed for FA can be translated to make inferences about the presumed underlying developmental stability. In this paper I show that previous estimates of this repeatability are incorrect. I provide a new method and show by means of simulations that the hypothetical repeatability is in most cases even lower than previously thought. This has important consequences for the analysis of FA with respect to statistical power and the interpretation of patterns in FA.     

The variance of phase-resetting curves

Phase resetting curves (PRCs) provide a measure of the sensitivity of oscillators to perturbations. In a noisy environment, these curves are themselves very noisy. Using perturbation theory, we compute the mean and the variance for PRCs for arbitrary limit cycle oscillators when the noise is small. Phase resetting curves and phase dependent variance are fit to experimental data and the variance is computed using an ad-hoc method. The theoretical curves of this phase dependent method match both simulations and experimental data significantly better than an ad-hoc method. A dual cell network simulation is compared to predictions using the analytical phase dependent variance estimation presented in this paper. We also discuss how entrainment of a neuron to a periodic pulse depends on the noise amplitude.