Minimum Variance Linear Unbiased Estimator Research Articles

A Multi‐Model Ensemble Kalman Filter for Data Assimilation and Forecasting

AbstractData assimilation (DA) aims to optimally combine model forecasts and observations that are both partial and noisy. Multi‐model DA generalizes the variational or Bayesian formulation of the Kalman filter, and we prove that it is also the minimum variance linear unbiased estimator. Here, we formulate and implement a multi‐model ensemble Kalman filter (MM‐EnKF) based on this framework. The MM‐EnKF can combine multiple model ensembles for both DA and forecasting in a flow‐dependent manner; it uses adaptive model error estimation to provide matrix‐valued weights for the separate models and the observations. We apply this methodology to various situations using the Lorenz96 model for illustration purposes. Our numerical experiments include multiple models with parametric error, different resolved scales, and different fidelities. The MM‐EnKF results in significant error reductions compared to the best model, as well as to an unweighted multi‐model ensemble, with respect to both probabilistic and deterministic error metrics.

A comprehensive framework for assessing the accuracy and uncertainty of global above-ground biomass maps

Over the past decade, several global maps of above-ground biomass (AGB) have been produced, but they exhibit significant differences that reduce their value for climate and carbon cycle modelling, and also for national estimates of forest carbon stocks and their changes. The number of such maps is anticipated to increase because of new satellite missions dedicated to measuring AGB. Objective and consistent methods to estimate the accuracy and uncertainty of AGB maps are therefore urgently needed. This paper develops and demonstrates a framework aimed at achieving this. The framework provides a means to compare AGB maps with AGB estimates from a global collection of National Forest Inventories and research plots that accounts for the uncertainty of plot AGB errors. This uncertainty depends strongly on plot size, and is dominated by the combined errors from tree measurements and allometric models (inter-quartile range of their standard deviation (SD) = 30–151 Mg ha−1). Estimates of sampling errors are also important, especially in the most common case where plots are smaller than map pixels (SD = 16–44 Mg ha−1). Plot uncertainty estimates are used to calculate the minimum-variance linear unbiased estimates of the mean forest AGB when averaged to 0.1∘. These are used to assess four AGB maps: Baccini (2000), GEOCARBON (2008), GlobBiomass (2010) and CCI Biomass (2017). Map bias, estimated using the differences between the plot and 0.1∘ map averages, is modelled using random forest regression driven by variables shown to affect the map estimates. The bias model is particularly sensitive to the map estimate of AGB and tree cover, and exhibits strong regional biases. Variograms indicate that AGB map errors have map-specific spatial correlation up to a range of 50–104 km, which increases the variance of spatially aggregated AGB map estimates compared to when pixel errors are independent. After bias adjustment, total pantropical AGB and its associated SD are derived for the four map epochs. This total becomes closer to the value estimated by the Forest Resources Assessment after every epoch and shows a similar decrease. The framework is applicable to both local and global-scale analysis, and is available at https://github.com/arnanaraza/PlotToMap. Our study therefore constitutes a major step towards improved AGB map validation and improvement.

Estimation of the Shape Parameter of a Wear-Out Failure Period for a Three-Parameter Weibull Distribution in a Small Sample

We propose a method to estimate a shape parameter for a three-parameter Weibull distribution. The proposed method first derives an unbiased estimator for the shape parameter independent of the location and scale parameters and then estimates the shape parameter using a minimum-variance linear unbiased estimator. Since the proposed method is expressed using a hyperparameter, its optimal hyperparameter is searched using Monte Carlo simulations. The recommended hyperparameter used for estimating the shape parameter depends on the sample size, and this causes no problems since the sample size is known when data is obtained. The proposed method is evaluated using a bias and a root mean squared error, and the results are very promising when the population shape parameter is 2 or more in the Weibull distribution representing the wear-out failure period. A numerical dataset is analyzed to demonstrate the practical use of the proposed method.

Estimation of the threshold parameter of a wear-out failure period in the case of a three-parameter Weibull distribution

We aimed to estimate the threshold parameter for the wear-out failure period of a three-parameter Weibull distribution. In this paper, we propose the minimum-variance linear unbiased estimator based on order statistics, which is denoted by 'TBEST'. In Section 2, we verify the validity of TBEST by comparing it with the existing threshold parameter estimator, based on simulation studies of bias and mean squared error (MSE). Our results show that TBEST requires all order statistics, except for the case of an exponential distribution, in which TBEST is reduced to an unbiased estimator based on the smallest observation only. In Section 3, by simulation studies, we compare TBEST and other estimators known to have good performances. In the simulation results, bias and MSE of TBEST were the smallest in most cases. We also show a numerical example to measure fatigue lives in hours of ten bearings from McCool (1974).

Estimation of the threshold parameter of a wear-out failure period in the case of a three-parameter Weibull distribution

We aimed to estimate the threshold parameter for the wear-out failure period of a three-parameter Weibull distribution. In this paper, we propose the minimum-variance linear unbiased estimator based on order statistics, which is denoted by 'TBEST'. In Section 2, we verify the validity of TBEST by comparing it with the existing threshold parameter estimator, based on simulation studies of bias and mean squared error (MSE). Our results show that TBEST requires all order statistics, except for the case of an exponential distribution, in which TBEST is reduced to an unbiased estimator based on the smallest observation only. In Section 3, by simulation studies, we compare TBEST and other estimators known to have good performances. In the simulation results, bias and MSE of TBEST were the smallest in most cases. We also show a numerical example to measure fatigue lives in hours of ten bearings from McCool (1974).

Simple Kalman filter alternative: the multi‐fractional order estimator

This study is an extension of a presentation given at the 2012 Radar Conference in Glasgow, UK entitled: Simple Disambiguation of Orthogonal Projection in Kalman's Filter Derivation . It is shown therein that Kalman misunderstood and consequently misapplied orthogonal projection in his derivation. Shocking analysis shows the Kalman filter (KF) to be truly optimum only in the absence of specific additive statistical measurement noise it is explicitly designed to filter out, rendering it a deterministic curve-fitting algorithm. Also included are two examples demonstrating that the KF is not the universally asserted minimum variance linear unbiased estimator. These consequences are based on the newly introduced multi-fractional order estimator (MFOE). Included herein are an additional background derivation and supplementary material as well as several new examples to better illustrate internal MFOE mechanics. Examples also show major problems in fine tuning the KF with a trade-off between process and measurement noises. Finally, an example shows a very simple, automatic, seamless, smooth, optimal, non-recursive second to third-order estimator transition which successfully overcomes inadequate and complicated KF adaptivity, including thresholding, feedback, manoeuvre detection, deweighting matrices, manoeuvre probability, dual filter weighting, variable dimension models, state augmentation, multiple models, interactive multiple models, interactive acceleration compensation and so on.

Design-based Generalized Least Squares estimation of status and trend of soil properties from monitoring data

This paper introduces and demonstrates design-based Generalized Least Squares (GLS) estimation of spatial means at selected time points from data collected in repeated soil surveys with partial overlap, such as a rotating and a supplemented panel. The linear time trend of the spatial means can then be obtained as a linear combination of the estimated spatial means. The GLS estimator is the minimum variance linear unbiased estimator. Five space–time designs were compared under a first order autoregressive time series model for the spatial means, through the average and generalized sampling variances of the estimated spatial means, and through the sampling variance of the estimated linear trend. None of the designs scored best on all three quality measures. If the aim of soil monitoring is estimation of both status and trend, then these two conflicting aims must be prioritized in order to choose an efficient space–time design. The methodology is demonstrated by a case study on eutrophication and acidification of forest soils. The linear trends in the spatial means of pH and the ammonium and nitrate concentrations at three depths in the soil profile, as estimated from a rotational design with four sampling times at an interval of one year, were small and not significant. Exceptions were pH in the subsoil (− 0.06 pH units yr − 1 ) and ammonium in the middle soil horizon (− 0.086 mg N kg − 1 yr − 1 ). The linear trend is here defined as a linear combination of the true and unknown, but fixed spatial means. In quantifying the uncertainty of the estimated trend, only the sampling error in the estimated spatial means is accounted for. If there is a need to include uncertainty due to fluctuations of the true spatial means around a linear trend, then a super-population or time series model for the spatial means must be postulated which comprises a model error term. The linear trend is then defined as a model parameter, that can be estimated by Generalized Least Squares as in generalized linear modeling.

A note on the shrinkage techniques for estimating the mean

Searls (1964) proposed an improved estimator for the mean when the coefficient of variation is known. Thompson (1968) considered a method to reduce the mean square error (MSE) of the minimum variance unbiased linear estimator of the mean. The latter work obtained a new improved estimator for the mean. The purpose of the present note is to find a relationship between Searls and Thompson’s estimators using a pivotal quantity for location parameter. Key words: Improved estimation, mean square error, pivotal quantity; shrinkage technique, location parameter.

Optimal diffusion MRI acquisition for fiber orientation density estimation: An analytic approach

An important challenge in the design of diffusion MRI experiments is how to optimize statistical efficiency, i.e., the accuracy with which parameters can be estimated from the diffusion data in a given amount of imaging time. In model-based spherical deconvolution analysis, the quantity of interest is the fiber orientation density (FOD). Here, we demonstrate how the spherical harmonics (SH) can be used to form an explicit analytic expression for the efficiency of the minimum variance (maximally efficient) linear unbiased estimator of the FOD. Using this expression, we calculate optimal b-values for maximum FOD estimation efficiency with SH expansion orders of L = 2, 4, 6, and 8 to be approximately b = 1,500, 3,000, 4,600, and 6,200 s/mm(2), respectively. However, the arrangement of diffusion directions and scanner-specific hardware limitations also play a role in determining the realizable efficiency of the FOD estimator that can be achieved in practice. We show how some commonly used methods for selecting diffusion directions are sometimes inefficient, and propose a new method for selecting diffusion directions in MRI based on maximizing the statistical efficiency. We further demonstrate how scanner-specific hardware limitations generally lead to optimal b-values that are slightly lower than the ideal b-values. In summary, the analytic expression for the statistical efficiency of the unbiased FOD estimator provides important insight into the fundamental tradeoff between angular resolution, b-value, and FOD estimation accuracy.

The Applicability of Ordinary Least Squares to Consistently Short Distances between Taxa in Phylogenetic Tree Construction and the Normal Distribution Test Consequences

Short phylogenetic distances between taxa occur, for example, in studies on ribosomal RNA-genes with slow substitution rates. For consistently short distances, it is proved that in the completely singular limit of the covariance matrix ordinary least squares (OLS) estimates are minimum variance or best linear unbiased (BLU) estimates of phylogenetic tree branch lengths. Although OLS estimates are in this situation equal to generalized least squares (GLS) estimates, the GLS chi-square likelihood ratio test will be inapplicable as it is associated with zero degrees of freedom. Consequently, an OLS normal distribution test or an analogous bootstrap approach will provide optimal branch length tests of significance for consistently short phylogenetic distances. As the asymptotic covariances between branch lengths will be equal to zero, it follows that the product rule can be used in tree evaluation to calculate an approximate simultaneous confidence probability that all interior branches are positive.

Estimation of the Length Distribution of Marine Populations in the Gaussian-multinomial Setting using the Method of Moments

In this paper, the problem of estimation of the length distribution of marine populations in the Gaussian-multinomial model is considered. For the purpose of the mean and covariance parameter estimation, the method of moments estimators are developed. That is, minimum variance linear unbiased estimator for the mean frequency vector is derived and a consistent estimator for the covariance matrix of the length observations is presented. The usefulness of the proposed estimators is illustrated with an analysis of real cod length measurement data.

Estimation of location and scale parameters for the Burr XII distribution using generalized order statistics

The minimum variance linear unbiased estimators (MVLUE), the best linear invariant estimators (BLIE) and the maximum likelihood estimators (MLE) based on n-selected generalized order statistics are presented for the parameters of the Burr XII distribution.

Estimation of the parameters of Gumbel and Burr distributions in terms of kth record values

The minimum variance linear unbiased estimators (MVLUE), the best linear invariant estimators (BLIE) and the maximum likelihood estimators (MLE) based on $m$ selected $k$th record values are presented for the parameters of the Gumbel and Burr distribution

Some estimation problems in multistage linear models

The paper deals with a multistage linear model that is a special case of a linear model with variance–covariance parameters. Within this setting questions related to estimation of linear functions of the mean parameter are studied. Necessary and sufficient conditions for the existence of uniformly minimum variance linear unbiased estimators of linear function of the mean are presented under rather general assumptions. For illustration, a special case of a two-stage model is shown.

One-Level Rotation Design Balanced on Time in Monthly Sample and in Rotation Group

We introduce the two-way balanced one-level rotation design for which balancing is done on interview time in sample and in rotation group and provide the necessary and sufficient condition for the two-way balancing and an algorithm to construct such design. Using this design, we obtain generalized composite estimators (GCE) and the minimum variance linear unbiased estimator (MVLUE). We then calculate their variance and mean squared error (MSE) when there exist two types of correlations and the rotation group bias. Minimizing the weighted sum of variances (or MSEs), we derive one set of the compromise coefficients for all important estimators and characteristics to retain the consistency in total. We compared the GCE with the compromise coefficients to MVLUE and other GCEs with the previous coefficients. We show that the second-order correlations have very significant effects on the variance and MSE. We also investigate the design efficiencies for some selected two-way balanced designs.

Generalized order statistics from exponential distribution

In this paper some distributional properties of the generalized order statistics from two parameter exponential distribution are given. The minimum variance linear unbiased estimators of the parameters and an important characterization of the exponential distribution are presented.

MVLUE of population parameters based on ranked set sampling

The minimum variance linear unbiased estimators (MVLUEs) of population parameters based on ranked set sampling are discussed for some distributions. Also values of the coefficients required for obtaining MVLUEs of the location and scale parameters are presented. For the same distributions, MVLUEs of the population mean are also obtained and the required coefficients are supplied. The MVLUEs of the population mean are compared with the other popular simple random sample (SRS) and ranked set sample (RSS) estimators of the population mean. The MVLUEs of the population mean are found to be more precise than the usual SRS or RSS estimators of the population mean for all cases considered.

Paired ranked set sampling: a more efficient procedure

The usual ranked set sampling (RSS) scheme is modified and a paired ranked set sampling (PRSS) method is suggested. Estimators for population mean and standard deviation using the PRSS method are obtained. The estimator of population mean based on the PRSS is compared with simple random sampling (SRS) mean, the RSS estimator of population mean and also with the minimum variance linear unbiased estimator (MVLUE) of population mean based on RSS. The estimator of standard deviation based on the PRSS is also compared with the MVLUE of population standard deviation. The estimators of population mean and standard deviation based on PRSS are found to be more efficient than their respective counterparts for six probability distributions considered. The estimators of population mean and standard deviation based on PRSS are compared with those based on the SRS and the RSS for normal distribution where errors in ranking are present. The estimators using the PRSS method are found to be more efficient than their respective counterparts for normal distribution, even in presence of errors in ranking. Copyright © 1999 John Wiley & Sons, Ltd.

Estimators of the unknown mean of a homogeneous and isotropic random field

Least-squares estimators and minimum-variance linear unbiased estimators of the unknown mean are studied for homogeneous and isotropic random fields observed on sets of a special kind (a system of concentric circles, a ring).

Optimal linear estimators of the unknown mean of a random field observed on a cross

Explicit formulas of minimum-variance linear unbiased estimators of the unknown mean are derived for homogeneous random fields with correlation functions observed on the set (a “cross”).