Quadratic Estimation Research Articles

On estimation of nonlinear functionals from discrete noisy measurements

The principal objective of this paper is to estimate a nonlinear functional of state vector (NFS) in dynamical system. The NFS represents a multivariate functional of state variables which carries useful information of a target system for control. The paper focuses on estimation of the NFS in linear continuous-discrete systems. The optimal nonlinear estimator based on the minimum mean square error approach is derived. The estimator depends on the Kalman estimate of a state vector and its error covariance. Some challenging computational aspects of the optimal nonlinear estimator are solved by usage of the unscented transformation for implementation of the nonlinear estimator. The special quadratic functional of state vector (QFS) is studied in detail. We derive effective matrix formulas for the optimal quadratic estimator and mean square error. The quadratic estimator has a simple closed-form calculation procedure and it is easy to implement in practice. The obtained results we demonstrate on theoretical and practical examples with different types of an nonlinear functionals. Comparison analysis of the optimal and suboptimal estimators is presented. The subsequent application of the proposed optimal nonlinear and quadratic estimators demonstrates their effectiveness.

Threonine supplementation reduces dietary protein and improves lipid metabolism in Pekin ducks

ABSTRACT1. This study was conducted to investigate the efficiency of threonine (Thr) supplementation on reducing dietary crude protein (CP) content and the effects of Thr on lipid metabolism in Pekin ducks. The effects of dietary CP concentration (160, 190 and 220 g/kg) and Thr supplemental concentration (0, 0.7, 1.4, 2.1 and 2.8 g/kg) on growth performance, carcass, liver lipid and plasma profiles were determined in Pekin ducks from 1–21 d of age.2. A total of 720-d-old male Pekin ducks were randomly allotted to 1 of 15 dietary treatments with 6 replicate cages of 8 birds per cage for each treatment according to average body weight.3. Dietary Thr supplementation improved growth performance and breast muscle percentage at all CP diets, and ducks fed Thr-supplemented diets had higher plasma concentrations of some plasma amino acids. Thr supplementation reduced the concentrations of total lipid, triglyceride, cholesterol in liver, and plasma low density lipoprotein cholesterin concentration at 160 and 190 g/kg CP, whereas it increased triglyceride concentration at 160 g/kg CP.4. Thr requirements based on quadratic broken-line model estimation were 6.6 and 7.0 g/kg for optimal average daily gain (ADG), and 6.7 and 7.3 g/kg for breast muscle percentage of Pekin ducks from 1–21 d of age at 190 and 220 g/kg CP, respectively. The dietary Thr requirements and estimated ADG (55.18 vs. 55.86 g/d/bird) and breast muscle percentage (2.79% vs. 2.75%) of Pekin ducks did not differ between 190 and 220 g/kg CP according to the t-test results.5. Dietary CP level could be reduced to 190 g/kg in Pekin ducks from 1–21 d of age with Thr supplementation to balance dietary amino acids, and Thr supplementation prevented excess liver lipid deposition in this instance.

Best unbiased estimates for parameters of three-level multivariate data with doubly exchangeable covariance structure

The article addresses the best unbiased estimators of doubly exchangeable covariance structure, an extension of block exchangeable covariance structure, for three-level multivariate data. Under multivariate normality, the free-coordinate approach is used to obtain linear and quadratic estimates for the model parameters that are unbiased, sufficient, complete and consistent. Data from a clinical trial study is analyzed to illustrate the application of the obtained results.

Measuring galaxy cluster masses with CMB lensing using a Maximum Likelihood estimator: statistical and systematic error budgets for future experiments

We develop a Maximum Likelihood estimator (MLE) to measure the masses of galaxy clusters through the impact of gravitational lensing on the temperature and polarization anisotropies of the cosmic microwave background (CMB). We show that, at low noise levels in temperature, this optimal estimator outperforms the standard quadratic estimator by a factor of two. For polarization, we show that the Stokes Q/U maps can be used instead of the traditional E- and B-mode maps without losing information. We test and quantify the bias in the recovered lensing mass for a comprehensive list of potential systematic errors. Using realistic simulations, we examine the cluster mass uncertainties from CMB-cluster lensing as a function of an experiment's beam size and noise level. We predict the cluster mass uncertainties will be 3 - 6% for SPT-3G, AdvACT, and Simons Array experiments with 10,000 clusters and less than 1% for the CMB-S4 experiment with a sample containing 100,000 clusters. The mass constraints from CMB polarization are very sensitive to the experimental beam size and map noise level: for a factor of three reduction in either the beam size or noise level, the lensing signal-to-noise improves by roughly a factor of two.

KiDS-450: the tomographic weak lensing power spectrum and constraints on cosmological parameters

We present measurements of the weak gravitational lensing shear power spectrum based on $450$ sq. deg. of imaging data from the Kilo Degree Survey. We employ a quadratic estimator in two and three redshift bins and extract band powers of redshift auto-correlation and cross-correlation spectra in the multipole range $76 \leq \ell \leq 1310$. The cosmological interpretation of the measured shear power spectra is performed in a Bayesian framework assuming a $\Lambda$CDM model with spatially flat geometry, while accounting for small residual uncertainties in the shear calibration and redshift distributions as well as marginalising over intrinsic alignments, baryon feedback and an excess-noise power model. Moreover, massive neutrinos are included in the modelling. The cosmological main result is expressed in terms of the parameter combination $S_8 \equiv \sigma_8 \sqrt{\Omega_{\rm m}/0.3}$ yielding $S_8 = \ 0.651 \pm 0.058$ (3 z-bins), confirming the recently reported tension in this parameter with constraints from Planck at $3.2\sigma$ (3 z-bins). We cross-check the results of the 3 z-bin analysis with the weaker constraints from the 2 z-bin analysis and find them to be consistent. The high-level data products of this analysis, such as the band power measurements, covariance matrices, redshift distributions, and likelihood evaluation chains are available at http://kids.strw.leidenuniv.nl/

A green method for production of nanobiochar by ball milling- optimization and characterization

Environmental considerations along with the technological challenges have led to search for green and energy-efficient processes for advanced nanostructured materials. In this study, nanobiochar was produced from pine wood biochar using a planetary ball mill. A central composite experimental design and response surface methodology was employed to optimize the ball milling parameters including time, rotational speed and ball to powder mass ratio to obtain nanoparticles in short time and at lower energy consumption. ANOVA results showed that the linear and quadratic effect estimates of time and the interaction effect of time and rotational speed were significant contributors to the size of particles during milling (p < 0.05). Based on the developed statistical model, the optimum conditions for obtaining the smallest particles, around 60 nm, were found to be 1.6 h, 575 rpm and 4.5 g/g. However, the size measurements indicated that particles had a great tendency to agglomerate. Further study showed that the conditioning of biochar at cryogenic temperatures prior to milling inhibits the agglomeration of nanoparticles which is essential in industrial processes. The adsorption test proved that the nanobiochar produced using green method is promising in the removal of micropollutants from aqueous media by removing up to 95% of carbamazepine from water. At the optimum milling parameters and conditioning for 24 h at −80 °C, nanobiochar with the average size of around 60 nm was obtained. The produced nanobiochar was characterized by Brunauer-Emmett-Teller (BET) gas porosimetry, scanning electron microscopy (SEM) and Fourier transform infrared spectroscopy (FTIR). Also, physical and chemical properties, such as water holding capacity, organic matter, oxidation-reduction potential (ORP), elemental composition, polycyclic aromatic hydrocarbons (PAHs) and heavy metals were analyzed.

Two-season Atacama Cosmology Telescope polarimeter lensing power spectrum

We report a measurement of the power spectrum of cosmic microwave background (CMB) lensing from two seasons of Atacama Cosmology Telescope Polarimeter (ACTPol) CMB data. The CMB lensing power spectrum is extracted from both temperature and polarization data using quadratic estimators. We obtain results that are consistent with the expectation from the best-fit Planck LCDM model over a range of multipoles L=80-2100, with an amplitude of lensing A_lens = 1.06 +/- 0.15 (stat.) +/- 0.06 (sys.) relative to Planck. Our measurement of the CMB lensing power spectrum gives sigma_8 Omega_m^0.25 = 0.643 +/- 0.054; including baryon acoustic oscillation scale data, we constrain the amplitude of density fluctuations to be sigma_8 = 0.831 +/- 0.053. We also update constraints on the neutrino mass sum. We verify our lensing measurement with a number of null tests and systematic checks, finding no evidence of significant systematic errors. This measurement relies on a small fraction of the ACTPol data already taken; more precise lensing results can therefore be expected from the full ACTPol dataset.

A smooth simultaneous confidence band for correlation curve

A plug-in estimator is proposed for a local measure of variance explained by regression, termed correlation curve in Doksum et al. (J Am Stat Assoc 89:571–582, 1994), consisting of a two-step spline–kernel estimator of the conditional variance function and local quadratic estimator of first derivative of the mean function. The estimator is oracally efficient in the sense that it is as efficient as an infeasible correlation estimator with the variance function known. As a consequence of the oracle efficiency, a smooth simultaneous confidence band (SCB) is constructed around the proposed correlation curve estimator and shown to be asymptotically correct. Simulated examples illustrate the versatility of the proposed oracle SCB which confirms the asymptotic theory. Application to a 1995 British Family Expenditure Survey data has found marginally significant evidence for a local version of Engel’s law, i.e., food budget share and household real income are inversely related (Hamilton in Am Econ Rev 91:619–630, 2001).

Internal delensing of Planck CMB temperature and polarization

We present a first internal delensing of CMB maps, both in temperature and polarization, using the public foreground-cleaned (SMICA) Planck 2015 maps. After forming quadratic estimates of the lensing potential, we use the corresponding displacement field to undo the lensing on the same data. We build differences of the delensed spectra to the original data spectra specifically to look for delensing signatures. After taking into account reconstruction noise biases in the delensed spectra, we find an expected sharpening of the power spectrum acoustic peaks with a delensing efficiency of 29 % (TT) 25 % (TE) and 22 % (EE). The detection significance of the delensing effects is very high in all spectra: 12 σ in EE polarization; 18 σ in TE; and 20 σ in TT. The null hypothesis of no lensing in the maps is rejected at 26 σ. While direct detection of the power in lensing B-modes themselves is not possible at high significance at Planck noise levels, we do detect (at 4.5 σ {under the null hypothesis}) delensing effects in the B-mode map, with 7 % reduction in lensing power. Our results provide a first demonstration of polarization delensing, and generally of internal CMB delensing, and stand in agreement with the baseline ΛCDM Planck 2015 cosmology expectations.

Bayesian One- Way Repeated Measurements Model Based on Bayes Quadratic Unbiased Estimator

In this paper, bayesian approach based on Bayes quadratic unbiased estimator is employed to the linear one- way repeated measurements model which has only one within units factor and one between units factor incorporating univariate random effects as well as the experimental error term. The prior information obtained by using variance analysis technique to represent prior estimates of the parameters of the model. Then, the prior distribution is considered as a uniform distribution.

Diurnal atmosphere‐ocean signals in Earth's rotation rate and a possible modulation through ENSO

AbstractSpace geodetic determinations of a 6 μs length‐of‐day (LOD) anomaly at the diurnal S1 frequency are reconciled with excitation estimates from geophysical fluid models. Preference is given to a hybrid excitation scheme that combines atmospheric torques with oceanic angular momentum (OAM) terms from hydrodynamic forward modeling. A joint inversion of all data sets yields an LOD in‐phase and quadrature estimate of (5.91, −0.22) μs, matching space geodetic S1 terms well within their formal uncertainties. Non‐harmonic LOD excitations, while less than 30% of the time‐averaged rotation rate contribution, are conclusively linked to El Niño–Southern Oscillation (ENSO) as the main perturbation of diurnal cycle characteristics in the troposphere. ENSO modulations of particular relevance are those in OAM, associated with the barotropic ocean response to regional modifications in the diurnal atmospheric pressure wave. The study thus highlights previously unexplored aspects of non‐tidal mass‐field variability in the Earth system.

Refined quadratic estimations of Shafer\u2019s inequality

We establish an inequality by quadratic estimations; the double inequality \t\t\tπ2x4+(π2−4)2+(2πx)2<arctanx<π2x4+32+(2πx)2\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\frac{\\pi^{2} x}{4 +\\sqrt{(\\pi^{2} -4)^{2} + (2\\pi x)^{2}}} < \\arctan{x} < \\frac{\\pi^{2} x}{4 +\\sqrt{32+ (2\\pi x)^{2}}} $$\\end{document} holds for x>0, where the constants (pi^{2} -4)^{2} and 32 are the best possible.

A GPU-Accelerated 3-D Coupled Subsample Estimation Algorithm for Volumetric Breast Strain Elastography.

Our primary objective of this paper was to extend a previously published 2-D coupled subsample tracking algorithm for 3-D speckle tracking in the framework of ultrasound breast strain elastography. In order to overcome heavy computational cost, we investigated the use of a graphic processing unit (GPU) to accelerate the 3-D coupled subsample speckle tracking method. The performance of the proposed GPU implementation was tested using a tissue-mimicking phantom and in vivo breast ultrasound data. The performance of this 3-D subsample tracking algorithm was compared with the conventional 3-D quadratic subsample estimation algorithm. On the basis of these evaluations, we concluded that the GPU implementation of this 3-D subsample estimation algorithm can provide high-quality strain data (i.e., high correlation between the predeformation and the motion-compensated postdeformation radio frequency echo data and high contrast-to-noise ratio strain images), as compared with the conventional 3-D quadratic subsample algorithm. Using the GPU implementation of the 3-D speckle tracking algorithm, volumetric strain data can be achieved relatively fast (approximately 20 s per volume [2.5 cm ×2.5 cm ×2.5 cm]).

Trigonometric Interpolation and Quadrature in Perturbed Points

The trigonometric interpolants to a periodic function $f$ in equispaced points converge if $f$ is Dini-continuous, and the associated quadrature formula, the trapezoidal rule, converges if $f$ is continuous. What if the points are perturbed? With equispaced grid spacing $h$, let each point be perturbed by an arbitrary amount $\le \alpha h$, where $\alpha\in [\kern .5pt 0,1/2)$ is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests there may be trouble for $\alpha\ge 1/4$. We show that convergence of both the interpolants and the quadrature estimates is guaranteed for all $\alpha<1/2$ if $f$ is twice continuously differentiable, with the convergence rate depending on the smoothness of $f$. More precisely, it is enough for $f$ to have $4\alpha$ derivatives in a certain sense, and we conjecture that $2\alpha$ derivatives are enough. Connections with the Fejer--Kalmar theorem are discussed.

Genetic parameters for some growth traits of Nigerian local chickens

Genetic parameters were estimated for bodyweight (BWT), shank length (SHL), and wing length (WL) of Nigerian local chicken (NLC) from 4 to 20 weeks of age by fitting dyadic mixed model (dmm) equations which yield estimates of variance components equivalent to minimum norm quadratic unbiased estimator (MINQUE). Data obtained from 600 chicks, progenies of 300 hens and 30 cocks were used for the analysis. The heritability estimates range from 0.08 to 0.80 for BWT, 0.03 to 0.69 for SHL and 0.22 to 0.47 for WL. The genetic correlations among BWT, SHL and WL at different ages were high and positive and range from 0.18 to 0.96 with the exemption of SHL and WL at 16 weeks (- 0.06). The phenotypic correlations were positive and range from 0.10 to 0.91. The results imply that NLC could be improved on any of the studied traits through mass selection and that improvement in one trait will result to correlated improvement in the others.

Forecasting method for global radiation time series without training phase: Comparison with other well-known prediction methodologies

Integration of unpredictable renewable energy sources into electrical networks intensifies the complexity of the grid management due to their intermittent and unforeseeable nature. Because of the strong increase of solar power generation the prediction of solar yields becomes more and more important. Electrical operators need an estimation of the future production. For nowcasting and short term forecasting, the usual technics based on machine learning need large historical data sets of good quality during the training phase of predictors. However data are not always available and induce an advanced maintenance of meteorological stations, making the method inapplicable for poor instrumented or isolated sites. In this work, we propose intuitive methodologies based on the Kalman filter use (also known as linear quadratic estimation), able to predict a global radiation time series without the need of historical data. The accuracy of these methods is compared to other classical data driven methods, for different horizons of prediction and time steps. The proposed approach shows interesting capabilities allowing to improve quasi-systematically the prediction. For one to 10 h horizons Kalman model performances are competitive in comparison to more sophisticated models such as ANN which require both consistent historical data sets and computational resources.

Green Environment Monitoring System (Gems) for Industries using Li-Fi Technology

Objective: This work assures to design a green environment monitoring system for industries using Li-Fi technology to provide secured communication, high data rate transfer and pollution free environment. Methods: The LPC 2148 microcontroller with ARM 7 core is chosen for implementation. Light-emitting diodes are used as the source for data transmission and they are preferred as they have a longer life-time. Different sensors are deployed to monitor various parameters namely temperature, Gas and light intensity in an industrial environment and these sensors transmit the sensed information using Li-Fi technology in terms of flickering of LEDs. Linear quadratic estimation algorithm is implemented to improve the accuracy of sensor values. Findings: The proposed system is implemented in ARM 7 Core (LPC2148) that consumes less power. The designed system provides communication at the rate of 1.5 Mb/s over a distance of 5 meters and they are free from electromagnetic interference. They are not affected by artificial or natural light sources. Improvement/Applications: The implementation of Linear Quadratic estimator improved the accuracy of sensor output values. This technology is eco-friendly and the working atmosphere is conserved thereby ensuring a green environment. The sensors used in industrial applications communicate with low cost and also in a secured manner.

A new PI optimal linear quadratic state-estimate tracker for continuous-time non-square non-minimum phase systems

ABSTRACTBased on the proportional-integral-derivative (PID) filter-shaping approach, this paper presents a new proportional-plus-integral (PI) optimal linear quadratic state estimator (LQSE) for the continuous-time non-square and non-minimum phase (NMP) multivariable systems. Together with the recently developed optimal linear quadratic tracker (LQT), the proposed LQSE-based tracker is able to optimally achieve good minimum phase-like tracking performances for a non-square NMP multivariable system with unmeasurable states and arbitrary command inputs.

Application of beam deconvolution technique to power spectrum estimation for CMB measurements

We present two novel methods for the estimation of the angular power spectrum of cosmic microwave background (CMB) anisotropies. We assume an absolute CMB experiment with arbitrary asymmetric beams and arbitrary sky coverage. The methods differ from the earlier ones in that the power spectrum is estimated directly from the time-ordered data, without first compressing the data into a sky map, and they take into account the effect of asymmetric beams. In particular, they correct the beam-induced leakage from temperature to polarization. The methods are applicable to a case where part of the sky has been masked out to remove foreground contamination, leaving a pure CMB signal, but incomplete sky coverage. The first method (deconvolution quadratic maximum likelihood) is derived as the optimal quadratic estimator, which simultaneously yields an unbiased spectrum estimate and minimizes its variance. We successfully apply it to multipoles up to ℓ = 200. The second method is derived as a weak-signal approximation from the first one. It yields an unbiased estimate for the full multipole range, but relaxes the requirement of minimal variance. We validate the methods with simulations for the 70 GHz channel of Planck surveyor, and demonstrate that we are able to correct the beam effects in the TT, EE, BB and TE spectra up to multipole ℓ = 1500. Together, the two methods cover the complete multipole range with no gap in between.

Least Squares Approximation to the Distribution of Project Completion Times with Gaussian Uncertainty

This paper is motivated by the following question: How to construct good approximation for the distribution of the solution value to linear optimization problem when the random objective coefficients follow a multivariate normal distribution? Using Stein’s Identity, we show that the least squares normal approximation of the random optimal value can be computed by estimating the persistency values of the corresponding optimization problem. We further extend our method to construct a least squares quadratic estimator to improve the accuracy of the approximation; in particular, to capture the skewness of the objective. Computational studies show that the new approach provides more accurate estimates of the distributions of project completion times compared to existing methods.