Conditioning and preconditioning of the optimal state estimation problem

Optimal state estimation is a method that requires minimising a weighted, nonlinear, least-squares objective function in order to obtain the best estimate of the current state of a dynamical system. Often the minimisation is non-trivial due to the large scale of the problem, the relative sparsity of the observations and the nonlinearity of the objective function. To simplify the problem the solution is often found via a sequence of linearised objective functions. The condition number of the Hessian of the linearised problem is an important indicator of the convergence rate of the minimisation and the expected accuracy of the solution. In the standard formulation the convergence is slow, indicating an ill-conditioned objective function. A transformation to different variables is often used to ameliorate the conditioning of the Hessian by changing, or preconditioning, the Hessian. There is only sparse information in the literature for describing the causes of ill-conditioning of the optimal state estimation problem and explaining the effect of preconditioning on the condition number. This paper derives descriptive theoretical bounds on the condition number of both the unpreconditioned and preconditioned system in order to better understand the conditioning of the problem. We use these bounds to explain why the standard objective function is often ill-conditioned and why a standard preconditioning reduces the condition number. We also use the bounds on the preconditioned Hessian to understand the main factors that affect the conditioning of the system. We illustrate the results with simple numerical experiments.

Short communication: estimation of the financial benefit of using Jersey milk at different inclusion rates for Cheddar cheese production using partial budgeting

Partial budgeting was used to estimate the net benefit of blending Jersey milk in Holstein-Friesian milk for Cheddar cheese production. Jersey milk increases Cheddar cheese yield. However, the cost of Jersey milk is also higher; thus, determining the balance of profitability is necessary, including consideration of seasonal effects. Input variables were based on a pilot plant experiment run from 2012 to 2013 and industry milk and cheese prices during this period. When Jersey milk was used at an increasing rate with Holstein-Friesian milk (25, 50, 75, and 100% Jersey milk), it resulted in an increase of average net profit of 3.41, 6.44, 8.57, and 11.18 pence per kilogram of milk, respectively, and this additional profit was constant throughout the year. Sensitivity analysis showed that the most influential input on additional profit was cheese yield, whereas cheese price and milk price had a small effect. The minimum increase in yield, which was necessary for the use of Jersey milk to be profitable, was 2.63, 7.28, 9.95, and 12.37% at 25, 50, 75, and 100% Jersey milk, respectively. Including Jersey milk did not affect the quantity of whey butter and powder produced. Althoug further research is needed to ascertain the amount of additional profit that would be found on a commercial scale, the results indicate that using Jersey milk for Cheddar cheese making would lead to an improvement in profit for the cheese makers, especially at higher inclusion rates.

Improving the estimation of probability of bidder participation in procurement auctions

Anticipating the number and identity of bidders has significant influence in many theoretical results of the auction itself and bidders' bidding behaviour. This is because when a bidder knows in advance which specific bidders are likely competitors, this knowledge gives a company a head start when setting the bid price. However, despite these competitive implications, most previous studies have focused almost entirely on forecasting the number of bidders and only a few authors have dealt with the identity dimension qualitatively. Using a case study with immediate real-life applications, this paper develops a method for estimating every potential bidder's probability of participating in a future auction as a function of the tender economic size removing the bias caused by the contract size opportunities distribution. This way, a bidder or auctioner will be able to estimate the likelihood of a specific group of key, previously identified bidders in a future tender.

CO2 emission estimation in the urban environment: measurement of the CO2 storage term

Eddy covariance has been used in urban areas to evaluate the net exchange of CO2 between the surface and the atmosphere. Typically, only the vertical flux is measured at a height 2–3 times that of the local roughness elements; however, under conditions of relatively low instability, CO2 may accumulate in the airspace below the measurement height. This can result in inaccurate emissions estimates if the accumulated CO2 drains away or is flushed upwards during thermal expansion of the boundary layer. Some studies apply a single height storage correction; however, this requires the assumption that the response of the CO2 concentration profile to forcing is constant with height. Here a full seasonal cycle (7th June 2012 to 3rd June 2013) of single height CO2 storage data calculated from concentrations measured at 10 Hz by open path gas analyser are compared to a data set calculated from a concurrent switched vertical profile measured (2 Hz, closed path gas analyser) at 10 heights within and above a street canyon in central London. The assumption required for the former storage determination is shown to be invalid. For approximately regular street canyons at least one other measurement is required. Continuous measurements at fewer locations are shown to be preferable to a spatially dense, switched profile, as temporal interpolation is ineffective. The majority of the spectral energy of the CO2 storage time series was found to be between 0.001 and 0.2 Hz (500 and 5 s respectively); however, sampling frequencies of 2 Hz and below still result in significantly lower CO2 storage values. An empirical method of correcting CO2 storage values from under-sampled time series is proposed.

A systematic method of parameterisation estimation using data assimilation

In numerical weather prediction, parameterisations are used to simulate missing physics in the model. These can be due to a lack of scientific understanding or a lack of computing power available to address all the known physical processes. Parameterisations are sources of large uncertainty in a model as parameter values used in these parameterisations cannot be measured directly and hence are often not well known; and the parameterisations themselves are also approximations of the processes present in the true atmosphere. Whilst there are many efficient and effective methods for combined state/parameter estimation in data assimilation (DA), such as state augmentation, these are not effective at estimating the structure of parameterisations. A new method of parameterisation estimation is proposed that uses sequential DA methods to estimate errors in the numerical models at each space-time point for each model equation. These errors are then fitted to pre-determined functional forms of missing physics or parameterisations that are based upon prior information. We applied the method to a one-dimensional advection model with additive model error, and it is shown that the method can accurately estimate parameterisations, with consistent error estimates. Furthermore, it is shown how the method depends on the quality of the DA results. The results indicate that this new method is a powerful tool in systematic model improvement.

Sparse density estimation on multinomial manifold combining local component analysis

A new sparse kernel density estimator is introduced based on the minimum integrated square error criterion combining local component analysis for the finite mixture model. We start with a Parzen window estimator which has the Gaussian kernels with a common covariance matrix, the local component analysis is initially applied to find the covariance matrix using expectation maximization algorithm. Since the constraint on the mixing coefficients of a finite mixture model is on the multinomial manifold, we then use the well-known Riemannian trust-region algorithm to find the set of sparse mixing coefficients. The first and second order Riemannian geometry of the multinomial manifold are utilized in the Riemannian trust-region algorithm. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with competitive accuracy to existing kernel density estimators.

Gaze stabilization testing: The efficiency and test-retest reliability of user operated control versus the parameter estimation (PEST) algorithm

The primary objective of this research study is to determine which form of testing, the PEST algorithm or an operator-controlled condition is most accurate and time efficient for administration of the gaze stabilization test

DETECTABILITY AND ERROR ESTIMATION IN ORBITAL FITS OF RESONANT EXTRASOLAR PLANETS

We estimate the conditions for detectability of two planets in a 2/1 mean-motion resonance from radial velocity data, as a function of their masses, number of observations and the signal-to-noise ratio. Even for a data set of the order of 100 observations and standard deviations of the order of a few meters per second, we find that Jovian-size resonant planets are difficult to detect if the masses of the planets differ by a factor larger than similar to 4. This is consistent with the present population of real exosystems in the 2/1 commensurability, most of which have resonant pairs with similar minimum masses, and could indicate that many other resonant systems exist, but are currently beyond the detectability limit. Furthermore, we analyze the error distribution in masses and orbital elements of orbital fits from synthetic data sets for resonant planets in the 2/1 commensurability. For various mass ratios and number of data points we find that the eccentricity of the outer planet is systematically overestimated, although the inner planet`s eccentricity suffers a much smaller effect. If the initial conditions correspond to small-amplitude oscillations around stable apsidal corotation resonances, the amplitudes estimated from the orbital fits are biased toward larger amplitudes, in accordance to results found in real resonant extrasolar systems.

Bayesian Estimation for Performance Measures of Two Diagnostic Tests in the Presence of Verification Bias

Sensitivity and specificity are measures that allow us to evaluate the performance of a diagnostic test. In practice, it is common to have situations where a proportion of selected individuals cannot have the real state of the disease verified, since the verification could be an invasive procedure, as occurs with biopsy. This happens, as a special case, in the diagnosis of prostate cancer, or in any other situation related to risks, that is, not practicable, nor ethical, or in situations with high cost. For this case, it is common to use diagnostic tests based only on the information of verified individuals. This procedure can lead to biased results or workup bias. In this paper, we introduce a Bayesian approach to estimate the sensitivity and the specificity for two diagnostic tests considering verified and unverified individuals, a result that generalizes the usual situation based on only one diagnostic test.

Methods of Estimation in Multiple Linear Regression: Application to Clinical Data

Nesse artigo, tem-se o interesse em avaliar diferentes estratégias de estimação de parâmetros para um modelo de regressão linear múltipla. Para a estimação dos parâmetros do modelo foram utilizados dados de um ensaio clínico em que o interesse foi verificar se o ensaio mecânico da propriedade de força máxima (EM-FM) está associada com a massa femoral, com o diâmetro femoral e com o grupo experimental de ratas ovariectomizadas da raça Rattus norvegicus albinus, variedade Wistar. Para a estimação dos parâmetros do modelo serão comparadas três metodologias: a metodologia clássica, baseada no método dos mínimos quadrados; a metodologia Bayesiana, baseada no teorema de Bayes; e o método Bootstrap, baseado em processos de reamostragem.

Bayesian nonlinear regression models with scale mixtures of skew-normal distributions: Estimation and case influence diagnostics

The purpose of this paper is to develop a Bayesian analysis for nonlinear regression models under scale mixtures of skew-normal distributions. This novel class of models provides a useful generalization of the symmetrical nonlinear regression models since the error distributions cover both skewness and heavy-tailed distributions such as the skew-t, skew-slash and the skew-contaminated normal distributions. The main advantage of these class of distributions is that they have a nice hierarchical representation that allows the implementation of Markov chain Monte Carlo (MCMC) methods to simulate samples from the joint posterior distribution. In order to examine the robust aspects of this flexible class, against outlying and influential observations, we present a Bayesian case deletion influence diagnostics based on the Kullback-Leibler divergence. Further, some discussions on the model selection criteria are given. The newly developed procedures are illustrated considering two simulations study, and a real data previously analyzed under normal and skew-normal nonlinear regression models. (C) 2010 Elsevier B.V. All rights reserved.

On estimation and influence diagnostics for log-Birnbaum-Saunders Student-t regression models: Full Bayesian analysis

The purpose of this paper is to develop a Bayesian approach for log-Birnbaum-Saunders Student-t regression models under right-censored survival data. Markov chain Monte Carlo (MCMC) methods are used to develop a Bayesian procedure for the considered model. In order to attenuate the influence of the outlying observations on the parameter estimates, we present in this paper Birnbaum-Saunders models in which a Student-t distribution is assumed to explain the cumulative damage. Also, some discussions on the model selection to compare the fitted models are given and case deletion influence diagnostics are developed for the joint posterior distribution based on the Kullback-Leibler divergence. The developed procedures are illustrated with a real data set. (C) 2010 Elsevier B.V. All rights reserved.

Transformed generalized linear models

The estimation of data transformation is very useful to yield response variables satisfying closely a normal linear model, Generalized linear models enable the fitting of models to a wide range of data types. These models are based on exponential dispersion models. We propose a new class of transformed generalized linear models to extend the Box and Cox models and the generalized linear models. We use the generalized linear model framework to fit these models and discuss maximum likelihood estimation and inference. We give a simple formula to estimate the parameter that index the transformation of the response variable for a subclass of models. We also give a simple formula to estimate the rth moment of the original dependent variable. We explore the possibility of using these models to time series data to extend the generalized autoregressive moving average models discussed by Benjamin er al. [Generalized autoregressive moving average models. J. Amer. Statist. Assoc. 98, 214-223]. The usefulness of these models is illustrated in a Simulation study and in applications to three real data sets. (C) 2009 Elsevier B.V. All rights reserved.

Identification of geodesic chirping Alfven modes and q-factor estimation in hot core tokamak plasmas in ASDEX Upgrade

Alfven eigenmodes (AE) driven by ion cyclotron resonance heating are usually registered by different diagnostic channels in the hot core plasmas of large tokamaks like JET and ASDEX Upgrade. These AE appear very near to the extremum points of Alfven wave continuum, which is modified by the geodesic effect due to poloidal mode coupling. It is shown that the AE spectrum may be explored as the magnetic spectroscopy (like Alfven cascades by Sharapov et al 2001 Phys. Lett. A 289 127) to determine the q-factor minimum and geodesic frequency at the magnetic axis in standard sawtoothed discharges without reversed shear.

Coherent states of a particle in a magnetic field and the Stieltjes moment problem

A solution to a version of the Stieltjes moment. problem is presented. Using this solution, we construct a family of coherent states of a charged particle in a uniform magnetic field. We prove that these states form an overcomplete set that is normalized and resolves the unity. By the help of these coherent states we construct the Fock-Bergmann representation related to the particle quantization. This quantization procedure takes into account a circle topology of the classical motion. (C) 2009 Elsevier B.V. All rights reserved.