Prediction of coagulation properties, titratable acidity and pH of bovine milk using mid-infrared spectroscopy

This study investigated the potential application of mid-infrared spectroscopy (MIR 4,000–900 cm−1) for the determination of milk coagulation properties (MCP), titratable acidity (TA), and pH in Brown Swiss milk samples (n = 1,064). Because MCP directly influence the efficiency of the cheese-making process, there is strong industrial interest in developing a rapid method for their assessment. Currently, the determination of MCP involves time-consuming laboratory-based measurements, and it is not feasible to carry out these measurements on the large numbers of milk samples associated with milk recording programs. Mid-infrared spectroscopy is an objective and nondestructive technique providing rapid real-time analysis of food compositional and quality parameters. Analysis of milk rennet coagulation time (RCT, min), curd firmness (a30, mm), TA (SH°/50 mL; SH° = Soxhlet-Henkel degree), and pH was carried out, and MIR data were recorded over the spectral range of 4,000 to 900 cm−1. Models were developed by partial least squares regression using untreated and pretreated spectra. The MCP, TA, and pH prediction models were improved by using the combined spectral ranges of 1,600 to 900 cm−1, 3,040 to 1,700 cm−1, and 4,000 to 3,470 cm−1. The root mean square errors of cross-validation for the developed models were 2.36 min (RCT, range 24.9 min), 6.86 mm (a30, range 58 mm), 0.25 SH°/50 mL (TA, range 3.58 SH°/50 mL), and 0.07 (pH, range 1.15). The most successfully predicted attributes were TA, RCT, and pH. The model for the prediction of TA provided approximate prediction (R2 = 0.66), whereas the predictive models developed for RCT and pH could discriminate between high and low values (R2 = 0.59 to 0.62). It was concluded that, although the models require further development to improve their accuracy before their application in industry, MIR spectroscopy has potential application for the assessment of RCT, TA, and pH during routine milk analysis in the dairy industry. The implementation of such models could be a means of improving MCP through phenotypic-based selection programs and to amend milk payment systems to incorporate MCP into their payment criteria.

State estimation using model order reduction for unstable systems

The problem of state estimation occurs in many applications of fluid flow. For example, to produce a reliable weather forecast it is essential to find the best possible estimate of the true state of the atmosphere. To find this best estimate a nonlinear least squares problem has to be solved subject to dynamical system constraints. Usually this is solved iteratively by an approximate Gauss–Newton method where the underlying discrete linear system is in general unstable. In this paper we propose a new method for deriving low order approximations to the problem based on a recently developed model reduction method for unstable systems. To illustrate the theoretical results, numerical experiments are performed using a two-dimensional Eady model – a simple model of baroclinic instability, which is the dominant mechanism for the growth of storms at mid-latitudes. It is a suitable test model to show the benefit that may be obtained by using model reduction techniques to approximate unstable systems within the state estimation problem.

Model estimation of cerebral hemodynamics between blood flow and volume changes: A data-based modeling approach

It is well known that there is a dynamic relationship between cerebral blood flow (CBF) and cerebral blood volume (CBV). With increasing applications of functional MRI, where the blood oxygen-level-dependent signals are recorded, the understanding and accurate modeling of the hemodynamic relationship between CBF and CBV becomes increasingly important. This study presents an empirical and data-based modeling framework for model identification from CBF and CBV experimental data. It is shown that the relationship between the changes in CBF and CBV can be described using a parsimonious autoregressive with exogenous input model structure. It is observed that neither the ordinary least-squares (LS) method nor the classical total least-squares (TLS) method can produce accurate estimates from the original noisy CBF and CBV data. A regularized total least-squares (RTLS) method is thus introduced and extended to solve such an error-in-the-variables problem. Quantitative results show that the RTLS method works very well on the noisy CBF and CBV data. Finally, a combination of RTLS with a filtering method can lead to a parsimonious but very effective model that can characterize the relationship between the changes in CBF and CBV.

Sparse model construction using coordinate descent optimization

We propose a new sparse model construction method aimed at maximizing a model’s generalisation capability for a large class of linear-in-the-parameters models. The coordinate descent optimization algorithm is employed with a modified l1- penalized least squares cost function in order to estimate a single parameter and its regularization parameter simultaneously based on the leave one out mean square error (LOOMSE). Our original contribution is to derive a closed form of optimal LOOMSE regularization parameter for a single term model, for which we show that the LOOMSE can be analytically computed without actually splitting the data set leading to a very simple parameter estimation method. We then integrate the new results within the coordinate descent optimization algorithm to update model parameters one at the time for linear-in-the-parameters models. Consequently a fully automated procedure is achieved without resort to any other validation data set for iterative model evaluation. Illustrative examples are included to demonstrate the effectiveness of the new approaches.

Complex-valued B-spline neural networks for modeling and inverting Hammerstein systems

Many communication signal processing applications involve modelling and inverting complex-valued (CV) Hammerstein systems. We develops a new CV B-spline neural network approach for efficient identification of the CV Hammerstein system and effective inversion of the estimated CV Hammerstein model. Specifically, the CV nonlinear static function in the Hammerstein system is represented using the tensor product from two univariate B-spline neural networks. An efficient alternating least squares estimation method is adopted for identifying the CV linear dynamic model’s coefficients and the CV B-spline neural network’s weights, which yields the closed-form solutions for both the linear dynamic model’s coefficients and the B-spline neural network’s weights, and this estimation process is guaranteed to converge very fast to a unique minimum solution. Furthermore, an accurate inversion of the CV Hammerstein system can readily be obtained using the estimated model. In particular, the inversion of the CV nonlinear static function in the Hammerstein system can be calculated effectively using a Gaussian-Newton algorithm, which naturally incorporates the efficient De Boor algorithm with both the B-spline curve and first order derivative recursions. The effectiveness of our approach is demonstrated using the application to equalisation of Hammerstein channels.

Conditioning of the weak-constraint variational data assimilation problem for numerical weather prediction

4-Dimensional Variational Data Assimilation (4DVAR) assimilates observations through the minimisation of a least-squares objective function, which is constrained by the model flow. We refer to 4DVAR as strong-constraint 4DVAR (sc4DVAR) in this thesis as it assumes the model is perfect. Relaxing this assumption gives rise to weak-constraint 4DVAR (wc4DVAR), leading to a different minimisation problem with more degrees of freedom. We consider two wc4DVAR formulations in this thesis, the model error formulation and state estimation formulation. The 4DVAR objective function is traditionally solved using gradient-based iterative methods. The principle method used in Numerical Weather Prediction today is the Gauss-Newton approach. This method introduces a linearised `inner-loop' objective function, which upon convergence, updates the solution of the non-linear `outer-loop' objective function. This requires many evaluations of the objective function and its gradient, which emphasises the importance of the Hessian. The eigenvalues and eigenvectors of the Hessian provide insight into the degree of convexity of the objective function, while also indicating the difficulty one may encounter while iterative solving 4DVAR. The condition number of the Hessian is an appropriate measure for the sensitivity of the problem to input data. The condition number can also indicate the rate of convergence and solution accuracy of the minimisation algorithm. This thesis investigates the sensitivity of the solution process minimising both wc4DVAR objective functions to the internal assimilation parameters composing the problem. We gain insight into these sensitivities by bounding the condition number of the Hessians of both objective functions. We also precondition the model error objective function and show improved convergence. We show that both formulations' sensitivities are related to error variance balance, assimilation window length and correlation length-scales using the bounds. We further demonstrate this through numerical experiments on the condition number and data assimilation experiments using linear and non-linear chaotic toy models.

Tests of sunspot number sequences: 3. Effects of regression procedures on the calibration of historic sunspot data

We use sunspot group observations from the Royal Greenwich Observatory (RGO) to investigate the effects of intercalibrating data from observers with different visual acuities. The tests are made by counting the number of groups RB above a variable cut-off threshold of observed total whole-spot area (uncorrected for foreshortening) to simulate what a lower acuity observer would have seen. The synthesised annual means of RB are then re-scaled to the full observed RGO group number RA using a variety of regression techniques. It is found that a very high correlation between RA and RB (rAB > 0.98) does not prevent large errors in the intercalibration (for example sunspot maximum values can be over 30 % too large even for such levels of rAB). In generating the backbone sunspot number (RBB), Svalgaard and Schatten (2015, this issue) force regression fits to pass through the scatter plot origin which generates unreliable fits (the residuals do not form a normal distribution) and causes sunspot cycle amplitudes to be exaggerated in the intercalibrated data. It is demonstrated that the use of Quantile-Quantile (“Q  Q”) plots to test for a normal distribution is a useful indicator of erroneous and misleading regression fits. Ordinary least squares linear fits, not forced to pass through the origin, are sometimes reliable (although the optimum method used is shown to be different when matching peak and average sunspot group numbers). However, other fits are only reliable if non-linear regression is used. From these results it is entirely possible that the inflation of solar cycle amplitudes in the backbone group sunspot number as one goes back in time, relative to related solar-terrestrial parameters, is entirely caused by the use of inappropriate and non-robust regression techniques to calibrate the sunspot data.

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.

The relationship between diffuse spectral reflectance of the soil and its cation exchange capacity is scale-dependent

Diffuse reflectance spectroscopy (DRS) is increasingly being used to predict numerous soil physical, chemical and biochemical properties. However, soil properties and processes vary at different scales and, as a result, relationships between soil properties often depend on scale. In this paper we report on how the relationship between one such property, cation exchange capacity (CEC), and the DRS of the soil depends on spatial scale. We show this by means of a nested analysis of covariance of soils sampled on a balanced nested design in a 16 km × 16 km area in eastern England. We used principal components analysis on the DRS to obtain a reduced number of variables while retaining key variation. The first principal component accounted for 99.8% of the total variance, the second for 0.14%. Nested analysis of the variation in the CEC and the two principal components showed that the substantial variance components are at the > 2000-m scale. This is probably the result of differences in soil composition due to parent material. We then developed a model to predict CEC from the DRS and used partial least squares (PLS) regression do to so. Leave-one-out cross-validation results suggested a reasonable predictive capability (R2 = 0.71 and RMSE = 0.048 molc kg− 1). However, the results from the independent validation were not as good, with R2 = 0.27, RMSE = 0.056 molc kg− 1 and an overall correlation of 0.52. This would indicate that DRS may not be useful for predictions of CEC. When we applied the analysis of covariance between predicted and observed we found significant scale-dependent correlations at scales of 50 and 500 m (0.82 and 0.73 respectively). DRS measurements can therefore be useful to predict CEC if predictions are required, for example, at the field scale (50 m). This study illustrates that the relationship between DRS and soil properties is scale-dependent and that this scale dependency has important consequences for prediction of soil properties from DRS data

Data assimilation for marine monitoring and prediction: The MERCATOR operational assimilation systems and the MERSEA developments

During the past 15 years, a number of initiatives have been undertaken at national level to develop ocean forecasting systems operating at regional and/or global scales. The co-ordination between these efforts has been organized internationally through the Global Ocean Data Assimilation Experiment (GODAE). The French MERCATOR project is one of the leading participants in GODAE. The MERCATOR systems routinely assimilate a variety of observations such as multi-satellite altimeter data, sea-surface temperature and in situ temperature and salinity profiles, focusing on high-resolution scales of the ocean dynamics. The assimilation strategy in MERCATOR is based on a hierarchy of methods of increasing sophistication including optimal interpolation, Kalman filtering and variational methods, which are progressively deployed through the Syst`eme d’Assimilation MERCATOR (SAM) series. SAM-1 is based on a reduced-order optimal interpolation which can be operated using ‘altimetry-only’ or ‘multi-data’ set-ups; it relies on the concept of separability, assuming that the correlations can be separated into a product of horizontal and vertical contributions. The second release, SAM-2, is being developed to include new features from the singular evolutive extended Kalman (SEEK) filter, such as three-dimensional, multivariate error modes and adaptivity schemes. The third one, SAM-3, considers variational methods such as the incremental four-dimensional variational algorithm. Most operational forecasting systems evaluated during GODAE are based on least-squares statistical estimation assuming Gaussian errors. In the framework of the EU MERSEA (Marine EnviRonment and Security for the European Area) project, research is being conducted to prepare the next-generation operational ocean monitoring and forecasting systems. The research effort will explore nonlinear assimilation formulations to overcome limitations of the current systems. This paper provides an overview of the developments conducted in MERSEA with the SEEK filter, the Ensemble Kalman filter and the sequential importance re-sampling filter.

Harmonic force fields from scaled SCF calculations: Program ASYM40

We report an extended version of our normal coordinate program ASYM40, which may be used to transform Cartesian force constants from ab initio calculations to a force field in nonredundant internal (symmetry) coordinates. When experimental data are available, scale factors for the theoretical force field may then be optimized by least-squares refinement. The alternative of refining an empirical force field to fit a wide variety of data, as with the previous version ASYM20, has been retained. We compare the results of least-squares refinement of the full harmonic force field with least-squares refinement of only the scale factors for an SCF calculated force field and conclude that the latter approach may be useful for large molecules where more sophisticated calculations are impractical. The refinement of scale factors for a theoretical force field is also useful when there are only limited spectroscopic data. The program will accept ab initio calculated force fields from any program that presents Cartesian force constants as output. The program is available through Quantum Chemistry Program Exchange.

Vibrational energy levels of water

Several quartic force fields and a full sextic anharmonic force field for H,O have been determined from high-quality ab initio calculations, the highest at the aug-cc-pVQZ CCSD(T) level of theory. These force fields have been used to determine vibrational excited state band origins up to 15 000 cm - ’ above the zero-point level, using both a perturbation-resonancea pproach and a variational approach. An optimisedq uartic force field hasb eeno btained by least squares refinement of our best ab initio results to fit the observed overtone levels of 5 symmetrically substituted isotopomers of water (Hi60, Hi70, HisO, D,O, and T,O) with an rms error of less than 10 cm-‘, using the perturbation-resonancem odel for the vibrational calculation. Predicatel east squaresr efinement was usedt o provide a loose constraint of the refined force field to the ab initio results. The results obtained prove the viability of the perturbation-resonancem odel for usei n larger molecular systemsa nd also highlight someo f its weaknesse

Anharmonic force field of acetylene

The harmonic and anharmonic force field of acetylene has been determined in a least-squares calculation from recently determined data on the spectroscopic constants of various isotopic species (including the vibrational l-doubling constant). A general quadratic and cubic force field was used, but a constrained quartic force field containing only 8 of the 23 possible quartic constants. The results are discussed and compared with earlier work.

Vibration–rotation variational calculations: precise results on HCN up to 25 000 cm−1

Variation calculations of the vibration–rotation energy levels of many isotopomers of HCN are reported, for J=0, 1, and 2, extending up to approximately 8 quanta of each of the stretching vibrations and 14 quanta of the bending mode. The force field, which is represented as a polynomial expansion in Morse coordinates for the bond stretches and even powers of the angle bend, has been refined by least squares to fit simultaneously all observed data on the Σ and Π state vibrational energies, and the Σ state rotational constants, for both HCN and DCN. The observed vibrational energies are fitted to roughly ±0.5 cm−1, and the rotational constants to roughly ±0.0001 cm−1. The force field has been used to predict the vibration rotation spectra of many isotopomers of HCN up to 25 000 cm−1. The results are consistent with the axis‐switching assignments of some weak overtone bands reported recently by Jonas, Yang, and Wodtke, and they also fit and provide the assignment for recent observations by Romanini and Lehmann of very weak absorption bands above 20 000 cm−1.

The anharmonic force field of HCN and HCP

The quadratic, cubic, and quartic force field of HCN has been calculated by a least squares refinement to fit the most recent observed data on the vibration-rotation constants of HCN, DCN and H13CN. All of the observed parameters are fitted within their standard errors of observation. The corresponding parameters for other isotopic species are calculated. For HCP and DCP the more limited data available have been fitted to an anharmonic force field using constraints based on comparison with HCN. Using this force field the zero-point rotational constants B0 have been corrected to obtain the equilibrium constants Be, and hence the equilibrium structure has been determined to be re(CH) = 1•0692(7)A, and re(CP) = 1•5398(2)A.