Data methods in optometry. Part 10: non-linear regression analysis

1. The techniques associated with regression, whether linear or non-linear, are some of the most useful statistical procedures that can be applied in clinical studies in optometry. 2. In some cases, there may be no scientific model of the relationship between X and Y that can be specified in advance and the objective may be to provide a ‘curve of best fit’ for predictive purposes. In such cases, the fitting of a general polynomial type curve may be the best approach. 3. An investigator may have a specific model in mind that relates Y to X and the data may provide a test of this hypothesis. Some of these curves can be reduced to a linear regression by transformation, e.g., the exponential and negative exponential decay curves. 4. In some circumstances, e.g., the asymptotic curve or logistic growth law, a more complex process of curve fitting involving non-linear estimation will be required.

Accurate WKB Approximation for a 1D Problem with Low Regularity

2000 Mathematics Subject Classification: 34L40, 65L10, 65Z05, 81Q20.

Representing spray zone with cross flow as a well-mixed compartment in a high shear granulator

The spray zone is an important region to control nucleation of granules in a high shear granulator. In this study, a spray zone with cross flow is quantified as a well-mixed compartment in a high shear granulator. Granulation kinetics is quantitatively derived at both particle-scale and spray zone-scale. Two spatial decay rates, DGSDR (droplet-granule spatial decay rate) ζDG and DPSDR (droplet-primary particle spatial decay rate) ζDP, which are functions of volume fraction and diameter of particulate species within the powder bed, are defined to simplify the deduction. It is concluded that in cross flow, explicit analytical results show that the droplet concentration is subject to exponential decay with depth which produces a numerically infinite depth of spray zone in a real penetration process. In a well-mixed spray zone, the depth of the spray zone is 4/(ζDG + ζDP) and π2/3(ζDG + ζDP) in cuboid and cylinder shape, respectively. The first-order droplet-based collision rates of, nucleation rate B0 and rewetting rate RW0 are uncorrelated with the flow pattern and shape of the spray zone. The second-order droplet-based collision rate, nucleated granule-granule collision rate RGG, is correlated with the mixing pattern. Finally, a real formulation case of a high shear granulation process is used to estimate the size of the spray zone. The results show that the spray zone is a thin layer at the powder bed surface. We present, for the first time, the spray zone as a well-mixed compartment. The granulation kinetics of a well-mixed spray zone could be integrated into a Population Balance Model (PBM), particularly to aid development of a distributed model for product quality prediction.

Periphyton light transmission relationships in Florida Bay and the Florida Keys, USA

Light transmission was measured through intact, submerged periphyton communities on artificial seagrass leaves. The periphyton communities were representative of the communities on Thalassia testudinum in subtropical seagrass meadows. The periphyton communities sampled were adhered carbonate sediment, coralline algae, and mixed algal assemblages. Crustose or film-forming periphyton assemblages were best prepared for light transmission measurements using artificial leaves fouled on both sides, while measurements through three-dimensional filamentous algae required the periphyton to be removed from one side. For one-sided samples, light transmission could be measured as the difference between fouled and reference artificial leaf samples. For two-sided samples, the percent periphyton light transmission to the leaf surface was calculated as the square root of the fraction of incident light. Linear, exponential, and hyperbolic equations were evaluated as descriptors of the periphyton dry weight versus light transmission relationship. Hyperbolic and exponential decay models were superior to linear models and exhibited the best fits for the observed relationships. Differences between the coefficients of determination (r2) of hyperbolic and exponential decay models were statistically insignificant. Constraining these models for 100% light transmission at zero periphyton load did not result in any statistically significant loss in the explanatory capability of the models. In most all cases, increasing model complexity using three-parameter models rather than two-parameter models did not significantly increase the amount of variation explained. Constrained two-parameter hyperbolic or exponential decay models were judged best for describing the periphyton dry weight versus light transmission relationship. On T. testudinum in Florida Bay and the Florida Keys, significant differences were not observed in the light transmission characteristics of the varying periphyton communities at different study sites. Using pooled data from the study sites, the hyperbolic decay coefficient for periphyton light transmission was estimated to be 4.36 mg dry wt. cm−2. For exponential models, the exponential decay coefficient was estimated to be 0.16 cm2 mg dry wt.−1.

Quantitative and Qualitative Aspects of Dissolved Organic Carbon Leached from Senescent Plants in an Oligotrophic Wetland

We conducted a series of experiments whereby dissolved organic matter (DOM) was leached from various wetland and estuarine plants, namely sawgrass (Cladium jamaicense), spikerush (Eleocharis cellulosa), red mangrove (Rhizophora mangle), cattail (Typha domingensis), periphyton (dry and wet mat), and a seagrass (turtle grass; Thalassia testudinum). All are abundant in the Florida Coastal Everglades (FCE) except for cattail, but this species has a potential to proliferate in this environment. Senescent plant samples were immersed into ultrapure water with and without addition of 0.1% NaN3 (w/ and w/o NaN3, respectively) for 36 days. We replaced the water every 3 days. The amount of dissolved organic carbon (DOC), sugars, and phenols in the leachates were analyzed. The contribution of plant leachates to the ultrafiltered high molecular weight fraction of DOM (>1 kDa; UDOM) in natural waters in the FCE was also investigated. UDOM in plant leachates was obtained by tangential flow ultrafiltration and its carbon and phenolic compound compositions were analyzed using solid state 13C cross-polarization magic angle spinning nuclear magnetic resonance (13C CPMAS NMR) spectroscopy and thermochemolysis in the presence of tetramethylammonium hydroxide (TMAH thermochemolysis), respectively. The maximum yield of DOC leached from plants over the 36-day incubations ranged from 13.0 to 55.2 g C kg−1 dry weight. This amount was lower in w/o NaN3 treatments (more DOC was consumed by microbes than produced) except for periphyton. During the first 2 weeks of the 5 week incubation period, 60–85% of the total amount of DOC was leached, and exponential decay models fit the leaching rates except for periphyton w/o NaN3. Leached DOC (w/ NaN3) contained different concentrations of sugars and phenols depending on the plant types (1.09–7.22 and 0.38–12.4 g C kg−1 dry weight, respectively), and those biomolecules comprised 8–34% and 4–28% of the total DOC, respectively. This result shows that polyphenols that readily leach from senescent plants can be an important source of chromophoric DOM (CDOM) in wetland environments. The O-alkyl C was found to be the major C form (55±9%) of UDOM in plant leachates as determined by 13C CPMAS NMR. The relative abundance of alkyl C and carbonyl C was consistently lower in plant-leached UDOM than that in natural water UDOM in the FCE, which suggests that these constituents increase in relative abundance during diagenetic processing. TMAH thermochemolysis analysis revealed that the phenolic composition was different among the UDOM leached from different plants, and was expected to serve as a source indicator of UDOM in natural water. Polyphenols are, however, very reactive and photosensitive in aquatic environments, and thus may loose their plant-specific molecular characteristics shortly. Our study suggests that variations in vegetative cover across a wetland landscape will affect the quantity and quality of DOM leached into the water, and such differences in DOM characteristics may affect other biogeochemical processes.

Ablation of Steel by Microsecond Pulse Trains

Laser micromachining is an important material processing technique used in industry and medicine to produce parts with high precision. Control of the material removal process is imperative to obtain the desired part with minimal thermal damage to the surrounding material. Longer pulsed lasers, with pulse durations of milli- and microseconds, are used primarily for laser through-cutting and welding. In this work, a two-pulse sequence using microsecond pulse durations is demonstrated to achieve consistent material removal during percussion drilling when the delay between the pulses is properly defined. The light-matter interaction moves from a regime of surface morphology changes to melt and vapour ejection. Inline coherent imaging (ICI), a broadband, spatially-coherent imaging technique, is used to monitor the ablation process. The pulse parameter space is explored and the key regimes are determined. Material removal is observed when the pulse delay is on the order of the pulse duration. ICI is also used to directly observe the ablation process. Melt dynamics are characterized by monitoring surface changes during and after laser processing at several positions in and around the interaction region. Ablation is enhanced when the melt has time to flow back into the hole before the interaction with the second pulse begins. A phenomenological model is developed to understand the relationship between material removal and pulse delay. Based on melt refilling the interaction region, described by logistic growth, and heat loss, described by exponential decay, the model is fit to several datasets. The fit parameters reflect the pulse energies and durations used in the ablation experiments. For pulse durations of 50 us with pulse energies of 7.32 mJ +/- 0.09 mJ, the logisitic growth component of the model reaches half maximum after 8.3 us +/- 1.1 us and the exponential decays with a rate of 64 us +/- 15 us. The phenomenological model offers an interpretation of the material removal process.

La metrica di Agmon ed il decadimento esponenziale delle soluzioni di equazioni alle derivate parziali

The aim of this master thesis is to study the exponential decay of solutions of elliptic partial equations. This work is based on the results obtained by Agmon. To this purpose, first, we define the Agmon metric, that plays an important role in the study of exponential decay, because it is related to the rate of decay. Under some assumptions on the growth of the function and on the positivity of the quadratic form associated to the operator, a first result of exponential decay is presented. This result is then applied to show the exponential decay of eigenfunctions with eigenvalues whose real part lies below the bottom of the essential spectrum. Finally, three examples are given: the harmonic oscillator, the hydrogen atom and a Schrödinger operator with purely discrete spectrum.

Unravelling the nanostructure of strawberry fruit pectins by atomic force microscopy

Atomic force microscopy (AFM) allows the analysis of individual polymers at nanostructural level with a minimal sample preparation. This technique has been used to analyse the pectin disassembly process during the ripening and postharvest storage of several fleshy fruits. In general, pectins analysed by AFM are usually visualized as isolated chains, unbranched or with a low number of branchs and, occasionally, as large aggregates. However, the exact nature of these structures is unknown. It has been suggested that pectin aggregates represent a mixture of rhamnonogalacturonan I and homogalacturonan, while isolated chains and their branches are mainly composed by polygalacturonic acid. In order to gain insight into the nature of these structures, sodium carbonate soluble pectins from ripe strawberry (Fragaria x ananassa, Duch.) fruits were subjected to enzymatic digestion with endo-Polygalacturonase M2 from Aspergillus aculeatus, and the samples visualized by AFM at different time intervals. Pectins isolated from control, non-transformed plants, and two transgenic genotypes with low level of expression of ripening-induced pectinase genes encoding a polygalacturonase (APG) or a pectate lyase (APEL) were also included in this study. Before digestion, isolated pectin chains from control were shorter than those from transgenic fruits, showing number-average (LN) contour length values of 73.2 nm vs. 95.9 nm and 91.4 nm in APG and APEL, respectively. The percentage of branched polymers was significantly higher in APG polyuronides than in the remaining genotypes, 33% in APG vs. 6% in control and APEL. As a result of the endo-PG treatment, a gradual decrease in the main backbone length of isolated chains was observed in the three samples. The minimum LN value was reached after 8 h of digestion, being similar in the three genotypes, 22 nm. By contrast, the branches were not visible after 1.5-2 h of digestion. LN values were plotted against digestion time and the data fitted to a first-order exponential decay curve, obtaining R2 values higher than 0.9. The half digestion time calculated with these equations were similar for control and APG pectins, 1.7 h, but significantly higher in APEL, 2.5 h, indicating that these polymer chains were more resistant to endo-PG digestion. Regarding the pectin aggregates, their volumes were estimated and used to calculate LN molecular weights. Before digestion, control and APEL samples showed complexes of similar molecular weights, 1722 kDa, and slightly higher than those observed in APG samples. After endo-PG digestion, size of complexes diminished significantly, reaching similar values in the three pectin samples, around 650 kDa. These results suggest that isolated polymer chains visualized by AFM are formed by a HG domain linked to a shorter polymer resistant to endo-PG digestion, maybe xylogalacturonan or RG-I. The silencing of the pectate lyase gene slightly modified the structure and/or chemical composition of polymer chains making these polyuronides more resistant to enzymatic degradation. Similarly, polygalacturonic acid is one of the main component of the aggregates.

Dissipation of triadimefon on the solid/gas interface

The dissipation or triadimefon, as pure solid and in the Bayleton 5 commercial formulation, was studied under controlled and natural conditions. Volatilization and photodegradation were shown to be the main dissipation processes. The volatilization results can be described by an empirical model assuming exponential decay of the volatilization rate. The filler of the commercial formulation is determinant for the volatilization but has little effect on the photodegradation rates. The main photoproducts were identified and a reaction mechanism proposed. (C) 2001 Elsevier Science Ltd. All rights reserved.

Comparative study of the dissipation of triadimefon in greenhouse and field conditions

The dissipation of triadimefon, {1-(4-chlorophenoxy)-3,3-dimethyl-1-(1H-1,2,4-triazol-1-yl)butanone}, was studied after its application to melon leaves, glass and paper, both in greenhouse and field conditions. The dissipation rate of triadimefon in its commercial formulation Bayleton 5 was found to be lower in greenhouse than field. The results for different samples in the same conditions show that the dissipation of triadimefon was found to be biphasic. This result can be accounted by a semi-empirical model which assumes an initial fast decline of the dissipation rate, attributed to an exponential decay of the volatilization rates, followed by a second phase where the dissipation is due to a first order degradation processes.The dissipation of triadimefon, {1-(4-chlorophenoxy)-3,3-dimethyl-1-(1H- 1,2,4-triazol-1-yl)butan-one}, was studied after its application to melon leaves, glass and paper, both in greenhouse and field conditions. The dissipation rate of triadimefon in its commercial formulation Bayleton 5 was found to be lower in greenhouse than field. The results for different samples in the same conditions show that the dissipation of triadimefon was found to be biphasic. This result can be accounted by a semi-empirical model which assumes an initial fast decline of the dissipation rate, attributed to an exponential decay of the volatilization rates, followed by a second phase where the dissipation is due to a first order degradation processes.

Intravoxel incoherent motion diffusion-weighted imaging in the liver: comparison of mono-, bi- and tri-exponential modelling at 3.0-T.

PURPOSE: To determine whether a mono-, bi- or tri-exponential model best fits the intravoxel incoherent motion (IVIM) diffusion-weighted imaging (DWI) signal of normal livers. MATERIALS AND METHODS: The pilot and validation studies were conducted in 38 and 36 patients with normal livers, respectively. The DWI sequence was performed using single-shot echoplanar imaging with 11 (pilot study) and 16 (validation study) b values. In each study, data from all patients were used to model the IVIM signal of normal liver. Diffusion coefficients (Di ± standard deviations) and their fractions (fi ± standard deviations) were determined from each model. The models were compared using the extra sum-of-squares test and information criteria. RESULTS: The tri-exponential model provided a better fit than both the bi- and mono-exponential models. The tri-exponential IVIM model determined three diffusion compartments: a slow (D1 = 1.35 ± 0.03 × 10(-3) mm(2)/s; f1 = 72.7 ± 0.9 %), a fast (D2 = 26.50 ± 2.49 × 10(-3) mm(2)/s; f2 = 13.7 ± 0.6 %) and a very fast (D3 = 404.00 ± 43.7 × 10(-3) mm(2)/s; f3 = 13.5 ± 0.8 %) diffusion compartment [results from the validation study]. The very fast compartment contributed to the IVIM signal only for b values ≤15 s/mm(2) CONCLUSION: The tri-exponential model provided the best fit for IVIM signal decay in the liver over the 0-800 s/mm(2) range. In IVIM analysis of normal liver, a third very fast (pseudo)diffusion component might be relevant. KEY POINTS: ? For normal liver, tri-exponential IVIM model might be superior to bi-exponential ? A very fast compartment (D = 404.00 ± 43.7 × 10 (-3)  mm (2) /s; f = 13.5 ± 0.8 %) is determined from the tri-exponential model ? The compartment contributes to the IVIM signal only for b ≤ 15 s/mm (2.)

Short-term temporal changes of soil carbon losses after tillage described by a first-order decay model

Tillage stimulates soil carbon (C) losses by increasing aeration, changing temperature and moisture conditions, and thus favoring microbial decomposition. In addition, soil aggregate disruption by tillage exposes once protected organic matter to decomposition. We propose a model to explain carbon dioxide (CO2) emission after tillage as a function of the no-till emission plus a correction due to the tillage disturbance. The model assumes that C in the readily decomposable organic matter follows a first-order reaction kinetics equation as: dC(sail)(t)/dt = -kC(soil)(t) and that soil C-CO2 emission is proportional to the C decay rate in soil, where C-soil(t) is the available labile soil C (g m(-2)) at any time (t). Emissions are modeled in terms soil C available to decomposition in the tilled and non-tilled plots, and a relationship is derived between no-till (F-NT) and tilled (F-Gamma) fluxes, which is: F-T = a1F(NT)e(-a2t), where t is time after tillage. Predicted and observed fluxes showed good agreement based on determination coefficient (R-2), index of agreement and model efficiency, with R-2 as high as 0.97. The two parameters included in the model are related to the difference between the decay constant (k factor) of tilled and no-till plots (a(2)) and also to the amount of labile carbon added to the readily decomposable soil organic matter due to tillage (a,). These two parameters were estimated in the model ranging from 1.27 and 2.60 (a(1)) and - 1.52 x 10(-2) and 2.2 x 10(-2) day(-1) (a(2)). The advantage is that temporal variability of tillage-induced emissions can be described by only one analytical function that includes the no-till emission plus an exponential term modulated by tillage and environmentally dependent parameters. (C) 2008 Elsevier B.V. All rights reserved.

Short-term temporal changes of bare soil CO2 fluxes after tillage described by first-order decay models

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

LOSS of EXPONENTIAL STABILITY FOR A THERMOELASTIC SYSTEM WITH MEMORY

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

First-order decay models to describe soil C-CO2 Loss after rotary tillage

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)