Statistical Curve Fitting Research Articles

Modeling and Experimental Verification of Pilot-Scale Hollow Fiber, Direct Flow Microfiltration with Periodic Backwashing

A mechanistic model was derived for direct flow microfiltration (MF), one which lends more insight into the factors that control MF performance compared to a statistical curve-fitting procedure that is often used to estimate chemical cleaning intervals at pilot-scale. This theoretical approach to MF fouling was based on fundamental feed water, membrane, and cake characteristics and was shown to predict well pilot-scale experimental specific flux profiles when backwash effectiveness is used as a fitting parameter. Long-term experimental specific flux profiles obtained during the constant flux filtration of natural colloidal materials are reported under conditions typical of water treatment applications. Cakes formed during the filtration of untreated surface waters were found to be highly resistant and compressible. Longer chemical cleaning intervals (and higher values of the backwash ef fectiveness parameter) were obtained during the constant flux filtration of natural colloidal materials by a decrease in flux, feed water turbidity, and/or backwash interval.

Prognostics for a reciprocating air compressor

The health of a reciprocating air compressor can affect the time of occurrence of valve impacts within the compressor. A computer simulation reveals useful insight into the nature of the timing changes, which provides a basis for automated prognosis of the machine health. As an example, consider that the opening impact of a leaky discharge valve is delayed by an amount proportional to the size of the leak. The effect that the leak has on the efficiency of the compressor can be determined using the simulation, and a threshold for lost efficiency can be tied to the time of occurrence of the valve impact. Using the computer simulation as an aid, the vibration transients generated by the impacts can be clearly identified in the vibration signal measured on the exterior of the compressor. The time of occurrence of the impacts can then be accurately determined from the measured vibration using minimum-phase processing, a sliding exponential window, and statistical curve fitting. [See NOISE-CON Proceedings for full paper.]

Measurement of the thermal inertia of the skin using successive thermograms taken at a stepwise change in ambient radiation temperature

Skin thermal properties are difficult to measure in vivo in the steady state because there is a constant temperature gradient across the skin surface. However, measurement of skin thermal properties is postulated in quantitative evaluation for thermographic observation. Here, imaging of the thermal inertia of the skin was attempted by thermographic measurements at a stepwise change in ambient radiation temperature achieved by quickly switching 2 hoods maintained at different temperatures. Using this technique, a total of 65 thermograms were sequentially recorded at intervals of 0.5 s beginning 2 s before the stepwise change. The image of skin thermal inertia was estimated by applying statistical curve fitting at each pixel of the thermograms. In addition, the emissivity and true temperature of the skin were also determined, together with thermal inertia, in a single measurement. Measurements were made at different sites on 10 subjects. The average values of thermal inertia of normal skin were scattered throughout a range from 1.4*103 to 2.1*103 W s1/2 m-2 K-1. Investigations of the relationship between skin blood flow and thermal inertia were also made by imaging thermal inertia when skin blood flow was changed by applying a vasodilator or vasoconstrictor on the skin surface. In a comparison with the data measured by laser Doppler flowmetry, the average slope of skin blood flow versus thermal inertia was 2.88*10-4 V per W s1/2 m-2 K-1, and the thermal inertia of the skin with no blood flow was 1.03*103 W s1/2 m-1 K-1. The results also show an almost linear correlation between skin blood flow and thermal inertia in each individual, but interindividual differences were also observed. The results suggest that skin blood flow distribution can be estimated by noncontact imaging of thermal inertia.

Data reduction for k and α from transient heating of slab

For the transient heat conduction of a thin slab, the temperature—time history of its mid-point can be determined from a plane wall theory, provided that a slab thickness to length ratio of 0.2 or smaller is used in the experiments. Using the theory, two data reduction methods are described for deducing the thermal properties of conductivity ( k) and diffusivity (α) simultaneously from a single transient heating experiment. The first method uses an analytical technique to fit the inflection point on the temperature-√t, history plot to the theoretical inflection point. The second method, which was earlier proposed by the authors, uses a statistical curve fitting procedure to obtain the best fit between theory and experiment over seven points on the temperature-time history. Comparisons of the k values obtained by these methods against steady-state heat conduction measurements of the same slabs for various building construction materials indicate good agreement, and give support to the validity of these procedures. The values of k and α obtained by these procedures also compare favourably with reported literature data.

Shapes of rate-versus-level functions of primary auditory nerve fibres: Test of the basilar membrane mechanical hypothesis

Rate-versus level functions (RI functions) for characteristic frequency (CF) stimulation were measured from primary auditory nerve fibres from different spontaneous rate categories in the guinea pig cochlea. Attention was focussed on those fibres that showed clear breakpoints in their RI functions (sloping-saturation fibres). A statistical curve fitting procedure to an empirical equation was used to provide a quantitative estimate of the breakpoint position in individual fibres. It was found that, within the limits of reliability of the curve fitting procedure, the breakpoint position was the same in fibres from the same CF regions in any given animal. This result is consistent with the notion that the breakpoint position is determined by global basilar membrane mechanics and not by processes private to each nerve fibre. However, a subgroup of fibres not easily classifiable as sloping-saturation, showed features of their RI functions suggesting that factors other than basilar membrane mechanics could lead to fibre-to-fibre differences in rate-versus-level behaviour.

Growth path envelope analysis of ostwald ripening

Optimized procedures for performing growth path envelope (GPE) analysis of processes involving growth or shrinkage of particles have been developed, and the process has been automated using computerized statistical curve fitting and numerical analysis. The Weibull function was found to represent accurately the initial cumulative size distribution data, while a nonlinear function was found best for fitting ther-tgrowth paths for coarsening. A strategy for efficient selection of constant Nv> values for the analysis is described. The optimized methods have been applied to the analysis of Ostwald ripening during liquid-phase sintering of Cu-20Co alloy, and means for obtainingr * and growth velocities and testing coarsening theories are presented. Results from the automated procedure compare favorably with those from manual analysis but are obtained with considerably reduced effort.

The mathematical expression of rate constants in empirical rate of dyeing equations in disperse dyeing

The relationship between the rate constants in empirical rate of dyeing equations and the diffusion coefficient of dye in the fibre have been clarified by means of a statistical curve fitting method, where the rate of dyeing curves calculated from the theoretical equations (those of Wilson and Newman) have been compared with those from the empirical equations in the range Mt/M∞ from zero to about 0.8. The comparisons revealed that many of the empirical equations are suitable for describing the dyeing rate when there is a delay of dye sorption in the early stages, and one of them is useful for determining the dyeing rate when there is rapid dye sorption at the initial stages. The fact that these empirical equations fit well to practical dyeing can be explained in terms of the sorption delay caused by the stagnant solution layer surrounding the fibres.

Reformulating the Cube Law for Proportional Representation Elections

The cube law was proposed around 1910 to express the conversion of a party's vote shares into its seat share in two-party plurality elections with single-seat districts. This article develops predictive seat-vote equations for a much wider range of elections, including those involving many parties, single- and multi-seat districts, and diverse seat allocation rules such as plurality and list proportional representation (PR). Without any statistical curve fitting based on the seat and vote shares themselves, the basic features of the conversion are predicted using exogenous parameters: magnitude and number of districts, number of parties, and total size of the electorate and of the assembly. The link between the proposed equations and the original cube law is explicated. Using an existing data base, the fit of the predictive model is examined. On balance, this model accounts well for the conversion of votes to seats, and for the deviation from proportionality in PR systems.

Ｓ‐Ｎ試験データの折れ線近似によるパラメータ表示および解析

A simple and new statistical curve fitting method for the S-N type fatigue test, data has been proposed. The method utilizes a bi-linear type S-N curve comprising inclined and horizontal straight lines, and thus the data can be represented by 4 parameters, i.e., slope, knee, fatigue limit, and range of scatter. The principle of curve fitting for the inclined part is based on the principal component analysis technique and that for the horizontal part on the Probit analysis technique. 213 sets of small sample S-N data from NRIM Fatigue Data Sheets were analysed by the proposed method and correlations found between S-N parameters were discussed.

A two wavelength holographic technique for the study of two-dimensional thermal boundary layers

A two wavelength holographic interferometer is used to obtain heat transfer coefficient in a turbulent two-dimensional thermal boundary layer. This type of flow problem imposes some restrictions on the interpretation of interferometric data particularly in regions of the flow where large turbulence eddies exist. However, in the region of the flow near a heated wall, optical measurements of both temperature and temperature gradient are obtained by utilising different wavefront reconstruction optics. The heat transfer coefficients are inferred from this data by a statistical curve fitting procedure which utilizes redundant data to minimize any dependence on individual measurement error. The resulting curves are used to perform the extrapolations to obtain wall values.

Ultrasonic evaluation of fetal ventricular growth.

The ratio of lateral ventricle to hemispheric width was measured in 200 normal pregnancies. Statistical curve fitting was tried to predict normal value and 5 and 95 percentiles. Anatomical sections of fetal head are provided to correlate the ultrasonic pictures. A typical case of application is described.

Dynamic Stiffness of Rice Grain

ABSTRACT EXPERIMENTS to determine dynamic mechanical properties of brown rice, variety IR-8, were car-ried out with cylindrically shaped specimens tested uniaxially along the cylindrical axis. Storage and loss moduli were determined from 100 to 1000 Hz forcing frequency at four different moisture levels from 12 to 29 percent (dry basis). Statistical analyses of the test data revealed the stress-strain relations for the rice grain to be frequency as well as moisture dependent with the moduli increasing with frequency of loading and decreasing with moisture content. Statistical curve fitting resulted in a mathematical relaxation modulus expression at each moisture level that was equivalent to a Maxwell model mechanical repre-sentation. General equations for storage and loss moduli as functions of frequency and moisture content were also developed.

Accelerated Lifetest Results on Submarine-cable Transistors

A computerized analysis is outlined; it uses a lognormal distribution that is fitted to the observed data by the maximum likelihood method. A typical computer output illustrates the failure levels at operational temperatures. The statistical curve fitting technique aids the analysis of accelerated lifetest data. Failure estimates at junction temperatures, much lower than those under accelerated lifetest, are obtained by extrapolation. However, it is up to user whether such an extrapolation is justified. Due to the quantized nature of the data, an optimum exists for the time of device inspection, whereby statistical information is obtained at minimum testing expense. For example, inspecting the 250°C sample at 80 and 160 hours would be a waste of time since no failures should occur.

Circular dichroism measurements of benzyl L‐aspartate–nitrobenzyl L‐aspartate copolymers and their use in detecting and characterizing preferred polymer conformations

AbstractWe have examined the nature of the circular dichroism band at 330 mμ for a series of copolymers of β‐p‐nitrobenzyl L‐aspartate with β‐benzyl L‐aspartate. The circular dichroism band arises from an electronic transition in the nitroaromatic group. In order to interpret the effect quantitatively, we employed a simplified statistical treatment and curve fitting for six copolymers. Both approaches gave consistent results, which indicates that the dichroism comes from pairwise interactions between two nitrobenzyl groups. We constructed a molecular model that meets the constraints and requirements of the analyses developed in this paper. In this model, it is proposed that the main chain forms a right‐handed α‐helix and that nitrobenzyl groups separated by four residues interact with each other.