Curve fitting

We present a case study of Machine Learning and Data Mining in intensive care medicine. In the study, we compared different methods of measuring pressure-volume curves in artificially ventilated patients suffering from the Adult Respiratory Distress Syndrome (ARDS).

This paper presents an obstacle detection and avoidance of mobile robot using stereo camera for indoor environment. Block matching algorithm is solved the correspondence problem occurred in comparing stereo images (left and right sensors of the camera). The algorithm uses Sum of Absolute Differences (SAD). Left image works as a reference block to the right image and the output is disparity mapping or depth maps with the left coordinate system. A pair of camera or stereo vision baseline is based on horizontal configuration. The block matching technique is briefly described with the performance of its output. The curve fitting tool would determine the range of each obstacle detected in disparity mapping. The programming activities are using Matlab software starting from capturing images until navigation of mobile robot.

Structural relaxation of polyvinyl acetate (PVAc) was studied by mercury dilatometry and differential scanning calorimetry. Tool-Narayanaswamy-Moynihan (TNM) and Adam-Gibbs-Scherer (AGS) models were used to fit numerous experimental data. Single set of volume and enthalpy relaxation parameters describing all performed experiments was found for each model. Comparison of volume and enthalpy relaxation was made on the basis of TNM parameter values. The famous volume relaxation measurements made by Kovacs [Fortschr Hochpolym Forsch 1963;3:394-507] were successfully fitted using the same set of parameters. Furthermore, several non-fitting methods of TNM parameters' estimation (including, e.g., peak-shift method or inflectional analysis) were applied to our data and their critical comparison is outlined. For most methods a good agreement with curve-fitting results was achieved.

The secondary structure of six bovine lens protein fractions (two a, three f3 and one y-crystallin) are examined in solution and in solid forms for the first time using FTIR spectroscopy. Films of the nuclear and cortical regions of the bowine lens are also examined. The structure is quantitatively estimated from the vibrational analysis of the resolution-enhanced amide-I profile achieved by Fourier self-deconvolution and linear least-squares curve-fit algorithm. All the protein fractions fold predominantly in a f3-pleated sheet structure with little or no a-helical domains in solution or in lyophilized solid form. These proteins also retain their predominant f3-sheet conformation in the cellular phospholipid environment of the lens, in conformity with the structure obtained for all the mammalian species examined to date. Despite structural homology, vibrational data indicate 'subtle structural differences within each class of the crystallins probably due to presence of several minor substructures/subconforma tions. Substantial high amounts of turns (approx. 40%) observed in the f3-fractions may have a fundamental implication in stabilizing the tertiary structure of the uniquely folded-proteins vital for the transparency of the lens. These proteins in solid KBr-matrix undergo a major structural change, induced primarily by ionic interactions which refold them in a helical conformation. IR spectroscopy together with band-narrowing procedures has proven to be an effective tool to obtain structural information of proteins in solution, as solid substrates or in a complex biological tissue, such as ocular lens.

A major goal of NASA's In-Space Propulsion Program is to shorten trip times for scientific planetary missions. To meet this challenge arrival speeds will increase, requiring significant braking for orbit insertion, and thus increased deceleration propellant mass that may exceed launch lift capabilities. A technology called aerocapture has been developed to expand the mission potential of exploratory probes destined for planets with suitable atmospheres. Aerocapture inserts a probe into planetary orbit via a single pass through the atmosphere using the probe's aeroshell drag to reduce velocity. The benefit of an aerocapture maneuver is a large reduction in propellant mass that may result in smaller, less costly missions and reduced mission cruise times. The methodology used to design rigid aerocapture aeroshells will be presented with an emphasis on a new systems tool under development. Current methods for fast, efficient evaluations of structural systems for exploratory vehicles to planets and moons within our solar system have been under development within NASA having limited success. Many systems tools that have been attempted applied structural mass estimation techniques based on historical data and curve fitting techniques that are difficult and cumbersome to apply to new vehicle concepts and missions. The resulting vehicle aeroshell mass may be incorrectly estimated or have high margins included to account for uncertainty. This new tool will reduce the guesswork previously found in conceptual aeroshell mass estimations.

We studied the effect of post-deposition annealing on a HfO 2 /Si interface of by extremely asymmetric X-ray diffraction. Comparing the rocking curves before annealing the sample with those of the annealed sample, it is found that an interfacial layer with a density of 3 g/cm 3 grows at the interface between the HfO 2 layer and the substrate during post-deposition annealing. The wavelength dependency of the integrated intensities of the rocking curve for the as-deposited sample fluctuated with the observation position. This fluctuation was suppressed by annealing. From these results we concluded that the strain introduced into the substrate becomes homogeneous by annealing. Moreover, a quantitative estimation of the strain by curve fitting reveals the existence of compressive strain under the HfO 2 layer.

The problem of specifying the two free parameters that arise in spatial Pythagorean-hodograph (PH) quintic interpolants to given first-order Hermite data is addressed. Conditions on the data that identify when the "ordinary" cubic interpolant becomes a PH curve are formulated, since it is desired that the selection procedure should reproduce such curves whenever possible. Moreover, it is shown that the arc length of the interpolants depends on only one of the parameters, and that four (general) helical PH quintic interpolants always exist, corresponding to extrema of the arc length. Motivated by the desire to improve the fairness of interpolants to general data at reasonable computational cost, three selection criteria are proposed. The first criterion is based on minimizing a bivariate function that measures how "close" the PH quintic interpolants are to a PH cubic. For the second criterion, one of the parameters is fixed by first selecting interpolants of extremal arc length, and the other parameter is then determined by minimizing the distance measure of the first method, considered as a univariate function. The third method employs a heuristic but efficient procedure to select one parameter, suggested by the circumstances in which the "ordinary" cubic interpolant is a PH curve, and the other parameter is then determined as in the second method. After presenting the theory underlying these three methods, a comparison of empirical results from their implementation is described, and recommendations for their use in practical design applications are made.

Videokeratometers and Scheimpflug cameras permit accurate estimation of corneal surfaces. From height data it is possible to adjust analytical surfaces that will be later used for aberration calculation. Zernike polynomials are often used as adjusting polynomials, but they have shown to be not precise when describing highly irregular surfaces. We propose a combined zonal and modal method that allows an accurate reconstruction of corneal surfaces from height data, diminishing the influence of smooth areas over irregular zones and vice versa. The surface fitting error is decreased in the considered cases, mainly in the central region, which is more important optically. Therefore, the method can be established as an accurate resampling technique.

Artificial Neural Network (ANN) and regression models were developed using watershed-scale geomorphologic parameters to predict surface runoff and sediment losses of the St. Esprit watershed, Quebec, Canada. Ge o- morphological parameters describing the land surface drainage characteristics and surface water flow behav- iour were empirically associated with measured rainfall and runoff data and used as input to a three-layered back-propagation

This work proposes a methodology of identifying linear parameter varying (LPV) models for nonlinear systems. First, linear local models in some operating points, by applying standard identifications procedures for linear systems in time domain, are obtained. Next, a LPV model with linear fractional dependence (LFR) with respect to measured variables is fitted with the condition of containing all the linear models identified in previous step (differential inclusion). The fit is carried out using nonlinear least squares algorithms. Finally, this identification methodology will then be applied to a nonlinear turbocharged diesel engine.

Analysis of Venera 15 and 16 radar images and topographic data from the Freyja Montes region on Venus suggest that this mountain belt formed as a result of a sequence of underthrusts of the lithosphere of the North Polar Plains beneath the highlands of Ishtar Terra. The Freyja Montes deformation zone consists, south to north, of a linear orogenic belt, an adjacent plateau, a steep scarp separating the plateau from the North Polar Plains, a linear depression at the base of the scarp, and an outer rise. The topographic profile of the depression and outer rise are remarkably similar to that of a foreland deep and rise formed by the flexure of the underthrusting plate beneath a terrestrial mountain range. We test the lithospheric flexure hypothesis and we estimate the effective thickness T e of the elastic lithosphere of the underthrusting portion of the North Polar Plains by fitting individual topographic profiles to deflection curves for a broken elastic plate. The theoretical curves fit the observed topographic profiles to within measurement error for values of flexural rigidity D in the range (0.8-3) x 1022 N m, equivalent to T e in the range 11-18 km. Under the assumption that the base of the mechanical lithosphere is limited by the creep strength of olivine, the mean lithospheric thermal gradient is 14-23 K/km. The portion of the North Polar Plains to the immediate north of Freyja Montes stands at a regional elevation very near the modal elevation for the planet, so the inferred elastic !ithosphere thickness and thermal gradient may be at least approximately representative of a significant fraction of the Venus lithosphere. That the inferred thermal gradient is similar to the value expected for the global mean gradient on the basis of scaling from Earth provides support for the hypothesis that simple conduction dominates lithospheric heat transport on Venus relative to lithospheric recycling and volcanism.

Measurements of real and imaginary parts of the relative complex permittivity of glycerol were carried out in the frequency range 1 mHz-1 MHz at different temperatures between 188 and 263 K. The permittivity data have been analyzed thoroughly by a new data curve-fitting approach that involves the so-called eigen-coordinates method in conjunction with a separation procedure and the inverse permittivity formulas. A new single permittivity function, based on the so-called recap element picture for a self-similar (fractal) structure, has been recognized to describe well such data over the entire frequency range studied. The recognized dielectric function enabled us to infer an electrical equivalent-circuit network for the glycerol sample studied that involves a series combination of two recap elements, reflecting the existence of two different dielectric relaxation processes in glycerol. The temperature dependence of the relaxation times s 1 ðT Þ and s 2 ðT Þ entering into the identified permittivity function was found to obey nearly an Arrhenius behaviour with activation energies E 1 % 114 kJ/mol and E 2 % 94 kJ/mol. The recognized permittivity function can be justified by presuming that the processes represented by the recap elements characterized by the parameters (m 1 , s 1 , E 1) and (m 2 , s 2 , E 2) are linked to 'donor-like' and 'acceptor-like' charges formed from the infinite hydroxyl hydrogen bonds.

This paper addresses the problem of parametric representation and estimation of complex planar curves in 2-D, surfaces in 3-D and nonplanar space curves in 3-D. Curves and surfaces can be defined either parametrically or implicitly, and we use the latter representation. A planar curve is the set of zeros of a smooth function of two variables X-Y, a surface is the set of zeros of a smooth function of three variables X-~-Z, and a space curve is the intersection of two surfaces, which are the set of zeros of two linearly independent smooth functions of three variables X-!/-Z. For example, the surface of a complex object in 3-D can be represented as a subset of a single implicit surface, with similar results for planar and space curves. We show how this unified representation can be used for object recognition, object position estimation, and segmentation of objects into meaningful subobjects, that is, the detection of "interest regions" that are more complex than high curvature regions and, hence, more useful as features for object recognition. Fitting implicit curves and surfaces to data would be ideally based on minimizing the mean square distance from the data points to the curve or surface. Since the distance from a point to a curve or surface cannot be computed exactly by direct methods, the approximate distance, which is a first-order approximation of the real distance, is introduced, generalizing and unifying previous results. We fit implicit curves and surfaces to data minimizing the approximate mean square distance, which is a nonlinear least squares problem. We show that in certain cases, this problem reduces to the generalized eigenvector fit, which is the minimization of the sum of squares of the values of the functions that define the curves or surfaces under a quadratic constraint function of the data. This fit is computationally reasonable to compute, is readily parallelizable, and, hence, is easily computed in real time. In general, the generalized eigenvector lb provides a very good initial estimate for the iterative minimization of the approximate mean square distance. Although we are primarily interested in the 2-D and 3-D cases, the methods developed herein are dimension independent. We show that in the case of algebraic curves and surfaces, i.e., those defined by sets of zeros of polynomials, the minimizers of the approximate mean square distance and the generalized eigenvector fit are invariant with respect to similarity transformations. Thus, the generalized eigenvector lit is independent of the choice of coordinate system, which is a very desirable property for object recognition, position estimation, and the stereo matching problem. Finally, as applications of the previous techniques, we illustrate the concept of "interest regions"

b-Cyclodextrin (b-CD) hydrate was dehydrated under vacuum at room temperature (293 K) and the changes of its IR n(OH) band profile were observed. A curve fitting with Voigt functions to the band profile was able using information obtained from difference spectra. Four different fitting strategies gave similar band components. The behavior of the integrated intensities of these components upon dehydration was the basis for an assignment of these band components. The component a 3580 cm À1 is due to water molecules in the b-CD cavity and they leave very fast the macrocycle upon dehydration. Two components at 3530 and 3289 cm À1 are assigned to OH groups retained in b-CD despite dehydration, i.e. primary and secondary OH groups. The components at 3422 and 3185 cm À1 may be associated with water molecules in the interstices among b-CD molecules and linked by hydrogen bonds with them. Nevertheless, additional contributions from primary and secondary OH groups were found for these two bands.

Curved cross-sections extracted from medical volume images are useful for analyzing nonplanar anatomic structures such as the aorta arch or the pelvis. For visualization and for performing distance measurements, extracted surface sections need to be adequately flattened. We present two different distance preserving surface flattening methods which preserve distances according to a user-specified center of interest and according to user-specified orientations. The first method flattens surface sections by preserving distances along surface curves located within planes having a user specified constant orientation. The second method flattens surfaces along curves located within radial planes crossing the center of interest. We study and compare the properties of the two flattening methods by analyzing their distortion maps. Thanks to a multiresolution approach, we provide surface flattening at interactive rates, allowing users to displace their focus point while visualizing the resulting flattened surface. These distance preserving flattening methods provide new means of inspecting curved cross-sections extracted from medical images.

The current paper focuses on the influence of the process parameters on the peak values of the interfacial heat transfer coefficient (IHTC) at metal/die interface during high pressure die casting (HPDC) process. A "step shape" casting and AM50 alloy were used during the experiment. The IHTC was determined by solving the inverse thermal problem based on the measured temperature inside the die. Results show that the initial die surface temperature (IDST, T DI ) has a dominant influence while the casting pressure and fast shot velocity have a secondary influence on the IHTC peak values. By curve fitting, it was found that the IHTC peak value (h max ) changes as a function of the IDST in a manner of h max =e α T γ DI . Such relationship between the IHTC peak value and the IDST can also be found when the casting alloy is ADC12, indicating that this phenomenon is a common characteristic in the HPDC process.

Aim The aims of this study are to resolve terminological confusion around different types of species-area relationships (SARs) and their delimitation from species sampling relationships (SSRs), to provide a comprehensive overview of models and analytical methods for SARs, to evaluate these theoretically and empirically, and to suggest a more consistent approach for the treatment of species-area data.

Two-stage process described in the present work is a combination of dark and photofermentation in a sequential batch mode. In the first stage glucose is fermented to acetate, CO 2 and H 2 in an anaerobic dark fermentation by Enterobacter cloacae DM11. This is followed by a successive second stage where acetate is converted to H 2 and CO 2 in a photobioreactor by photosynthetic bacteria, Rhodobacter sphaeroides O.U. 001. The yield of hydrogen in the first stage was about 3:31 mol H 2 ðmol glucoseÞ À1 (approximately 82% of theoretical) and that in the second stage was about 1:521:72 mol H 2 ðmol acetic acidÞ À1 (approximately 37-43% of theoretical). The overall yield of hydrogen in two-stage process considering glucose as preliminary substrate was found to be higher compared to a single stage process. Monod model, with incorporation of substrate inhibition term, has been used to determine the growth kinetic parameters for the first stage. The values of maximum specific growth rate (m max Þ and K s (saturation constant) were 0:398 h À1 and 5:509 g l À1 , respectively, using glucose as substrate. The experimental substrate and biomass concentration profiles have good resemblance with those obtained by kinetic model predictions. A model based on logistic equation has been developed to describe the growth of R. sphaeroides O.U 001 in the second stage. Modified Gompertz equation was applied to estimate the hydrogen production potential, rate and lag phase time in a batch process for various initial concentration of glucose, based on the cumulative hydrogen production curves. Both the curve fitting and statistical analysis showed that the equation was suitable to describe the progress of cumulative hydrogen production.

A theory which describes the share price evolution at financial markets as a continuoustime random walk Scalas (2000); Mainardi (2000); Sabatelli (2002); Raberto (2002) has been generalized in order to take into account the dependence of waiting times t on price returns x. A joint probability density function (pdf) φ X,T (x, t) which uses the concept of a Lévy stable distribution is worked out. The theory is fitted to high-frequency US $/Japanese Yen exchange rate and low-frequency 19th century Irish stock data. The theory has been fitted both to price return and to waiting time data and the adherence to data, in terms of the χ 2 test statistic, has been improved when compared to the old theory Sabatelli (2002).

An empirical (target-) BRDF normalization method has been implemented for Imaging Spectrometry data processing, following the approach of Kennedy, published in 1997. It is a simple, empirical method with the purpose of a rapid technique, based on a least-squares quadratic curve fitting process. The algorithm is calculating correction factors in either multiplicative or additive manner for each of the identified land cover classes, per spectral band and view angle unit. Image pre-classification is essential for successful anisotropy normalization. This anisotropy normalization method is a candidate to be used as baseline correction for future data products of APEX, a new airborne Imaging Spectrometer suitable for simulation and inter-calibration of data from various other sensors. A classification algorithm, being able to provide anisotropy class indexing that is optimized for the purpose of BRDF normalization has to be used. In this study, the performance of the standard Spectral Angle Mapper (SAM) approach with RSL's spectral database SPECCHIO attached is investigated. Due to its robustness regarding directional effects, SAM classification is estimated to be the most efficient. Results of both the classification and the normalization process are validated using two airborne image datasets from the HyMAP sensor, taken in 2004 over the "Vordemwald" test site in northern Switzerland.

A series of coals varying in rank from peat to semi-anthracite was studied by Fourier Tramform infrared spectroscopy (FTIR) and curve-fitting analysis in order to gain additional information on coal structure and the main structural changes that take place during the coalification process. Several structural parameters based on FTIR data and curve-resolved bands were calculated. These parameters provide quantitative evidence of the increase of aromaticity and the loss of aliphatic and oxygen-containing structures with increasing maturation. The analysis of the aromatic C-H out-of-plane region (900-700 cm-~) reveals the loss of aromatic substituents up to the stage of bituminous coal and the subsequent increase in the degree of aromatic condensation with increasing coalification. The ratio of aromatic carbons to carboxyl groups (1605 cm -1/1605 cm -1 + 1705 cm -1) appears to be a suitable parameter for assessing the natural maturation of coal.

This article puts forward a method for codebook design to support beamforming mechanism in a 60GHz millimeter-wave wireless communication environment. The codebook is designed with only phase shifting but not any amplitude adjustment to meet the low power consumption requirement. IEEE802.15.3c beam MRA (main response axis) directions are taken as the original version to make curve fitting, so that weighting vector based on uniform weighting can be obtained to generate a codebook matrix. The simulation results show that, compared with the codebook in IEEE802.15.3c, similar main lobe patterns can be obtained in the proposed codebook, while the side lobe level is lower and the antenna gain at the direction of MRA is higher. Meanwhile, the interference decreases obviously with an improved network channel capacity in the practice environment, and the codebook is also robust to the beam shifting deriving from phase errors.

A simulation based comparative study of the polarization hysteresis of the ferroelectric capacitor using various ferroelectric models is presented. A 2-dimensional finite element device-level model was implemented using SILVACO's ATLAS device simulator to generate the polarization hysteresis characteristics for Au/Poly(vinylidene fluoride-trifluoroethylene)/Au metal-insulator-metal (MIM) device. Landau free energy expression for electric field in terms of polarization is also implemented in the MATLAB to produce the polarization hysteresis curve of polycrystalline ferroelectrics. The main drawback of these models was their inability to predict polarization saturation at the same electric field limits for which the experimental device was saturating. A new model for ferroelectric hysteresis based on curve fitting algorithm is derived that forces the polarization to be saturated at desired electric field. The MATLAB simulation of this model and the experimental hysteresis is compared which shows an excellent level of agreement.

This paper introduces the rational forward difference operator for differential computation on a rational Bézier patch based on its control mesh. With this rational version of the forward difference operator, and by ignoring the appropriate dependent (on lower order derivatives) terms of various derivatives, the derivatives themselves have the same expressions as their polynomial counterparts; the curvature and torsion, and the first and second fundamental forms, all have very similar expressions as those equations from classical differential geometry. This new approach also allows straightforward generalization to higher dimensional rational Bézier patches, the special co-dimension 1 case of which is treated with the result of the first and second fundamental forms.

Objective: Calibration with LC-MS data may be accomplished with different combinations of curve fit, weighting, and origin treatment. Agilent's Mass Hunter Quantitative Analysis software contains a Curve Fit Assistant, which allows comparison of over 100 different calibration options. Curve fit choices include linear, quadratic, power, logarithmic, second order logarithmic, and average of response factors. Weighting choices include equal, inverse x

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To verify a previously reported three-dimensional (3D) ultrasound method for the measurement of time-average volumetric blood flow, experiments were performed under pulsatile flow conditions, including in vivo investigations, and results were compared with accepted, but invasive, ''gold standard'' techniques. Results showed that volume averaging results in the correct time-average volume flow without the need for cardiac gating. Unlike other currently employed methods, this method is independent of Doppler angle, flow profile and vessel geometry. A GE Logiq 9 ultrasound system (GE Medical Systems, Milwaukee, WI, USA) and a four-dimensional (4D) 10L and 4D 16L probe were used to acquire 3D Doppler measurements in the femoral and carotid arteries of four canines. Two invasive blood flow meters were used (electromagnetic for one canine and ultrasonic for three canines) as the gold standard techniques. Transcutaneous color flow measurements were taken to obtain 3D volume data sets encompassing the vessel. Constant depth planes were used to integrate color flow pixels encompassing the entire vessel cross-section. Power Doppler data were used to correct for partial volume effects. An artificial stenosis was induced to vary the ambient volume flow. Unrestricted, bidirectional flow was measured as high as 400 mL min -1 . Several flow restrictions were imposed that decreased the measured volumetric flow rate to as low as 30 mL min -1 . All flow rate estimates (n 5 38) were plotted against results obtained via the gold standards. A general line fit resulted in y 5 0.926 x -0.87 (r 2 5 0.95), which corresponds to a 0.6% flow offset relative to the average flow rate of 142 mL min -1 , as well as a 7.4% error in the linearity of our estimate. A secondary curve fit was performed that required the slope to be 1 and the intercept to be 0, which yielded an r 2 -value of 0.93. The percent-error distribution was computed and fitted to a Gaussian function, which yielded m 5 -7.04% and s 5 9.52%. Theoretical studies were conducted to estimate the expected error in our volume flow measurements as a function of number of samples (N) used for averaging pulsatile waveforms. Experiments showed the same 1= ffiffiffiffi N p dependence as theory. Direct comparisons of volume flow rate estimates using volumetric color Doppler and independent standards showed that our method has good accuracy under in vivo pulsatile blood flow condi-

ABSTRACT In the companion paper [Z.T. Chang, B. Cherry, M. Marosszeky, Polarisation behaviour of steel bar samples in concrete in seawater, Part 1: Experimental measurement of polarisation curves of steel in concrete, Corrosion Science 50(2) (2008) 357–364], influences of the experimental procedure on measured polarisation curves of steel in concrete in seawater were investigated. It was found that an undistorted full polarisation curve of a steel sample in concrete can be obtained by the two-test procedure to conduct separate anodic and cathodic polarisation tests and combine the two partial curves into one curve. However, polarisation curves of steel samples in concrete in seawater were found not to fit with the theoretical curves based on the kinetics of charge transfer reactions. This was considered to be due in the main to the influence of a passive film on the steel surface in concrete. This paper proposes an empirical model for the polarisation behaviour of steel in concrete based on the assumption of two major electrochemical processes taking place at the interfaces of steel/passive-film/concrete: one is the active corrosion process and the other is the passive film growth or dissolution process. Typical curve-fit results are presented using the proposed model to simulate the polarisation behaviour and to evaluate the corrosion rate and Tafel parameters of three types of steel corrosion in seawater: steel bars in concrete, new steel bars and corroded steel bars.

This paper describes an analytical approach to determine the chemical composition of bio-oils in terms of macro-chemical families. Bio-oils from the vacuum pyrolysis of softwood bark and hardwood were first fractionated using solvent extraction. Fractions obtained were then characterized using GC-MS, thermogravimetric techniques (TG) and Gel Permeation Chromatography (GPC). Thermogravimetric and molar mass distribution curves of each fraction were interpreted in terms of macro-families applying curvefitting procedures. The composition of the different macro-fractions obtained was in agreement using both methods. The proposed procedure enables a thorough description of bio-oil composition as a mixture of water, monolignols, polar compounds with moderate volatility, sugars, extractive-derived compounds, heavy polar and non-polar compounds, MeOH-toluene insolubles and volatile organic compounds. r

The spline under tension was introduced by Schweikert in an attempt to imitate cubic splines but avoid the spurious critical points they induce. The defining equations are presented here, together with an e fficient method for determining the necessary parameters and computing the resultant spline. The standard scalar-valued curve fitting problem is discussed, as well as the fitting of open and closed curves in the plane. The use of these curves and the importance of the tension in the fitting of contour lines are mentioned as application.

The application of a trigonometric polynomial and an exponential fitting approach is compared for a three-point formula for second-order derivatives, for Simpson's quadrature rule and for Numerov's scheme for second-order differential equations. The expressions for the occurring parameters are constructed in both the approaches and the behaviour of these parameters with respect to the introduced frequency is studied. The errors for specific problems obtained in both the approaches as a function of the frequency are compared.

The effects of extended source-to-surface distance (SSD) on the electron beam dose profiles were evaluated for various electron beam energies-6, 9, 12, 15 and 20 MeV-and the accuracy of various output correction methods was analysed on a Mitsubishi linear accelerator using a radiation field analyser (RFA). The dose fall-off region of the central axis depth-dose curves was nearly independent for SSDs up to 120 cm where as in the build-up region, a marginal reduction of surface dose was observed, particularly for lower energies and for smaller field sizes. Effective SSDs and virtual source distances were evaluated for field sizes ranging from 5 × 5 to 15 × 15 cm 2 for various energies. Curve fitting was done with the measured outputs with various equations and coefficients were evaluated. The accuracy of the derived output correction factors by effective SSD, virtual source distance and curve-fit methods was assessed by evaluating correlation coefficients between the calculated and the measured values. The correlation coefficient was best with the linear-quadratic equation followed by the effective SSD method and the virtual source method. The output correction based on the linear-quadratic equation showed the best estimate of electron beam output at extended SSDs with an accuracy well within ±1%. The rapid reduction of dose due to the applicator-scattered component at d max point with an extended SSD was significant for the 5 × 5 cm 2 applicator and lower energies.

The most stable formula for a rational interpolant for use on a finite interval is the barycentric form [1, 2]. A simple choice of the barycentric weights ensures the absence of (unwanted) poles on the real line [3]. In [4] we indicate that a more refined choice of the weights in barycentric rational interpolation can guarantee comonotonicity and coconvexity of the rational interpolant in addition to a polefree region of interest.In this presentation we generalize the above to the multivariate case. We use a product-like form of univariate barycentric rational interpolants and indicate how the location of the poles and the shape of the function can be controlled. This functionality is of importance in the construction of mathematical models that need to express a certain trend, such as in probability distributions, economics, population dynamics, tumor growth models etc.

An explicit analytical formula for a short kernel fifth-order polynomial interpolator is obtained. It is also possible to obtain the explicit forms of even higher order interpolation kernels with the method of calculation used, but it is seen that these local kernels become "remainderdominated" as the order increases. The frequency domain properties and the accuracies of the obtained kernel and the known convolution kernels are compared. Frequency domain comparison with the cubic B-spline interpolators is also given. Some cases of proper use of the calculated kernels have been pointed out.

https://towardsdatascience.com/data-smoothing-for-data-science-visualization-the-goldilocks-trio-part-1-867765050615

Making peace with piecewise. We've all heard that information is beautiful. But then life happens, and we find ourselves trying to prettify the mess that often is reality…

Three thin TiO2 films obtained by magnetron sputtering onto glass in different deposition conditions were compared with respect of their morphologies by means of AFM investigations. Acquired AFM data were processed in order to get the Power Spectral Densities (PSD) ofthe films and glass substrate. For each sample the PSDs computed at two scan sizes were combined in a single PSD curve with an extended range ofspatialfrequencies. The obtained PSD curves were fitted with a suitable theoretical model and characteristic parameters were deduced and discussed in order to compare the film morphologies in conjunction with deposition conditions.

Thermoluminescence (TL) emission spectra and TL glow curves of powder samples of YPO 4 :Ce 3+ , Ln 3+ (Ln 3+ = Nd 3+ , Sm 3+ , Dy 3+ , Ho 3+ , Er 3+ , Tm 3+ ) were measured in order to investigate the nature of the trapping centres and to locate the lanthanide energy levels relative to the valence and conduction bands of the YPO 4 host. Trap depths were determined by three different TL techniques (simple formula, various heating rate method and glow curve fitting) and compared with an earlier developed predictive energy level scheme. The nature of the trapping centres agrees with that predicted by the energy level scheme. Further, the trend of trap depths of the divalent lanthanides (Ln 2+ = Nd 2+ , Sm 2+ , Dy 2+ , Ho 2+ , Er 2+ , Tm 2+ ) agrees very well with the trend of levels determined by three different thermoluminescent techniques. It is concluded that the energy level scheme can help in the development of new TL materials suitable in dosimetry.

Symbolic computation of NURBS plays an important role in many areas of NURBS-based geometric computation and design. However, any nontrivial symbolic computation, especially when rational B-splines are involved, would typically result in B-splines with high degrees. In this paper we develop degree reduction strategies for NURBS symbolic computation on curves. The specific topics we consider include zero curvatures and critical curvatures of plane curves, various ruled surfaces related to space curves, and point/curve bisectors and curve/curve bisectors.

A method for determining the best fit of a complex M&batter spectrum is presented and applied to the room temperature 57Fe Mossbatter spectrum of HoBa,(Cu a,ssFea,as)sO,,,,. The method generates many '"good fits" as do previous methods. The "best fit", which is the goal of curve fitting, is selected from the class of good fits by means of a statistical test using the "F" function. The best fit for this compound represents 7 doublets. This is significantly larger than the previously reported number (3 or 4) of subspectra associated with REBa,(Cu,_,Fe,),O, compounds. Assuming no mixed valence, this implies 7 Fe sites. This result is also significantly larger than the previously reported number of Fe sites (l-4) in the REBa&Cul_nFe,)sO, compounds. For 6 of the 7 doublets, the isomer shift, quadrupole splitting and line width are strongly correlated. The valence state corresponding to these 6 doublets is high spin + 4. The seventh doublet is high spin + 3. The sum of Mossbatter site multiplicities is related to the number of accidental degeneracies by a sum rule.

The so-called ‘least squares regression’ for mathematical modeling is a widely used technique.  It’s so common that one might think nothing could be improved to the algorithm anymore.  But it can.  By minimizing the squares of the differences between measured and predicted values not only in the vertical, but also in the horizontal direction.  I call this ‘multidirectional regression’.  The difference is very significant, especially for power function models, often used in biomedical sciences.  And it makes the regression invariant if the dependent and independent variables are switched.  This was a neglected problem with the traditional method.  The Body Mass Index and the Corpulence Index and their correlation with body fat percentage are studied here as an example showing that this regression technique produces better results.

The aim of this study was to develop a mathematic model to describe the relationship between magnitude of applied force and rate of orthodontic tooth movement. Initially, data were extracted from experimental studies in dogs (beagles), in which controlled, standardized forces were used to move mandibular second premolars distally. Curve-fitting by nonlinear regression analysis provided an equation describing the relationship between force magnitude and rate of tooth movement in beagles. A similar equation was subsequently used to analyze the limited available data from the literature on human canine retraction. The maximum rates of tooth movement in humans and dogs are very similar. A threshold for force magnitude that would switch on tooth movement could not be defined. The model showed that a wide range of forces can be identified, all of which lead to a maximum rate of tooth movement. (Am J Orthod Dentofacial Orthop 2004; 125:71-7)

The current–voltage (I–V) characteristics of monocrystalline, polycrystalline and amorphous silicon solar cells are measured in the dark. A two diodes equivalent model is used to describe the electronic properties of solar cells. The non-linear curve fitting of the dark I–V curves obtain besides ...

A growing number of commentators are forecasting a near-term peak and subsequent terminal decline in the global production of conventional oil as a result of the physical depletion of the resource. These forecasts frequently rely on the estimates of the ultimately recoverable resources (URR) of different regions, obtained through the use of curve-fitting to historical trends in discovery or production. Curve-fitting was originally pioneered by M. King Hubbert in the context of an earlier debate about the future of the US oil production. However, despite their widespread use, curve-fitting techniques remain the subject of considerable controversy. This article classifies and explains these techniques and identifies both their relative suitability in different circumstances and the level of confidence that may be placed in their results. This article discusses the interpretation and importance of the URR estimates, indicates the relationship between curve fitting and other methods of estimating the URR and classifies the techniques into three groups. It then investigates each group in turn, indicating their historical origins, contemporary application and major strengths and weaknesses. The article then uses illustrative data from a number of oil-producing regions to assess whether these techniques produce consistent results as well as highlight some of the statistical issues raised and suggesting how they may be addressed. The article concludes that the applicability of curve-fitting techniques is more limited than adherents claim and that the confidence bounds on the results are wider than usually assumed. 4 These involve selecting and parameterising a mathematical function that has the ÔbestÕ statistical fit to a set of data points, possibly subject to constraints. 5 HubbertÕs original curve was hand drawn, but his later articles used more formal techniques. 209 1520-7439/10/0900-0209/0 Ó

Thirty empirically assessed utility functions on changes in wealth or return on investment were examined for general features and susceptability to fits by linear, power, and exponential functions. Separate fits were made to below-target data and above-target data. The usual “target” was the no-change point.The majority of below-target functions were risk seeking; the majority of above-target functions were risk averse; and the most common composite shape was convex-concave, or risk seeking in losses and risk averse in gains. The least common composite was concave-concave. Below-target utility was generally steeper than above-target utility with a median below-to-above slope ratio of about 4.8. The power and exponential fits were substantially better than the linear fits. Power functions gave the best fits in the majority of convex below-target and concave above-target cases, and exponential functions gave the best fits in the majority of concave below-target and convex above-target cases. Several implications of these results for decision making under risk are mentioned.