Extrapolation measured data

This requirement also makes good sense. A calibration is nothing more than a mathematical model that relates the behavior of the measureable data to the behavior of that which we wish to predict. We construct a calibration by finding the best representation of the fit between the measured data and the predicted parameters. It is not surprising that the performance of a calibration can deteriorate rapidly if we use the calibration to extrapolate predictions for... 

It should be pointed out that a finite residual entropy calculated for a substance from experimental data obtained at temperatures extending down to a certain temperature, with extrapolation below that point, may arise either from failure of the experimenter to obtain thermodynamic equilibrium in his calorimetric measurements or from error in the extrapolation. Measurements made under ideal conditions and extended to sufficiently... 

Owing to the necessity for extrapolating measurements on methyl acrylate to a conversion of less than 1 percent in order to avoid the pronounced autoacceleration occurring with this monomer, the data are of lower accuracy than for most of the other monomers investigated. 

Often, it will be found that currents for a given reaction cannot be measured at all metals at the same value of potential. At some metals the currents would be too low for a reliable, sufficiently accurate determination at others they might be too high for a satisfactory experimental realization. A comparison will then be possible only after an extrapolation of data obtained in a different region of potentials, to the value of selected for comparison. This extrapolation may not be sufficiently reliable where the Tafel section of the polarization curve is too short or indistinct. 

Densities were measured using a Paar DMA 60 meter equipped with DMA 512 and DMA 601 HP external cells. Values in the 50-150°C range were interpolated from measured data (3-5 points) values above 150°C were extrapolated and are less accurate. Interfacial tension measurements at the minimum density difference encountered (0.05 g/cm3) could be in error by as much as 10%, which is within the repeatability of measurements with heavy crude oil samples (see below). 

An indirect indication of service life is obtained simply by comparison of the performance of materials under given test conditions, the one which shows the smaller change being deemed to perform better. If one material is a standard with known service performance an estimate can be made of the other material s expected performance. Particularly with accelerated tests, this can be a dangerous assumption because the differences seen under the test conditions may not be similar to the differences realised in practice. To make a direct estimate of service life it is necessary to apply some form of extrapolation technique to measured data. 

When the form of the change in a parameter with time has been established and a suitable measure to represent that form selected, the relation with the level of the degradation agent is needed to allow extrapolation to the service level. Generally, measurements need to be made at several agent levels to establish a model with reasonable confidence. Typically five levels are considered satisfactory. However, it should be noted that when extrapolation is to be made over several decades of time the uncertainty of the prediction will be large, even if the measured data looked very consistent. Estimates of uncertainty should always be made (see Section 9.3). 

When extrapolation of measured data is carried out all the uncertainties become magnified, increasing as the degree of extrapolation is increased. Inevitably, extrapolations from accelerated tests to normal ambient conditions will be subject to enormous uncertainties, which is why the general advice for temperature is to extrapolate to 30-40 °C beyond the last data point at the very most. Additional to this, but generally not quantifiable, is the uncertainty of the validity of the extrapolation model. 

Small molecule crystallographers are familiar with these concepts, since it is routine to measure data at low temperature to improve precision by reduction of thermal motion, and structures are often done at multiple temperatures to assess the origins of disorder in atomic positions. Albertsson et al. (1979) have reported the analysis of the crystal structure of Z)(-l-)-tartaric acid at 295, 160, 105, and 35 K. Figure 22 shows the individual isotropic. S-factors for the atoms in the structure at each of these temperatures the smooth variation of B with T is apparent. Below 105 K, B is essentially identical for all atoms and is also temperature independent the value of B = 0.7 agrees well with the expected zero-point vibradonal value. However, even for this simple structure, not all of the atoms show B vs T behavior at high temperature which extrapolates to 0 A at 0 K. 

By extrapolating scattering data for each angle to zero concentration, the mean-square radius of gyration may be measured [9,10,15-18]... 

In this way an extrapolation of data to zero concentration seems avoidable. This is in accordance with an early expectation of Peterlin and Signer (114). A refinement of this view, however, will be given below and in Chapter 4. Furthermore, a fair reduction with respect to the molecular weight is noticed. All measuring points of Fig. 3.1, which were obtained at / values smaller than two fall within the limits of the theory indicated by the lines "N and F (non-draining and free-... 

NH4NO3 (aq.). The heat of neutralization of aqueous ammonia with aqueous nitric acid was measured by Hess,6 Andrews,1,2<11 Favre and Silbermann,8 Favre,3 Bouzat,1 Berthelot,3,4 and Thomsen.15. Thomsen s data are the most reliable, and yield N=12.402oo- Data on the heat of dilution of aqueous ammonium nitrate were obtained by Pratt,1 Jirsa and Diamant,1 Thomsen,15 Tollinger1 Dunnington and Hoggard,1 Mondain-Monval,2,3 Lemer-Steinberg,1 Riimelin,1 Winkelmann,1 and Fricke and Havestadt.1 We have extrapolated the data to infinite dilution. 

CH3COOH (aq.). The heat of solution of solid acetic acid in water was measured by Berthelot,3,9 and Petterson,1 and of liquid acetic acid in water by Berthelot,3,9 and Berthelot and Louguinine.1 Data on the heat of dilution of aqueous acetic acid were reported by Thomsen,15 Berthelot and Louguinine,1 Berthelot,3,9 Faucon,1 Payn and Perman,1 Bose,1 and Richards and Gucker.1 We have extrapolated the data of Richards and Gucker1 to infinite dilution to obtain a value for aqueous undissociated CH3COOH at infinite dilution in water. 

Finally sine-wave analysis/synthesis is also suitable for extrapolation of missing data [Maher, 1994], Situations occur, for example, where a data segment is missing from a digital data stream. Sine-wave analysis/synthesis can be used to extrapolate the data across the gap. In particular, the measured sine-wave amplitude and phase are interpolated using the linear amplitude and cubic phase polynomial interpolators, respectively. In this way, the slow variation of the amplitude and phase function are exploited, in contrast with rapid waveform oscillations. 

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