Data-point

The most reliable estimates of the parameters are obtained from multiple measurements, usually a series of vapor-liquid equilibrium data (T, P, x and y). Because the number of data points exceeds the number of parameters to be estimated, the equilibrium equations are not exactly satisfied for all experimental measurements. Exact agreement between the model and experiment is not achieved due to random and systematic errors in the data and due to inadequacies of the model. The optimum parameters should, therefore, be found by satisfaction of some selected statistical criterion, as discussed in Chapter 6. However, regardless of statistical sophistication, there is no substitute for reliable experimental data. 

The total number of experimental data points is N. Data points 1 through L and L+1 through M refer to VLB measurements (P, T,... 

X, y) for the 1-2 binary and 2-3 binary, respectively. Data points M+1 through N are ternary liquid-liquid equilibrium measurements (T, x, x, x, . The 1-rich phase is indicated... 

In the maximum-likelihood method used here, the "true" value of each measured variable is also found in the course of parameter estimation. The differences between these "true" values and the corresponding experimentally measured values are the residuals (also called deviations). When there are many data points, the residuals can be analyzed by standard statistical methods (Draper and Smith, 1966). If, however, there are only a few data points, examination of the residuals for trends, when plotted versus other system variables, may provide valuable information. Often these plots can indicate at a glance excessive experimental error, systematic error, or "lack of fit." Data points which are obviously bad can also be readily detected. If the model is suitable and if there are no systematic errors, such a plot shows the residuals randomly distributed with zero means. This behavior is shown in Figure 3 for the ethyl-acetate-n-propanol data of Murti and Van Winkle (1958), fitted with the van Laar equation. 

An additional advantage derived from plotting the residuals is that it can aid in detecting a bad data point. If one of the points noticeably deviates from the trend line, it is probably due to a mistake in sampling, analysis, or reporting. The best action would be to repeat the measurement. However, this is often impractical. The alternative is to reject the datum if its occurrence is so improbable that it would not reasonably be expected to occur in the given set of experiments. 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

The low-pressure extrapolated data points were generated by linear extrapolation of the lowest 4-6 points on a plot of... 

At pressures above the highest real data point, the extrapolated data were generated by the correlation of Lyckman et al. (1965), modified slightly to eliminate any discontinuity between the real and generated data. This modification is small, only a few percent, well within the uncertainties of the Lyckman method. The Lyckman correlation was always used within its recommended limits of validity--that is, at reduced temperatures no greater than 1.5 to 2.0. 

Appendix C-6 gives parameters for all the condensable binary systems we have here investigated literature references are also given for experimental data. Parameters given are for each set of data analyzed they often reflect in temperature (or pressure) range, number of data points, and experimental accuracy. Best calculated results are usually obtained when the parameters are obtained from experimental data at conditions of temperature, pressure, and composition close to those where the calculations are performed. However, sometimes, if the experimental data at these conditions are of low quality, better calculated results may be obtained with parameters obtained from good experimental data measured at other conditions. 

READ COMPONENT NAMES, NUMBE OF DATA POINTS, AND EXECUTION CODES... 

PRINTS THE results OF THE REGRESSION-OP VARRIANCE-COVARPIANCE MATRIX, CORRELAT AND THE VARRIANCE OF THE FIT. ALSO, PP DATA, THEIR ESTIMATED TRUE VALUES, AND ALL NN DATA POINTS. FINALLY, THF ROOT-DEVIATIONS ARE GIVEN FOR EACH 0 = THE... 

Figure 1. shows the measured phase differenee derived using equation (6). A close match between the three sets of data points can be seen. Small jumps in the phase delay at 5tt, 3tt and most noticeably at tt are the result of the mathematical analysis used. As the cell is rotated such that tlie optical axis of the crystal structure runs parallel to the angle of polarisation, the cell acts as a phase-only modulator, and the voltage induced refractive index change no longer provides rotation of polarisation. This is desirable as ultimately the device is to be introduced to an interferometer, and any differing polarisations induced in the beams of such a device results in lower intensity modulation. 

Fig. V-13. Composite x/ln a curve for 3-pentanol. The various data points are for different E values each curve for a given E has been shifted horizontally to give the optimum match to a reference curve for an E near the electrocapillary maximum. (From Ref. 134.)...

A8, which leads to D, = 1/(2A8). The factor of two arises because a minimum of two data points per period are needed to sample a sinusoidal wavefonn. Naturally, the broadband light source will detennine the actual content of the spectrum, but it is important that the step size be small enough to acconunodate the highest frequency components of the source, otherwise they... 

Figure C2.18.6. The coverages of fluorosilyl groups in tire reaction layer shown as a function of exposure. The coverages refer to monolayers of SiF groups. The smootli curves are drawn tlirough tire data points. Reproduced from 1411.

We now examine how a next-amplitude-map was obtained from tire attractor shown in figure C3.6.4(a) [171. Consider tire plane in tliis space whose projection is tire dashed curve i.e. a plane ortliogonal to tire (X (tj + t)) plane. Then, for tire /ctli intersection of tire (continuous) trajectory witli tliis plane, tliere will be a data point X (ti + r), X (ti + 2r))on tire attractor tliat lies closest to tire intersection of tire continuous trajectory. A second discretization produces tire set Xt- = k = 1,2,., I This set is used in tire constmction... 

Calculating points on a set of PES, and fitting analytic functions to them is a time-consuming process, and must be done for each new system of interest. It is also an impossible task if more than a few (typically 4) degrees of freedom are involved, simply as a consequence of the exponential growth in number of ab initio data points needed to cover the coordinate space. 

A further model Hamiltonian that is tailored for the treatment of non-adiabatic systems is the vibronic coupling (VC) model of Koppel et al. [65]. This provides an analytic expression for PES coupled by non-adiabatic effects, which can be fitted to ab initio calculations using only a few data points. As a result, it is a useful tool in the description of photochemical systems. It is also very useful in the development of dynamics methods, as it provides realistic global surfaces that can be used both for exact quantum wavepacket dynamics and more approximate methods. 

Fig. 3. Some representative pair potentials Uy(r), sealed to move their interesting range to [0,5]. The numbers above each potential denote the class label 7 and the iiinnber of data points available for the fit. (For example, elass 63 gives distanee 3 potentials for the amino acid pairs Lys-Asp, Arg-Lys and Glu-Tyr.) The spectrum below each potential consists of 50 lines pieked uniformly from the data.

Fig. 4. The average end-to-end-distance of butane as a function of timestep (note logarithmic scale) for both single-timestep and triple-timestep Verlet schemes. The timestep used to define the data point for the latter is the outermost timestep At (the interval of updating the nonbonded forces), with the two smaller values used as Atj2 and At/A (for updating the dihedral-angle terms and the bond-length and angle terms, respectively).

In misiipcrviscd learning, the network tries to group the input data on the basis of similarities between theses data. Those data points which arc similar to each other arc allocated to the same neuron or to eloscly adjacent neurons. 

Steps 2 and 3 are performed for all input objects. When all data points have been fed into the network one training epoch has been achieved. A network is usually trained in several training epochs, depending on the size of the network and the number of data points. 

The explorative analysis of data sets by visual data mining applications takes place in a three-step process During the first step (overview), the user can obtain an overview of the data and maybe can identify some basic relationships between specific data points. In the second step (filtering), dynamic and interactive navigation, selection, and query tools will be used to reorganize and filter the data set. Each interaction by the user will lead to an immediate update of the data scene and will reveal the hidden patterns and relationships. Finally, the patterns or data points can be analyzed in detail with specific detail tools. 

The DIM statement in Program QENTROPY sets aside 100 memory locations for the experimental data points. It is necessary for any data set having more than 12 data pairs. What is the entropy of Pb at 100 and 200 K Make a rough sketch of the curve of C, vs. T for lead. Sketch the curve of Cp/T vs. T for lead. 

To anyone who has carried out curve-fitting calculations with a mechanical calculator (yes, they once existed) TableCurve (Appendix A) is equally miraculous. TableCurve fits dozens, hundreds, or thousands of equations to a set of experimental data points and ranks them according to how well they fit the points, enabling the researcher to select from among them. Many will fit poorly, but usually several fit well. 

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