Discrete data set

In this appendix, we summarize how the discrete data sets can be represented in terms of averages, standard deviation, and moments in a concise way. In principle, under certain conditions, these quantities represent the same information represented through histograms. 

FIGURE 1 PSl and PS2 are discrete data sets because they have no datum point correlation. PSl = PS2. PS3 is different from PS4, although they have the exact same data points. PSS can be assigned onto PS4, but not PS6. 

Whenever the same parameters are available from two different curves (e.g., wq aiid t from Figure 1 or Figure 4a), there is some mathematical relation between the curves. For the "linear" system we have considered (i.e., displacement is proportional to driving amplitude Fq) the time-domain and frequency-domain responses are connected by a Fourier transform. Similarly, absorption and dispersion spectra both yield the same information, and are related by a Hilbert transform (see Chapter 4). In this Chapter, we will next develop some simple Fourier transform properties for continuous curves such as Figures 1-4, and then show the advantages of applying similar relations to discrete data sets consisting of actual physical responses sampled at equally-spaced intervals. 

Rigorous mathematical proof of the SOM algorithm is very difficult in general. In the case of a discrete data set and a fixed neighborhood function, the error function of the system may be defined as... 

Any data set that consists of discrete classification into outcomes or descriptors is treated with a binomial (two outcomes) or multinomial (tliree or more outcomes) likelihood function. For example, if we have y successes from n experiments, e.g., y heads from n tosses of a coin or y green balls from a barrel filled with red and green balls in unknown proportions, the likelihood function is a binomial distribution ... 

The conclusion is that for every particular set of basis functions and given data, there exists an appropriate size of G that can approximate both accurately and smoothly this data set. A decisive advantage would be if there existed a set of basis functions, which could probably represent any data set or function with minimal complexity (as measured by the number of basis functions for given accuracy). It is, however, straightforward to construct different examples that acquire minimal representations with respect to different types of basis functions. Each basis function for itself is the most obvious positive example. A Gaussian (or discrete points... 

Another problem that should be considered carefully is the fact that the evaluated phase shift data set is discrete, so that conventional numerical derivation methods... 

For longer boreholes the data may need to be synchronized by comparing the return temperature not with the current time step but with the time step— n, where n is the travel time. The error minimized is the sum square error of the difference between the calculated and measured borehole heat exchanger return temperature. We have set up the analyses procedure in such a way that it is easy to select discrete data-windows for the calibration. 

If the goal is to develop a predictive model, a training set with known class memberships is required to construct the model and the techniques are termed supervised." The techniques are designed to operate on data sets with discrete class membership. For example, oranges belong to the fruit class whereas broccoli belongs to the vegetable" class. If the measurements on the samples in a data set are continuous in nature (e.g., concentrations of... 

The loadings plot is also examined for inherent dimensionality and unusual variables. No conclusions can be drawn because of the limited utility of this diagnostic when examining data sets comprised of discrete measurement variables. 

Figure 1 graphically depicts the numerical data relevant to our application listed by Lundberg et al. Different sets of curves off vs. X are provided for individual values of d /d0. Discrete data were provided in the numerical tables of the original work to produce the continuous traces in Figure 1, a cubic spline fitting was used. 

The tunable filter technology also has distinct PAT advantages. Rapid timing of discrete wavelengths through software control and no moving parts enables the collection of data sets comprising approximately 80 000 spectra in a matter of a couple of minutes. For most... 

The boiling point is the last data herein sought out however, it is indeed the most important data to secure for a discrete pure component or a pseudo-crude cut component. Since the discrete pure components are generally a known type of molecular structure, their boiling points may readily be obtained or estimated from data sets such as Table 1.3. The crude oil components are left, unfortunately, undefined. Therefore, this section is dedicated to defining the boiling points of crude oil and its products. 

A probability distribution is a mathematical description of a function that relates probabilities with specified intervals of a continuous quantity, or values of a discrete quantity, for a random variable. Probability distribution models can be non-parametric or parametric. A non-parametric probability distribution can be described by rank ordering continuous values and estimating the empirical cumulative probability associated with each. Parametric probability distribution models can be fit to data sets by estimating their parameter values based upon the data. The adequacy of the parametric probability distribution models as descriptors of the data can be evaluated using goodness-of-fit techniques. Distributions such as normal, lognormal and others are examples of parametric probability distribution models. 

Numerical methods can be apphed to discrete (finite) data sets in order to carry ont such procedures as differentiation, integration, solution of algebraic and differential eqna-tions, and data smoothing. Analytical methods, which deal with continnons functions, are exact or at least capable of being carried ont to any arbitrary precision. Nirmerical methods applied to experimental data are necessarily approximate, being limited by the finite nnm-ber of data points employed and their precision. 

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

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