Correlation measure

Seam correlations, measurements of rank and geologic history, interpretation of petroleum (qv) formation with coal deposits, prediction of coke properties, and detection of coal oxidation can be deterrnined from petrographic analysis. Constituents of seams can be observed over considerable distances, permitting the correlation of seam profiles in coal basins. Measurements of vitrinite reflectance within a seam permit mapping of variations in thermal and tectonic histories. Figure 2 indicates the relationship of vitrinite reflectance to maximum temperatures and effective heating time in the seam (11,15). 

Fink et al. [17] correlated measurements from different authors and test methods to compare Young s modulus for cellulose of type 1 and II. Most of the authors determined higher characteristic values for type I than for type II (Table 7). 

In its place we can use the simplest correlation measure, the two point correlation function,... 

There are a number of additional local structures and properties that appear even in elementary CA systems. Grassberger [grass84b, for example, has observed that rule R22 actually harbors very complex long-range effects, similar to a critical phenomenon (see section 3.1.4). Since the majority of these findings require the use of more general and sophisticated correlation measures than we have defined thus far, we will pick up our discussion of them in chapter 4. 

A = 3/4 appears to represent a state of maximal disorder (as deduced from computing certain entropy and correlation measures see next section). 

This equation, however, does not adequately represent the experimental data when used to correlate measured values of k0, Ag and Qx. It was modified empirically, but the modified equation does not account for observed variations of conversion with bed height and pressure. The proposed model would therefore appear to rest on oversimplified assumptions. 

Otake and Okada correlated measurements of total holdup by the equation... 

There was thus the need for optical experiments showing the flaws of classical electrodynamics. An important difference between a wave and a particle is with respect to a beam splitter a wave can be split in two while a photon can not. An intensity correlation measurement between the two output ports of the beamsplitter is a good test as a wave would give a non zero correlation while a particle would show no correlation, the particle going either in one arm or the other. However, when one takes an attenuated source, such as the one used by Taylor, it contains single photon pulses but also a (small) fraction of two... 

Only a few publications in the literature have dealt with this problem. Almasy and Mah (1984) presented a method for estimating the covariance matrix of measured errors by using the constraint residuals calculated from available process data. Darouach et al. (1989) and Keller et al. (1992) have extended this approach to deal with correlated measurements. Chen et al. (1997) extended the procedure further, developing a robust strategy for covariance estimation, which is insensitive to the presence of outliers in the data set. 

General case. If some measurement errors are correlated, the covariance matrix is not diagonal. It is assumed that we know the sensors which are subjected to correlated measurement errors, for example because they share some common elements (e.g., power supplies). There are then s off-diagonal elements of d>,. .., w),... 

The decay of the structural correlations measured by the static structure factor can be studied by dynamic scattering techniques. From the simulations, the decay of structural correlations is determined most directly by calculating the coherent intermediate scattering function, which differs from Eq. [1] by a time shift in one of the particle positions as defined in Eq. [2] ... 

Fig. Tl.n. An optical microscope adapted for fluorescence correlation measurements.

The range of rjk is — 1 to +1 a value of +1 indicates a perfect linear relationship, a value of —1 indicates a perfect inverse linear relationship absolute values of approximately <0.3 indicate a poor or no linear relationship. The Pearson correlation coefficient is best suited for normally distributed variables however, it is very sensitive to outliers. This coefficient is the most used correlation measure as usual also throughout this book the term correlation coefficient will be used for the Pearson correlation coefficient. 

A more robust correlation measure, -y Vt, can be derived from a robust covariance estimator such as the minimum covariance determinant (MCD) estimator. The MCD estimator searches for a subset of h observations having the smallest determinant of their classical sample covariance matrix. The robust location estimator—a robust alternative to the mean vector—is then defined as the arithmetic mean of these h observations, and the robust covariance estimator is given by the sample covariance matrix of the h observations, multiplied by a factor. The choice of h determines the robustness of the estimators taking about half of the observations for h results in the most robust version (because the other half of the observations could be outliers). Increasing h leads to less robustness but higher efficiency (precision of the estimators). The value 0.75n for h is a good compromise between robustness and efficiency. 

R. G. W. Brown, K. D. Ridley, and J. G. Rarity, Characterization of silicon avalanche photodiodes for photon correlation measurements. 1 Passive quenching, Appl. Opt. 25, 4122-4126 (1986). 

A little work has been reported on time-correlated measurements with germanium APDs/104 105) showing the potential for extending single-photon APD fluorescence lifetime measurements up to 1.7 mwith picosecond resolution. 

SIMCA can be applied to the problem of classification when attempting to correlate measurable effect variables with composition of the classified samples. In correlation analyses one may wish to determine how other sample variables, such as sediment composition, organic content, lipid concentration, etc., influence the composition of measured residues or concentrations of PCBs. 

For the remainder of this book we shall always refer to g (l, 1) as the pan-correlation, and drop the superscript zero. We saw in Section 4.2. that this correlation measures the deviation of from i.e.. 

In thip appendix, a summary of the error propagation equations and objective functions used for standard characterization techniques are presented. These equations are Important for the evaluation of the errors associated with static measurements on the whole polymers and for the subsequent statistical comparison with the SEC estimates (see references 26 and 2J for a more detailed discussion of the equations). Among the models most widely used to correlate measured variables and polymer properties is the truncated power series model... 

A surprising aspect of SD is how rapidly C i) in highly polar solvents decays relative to other relaxation processes such as reorientation of solvent dipoles. This very rapid time scale cannot be ascribed to dynamical solvent-solvent correlations, which, as illustrated in Fig. 6, are modest even for the longest ranged A . Thus the key to imderstanding the reasons for the rapid decay of C i) is in examining how solvent-solvent correlations contribute to it and to what extent their contributions can be accounted for in terms of static correlations measured by ((5A ) ), Eq. (32). The initial cmvature of C(t), which characterizes its short-time Gaussian-like behavior is often characterized in terms of the solvation frequency co o/v... 

Brogniez, C J. Lenoble, R. Ramananaherisoa, K. H. Fricke, E. P. Shettle, K. W. Hoppel, R. M. Bevilacqua, J. S. Hornstein, J. Lumpe, M. D. Fromm, and S. S. Krigman, Second European Stratospheric Arctic and Midlatitude Experiment Campaign Correlative Measurements of Aerosol in the Northern Polar Atmosphere, J. Geophys. Res., 102, 1489-1494 (1997). 

The phase-dependent directionality of photocurrents produced by such a detector entails advantageous properties of the photocurrents cross correlations in nonoverlapping time intervals or spatial regions (considered in Section 4.2.2). These directional time-dependent correlations are measured with one detector only. They involve solely terms dependent on LO phases, in contrast to similar correlations measured by conventional photocounters, which inevitably contain terms depending on photon fluxes such as the LO excess noise. Owing to these properties, the mean autocorrelation function of the SL quadrature is shown in the schemes considered here to be measurable without terms related to the LO noise. LO shot noise, which affects the degree of accuracy to which this autocorrelation is measured (i.e., its variance) is easily obtainable from zero time delay correlations because the LO excess noise is suppressed. The combined measurements of cross correlations and zero time delay correlations yield complete information on the SL in these schemes. 

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