Correlation techniques

1 Autocorrelation, Autoregression, Partial Autocorrelation, and Cross-correlation Function 

Very useful tools for analyzing time series are the autocorrelation function, the crosscorrelation function, and the partial autocorrelation function [SCHLITTGEN and STREITBERG, 1989 DOERFFEL and WUNDRACK, 1986], The interpretation of the patterns of these functions provides the experienced user with substantially more information about the time series than plotting methods. 

The aim of correlation analysis is to compare one or more functions and to calculate their relationship with respect to a change of t (lag) in time or distance. In this way, memory effects within the time curve or between two curves can be revealed. Model building for the time series is easy if information concerning autocorrelation is available. 

In order to imagine such memory effects, it is useful to plot x t) against previous values, such as x(l - 1) (Fig. 6-14)  

It can be seen that the series is correlated with itself by a lag of one. The series has a memory for the first preceding value. All cases of the series are dependent on the preceding value. The statistical expression for a relationship between two variables is the correlation coefficient (see Section 2.4.2). The resulting correlation coefficient between the variables x(t) and x(t - 1) is the autocorrelation coefficient [KATEMAN, 1987] for the lag t = 1 in this case. If such a relationship between the values x t) and x(t— 1) exists, we can formulate a linear regression expression  

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Under certain conditions of temperature and pressure, and in the presence of free water, hydrocarbon gases can form hydrates, which are a solid formed by the combination of water molecules and the methane, ethane, propane or butane. Hydrates look like compacted snow, and can form blockages in pipelines and other vessels. Process engineers use correlation techniques and process simulation to predict the possibility of hydrate formation, and prevent its formation by either drying the gas or adding a chemical (such as tri-ethylene glycol), or a combination of both. This is further discussed in SectionlO.1. 

D. Leger, E. Mathieu and J. C. Perrin, Optical surface determmation usmg speckle correlation technique . Appl. Opt., Vol. 14, 872-877 (1975). 

Cross Correlation. Considerable research has been devoted to correlation techniques where a tracer is not used. In these methods, some characteristic pattern in the flow, either natural or induced, is computer-identified at some point or plane in the flow. It is detected again at a measurable time later at a position slightly downstream. The correlation signal can be electrical, optical, or acoustical. This technique is used commercially to measure paper pulp flow and pneumatically conveyed soHds. 

H. A. Walter, Jr. and D. E. Flanigan, "Detection of Atmosphere Pollutants A Correlation Technique," Appl Opt. 14, (June 1975). Available from National Technical Information Service, U.S. Department of Commerce, Springfield, Va. 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

An overview of some basic mathematical techniques for data correlation is to be found herein together with background on several types of physical property correlating techniques and a road map for the use of selected methods. Methods are presented for the correlation of observed experimental data to physical properties such as critical properties, normal boiling point, molar volume, vapor pressure, heats of vaporization and fusion, heat capacity, surface tension, viscosity, thermal conductivity, acentric factor, flammability limits, enthalpy of formation, Gibbs energy, entropy, activity coefficients, Henry s constant, octanol—water partition coefficients, diffusion coefficients, virial coefficients, chemical reactivity, and toxicological parameters. 

Generalized Correlations. Generalized correlations are often the only recourse when a property value cannot be determined from empirical correlations or by other means. Several powerful correlating techniques fall under this category, including the principle of corresponding states (3,17), reduced property models (1), and the Polanyi-type characteristic curve for microporous adsorbents (14). 

In the gas correlation techniques, gas-filled cells mounted on a rotating disk cross the analyzing infrared beam in turn. One correlation cell is filled with a gas that will not absorb infrared light, such as nitrogen (N2). The other cell (or cells) are filled with a high concentration of the gas to be measured. The wavelength range is selected at the absorption band of the gas to be measured by an optical band-pass filter. 

McLeman<7, ) used a method which was really a pre-cursor of the cross-correlation technique. He injected a pulse of air into the conveying line as a result of which there was a very short period during which the walls at any particular point were not subject to... 

Development in recent years of fast-response instruments able to measure rapid fluctuations of the wind velocity (V ) and of fhe tracer concentration (c ), has made it possible to calculate the turbulent flux directly from the correlation expression in Equation (41), without having to resort to uncertain assumptions about eddy diffusivities. For example, Grelle and Lindroth (1996) used this eddy-correlation technique to calculate the vertical flux of CO2 above a foresf canopy in Sweden. Since the mean vertical velocity w) has to vanish above such a flat surface, the only contribution to the vertical flux of CO2 comes from the eddy-correlation term c w ). In order to capture the contributions from all important eddies, both the anemometer and the CO2 instrument must be able to resolve fluctuations on time scales down to about 0.1 s. 

We have also developed a method of measurement for local temperature in microspace with a fluorescence correlation technique. Using this method, the temperature elevation at the optical trapping point due to absorption of the NIR trapping beam by solvent was quantitatively evaluated the temperature at the trapping point increased linearly with increase in the incident NIR light, and the temperature elevation coefficient was mainly dependent on two physical parameters of the solvent the absorption coefficient at 1064 nm and the thermal conductivity. 

Lockhart and Martinelli (1949) suggested an empirical void fraction correlation for annular flow based mostly on horizontal, adiabatic, two-component flow data at low pressures, Martinelli and Nelson (1948) extended the empirical correlation to steam-water mixtures at various pressures as shown in Figure 3.27. The details of the correlation technique are given in Chapter 4. Hewitt et al. (1962) derived the following expression to fit the Lockhart-Martinelli curve ... 

Indirect detection Method for the observation of an insensitive nucleus (e.g., 13C) by the transfer of magnetisation from an abundant nucleus (e.g., 1H). This method of detection offers great improvements in the sensitivity of proton-carbon correlated techniques. 

The radial velocities have been computed with the low resolution set-up (more spectral lines, no telluric line), using a cross-correlation technique. When excluding the seven outliers, the peak in centered at 83.0 0.4kms 1 with a dispersion of 1.9 0.2kms 1. Lithium abundance is being determined using Li i 6707.8 A. We used the B — V index to determined the ([3]), and the curve of growth from [7] to derive AT(Li). 

Surprisingly the first attempts to measure displacements originating from genuine periodic or quasi-periodic perturbations were carried out in vivo. Those studies took advantage of the internal pulsatile motions provided by heart beating. As soon as 1982 small displacements from aortic pulsations were visualized in liver on M-mode scans.69 It was shown that such displacements could be calculated from correlated successive A-scans,70 with an application on liver.71,72 Correlation techniques were also applied on M-mode images to quantitatively estimate motions and deformations of fetal lung.73,74... 

More recently, homonuclear correlation techniques relying on J-couplings have also been developed [86, 87, 89] and applied to the assignment of spin systems of amino-acid residues in uniformly labeled proteins and peptides [90]. These have, in some cases, a higher information content than the comparable dipolar-mediated experiments, as relayed correlations throughout the continuous 13C-13C network are more easily realized at high B0 fields [90]. 

Mansuy et al. [97] investigated the use of GC-C-IRMS as a complimentary correlation technique to GC and GC-MS, particularly for spilled crude oils and hydrocarbon samples that have undergone extensive weathering. In their study, a variety of oils and refined hydrocarbon products, weathered both artificially and naturally, were analyzed by GC, GC-MS, and GC-C-IRMS. The authors reported that in case of samples which have lost their more volatile n-alkanes as a result of weathering, the isotopic compositions of the individual compounds were not found to be extensively affected. Hence, GC-C-IRMS was shown to be useful for correlation of refined products dominated by n-alkanes in the C10-C20 region and containing none of the biomarkers more commonly used for source correlation purposes. For extensively weathered crude oils which have lost all of their n-alkanes,it has been demonstrated that isolation and pyrolysis of the asphaltenes followed by GC-C-IRMS of the individual pyrolysis products can be used for correlation purposes with their unaltered counterparts [97]. 

Correlation Chromatography (CC) can be considered a typical example of an active or on-line chemometrie technique. Impossible without computers, it shows promising results in (ultra) trace analysis. This paper will describe two directions that utilize correlation techniques a semi-continuous kind of chromatography ( ) and an extension of the limit of detection in trace analysis ( 2 ). Correlation Chromatography will be shown to be a powerful method for application in (ultra) trace analysis. 

Katchalski-katzir, E., Shariv, I., Eisenstein, M., Friesem, A.A., and Aelalo, C. Molecular surface recognition determination of geometric flt between proteins and their ligands by correlation techniques. Proc. Nat. Acad. Sci. U. S.A. 1992, 89, 2195-2199. 

The first stage includes the selection of a dataset for QSAR studies and the calculation of molecular descriptors. The second stage deals with the selection of a statistical data analysis and correlation technique, either linear or nonlinear such as PLS or ANN. Many different algorithms and computer software are available for this purpose in all approaches, descriptors serve as independent variables and biological activities serve as dependent variables. 

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