Correlation coefficient, defined

Accepting normal bivariate distribution, Pearson s correlation coefficient, defined... 

Similar arguments can be adduced and for an explanation of the different dependences of ACW values on the blood serum uric acid content (the biochemical analysis) received forthese models (Figure 18.5). The relative stoichiometric coefficient for uric acid in ABAP-luminol system makes about 2.0 [12], while in Hb-H O -luminol - less 1.0 [4]. Interacting with various radical intermediates, uric acid influences in the first model not only on the latent period, but substantially and on the CL intensity. Correlation of ACW with the uric acid content is much worse for the first model (r=0.583. Figure 18.5a), than for the second (n=0.745. Figure 18.5b). The values of similar correlation coefficients defined with uric acid on ABAP-luminol model in other works [ 12,13] are rather close to coefficient received in ours experiments. In work [2] using ABAP-luminol model on the basis of blood serum measurements for 45 donors it is shown that the contribution of uric acid to ACW makes 64%, and proteins of 5%. 

Entries above the diagonal represent the number of solvents for which both parameters were known. Entries on and below the diagonal are the correlation coefficients, defined by ... 

Compared to the A-matrix in RPA, Eq. (10.18), one obtains in second order two additional contributions, which consist of contractions of two-electron repulsion integrals with the first-order doubles correlation coefficients defined in Eq. (9.67), and one term that contains the second-order correction to the density matrix, Eqs. (9.116) and (9.117). The latter contribution... 

The idea behind this approach is simple. First, we compose the characteristic vector from all the descriptors we can compute. Then, we define the maximum length of the optimal subset, i.e., the input vector we shall actually use during modeling. As is mentioned in Section 9.7, there is always some threshold beyond which an inaease in the dimensionality of the input vector decreases the predictive power of the model. Note that the correlation coefficient will always be improved with an increase in the input vector dimensionality. 

CTj and (3 are correlated, with correlation coefficient c , defined through the relation ... 

The above correlation is valid for a bioreactor size of less than 3000 litres and a gassed power per unit volume of 0.5-10 kW. For non-coalescing (non-sticky) air-electrolyte dispersion, the exponent of the gassed power per unit volume in the correlation of mass transfer coefficient changes slightly. The empirical correlation with defined coefficients may come from the experimental data with a well-defined bioreactor with a working volume of less than 5000 litres and a gassed power per unit volume of 0.5-10 kW. The defined correlation is ... 

The mixing data were correlated by defining a thermal diffusion coefficient, a, such that... 

The description of the degree of retention data correlation is more complicated than it appears. For example, the 2D retention maps cannot be characterized by a simple correlation coefficient (Slonecker et al., 1996) since it fails to describe the datasets with apparent clustering (Fig. 12.2f). Several mathematical approaches have been developed to define the data spread in 2D separation space (Gray et al., 2002 Liu et al., 1995 Slonecker et al., 1996), but they are nonintuitive, complex, and use multiple descriptors to define the degree of orthogonality. 

Figure 3.4 shows a fair correlation between vo-2ot and the Hildebrand solubility parameter 8 (linear correlation coefficient = 0.930) which makes intuitive sense. The Hildebrand parameter, which is often used to characterize liquids, is defined as the square root of the cohesive energy density (Barton 1991), while vcr2o( can be viewed as reflecting how strongly a molecule interacts with others of the same kind (Murray et al. 1994). 

Figure 1-2 Projecting each point of the three-dimensional MND onto any of the planes defined by two axes of the coordinate system (or, more generally, any plane passing through the coordinate system) results in the projected points being represented by a two-dimensional MND). The correlation coefficients for the projections in all planes are needed to fully describe the original MND.

The final key point to note about the MND, which can also be seen from Figure 1-2, is the fact that when the MND is projected onto the plane defined by any two axes of the coordinate system the data may show some correlation (as does the data in Figure 1-2). In fact, the projection onto any of the planes defined by two of the axes will have some value for the correlation coefficient between the corresponding pair of variables. The amount of correlation between projections along any pair of axis can vary from zero, in which case the data would lie in a circular blob, to unity, in which case the data would all lie exactly on a straight line. 

Secondly, as a measure of nonlinearity, the calculation conforms more closely to that concept than the correlation coefficient does. As a contrast, we can consider terms such as precision and accuracy, where high precision and high accuracy mean data with small values of standard deviation> while low precision and low accuracy mean large values of the measure. Thus, for those two characteristics, the measured value changes in opposition to the concept. If we were to use the correlation coefficient calculation as the measure of nonlinearity, we would have the same situation. However, by defining the linearity calculation the way did, the calculation now runs parallel to the concept a calculated value of zero means no nonlinearity while increasing values of the calculation corresponds to increasing nonlinearity. 

Note also that we can use the correlation test statistic (described in the correlation coefficient section) to determine if the regression is significant (and, therefore, valid at a defined level of certainty. A more specific test for significance would be the linear regression analysis of variance (Pollard, 1977). To so we start by developing the appropriate ANOVA table. 

When the excitation light is polarized and/or if the emitted fluorescence is detected through a polarizer, rotational motion of a fluorophore causes fluctuations in fluorescence intensity. We will consider only the case where the fluorescence decay, the rotational motion and the translational diffusion are well separated in time. In other words, the relevant parameters are such that tc rp, where is the lifetime of the singlet excited state, zc is the rotational correlation time (defined as l/6Dr where Dr is the rotational diffusion coefficient see Chapter 5, Section 5.6.1), and td is the diffusion time defined above. Then, the normalized autocorrelation function can be written as (Rigler et al., 1993)... 

Matrix B consists of q loading vectors (of appropriate lengths), each defining a direction in the x-space for a linear latent variable which has maximum Pearson s correlation coefficient between y and jf for j = 1,..., q. Note that the regression coefficients for all y-variables can be computed at once by Equation 4.52, however,... 

Gifford and Hanna tested their simple box model for particulate matter and sulfur dioxide predictions for annual or seasonal averages against diffusion-model predictions. Their conclusions are summarized in Table 5-3. The correlation coefficient of observed concentrations versus calculated concentrations is generally higher for the simple model than for the detailed model. Hanna calculated reactions over a 6-h period on September 30, 1%9, with his chemically reactive adaptation of the simple dispersion model. He obtained correlation coefficients of observed and calculated concentrations as follows nitric oxide, 0.97 nitrogen dioxide, 0.05 and rhc, 0.55. He found a correlation coefficient of 0.48 of observed ozone concentration with an ozone predictor derived from a simple model, but he pointed out that the local inverse wind speed had a correlation of 0.66 with ozone concentration. He derived a critical wind speed formula to define a speed below which ozone prediction will be a problem with the simple model. Further performance of the simple box model compared with more detailed models is discussed later. 

There are several figures of merit that can be used to describe the quality of a linear regression model. One very common figure of merit is the correlation coefficient, r, which is defined as ... 

The correlation coefficient (in wavelength space) is especially suitable for constructing a general library as it has the advantage that it is independent of library size, uses only a few spectra to define each product and is not sensitive to slight instrnmental oscillations. This parameter allows the library to be developed and validated more rapidly than others. Correlation libraries can also be expanded with new prodncts or additional spectra for an existing prodnct in order to incorporate new sources of variability in an expeditious manner. 

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