Hidden correlations

Care has been taken to ensure that the fixed time at which the particle positions are specified differs from the fixed time at which the particle velocities are specified. Failure to take this precaution readily leads to incorrect results through the hidden correlations(26). 

Terms in which i,j, and i are not all unequal lead to contributions to Dm that are first order in concentration. If the precautions relating particle positions and velocities, which arise from hidden correlations(26) in the nominal Brownian velocity, had been neglected, the nonzero jqV [Dje] term would have been lost. 

G. D. J. Phillies. Hidden correlations and the behavior of the dynamic structure factor at short times. J. Chem. Phys., 80 (1984), 6234-6239. 

Bell s result, made all the more remarkable for the very few assumptions he makes to derive it, rather dramatically asserts that cither EPR s three premises are wrong or quantum mechanics is incorrect. However, recent experiments by A.spect, et.al. ([aspect82a], [aspect82b]). On and Mandel [01188], and others have shown, virtually conclusively, that nature satisfies the quantum mechanical prediction (equation 12.54) and not Bell s inequality (equation 12.55), thus strongly denying the possibility of local hidden variables. We are thus left with what is arguably one of the deepest mysteries in the foundations of physics the existence of a profoundly nonclassical correlation between spatially-far separated systems, or nonscparability. 

Figure 4.28. Correlation graph for file PROFILE.dat. The facts that (a) 23 out of 55 combinations yield probabilities of error below p = 0.04 (42% expected due to chance alone =8%) and (b) that they fall into a clear pattern makes it highly probable that the peak areas [%] of the corresponding chromatograms follow a hidden set of rules. This is borne out by plotting the vectors two by two. Because a single-sided test is used, p cannot exceed 0.5.

Because the results of all correlations are viewed simultaneously, patterns can emerge that are much more powerful indicators of hidden action than any single correlation would be. 

The literature is filled with empirical formulas for fire phenomena. In many cases, the dimensionless groups upon which the correlation was based have been hidden in favor of engineering utility, and specific dimensions are likely to apply. For example, a popular formula for flame height from an axisymmetric source of diameter D is... 

Often, the correlation is not good, and we need to search for the hidden variable that we have not yet discovered. But when we find a good correlation, it could prove useful in the reverse search for other untried compounds that may have higher or lower camphor smell, even if we do not understand the mechanism of how it works. There is always the hope that, if we know which parameters are important to smell, we may generate one or more hypotheses on the nature of camphor smell this would be followed by predictions and experiments that could lead us to future understanding. 

When we are truly clueless, we can nevertheless rely on intuition to propose an ad hoc set of structural and related property parameters for the correlation. We may be lucky and find the hidden variable by chance, and we may be inspired. An example is the topological index, which describes how carbon atoms are connected together, and was proposed in the hope that it would correlate a large range of molecular properties. 

Large data tables contain an amount of information which is partly hidden because the data complexity prevents ready interpretation. This is typical of NIR spectra collections. PCA is a projection method used to visualize all the information contained in the data table. It can be used to show in what respect one sample differs from another, which variables contribute most to this difference, and whether these variables contribute in the same way and are correlated or independent of each other. It also reveals sample patterns or groupings. In addition, it quantifies the amount of useful information, as opposed to noise or meaningless variation, contained in the data table. Principal components are defined only for the data set from which they were computed. They may also hold for other data of identical type, but this is not guaranteed, and it is certainly not true for different types of data. 

Miles et a/.418 to be of the n — it type and correlated with the B2u, Blu, and Elu bands of benzene. The absorption at 180 nm in uracil, according to Miles et aZ.,418,421 covers a fifth it -> n transition. A recent study405 of the influence of an electric field on the light absorption of uracil and thymine in solution confirms that in the region < 255 nm the long-wavelength band of uracil overlaps a second transition which is hidden in the absorption spectrum. Similarly, in thymine the second transition appears below 275 nm. In both cases, however, no conclusion as to the nature of the weak bands was given. 

An artificial neural network (ANN) model was developed to predict the structure of the mesoporous materials based on the composition of their synthesis mixtures. The predictive ability of the networks was tested through comparison of the mesophase structures predicted by the model and those actually determined by XRD. Among the various ANN models available, three-layer feed-forward neural networks with one hidden layer are known to be universal approximators [11, 12]. The neural network retained in this work is described by the following set of equations that correlate the network output S (currently, the structure of the material) to the input variables U, which represent here the normalized composition of the synthesis mixture ... 

The value of RAF obtained in the first approach is about 1.1 for both locations and is slightly higher (1.15) if determined in the second approach. This discrepancy may be due to some hidden factor which affects the erythemal irradiance and is correlated with total ozone but not included in UV-ERY versus total ozone regression. Overall, the values of RAF are in good agreement with other authors [1, 11, 13]. 

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