Random correlation

Analysis of substituent effects in the reaction revealed random correlations of rate with a values of substituents for the uncatalysed reaction, but for the acid-catalysed reaction, reasonable Hammett plots with p factors of —1.27 (HC104) and -1.12 (H3P04) were obtained (Table 253). 

Limiting expressions of r for large numbers of coordinates in WEG, FAN, and PAR were calculated from Equation 3.B4 and expressions described in Appendix 3B for a family of (3i and p2 values and were tested by simulation. Specifically, 500 (x, y) random correlated coordinates were distributed over WEG, FAN, and PAR for specific Pi and p2> and rwas calculated from the usual statistics equation (Eq. 3.B 1 in Appendix 3B). The simulation was repeated 1000 times, and the average r was compared to the limiting expressions. 

Figure 3.3a-c shows graphs of correlation coefficient r versus p2 f°r different pj in WEG, FAN, and PAR. The curves are limiting values of r evaluated as described in Appendix 3B. The symbols are the average r s from 1000 simulations of 500 random correlated coordinates for different pj and p2. The agreement is excellent, except for r < 0.1. The small deviation probably is caused by minor imperfections in the random number generator. 

Comparison with the highest possible values of a multivariate random correlation (see also Section 6.6.3) shows a correlation period of approximately two weeks and two days. This means that sampling of suspended dust at intervals of two weeks is sufficient for the characterization of the average impact of multielement emissions (for all investigated elements simultaneously) at that particular sampling point. 

Fig. 9-5. Autocorrelation functions of the investigated heavy metals, (a) Cd, (b) Cr, (c) Cu, (d) Ni, (e) Pb, (f) Zn. (The dashed lines correspond to the highest possible values of a random correlation for the probability of an error of the first kind of a = 0.05)...

The points of intersection between the highest value of a random correlation and the lower limit of the confidence interval of the corresponding empirical model function according to Eq. 9-4 correspond to the lower limits of the confidence range of the critical distances between the sampling points. These values are represented in Tab. 9-2. 

This validation technique is adopted to check models with chance correlation, i.e. models where the independent variables are randomly correlated to the response variables. The test is performed by calculating the quality of the model (usually F or, better, randomly modifying the sequence of the response vector y, i.e. by assigning to each object a response randomly selected from the true responses [Lindgren et al, 1996]. If the original model has no chance correlation, there is a significant difference in the quality of the original model and that associated with a mode obtained with random responses. The procedure is repeated several hundreds of times. 

Jouan-Rimbaud, D., Massart, D.L. andde Noord, O.E. (1996) Random correlation in variable selection for multivariate calibration with a genetic algorithm. Chemom. Intell. Lab. Syst., 35, 213—220. 

With the data available, two different models were developed for solubility prediction. In the first model, 90% of dataset A was randomly chosen as a training set, and the rest was used as test set. The molecules of set B were not included but used as an independent validation set. The procedure was repeated in a 10-fold cross-validation to avoid random correlations. 

A question that sometimes arises in QSAR analyses deals with the possibility of random correlations. In this study the regression equations were carefully and extensively analyzed for spurious effects due to random correlations. The regressions were repeated with randomly selected observations deleted, and no significant effects were observed in the regression equations. Further, the process of equation selection was repeated using random numbers in place of the chi indexes. No correlation obtained with the random numbers was found to be significant in comparison to that of Eq. [41]. These random number analyses have also been carried out for other QSAR investigations. ... 

D. Jouan-Rimbaud, D.L. Massart, O.E. de Noord, Random Correlation in Variable Selection for Multivariate Calibration with a Genetic Algorithm, Chemometrics and Intelligent Laboratory Systems, 35 (1996), 213-220. 

An alternative way to assess the significance of a model is to randomly reassign the activities, thereby associating the wrong activity with each set of grid values. When this is done, the predictive ability of the model should be significantly better for the true data set than for any of the randomised sets. This is a useful technique to check for random correlations when using descriptors that are not easy to interpret. 

For random correlations (7i—54) = 0 and the scattering is independent of This corresponds to a scattering pattern which is cylindrically symmetrical about the incident beam. For non-random correlations (7i—S4) is finite which leads to a pattern which has four-fold symmetry in as is often experimentally observed (Fig. 27). Thus in addition to the usual correlation function 7i(r), the coefficient T4.—S4) plays the role of another correlation function which characterises the shape of the correlated region. The evaluation of these correlation functions for perfect two-dimensional spherulites has been carried out and their substitution into eqn. (84) has been shown to lead to a scattering pattern which is identical with that which is directly calculated by the amplitude summation method. Thus, this formulation enables one in principle to describe scattering from systems ranging from random to highly ordered. 

The extension of the random orientation correlation theory to the description of the scattering from oriented polyethylene films was described by Stein and Hotta. This application is not strictly correct since it was made prior to the development of non-random orientation correlation theory and it is now realised that random theory is not adequate for the description of such films. In fact, it is now felt that random correlations in oriented systems are quite unlikely and that the degree of non-randomness increases with orientation. While the non-random correlation theory has been generalised to permit the description of oriented systems, it is still rather complex so that it has not been actually applied. In principle eqn. (118) could be used to describe oriented systems, by using the van Aartsen-Stein expression for / and the Stein-Hotta expression for /r in the oriented state. It is likely that 4>s and may also vary with orientation. So far, this has not been done. 

Three correlated aa-mother-daughter decays were observed that were assigned to the decay of and No as the decay products of Sg. The three correlated events have to be compared with an expectation value of 0.27 for random correlations. This gives a probability of 0.24% that the three events are random correlations. As the mother decays were not observed, it is important to note that and can only be observed if Sg... 

Because there are a finite number of samples in the set used for prediction, in many cases the number of factors that gives a minimum PRESS value can still be overfit for predicting unknown samples. In other words, there is a statistical possibility that some of the noise vectors from the spectral decomposition may be present in more than one sample. These vectors can appear to improve the calibration by a small amount when, by random correlation, they are added to the model. However, if these exact same noise vectors are not present in future unknown samples (and most likely they will not be), the predicted concentrations will have significantly larger prediction errors than if those additional vectors were left out of the model. 

Multiple-longitudinal-strip position-sensitive detectors are being replaced by double-sided strip detectors [359,428] in this application. These detectors have the potential for smaller detector pixels (higher granularity) and, consequentiy, a reduced random-correlation rate. Digital signal processing allows the observation of shorter sequential-decay intervals, down to 1 ps [429]. 

Hartzell S, Harmsen S, Frankel A (2010) Effects of 3D random correlated velocity perturbations on predicted ground motions. Bull Seismol Soc Am 100(4) 1415-1426... 

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