Errors terms Links

When specifying atomic coordinates, interatomic distances etc., the corresponding standard deviations should also be given, which serve to express the precision of their experimental determination. The commonly used notation, such as d = 235.1(4) pm states a standard deviation of 4 units for the last digit, i.e. the standard deviation in this case amounts to 0.4 pm. Standard deviation is a term in statistics. When a standard deviation a is linked to some value, the probability of the true value being within the limits 0 of the stated value is 68.3 %. The probability of being within 2cj is 95.4 %, and within 3ct is 99.7 %. The standard deviation gives no reliable information about the trueness of a value, because it only takes into account statistical errors, and not systematic errors. 

Another of our present aims is to continue investigating the theoretical properties of the 2- and 3-body terms. Thus, there are many exact relations linking the different 3-body terms arising from each of the 36 options of equation (9) as well as their holes counterparts [16]. These relations are interesting by themselves, because they widen our understanding of the problem and may be helpful for improving our approximations. Another important theoretical question is, as mentioned above, to investigate the reason why one Ai corrects the VCP error which in some cases is only due to the 3-body contribution but which may also involve other kind of correlation effects. 

Nakatsuji and Yasuda [56, 57] derived the 3- and 4-RDM expansions, in analogy with the Green function perturbation expansion. In their treatment the error played the role of the perturbation term. The algorithm that they obtained for the 3-RDM was analogous to the VCP one, but the matrix was decomposed into two terms one where two A elements are coupled and a higher-order one. Neither of these two terms can be evaluated exactly thus, in a sense, the difference with the VCP is just formal. However, the structure of the linked term suggested a procedure to approximate the A error, as will be seen later on. 

Agglomerative methods, such as single link and complete link, are stepwise procedures. The formulation in (5)-(7) allows us to define the hierarchical clustering problem in terms of combinatorial optimization. To do this, however, we need an appropriate internal clustering criterion. The most obvious is squared error. 

In this expression a common residual variance term is specified, although the residual variance can be indexed to the model, in which case the overall residual variance will be the sum of the contribution of the residual variance for each of the m candidate models. It has been found that chain mixing occurs faster when competing models are linked with a common parameter (e.g., the residual error) (42). It is common in the non-Bayesian model framework to address model selection as a... 

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