Weighting used observed variances

Our choice of model differs from that of Tschernitz et al. (1946), who preferred Model d over Model h on the basis of a better fit. The difference lies in the weightings used. Tschernitz et al. transformed each model to get a linear least-squares problem (a necessity for their desk calculations) but inappropriately used weights of 1 for the transformed observations and response functions. For comparison, we refitted the data with the same linearized models, but with weights Wu derived for each model and each event according to the variance expression in Eq. (6.8-1) for In 7. The residual sums of squares thus found were comparable to those in Table 6.5. confirming the superiority of Model h among those tested. 

In the absence of replicate observations, replic te residuals were used. The observational variance was estimated as the smallest residual mean square. S(d)/ n — p). obtained when the weighted observations of og R,/pHPu) were fitted with detailed polynomial functions of the interfacial temperature and the three partial pressures. 

When the weights used in a weighted least squares fit do not reflect the variances of the observations, the result is more complicated, but qualitatively similar. 

When no replicates are available, common weights that use the observed data include 1/Y or 1/Y2. These two weighting schemes in essence assume that the variance model is proportional to the mean or mean squared, respectively, and then crudely use the observation itself to estimate the mean. Although using observed data has a tremendous history behind it, using observed data as weights is problematic in that observed data are measured with error. A better estimate might be 1/Y or 1 /Y2 where the predicted values are used instead. In this manner, any measurement error or random variability in the data are controlled. 

Once numerical estimates of the weight of a trajectory and its variance (2cr ) are known we are able to use sampled trajectories to compute observables of interest. One such quantity on which this section is focused is the rate of transitions between two states in the system. We examine the transition between a domain A and a domain B, where the A domain is characterized by an inverse temperature - (3. The weight of an individual trajectory which is initiated at the A domain and of a total time length - NAt is therefore... 

The weight of the ith observation is inversely proportional to the variance of the observation we will use Eq. (2-82) for this quantity, n being the number of observations. 

If the original model is sufficiently perfect, the linearization of the problem adequate, the measurements unbiased (no systematic error), and the covariance matrix of the observations, 0y, a true representation of the experimental errors and their correlations, then c2 (Eq. 21c) should be near unity [34], If 0y is indeed an honest assessment of the experimental errors, but a2 is nonetheless (much) larger than unity, model deficiencies are the most frequent source of this discrepancy. Relevant variables probably exist that have not been included in the model, and the experimental precision is hence better than can be utilized by the available model. Model errors have then been treated as if they were experimental random errors, and the results must be interpreted with great caution. In this often unavoidable case, it would clearly be meaningless to make a difference between a measurement with a small experimental error (below the useful limit of precision) and another measurement with an even smaller error (see ref. [41 ). A deliberate modification of the variance-covariance matrix y towards larger and more equal variances might then be indicated, which results in a more equally weighted and less correlated matrix. 

Uniform weighting (known as simple least squares) is appropriate wJien the expected variances of the observations are equal and is commonly used when these values are unknown. GREGPLUS uses this w eighting when called with JWT = 0 the values MwijJ = 1 are then provided automatically. 

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