Correlations between parameters

Correlations between parameters are not favorable and can lead to difficulty in making unique estimates of the parameters. 

In practice, the choice of parameters to be refined in the structural models requires a delicate balance between the risk of overfitting and the imposition of unnecessary bias from a rigidly constrained model. When the amount of experimental data is limited, and the model too flexible, high correlations between parameters arise during the least-squares fit, as is often the case with monopole populations and atomic displacement parameters [6], or with exponents for the various radial deformation functions [7]. 

Equations 27—29 with two parameters seem to show reasonable correlation. Although Equation 28 would be unacceptable because of unusually large parameter coefficients, the correlations are completely equivalent statistically. As far as compounds used for the analyses are concerned, it is impossible to choose the two significant parameters. In this case, besides a significant correlation between parameters nr and E8, there are mutual relationships among three parameters. Each parameter is expressed as a linear combination of the other two and is not separated from others. 

Values of the e.s.d.s of parameters can be obtained, as shown in Figure 10.13, in the least-squares refinement from values of the diagonals of the inverse matrix. Similarly, any correlations between parameters, such as is often found to occur between occupancy and atomic displacement parameters, can be identified and taken into account in the description of the resulting molecular structure. The e.s.d.s for the refined parameters can then be used to calculate e.s.d.s of derived parameters, such as distances, angles, and torsion angles. ... 

Correlation between parameters A correlation is a measure of the extent to which two mathematical variables are dependent on each other. In the least-squares refinement of a crystal structure, parameters related by symmetry are completely correlated, and temperature factors and occupancy factors are often highly correlated. 

It is fairly evident that because of the complex interactions of deposi-tionally influenced and metamorphically influenced properties, the fundamental chemical-structural properties will need to be related to each other in a complex statistical fashion. A multivariate correlation matrix such as that pioneered by Waddell (8) appears to be an absolute requirement. However, characterization parameters far more sophisticated than those employed by Waddell are required. One can hope that, as correlations between parameters become evident, certain key properties will be discovered that will allow coal scientists and technologists to identify and classify vitrinites uniquely. Measurement of reflectance or other optical properties, if carried out properly, possibly on somewhat modified samples, might prove valuable in this respect. It then would not be necessary for every laboratory to have supersophisticated analytical equipment at its disposal in order to classify a coal properly. By properly identifying and classifying the vitrinite in a coal, one then could estimate accurately the many other vitrinite properties available in the multivariate correlation matrix. 

This type of rate expression is often used in models for water treatment, and many environmental factors can be included (the effect of, e.g., phosphate, ammonia, volatile fatty acids, etc.). The correlation between parameters in such complicated models is, however, severe, and very often a simple Monod model (7-92) with only one limiting substrate is sufficient. 

Provided r] values are deemed reliable, they can be used in scatter matrix plots to investigate correlations between parameters or in QQ plots or histograms to assess appropriateness of the selected parametric shape of parameter variability. Models... 

In modeling intersubject variability, it is advisable to start with the diagonal elements of the covariance matrix instead of starting with the full covariance matrix. If NONMEM is used for modeling, the posthoc parameter (or p) values should be subjected to a pairs plot to determine if there are any relationships (correlations) between parameters. A correlation test should be performed for parameters that... 

An alternative way to reach the minimum of is the use of the Bayesian approach, which deals with the direct determination of the Probability Distribution Function (PDF) for the final parameters. This method has the advantage that all the parameter space compatible with the experimental error is explored and therefore correlation between parameters and multimodal minima in the parameter-x space are naturally taken into account. For a review on Bayesian methods the reader is refereed to the excellent monograph of Sivia et al. (Ref ), while in this work we will only briefly explain the method used for our specific problem. 

Although it may well be true that the method of least squares is widely misused because of its apparent objectivity and general availability, it was clearly also true that much of the information obtainable from least squares is not used as completely as it could be. The problem of correlation between model parameters illustrates this clearly. High correlation between parameters amounts only to a statement about the data structure as opposed to the data values. The essential issue is the nature of the dependence of the parameter being determined on the data set. If two parameters have similar dependences, then their estimates are going to be correlated. Measuring more data points or a different set of data points would result in a different correlation matrix. The physical limitations of the experimental method, such as the inability to measure spectral characteristics of weak transitions or transitions that fall in inaccessible frequency regions, make it impractical to avoid correlations. 

Although correlation between parameters is a function of the data structure and has nothing to do with deficiencies in the model, it has implications for both the choice of the model and the design of the experiment. EVANS described his experiences with the determination of the crystal structure of tetragonal barium titanate (BaTiOa). The problem was ample in that it involved only three atomic positional parameters (one for Ti and two for 0), plus nine thermal parameters. There was considerable interest in the details of the structure because of the ferroelectric properties of the material. The proposed model was essentially a simple cubic arrangement of atoms, but with Ti displaced slightly from the center of an octahedron. By ordinary x-ray standards, this distortion (which was expected to be on the order of 0.15 A) could be measured with a standard error of 0.01-0.02 A if... 

GC6H5NHS02Ph and 4-GC6H4CH2S02Ph have also been explained. In each case the correlation between parameters is ascribed to an underlying molecular connection. 

Small variances guarantee that the parameter is accurately estimated and small correlation coefficients indicate that the parameters camiot be mutually compensated. The variances are very much dependent on the precision of the experiments since they are directly proportional to the weighted sum of residual squares (Q), while the correlation coefficient depends heavily on the model structure as such. Special tricks to suppress the correlation between parameters exist, and should always be used. 

Suppression of the mutual correlation between parameters will be illustrated by a case study. Rates of catalytic reactions are very frequently measured in gradientless and differential reactors, and the rate expressions of the Langmuir-Hinshelwood type are frequently used in the interpretation of experimental data. The rate expression has the general form... 

It has to be concluded that there are physically justified correlations between parameters which characterize the bonding strength and the phase stability on the one hand, and the macroscopic phase behavior on the other. However, such correlations represent complex functional relationships, and thus they are useful only for order-of-magnitude predictions. More quantitative predictions have to consider the character and strength of bonding in a more detailed way, i.e. they have to rely on quantum-mechanical calculations which are cumbersome and time consuming. 

---