Interpretation Methods

Local interpretation methods encompass a wide variety of approaches that resolve decisions about input data relative to annotated data or known features that cluster. By characterizing the cluster or grouping, it is possible to use various measures to determine whether an arbitary pattern of data can be assigned the same label as the annotated grouping. All approaches are statistical, but they vary in terms of measures, which include statistical distance, variance, probability of occurrence, and pattern similarity. 

If the probability distribution of the data is or assumed Gaussian, several statistical measures are available for interpreting the data. These measures can be used to interpret the latent variables determined by a selected data analysis method. Those described here are a combination of statistical measures and graphical analysis. Taken together they provide an assessment of the statistical significance of the analysis. 

The g-statistic or square of predicted errors (SPE) is the sum of squares of the errors between the data and the estimates, a direct calculation of variability  

Here xik is an estimated value of a variable at a given point in time. Given that the estimate is calculated based on a model of variability, i.e., PCA, then Qi can reflect error relative to principal components for known data. A given pattern of data, x, can be classified based on a threshold value of Qi determined from analyzing the variability of the known data patterns. In this way, the -statistic will detect changes that violate the model used to estimate x. The 0-statistic threshold for methods based on linear projection such as PCA and PLS for Gaussian distributed data can be determined from the eigenvalues of the components not included in the model (Jack-son, 1992). 

The Mahalanobis distance measures the degree to which data fit the calibration model. It is defined as 

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Interpretive methods Involve modeling the retention surface (as opposed to the response surface) on the basis of experimental retention time data [478-480,485,525,541]. The model for the retention surface may be graphical or algebraic and based on mathematical or statistical theories. The retention surface is generally much simpler than the response surface and can be describe by an accurate model on the basis of a small number of experiments, typically 7 to 10. Solute recognition in all chromatograms is essential, however, and the accuracy of any predictions is dependent on the quality of the model. 

Both inputs and outputs, as in input-output analysis and interpretation methods... 

Data interpretation methods can be categorized in terms of whether the input space is separated into different classes by local or nonlocal boundaries. Nonlocal methods include those based on linear and nonlinear projection, such as PLS and BPN. The class boundary determined by these methods is unbounded in at least one direction. Local methods include probabilistic methods based on the probability distribution of the data and various clustering methods when the distribution is not known a priori. 

Work on dimension reduction methods for both input and input-output modeling and for interpretation has produced considerable practical interest, development, and application, so that this family of nonlocal methods is becoming a mainstream set of technologies. This section focuses on dimension reduction as a family of interpretation methods by relating to the descriptions in the input and input-output sections and then showing how these methods are extended to interpretation. 

TABLE 10 Effects on the Responses of Table 8 and Critical Effects According to Different Statistical Interpretation Methods... 

Eor the two examples, described in references 22 and 23, the estimated effects on the responses (Tables 8 and 9) are given in Tables 10 and 11, respectively. Their significance according to the different statistical interpretation methods was determined when possible. 

Eortier, S., Chiverton, A., Glasgow, J. and Leherte, L. (1997). Critical-point analysis in protein electron-density map interpretation. Method Enzymol. Ill, 131-157. 

A ruggedness test is a part of method validation (Table 3.1) and can be considered as a part of the precision evaluation [2,4,5]. Ruggedness is related to repeatability and reproducibility. Some definitions for ruggedness come very close to those for reproducibility. Certain interpretation methods to identify the significant factors in a ruggedness test use criteria based on results for repeatability or reproducibility. These two items will be considered in Section 3.4.7. 

To determine whether an effect is statistically significant or not, a statistical interpretation method is used. Different possibilities have been described. An overview of them is given below. 

Some authors [29,34] present the statistical interpretation method as an ANOVA table. A general example for a 2 full factorial design is given in Table 3.19. The sums of squares (55x) are obtained with the effect values (Ex) and the number of experiments in the design (N). The mean square... 

A subset of simultaneous methods which overcomes the difficulty of mapping complex response surfaces by an exhaustive series of experiments are the interpretive methods, in which retention surfaces are modeled using a minimum number of experimental data points. Retention surfaces thus obtained for the individual solutes are then used to calculate (via computer) the total response surface according to some predetermined criterion. The total response surface is then searched for the optimum. 

A disadvantage of simple interpretive methods is that the model to which the retention data (or other data) are fit must be fairly accurate. In other words, an interpretive approach may fail if one or more sample components exhibits anomalous retention. Although rare in SFC, such retention behavior is observed occasionally and is difficult to predict intuitively. Note, however, that by anomalous retention we do not mean behavior that is merely unusual, e.g., retention that decreases smoothly with increasing density (at constant temperature). Retention that varies in a regular (continuous) manner, even if unusual, can usually be modeled with a high degree of accuracy (vide infra). 

Although it is beyond the scope of this chapter, more sophisticated interpretive methods can be employed to compensate for the anomalous retention behavior. That is, the complex retention surfaces are broken down into smaller parts in an iterative fashion the smaller retention surfaces are more accurately modeled by simple functions. This iterative interpretive approach has recently been applied in micellar LC (MLC) for the optimization of organic modifier, surfactant concentration, and pH (57,58). 

Optimization Criteria for Interpretive Methods. As noted earlier in our discussion of the simplex methods, there are many chromatographic response functions (CRFs) for the evaluation and comparison of chromatograms during an optimization process. Here we discuss two CRFs that we employed successfully with this interpretive method of optimization. Since the retention behavior of every solute must be modeled prior to optimization, the number of sample components is known beforehand it is thus unnecessary to include the number of peaks in these CRFs as was done in CRF-3 (equation 8) for the simplex. 

This example describes the approach and interpretative methods used in a SQT conducted in 1998 for a former shipyard located within a busy urban harbour. Site facilities included metal-smithing shops, electrical equipment shops, sandblasting and painting berths, and fuel storage facilities, which led to the introduction of numerous contaminants to the environment. This particular case study was chosen as illustrative due to its complexity coupled with the fact that it has neither been previously published, nor is it subject to client confidentiality (which is the case for other, more recent SQT studies). While a marine case study is presented, the SQT is equally applicable to freshwater environments. 

More sophisticated interpretation methods have been developed, e.g. application of semi-empirical complexation isotherms or a priori affinity spectra (Karush and Sonenberg, 1949 Posner, 1966 Perdue and Lytle, 1983a, b Allison and Perdue, 1994 Grzyb, 1995 Ruzic, 1996). All these methods have in common a conceptual approach similar to that used in the study of metal complexation by... 

Hence, in combination with interpretive methods the use of Rs or S as the resolution criterion appears to be always advantageous. 

Although the value of Rs for a given pair of peaks can be quickly transferred from one column to another by using the proportionality of Rx and Vn, this is not the case for the threshold criterion of eqn.(4.23). The problem is that if we know the boundaries of the area for which Rs min > 1 using a column of 10,000 plates, we only know the boundaries of the area for which Rs min > 0.5 for a column with 2,500 plates. We do not know what the boundaries for Rs min > 1 are in the latter case, because we have no information on how the value of Rs min changes with variations in the parameter settings. Only if the variation of the capacity factors as a function of the relevant parameters is known, can the boundaries of the area in which the resolution is adequate be calculated for different columns with different numbers of theoretical plates. Optimization methods in which this is the case (so-called interpretive methods ) will be discussed in section 5.5. 

This section deals with interpretive optimization methods. In these. methods, the extent of chromatographic separation is predicted indirectly from the retention behaviour of the individual solutes. The data are interpreted to locate the optimum in terms of the complete chromatogram. The interpretive methods may involve a limited number of experiments according to a pre-planned experimental design (section 5.5.1) or may start with a minimum number of experiments in order to try and locate the optimum by an iterative process (section 5.5.2). 

For all the interpretive methods described in section 5.5 it is essential to know the retention data of all the individual components in a sample. Section 5.6 deals with possibilities to obtain all this chromatographic information. 

We will refer to methods that try to characterize the response surface indirectly through the retention surfaces of the individual solutes as interpretive methods. They will be discussed extensively in section 5.5. 

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