Statistics data representation

P.J. Lewi, Multidimensional data representation in medicinal chemistry. In Chemometrics. Mathematics and Statistics in Chemistry (B.R. Kowalski, Ed.), Reidel, Dordrecht, 1984, pp. 351-376. 

The statistical measure of the quality of the regression is used to determine whether the model provides a meaningful representation of the data. The parameter estimates are reliable only if the model provides a statistically adequate representation of the data. The evaluation of the quality of the regression requires an independent assessment of the stochastic errors in the data, information that may not be available. In such cases, visual inspection of the fitting results may be useful. Issues associated with assessment of regression quality are discussed further in Section 19.7.2 and Chapter 20. 

A refined philosophical approach toward the use of impedance spectroscopy is outlined in Figure 23.1, where the triangle evokes the concept of an operational amplifier for which the potential of input channels must be equal. Sequential steps are taken until the model provides a statistically adequate representation of the data to within the independently obtained stochastic error structure. The different aspects that comprise the philosophy are presented in this section. 

Statistical data for evaluations. To aid interpretation of data, the imprecision is shown by means of graphic representations, as far as this has been possible. 

The representation of this equation for anything greater than two variates is difficult to visualize, but the bivariate form (m = 2) serves to illustrate the general case. The exponential term in Equation (26) is of the form x Ax and is known as a quadratic form of a matrix product (Appendix A). Although the mathematical details associated with the quadratic form are not important for us here, one important property is that they have a well known geometric interpretation. All quadratic forms that occur in chemometrics and statistical data analysis expand to produce a quadratic smface that is a closed ellipse. Just as the univariate normal distribution appears bell-shaped, so the bivariate normal distribution is elliptical. 

The data should be reported as specified in the protocol with the requested significant figures. Valid data (those free of gross errors and produced following the protocol) should be submitted to various statistical treatment for outlier detection of mean and variance, and an ANOVA treatment to establish the repeatability and reproducibility figures. All these treatments and their sequence are specified in the lUPAC protocol [2]. The final report should contain all individual and statistical data additional graphical representation e.g. Youden-plots, bar-graphs etc may also be added. 

Multi-layer feedforward networks contain an input layer connected to one or more layers of hidden neurons (hidden units) and an output layer (Figure 3.5(b)). The hidden units internally transform the data representation to extract higher-order statistics. The input signals are applied to the neurons in the first hidden layer, the output signals of that layer are used as inputs to the next layer, and so on for the rest of the network. The output signals of the neurons in the output layer reflect the overall response of the network to the activation pattern supplied by the source nodes in the input layer. This type of network is especially useful for pattern association (i.e., mapping input vectors to output vectors). 

There are cases for which the Master Curve does not provide a statistically valid representation of the experimental data and procedures have been developed to analyze the so-called inhomogeneous data. At this time, none of the procedures has been incorporated in the ASTM E1921 test standard, but they are utilized in some cases with example procedures described by Wallin et al. (2004), Scibetta (2012) and Choi et al. (2012). 

D-QSAR studies demonstrated that there could be more than one way to fit structure-activity data within a QSAR methodology. A receptor-independent 4D-QSAR study identified the hydrophobic nature of a HIV protease receptor site and helped in structural modification to improve the potency of the AHPBA inhibitors [241], A 4D-fingerprint-based QSAR model developed for AHPBA inhibitors of HIV was generated independent of any receptor structure or alignment information [126]. These models exhibited comparable statistical data with CoMFA, CoMSIA and H-QSAR approaches. This study proved that genuine representation of 3D and conformational properties of compounds is possible using this approach. 

One of the milestone references for EDA is the comprehensive book by Tukey [1]. Tukey, in his work, aimed to create a data analysis framework where the visual examination of data sets, by means of statistically significant representations, plays the pivotal role to aid the analyst to formulate hypotheses that could be tested on new data sets. The stress on two concepts such as dynamic experimenting on data (e.g. evaluating the results on different subsets of a same data set, under different data-preprocessing conditions) and exhaustive visualization capabilities offers researchers the possibility to identify outliers, trends and patterns in data, upon which new theories and hypothesis can be built. Tukey s first view on EDA was based on robust and nonparametric statistical concepts such as the assessment of data by means of empirical distributions, hence the use of the so-called five-number summary of data (range extremes, median and quartiles), which led to one of his most known graphical tools for EDA, the box plot. 

It is hoped that the more advanced reader will also find this book valuable as a review and summary of the literature on the subject. Of necessity, compromises have been made between depth, breadth of coverage, and reasonable size. Many of the subjects such as mathematical fundamentals, statistical and error analysis, and a number of topics on electrochemical kinetics and the method theory have been exceptionally well covered in the previous manuscripts dedicated to the impedance spectroscopy. Similarly the book has not been able to accommodate discussions on many techniques that are useful but not widely practiced. While certainly not nearly covering the whole breadth of the impedance analysis universe, the manuscript attempts to provide both a convenient source of EK theory and applications, as well as illustrations of applications in areas possibly u amiliar to the reader. The approach is first to review the fundamentals of electrochemical and material transport processes as they are related to the material properties analysis by impedance / modulus / dielectric spectroscopy (Chapter 1), discuss the data representation (Chapter 2) and modeling (Chapter 3) with relevant examples (Chapter 4). Chapter 5 discusses separate components of the impedance circuit, and Chapters 6 and 7 present several typical examples of combining these components into practically encountered complex distributed systems. Chapter 8 is dedicated to the EIS equipment and experimental design. Chapters 9 through 12... 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

From a statistical viewpoint, there is often little to choose between power law and hyperbohc equations as representations of data over an experimental range. The fact, however, that a particular hyperbolic equation is based on some land of possible mechanism may lead to a belief that such an equation may be extrapolated more safely outside the experimental range, although there may be no guarantee that the controlling mechanism will remain the same in the extrapolated region. 

In some instances, all one is interested in is an accurate numerical representation of data, without any intent of physicochemical interpretation of the estimated coefficients a simple polynomial might suffice the approximations to tabulated statistical values in Chapter 5 are an example. 

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