Univariate data

The title implies that in this first chapter techniques are dealt with that are useful when the observer concentrates on a single aspect of a chemical system, and repeatedly measures the chosen characteristic. This is a natural approach, first because the treatment of one-dimensional data is definitely easier than that of multidimensional data, and second, because a useful solution to a problem can very often be arrived at in this manner. 

A scientist s credo might be One measurement is no measurement. Thus, take a few measurements and divine the truth This is an invitation for discussions, worse yet, even disputes among scientists. Science thrives on hypotheses that are either disproven or left to stand in the natural sciences that essentially means experiments are re-mn. Any insufficiency of a model results in a refinement of the existing theory it is rare that a theory completely fails (the nineteenth-century luminiferous ether theory of electromagnetic waves was one such, and cold fusion was a more shortlived case). 

Reproducibility of experiments indicates whether measurements are reliable or not under GMP regulations this is used in the systems suitability and the method validation settings. 

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Overdetermination of the system of equations is at the heart of regression analysis, that is one determines more than the absolute minimum of two coordinate pairs (xj/yi) and xzjyz) necessary to calculate a and b by classical algebra. The unknown coefficients are then estimated by invoking a further model. Just as with the univariate data treated in Chapter 1, the least-squares model is chosen, which yields an unbiased best-fit line subject to the restriction ... 

This is for univariate data what happens in the case of multivariate (multiwavelength) spectroscopic analysis. The same thing, only worse. To calculate the effects rigorously and quantitatively is an extremely difficult exercise for the multivariate case, because not only are the errors themselves are involved, but in addition the correlation stmcture of the data exacerbates the effects. Qualitatively we can note that, just as in the univariate case, the presence of error in the absorbance data will bias the coefficient(s) toward zero , to use the formal statistical description. In the multivariate case, however, each coefficient will be biased by different amounts, reflecting the different amounts of noise (or error, more generally) affecting the data at different wavelengths. As mentioned above, these... 

We will not repeat Anscombe s presentation, but we will describe what he did, and strongly recommend that the original paper be obtained and perused (or alternatively, the paper by Fearn [15]). In his classic paper, Anscombe provides four sets of (synthetic, to be sure) univariate data, with obviously different characteristics. The data are arranged so as to permit univariate regression to be applied to each set. The defining characteristic of one of the sets is severe nonlinearity. But when you do the regression calculations, all four sets of data are found to have identical calibration statistics the slope, y-intercept, SEE, R2, F-test and residual sum of squares are the same for all four sets of data. Since the numeric values that are calculated are the same for all data sets, it is clearly impossible to use these numeric values to identify any of the characteristics that make each set unique. In the case that is of interest to us, those statistics provide no clue as to the presence or absence of nonlinearity. 

In Chapter 4.1 Background to Least-Squares Methods, e.g. in Figure 4-3 and Figure 4-5, we have seen that for univariate data, the vector r of residuals and thus the sum of squares ssq, is a function of the measurement y and the parameters p of the model of choice. 

Some of the above plots can be combined in one graphical display, like onedimensional scatter plot, histogram, probability density plot, and boxplot. Figure 1.7 shows this so-called edaplot (exploratory data analysis plot) (Reimann et al. 2008). It provides deeper insight into the univariate data distribution The single groups are... 

More complex than vectors or matrices (X, X andy, X and Y) are three-way data or multiway data (Smilde et al. 2004). Univariate data can be considered as one-way data (one measurement per sample, a vector of numbers) two-way data are obtained for instance by measuring a spectrum for each sample (matrix, two-dimensional array, classical multivariate data analysis) three-way data are obtained by measuring a spectrum under several conditions for each sample (a matrix for each sample, three-dimensional array). This concept can be generalized to multiway data. 

For univariate data, only one variable is measured at a set of objects (samples) or is measured on one object a number of times. For multivariate data, several variables are under consideration. The resulting numbers are usually stored in a data matrix X of size ii x in where the n objects are arranged in the rows and the m variables in the columns. In a geometric interpretation, each object can be considered as a point in an m-dimensional variable space. Additionally, a property of the objects can be stored in a vector y (nx 1) or several properties in a matrix Y nxq) (Figure 2.19). 

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