Hat notation

In many cases, one may measure spectra of solutions of the pure components directly, and the above estimation procedure is not needed. For the further development of the theory of multicomponent analysis we will therefore abandon the hat-notation in K. Given the pure spectra, i.e. given K (pxq), one may try and estimate the vector of concentrations (pxl) of a new sample from its measured... 

If we observe X, can we then predict C and SI In previous chapters we have used a hat notation to indicate a prediction, so it is also possible to write the equation above as... 

The hat notation previously used to indicate a complex amplitude has been omitted for simplicity. 

The extra factor of g, depends on the correlation of successive MC configurations. We expect Ht(Ej) Ht Ej). The hat notation indicates that Ht(Ej) is sample statistic [vs. the population statistic H Ej)]. If the successive configurations are not correlated, then gt = 1. However, this is rarely the case. When the successive MC configurations are correlated the correct value for gt is given by... 

We will use the hatted symbols without a hat as, in practice, it does not lead to confusion. With these notations we have g z) = Qxpi v z)/2) and the Poisson equation is... 

Shorthand notations such as ET (electron transfer), HAT (hydrogen atom transfer), BDE (bond dissociation energy), NHE (normal hydrogen electrode), CV (cyclic voltammetry), LFP (laser flash photolysis), EPR (electron paramagnetic resonance) and KIE (kinetic isotope effect) will be used throughout the chapter. In addition, recurring chemical compounds such as TEMPO (2,2,6,6-tetramethylpiperidine-Ai-oxyl), HBT (1-hydroxyben-zotriazole), BTNO (benzotriazole-A-oxyl), HPI (iV-hydroxyphthalimide), PINO (phthal-imide-iV-oxyl), NHA (A-hydroxyacetanilide) and a few others will be referred to by means of the capital-letter acronym. 

Matrices are shown in bold capital letters e.g. R), column vectors in bold lowercase letters (e.g. c) (row vectors are transposed column vectors) and scalars in italic characters (e.g. q). True values are indicated by Greek characters or the subscript true . Calculated or measured values are indicated by Roman characters. The hat ( ), used in the literature to indicate calculated, has been dropped to simplify the notation whether the magnitude is measured or calculated can be deduced from the context. The running indexes in multivariate calibration are as follows k = 1 to Al analytes are present in z = 1 to I... 

In fact we obtain estimates of b from regression, so strictly there should be a hat on top of the b, but in order to simplify the notation we ignore the hat and so the approximation sign becomes an equals sign. 

The costate at the minimum is denoted by A without the hat Using the above notation, the state equations can be written as... 

Choosing a coordinate system appropriate to the symmetry of the problem at hand can make problems easier to solve. Cartesian coordinates are usually easiest for problems involving planes and Hat surfaces. For problems involving spheres or cylinders, spherical or cylindrical coordinates are most convenient. The advantage of the operator notation, V, V , and V, is that it is independent of the coordinate system. But when it comes to performing specific calculations, you can use Table 17.1 (pages 312 and 313) to translate from the vector notation, such as VT, to the specific expressions that you need for a particuiar problem. 

Figure 23 presents a composite of four computer plots showing linear and quartic fits applied to the synthetic data with the corresponding residuals. The line in Fig. 23a is the best-fit, least-squares linear regression drawn through die 31 data points. The corresponding residuals pattern is shown in Fig. 23b. A residual is defined as the deviation of the actual data point along the y axis away from the calculated y value based on the least squares fit. The calculated y value is sometimes referred to as y "hat" (which we write as YH), hence our notation of Y-YH for residuals. Residuals are normalized by dividing actual y value differences by the standard error of the calculated y values, SEYH.

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