Redundant transformations

Subtraction of reference Linear drift subtraction Normalization Averaging Linearization Differential measurements Baseline correction Relative signals Signal quality redundancy Transform functions, e.g., antilog... 

A virtually endless stream of new investment products is paraded forth by financial service companies each year. Although promoted as innovative with superlative benefits, in reality most such creations are often narrowly focused with high risk or are reconfigured and redundant transformations of assets already available. Sorting through the muck and discovering innovative products that shift the efficient frontier outward is not an easy task... 

The CWT is compactly described by Eqs. [4] and [6], but this definition allows for infinitely redundant transformations.There is no limit to the number of dilated and translated wavelets (4 (a ), where a and b are real numbers) used in the transform. This unrestricted and unguided use of wavelets to convert a signal into wavelet space often prevents the use of an inverse wavelet transformation because of violations of the conditions required by Eq. [7]. Even though these transforms are redundant and nonreversible, they still reveal information about the character of a particular signal. 

All the CT (current transformer) secondaries must be shorted and grounded while conducting the test on the main circuit. This condition is redundant while conducting test on the auxiliary circuit. 

The characters Xj for the examples in the previous section were calculated following the method described in Section 8.9, that is, on the basis of Cartesian displacement coordinates. Alternatively, it is often desirable to employ a set of internal coordinates as the basis. However, they must be well chosen so that they are sufficient to describe the vibrational degrees of freedom of the molecule and that they are linearly independent The latter condition is necessary to avoid the problem of redundancy. Even when properly chosen, the internal coordinates still do not usually transform following the symmetry of the molecule. Once again, the water molecule provides a very simple example of this problem. 

Resorting to the definition of the Fourier transform, Eq. (2.9), we notice that for the redundant coordinates the harmonic kernel degenerates and becomes exp (0) = 1. Thus for the redundant coordinates the Fourier transform turns into a simple integration with respect to the respective reciprocal coordinate20 - a projection . 

Generally, a significant amount of the information contained in the experimentally measured variables is redundant (correlated with other variables) or nonrele-vant. PCA allows the transformation of this set of variables into a new set of noncorrelated variables, which is more easily interpretable, and underlines the... 

The classical spin motion that follows from the above considerations can be viewed as a dynamics on the sphere S2 driven by the Hamiltonian dynamics in phase space. To see this one first transforms the (redundant) fourdimensional representation of the matrix degrees of freedom, corresponding... 

Also, since each measurement corresponding to yi was redundant for the original system, each row of Ci was linearly dependent on some other rows of [ ]. But, any row in C must be linearly independent of any row in C2 otherwise y2 would be redundant. Hence, any row in Ci is dependent on the other rows in [ , ]. The dependency is unchanged by the transformation F thus, yi is also redundant in the system [ "]. ... 

Thus, we have identified the subset of redundant equations containing only the redundant process variables fa, fa, fu, fH, and fa5. Furthermore, the rank of Ru is equal to 6, which means that at least one of the unmeasured variables is indeterminable. The remaining ones can be written in terms of it, as indicated by Eq. (4.15). In this case, from the orthogonal transformation, the subsets of u are defined as... 

The evaluation of the contribution of cycle diagrams to the activity coefficient is formally complete once the eigenvalues of SI have been found. Let us write the result explicitly in terms of the Fourier transforms for the important case of two defects a and / each of which is allowed to occupy a particular one of the two positions in the unit cell. In this simple system the labelling of positions in the unit cell by superscripts on Fourier transforms becomes redundant and can be omitted. The result is... 

The use of hybrid atomic orbitals in qualitative valence theory has, in the past, rested on two points (i) an empirical justification of their use involving the concept of the valence state of an atom and (ii) a simple linear transformation technique for the construction of the explicit forms of the orbitals. In this section we show that both of these points can be replaced. The justification can be replaced by a derivation and this derivation can be used to suggest variational forms which render the linear transformation technique redundant. 

One may question the need for a four parameter enthalpy equation, i.e., whether describing an acid or base by two parameters each is redundant. The following simple matrix algebra shows the conditions whereby a four parameter model reverts to a less redundant two parameter equation. Letting A be the transformation matrix, E and C represent the parameters for the four parameter model, and a represent the acid parameters for the two parameter model, the following equation results ... 

It is reasonable to hope to assemble a complete set of representations to provide a full and non-redundant description of the symmetry species compatible with a point group The problem is that there are far too many representations of any group. On the one hand, matrices in representations derived from expressing symmetry operations in terms of coordinates - as in problem 5-18 - depend on the coordinate system. Thus there are an infinite number of matrix representations of C2v equivalent to example 7, derivable in different coordinate systems. These add no new information, but it is not necessarily easy to recognize that they are related. Even in the cases of representations not derived from geometric models via coordinate systems, an infinite number of other representations are derivable by similarity transformations. 

Spectral data are highly redundant (many vibrational modes of the same molecules) and sparse (large spectral segments with no informative features). Hence, before a full-scale chemometric treatment of the data is undertaken, it is very instructive to understand the structure and variance in recorded spectra. Hence, eigenvector-based analyses of spectra are common and a primary technique is principal components analysis (PC A). PC A is a linear transformation of the data into a new coordinate system (axes) such that the largest variance lies on the first axis and decreases thereafter for each successive axis. PCA can also be considered to be a view of the data set with an aim to explain all deviations from an average spectral property. Data are typically mean centered prior to the transformation and the mean spectrum is used a base comparator. The transformation to a new coordinate set is performed via matrix multiplication as... 

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