Response vector

The covariance matrix COV(k ) is obtained by Equation 3.30. Let us now concentrate on the expected mean response of a particular response variable. The (l-a)100% confidence interval of yl0 (i=l.,w). the i,h element of the response vector y0 at x0 is given below... 

As the above example illustrates, PCA can be an effective exploratory tool. However, it can also be used as a predictive tool in a PAT context. A good example of this nsage is the case where one wishes to determine whether newly collected analyzer responses are normal or abnormal with respect to previously collected responses. An efficient way to perform snch analyses wonld be to construct a PCA model using the previously collected responses, and apply this model to any analyzer response (Xp) generated by a subse-qnently-collected sample. Such PCA model application involves hrst a mnltiplication of the response vector with the PCA loadings (P) to generate a set of PCA scores for the newly collected response ... 

Assume also that a validation sample has been collected with concentrations for Sj, Sy, and ij of 1, 3, and 2 respectively (c = [1 3 2]). Assuming linear additivity holds, the resulting response vector for this mixture sample is r= (12 8 10]. When validating tire models using this sample, the known information is the measured spectrum, r= [12 8 10], and the component amcentrations for the known analytes c = [1 3]- The steps for validating the CLS model arc shown in Figure 5.62 and include (a) formulating... 

FIGURE 5.62. Example of calibration and validation using the classical calibration approach, (a) Initial classical model form (b) estimating concentrations (c) reconstructing the response vector (d) calculating the spectral residual (e) calculating the concentrational residual. 

Arbitrarily chosen intensive vectors Rai) are in general neither orthogonal nor normalized. It is therefore of interest to identify a particular choice of A for which these vectors become an orthonormal set. Such unit-like response vectors, denoted as E ), can be expressed as usual as linear transformations of standard reference intensities... 

Shibata, T., Giaccia, A. J. and Brown, J.M. (2000) Development of a hypoxia-responsive vector for tumor-specific gene therapy. Gene Then, 7, 493 198. 

The coefficients a0 to a7 are calculated by multiplication of the vector xt by the response vector y, and division by the number of solutions (experiments), for example ... 

In matrix notation, the model is written by augmenting the instrument response vector, x, with a column of ones, producing the response matrix as shown in the middle of Figure 5.2 (y = Xb + e). Least-squares estimates of the model parameters bQ and bl are computed by... 

An important feature of any subset selection procedure is that it should be able to identify the situation where no effects are active. In order to explore this, the 12 values in the response vector Y were randomly shuffled in the order 6, 2, 7, 8, 9, 1, 4, 12, 5, 3, 11, 10, and rows of X were not changed. The analysis was re-run and the marginal probabilities plotted in Figure 6(b). Although a few factors have some probability of activity, there is nothing quite as convincing as in the original analysis, see below. 

In an experimental design, the settings of the variables are given by the row vector x = [x1. .. xk] in the design matrix. When there are several responses, the result will define a response vector, y = [yt. .. ym]. For the whole set of experiments these vectors define a response matrix, Y. 

Note that each component ofp, the physical response vector, in Eq. 6.1 is related to aU three components of q, the driving force, which is an applied or stimulus vector. Thus, each component of the tensor (the physical property) is associated with a pair of axes. For example, T32 gives the component of p parallel to the third axis when q is parallel to the second axis. In general, the number of indices assigned to a tensor component is equal to the rank of the tensor. Tensors of all ranks, like vectors, are defined by their transformation laws. For this discussion, they need not be considered here. 

An alteniate means of expressing the response data, which is sometimes chosen for ease of mathematical manipulation, is in terms of the real and imaginary component of a vector in phase space. For such a response vector, expressed S = a + yh, the magnitude of the signal is given by 5 = (a + and its phase angle is = tan" (ft/a). 

The leverage matrix is related to the response vector y by the following relationship ... 

Model complexity is defined as the ratio between the multivariate entropy Sx of the X-block n objects and p variables) of the model and - Shannon s entropy Hyoi the y response vector, thus also accounting for the information content of the y response [Authors, This Book] ... 

This validation technique is adopted to check models with chance correlation, i.e. models where the independent variables are randomly correlated to the response variables. The test is performed by calculating the quality of the model (usually F or, better, randomly modifying the sequence of the response vector y, i.e. by assigning to each object a response randomly selected from the true responses [Lindgren et al, 1996]. If the original model has no chance correlation, there is a significant difference in the quality of the original model and that associated with a mode obtained with random responses. The procedure is repeated several hundreds of times. 

Numerical evaluations of Kohn-Sham matrix elements and exchange-correlation (xc) contributions to response vectors follow the same scheme. In contrast to the Coulomb and exact Hartree-Fock exchange contributions which are usually evaluated by summing analytically computed integrals between basis functions... 

Computing linear response of a molecular property from Eq. (8(() requires knowledge of a response vector k" corresponding to the perturbation This response vector can be found by solving a linear set of equations in Eq. (76). This set of equations is large and tlierefore is solved iteratively for x N " variables where... 

Logistic regression using a binary response vector. 

Similar to graph potentials, another set of LOVIs was proposed based on the —> geometry matrix G, using as the diagonal terms the —> Balaban distance connectivity index and as the response vector the adjacency matrix A multiplied by the column vector z collecting the atomic numbers of all the non-hydrogen atoms [Beteringhe, Filip et al, 2005] ... 

Regression leads to a model estimating the relation between the Ai x 1 response vector y, and the Ai x r model matrix X (7,17,116) (Eq. 2.27). N is the number of design experiments, and t the number of terms included in the model. For example, in Equation 2.26, the number of terms equals six, since one intercept, two main effect terms, one interaction term, and two quadratic effect terms were included. The model matrix X is obtained by adding a row of ones before the Aix (r-1) design matrix, which consists of the coded factor levels and columns of contrast coefficients, as defined by the chosen experimental design. 

We want only the first output-layer unit (which corresponds to "A") to fire if this pattern is presented. That is, in terms of l s and 0 s, we would want the output-layer s response vector to be... 

---