Inner relation

Of the three aspects inherent to the covariance criterion (35.26), CCA just considers the so-called inner relation between t and u as expressed by RRR entirely neglects the var(t) aspect, whereas PCR emphasizes this var(t) component. One might maintain that PLS forms a well-balanced compromise between the methods treated thus far. PLS neither emphasizes one aspect of the X4-4Y relation unduly, nor does it completely neglect any. 

S. Wold, Non-linear partial least squares modelling. II. Spline inner relation. Chemom. Intell. Lab. Syst., 14(1992)71-84. 

A.P. de Weyer, L.M.C. Buydens, G. Kateman and H.M. Heuvel, Neural networks used as a soft modelling technique for quantitative description of the inner relation between physical properties and mechanical properties of poly ethylene terephthalate yams. Chemom. Intell. Lab. Syst., 16(1992) 77-82. 

This class of methods transforms the inputs in a nonlinear manner. The distinction is readily seen by referring once again to Eq. (22). This family of methods makes use of nonlinear functions both in the bracketed term and in the inner relation. Most of the popular methods project the inputs on a localized hypersurface such as a hypersphere or hyperrectangle. 

There are also forms of nonlinear PCR and PLS where the linear PCR or PLS factors are subjected to a nonlinear transformation during singular value decomposition the nonlinear transformation function can be varied with the nonlinearity expected within the data. These forms of PCR/PLS utilize a polynomial inner relation as spline fit functions or neural networks. References for these methods are found in [7], A mathematical description of the nonlinear decomposition steps in PLS is found in [8],... 

FIGURE 4.25 PLS2 works with X- and K-matrix in this scheme both have three dimensions. t and u are linear latent variables with maximum covariance of the scores (inner relation) the corresponding loading vectors are p und q. The second pair of x- and y-components is not shown. A PLS2 calibration model allows a joint prediction of all y-variables from the x-variables via x- and y-scores. 

Linear inner relation (Equation 4.65) is changed to a nonlinear inner relation, i.e., the y-scores have no longer a linear relation to the x-scores but a nonlinear one. Several approaches for modeling this nonlinearity have been introduced, like the use of polynomial functions, splines, ANNs, or RBF networks (Wold 1992 Wold et al. 1989). 

If severe nonlinearities might be present, the linear inner relation can be modified to a quadratic or cubic one. This strong nonlinear situation might arise whenever problems occur on the detector or monochromator, malfunction of the automatic sampler in ETAAS, strong influence of the concomitants on the signal, when the linear range for the analyte is too short or when LIBS, LA-ICP-MS measurements or isotope dilution are carried out (see Chapter 1 for more possibilities). 

Figure 6 JO. Illustration of the matrices and vectors used in a PLS model. In this example, two PLS dimensions are indicated by the presence of two vectors each of t, u, w, p and c, as well as two inner relation regression coefficients, b.

The above model is improved by developing the so-called inner relationship. Because latent (basis) vectors are calculated for both blocks independently, they may have only a weak relation to each other. The inner relation is improved by exchanging the scores, T and U, in an iterative calculation. This allows information from one block to be used to adjust the orientation of the latent vectors in the other block, and vice versa. An explanation of the iterative method is available in the literature [42, 51, 52], Once the complete model is calculated, the above equations can be combined to give a matrix of regression vectors, one for each component in Y ... 

Predictions for a new object by the PLS model can be understood from Fig. 11 as follows A new object defines a point, i in the X space and its projection on the PLS(X) component gives the score tu. The corresponding score, uu, along the PLS(Y) component is obtained from the correlation (called the inner relation) between and uv The calculated uu score corresponds to a point along the PLS(Y) component, and the coordinates of this point yield the predicted values j>j of each response. 

T and U are matrices of score vectors, P and C are the transposed matrices of the loading vectors, E and F are matrices of residuals. The inner relation which describes the correlation between the scores is defined by D which is a diagonal matrix in which the elements, dH, are the correlation coefficients of the linear relation between the scores. H is the matrix of residuals from the correlation fit. 

Recently, PLS modelling involving non-linear inner relation has been described [76]. 

The different spheres of production. .. constantly fend to an equilibrium for, on the one hand, while each producer of a commodity is bound to produce a use-value, to satisfy a particular social want, and while the extent of these wants differs quantitatively, still there exists an inner relation which settles their proportions into a regular system, and that system one of spontaneous growth and, on the other hand, the law of the value of the ccHnmodities ultimately determines how much of the disposable working-time sodety can spend on each particular class of commodities. But this constant tendency to equilibrium, of the various spheres of production, is exercised only in the shape of a reaction against the constant upsetting of the equilibrium. ... 

Predictions by the PLS model, as illustrated in Fig. 17.3, can be explained as foUows For an object "z", the corresponding x variables in the X matrix define a point in the X space. This point is projected on the first PLS(X) component to afford the score Zn. From the correlation, called the inner relation, the corresponding score, iZjj, along the PLS(Y) component of the Y block is determined. This score corresponds to a point in the Y space, and this point in turn corresponds to the predicted values of each response variable. These can then be compared to the observed responses in the Y matrix. 

Compute the first correlation coefficient djj of the inner relation, diagfdy ... 

A plot of Uj aginst tj shows the linear relation between the PLS components. The slope is the coefficient of the inner relation. 

The diagonal matrix which defines the inner relation between the score vectors is therefore ... 

The fji score values are then entered into the inner relation to predict the corresponding Mjj scores. These are then used for predictions of the responses. 

Another structure for expressing a nonlinear relationship between X and Y is splines [333] or smoothing functions [75]. Splines are piecewise polynomials joined at knots (denoted by Zj) with continuity constraints on the function and all its derivatives except the highest. Splines have good approximation power, high flexibility and smooth appearance as a result of continuity constraints. For example, if cubic splines are used for representing the inner relation ... 

S Wold. Nonlinear partial least squares modelling II. Spline inner relation. Chemometrics Intell. Lab. Sys., 14 71-84, 1992. 

Figure 33 Representation of a PLS regression through the inner relation u = b.t. The solid lines in X- and Y-space are the principal components and the dashed lines are the PLS vectors. These are slightly skewed to account for the correlation between the two data blocks (redrawn from Figure 9 of ref [487] with permission from Pergamon Press Ltd., Headington Hill Hall, Oxford 0X3 OBW, UK).

Quality risk accidents, safety risk accidents and investment risk accidents of deep foundation pit are not independent from each other, when consider one kind of risk accidents and its factors separately we may leave out some inner relation of them and then get a distorted risk recognition result. Via combing accident cases of deep foundation pit, the article carried out an analysis on the relationship among quality risk, security risk and investment risk and then established a macroscopic logic diagram of the influence rule of the three objective dimensions, reached a tentative conclusion as follows ... 

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