Set of regression coefficients

This b (also known as p = the prediction vector) is often referred to as the regression vector or set of regression coefficients. Note that (A A)-1 A is referred to as the pseudoinverse of A designated as A+. Note that there is one regression coefficient for each frequency (or data channel). 

The PCR model can be easily condensed into a set of regression coefficients (Ijpcr) using the following eqnation ... 

Depending on what kind of information is needed, different possibilities exist for using a multilinear partial least squares regression model on new data. If only the prediction of y is wanted, it is possible to obtain a set of regression coefficients that relates X directly to y. This has been shown in detail by Smilde and de Jong for the multilinear PLS1 model [de Jong 1998, Smilde 1997] as follows. The first score vector ti is... 

Then, it is possible to predict the values of the responses on new samples in a way that is completely identical to what is already described in Equation (23), the only exception being that the set of regression coefficients would be the one computed by PCR, instead of MLR, that is, by Equation (28). 

Here, B ls = K (KK )" is the final qxp matrix of regression coefficients for converting a spectral measurement into concentration estimates. For KK (pxp) to be invertible K should be of full rank. A first requirement for this is that pchemical components should not exceed the number of wavelengths. Furthermore, the set of pure spectra in K should be independent, i.e. no pure spectrum may be an exact linear combination of the other pure spectra. 

The Method of steepest ascent has proved to be successful since in the third trial the yield of 85.0% has been reached, which is 30% more than the best yield from FUFE. A new FUFE 24 has been set up near trial No. 3, with conditions and outcomes of the first 16 trials from Table 2.243. A calculation of regression coefficients of some interactions has shown that the mentioned effects are even more significant. A decision has therefore been made to upgrade FUFE to CCOD, as shown in Table 2.242. 

To compare the selection of E-optimal subsets with a complete data set, an E-optimal subset of ten points was generated. The respective condition numbers, based on 1 to 10 latent variables, were calculated and are shown in Figure 8.29. We see that the condition numbers of the E-optimal subsets are lower and are more stable than the condition number for the whole set. We also observe that at six or more latent variables, the condition number of the complete set increases much more rapidly than the condition numbers of the E-optimal subsets. Calculation of regression coefficients at six or more latent variables may be considerably more stable by using the E-optimal subset of 10 points compared with the whole set of 102 points. 

Subsequent to publication of the original work done by Udvardy (17), a multiple correlation analysis of all replicates of the data points was conducted at Mobay. A set of regression equations was generated for each binder studied. Each equation predicted one of the board properties in terms of a number of processing variables including panel density. Correlation coefficients were 0.8 for internal bonds for the Mondur E-lll panels with all others being 0.9 or greater. 

After the deconvolution step, giving the contribution coefficients of reference spectra (see Chapter 2), the parameters calculation is possible by using the same coefficients and a corresponding calibration file (Fig. 11). This calibration file includes the corresponding concentrations for specific compounds (nitrate, nitrite, anionic surfactants, etc.) and the values related to the reference spectra of mixtures (Table 3). The latter are statistically calculated for the purpose, through a preliminary stepwise regression study, from a set of at least 30 samples (with 30 corresponding values of parameters and 30 sets of contribution coefficients). 

Caveats. In complicated situation, statistics may zero in on a false optimum, that is, a local rather than absolute minimum of the objective function. Moreover, statistics by itself delivers uncritically the set of variables that gives the best numerical fit to the not entirely accurate input data, and it may happen that the set of correct coefficient values is different and is disregarded because its fit is not quite as good. Therefore, it is good practice to run regression programs with any constraints on the variables that the physics of the system may suggest. 

In addition to the set of adjustable coefficients and b contained in Eq. (13.7), the method CMF also requires ealeulation of a certain nmnber of adjustable parameters. Among them are the parameter v for the support vector regression method v-S VR and the ridge parameter y for KRR. Their values should be optimized with the aim to improve the predictive eapability of the model constracted. In addition, for each molecular field one can adjust the values of up to two parameters (attenuation factor, which is related to the width of the Gaussian function) and hj (mixing coefficient, which has the meaning of the relative contribution of molecular field of the ythtype). 

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