Variables and predicting functions

Supervised learning starts with a set of m observations, n independent variables Xj and one dependentvariable Y. The terms predictors for Xj and targetvariable for Y are perhaps more informative. Each observation i e m gives values for Xj emd y, for Y. Let us imagine these values as matrices X = (x ) = (x ) is an m x n-matrix whose rows are x,, Y = (yj) is an m x 1-matrix. Usually, in both statistics and in linear algebra, rows of an m X n-matrix are indicated by i = 1. m, columns by = 1. n. For conformity with the earlier chapters of this book we choose here also indices ( em = 0.m-l and n = 0. n - 1.  

For our purposes, the predictors are assumed to be continuous, i.e. they have real values. The target variable is either continuous or discrete. The first aim of supervised learning is to find a fitting (and hopefully predictive) function / called a predicting function whose values /(xj) are in as close as reasonable agreement with target values yj for all observations i cm. k predicting function, despite its name, is not predictive per se, as is seen from its method of construction. A better name therefore may be fit- 

1 Example In molecular structure elucidation the observations are pairs of spectra and compounds. Predictors used are spectral predictors, functions that map spectra onto real numbers. The target variable is, for example, a binary molecular descriptor of a structural property SP, equal to 1 if a compound has property SP, and equal to 0 otherwise. The search is for a function able to predict whether or not the corresponding compound has property SP for a given spectrum. We will calculate such predicting functions in Section 8.5. 

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The film properties required for some appHcation can only be deterrnined by the performance of the appHed coating in practice. Because requirements and exposure conditions vary widely, devising laboratory tests to predict film performance is difficult and frequendy not possible. Data banks of actual field performance as functions of coating compositions, appHcation variables, and environmental factors can be very usehil. 

A typical break-even chart is used with production models to predict optimum production levels, break-even points, and shutdown conditions under various scenarios. These models tend to involve a reasonable amount of approximation. For example, sales revenue as a function of production level involves numerous variables and relationships that are not always weU known. Such charts, however, provide useful guides for production operations. 

Some emphasis is given in the first two chapters to show that complex formation equilibria permit to predict quantitatively the extent of adsorption of H+, OH , of metal ions and ligands as a function of pH, solution variables and of surface characteristics. Although the surface chemistry of hydrous oxides is somewhat similar to that of reversible electrodes the charge development and sorption mechanism for oxides and other mineral surfaces are different. Charge development on hydrous oxides often results from coordinative interactions at the oxide surface. The surface coordinative model describes quantitatively how surface charge develops, and permits to incorporate the central features of the Electric Double Layer theory, above all the Gouy-Chapman diffuse double layer model. 

For comparison purposes, regression parameters were computed for the model defined by Equations 6, 7, 8, and 10 and the model obtained by replacing In (1/R) in those equations by R. The dependent variable (y) is particulate concentration because it is desired to predict particulate content from reflectance values. Data from Tables I and II were also fitted to exponential and power functions where the independent variable (x) was reflectance but the fits were found to be inferior to that of the linear relationship. 

The mass flow of the conversion gas, its molecular composition, temperature and stoichiometry, are a complex function of volume flux of primary air, primary air temperature, type of solid fuel, conversion concept, etc. Several workers have tried to mathematically model these relationships, which are commonly referred to as bed models [12,33,14,51,52]. It is an extremely difficult task to obtain a predictive bed model, which is discussed in the introduction of this ew. The review of the thermochemical conversion processes below will outline the complex relationships between these variables and their effect on the conversion gas in sections B 4.4-B 4.6. 

A correlation is based on the proposition that, for a particular population S, the property y is related to one or several independent variables, often called predictors (xi, X2,..., xt), the correlation is based on a particular function y = f(x, X2,. , x. ) and a set of adjustable parameters (co, ci, C2,...). The optimal parameter values for ci are extracted from the data, which minimizes the differences between data and predictions. The principal usefulness of a regression formula, which has been proven valid only in S, is in its success in estimating the property in the bigger population of P (table 5.3). Some investigators would immediately proceed to export this correlation from the training set S to the larger population P without further assurances that it would still work. 

To predict the performance of equipment we must know (a) the rate at which fluid is modified as a function of the pertinent variables, and (b) the way fluid passes through the equipment. 

Root Mean Square Error of Prediction (RMSEP) Plot (Model Diagnostic) The validation set is employed to determine the optimum number of variables to use in the model based on prediction (RMSEP) rather than fit (RMSEO- RM-SEP as a function of the number of variables is plotted in Figure 5.7S for the prediction of the caustic concentration in the validation set, Tlie cuive levels off after three variables and the RMSEP for this model is 0.053 Tliis value is within the requirements of the application (lcr= 0.1) and is not less than the error in the reported concentrations. 

FIGURE 2.18 Typical profile of calibration and prediction errors as a function of the PLS model complexity (number of latent variables). The examination of such a plot may be helpful in selecting the optimal model complexity. 

For each of the four cases an expression for the rate of polymer deposition was derived in terms of the gas pressure, p, the plasma current, I, the measure of the average electron energy, V/p, and the ratio of the monomer pressure to its saturation vapor pressure, x. The functional dependencies of rp on these variables are given in Table 3. The constants a, c, and n, are taken to be parameters whose values are adjusted to obtain a fit between the measured and predicted values... 

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