Random training set

Figure 7. Random training set manually augmented with factorially designed samples.

First, one can check whether a randomly compiled test set is within the modeling space, before employing it for PCA/PLS applications. Suppose one has calculated the scores matrix T and the loading matrix P with the help of a training set. Let z be the characteristic vector (that is, the set of independent variables) of an object in a test set. Then, we first must calculate the scores vector of the object (Eq. (14)). 

A data set can be split into a training set and a test set randomly or according to a specific rule. The 1293 compounds were divided into a training set of 741 compounds and a test set ot 552 compounds, based on their distribution in a K.NN map. From each occupied neuron, one compound was selected and taken into the training set, and the other compounds were put into the test set. This selection ensured that both the training set and the test set contained as much information as possible, and covered the chemical space as widely as possible. 

Figure 5. Randomly designed training set employing uniform distribution.

We will now construct the concentration matrices for our training sets. Remember, we will simulate a 4-component system for which we have concentration values available for only 3 of the components. A random amount of the 4th component will be present in every sample, but when it comes time to generate the calibrations, we will not utilize any information about the concentration of the 4th component. Nonetheless, we must generate concentration values for the 4th component if we are to synthesize the spectra of the samples. We will simply ignore or discard the 4th component concentration values after we have created the spectra. 

We will also create validation data containing samples for which the concentrations of the 3 known components are allowed to extend beyond the range of concentrations spanned in the training sets. We will assemble 8 of these overrange samples into a concentration matrix called C4. The concentration value for each of the 3 known components in each sample will be chosen randomly from a uniform distribution of random numbers between 0 and 2.5. The concentration value for the 4th component in each sample will be chosen randomly from a uniform distribution of random numbers between 0 and 1. 

It would be interesting to see how well CLS would have done if we hadn t had a component whose concentration values were unknown (Component 4). To explore this, we will create two more data sets, A6, and A7, which will not contain Component 4. Other than the elimination of the 4randomly structured training set, and A7 will be identical to A3, the normal validation set. The noise levels in A6, A7, and their corresponding concentration matrices, C6 and C7, will be the same as in A2, A3, C2, and C3. But, the actual noise will be newly created—it won t be the exact same noise. The amount of nonlinearity will be the same, but since we will not have any absorbances from the 4 component, the impact of the nonlinearity will be slightly less. Figure 24 contains plots of the spectra in A6 and A7. 

Structure Training Set Random Distribution Training Set Same as A2, but with only 3 components A1 spectra condensed into 10 bins ... 

Two models of practical interest using quantum chemical parameters were developed by Clark et al. [26, 27]. Both studies were based on 1085 molecules and 36 descriptors calculated with the AMI method following structure optimization and electron density calculation. An initial set of descriptors was selected with a multiple linear regression model and further optimized by trial-and-error variation. The second study calculated a standard error of 0.56 for 1085 compounds and it also estimated the reliability of neural network prediction by analysis of the standard deviation error for an ensemble of 11 networks trained on different randomly selected subsets of the initial training set [27]. 

As described in Section 44.5.5, the weights are adapted along the gradient that minimizes the error in the training set, using the back-propagation strategy. One iteration is not sufficient to reach the minimum in the error surface. Care must be taken that the sequence of input patterns is randomized at each iteration, otherwise bias can be introduced. Several (50 to 5000) iterations are typically required to reach the minimum. 

CV or bootstrap is used to split the data set into different calibration sets and test sets. A calibration set is used as described above to create an optimized model and this is applied to the corresponding test set. All objects are principally used in training set, validation set, and test set however, an object is never simultaneously used for model creation and for test. This strategy (double CV or double bootstrap or a combination of CV and bootstrap) is applicable to a relatively small number of objects furthermore, the process can be repeated many times with different random splits resulting in a high number of test-set-predicted values (Section 4.2.5). 

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