Class membership

For example, the objects may be chemical compounds. The individual components of a data vector are called features and may, for example, be molecular descriptors (see Chapter 8) specifying the chemical structure of an object. For statistical data analysis, these objects and features are represented by a matrix X which has a row for each object and a column for each feature. In addition, each object win have one or more properties that are to be investigated, e.g., a biological activity of the structure or a class membership. This property or properties are merged into a matrix Y Thus, the data matrix X contains the independent variables whereas the matrix Ycontains the dependent ones. Figure 9-3 shows a typical multivariate data matrix. 

Class identifier this gives the column number which contains information about the class membership. 

In the original kNN method, an unknown object (molecule) is classified according to the majority of the class memberships of its K nearest neighbors in the training set (Fig. 13.4). The nearness is measured by an appropriate distance metric (a molecular similarity measure as applied to the classification of molecular structures). It is implemented simply as follows ... 

Fig. 44.6. The successive boundaries (wO to w4) as found by the LLM, during the training procedure described in Table 44.1. The crosses indicate the positions of the objects (1 to 4) their class membership (A or B) is given between parentheses.

The primary purpose of pattern recognition is to determine class membership for a set of numeric input data. The performance of any given approach is ultimately driven by how well an appropriate discriminant can be defined to resolve the numeric data into a label of interest. Because of both the importance of the problem and its many challenges, significant research has been applied to this area, resulting in a large number of techniques and approaches. With this publication, we seek to provide a common framework to discuss the application of these approaches. 

In brief, the Bayesian approach uses PDFs of pattern classes to establish class membership. As shown in Fig. 22, feature extraction corresponds to calculation of the a posteriori conditional probability or joint probability using the Bayes formula that expresses the probability that a particular pattern label can be associated with a particular pattern. 

Class membership is assigned using some decision rule that is typically some inequality test performed on P(co,. r ). 

Exploratory data analysis shows the aptitude of an ensemble of chemical sensors to be utilized for a given application, leaving to the supervised classification the task of building a model to be used to predict the class membership of unknown samples. 

Exploration analysis is not adequate when the task of the analysis is clearly defined. An example is the attribution of each measurement to a pre-defined set of classes. In these cases it is necessary to find a sort of regression able to assign each measurement to a class according to some pre-defined criteria of class membership selection. This kind of analysis is called supervised classification. The information about which classes are present have to be acquired from other considerations about the application under study. Once classes are defined, supervised classification may be described as the search for a model of the following kind ... 

Additionally to the x-data, a property y may be known for each object (Figure 2.3). The property can be a continuous number, such as the concentration of a compound, or a chemical/physical/biological property, but may also be a discrete number that encodes a class membership of the objects. The properties are usually the interesting facts of the objects, but often they cannot be determined directly or only with high cost on the other hand, the x-data are often easily available. Methods from... 

FIGURE 2.3 Variable (feature) matrix X and a property vector y. The property may be a continuous number (a physical, chemical, biological, or technological property), as well as a discrete number or categorical variable defining a class membership of the objects. 

For k 1 (1 -NN), a new object would always get the same class membership as its next neighbor. Thus, for small values of k, it is easily possible that classes do no longer form connected regions in the data space, but they can consist of isolated clouds. The classification of new objects can thus be poor if k is chosen too small or too large. In the former case, we are concerned with overfitting, and in the latter case with underfitting. 

In Section 4.8.3.3, we already mentioned regression trees which are very similar to classification trees. The main difference is that the response y-variable now represents the class membership of the training data. The task is again to partition the... 

The outcome from the neural network is a prediction of the class membership for each object (either training objects or test objects). It is a matrix Y with the same dimensions as Y, its elements v(/- are in the interval [0, 1], and they can be seen as somewhat like a probability for the assignment of the th object x, to the /th group. 

Comparison of the success of different classification methods requires a realistic estimation of performance measures for classification, like misclassification rates (% wrong) or predictive abilities (% correct) for new cases (Section 5.7)—together with an estimation of the spread of these measures. Because the number of objects with known class memberships is usually small, appropriate resampling techniques like repeated double CV or bootstrap (Section 4.2) have to be applied. A difficulty is that performance measures from regression (based on residuals) are often used in the development of classifiers but not misclassification rates. 

Unlike the straightforward methods for solubility, dissolution, and gastric stability, the BCS guidance recommends several methods to determine the permeability class membership of a drug substance (Table 28.1). 

Faustino PJ, Volpe DA, Knapton AD, Ellison CD, Hussain AS (1999) Value of an internal standard approach for determining internal permeability class membership of drugs. AAPS PharmSci 1(4) abstract. 

This section will focus on classification methods, or supervised learning methods, where a method is developed using a set of calibration samples and complete prior knowledge about the class membership of the samples. The development of any supervised learning method involves three steps ... 

Developing a classification rule, using objects with known class membership. 

Developing a classification rule This step requires the known class membership values for all calibration samples. Classification rules vary widely, but they essentially contain two components ... 

This classification method [78,79] actually uses the quantitative regression method of PLS (described earlier) to perform qualitative analysis. This is done by populating one or more y variables not with reference property values, but rather with zeros or ones, depending on the known class membership of the calibration samples. For example, if there are only two possible classes and four calibration samples, a single y variable can be constructed as follows ... 

Unlike the methods discussed above, which strive to find directions in a common space that separate known classes, the SIMCA method [81] works on a quite different principle define a unique space for each class, define class-specific models using each of these spaces, and then apply any unknown sample to all of these models in order to assess class membership. 

Figure 2. Two linear clusters, showing starting class membership assignments.

This criterion for selection of features leads to a set of descriptors that contain optimal Information about class membership as (opposed to information about class differences. 

In contras to unsupervised methods, supervised pattern-recognition methods (Section 4.3) use class membership information in the calculations. The goal of these methods is to construct models using analytical measurements to predict class membership of future samples. Class location and sometimes shape are used in the calibration step to construct the models. In prediction, these moddsare applied to the analytical measurements of unknowu samples to predict dsss membership. 

It is used to examine the similarities and differences between samples without imposing a priori information regarding class membership. 

Same description as Dendrogram without class labels above except the class membership is included. 

The preprocessed data and class membership information is submitted to the analysis software. Euclidean distance and leave-one-out cross-validation is used to determine the value for K and the cutoff for G. 

Supervised versus Unsupervised Pattern Recognition In some situations the class membership of the samples is unknown. For example, an analyst may simply want to examine a data set to see what can be learned. Are there any groupings of samples Are there any outliers (i.e., a small number of samples that are not grouped with the majority) Even if class information is known, the analyst may want to identify and display natural groupings in the data without imposing class membership on the samples. For example, assume a series of spectra have been collected and the goal is to... 

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