Maximum likelihood classifie

Bayes- and Maximum Likelihood Classifiers for Binary Encoded Patterns... 

Successful and economical applications of maximum likelihood classifiers have been reported for binary encoded infrared spectra C356T and nuclear magnetic resonance spectra C3573. An extensive examination of a maximum likelihood algorithm for the interpretation of mass spectra was made by Franzen et.al. C86, 87, 1083 and others C200, 2483. Approximation of the probability density by a mathematical function has found up to now only little interest C3173. 

Those features with the largest values of the mutual information are the ones that should be most useful for classification. The square root of the mutual information was found to correlate quite well with the recognition ability of maximum likelihood classifiers in an application to infrared spectra C2443. 

Fig. 44 Maximum likelihood classified image, Kuwait city...

Fig. 47 Maximum likelihood classified image, showing farm in north Kuwait...

In general, several tools have been developed to match reference spectra with those measured by an imaging spectrometer. The most common approach is based on the use of standard supervised classification techniques, where known spectra are used to determine the statistical properties of each class based on spectral characteristics. Examples of supervised classification approaches applied to hyperspectral data are described in McKeown et al. (1999) and Roessner et al. (2001), where the maximum likelihood classifier (MLC) was applied to map urban land cover. Other techniques are based on the use of support vector machines (SVM) (Melgani and Bruzzone 2004) and neural networks (NNs) (Licciardi et al. 2009, 2012). Other approaches have been designed explicitly for the analysis of imaging spectrometry data, such as the Spectral Angle Mapper (SAM Kruse et al. 1993). 

Tel Aviv, Israel. Data acquired by the Digital Airborne Imaging Spectrometer (DAIS) have been used by Roessner et al. (2001) to obtain a map of urban materials in the city of Dresden, Germany. In this study, a maximum likelihood classifier has been used to derive a first map of pure spectral features and then used these feamres to unmix the other spectra. 

Table 2.3 is used to classify the differing systems of equations, encountered in chemical reactor applications and the normal method of parameter identification. As shown, the optimal values of the system parameters can be estimated using a suitable error criterion, such as the methods of least squares, maximum likelihood or probability density function. 

The maximum-likelihood rule classifies an observation x IRp into na if ln(pja(x)) is the maximum of the set In(pfjix)) j = 1,. .., Z. If we assume that the density f for each group is Gaussian with mean pj and covariance matrix Xy, then it can be seen that the maximum-likelihood rule is equivalent to maximizing the discriminant scores df(x) with... 

Assumptions may be made or models adopted (often by implication) about a system being measured that are not consistent with reality. The selection of the method of data reduction may be partly on the basis of the model adopted and partly on the basis of features such as computation time and simplicity. Kelly classified data processing methods as direct, graphical, minmax, least squares, maximum likelihood, and bayesian. Each method has rules by which computations are made, and each produces an estimate (or numerical result) of reality. 

If all classes have equal a priori probabilities, and if the loss functions are symmetrical, the classification of an unknown pattern x needs only the determination of probability densities p(x m) for all classes m. The maximum value gives the class to which x is classified This method is called a maximum likelihood decision. 

Once the parameters of the Gaussian probability density functions for all classes are known, the density at any location can be calculated and an unknown pattern can be classified by the Bayes rule or by the maximum likelihood method. A binary classification with equal covariance matrices for both classes can be reduced in this way to a linear classifier C87, 317, 396D. 

This classification method was first applied to chemical problems by Franzen and Hillig C86, 87, 108D. Although many simplifications have been introduced into this maximum likelihood method a considerable computational effort is necessary for the training and application of such parametric classifiers. However, the effort is much smaller if binary encoded patterns are used (Chapter 5.4). 

To classify the measurement vector v of an unknown pixel into a class, the maximum likelihood decision rule computes the value p for each class. Then it assigns the pixel to the class that has the maximum value. 

The probabUistic model for a naive Bayes classifier is a conditional model P(T Xi, X2,..., X ) over a dependent class variable F, conditional on features Xi, X2, X. Using Bayes s theorem, F(F Xj,..., X ) oc P(F) 7(Xi,..., X F). The prior probability F(F = j) can be calculated based on the ratio of the class j samples such as P(F = 7) = (number of class j samples)/(total number of samples). Having formulated the prior probabihties, the likelihood function p(Xi, X2,..., X F) can be written as ]/[ j p(Xi F) under the naive conditional independence assumptions of the feature X, with the feamre Xj for j i. A new sample is classified to a class with maximum posterior probability, which is argmaxr erF (r7)nr ( i 1 /)- If the independence assumption is correct, it is the Bayes optimal classifier for a problem. Extensions of the naive Bayes classifier can be found in Demichelis et al. (2006). 

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