The classification of large dimensional data sets arising from the merging of remote sensing data with more traditional forms of ancillary data causes a significant computational problem. Decision tree classification is a popular approach to the problem. This type of classifier is characterized by the property that samples are subjected to a sequence of decision rules before they are assigned to a unique class. If a decision tree classifier is well designed, the result in many cases is a classification scheme which is accurate, flexible, and computationally efficient. This correspondence provides an automated technique for effective decision tree design which relies only on a priori statistics. This procedure utilizes canonical transforms and Bayes table look-up decision rules. An optimal design at each node is derived based on the associated decision table. A procedure for computing the global probability of correct classification is also provided. An example is given in which class statistics obtained from an actual Landsat scene are used as input to the program. The resulting decision tree design has an associated probability of correct classification of 0.75 compared to the theoretically optimum 0.79 probability of correct classification associated with a full dimensional Bayes classifier. Recommendations for future research are included.

An Automated Approach to the Design of Decision Tree Classifiers

A survey of the theory of fixed points of mappings of metric spaces, containing some unpublished results of the author, is presented.

Fixed points of mappings of metric spaces

In this research, feature extraction and classification algorithms for high dimensional data are investigated. Developments with regard to sensors for Earth observation are moving in the direction of providing much higher dimensional multispectral imagery than is now possible.
In analyzing such high dimensional data, processing time becomes an important factor. With large increases in dimensionality and the number of classes, processing time will increase significantly. To address this problem, a multistage classification scheme is proposed which reduces the processing time substantially by eliminating unlikely classes from further consideration at each stage. Several truncation criteria are developed and the relationship between thresholds and the error caused by the truncation is investigated.
Next a novel approach to feature extraction for classification is proposed based directly on the decision boundaries. It is shown that all the features needed for classification can be extracted from decision boundaries. A novel characteristic of the proposed method arises by noting that only a portion of the decision boundary is effective in discriminating between classes, and the concept of the effective decision boundary is introduced. The proposed feature extraction algorithm has several desirable properties: (1) it predicts the minimum number of features necessary to achieve the same classification accuracy as in the original space for a given pattern recognition problem; (2) it finds the necessary feature vectors. The proposed algorithm does not deteriorate under the circumstances of equal means or equal covariances as some previous algorithms do. In addition, the decision boundary feature extraction algorithm can be used both for parametric and non-parametric classifiers.
Finally, we study some problems encountered in analyzing high dimensional data and propose possible solutions. We first recognize the increased importance of the second order statistics in analyzing high dimensional data. By investigating the characteristics of high dimensional data, we suggest the reason why the second order statistics must be taken into account in high dimensional data. Recognizing the importance of the second order statistics, there is a need to represent the second order statistics. We propose a method to visualize statistics using a color code. By representing statistics using color coding, one can easily extract and compare the first and the second statistics.

/pdf/feature-extraction-and-classification-algorithms-for-high-35eoz5denz.pdf

Feature extraction and classification algorithms for high-dimensional data

/pdf/an-automated-approach-to-the-design-of-decision-tree-52imn6efuf.pdf

Linear dimension reduction and Bayes classification with unknown population parameters

Linear feature selection with applications

Distance preserving linear feature selection

A mapping f from a metric space (X,d) into itself is said to be contractive if x≠y implies d(f(x),f(y))<d(x,y). Theorems of Edelstein [2] state that a contractive selfmapfofa metric space X has a fixed point if, for some x 0, the sequence {fn(x0)} of iterates at x0 has a convergent subsequence; moreover, the sequence {fn(x0)} converges to the unique fixed point of f. Nadler [3] observes that, from the point of view of applications, it is usually as difficult to verify the condition (for some x0 …) as it is to find the fixed point directly.

Extensions of contractive mappings and Edelstein's iterative test

Many known results on the existence of fixed or periodic points of self-maps of a metric space are simple consequences of the following lemma: Lemma. Let (X, $d$) be a metric $sPace$ and let $f:X\rightarrow X$ be a continuous maPping, and consider $\phi.(x, y)=d(f^{n}(x),f^{n}(y))$ . If (1) there exist $x_{0},$ $y_{0}$ with $f^{n_{i}}(x_{0})\rightarrow u,$ $f^{n_{i}}(y_{0})\rightarrow v$ , and (2) $L=\lim_{n}\phi_{n}(x_{0}, y_{0})$ exists, then $\phi_{m}(u, v)=L$ for each $m\geq 0$ . Proof. For each $m\geq 0,$ $f^{m}$ is continuous; also $\lim_{i}\phi_{n_{i}+m}(x_{0}, y_{0})=L$ . Hence

L. F. Guseman

Papers

Linear feature selection with applications

Distance preserving linear feature selection

Extensions of contractive mappings and Edelstein's iterative test

Sequential conditions for fixed and periodic points