Mathematical Sciences: Adaptive Spatial Regression and Classification

[403804]弗里德曼预测是应用最广泛的统计方法之一。目的是预测（估计）一个或多个属性{y(1)，…，y(q)}（“响应”变量）与一个对象（观察）相关联，给定另一组属性{x(1)，…，x(n)}（“预测器”变量）与同一对象关联。预测规则来源于一组“训练”观测值，其中所有属性（预测器和响应）的值都已被测量。当响应变量假设为实数（有序）值时，预测问题被称为“回归”。当一个（单一）响应有K个无序的分类值时，这个问题被称为“分类”。在这种情况下，响应值可以看作是一个标签，它将观测值分配给K个组或类中的一个，每个类与一个标签值相关联。本提案下的研究解决了预测问题的一个重要子类，其中预测变量{x(t)}被一个指标t标记，该指标t在d维欧几里德空间中取实值，并且将这些变量之间的距离与其对应的指标值之间的距离相关联并不是不自然的。在这些应用中，所有预测变量通常是对指标t的不同值的相同数量的测量。当t是一维时，{x(t)}表示t的不同值称为“信号”或“频谱”。类似地，二维索引产生“模式”，而三维或更高维度的索引产生更一般的“空间模式”。本研究的目标是推导出利用预测变量指数的空间性质的方法，这种方法比过去（用线性方法）所做的更一般（也更强大）。这些新方法将基于自适应（回归）样条，这是线性平滑到非线性柔性建模的最有前途的扩展之一，特别是在高维中。如果成功，结果将是一种更通用的空间预测程序，在许多情况下将比目前的线性方法获得更高的精度。这项研究的目的是开发新的、更强大的计算机自动模式识别方法。接收信号、光谱或图像等模式，目的是预测产生该模式的特定（未知）系统的身份，或该系统的某些特性。例如，从心电图或脑电图测量中进行疾病诊断，从数字化电子语音模式中识别特定单词（或说话者），从数字化图像中识别印刷或手写字符，或从卫星图像中识别地面上的物体。要开发的方法是基于从过去成功破案的经验中学习。该方法给出了一系列的模式，以及从过去的经验中得到的每个模式对应的正确答案。利用这些数据，该方法试图自动学习规则，以预测正确答案未知的未来模式。本研究的重点是开发比目前使用的模式识别学习方法具有更大的灵活性和适应性。如果成功，这项研究将产生新的程序，为许多应用提供比过去更高的预测精度。

9403804 Friedman Prediction is one of the most widely applied statistical procedures. The purpose is to predict (estimate) the value(s) of one or more attributes {y(1),...,y(q)} ("response" variables) associated with an object (observation), given the simultaneous values of another set of attributes {x(1),...,x(n)} ("predictor" variables) associated with the same object. The prediction rule is derived from a set of "training" observations for which the values of all attributes (predictor and response) have been measured. When the response variables assume real (orderable) values the prediction problem is referred to as "regression". When a (single) response takes on K unorderable categorical values the problem is called "classification". In this case the response values can be viewed as a label that assigns the observation to one of K groups or classes, each class associated with one of the label values. The research under this proposal addresses an important subclass of prediction problems in which the predictor variables {x(t)} are labeled by an index t that takes on real values in a d - dimensional Eucludean space, and it is not unnatural to associate a distance between such variables with the distance between their corresponding index values. In these applications all of the predictor variables are generally measurements of the same quanity for different values of the index t. When t is one - dimensional, {x(t)} for varying values of t is call a "signal" or "spectrum". Similarly, a two-dimensional index gives rise to a "pattern", whereas dimensionalities of three or higher give rise to more general "spatial patterns". The goal of this research is to derive methods that exploit the spatial nature of the predictor variable index in more general (and more powerful) ways than have been done (with linear methods) in the past. These new methods will be based on adaptive (regression) splines, which have been among the most promising extensions of linear smoot hing to nonlinear flexible modeling, especially in higher dimensions. If successful, the result will be a more general class of spatial prediction procedures that will achieve higher accuracy in many situations than present day linear methods. The purpose of the research proposed under this grant is to develop new, more powerful, methods for computer automated pattern recognition. Patterns such as signals, spectra, or images are received, and the purpose is to predict the identity of the particular (unknown) system that produced the pattern, or some property of that system. Examples are disease diagnosis form EKG or EEG measurements, recognition of specific words (or speakers) from digitized electronic patterns of spoken speech, recognition of printed or handwritten characters from their digitized images, or identification of objects on the ground from satellite images. The methods to be developed are based on learning through experience from past successfully solved cases. The method is presented with a series of patterns, along with the corresponding correct answer for each one, obtained from past experience. Using this data, the method attempts to automatically learn rules for predicting future patterns for which the correct answer is unknown. This research focuses on developing pattern recognition learning methods that have greater flexibility and adaptibility than those presently in use. If successful, this research should produce new procedures that provide higher prediction accuracy for many applications than has been achieveable in the past.