Mathematical Sciences: Adaptive Spatial Regression and Classification

数学科学：自适应空间回归和分类

• 批准号: 9403804
• 负责人: Jerome Friedman
• 金额: $ 11万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9403804&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Adaptive; Spatial; Regression

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.

9403804弗里德曼预测是应用最广泛的统计程序之一。其目的是在给定与同一对象关联的另一组属性{x(1)，...，x(N)}(预测器变量)的同时值的情况下，预测(估计)与对象(观测)关联的一个或多个属性{y(1)，...，y(Q)}(“响应”变量)的值(S)。预测规则是从一组已经测量了所有属性(预测值和响应值)的“训练”观测中得出的。当响应变量采用真实(可排序)值时，预测问题称为“回归”。当一个(单一的)响应具有K个无序的类别值时，这个问题被称为“分类”。在这种情况下，响应值可以被视为将观测分配给K个组或类中的一个的标签，每个类与标签值之一相关联。在这一建议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)}由指数t标记，该指数t在d维欧氏空间中呈现实值，并且将这些变量之间的距离与其相应索引值之间的距离相关联并不是不自然的。在这些应用中，所有预测器变量通常是对指数t的不同值的相同量度的测量。当t是一维时，t的变化值的{x(T)}称为“信号”或“谱”。同样，二维指数会产生一种“模式”，而三维或更高的维度则会产生更一般的“空间模式”。这项研究的目标是推导出比过去(用线性方法)更一般(也更强大)地利用预测变量指数的空间性质的方法。这些新方法将基于自适应(回归)样条线，这是线性Smoot Hing到非线性灵活建模的最有前途的扩展之一，特别是在更高的维度。如果成功，结果将是一类更通用的空间预测程序，在许多情况下将实现比目前的线性方法更高的精度。根据这项拨款提议的研究的目的是为计算机自动模式识别开发新的、更强大的方法。接收诸如信号、光谱或图像之类的图案，其目的是预测产生该图案的特定(未知)系统的身份，或该系统的某些属性。例如，通过EKG或EEG测量进行疾病诊断，从数字化的语音电子模式识别特定的单词(或说话者)，从其数字化图像识别印刷或手写字符，或从卫星图像识别地面物体。待开发的方法是根据从过去成功解决的案例中吸取的经验而制定的。该方法根据过去的经验提出了一系列模式，并对每一种模式给出了相应的正确答案。使用该数据，该方法试图自动学习用于预测正确答案未知的未来模式的规则。本研究致力于开发比目前使用的模式识别学习方法更具灵活性和适应性的模式识别学习方法。如果成功，这项研究应该会产生新的程序，为许多应用提供比过去可以实现的更高的预测精度。