Mathematical Sciences: Methods for Smoothing Bivariate, Irregularly Spaced Data

9510435卡法达尔摘要 该项目下的研究将开发用于平滑双变量，不规则间隔数据的方法，例如环境，地质或与地理定义的坐标有关的健康数据。因为这样的数据通常来自具有潜在非平稳噪声的非高斯分布（例如，气压数据中的高度偏斜值、由于地球物理数据中的地震而引起的奇异值、由于地质数据中的故障而引起的不连续性）、线性平滑器或假定平稳白色噪声而导出的平滑器在捕获潜在趋势时可能不能执行得与更鲁棒的非线性平滑器一样好。 本研究将尝试确定（1）非线性平滑器与线性平滑器应产生良好性能的情况;（2）应如何定义边界处的平滑值（通常是估计趋势时的显著偏差的来源）;以及（3）如何显示所产生的平滑趋势。 1974年，美国国家癌症研究所出版了《美国各县癌症死亡率地图集：1950-1969》。 它由美国地图组成，每个地图代表35个癌症部位，根据所描绘的癌症部位的死亡率水平，县被涂上阴影。 例如，这些地图显示了墨西哥湾周围肺癌的高发病率以及特拉华州和新泽西周围膀胱癌的高发病率;随后，这些高发病率的环境原因被确定。 更多的癌症死亡率地图集已经出版，国家卫生统计中心即将发布癌症以外原因的死亡率地图集。 由于一些县的人口非常少，报告的死亡率非常不确定;风险升高的地区可能难以发现。这项研究的目的是开发方法，突出数据中的地理模式，如美国各县的癌症死亡率。人口的地域流动往往是造成风险扩散的原因。 因此，重要的是不仅要确定风险较高的孤立国家，而且要确定可能表明引起关注的环境原因的广泛区域。 相反，低风险区域可作为疾病预防和控制措施的模式。 这些方法可以应用于其他类型的数据，以回答类似的问题，例如哪些地区显示出重大的地震活动，或者哪些地方的臭氧层消耗最快。

9510435 Kafadar Abstract The research under this project will develop methods for smoothing bivariate, irregularly spaced data such as environmental, geological, or health-related data with geographically-defined coordinates. Because such data often arise from non-Gaussian distributions with potentially non-stationary noise (e.g., highly skewed values in barometric pressure data, exotic values due to earthquakes in geophysical data, discontinuities due to faults in geological data), linear smoothers, or smoothers derived assuming stationary white noise, may not perform as well as more robust, nonlinear smoothers in capturing the underlying trend. This research will attempt to identidy (1) those situations where nonlinear versus linear smoothers should yield good performance; (2) how smoothed values at the border should be defined (often a source of significant bias in estimating the trend); and (3) how the resulting smoothed trend can be displayed. In 1974, the National Cancer Institute published the Atlas of Cancer Mortality for U.S. Counties: 1950-1969. It consisted of U.S. maps, one for each of 35 sites of cancer, where counties were shaded according to the level of the mortality rate for the cancer site being depicted. These maps illustrated, for example, high rates of lung cancer around the Gulf of Mexico and high rates of bladder cancer around Delaware and New Jersey; subsequently, environmental causes for these high rates were identified. Further atlases of cancer mortality were published, and an atlas of mortality from causes other than cancer is forthcoming from National Center for Health Statistics. Because some counties have very small populations, reported mortality rates are very uncertain; regions of elevated risks may be difficult to detect. The objective of this research is to develop methods which will highlight geographical patterns in data such as cancer mortality in U.S. counties. Geographical movement of populations often is responsible for spreading the risk around. Thus it is important to identify not just isolated counties of elevated risk but broad regions which may indicate environmental causes for concern. Conversely, regions of low risk may serve as models for measures of disease prevention and control. These methods can be applied to other sorts of data to answer similar questions, such as which regions indicate significant seismic activity, or in which places the ozone layer is depleting most rapidly.