Mathematical Sciences: Methods for Smoothing Bivariate, Irregularly Spaced Data

摘要该项目的研究将开发具有地理定义坐标的双变量、不规则间隔数据（如环境、地质或健康相关数据）的平滑方法。因为这些数据通常来自具有潜在非平稳噪声的非高斯分布（例如，大气压力数据中的高度偏斜值，地球物理数据中地震引起的奇异值，地质数据中断层引起的不连续），线性平滑或假设平稳白噪声导出的平滑，在捕捉潜在趋势方面可能不如更稳健的非线性平滑。本研究将试图确定(1)非线性与线性平滑器应该产生良好性能的情况；(2)边界处的平滑值应该如何定义（通常是估计趋势的重大偏差的来源）；(3)如何显示得到的平滑趋势。1974年，美国国家癌症研究所出版了《1950-1969年美国各县癌症死亡率地图集》。它由美国的地图组成，每35个癌症地点各一张，其中各县根据所描绘的癌症地点的死亡率水平被阴影化。例如，这些地图表明，墨西哥湾附近肺癌发病率高特拉华州和新泽西州附近膀胱癌发病率高；随后，确定了这些高比率的环境原因。癌症死亡率的进一步地图集已经出版，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。由于一些县人口很少，报告的死亡率非常不确定；风险较高的地区可能难以发现。这项研究的目的是开发一种方法，以突出美国各县癌症死亡率等数据中的地理模式。人口的地理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县，还要确定可能表明环境问题的广泛地区。相反，低风险地区可以作为疾病预防和控制措施的样板。这些方法可以应用于其他类型的数据，以回答类似的问题，例如哪些地区表明有明显的地震活动，或者在哪些地方臭氧层消耗得最快。

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.