Constrained Nonparametric Inference and Data Visualization through Data Sharpening

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The proposed research seeks to improve upon methods for smoothing data in order to more clearly visualize the information contained therein.  In order to extract useful information from data, it is necessary to explore it graphically, especially in situations where there are many observations.  Kernel smoothing methods provide graphical summaries of data which allow a data analyst to detect patterns and possible relationships between variables, and the analysis is done in such a way that very few assumptions need to be made, allowing the data to "speak for themselves''.    In many circumstances, additional information about the data is available.  For example, the area burned by a forest fire will not decrease over the time that the fire is burning.  Therefore, when modelling the growth of a wildfire, we could use this information in addition to observed measurements.  Standard techniques, without adjustment, usually ignore this information, and the resulting models can sometimes produce anomalies that do not match reality.  A particular focus of the proposed research will be on data sharpening which is a data-adjustment technique where extra information about the data, or the process that gave rise to the data, is exploited.  Standard statistical techniques are then applied to the adjusted data, so that the given information is incorporated, without compromising statistical performance. The plan is to optimize the methodology for smoothing and to launch a full-scale investigation into statistical inference methods that use data sharpening. The latter techniques will lead to improved uncertainty quantification in applications such as wildfire prediction.  Data sharpening falls into two categories: "supervised" and "unsupervised". In the former, data are moved a minimal distance, subject to the extra information being satisfied by the resulting model.  For example, we might want to move fire size measurements minimally, subject to the requirement that the resulting model only predicts that fire size increases over time.  Unsupervised data sharpening does not directly incorporate external information but rather works with properties of the given statistical technique to come up with rules for adjusting the data so that accuracy of the resulting estimate is improved.  A thorough investigation of data sharpening and its variants is overdue, since the methodology was proposed some time ago but not at all fully explored,  and since it holds considerable promise in terms of improving performance of a wide variety of statistical methods.  The methodologies developed in the proposal will also immediately be put to use in important applications including: a study of factors underlying airtanker pilot fatigue, where reaction time data, physiological variables, and environmental variables are studied; problems in financial credit risk; and, as alluded to above,  environmental risk due to phenomena such as wildfire.