'Smoothness Properties of Local Polynomial Regression and Its Applications'
Smoothing is a primary tool to explore functional dependence among variables through data. A recent proposed smoothing technique called LOESS (Local Polynomial Regression) by W. S. Cleveland is a breakthrough in the history of smoothing.
It is not only theoretically interesting but also has various successful applications reported in practice. The LOESS is particularly useful for decomposing given time series into several components by its sequential applications to the residual. My recent paper "Analysis of Bird Count Series by Local Regression to Explore Environmental Changes" (submitted to Journal of Japan Statistical Society) with Prof. R. Shibata is one of such successful applications to real data. Each five bird count series observed on a monthly basis are decomposed into three components "long trend", "short trend" and "irregular" by two-step LOESS smoothing. The decomposition explains well the relationship between bird count and environmental changes. In fact, each five long trend very similarly moves with one of environmental factors. Turtledove (Streptopelia orientalis), Browneared Bulbul (Hypsipetes amaurotis) and Great Tit (Parus major) increased its number to link with the enlargement of housing area. On the other hand, Tree Sparrow (Passer montanus) and Gray Starling (Sturnus cineraceus) gradually decreased its number to link with the reduction of field space. The difference between these two groups is perfectly described through the environmental preference of birds. Furthermore, variation of each short trend is corresponding to the effect of breeding season or of winter wandering.
I am now working in theoretical justification of LOESS and development of powerful way of selecting window width. A key idea is "Local Mean Square Error (LoMSE)" which measures how well a smoothing technique worked. A good smoothing technique is to extract any slower movement than the window width but retain any faster movement in the residual. I am now getting a preliminary mathematical theory and expecting that it will develop to a general theory of smoothing beyond LOESS. I believe that my investigation will very much contribute to the COE programme "Analysis of nonlinear phenomena in the framework of data science" by providing a base for this programme.