DUGi

Compositional Loess Modelling

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Cleveland (1979) is usually credited with the introduction of the locally weighted regression,Loess. The concept was further developed by Cleveland and Devlin (1988). The general idea isthat for an arbitrary number of explanatory data points xi the value of a dependent variable isestimated ˆyi. The ˆyiis the fitted value from a dth degree polynomial in xi. (In practice oftend = 1.) The ˆyiis fitted using weighted least squares, WLS, where the points xk (k = 1, . . . , n)closest to xi are given the largest weights.We define a weighted least squares estimation for compositional data, C-WLS. In WLS thesum of the weighted squared Euclidean distances between the observed and the estimated values isminimized. In C-WLS we minimize the weighted sum of the squared simplicial distances (Aitchison,1986, p. 193) between the observed compositions and their estimates.We then define a compositional locally weighted regression, C-Loess. Here a composition isassumed to be explained by a real valued (multivariate) variable. For an arbitrary number of datapoints xi we for each xi fit a dth degree polynomial in xi yielding an estimate ˆyi of the compositionyi. We use C-WLS to fit the polynomial giving the largest weights to the points xk (k = 1, . . . , n)closest to xi.Finally the C-Loess is applied to Swedish opinion poll data to create a poll-of-polls time series.The results are compared to previous results not acknowledging the compositional structure of thedata

Universitat de Girona. Departament d’Informàtica i Matemàtica Aplicada