Compositional Loess modeling

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 is that for an arbitrary number of explanatory data points x_i the value of a dependent variable is estimated ŷ_i. The ŷ_i is the fitted value from a dth degree polynomial in x_i. (In practice often d = 1.) The ŷ_i is fitted using weighted least squares, WLS, where the points x_k (k = 1, ..., n) closest to x_i are given the largest weights.

We define a weighted least squares estimation for compositional data, C-WLS. In WLS the sum of the weighted squared Euclidean distances between the observed and the estimated values is minimized. 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 is assumed to be explained by a real valued (multivariate) variable. For an arbitrary number of data points x_i we for each x_i fit a dth degree polynomial in x_i yielding an estimate ŷ_i of the composition y_i. We use C-WLS to fit the polynomial giving the largest weights to the points x_k (k = 1, ..., n) closest to x_i.

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 the data.