Compositional Loess modeling
Author
Editor
- J.J. Egozcue
- R. Tolosana-Delgado
- M.I. Ortego
Summary, in English
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<sub>i</sub> the value of a dependent variable is estimated ŷ<sub>i</sub>. The ŷ<sub>i</sub> is the fitted value from a dth degree polynomial in x<sub>i</sub>. (In practice often d = 1.) The ŷ<sub>i</sub> is fitted using weighted least squares, WLS, where the points x<sub>k</sub> (k = 1, ..., n) closest to x<sub>i</sub> 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<sub>i</sub> we for each x<sub>i</sub> fit a dth degree polynomial in x<sub>i</sub> yielding an estimate ŷ<sub>i</sub> of the composition y<sub>i</sub>. We use C-WLS to fit the polynomial giving the largest weights to the points x<sub>k</sub> (k = 1, ..., n) closest to x<sub>i</sub>.
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.
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<sub>i</sub> we for each x<sub>i</sub> fit a dth degree polynomial in x<sub>i</sub> yielding an estimate ŷ<sub>i</sub> of the composition y<sub>i</sub>. We use C-WLS to fit the polynomial giving the largest weights to the points x<sub>k</sub> (k = 1, ..., n) closest to x<sub>i</sub>.
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.
Department/s
Publishing year
2011
Language
English
Publication/Series
Proceedings of the 4th International Workshop on Compositional Data Analysis
Full text
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Links
Document type
Conference paper
Topic
- Probability Theory and Statistics
Conference name
CoDaWork'11
Conference date
2011-05-10 - 2011-05-13
Conference place
Sant Feliu de Guixols, Girona, Spain
Status
Published
ISBN/ISSN/Other
- ISBN: 978-84-87867-76-7