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Compositional Loess modeling

Bergman, Jakob LU and Holmquist, Björn LU (2011) CoDaWork'11 In Proceedings of the 4th International Workshop on Compositional Data Analysis
Abstract
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... (More)
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. (Less)
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author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
Proceedings of the 4th International Workshop on Compositional Data Analysis
editor
Egozcue, J.J.; Tolosana-Delgado, R. and Ortego, M.I.
pages
11 pages
conference name
CoDaWork'11
ISBN
978-84-87867-76-7
language
English
LU publication?
yes
id
c5739588-81c9-477d-b33e-fb659c734d3b (old id 1963153)
alternative location
http://congress.cimne.com/codawork11/Admin/Files/FilePaper/p26.pdf
date added to LUP
2011-05-24 10:35:15
date last changed
2016-04-16 11:29:10
@misc{c5739588-81c9-477d-b33e-fb659c734d3b,
  abstract     = {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&lt;sub&gt;i&lt;/sub&gt; the value of a dependent variable is estimated ŷ&lt;sub&gt;i&lt;/sub&gt;. The ŷ&lt;sub&gt;i&lt;/sub&gt; is the fitted value from a dth degree polynomial in x&lt;sub&gt;i&lt;/sub&gt;. (In practice often d = 1.) The ŷ&lt;sub&gt;i&lt;/sub&gt; is fitted using weighted least squares, WLS, where the points x&lt;sub&gt;k&lt;/sub&gt; (k = 1, ..., n) closest to x&lt;sub&gt;i&lt;/sub&gt; are given the largest weights. <br/><br>
<br/><br>
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. <br/><br>
<br/><br>
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&lt;sub&gt;i&lt;/sub&gt; we for each x&lt;sub&gt;i&lt;/sub&gt; fit a dth degree polynomial in x&lt;sub&gt;i&lt;/sub&gt; yielding an estimate ŷ&lt;sub&gt;i&lt;/sub&gt; of the composition y&lt;sub&gt;i&lt;/sub&gt;. We use C-WLS to fit the polynomial giving the largest weights to the points x&lt;sub&gt;k&lt;/sub&gt; (k = 1, ..., n) closest to x&lt;sub&gt;i&lt;/sub&gt;.<br/><br>
<br/><br>
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.},
  author       = {Bergman, Jakob and Holmquist, Björn},
  editor       = {Egozcue, J.J. and Tolosana-Delgado, R. and Ortego, M.I.},
  isbn         = {978-84-87867-76-7},
  language     = {eng},
  pages        = {11},
  series       = {Proceedings of the 4th International Workshop on Compositional Data Analysis},
  title        = {Compositional Loess modeling},
  year         = {2011},
}