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| Titel |
Interpretation of a compositional time series |
| VerfasserIn |
R. Tolosana-Delgado, K. G. van den Boogaart |
| Konferenz |
EGU General Assembly 2012
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| Medientyp |
Artikel
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| Sprache |
Englisch
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| Digitales Dokument |
PDF |
| Erschienen |
In: GRA - Volume 14 (2012) |
| Datensatznummer |
250066002
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| Zusammenfassung |
| Common methods for multivariate time series analysis use linear operations, from the
definition of a time-lagged covariance/correlation to the prediction of new outcomes.
However, when the time series response is a composition (a vector of positive components
showing the relative importance of a set of parts in a total, like percentages and
proportions), then linear operations are afflicted of several problems. For instance, it
has been long recognised that (auto/cross-)correlations between raw percentages
are spurious, more dependent on which other components are being considered
than on any natural link between the components of interest. Also, a long-term
forecast of a composition in models with a linear trend will ultimately predict negative
components.
In general terms, compositional data should not be treated in a raw scale, but after a
log-ratio transformation (Aitchison, 1986: The statistical analysis of compositional data.
Chapman and Hill). This is so because the information conveyed by a compositional data is
relative, as stated in their definition. The principle of working in coordinates allows to apply
any sort of multivariate analysis to a log-ratio transformed composition, as long as
this transformation is invertible. This principle is of full application to time series
analysis.
We will discuss how results (both auto/cross-correlation functions and predictions) can be
back-transformed, viewed and interpreted in a meaningful way. One view is to use the
exhaustive set of all possible pairwise log-ratios, which allows to express the results into
D(D - 1)-2 separate, interpretable sets of one-dimensional models showing the behaviour of
each possible pairwise log-ratios. Another view is the interpretation of estimated coefficients
or correlations back-transformed in terms of compositions. These two views are compatible
and complementary.
These issues are illustrated with time series of seasonal precipitation patterns at different
rain gauges of the USA. In this data set, the proportion of annual precipitation falling in
winter, spring, summer and autumn is considered a 4-component time series. Three invertible
log-ratios are defined for calculations, balancing rainfall in autumn vs. winter, in summer vs.
spring, and in autumnÃwinter vs. springÃsummer. Results suggest a 2-year correlation
range, and certain oscillatory behaviour in the last balance, which does not occur in the other
two. |
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