Geoscience Reference
In-Depth Information
“chronological series analysis”. A time series is denoted by Yt, {with t in Dt}. The
analysis of time series is a classical field that has given rise to numerous methods
seeking to decompose the series to identify its different components and make
forecasts. They organize themselves into families whose genealogy dates back to the
autoregressive (AR) models (models where Yt is a linear function of Yt-1
and Yt-2) and to the mobile averages over time (MA) [GOU 95]. The models
are differentiated according to whether they process states with continuous values
(family of models based on AR-MA logics), or they deal with discrete
states (Markov chains). Without giving much detail about the different families, it is
interesting to note the difference in perspective that these models propose: time is
“unfolded” and oriented there; a state at time t is explained by the prior states. But in
this “unfolded time” perspective, a date (t) has a specific status compared to earlier
dates. It thus differs from the “snapshot” approach where each date t has the same
status and is interpreted in its succession. It is to be noted that these families of
models have also integrated the spatial dimension: a state at time t may be explained
by the prior states, and most often for geographical issues, the prior states of the
entities in the neighborhood.
To finish this panorama, we develop the example of a model based on Markov
chains, a model where the states space is discrete. This method is particularly
suitable to model the process of change in land cover, and therefore complements
the examples presented above. The example that serves as a support here refers to a
prospective analysis framework of the urban sprawl around the city of Belfort (east
of France) developed by Antoni [ANT 06]. The data on which the model is based
are of
field
type: the land cover in 13 categories at three dates 1955, 1975 and 1995,
around the city of Belfort. They have been explicitly produced to be comparable.
They have allowed calculating that the city has spread-out by 62% over the period,
growth that is rather concentrated on the second period.
The principle of a Markov chain is based on the property that the future
evolution solely depends on the current value (the present), with the hypothesis that
the effect of the past on the future is summarized in this present state. In such a
process, the probability that a cell is in a certain state at t+1 therefore depends on its
status at t (Markov chain). To define a Markov process, the whole set of transition
probabilities between two states must be assessed. It refers to the conditional
probabilities p
k,k'
to move from the state k to the state k':
p
k,k′
= P(E
i,t+1
=k/E
i,t
=k′)
where E
i,t+1
is the state of the cell i at time t+1, and k and k′ are two modalities
among the set of possible modalities (that is for this example, the 13 categories of
land occupation).
