ARIMA PROCESSES FOR MODELLING DIGITAL TERRAIN PROFILES
Joachim Lindenberger
Stuttgart University
Stuttgart, Federal Republic of Germany
Abstract
The paper presents the theory of a particular type of
stochastic time series, the autoregressive integrated
moving average (ARIMA) processes and shows their
application in modelling digital terrain profiles.
1. Introduccion
In this paper a class of stochastic processes, known as the
autoregressive integrated moving average (ARIMA) processes
is presented and their application to modelling digital
terrain profiles is discussed. ARIMA processes represent a
general class of stochastic processes. It will be shown
that other concepts for describing stochastic processes
like those operating with spectral analysis or with auto-
covariance functions are related with them and can be
derived from them.
The paper outlines the theory of ARIMA processes in chapter
2 and 3. It follows a discussion of the problems of process
identification in chapters 4 and 5. Finally ARIMA processes
will be applied for modelling digital terrain profiles
(chapter 6). ' ;
2. Definition of ARIMA Processes
A set of observations made sequentially in time or
equidistantly in space can be regarded as a time series. A
discrete time series
(x_) = {x(1), x(2), .., x(N))
denotes a set of observations generated by sampling a
continuous signal x(t) at equidistant intervals. .^ A time
series is said -to be deterministic if future values of the
series can be exactly determined by some mathematical
functions. The future values of a statistical time series -
a so-called stochastic process - can be described only in
terms of a probability distribution. Typical time series
are neither completely deterministic nor completely random.
The concept of modelling a time series by an autoregressive
moving average (ARMA) process contains both random and
non-random elements.
In addition to deterministic terms the stochastic proper-
ties of an ARMA process are resulting from a random uncor-
related innovation series (e,). It is thereby assumed that
the innovation series (e,) is a white Gaussian noise
- 427 -
