Without explicit reference to these conditions the statio-
narity of ARMA processes is declared if the stationarity
condition is satisfied: Every ARMA process is stationary if
all zeros of A(z), called the poles of the ARMA process,
are placed inside the unit circle of the complex plane.
Non-stationarity is often caused by a trend function in the
time series which may be determined and removed. A trend
can be removed, for instance, by using derivatives of the
time series /4/. A time series is called an autoregressive
integrated moving average (ARIMA) process of order (p,d.q),
if the d-th derivatives of the series are stationary and
can be described by an ARMA (p,q) process.
3. Further Representations of ARMA Processes
The theory of ARMA processes is related to other concepts
which are used for describing and analyzing time series.
Especially the linear filtering technique, the spectral
analysis and the methods using autocorrelation functions
may be considered as further representations of ARMA
processes. These relations are derived in the following.
3.1 Linear Filtering
Linear filtering techniques describe the output: x(t) of a
linear filter by convoluting the input e(t) with the
impulse response of the filter ht).
x(t) = h(t) % e(t) = Z h(i)-e(t-1i)
Transforming this equation from the time/space domain into
the complex z-domain applying the z-transform, the
convolution operation becomes a simple multiplication
X(z).-.H(z) -.E(z)
In terms of linear filter technique an ARMA process may be
considered as the output of a linear feedback-feedfront
system with a white noise process as input /14/. The trans-
formed impulse response, called the transfer function, is
described by the polynomials in z of the ARMA process
H(z) = D(z) / Az)
3.2 Spectral Analysis
Many applications of time series analysis are based on
spectral analysis. A function x(t) defined in the time
domain can be transformed in a function X(u) defined in the
frequency domain by applying the Fourier transformation
XK(u):;- FU x(tc) )
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