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out. Their purpose of use, specifically, to identify the order of an autoregressive
model. The sample partial autocorrelation plot is examined to decide on the order
of the model if the sample autocorrelation plot indicates that an autoregressive
model may be appropriate.
Denote the k-th partial correlation by k in autoregressive model:
AR(p) : μ(t|1, . . . , t−1) = 1t−1 + 2t−2 + pt−p
Define the null hypothesis, H0 : k = 0. Then, if rk is the k-th partial autocorrelation
coefficient, then S E(rk)  1/pn and rk is approximately normally distributed.
The approximate 95% confidence interval for the partial autocorrelations are at
(+/−)2/pn. Therefore, we reject H0 if |rk| > 2/pn. If there are p significant partial
autocorrelations, then the order should be AR(p).
To sum up, if the process is autoregressive, i.e, its autocorrelation coefficients decline
to 0 exponentially, the partial autocorrelations can be examined to determine
the order of the process. That order is equal to the number of significant partial
autocorrelations.
3.1.5 Examining Stationarity and Seasonality in a Time-Series
The term stationarity is used when there is no significant growth or decline in the
data, i.e, the data vary around a constant mean, independent of time, and the variance
of the fluctuation remains essentially constant over time.
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The stationarity or the nonstationarity is often recognized by using the plot of the
row data. The autocorrelation plot can also show the nonstationarity quite readily.
The autocorrelation coefficients of nonstationary data are significantly different
from 0 for several time periods, while the ones for a stationary series drop to 0 after
the second or the third time lag.
Positively correlated succesive values indicate that there is a trend (linear or nonlinear)
in the data.
Seasonality is defined as a pattern that repeats itself over fixed intervals of time.
If the pattern is consistent over the fixed intervals of length i, the autocorrelation
coefficient of length i lags will have a high positive value indicating the existence
of seasonality. If the autocorrelation coeffient is not significantly different from 0,
one concludes that there is no consistent pattern one interval of length i to the other,
which indicates the intervals length i apart are unrelated.
3.2 APPLICATIONS WITHKLMGROUND HANDLING
PROCESS
TIMES DATA
In this section, we applied the above mentioned methods to KLM ground handling
process data. First we determined the  quantiles,  2 {0.1, 0.2, . . . , 0.9}. For this
purpose, we used 2 different data sets, one is for Boeing 737-400 type aircrafts
used for intercontinental departure flights from Amsterdam Schiphol Airport and
the other is for Boeing 737-400 type aircrafts used for Europe flights again from
the same airport.
First, we plotted the behaviours of the  quantiles over operation plan periods.
Figure 3.1 and Figure 3.2 belong to the former group with  = 0.1 and  = 0.2,
respectively.
Due to few number of data available, it is difficult to identify if there is an trend
and/or seasonality from the plots of raw data.
Next, we computed the autocorrelation coefficients.
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Figure 3.1: Graphical Representation of The 0.1 Quantiles of GHP Time of Boeing
737-400(ICA) over OP Periods
The following analysis are conducted for the first group. Observe that for  = 0.1,
we have the time-series:
Yt = {79, 81, 82, 81, 79, 79, 81.7, 77, 77, 79, 80, 77}
Hence,
Yt−1 = {81, 82, 81, 79, 79, 81.7, 77, 77, 79, 80, 77},
Yt−2 = {82, 81, 79, 79, 81.7, 77, 77, 79, 80, 77},
Yt−3 = {81, 79, 79, 81.7, 77, 77, 79, 80, 77},
. . .
First we compute the means, μt−k for Yt−k, k = 1, 2, . . . , n.
μt = 79.39167
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Figure 3.2: Graphical Representation of The 0.2 Quantiles of GHP Time of Boeing
737-400(ICA) over OP Periods
μt−1 = 79.42727
μt−2 = 79.27
μt−3 = 78.9667
. . .
Remember the autocorrelation coeeficient formula for 1, 2, . . . , k time lags:
rk = Pn−k
t=1 (Yt − ¯Y )(Yt+k − ¯Y)
Pnt
=1 (Yt − ¯Y )2
Below we apply this formula to Y for k = 1 and k = 2, respectively:
r1 = Pn−1
t=1 (Yt − ¯Y )(Yt+1 − ¯Y )
Pnt
=1 (Yt − ¯Y )2
= 0.1642473
r2 = Pn−2
t=1 (Yt − ¯Y )(Yt+2 − ¯Y )
Pnt
=1 (Yt − ¯Y )2
= −0.1454469
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Figure 3.3: Autoregression Coefficients Graph for 0.1 Quantiles of GHP Times
over Lags
The autocorrelation coefficient graph with respect to number of lags is given in the
　
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