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year 2006 to 2008. Each year is divided into 4 operation plan periods, so we have
12 observations for each quantile.
First of all, in order to see if there is a trend/season effect on our data, let’s examine
the quantile behaviours. The next section explains the methods to analyze a
time series and applications.
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3.1 METHODOLOGICAL TOOLS FOR ANALYZING
TIME-SERIES DATA
Although our observations do not have time-series characteristics, if we calculate
the  quantiles,  2 {0.1, 0.2, . . . , 0.9}, for each operation plan period for the same
subgroup, we have time-series data constructed by the  quantiles of our samples.
There are certain methods which are used to determine the statistical properties of
time-series and therefore gives an insight about the appropriate model that can be
applied to the data. In this section, we will concentrate on those methods. Denote
the time-series by Y = Yt = {Y1, Y2, . . . , Yn}.
3.1.1 Plotting the Data
A graphical plot of the data is a classical starting point which gives the first insight
about the data. It visually helps to identify the existence of the trends and the
seasonality.
3.1.2 The Autocorrelation Coefficient
The auto-correlation coefficient is the key statistics in the time-series analysis which
indicates the relation between the values in the time-series Y.
The first-order autocorrelation coefficient is the simple correlation coefficient of
the first n − 1 observations, Yt, t = 1, 2, . . . , n − 1 and tehe next n − 1 observations,
Yt, t = 2, 3, . . . , n. The correlation between Yt and Yt+1 is given by
r1 = Pn−1
t=1 (Yt − ¯Y(1))(Yt+1¯Y(2))
qPn−1
t=1 (Yt − Y(¯1))2 qPn−1
t=1 (Yt − ¯Y(2))2
where ¯Y(1) is the mean of the first n − 1 observations and ¯Y(2) is the mean of last
n − 1 observations. For sufficiently large n, assuming the stationarity in the mean,
one can simplify and generalize the formula, for 1, 2, . . . , k time lags, as follows:
rk = Pn−k
t=1 (Yt − ¯Y )(Yt+k − ¯Y)
Pnt
=1 (Yt − ¯Y )2
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where rk = rYtYt−k , and ¯Y = Pnt
=1 Yt is the overall mean.The plot of the autocorrelation
function as a function of lag is also called the correlogram.
In order to investigate the properties of an empirical time-series, the autocorrelation
coefficient is a useful tool. In theory, if we assume an infinite sample, all
autocorrelation coefficients of a series with random numbers should be 0. In practise,
this situation is not observed. However, we can determine the properties of
the distribution of the autocorrelation coefficients of a series.
As Anderson (1942), Bartlett (1946), and others showed, the autocorrelations of
random data have a sampling distribution that can be approximated by a normal
curve with mean 0 and standard error 1/pn. One can use this information to develop
tests similar to F-test or t-test and determine the confidence intervals for
which the values of the series are random. For this purpose, the area under the
normal curve is used.
3.1.3 The Periodogram and Spectral Analysis
Another way to analyse a time-series is to decompose it into a set of sine waves
of different frequences. This method can be useful to identify the randomness and
seasonality in the time-series.
In a discrete time-series, because there are noangles to deal with, one uses the time
units instead of wavelength. As stated in Makridakis, Wheelwright and McGee
(1983), any time-series, composed of n equally spaced observations, can be decomposed
by least-squares fitting into a number of sine waves of given frequency,
amplitude, and phase, subject to the following conditions:
• if n is an odd number, a maximum of (n − 1)/2 sine waves can be fitted.
• if n is an even number, a maximum of (n − 2)/2 sine waves can be fitted.
This method was originally known as periodogram analysis (Schuster, 1898) and
is variously known as harmonic analysis, Fourier analysis, or spectarl analysis. It
helps to identify
• randomness in the data,
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• seasonality in a time series,
• the predominance of positive or negative autocorrelation (nasil yani? and
eee).
3.1.4 The Partial Autocorrelation Coefficient
The partial autocorrelation at lag k is the autocorrelation between Yt and Yt−k that
is not accounted for by lags 1 through k − 1.
Partial autocorrelations are used to measure the degree of association between Yt
and Yt−k, when the effects of other time lags - 1, 2, . . . , k−1 - are somehow partialled
　
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