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After we conclude the distribution of the sample is not a Gamma distribution, in
this section we tested if the sample comes from a Weibull Distribution.
Remember that the probability density function of 2-parameterWeibull distribution
with shape k and scale is
f (x) =
k
x
k−1
e−(x/)k
, x 0, k > 0, > 0
The corresponding cumulative distribution function is
F(x) = 1 − e−(x/)k
As stated in Seciton A.3 the MLE estimators,ˆk,ˆ of the shape and the scale,respectively,
are
1
k = Pni
=1 xk
i log xi
Pni
=1 xk
i −
1
n
nX
i=1
log xi (2.1)
ˆ
=0BB@Pn
i=1 xk
i
n 1CCA
1/k
(2.2)
Applying the above formulas in the statistical software R, one gets
ˆk
= 3.445772 (2.3)
ˆ
= 46.0154 (2.4)
27
Lets first look at the q-q plot of the sample quantiles and the estimated Weibull
quantiles. Figure 2.8 shows clearly that the distribution of the sample is notWeibull.
Next, we apply the optimization approach with Weibull Distribution. In this case,
Figure 2.8: Quantile-Quantile Plot of the Theoretical Weibull Quantiles with MLE
parameters and the Sample Quantiles
the minimization problem becomes
For 2 {0.1, 0.2, . . . , 0.9}
Given quantiles μ
min
s,k,
d =X
(μ − s − F−1( , k, ))2
subject to:
s 0, k 0, 0
F cumulative Weibull distribution function.
28
We used the statistical software R for this simple optimization problem. You can
find the corresponding R code below:
>p<-function(x) kuantile(x,seq(0.1, 0.9, 0.1))
>i<-seq(0.1,0.9,0.1)
>fh<-function(theta,x)sum((p(x)-theta[1]-qweibull(i,shape=theta[2],
>+scale=theta[3],log=F))**2)
>theta.start<- c(1,1,1)
>out<-function(x)nlm(fh,theta.start,x=x)
The above code is the same as defined for Gamma distribution. We only change R
function qgamma to qweibull to find the theoreticalWeibull distribution quantiles.
Figure 2.9 shows the fit. Now, one would like to see how good this fit is; therefore,
Figure 2.9: Weibull Distribution Fit by Optimizing the Distance between Theoretical
and Sample Quantiles
29
we applied 2 Goodness of Fit Test with
H0 = The censored part of the sample comes from a Weibull distribution with the
estimated parameters.
H1 = The distribution of the censored part of the sample is not Weibull with the
estimated parameters.
Applying the defined 2 formula
2 =
nX
i=1
(Oi − Ei)2
Ei
= 271.4117
The corresponding p value is very close to 0 with 7 degrees of freedom. Hence we
conclude that the censored observations do not follow Weibull distribution.
Since the determination of the distribution of the ground handling process is not
straightforward, we suspect it has a mixed distribution. Examining the distribution
and afterwords conducting the forecasting process are time consuming. Instead,
one can predict each quantile for the next operation plan period using the historical
sample quantiles. For this procedure, there are many methods, parametric and
non-parametric, such as linear, quantile, non-parametric regression or exponential
smoothing. In this paper, we first analyse the properties of the quantile time-series
we have, then describe some of the prediction methods above mentioned and at the
end develop a robust recursive nonparametric curve estimatior in order to estimate
the regression function.
30
Chapter 3
FORECASTING
Many business and economics data are non-stationary and may contain trend and/or
seasonal variations. Periodic and recurrent pattern seasonality may be caused by
weather, holidays, or repeating promotions. A stochastic trend is often accompanied
with the seasonal variations and can have a significant impact on various
forecasting methods. In this section, we will compare some forecasting methods
for time-series with and without seasonal and/or trend patterns. These methods are
Regression, Exponential Smoothing Algorithms and Robust Recursive Nonparametric
Curve Estimation. We aim to find the effects of seasonality on our data and
we study on the efficiency of the forecasting performance, in particular to answer
if a forecasting method concerning trend and/or seasonality yields better forecasts
for the ground handling process time data.
We are interested in forecasting the quantiles of ground handling process time data;
therefore, our series will be constructed by 10% to 90% quantiles of the data from
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