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CORRELATION AND REGRESSION ANALYSIS OF SPRING STATISTICAL DATA OF
MARITIME FERRY OPERATION PROCESS
M.S. Habibullah, Fu Xiuju
Institute of High Performance Computing, Singapore e-mail: mohdsh@ihpc.a-star.edu.sg
K. Kolowrocki, J. Soszynska
Gdynia Maritime University, Gdynia, Poland e-mail: katmatkk@am.gdynia.pl, joannas@am.gdynia.pl
ABSTRACT
These are presented statistical methods of correlation and regression analysis of the operation processes of complex technical systems. The collected statistical data from the Stena Baltica ferry operation process are analysed and used for determining correlation coefficients and linear and multiple regression equations, expressing the influence of the operation process conditional sojourn times in particular operation states on the ferry operation process total conditional sojourn time.
1 INTRODUCTION
Many real transportation systems belong to the class of complex systems. First, and foremost, these systems are concerned with the large numbers of components and subsystems they are built and with their operating complexities. Modeling of these complicated system operation processes is, first of all, difficult because of the large number of the operation states, impossibility of their precise definition as well as the impossibility of the exact description of the transitions between these states. Generally, the change of the operation states of the system operations processes causes the changes of these systems reliability structures and their components reliability functions. Therefore, the system operation process and its operation states proper definition and accurate identification of the interactions between the particular operation states and their influence on the entire system operation process is very important.
The model of the operation processes of the complex technical systems (Blokus et al. 2008) with distinguishes their operation states is proposed in (Kolowrocki & Soszynska 2008). The semi-markov process (Grabski 2002) is used to construct a general probabilistic model of the considered complex industrial system operation process. To apply this model in practice its unknown parameters have to be identified. Namely, the vector of the probabilities of the system initial operation states, the matrix of the probabilities of transitions between the operation states and the matrix of the distribution functions or equivalently the matrix of the density functions of the conditional sojourn times in the particular operation states, needs to be estimated on the basis of the statistical data. The methods of these unknown parameters evaluation are developed and presented in details in (Kolowrocki & Soszynska 2009A-B). In addition to these methods the simple data mining techniques such as correlation coefficient, linear and multiple regression as well as root mean square error can be used on the statistical data samples to perform the analyses. The results of that analysis as well as relevant conclusions that can be reached from the studies may give practically important information in the operation processes of the complex technical systems investigation.
The aim of this report is to use these techniques in studying the patterns that can be derived, from realizations of the conditional sojourn times, obtained from the Stena Baltica ferry operation process, for the early spring data (Kolowrocki et al. 2009A-B).
The report is organized is the following way. In Section 1, some general comments on complex technical systems operation processes modeling are given and the problem considered in this report is defined. In Section 2, the general assumptions on the complex system operation process are presented. In Section 3, the Stena Baltica ferry operation process is described. In Section 4, the formulae for the total conditional sojourn time its mean and standard deviation are presented and applied to the spring statistical data of the Stena Baltica ferry operation process. This is then followed by determining the correlation coefficient, linear and multiple regression and root mean square error for the ferry operation process spring data. In Section 5, the report summary is given.
2 SYSTEM OPERATION PROCESS
We assume, similarly as in (Blokus et al. 2008, Kolowrocki & Soszynska 2008), that a system during its operation at the fixed moment t, t e <0, + » >, may be in one of v, v e N, different operations states zb, b = 1,2, ..., v. Next, we mark by Z(t), t e <0, + <x> >, the system operation process, that is a function of a continuous variable t, taking discrete values in the set Z =
{z1,z2,...,zv } of the operation states. We assume a semi-markov model (Blokus et al. 2008, Grabski 2002, Kolowrocki & Soszynska 2008) of the system operation process Z(t) and we mark by 6bl its random conditional sojourn times at the operation states zb, when its next operation state is
z,, b, l = 1,2,..., v, b * l.
Under these assumptions, the operation process may be described by the vector [pb (0)]1xv of probabilities of the system operation process staying in particular operations states at the initial moment t = 0, the matrix [pbl (t)]vxv of the probabilities of the system operation process transitions
between the operation states and the matrix [Hbl (t)]vxv of the distribution functions of the conditional sojourn times dbl of the system operation process at the operation states or equivalently by the matrix [hbl (t)]vxv of the density functions of the conditional sojourn times dbl, b, l = 1,2,..., v, b * l, of the system operation process at the operation states.
To estimate the unknown parameters of the system operations process, the first phase in the experiment, is to collect necessary statistical data. This is performed in the following steps (Kolowrocki et al. 2009A-B):
i) To analyze the system operation process and either to fix or to define the following general parameters:
- the number of the operation states of the system operation process v ;
- the operation states of the system operation process z1, z2, ..., zv;
ii) To fix and collect the following statistical data necessary in evaluating the probabilities of the initial states of the system operations process:
- the duration time of the experiment ©;
- the number of the investigated (observed) realizations of the system operation process n (0);
- the numbers of staying operation process respectively in the operations states z1, z2, ..., zv, at the initial moment t = 0 of all n(0) observed realizations of the system operation process n1 (0), n2(0), ..., nv(0), where n1(0) + n2(0)+ nv(0) = n(0);
iii) To fix and collect the following statistical data necessary to evaluating the transient probabilities between the system operation states:
- the numbers nbl, b, l = 1,2,...,v, bW, of the transitions of the system operation process from the operation state zb to the operation state zl during all observed realizations of the system operation process;
- the numbers nb, b = 1,2,...,v, of departures of the system operation process from the
v
operation states zb, where nb = £ nbl;
i=1
iv) To fix and collect the following statistical data necessary in evaluating the unknown parameters of the distributions of the conditional sojourn times of the system operation process in the particular operation states:
- the realizations 0 k , k = 1,2, ..., nbi, b, l = 1,2,...,v, b ^ l, of the conditional sojourn times 0bl of the system operations process at the operation state zb when the next transition is to the operation state zl during the observation time;
After collecting the above statistical data it is possible to estimate the unknown parameters of the system operation process (Kolowrocki & Soszynska 2009A-B). It is also possible to analyze rather accurately the system operation process sojourn times in the particular operation states and their influence on the entire system operation process total sojourn time (Kolowrocki et al. 2009B).
3 STENA BALTICA FERRY OPERATION PROCESS
The problem considered in this report is based on real maritime statistical data, obtained from Stena Baltica ferry operation process, whereby the ferry performs continuous journeys from Gdynia in Poland to Kalskrona in Sweden. Table 1 show the operation states that the Stena Baltica ferry undertakes, beginning with loading at Gdynia then passing through the Traffic Separation Scheme to Karlskrona for unloading/loading and back to Gdynia for unloading/loading. This operation process is repeated continuously and it is assumed that one voyage from Gdynia to Kalskrona and back to Gdynia is a single realization of its operation process. For the voyage described, time-series data were collected for the realization of the conditional sojourn times 0bl of the system operations
process at the operation state zb when the next transition is to the operation state zl for spring
conditions. These data are shown in the Appendix in Tables A1-A4 coming from (Kolowrocki et al. 2009B).
Table 1. Stena Baltica ferry operation states
Operation state Description Operation State Description
z1 Gdynia: Loading Z10 Karlskrona: Unmooring
Z 2 Gdynia: Unmooring zii Karlskrona: Turning
Z 3 Gdynia: Navigating to GD buoy Z12 Karlskrona: Navigating to Angoring buoy
Z 4 Gdynia: Navigating to TSS Z13 Karlskrona: Navigating to TSS
Z 5 Gdynia: Navigating to Angoring buoy Z14 Karlskrona: Navigating to GD buoy
Z 6 Karlskrona: Navigating to Verko berth Z15 Karlskrona: Navigating to Turning Area
Z 7 Karlskrona: Mooring Z16 Gdynia: Ferry Turning
Operation state Description Operation State Description
Z 8 Karlskrona: Unloading Z17 Gdynia: Mooring
Z 9 Karlskrona: Loading Z18 Gdynia: Unloading
It is also important to note that the operation process is very regular and cyclic, in the sense that the operation states changes from the particular state zb, where b = 1,2.... 17 to the neighbouring state
zb+1, where b = 1,2 ... 17 only and from z18 to z1. Therefore, based on this definition the spring realization of the ferry conditional sojourn times 6bb+1, where b = 1,2 ... 17 and 9 x for k = 1,2,..., nbl, where nbl = 42, are given in Tables A1-A4. Also included in Tables A1-A4 are the values of the total conditional sojourn times for each realization, 9T, for k = 1,2,..., nbl, where nbl = 42. In our analysis the values of 9T are important in analyzing the behaviour of the Stena Baltic ferry operation process.
4 DATA ANALYSIS ON STENA BALTICA OPERATION PROCESS
In this section, the use of several data mining techniques on the system total conditional sojourn time is described. The techniques adopted are namely, correlation coefficient, linear and multiple regression and root mean square error. These techniques are applied on the early spring data from the Stena Baltica ferry operation process.
4.1 Total conditional sojourn time
As discussed above, the Stena Baltica ferry operation process data for spring is shown in the Appendix in Tables A1-A4 for spring. In analyzing the behavior of the data patterns, this report examines the ferry total conditional sojourn time (the time length of one ferry voyage) 9T by analyzing its successive realizations 9T, defined as
9T = £ 9bb+1 +91k8 1 (1)
b=1
for k = 1,2,..., nbl, where nbl = 42 for spring data. Using equation (1), the total conditional sojourn
times were then calculated for both spring with the values shown in Tables A1-A4. These values form the basis of our conjecture in this paper.
g
1500
I Total Conditional Sojourn time ----Mean
Number of realizations 20 25
Figure 1. Plot of realizations 9T of total conditional sojourn time 9T for spring data
Figure 1 shows the plot of the realizations 9T of the ferry total conditional sojourn time 9T against the realization number k for spring data. In the picture, by STD there are marked 1-sigma lower 9T - gt and upper 9t + aT bounds for the ferry total conditional sojourn time 9T . Although the ferry operation process is regular and cyclic, i.e. the operation states follows the process in Table 1, it can be observed that the values of 9T are not constant. Furthermore, by using the mean total conditional sojourn time 9T, evaluated from the following equation
0T —
1 nbl
- Z «
nbl k—1
(2)
and the standard deviation defined as
V
1 bl ^ _ _
-Z («T -«t )2
(3)
lbl k—1
it was found that nearly 26% of the 9T values fall outside of the interval <9T -aT, 9T +aT >. The results in Figures 1 seem to indicate a pattern whereby in each realization the contribution of the ferry conditional sojourn time 9 for some operation states towards 9£ is more for some than
that for others. Thus, identifying the conditional sojourn time for such operation states, which has major effect on the ferry total operation process times enable the total conditional sojourn time for the operation process to be studied, analysed and predicted. These are discussed in the following sections where the use of data mining techniques to understand the behaviour of 9£ is presented.
4.2 Correlation
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Correlation analysis is a method commonly used to establish, with certain degree of probability, whether a linear relationship exists between two measured quantities. This means that when there is correlation it implies that there is a tendency for the values of the two quantities to effect one another. Vice-versa also holds true if there is no correlation which implies no effect on
each other. Furthermore, using the values of the correlation coefficient, a positive or negative relationship can also be identified. If the coefficient values are close to 1, it implies positive linear relationship, whilst values close to 0 imply no linear relationship. Thus, based on the values of the correlation coefficient, the relationship between two measured quantities can be determined. The adopted formula for evaluating the correlation coefficient rbl between the ferry conditional sojourn
time 0bl in particular operation states and the ferry total conditional sojourn time 0T is given by
nbl , — , —
z (0k -0bl )(0T -0t )
k=1
nbl , — n \"bl , — „ zO-Ou)\ z(okT -eT)2
k=1 V k=1
(4)
for b = 1,2,...,17, l = b +1 and b = 18, where nbl = 42 is the number of realizations, 0bl is the k-th realization of the conditional sojourn time 0bl, 0'k is the k-th realization of the total conditional sojourn time 0T evaluated from (1), 0T is the mean total conditional sojourn time evaluated from the equation (2) and 0bl is the mean conditional sojourn time obtained from
Obl =
1 nbl
- z Ok.
nbl k=1
(5)
Thus, using the values from Tables A1-A4, the correlation coefficient, rbl, were then evaluated using equation (4). Table 2 shows the values of rb l for the spring data.
Table 2. Correlation coefficient rbl values for spring data
Operation State Correlation coefficient Operation state Correlation coefficient
z1 0.221169 z10 0.401463
Z 2 0.298071 z11 0.324054
Z 3 -0.13934 Z12 0.306238
Z 4 0.642635 Z13 0.640848
Z 5 0.738339 Z14 0.365648
Z 6 0.020627 Z15 0.099242
Z 7 -0.04948 Z16 0.142937
Z 8 0.2035 Z17 0.149159
Z 9 0.1559 Z18 0.057029
Figure 2 shows the plot of the correlation coefficient rbl against the number b of the operation state zb. It can be seen that 6A5, 056 and 6>1314 has the strongest positive linear relationship, as compared to the conditional sojourn times in the remaining operation states, where 056 and 6>1314 coincides
with the longest parts of the voyage. This implies that any variations in the conditional sojourn times 9bb+1 associated with these 3 operation states, namely z4, z5 and z13, will significantly
effect the total conditional sojourn time 0T .
sojourn time for spring data
The plots given in Figure 2 also shows that most of the rbl values are more than 0, which seems to
indicate a positive linear relationship, albeit weak linear relationship for some. Thus, from the correlation coefficient values, it can be deduced that the values of the total conditional sojourn time 9T is strongly dependent on the conditional sojourn times 9bl for some operation states. In the
following section, this understanding of the data behaviour will be used in the regression model to predict the values of the total conditional sojourn time 9T .
4.3 Regression
Regression analysis is a data mining technique used in modeling, analyzing and predicting numerical data. In linear regression, input statistical data are necessary, whereby the data is modeled as a function, in coming out with the model parameters. These parameters are then estimated so as to give a "best fit" of the data, which are then used to predict future data behaviour. Multiple regression is another type of regression model. It is similar to linear regression but in this model the interest is on examining more than one predictor variables. In this technique the aim is to determine whether the inclusion of additional predictor variables leads to increased prediction of the outcome. Here, the use of both linear and multiple regression models on the spring data are described.
From the above discussions, it can be seen that the aim of using the linear regression technique is to use initial sample data of the conditional sojourn times 9b to predict subsequent behavior of the
total conditional sojourn time 9T . In the paper the equation adopted is given by
9t =a +Pb9bi +£b (6)
for b = 1,2,...,17, l = b +1 and b = 18, l = 1, where ab, 3b are the unknown regression coefficients and sb is the random noise.
Before predicting the subsequent behavior, the values of ab and /3b based on varying realizations of the operation process need to be evaluated. Here, the unknown regression coefficients ab and 3b are evaluated by minimizing the functions
N , k 2
A(«b,3b) = Z0 -(ab +3b0kbi)]2 k=1
(7)
for b = 1,2,...,17, l = b +1 and b = 18, defined as the measure of divergences between the empirical values dT and defined by (6) the predicted values 0T (dku) = ab + 3b@ki of the total conditional sojourn time 9T .
From the necessary condition, i.e. after finding the first partial derivatives of A(ab,3b) with respect to ab and 3b and putting them equal to zero, we get the system of equalities involving the realizations 6T of the total conditional sojourn time 9T and the realizations 0^ of the conditional sojourn times dbl defined as follows
N N .
Nab + zekt ßb = tOk
k=l k=l
(8)
T oka + T oki Yßb = T okok
k=1 k=1 k=1
for b = 1,2,...,17, l = b +1 and b = 18, l = 1 and N = 1,2,...,nb!.
The remaining question that needs to be addressed is how many realizations marked by N does it take to obtain a reasonable representation of ab and 3b. By using Matlab and putting the values from Tables A1-A4 into the system of equations (6), the varying ab and 3b values were calculated for N = 1,2,...,nbl.
Figure 3 shows the plot of the regression coefficient 3b against N, for the operation states of z5 and z13. From the discussions in Section 4.2, these 2 operation states represents among the longest part of the voyage and has major influence on the total conditional sojourn time. From the plot, it can be observed that other than the initial instability for low values of N, the values of 3b seems to
stabilize for larger N. In our analyses, it was discovered that the value of 3b stabilizes at N = 30 . Although not shown in the paper this behavior also holds true for all the other operation states.
- Regression Coefficient for operation state 56
- Regression Coefficient for operation state 1314
2.5
Regression Coefficient 2
4
3
1
0.5
N= Number of realizations -1
Figure 3. Plot of regression coefficient 3b for spring data
0
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40
Thus, based on the above observations, the predicted total conditional sojourn time 9*T can then be evaluated using /3b values at N = 30. In evaluating 9T , the formulations in the system of equations
(8), lead to
9T = ab +39bi
(9)
for b = 1,2,...,17, l = b +1 and b = 18, l = 1, where a"b and 31 are respectively the value of ab and 3b at N = 30 .
Figure 4. Plots of empirical realizations and predicted from linear regression values of total
conditional sojourn time for spring data
Figure 4 shows the comparison plots of the values of the empirical realizations 0T of the total conditional sojourn time 0T and the predicted values 0T* of the total conditional sojourn time 0*T defined by the equation (9) against the number of realizations k for summer data. It can be observed that for both the operation states of z 5 and z13, the predicted 0T* values are not close to
the empirical 0T values. Similar pattern of behaviour were also seen when the values of 0T* for other operation states, were considered. These results seem to indicate that linear regression does not provide an accurate means of predicting the behaviour of the Stena Baltica ferry operation process.
Since linear regression does not provide an accurate prediction of the total conditional sojourn time the multiple regression technique is explored instead. As described earlier, the difference in the multiple regressions technique is that in this method, more than one predictor variables are considered. It is envisaged that the inclusion of additional predictor variables will lead to increased prediction of the total conditional sojourn time. Thus, for multiple regressions, the equation adopted is given by
OT =aB + tßbObi +Sb
b=1
(10)
for b = 1,2,...,17, l = b +1 and b = 18, l = 1 and B = 1,2,..., v, v = 18, where aB , 3, 32, ..., 3B are the unknown regression coefficients and sb is the random noise.
Before predicting the subsequent behaviour of aB, 3, 32, • •, 3B values based on varying realizations of the operation process need to be evaluated. The unknown regression coefficients aB, 3, 32, 3B are obtained by minimizing the functions,
k2
MaB,ßi,ß2ßB) = TOk -K + tßbOkt)]
k=1 b=1
(11)
for b = 1,2,...,17, l = b +1 and b = 18, l = 1 and B = 1,2,..., v, v= 18, that is the measure of divergences between the empirical values 0T and predicted values
0T (01kl,0ki,—,@bi) = aB + ZPb0bi of the total conditional sojourn time 0T defined by (8).
b=1
From the necessary condition, i.e. after finding the first partial derivatives of A(aB,3,P2,...,PB) with respect to aB, 31, 32, —, 3B and putting them equal to zero, we get the system of equalities involving the realizations 0T of the total conditional sojourn time 0T and the realizations 0^ of the conditional sojourn times 0bl defined as follows,
B N N
naB + ttOkbl ßb = tOk
b=1k=1 k=1
(12)
N . BN.. N . .
t okaB + t t okuokbl ßb = t okok
k=1 b=1 k=1 k=1
N k B N k k N k k
t9BiaB + £ £9Bl9bl3 = t9Bi9T
k=1 b=1 k=1 k=1
for b = 1,2,...,17, l = b +1 and b = 18, l = 1 and B = 1,2,..., v, v = 18 and N = 1,2,...,nbl. The remaining question that needs to be addressed here is that how many realizations marked by N in (12) does it take to obtain a reasonable representation of aB, 3, 32, • • •, 3B. Thus, by using Matlab and putting the values from Tables A1-A4 into the system of equations (10) for N = 1,2,... ,nbl, the varying aB, 3i, 32, • • •, 3B values were calculated.
In our analyses on the values of aB, 31, 32, 3B, the observation is that the values of aB, 31, 32, •.., 3B stabilizes at N = 30. It was also observed that aB, 31, 32, •••, 3B vary with respect to the number B, B = 1,2,..., v, v = 18, of predictor variables considered changing 1 to 18. The argument for this method is that by using more than one predictor variables, better results will be obtained. The aim is also to use as minimal number of predictor variables to generate accurate results, within as short period of time. Thus, based on the above observations, the predicted total conditional sojourn time, 9T , can then be evaluated using aB, 31, 32, • • •, 3B values at N = 30 . In evaluating 9T , the formulations in the system of equations (10) lead to
9*T =a*B + tßl9bl b=l
(13)
for b = 1,2,...,17, l = b +1 and b = 18, l = 1 and B = 1,2,..., v, v= 18, where a*B, 3;, 31, ..., 3B are respectively the value of aB, 31, 32, 3B at N = 30.
Total Conditional Sojourn Time 2200
—Total Conditional Sojourn Time
• • -Total Conditional Sojourn Time (Predicted with values from operation states 12 to 23) - - Total Conditional Sojourn Time (Predicted with values from operation states 12 to 1415)
k ♦
. .. .
1400 t ♦*»*,»*♦ %
A W
Number of Realizations
Figure 5. Plots of empirical realizations and predicted from multiple regression values of total conditional sojourn times 9^ and 9^* for spring data
2400
2000
1800
1600
1200
1000
0
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600
Figure 5 shows the comparison plots of the values of the empirical realizations 0T of the total conditional sojourn time 0T and the predicted values 0T* of the total conditional sojourn time 0*T defined by the equation (11) against the number of realizations k for summer data. It can be seen that if only 2 predictor variables 012 and 023 (B = 2) are used in the equation (11), then the
predicted values differ much from the empirical values 0T* and are not accurate at all. It was discovered that as we increased the number of predictor variables, the accuracy improves, leading to the best accuracy at B = 14 predictor variables 012, 023, ..., 01415. It was also observed that if
more than 14 predictor variables were used, the results doesn't change much, indicating that 14 predictors variables provides a good representation of the prediction. The analyses also show that multiple regression is a better method of predicting the behaviour of the Stena Baltica ferry data than the linear regression.
4.4 Accuracy
To further access the accuracy of the predicted data the root mean square error s is applied. The root mean square error is commonly used to calculate the error and is often used to measure the success of numerical prediction. If the value of s is 0 it simply means that there is no error to the prediction and the prediction is accurate. The greater values of s mean that the more inaccurate is the prediction. Here, the values of the root mean square errors for both the linear and multiple regressions are calculated. The adopted for the root mean square error equation is given by
s =
1
1 nbl
—x (0;k -0kT )2, (14)
nbl k=1
where 0*k =0"T(0kbl) for linear regression, 0*k =0*T(01kl,02kl,...,0kBl) for multiple regression and
nbl = 42 in the case of spring data. By using the predicted values 0T for both linear and multiple
regressions and the empirical value of 0£ from the spring data the values of s were calculated. It was found for spring data that for instance for linear regression with one predictor variable 056 that s = 77.9 and for multiple regression with 14 predictor variables 012, 023, ..., 01415 this value was
s = 5.3. These values of the the root mean square errors validate the results obtained from the regression analyses, indicating the accuracy of multiple regressions as compared to linear regression.
5 SUMMARY
This report has described the use of simple data mining techniques on the Stena Baltica ferry operation process statistical data given in Tables A1-A4. The aim is to observe the behaviour of the ferry operation process total conditional sojourn time and use it to predict future behaviours. In our analyses, we applied the correlation coefficient, linear and multiple regressions and root mean square error on spring data. From the results, it can be concluded that multiple regressions technique provides an accurate of predicting the ferry total conditional sojourn time.
8 REFERENCES
Blokus-Roszkowska, A., Guze, S., Kolowrocki, K., Kwiatuszewska-Sarnecka, B., Soszynska, J. 2008. Models of safety, reliability and availability evaluation of complex technical systems related to their operation processes. WP 4 - Task 4.1 - English - 31.05.2008. Poland-Singapore Joint Project.
Grabski, F. 2002. Semi-Markov Models of Systems Reliability and Operations. Monograph. Analysis. Monograph. System Research Institute, Polish Academy of Science, (in Polish).
Kolowrocki, K., Soszynska, J. 2008. A general model of technical systems operation processes related to their environment and infrastructure. WP 2 - Task 2.1 - English - 31.05.2008. Poland-Singapore Joint Project.
Kolowrocki, K., Soszynska, J. 2009A. Methods and algorithms for evaluating unknown parameters of operation processes of complex systems. Proc. Summer Safety and Reliability Seminars -SSARS 2009, Vol. 2, 211-221.
Kolowrocki, K., Soszynska, J. 2009B. Data mining for identification and prediction of safety and reliability characteristics of complex systems and processes. Proc. European Safety and Reliability Conference - ESREL 2009, Vol. 2, 853-863.
Kolowrocki, K., Soszynska, J., Kaminski, P., Jurdzinski, M., Guze, S., Milczek, B., Golik, P. 2009A. Data mining for identification and prediction of safety and reliability characteristics of complex industrial systems and processes.WP6 - Task 6.2. Preliminary statistical data collection of the Stena Baltica ferry operation process and its preliminary statistical identification. WP6 -Sub-Task 6.2.5 - Appendix 5A - English - 31.10.2009. Poland-Singapore Joint Project.
Kolowrocki, K., Soszynska, J., Salahuddin Habibullah, M., Xiuju, F. 2009B. Data mining for identification and prediction of safety and reliability characteristics of complex industrial systems and processes.WP6 - Task 6.1.3. Experimental statistical data correlation and regression analysis - Correlation and regression analysis of experimental statistical data of the operation process of the Stena Baltica ferry. Task 6.1.3 - Section 4 and Section 5.5.4 - English -31.08.2009. Poland-Singapore Joint Project.
Appendix
Statistical summer data collection of the Stena Baltica ferry operation process
In the Tables A1-A4 there are given realizations of the conditional sojourn times in particular operation states on the basis of a sample composed of n = 42 realizations of the Stena Baltica ferry operation process. It is assumed that one voyage from Gdynia to Kalskrone and back to Gdynia of the ferry is a single realization of its operation process. The conditional sojourn times in particular operation states of each single realization of the ferry operation process are given in separate columns. The operation process is very regular in the sense that the operation state changes are from the particular state zb, b = 1,2,...,17,to the neighboring state zb+1, b = 1,2,...,17, only and from z18 to z1. Therefore the realizations of the conditional sojourn times 9j+1, b = 1,2,...,17, j = 1,2,...,42, are given in the Tables b-th row and the realizations of the conditional sojourn time 91, b = 1,2,...,17, are given in the Tables 18-th row.
Appendix 5A
5A. 1. Statistical summer data collection of the Stena Baltica ferry operation process
Date/2008 24/25 26/27 27/28 11/12 12/13 26/27 27/28 28/01 01/02 02/03 11/12 12/13
Jan Jan Jan Feb Feb Feb Feb Mar Mar Mar Mar Mar
In the Tables A1-A4 there are given realizations of the conditional sojourn times in particular operation states on the basis of a sample composed of n = 42 realizations of the Stena Baltica ferry operation process. It is assumed that one voyage from Gdynia to Kalskrone and back to Gdynia of the ferry is a single realization of its operation process. The conditional sojourn times in particular operation states of each single realization of the ferry operation process are given in separate columns. The operation process is very regular in the sense that the operation state changes are from the particular state zb, b = 1,2,...,17,to the neighboring state zb+1, b = 1,2,...,17, only and from z18 to z1. Therefore the realizations of the conditional sojourn times 0j b = 1,2,...,17, j = 1,2,...,42, are given in the Tables b-th row and the realizations of the conditional sojourn time 01 b = 1,2,...,17, are given in the Tables 18-th row.
Table A1: Realization of conditional sojourn times in operations states (early spring)
/18 1 •
Realization number k i 2 3 4 5 6 7 8 9 i0 ii i2
Realization of conditional sojourn times in operations states (in minutes)
Operation state Obb +i O°b +i Obb+i Obb+i Obb +i Obb+i Obb+i Obb+i Obb+i Obb +i Oii+i /3 i2 Obb +i
zb
zi 55 52 47 75 60 60 62 43 50 6i 65 63
z 2 4 3 3 2 2 2 2 3 3 4 3 2
z 3 28 3i 32 35 37 48 33 38 39 43 40 42
Z 4 52 46 48 65 53 47 49 62 45 46 5i 47
z 5 598 635 539 572 499 507 62i 580 507 5ii 497 496
z 6 35 42 42 44 35 37 34 40 36 33 38 38
z 7 7 9 8 7 7 5 5 5 5 5 8 7
z 8 25 20 23 27 20 3i i5 i7 i6 2i 33 34
z 9 75 59 56 40 66 47 26 60 65 25 55 40
zio 5 3 2 3 2 3 5 6 3 4 4 2
zil 6 5 4 5 4 5 4 4 4 6 4 5
zi2 25 22 25 25 23 25 20 33 24 24 22 22
zi3 574 427 46i 50i 498 490 438 56i 49i 5i3 496 500
zi4 6i 43 43 46 49 52 42 63 46 60 50 50
zi5 33 32 33 36 35 33 35 34 3i 33 34 36
zi6 4 4 5 4 4 4 3 4 4 4 4 4
zi7 8 io 6 5 5 6 4 5 8 7 6 7
zi8 26 26 30 20 i6 i7 i6 22 i7 8 i7 i7
Total Ok i62i i469 i407 i5i2 i4i5 i4i9 i4i4 i580 i394 i408 i427 i4i2
Table A2: Realization of conditional sojourn times in operations states (early spring)
Date/2008 13/15 Mar 15/16 Mar 16/17 Mar 17/18 Mar 18/19 Mar 19/20 Mar 20/21 Mar 21/22 Mar 22/23 Mar 23/24 Mar 08/09 Apr 09/10 Apr
Realization number k 13 14 15 16 17 18 19 20 21 22 23 24
Realization of conditional sojourn times in operations states (in minutes)
Operation state /314 8bb+1 8bb5+1 /316 8bb+1 8bb7+1 /318 8bb+1 ,319 8bb +1 /3 20 8bb+1 21 8bb+1 22 8bb +1 ,3 23 8bb+1 24 8bb+1
zb
45 45 40 20 33 50 43 15 45 57 97 68
Z 2 2 2 2 2 2 3 2 2 3 2 2 3
Z 3 35 36 36 36 37 35 34 34 36 36 39 36
Z 4 51 51 51 49 53 44 51 52 50 53 53 54
Z 5 595 495 504 507 498 483 497 504 507 503 500 492
Z 6 34 39 38 39 38 35 37 36 37 34 38 40
Z 7 7 8 7 10 8 8 7 8 8 8 7 9
Z 8 18 16 13 3 15 6 9 25 19 31 30 35
Z 9 75 77 60 73 82 118 71 55 30 24 34 41
Z10 5 2 2 2 3 4 2 2 3 3 2 5
Z11 4 4 4 4 4 4 4 4 4 4 4 4
Z12 24 24 25 24 23 22 23 22 22 22 26 22
Z13 522 491 499 488 464 484 498 496 505 595 483 499
Z14 72 50 48 50 48 52 47 53 51 61 61 48
Z15 34 35 35 34 35 34 31 32 33 46 34 34
Z16 5 5 5 4 4 4 5 5 3 4 6 6
Z17 7 7 6 4 4 7 5 5 7 5 4 5
Z18 26 40 21 34 40 35 28 22 8 2 12 13
Total 8 1561 1427 1396 1383 1391 1428 1394 1372 1371 1490 1432 1414
Table A3: Realization of conditional sojourn times in operations states (early spring)
Date/2008 10/12 Apr 12/13 Apr 13/14 Apr 14/15 Apr 15/16 Apr 16/17 Apr 18/19 Apr 19/20 Apr 20/21 Apr 05/06 May 06/07 May 07/08 May
Realization numbr k 25 26 27 28 29 30 31 32 33 34 35 36
Realization of conditional sojourn times in operations states (in minutes)
Operation state zb /3 25 0bb +1 /3 26 0bb +1 /3 27 0bb+1 /3 28 0bb+1 /3 29 0bb +1 /3 30 0bb+1 0bb+1 32 0bb+1 33 0bb+1 /3 34 0bb +1 0bb5+1 36 0bb +1
58 35 45 75 72 62 37 44 46 78 59 65
Z 2 3 4 3 3 2 3 6 3 2 2 2 2
Z 3 37 36 35 39 37 36 37 36 36 37 36 36
Z 4 67 51 50 62 49 48 64 51 53 63 55 53
Z 5 573 498 506 576 494 505 576 495 502 574 492 497
Z 6 36 37 35 38 38 36 35 39 37 36 38 37
Z 7 8 7 5 7 10 9 10 6 7 7 6 6
Z 8 25 11 17 31 23 25 23 15 18 19 18 24
Z 9 55 55 43 45 52 48 50 58 53 30 30 45
Z10 3 3 3 3 2 3 2 2 3 3 2 2
Z11 4 4 5 5 4 5 4 5 4 5 4 4
Z12 23 22 23 26 23 23 24 23 24 23 28 24
Z13 573 497 531 500 492 496 590 508 520 502 508 508
Z14 58 51 54 47 40 51 47 47 56 47 46 42
Z15 34 35 33 35 35 34 33 34 35 36 35 35
Z16 5 5 6 5 4 6 5 5 4 4 5 4
Z17 4 5 5 5 7 6 5 6 6 10 5 4
Z18 18 20 11 10 16 18 25 18 12 12 17 14
Total 00 1584 1376 1410 1512 1400 1414 1573 1395 1418 1488 1386 1402
Table A4: Realization of conditional sojourn times in operations states (early spring)
Date/2008 08/09 May 10/11 May 11/12 May 12/13 May 13/14 May 14/15 May
Realization number k 37 38 39 40 41 42
Realization of conditional sojourn times in operations states (in minutes)
Operation state 8b37+1 /3 38 8bb+1 /3 39 8bb+1 n 40 8bb+1 41 8bb +1 /3 42 8bb+1
zb
53 25 55 84 71 67
Z 2 2 2 3 2 2 2
Z 3 38 37 40 36 37 34
Z 4 60 49 46 57 53 51
Z 5 584 504 505 573 494 495
Z 6 38 35 36 39 36 36
Z 7 5 7 5 5 6 6
Z 8 15 6 40 28 32 28
Z 9 70 35 35 47 40 50
Z10 2 2 3 3 3 2
Z11 5 4 5 5 4 4
Z12 25 25 24 23 26 24
Z13 595 506 535 506 503 503
Z14 42 45 47 46 51 43
Z15 34 35 34 34 33 33
Z16 6 4 4 5 5 4
Z17 5 3 4 5 3 5
Z18 20 11 11 10 13 18
Total 8 1599 1335 1432 1508 1412 1405
