Regression in Personal Air Transport of Passengers Evolution at Selected Airport Time Series Method

KLJUČNE RIJEČI putnik u zračnom prijevozu regionalni zračni prijevoznik kvadratna matrica mjesečne varijacije razdoblja deterministički sustav statički model regresijski model Tobiáš Lazar Technical University of Košice Faculty of Aeronautics Slovakia e-mail: tobias.lazar@tuke.sk Alena Novák Sedláčková Technical University of Košice Faculty of Aeronautics Slovakia e-mail: alena.sedlackova@fpedas.uniza.sk Róbert Bréda Technical University of Košice Faculty of Aeronautics Slovakia e-mail: robert.breda@tuke.sk


INTRODUCTION
The significance of air transport is based on its speed, operation ability, comfort and possibility to cross big distances as well as quick technical development.Air transport has to react to the development of production force, development of aircraft technology, ground security systems by the modernisation of airport equipment within an appropriate timeframe.
The problem of number transported passengers, their diversity, as well as following their customs is a problem which has been solved in academic and scientific communities along with being solved in practice itself.Air transport providers and airports follow the numbers of transported passengers; define their predictions, which are necessary in the creation of long-term intentions of the development of the transport and airports.This indicator is one of the most important factors for air transport providers to decide about the number and frequency of lines among different destinations as well as about the fact, if those lines would stay in the flight plan or be cancelled.Equally important, this indicator plays a role in making code-share contracts among different providers.These contracts serve the providers as means to enlarge the line networks or they are enabled to use the network of contracting partners and sell tickets to them as if they were their own, increasing the number of transported passengers.Based on these types of contracts and understanding their benefits, there were several attempts to create multilateral partnerships, known as world flight alliances, in the period of 1989-2002.Historically it can be stated that only those world alliances, which have had wide participation in north-Atlantic market, Europe and recently also in Asia have become successful.It has been north-Atlantic region which has prevailed in the transport of passengers in intercontinental and intracontinental connections in long term.[6] The most important indicator which influences further strategic control and development is the number of interested people in air transport as well as the prediction of its decrease or increase for the airports.One example could be M. R. Stefanik airport in Bratislava, when the intentions to build a new terminal got its realistic shape around 2003, when air traffic provider Sky Europe Airlines PLC established itself on Slovak market.This fact was at the same time one of the impulses of the development of air transport in Slovakia when the number of passengers grew in jumps annually.[10] In 2002 the airport recorded 371 631 a year, in 2003 it was 493 917 passengers, in 2004 their number increased to 893 614 passengers (yearly growth 1,86), in 2005 it was 1 326 493 passengers (yearly growth 1,48) and in 2006 it was almost 2 million of passengers -exactly 1 937 642 (yearly growth 1,41), of which 47,52% of transported passengers of this airport used the services of SkyEurope Airlines.It can be said that the number of passengers has become the existential question according to the experience of other international and regional airports in Slovakia, therefore the application of the below mentioned method can significantly speed up and ease their decision-making process in the area of their development or direction.This method clearly shows the fact that management of airlines, as well as airports, does not rely on blind trust or probable expectation of the number of transported passengers and they could support their intentions by a clear mathematical method in future [9].
Time causality and cyclic changes in the number or air transported passengers in each calendar year is the rule, which is based on yearly (seasonal) periods.Accidental manifestations, which have importance of non-stationary influences, have supported the motivation to devote attention to mathematically describable expressions of such periodical changes on the air transport effectiveness.In most cases it is possible to create mathematical models which can existing changes to approximate on the bearable values.This is the reason why existing methods based on the theories of time series enable to predict the following state at the beginning [3].Problems of investigate to reflect basic notions of time series in connection with their practical use in the area of identification of personal air transport quality and its model.Time functions, which are basic analytical tools, enable us to substitute numeric form by which the number of passengers is expressed by successiveness of time impulses.It means that impulses ordered in time into a series shifted in time increase enable to create a function of their peaks.Programme MATLAB: Title Time Series Regression of Airline Passenger [1] has been used in the following procedures [2], [3].

TIME SERIES METHOD IN THE EVOLUTION OF PASSENGER AIR TRANSPORTATION
In the observed case, time series are created by values, the set of which is the result of observation in the following direction: calendar past -present.In the following description of the evolution of air transport it is supposed that time interval between measurements is the same while discretion with equidistant step is accepted.The main task of the performed analytical -synthetic research is to show the possibility of actual use of time series to predict future values [2].Seasonal element of time series, the period of which equals standard element of calendar time (hour, day, week, month, year) or its constant multiple is important to solve the task.Operations with time series has been thoroughly described [8].The method of time series in the synthesis of the model of transported passengers by air effectiveness accepts concrete knowledge of the number of passengers in months and years.It is supposed that a regional airline provider transported in the period 2000, 2001 to 2011 in different months the numbers of passengers which create a square matrix.The data has been adapted from the statistic records of Kosice airport [4] The method of transfer of depicted set into time series has been performed in programme environment MATLAB according to the following methodology [1]: Matrix columns are matched with years: yr=repmat ((2000:2011),12,1) Sequence (time) of calendar moths in years is determined by command: mo=repmat((1:12)' ,1,12) Time series (calendar) of transported passengers is then: time = datestr(datenum(yr(:),mo(:),1)) Numbers of passengers are discrete values of time series (sequence) expressed by the command: ts = timeseries(y(:),time,'name' ,'AirlinePassengers') By standardisation of time series (ts-time series) into normative calendar form is used for formal calendar form: dd-mmm-yyyy (days -mouths -years) and command is sent: ts.TimeInfo.Format = 'dd-mmm-yyyy' time series creates a collection which can be used as a command: tscol = tscollection(ts); Shows: plot(ts), with the use of Tools in the figure Matlab, straight line is found and its equation Basic Fitting is formed: a straight line is the output of the solution: y= -0.16*x+480The difference of input (x=0) and output numbers of passengers show the decrease (degradation) of transported people.
Fig. 1 draws attention to the beginning of the series ('0') which is presented by calendar year 2000.

TRANSPORT TRENDS
Transport trend can have gradation or degradation character which can be expressed by a straight line, exponent line or other function.Monthly magnitudes of transported passengers are variable and differ in size considerably.Because the equation of the straight line requires work with large numbers, it is suitable to differ between magnitudes with the help of logarithms [7].Logarithmic coordinates are adjusted by a scale to the number of passengers by the following command: set(gca,'YScale' ,'log') The set is matched to the time series as constants which accept the collection in stablished logarithms: tscol = addts(tscol, log(ts.data),'logAirlinePassengers') Next, monthly variations of transported passengers are considered in observed years.Calendar time is dated backwards by the following command: t = reshape(datenum(time),12,12) Command 'reshape' enables the user to transform the matrix of 'calendar' years according to selected month, which shows the required number of passengers.In the given case the selected month is July.July is comparative month for numeric evaluation of effectiveness.Meaning of matrix 't' elements: first element 730486 : 7 -July,30 -calendar day of July, 486 -number of days from the beginning of coordinate system (year 2000) towards the past, i.e. end of August, year 1998.Total: 730486+31=730517,... Columns 't i ' (i -serial column of matrix) expressing growing time.After filling the year with calendar days the beginning of the following year is obtained: 731003, which is in the second column ('t i ' , where i=2).Differences of days in lines present growing years.Example: 730486 -730852 = 366 days.Meaning: command 't' present actual calendar time which can be divided into periods.In the following step, matrix ‚y' is taking a logarithm due to more suitable graphic description: Knowledge of daily number of passengers is necessary for the model synthesis, e.g.monthly effectiveness, see Fig. 2    title (' Distribution of the relative values of passengers on days of January,'FontSize' ,15), ylabel(' The magnitude of incremental rate of passenger numbers' ,'FontSize' ,13), xlabel(‚Sequence of selected days of January of the year2000' , 'FontSize' ,13), Equation Y=0.0044*x + 0.18 is the model of the average of the increase of passenger growth average in the given month and calendar year.It can be seen that in the month of January there exists progress of the increase of air transport.This problem can be expanded by further considerations in which it is possible to prognoses the development of air transport from the wider angle of view, in complex range of the use by its users.[1]

CONCLUSION
On the basis of above mentioned method, after its practical application on an actual set of transported passengers at selected regional airport, it is possible to show that the change of the number of transported passengers, its decrease or increase is the phenomenon, which is influenced by accident and has impact on all area of air transport.A mathematical model which has been obtained by the analysis of the reason of the decrease does not research it, it only statistically evaluates the existing state, however, it is necessary to search for other concrete reason or actual event which has caused this state.From the viewpoint of the evaluation of outputs of the used method it can be confirmed that with correct interpretation Figure 4 Incremental rate of passenger number in the period of 7 days in January 2000 and its possibilities, the programme MATLAB enables the ability to create mathematical models of such a complex system [3] as passenger air transport is.Obtained outputs, which are presented by mathematical models, can be used in the creation of the strategies of management of air transport providers and also enable the transfer to a posterior evaluation of the quality of passenger air transport.This problem is the subject of further detailed investigation of continuing research, when obtained results and used method can open new dimension of looking into this problem in scientific and academic communities.

Figure 1
Figure 1 Graphic ordering of passenger transport set into time series