A Multiplicative Time Series Model For Air Transport Demand Forecasting Presented By Mohammed Salem Awad Consultant YEMEN
A Multiplicative Time Series Model For Air Transport Demand ForecastingJairam Singh1 A. A. Bashaswan2 Mohd Salem Awad3Abstract:The air travel demand is influenced by a variety of factors which renders the numberof passengers to vary as a seasonality fluctuating non-stationary time series. For non-seasonal series it is possible to obtain a parsimonious representation in the form ofARMA model but seasonality makes the model cumbersome. In the present paper amultiplicative model of the type (p,d,q)lx(P,D,Q)s has been used to represent a non-stationary time series displaying seasonality at an interval of S observations. The dataof the International Passengers from the records of Yemenia. Yemen Airways havebeen used to estimate the parameters of this model. The model gives a fairly goodforecast values. 1. Introduction: There are several modes of transportations for passengers and freight and airtransport is a part of a larger product. Air travel is not an end in it self. Air traveldemand is dependent on the demand for the other product of leisure and business.Airline product is characterized by the passengers and the freight mix strategy. A seatin the aircraft is very much like another. Similarly the freight product is alsohomogeneous but the role airfreight tends to be underestimated although it amounts toone quarter of the RTKs output and one tenth of the total revenue. Therefore, thedeterminants of air travel demand are many such as personal income, cost of airtravel, convenience and speed level of trade, population distribution and changes andthe of economic activity. The institutional factors such as festivals working practiceand school holidays cause the seasonal fluctuation in the demand.Figure 1. Shows a time series of the total of international airline passengers quoted byYemenia / Aden Center  given in Appendix A.1 Prof. of Industrial Management, Department of Mechanical Engineering, Faculty of Engineering,Aden University.2 Associate Prof. of Mechanical Engineering, Faculty of Engineering, Aden University3 Mechanical Engineer, YEMENIA Yemen Airways.
This series exhibits a periodic behavior with period S = 12 months, in the late summermonths a secondary peak occurs in the spring.One of the deficiencies in the analysis of a time series has been the confusion offitting a series and forecasting it. A common method of analyzing a time series is todecompose it arbitrarily into three components – a "trend", "seasonal component", anda "random component". The trend can be fitted by a polynomial and the seasonalcomponents by a Fourier series. A forecast can then be made by projecting these fittedfunctions. Such methods can give an extremely misleading results in cases where apart of the time series may look to be quadratic some times due to some of the randomdeviates which is taken to be the characteristics of the demand data, if they are fittedto it.Another model involving sines and cosines may be used to present seasonal variationswhich may be written as Zt j1 π j Zt j a t ψ a j 1 j t j at …………………………. ( 1 )With suitable values of the coefficients π j and ψ j , is entirely adequate to describemany seasonal time series analysis. The problem is to choose a suitable system ofparsimonious parameterization for such models involving a mixture of sines andcosines and polynomial terms to allow the changes in the level of series. For non-seasonal series, it is usually possible to obtain a useful andparsimonious representation in the form of ARMA model as Φ (B) Zt Θ (B) a t …………………………………………. ( 2 )Where B is backward shift operatorMoreover, the generalized autoregressive operators (B) determines the eventualforecast function which is the solution of the difference equation (B) Zt ( l ) 0(1 Φ1 B - Φ2 B2 - Φ 3B3 .......... p Bb ) ZT (l) 0 ………………. ( 3 ) Φ
Where B is understood to operate on l . In representing a seasonal behavior we shallwant the forecast function to trace out a periodic pattern. If (B) represents aforecast a forecast function which is a sine wave with a twelve month period, adaptivewith phase and amplitude, will satisfy the difference equation.(1 - 3 B B 2 ) Z t ( l ) 0However, it is not sure that the periodic behavior is truly represented. It might needmany sine – cosine componentsIf we give some thought as to what happens when we try to induce stationary bydifferencing d times and we write (B) (B) (1 - B) dWhich is equivalent to setting d roots of equation (B) 0 , equal to unity. When such a presentation proved adequate 1 - Bwas used as a simplifying operators to convert a non-stationary series into a stationaryseries. The fundamental fact that a seasonal time series will have observations similarto each other after a certain intervals which is called a period. Therefore, the operationB s Z t will represent an observation before s interval i.e B s Z t Z t -sand then the series which exhibits seasonally will be Z t , Z t -s , Z t -2s , Z t -3s ,......This series may also be expected to be non-stationary, therefore, a simplifyingoperator , Z t (1 - B s )Z t Z t Z t -s might be useful to make it stationary [2,3,4]. 2. The Multiplicative ModelThe seasonal effect implies that an observation for a particular month, say April isrelated to the observations for previous Aprils. Suppose the t th observation Z t is the
month of April. We might be able to link this observation Z t , with observations inprevious April by a model of the form (B s ) S Z t (B S ) t …………………………………….. (4) DWhere S=12, S 1 B S and (B s ) , (B S ) , are polynomial in BS of degree P andQ, respectively and satisfying the stationary and invertibility criteria. Similarly amodel of the form [5,6,7,8] (B s ) S Z t -1 (B S ) t -1 …………………………..………( 5 ) D,might be used to link the current behavior for March with the previous Marchobservations, and so on, for each of the twelve months.Now the error components t , t -1 , t - 2 ,........would not in general is uncorrelated. Forexample, the total airline passengers in April 1990, while related to April totals,would also relate to totals in March 1990, February of 1990 and January of 1990, etc.Thus, we would expect that t , in eq. (4) would be related to t -1 , in eq.(5) and so on.A second model may be introduced to take care of such relationships, (B) d t q (B) a t ……..………………………………… ( 6 ),where at is a white noise process and (B) and q (B) , are the polynomials of degreep and q respectively, and 1 1 B .Substituting Eq. (6) in (4), we get p (B) (B S ) d S Z t q (B)(B S ) a t ……………………….. ( 7 ) D,where for this particular example, S=12. The resulting multiplicative process willbe said to be the order of (p,d,q)x(P,D,Q)S
3. Choice of Transformation of the Data from Yemen Airways ( YEMENIA Aden Center ) It is particularly true for seasonal model that the weighted averages of theprevious data values, which comprise the forecasts, may extend far back into theseries; care is therefore needed in choosing a transformation in terms of which aparsimonious linear model will closely apply over a sufficient stretch of the series. Adata based transformation may help determine in what metric the amplitude of theseasonal components is roughly independent of the level of the series. Let us assumethat some power transformation, z = x for ≠ 0 , z = ln x for = 0, may beneeded to make the model (7) appropriate. The approach of Box and Cox  may befollowed and the maximum likelihood value obtained by fitting the model to x() = (x- 1)/ x-1 for various values of which results in the smallest residual sum ofsquares S , In this expression x is the geometric mean of the series. It was shown byBox and Jenkins  by the airline data, the maximum likelihood value is thus close to = 0 confirming for this particular example, the appropriateness of the logarithmictransformation. The monthly totals of the passengers in international travel shows a seasonalbehavior with period S = 12. The data are shown in Table 1. , which represents thelogarithms of the airline data. Table 1. Natural Logarithmic of Monthly Passengers Total in international Air Travel By Yemenia, Aden Center (Using Aden as a hub).
4. Representation of the Airline Data by Multiplicative by (p,d,q) x ( P,D,Q)12 ModelThe arrangement of Table 1. emphasizes the fact that in periodic data there are twomain intervals which are important. We expect relationships to occur: (a) between the observations of the same month in the successive years (b) between the observations in the successive months in a particular years. Identification of Multiplicative Model A tentative identification of time series model is done by analysis of historicaldata. Usually at least 50 observations are required to achieve satisfactory results. Theprimary tool used in this analysis is the autocorrelation function.Consider the time Z1, Z2 , ………………..ZNThe theoretical autocorrelation function is EZ t μ Z t -κ μ ρκ , κ 0,1,2, .......... K , (8) σ2 z Where σ 2 is the variance of the series. The quantity ρ κ , is called autocorrelation zat lag k. Obviously ρ 0 1. The theoretical autocorrelation function is never knownwith a certainty, and must be estimated. Satisfactory estimate of ρ κ is the sampleautocorrelation function Z N -k 1 N-k t Z Z t -κ Z t 1 ρκ N , ………………….………….. (9) Z 1 2 t Z N t 1 NFor useful results, we would usually compute the first k , autocorrelations 4 As a supplemental aid the partial autocorrelation function often proves useful. Weshall define partial autocorrelation coefficient kk as the kth , coefficients in anautoregressive process of order k. It can be shown ( Ref. 9 chapter 3 ) that the partialautocorrelation coefficients satisfy the following Yule-Walker equations.
ρ j k1 ρ j-1 k2 ρ j- 2 k3 ρ j- 3 .......... kk ρ j-k ... ……….……… (10) j 1,2, .......... k ˆ This partial autocorrelation coefficient may be estimated by substituting ρ j , forρ j , in Eq. (10), Yielding ρ j k1 ρ j-1 k2 ρ j- 2 k3 ρ j- 3 .......... kk ρ j- k ˆ ˆ ˆ ˆ ... ˆ ………… ( 11 ) j 1,2, .......... k And solving the resulting equation for k 1,2, ......K to obtain 11 , 22 , ....,kk, the sample partial correlation function. From the estimated autocorrelation function, which can be convenientlyexhibited by a graph, a tentative model autocorrelation function patterns. Thesepatterns are discussed in reference . A sample autocorrelation and partial autocorrelations function of non-stationarytime series die down extremely slowly from a value of one. If this type of behavior isexhibited, the usual approach is to compute the autocorrelation and the partialautocorrelation functions for the first difference of the series. If these functionsbehave according to the characteristics of a stationary series. Then one degree ofdifferencing is necessary to achieve stationary. In case of Yemenia ( Aden Center) data after first differencing, theautocorrelations for all lags beyond the first is zero ( see table 2 ). Therefore, anIMA(0,1,1) model is appropriate. This contains no seasonal components.Suppose, we want to use this model to link the data 12 months apart then the modelwould be 12 Z t (1 - B 12 ) t ……………………………….. (12)Further, we want to employ a similar model using a linear filter to link the data onlyone month apart. This gives a model t Z t (1 - B) a t ……………………..………… (13) Where and will have different values.
Then on combining expression (12) and (13), we would obtained the seasonalmultiplicative model 12 Z t (1 - θB) (1 - B 12 ) a t ……………………………..…………….. ( 14)of the order (0,1,1)x(0,1,1)12 . The model written explicitly is Z t - Z t 1 - Z t -12 Z t -13 a t - θa t -1 θa t -13 ………………….. (15)The invertibility region of this model is defined by 1 θ 1 and 1 1From the table 2, we see that the autocorrelation function does not die down rapidlyand it can be concluded from this that the logged data of the time series is non-stationary. Therefore, some degree of differencing will be necessary to producestationary. The first difference of time series is taken and its autocorrelation functionis calculated as shown in table 2. It appears that the simple differencing reduces theauto correlations in general but a heavy periodic components remains, this is evidentparticularly at large lags. Sample differencing with respect to period 12 results incorrelations which are firstly persistently positive and then persistently negative. Bycontrast the differencing 12 Z markedly reduces the correlation coefficientthroughout.
Table 2. Estimated Autocorrelations of Various Differencing of the Logged Airlines Data.Autocorrelations ˆ ˆ Estimation: Iterative Calculation of least squares estimates θ , and .An iterative linearization technique may be used in straight forward situation tosupply the least squares estimates and their standard approximates errors. For thepresent examples we can write approximatelya t,0 θ θ 0 χ 0 χ at 1,t 2,tWhere a a χ 1,t - χ 2,t - θ θ 0Θ 0 Θ θ 0Θ 0 ,and where θ 0 , and Θ 0 are guessed values and a t,0 at θ0Θ0 , The derivative aremostly easily computed numerically .For (p,d,q) x ( P,D,Q)12 model the preliminary estimates of autocorrelationsfunctions would be
-θ -Θ ρ1 ρ 12 1 θ 2 , and 1 Θ 2 ………………. (16)On substitution the sample estimates r1 0.337 , and r12 0.189 in (16) , we obtain 2 20.337(1 θ ) θ and 0.189(1 Θ ) Θ . 2 2Or …… θ 2.967 θ 1 0 , and Θ 5.291 Θ 1 0 ˆ ˆFrom these a rough estimates of θ 0.3876 , and Θ 0.1962 …… 4.2.1. Iterative Estimation of θ , and Θχ 1,t , can be written asχ1,t at ω, β1,0, .....,β.....βk,0 at ω, β1,0, .....,β1,0 δ1 .....βk,0 δ1 ….. (17) Consider the fitting of the airline data to (0,1,1)x(0,1,1) 12 processω t 12 Z t (1 - θB)(1 - ΘB 12 ) a tThe beginning of the calculation is shown in table (3) for the estimated value ofˆ ˆθ 0.3876 , and Θ 0.1962 The back – forecasted values of [at] were actuallyobtained using the expression as worked out in Appendix A.e t 12 Z t θe t-1 Θe t -12 θΘe t -13 The value of [at] can be obtained bya t 12 Z t θa t-1 Θa t-12 θΘa t -13 ˆ ˆχ 1,t , can be evaluated by eq. (17) giving an increments in θ and Θ till S(,)becomes the minimum. After many iterations improves to 0.3225 and , 0.3712.
5. Forecasting:Forecasting are best computed directly from the difference equation it self. Thus usingthe seasonal model (15) for forecasting at a lead time , and origin t is given byZ t Z t 1 Z t 12 Z t 13 a t θa t 1 Θa t 12 θΘa t 13 ……..(18)After setting = 0.3225, and = 0.3712 , the minimum means squared errorsforecast at lead time , and origin t is given byZ t Z t 1 Z t 12 Z t 13 a t θa t 1 Θa t 12 θΘa t 13 …….ˆ (19)Here we refer toZ t EZ t θ, Θ, Z t , Z t ,......... .......... .... ……... (20)As the conditional expectation of Z t , taken at origin t, In this expression theparameters are supposed exactly known and knowledgeable of series Z t , Z t 1 , issupported to the extend into the remote past. This practical application depends uponThe facts that.a) Invertible models fitted to the actual data usually yield forecasts which depends appreciably only on the recent values of the series.b) The forecasts are insensitive to small changes in parameter values such as are introduced by estimation errors.Now Z t 1 j 0 Z t 1 ………………. (21) Z j ˆ j 0 t a t 1 j 0 a t 1 …………….. (22) 0 j 0
Thus to obtain the forecasts, we simply replace unknown ZS, by forecast, andunknown a S, by zeros.The known aS , are of course the one step ahead forecast errors already computed, ˆthat is, a t Z t Z t -1 …… (1)For example, to obtain the three months ahead forecast, we haveZ t 3 Z t 2 Z t 9 Z t -10 a t 3 0.3225 a t 2 0.3712 a t 9 0.1197a t 10Taking conditional expectation at origin,Z t 3 Z t 2 Z t 9 Z t -10 0.3712a t 9 0.1197a t 10ˆ ˆThat is Z t 3 Z t 2 Z t 9 Z t -10 0.3712 Z t -9 Z t -10 (1) 0.1197 Z t -10 Z t -11 (1) …ˆ ˆ ˆ ˆHenceZ t 3 Z t 2 0.6288 Z t 9 0.8803 Z t -10 0.3712 Z t -10 (1) 0.1197 Z t -11 (1)ˆ ˆ ˆ ˆ (1)This expresses the forecasts in terms of ZS and previous forecast of ZS
Conclusions: It is obvious from the forecast values shown in Fig. 1., that the simple modelcontaining only two parameters faithfully reproduces the seasonal pattern andsupplies excellent forecasts. It is to be remembered that like all predications obtainedfrom the general pattern linear stochastic model, the forecast function is adaptive.When the seasonal pattern changes, these will be appropriately projected into theforecast. Of course, a forecast for a lead time of 36 may necessarily contains a fairlylarger error. However, in practice, an initially remote forecast will be continuallyupdated and as the lead shortens, greater accuracy will be possible.The model presented here is robust to moderate changes in the values of values of theparameters. Thus, if = 0.35 , and = 0.4 , instead of 0.3225 and 0.37, the forecastwould not be greatly affected. This is true for the forecasts made several steps aheade.g 12 months. This has been seen by studying the sum of squares surfaces ofmodifying the values of the parameters by one step ahead forecasts.References:1- Brown, R.G. "Smoothing, Forecasting and Predication of Discrete Time Series", Prentice Hall, New Jersey, 1962.2- Box, G.E.P. and Jenkins G.M.. "Some Statistical Aspects of Adaptive Optimization and Control", Jour. Royal stat. Soc. B24,297,1962.3- Box, G.E.P. and Jenkins G.M.. "Further Contribution to Adaptive Quality: Control Simultaneous estimation of Dynamos ; non zero costs", Bull Ins. Stat. 34th Seminars, 1963.4- Box, G.E.P. and Jenkins G.M.. "Mathematical Models for Adaptive Control and Optimization", A.I. Ch. E.-J. Chem. E. Symposium Series, 4, 61, 1965.5- Box, G.E.P. and Jenkins G.M.. "Models for Forecasting Seasonal and Non Seasonal Time Series" , Advanced Seminar On Spectral Analysis of Time Series, ed B. Harris, 271, John Wiley, New York, 1967.6- Box, G.E.P. and Jenkins G.M.. "Some Recent Advances in Forecasting and Control, I". Applied Statistics, 17,91,1968.7- Box, G.E.P. and Jenkins G.M.. "The Time Series Analysis Forecasting and Control", Holden.-Day Singapore 1976.8- Yule, G. U. , "On the Method of Investigating Periodicity in Disturbed Series with Special Reference to Walfers Sunspot Number". Phil. Trans A226, 267, 1927.9- Daniels , H.E. " Approximate Distribution of Serial Correlation Coefficients" Biometrica, 43, 169, 1956.10- Yemenia: Records of International Passengers, 1996.11- Box, G.E.P. and Cox D.R., "An Analysis of Transformation". Jour. Of Rayal Stat. Soc. B26, 211. 1964.
APPENDIX A :Data International Airline Passenger quoted by Yemenia / Aden Centre
APPENDIX B :CALCULATION OF THE UNCONDITIONAL SUM OF SQUARES FORTHE MODEL ωt 12 Zt (1 - θB) (1 - B12 ) at With ωt 12 Zt , the model ( 0,1,1) x (0,1,1)12 may be written in either theforward or backward form. ωt (1 - θB) (1 - B12 ) atOr ωt (1 - θF ) (1 - F12 ) etAnd where μ E ω t is assumed to be zero. Hence we can write e t ω t θ e t 1 e t 12 θΘe t 13 ……………… B1 a t ω t θ a t 1 a t 12 θΘa t 13 ……………… B2Where ω t ω t for t = 1, 2, ………, n and is the back forecast of ω t for t 0There are N = 48 observations in the airline series. Accordingly, in Table, these aredesignated as Z12 , Z11 , Z10 , Z9 ,……….. Z35 . The ω s obtained by differencingfrom the series ω 1 , ω 2 , ω 3 , ,......... ω n , where n = 35 we shall start calculation e 35 .......by setting unknowns e s equal to zero.Using (B.1) we gete 35 ω 35 θ 0 0 θΘ 0 e 34 ω 34 θ e 35 0 θΘ 0e 33 ω 33 θ e 34 e 35 θΘ 0………….…………..e 4 ω 4 θ e 2 e 3 θΘ e 14
Table B1. Computation at ..........ω -12 , then B.2 is used to calculate a t We can calculate ω 0 , ω -1 , ..........Since each a t is function of previously occurring ω and a -1 0 , j > 12Now 24Sθ, Θ a t 12 t 2The next iteration would start with e s starting of the iteration using forecast values ......... ,from a s already calculated.ω n1 , ω n 2 , ω n 3 ..........Table 3. Numerical Calculation of Derivative from Airline Data t Z a t,0 a t θ 0 a t θmδ a a δ t θ t θδ -12 8.1715 -0.0545 -0.0545 -0.0545 0 -11 7.9596 0.0995 0.0783 0.0778 0.0545 -1 0 1 35 9.8182 0.7071 0.6377 0.6361 0.1560
Using the whole series, the iteration can be proceeded. a χ n t,0 1t t 0θ - θ0 n χ t 0 1t 2Then changing the value θ 0 θ and repeating the same procedure, the values of θ and can be found out which minimizes nSθ, Θ a t0 t0 2