Tractors in Spain- A Dynamiv=c Reanalysis

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Tractors in Spain -- A Dynamic Reanalysis

Author(s): Dani Gamerman and Hlio S. MigonSource: The Journal of the Operational Research Society, Vol. 42, No. 2 (Feb., 1991), pp. 119-124Published by: Palgrave Macmillan Journalson behalf of the Operational Research SocietyStable URL: http://www.jstor.org/stable/2583176.

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J. OpI Res. Soc. Vol. 42, No. 2, pp. 119-124, 1991Printed in Great Britain. All rights reserved 0160-5682/91 $3.50 + 0.00Copyright ? 1991 Operational Research Society Ltd

Tractors in Spain A Dynamic ReanalysisDANI GAMERMAN and HELIO S. MIGONUniversidade Federal do Rio de Janeiro, Brazil

This paper presents a class of models obtained from the logistic curve to forecast stocks of goods. Thedynamic approach allows variation of the parameters with time, thus allowing the model to adapt forchanges. The observations are kept at the original scale and the model is reformulated in terms of theparameters.On-line estimates, forecasts and measures of performance are obtained. The methodology isapplied to the stock of tractors in Spain. Fitted figures are obtained and compared with those fromprevious work.Key words: dynamic forecasting, logistic curve, tractors, Spain

INTRODUCTIONThe data on the stock of tractors in Spain have been the subject of study of many articles in thisjournal. All the models proposed are derived from the logistic growth curve. This seems to providea good model for the stock of durable goods as it tends to reach a saturation level. The modifi-cations adopted always allow for such characteristics. Mar-Molinero' initially adopted the basiclogistic curve given byyt = At+ et where ut= 1

where et are normally distributed errors with variance a2 and a, b, 0 > 0. Upon inspection of theautocorrelation of fitted residuals, the model was extended to an AR(1) for et. Oliver2 extended theabove model by including an additive intercept. Later, Harvey3 suggested the use of a variation ofthe model employing difference operations. Although described as a local trend model, it keepsone of the key parameters of the model, 0, constant as in (1). Oliver4 studied the problem with adifferent model. It assumed a fixed number of buyers deciding to buy the considered item (in thiscase, tractors) with common probability function F(t). It is not difficult to show that a logisticmean response is obtained for this binomial model with F(t) = a/(a + bqt). However, Oliver used aslightly different form for F. All these models have in common the assumption that the parametersdo not change with time.-Our approach is based on dynamic models which are obtained by allowing the parameters in(1) to be time-dependent. A simple relationship is established between parameters in successivetimes and inference can be carried out. Forecasts are produced and the fit can be assessed. It willbe shown that simple models provide a better fit than do all models mentioned previously.Another approach to dynamic modelling was proposed by Meade.' As in Harvey, the meanresponse function is algebraically manipulated and described iteratively. Upon identification ofobservations with their mean responses, different parametrizationsare obtained. Our approach isalso based on transformations but these are performed only on the parameters; the observationsare unchanged.The main advantage of dynamic modelling is the allowance for changes. It may well be that 0changes (probably smoothly) with time as the environment changes (see Meade6). More generally,all the model parameters are subject to changes with time. The saturation level of the stock ofgoods (or the number of buyers in Oliver's terminology) is depressed if the economy is undergoinga period of recession. Models of such economic series should in general be capable of adaptation.Dynamic models are designed for this situation. See Migon and Gamerman7for an application.The next section outlines the basic ideas of the model, with a more detailed description of theinference procedure left to the appendix. Improvements on and generalizations of the basic logisticmodels are described and, after a brief discussion on measures of performance, the data on thestock of tractors in Spain are reanalysed.Correspondence:D. Gamerman,nstitute de Matematica, UFRJ, Caixa Postal 68530, 21944 Rio de Janeiro, RJ, Brazil

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Journal of the Operational Research Society Vol. 42, No. 2DYNAMIC MODELLING

The theory of dynamic models in forecasting was put forward by Harrison and Stevens.8 Itinvolves the specification of two components: the observation equation and the system equation.The former is given for logistic curves by (1) with the important difference that the parameters (a,b, 4)) re indexed by t. The local description is evident now, since for each time t a different logisticcurve is considered. The second component of the model provides the link between the differentcurves. In general, the curves for each time period are not entirely different but similar in thatsmall perturbations are expected to promote (minor) changes in the form of successive logisticcurves. This is parsimoniously modelled by a random walk in the parameters as follows:

at = at1 + V1t,bt= bt- 1 + V2t, (2)At = Ot-1 + v3t,

where v1t, v2t and v3t are the perturbations to the parameters. They are generally zero meanrandom variables with some known covariance structure. The larger the variances of the vts themore different are the logistic curves. If the variances are zero, the parametersremain constant intime and the global model (1) is obtained.The model in (1) is typically non-linear in the parameters. Exact results are not possible toobtain and alternative procedures should be pursued. One possibility is to seek parametric trans-formations to simplify the structure of the model. This is described briefly in the appendix alongwith an outline of the inferenceprocedure.A fullerdescription is given in Migon and Gamerman.]The main outputs are:(a) one- (or many-) steps-ahead predictions from any given time point. This is based on the meanof predictive distribution of Yt+kIDt, k = 1, 2, . . ., where Dt is the information set containingall observations up to time t;(b) on-line parametric trajectories based on the mean and variance of the posterior distribution ofthe parametersat all time points given in the appendix by equation (A4). The trajectoryof ( isexamined via the means of the distributions of AtIDt, Vt;(c) fitted values of the observed data series given by the mean response estimate based on all datainformation. This estimate is obtained from the mean value of the parameters given in equa-tion (A5).It is important to note that, despite the local behaviour, the saturation level a- can still beestimated at all times. In the dynamic setting the saturation level is allowed to change in time andthereforedifferent estimates are obtained for each time. The main interest is to obtain the estimateat the end of the data series, i.e. given Dn. Another useful addition to the model adopted by

Mar-Molinero is the use of variance laws to give different weight to the observations. There aremany possible weighting schemes and we adopted the Poisson law of variance where V[et] =ut 2. The variance increase with the mean is to be expected from count data, as is the case withthe stock of goods in general. Oliver4 showed the equivalence of his binomial model to a Poissonmodel for Yt As the data magnitude is large, it can be approximated by normal distributions withthe same variance law.The model also provides for an estimation of U2. As with the other parameters,it is allowed tochange with time owing to small perturbations. These are present to reflect the influence of theenvironment (or economic background)on the dispersion of the data.

GENERALIZED LOGISTIC CURVESThe logistic curve can be embedded in the more general family generated by

pi= a- b(Pt. (3)When i = -1, the logistic curve is obtained. The Gompertz curve (A= 0) is obtained by takingthe logarithm of the mean response and is similar in shape to the logistic curve. This gener-alization is given in Gregg et al.9 and is used here to assess the performance of the model as Avaries. In our assessment, A s taken as -1, 0 and 1. The latter is the modified exponential curve,

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D. Gamerman and H. S. Migon-Tractors in Spain-A Reanalysiswhich was not appropriate due to the nature of the data. Observe that as t -* oo, if 0 < + < 1,

t -* a. Therefore,all' (ea, when A= 0) is the limiting value or saturation level of the series provi-ded 0 < 0 < 1. Otherwise, no such limit exists.DATA ANALYSIS

The data consist of 26 yearly figures on tractor ownership in Spain given by Mar-Molinero.'The data were analysed with the three models described in the previous section. The prior dis-tribution for the parameters was centred around values within the range of the data with largevariances. The prior mean for 0 was 0.9, with an associated standard deviation of 0.2. Thevariances in equation (2) were set using the discount approach of Ameen and Harrison 0 withdiscount values 0.9.After analysing the data, final estimates for 0 and al/2 were obtained. These are reported inTable 1. The estimates for the logistic model are in the region of the estimates obtained by pre-vious authors. The estimate of 0 for the Gompertz model is larger implying a larger estimate forthe saturation level. The on-line estimates of 0 for the two models considered are shown in Figure1 with corresponding confidence limits. It shows that despite their common prior setting themodel learns from the data and processes the information in different ways. There is a suggestionthat the trajectories increase with the value of A.TABLE 1. Summary of estimation

Estimate Estimate of Sum of squaredModel of 0 saturation level fitted residualsLogistic (1) 0.823 52.08 4.98Logistic with 0.829 53.72 4.72additive interceptLogistic with 0.823 52.08 2.57

AR(1) errorsLogistic with 0.826 55.84 4.40differenceoperatorBinomial 0.840 55.49 n/aDynamic/Gompertz 0.932 93.44 1.88Dynamic/logistic 0.835 56.09 1.51

There are many ways to measure the performance of a model. One could use a long-termforecast as the basis, but a simple and effective scheme is to consider one-step-ahead forecasts.They can be compared with data forming a set of one-step-ahead forecast residuals utgiven byt = t- yt = Yt E[ytIDt-,], t = 1, 2, ..., n -1.

n-1It is common to use E 2 as a performancemeasure. Another approach is to consider the fit oft= 1the model to the data and measure the model by how well it adjusts to the observed data. A set offitted residuals fitgiven byfat= yt - t = yt -E[pt IDJ] t = 1, ..,n,

n-1is now formed. The model is then evaluated by consideration of E u2. Measures based on predic-t= 1tion, which is essentially an extrapolative fit, should be preferredin forecasting system. Neverthe-less, fitted figures are also presented here for the assessment of models relative to those previouslyused. The fits obtained with the two dynamic models are better than ever before, as can be seenfrom Table 1. The sums of squared forecast residuals are 14.26 and 13.18 for the Gompertz andlogistic models, respectively. These models have similar performance in terms of fit and predictionand therefore one cannot be ruled out in favour of the other despite the difference evidenced inestimation. The fitted figures are substantially lower than those obtained with forecast errors,which stresses the additional difficulty associated with prediction.

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Journal of the Operational Research Society Vol. 42, No. 2Logistic model1.10 _

1.00 _0.90_0.80 -0.70 -0.60 l l l l l l1950 1955 1960 1965 1970 1975 1980

Year

Gompertz model1.201.10 -1.00 '0.900.80 -0.70 _0.60 | I l l1950 1955 1960 1965 1970 1975 1980

YearFIG. 1. 4 estimates: mean, ; 2 s. d. limits,- -

Figure 2 shows the one-step-ahead forecasts for yt, t = 1, ..., n, with confidence bounds for theGompertz and logistic models. The performance is considered satisfactory for both, as all thehistorical data is included in the narrow confidence limits around the forecasts. The figure alsoshows forecasts up to six years ahead at the end of the data. The effect of the discount factors canbe appreciated here. They downweigh the data information and, when one predicts further intothe future,these are less important. As a result, the confidence band gets larger and larger.The residual autocorrelation was also analysed. This was also motivated by the improvementon the fit obtained by Mar-Molinero after allowing for an AR(1) error structure. Fitted andforecast residuals can be-considered in the dynamic approach. For the logistic model these wereestimated as 0.43 and 0.58 respectively,whereas for the Gompertz model these were 0.45 and 0.66.This again shows similarities in both models. The autocorrelations are not negligible, as wasexpected for cumulative series, but the fit of the model was considered satisfactory and in thename of parsimony the models remained unchanged.

SUMMARYThis paper provides an alternative way of analysing the data of the tractors in Spain. Thedynamic Bayesian methodology leads to an adaptive procedure which enables calculation ofparameter estimates, prediction and implementation of many additional features. This is easily

achieved within the context of the method.Simple models were used and results show the improvement obtained. The key element is therecognition that dynamic models are essential to the study of time series data. In fact, data setsbearing some indexation on time or any other variable subject to environmental changes shouldbe approached in the same way.The use of the logistic curve to model the purchase of goods in a population is well known inthe literature. Despite their non-linear behaviour, approximating techniques efficiently simplify thecomputations. (A simple APL program provides the results of the analysis in a few seconds on a122



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D. Gamerman and H. S. Migon-Tractors in Spain-A ReanalysisLogistic model60.00 -

40.00 - -n/

--

20.00 - X

0.001950 1955 1960 1965 1970 1975 1980 1985Year

Gompertz model60.00 -

/ /////0.00 _-,/

/X/'

20.00 _

0.001950 1955 1960 1965 1970 1975 1980 1985Year

FIG.2. Forecasts: data, x; forecasts, ; 1.5 s. d. limits, - -.

microcomputer.)Improvementsin the fit can be obtained through techniques such as intervention,incorporation of expert opinion and further elaboration of the model. These are features of thedynamic approach and despite not being required for this application can be readily accessedwhenevernecessary.APPENDIX

Reparametrization and Inference ProceduresConsider a vector sequence {01, 02, 03}t specifiedby

01,= t ,t -l + 02,t-1 (Al)02, = 03,t-102,t-1 (A2)03,, = 03, t- 1 (A3)

that can be concisely written as Ot = G(Ot i). 01 is the current level to which an increment 02 isadded. This increment is inflated or deflated by 03. Proceeding iteratively backward in time, by(Al) and (A2) at t - 1,01,t = [01,t-2 + 02,t-2] + 03,t-202,t-2

= 01,t-2 + 02,t-2(1 + 03,t-2)= [01,t-3 + 02,t-3] + [03,t-3 + 02,t-3](l + 03,t-3), by (Al), (A2) and (A3) at t -2= 01,t-3 + 02,t-3(l + 03,t-3 + 03,t-3).

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Journal of the OperationalResearch Society Vol. 42, No. 2If the process is continued up to time t = 0,

01, t = 01, 0 + 02, (1 + 03,0 + 03 + 3*+ 0g)t- 1

= 01,0 + 02,0 E O03,0j=0= 01, 0 + 02, 0 X 1 01 -03, 0= 01,0 + 2,0 - 1 0, 03,0

Upon identification of a = 01, + 02, /(1 - 03, ) b =_ 02, 0/( -03, o) and + = 03, 0, the originalparametrizationin (3) is obtained. The parametric description is completed by setting ut = 01, t toreplace equation (3). The system equation becomes dynamic by addition of a three-dimensionalperturbation ot to G(Ot-1).The relations (A1)-(A3) are assumed to hold on average.Under this parametrization,the observation equation becomesyt = pt + et where u' = FO and et .N(0, U21ut)where Ft = (1, 0, 0) is a row vector. Setting mt = E[OtIDt] and r2Ct = V[Ot Dt, a2], one canobtain a recursive relation to update mean and variance of parametric distribution by Kalmanfilter type equations:

Mt= G(mt-1) + stq 1(gt '4),Ct = Rt - s s(1 - pt/qt)/qt, (A4)where st = R F qt= R = G'(mt -1)Ct - 1 G'(m_1)]T + Wt. Wt= V[Cot], gt = E[jiI Dt] andPt= V[,ui Dt]. G' is the matrix of derivatives of G with element Gij= aGi/aOj,1 < i, j < 3. Esti-mates of a, b and 0 are calculated from components of mt.

The variance U72 is also sequentially updated via a gamma distribution with parameters ct=J Ot-1 + 2 and fit = 3vft-l + ui/2 where bvis the variance discount factor set in the applicationas 0.98 and it is the standardizedforecast residual at time t.The information is redistributed back via a smoothing backward algorithm. This algorithmsuccessively relatesm' = E[OtIDj and r2C,= V[Ot D, a2] viamn = mt + Lt[mn+ 1 -G(mt)],Cn = Ct - Lt[Rt- Cn+1]Lf, (A5)

whereLt = Ct[G'(mt + 1)] TR-+1-

Acknowledgements-We are grateful to the referees for useful comments. This research was supported by grants fromCNPq and FAPERJ.

REFERENCES1. C. MAR-MOLINERO (1980)Tractors in Spain: a logistic analysis. J. OplRes. Soc. 31, 141-152.2. F. R. OLIVER (1981) Tractors in Spain: a further logistic analysis. J. OplRes. Soc. 32, 499-502.3. A. C. HARVEY (1984)Time series forecasting based on the logistic curve. J. Opl Res. Soc. 35, 641-646.4. R. M. OLIVER (1987) A Bayesian model to predict saturation and logistic growth. J. OplRes. Soc. 38, 49-56.5. N. MEADE (1985) Forecasting using growth curves-an adaptative approach. J. OplRes. Soc. 36, 1103-1115.6. N. MEADE (1988) Forecasting with growth curves-the effect of errorstructure.J. of Forecasting 7, 235-244.7. H. S. MIGON and D. GAMERMAN (1989) Generalised exponential growth models: a Bayesian approach. TechnicalReport No. 41, LES/IM, Universidade Federal do Rio de Janeiro.8. P. J. HARRISON and C. F. STEVENS (1976) Bayesian forecasting (with discussion). J. R. Statist. Soc. B., 38, 205-247.9. J. V. GREGG, C. H. HASSEL and T. J. RICHARDSON (1964) Mathematical Trend Curve: An Aid to Forecasting. Oliver &Boyd, Edinburgh.10. J. R. M. AMEEN and P. J. HARRISON (1985) Normal discount Bayesian models. In Bayesian Statistics 2 (J. M. BER-

NARDO, M. H. DEGROOT, D. V. LINDLEY and A. F. M. SMITH, Eds.) University Press, Valencia.

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Tractors in Spain- A Dynamiv=c Reanalysis