Application of time series models

Forecasting is an ultimate aim in the study of time series analysis. Anyone who is engaged in planning, controlling and managing projects, personnel, finance and operations will be interested in knowing what will happen in future with the analysis of the available data.

shown in Table 2. The autocorrelations for z t 's are large and fail to die out at higher lags. While simple differencing reduces the correlation in general, a very heavy periodic component remains. This is evidenced particularly by very large autocorrelations at lags 12, 24, 36 and 48. Simple differencing with respect to period 12 results in autocorrelations which are first persistently positive and persistently negative. The differencing ∇∇ 12 markedly reduces autocorrelations throughout. The large autocorrelations at lag 1 & lag 12 indicate that the model will have one regular moving average operator and seasonal moving average operator using the hypothesis that autocorrelation function of a moving average process of order q has a cut off after lag q, while its partial autocorrelation fails off. The model can be written in the form w t = (1-θB) (1-ΘB 12 ) a t , w t = ∇∇ 12 z t = (1 -θB -ΘB 12 + θΘB 13 ) a t w t+k = a t+k -θ a t+k-1 -Θ a t+k-12 + θΘa t+k-13 The autocovariance at lag k is defined as It is found that for the airline data ρ 1 = -0.40, ρ 12 = -0.339 Substitute the values of ρ 1 and ρ 12 in 1.1 and 1.2 We obtain 0.40 θ 2 -θ + 0.40 = 0 0.339 Θ 2 -Θ + 0.339 = 0 Solving these equations we obtain the initial estimates for θ and Θ θ ) = 0.50, Θ ) = 0.3907

Estimation of parameters
Corresponding to N = nd -sD, where d is the order of regular differencing, D is the order of seasonal differencing with period `s' the unconditional loglikelihood for the multiplicative model (p,d,q) x (P,D,Q) 12 is given by Where β is a general symbol for k = p+q+P+Q parameters.
The unconditional sum of squares function is given by: [a t / β, w] denotes the expectation of a t conditional on β and w.

Iterative calculation of least square estimates
For the multiplicative and θ, Θ are assigned initial values. The derivatives x 1,t , x 2,t are calculated numerically.
For the average residential electricity usage for the data, taking the preliminary estimates θ ) = 0.50, Θ ) = 0.3907 and using the above process the iterative estimation of θ and Θ are tabulated as in Table 4.
The least square estimates are

Cumulative periodogram check
If the model were adequate and parameter known exactly, the plot of C(f j ) against f j would be scattered about a straight line joining the points (0,0) and (0.5,1). For the model identified using ARIMA multiplicative model C(f j )'s are found and the plot of C(f j ) against f j will be useful.
The Kolmogorov-Smirnov 5% and 25% probability limits supplying a very rough guide to the significance of apparent deviation fail in this instance to indicate any significant departure from the assumed model. The limit lines are drawn at distances ± K ε / √q. The multiplicative seasonal model (Box & Jenkins 1976) for the air-line data is written in the form z t-l = z t+l-1 + z t+l-12 -z t+l-13 + a t+l -θa t+l-1 -Θ a t+l-12 + θΘa t+l-13 where θ = 0.692, Θ = 0.8866 The minimum mean square error forecast at lead timel at origin t is given by ẑ t (l) = [z t+l-1 +z t+l-12 -z t+l-13 + a t+l -θ a t+l-1 -Θ a t+l-12 + θΘ a t+l-13 ] Also (i). [z t+l ] = E (z t+l / θ, Θ, zt, z t-1 ....] is the conditional expectation of z t+l taken at origin t. (ii).Invertible model fitted to actual data usually yield forecast which depend only on recent value of the series.
(iii). The forecast are insensitive to small changes in parameter values such as the one introduced by estimation errors.
The unknown z t 's are replaced by forecasts and the unknown a t 's are replaced by zeroes.   The logged airline data of UK of total miles flown for the period January 1964 to December 1970 is given in Table 5.

Identification of ARARMA model
Since the airline data is a non-stationary time series, we transform it into stationary time series using the transformation The most significant log τ is defined as the value of τ minimizing E rr (τ) given in equation Cumulative periodogram check C(f j )'s are found and the plot of C(f j ) against f j are drawn. The Kolmogorov-Smirnov 5% and 25% probability limits supply a very rough guide to the significance of apparent deviation and fail in this instance to indicate any significant departure from the assumed model. The limit lines are drawn at distances ± K ε /√q.
Identification of fractional ARIMA model It may be seen that the autocorrelations decay very slowly and it indicates the long-term persistence in the series. The value of d is estimated using equation 1.4.1 and it is found to be d = -0.0538 as given in Table 6. The transformed series can be modeled using either methods proposed by Box and Jenkins (1976) or autoregressive scheme proposed by Parzen (1982). The Kolmogorov-Smirnov 5% and 25% probability limits supply very rough guide to the significance of apparent deviations.

Analysis of Chennai city traffic accidents data
The number of accidents in Chennai city traffic for the period 1987-1997 is taken for analysis. The logged data of Chennai city traffic accidents is given in Table 7. Autoregressive scheme, ARIMA process, fractional ARIMA process, ARARMA models are identified simultaneously after applying the usual stochastic model building procedure.
The models for the Chennai city traffic accidents are listed below: The autoregressive model is φ(B)z t = a t , where