Applications of Mathematics

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1 Applications of Mathematics Tomáš Cipra Exponential smoothing for irregular data Applications of Mathematics Vol ) No Persistent URL: Terms of use: Institute of Mathematics AS CR 2006 Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library

2 5 2006) APPLICATIONS OF MATHEMATICS No EXPONENTIAL SMOOTHING FOR IRREGULAR DATA* Tomáš Cipra Praha Received March in revised version June ) Abstract. Various types of exponential smoothing for data observed at irregular time intervals are surveyed. Double exponential smoothing and some modifications of Holt s method for this type of data are suggested. A real data example compares double exponential smoothing and Wright s modification of Holt s method for data observed at irregular time intervals. Keywords: double exponential smoothing exponential smoothing Holt s method Holt- Winters method irregular time intervals missing observations time series MSC 2000 : 62M0 62M20 90A20 60G35. Introduction Various types of exponential smoothing seem to be a simple and practical instrument for smoothing and forecasting time series. Moreover modifications of exponential smoothing have been developed which can be used in special situations e.g. robust exponential smoothing for time series with outliers e.g. [] [3] [4]) exponential smoothing for exponential or damped trends e.g. [6]) exponential smoothing for multivariate time series e.g. [2] [6]) and others. Another special time series typical in practice are data observed or published) at irregular time intervals. One may denote them briefly as irregular data ) y t y t2 y t3... Such a situation can include as a special case time series with missing observations see also [2] [5] [8]). Simple exponential smoothing and Holt s method for irregular * This research has been supported by the Ministry of Education Youth and Sports of the Czech Republic Research Project MSM and by the Grant Agency of Charles University Grant No. 342/

3 data have been suggested by Wright see [9]). This approach is recalled in Section 2. Double exponential smoothing for irregular data is suggested in Section 3. Some modifications of Holt s method including the seasonal additive and multiplicative Holt-Winters methods are shown in Section 4. Finally a real data example in Section 5 demonstrates Holt s method from Section 2 and double exponential smoothing from Section 3 numerically for irregular data. 2. Simple exponential smoothing and Holt s method for irregular data Wright s simple exponential smoothing for irregular data i.e. the exponential smoothing method with one smoothing coefficient for local constant trends) has the form which generalizes in a natural way the regular case 2) 3) 4) S tn = y tn + )S tn β tn tn + ŷ tn+k ) = S tn k 0. It is easy to show that 2) and 3) are recursive versions of relations 5) 6) n S tn = β tn tj β tn tj y tj which generalize the well-known relations of the regular simple exponential smoothing. The discount coefficient β 0 < β < ) or the smoothing coefficient α = β would be used in the regular case but in the irregular case one must recalculate a nonconstant smoothing coefficient recursively according to 3). The initial values of the smoothing statistics S 0 and the smoothing coefficient α 0 are recommended as 7) 8) S 0 = S t0 = n 0 y tj n 0 α 0 = α t0 = = β q = α) q β q j j=0 where n 0 is the length of an initial block of observations e.g. n 0 = 6 or n 0 = n/4) and q is the average time spacing of the data. 598

4 Similarly Wright s modification of Holt s method for irregular data i.e. the ad hoc exponential smoothing method with two smoothing coefficients for local linear trends) has the form 9) 0) ) 2) S tn = y tn + )[S tn + )T tn ] S tn S tn T tn = γ tn + γ tn )T tn γ tn γ tn = α) tn tn + γ) tn tn + γ tn ŷ tn+k ) = S tn + kt tn k 0. The initial values of the level S 0 the slope T 0 and the smoothing coefficients α 0 and γ 0 are recommended similarly as for the irregular simple exponential smoothing i.e. 3) 4) 5) S 0 = S t0 = ˆb 0 0) T 0 = T t0 = ˆb 0) α 0 = α t0 = α) q γ 0 = γ t0 = γ) q where ˆb 0 0) and ˆb 0) are regression estimates of the linear trend b 0 +b t in an initial block of observations with length n 0 the smoothing coefficients α and γ 0 < α < 0 < γ < ) would be used in the regular case and q is again the average time spacing of the data. 3. Double exponential smoothing for irregular data Double exponential smoothing for irregular data i.e. the exponential smoothing method with one smoothing coefficient for local linear trends) is not an ad hoc procedure as Holt s method in the previous section and has the form 6) 7) 8) 9) S tn = y tn + )S tn S [2] = S tn + )S [2] = β tn tn + β tn tn + )β tn tn w z tn z tn = β tn tn + αtn z ztn ŷ tn+k ) = S tn + + z ) k S tn S [2] t α n ) k 0. tn One can see that in addition to the smoothing coefficient α one must recalculate recursively two auxiliary values w and z see 8)). It is not difficult to show that in 599

5 the regular case one obtains from 6) 9) the regular double exponential smoothing formulas the so called Brown s method). The initial values of the smoothing statistics S 0 and S [2] 0 and the values α 0 w 0 and z 0 can be recommended as 20) 2) 22) S 0 = S t0 = ˆb 0 0) qβq β q ˆb 0) S [2] 0 = S [2] t 0 = ˆb 0 0) 2 qβq β ˆb q 0) α 0 = α t0 = β q w 0 = w t0 = βq ) 2 qβ q z 0 = z t0 = βq ) 2 qβ q where ˆb 0 0) ˆb 0) β = α and q are as in the irregular Holt s method in Section 2. Derivation of double exponential smoothing for irregular data: One can proceed similarly as for the regular double exponential smoothing see e.g. [7]) in the model 23) y tj = b 0 + b t j + ε tj j = 2... where the residuals ε form a white noise. Let us denote 24) 25) 26) S tn = S [2] = β tn tj = n β tn tj y tj β tn tj S tj z tn = t j )β tn tj α tj w tj β tn tj for n >. Then it is easy to verify that the recursive relations 6) 8) hold. Moreover the expectation of the smoothing statistics S satisfies 27) ES tn ) = n = b 0 + b ) b 0 + b t j )β tn tj β tn tj b = b 0 + b t j b n = Ey tn ) b t j )β tn tj and similarly for the expectation of the double smoothing statistics S [2] 28) ES [2] ) = ES tn ) b z tn. 600

6 Now one can construct the unbiased estimator of b according to 28) as 29) ˆb ) = z S tn S [2] ) and the unbiased estimator of Ey tn ) according to 27) and 29) as 30) ŷ tn = S tn + ˆb ) α = S tn + z S tn S [2] ). Hence one obtains finally the unbiased predictor 9) as ztn 3) ŷ tn+k ) = ŷ tn + kˆb ) = S tn + + z ) k S tn S [2] t α n ) k 0. tn As the initial values 20) 2) are concerned one can use relations 32) 33) ˆb ) = z S tn S [2] ) ˆb0 ) = ŷ tn ) ˆb ztn ) = S tn + z a tn ) S tn S [2] ) for = 0 and e.g. the initial value w 0 in 22) follows as 34) w 0 = w t0 = = qjβ q j j=0 βq )2 qβ q. 4. Modifications of Holt s method for irregular data Further modifications of Holt s method for irregular data 9) 2) may be useful in practical situations. In the case of exponential trends one can use 35) 36) 37) S tn = y tn + )S tn T tn tn T tn = γ tn Stn S tn ) /tn ) ŷ tn+k ) = S tn T k k 0 + γ tn )T tn where S is the level and T is the base of the power. 60

7 In the case of damped linear trends one can use 38) 39) 40) t n S tn = y tn + ) S tn + ϕ i T tn ) T tn = γ tn ϕ tn tn i= i= ϕ i S tn S tn ) + γ tn )ϕ tn tn T tn ŷ tn+k ) = S tn + k ϕ i T tn k 0 i= where S is the level T is the slope and ϕ is the damping coefficient 0 < ϕ < ). The smoothing coefficients both for 35) 37) and for 38) 40) are calculated according to ). In the case of seasonality one can use the modification of Holt-Winters method i.e. the ad hoc exponential smoothing method with three smoothing coefficients for local linear trends combined with seasonality) for irregular data suggested by Cipra see [5]). The additive method is 4) 42) 43) S tn = y tn I t n ) + )[S tn + )T tn ] T tn = γ tn S tn S tn + γ tn )T tn I tn = δ tn y tn S tn ) + δ tn )I t n 44) γ tn γ tn = α) tn tn + γ) tn tn + γ tn δ tn = δ tn δ) tn tn + δ tn 45) 46) ŷ tn = S tn + I tn ŷ tn+k ) = S tn + kt tn + I tn+k) k > 0 where S is the level T is the slope I is the seasonal index p is the length of season t n is the largest value among 2... such that the time t n corresponds to the same seasonal period as the time and + k) is the largest value among... such that the time + k) corresponds to the same seasonal period as the time + k. 602

8 Analogously the multiplicative method includes the relations 42) 44) and 47) 48) 49) 50) S tn = y tn I t n + )[S tn + )T tn ] y tn I tn = δ tn + δ tn )I t S n tn ŷ tn = S tn I tn ŷ tn+k ) = S tn + kt tn )I tn+k) k > An example In Figs. and 2 one considers relative price changes of investment units in basis points 0.0 %) on 60 irregular dates for a Czech investment fund: 2 on January 3 on January 6 26 on January on June 30. In addition to smoothing predicted values for July 8 should be constructed. In Fig. one has used Wright s modification of Holt s method for irregular data according to 9) 2) with α = 0. γ = 0. n 0 = 6 q = 3. The mean square error for 60 historical observations is 63.9 i.e. MSE is 2.8 basis points). The predicted values decrease linearly since T t60 in 2) is 0.90) Basis points Days Figure. Relative price changes of investment units in basis points) on irregular dates for a Czech investment fund dotted line) and its smoothed and predicted values by means of Wright s modification of Holt s method for irregular data full line). In Fig. 2 one has used double exponential smoothing on irregular data according to 6) 9) with α = 0. n 0 = 6 q = 3. The mean square error for 60 historical observations is 58.6 i.e. MSE is only 7.7 basis points). In contrast to Fig. the predicted values increase linearly according to the substance of the method suggested 603

9 Basis points Days Figure 2. Relative price changes of investment units in basis points) on irregular dates for a Czech investment fund dotted line) and its smoothed and predicted values by means of double exponential smoothing for irregular data full line). and match better the real data. Similar conclusions hold also for other real data applications. References [] D. Bote: Exponential smoothing with missing observations and outliers. Proceedings of the 6th Annual Conference of Doctoral Students Part I J. Šafránková ed.). Faculty of Mathematics and Physics Charles University of Prague 997 pp [2] T. Cipra: Some problems of exponential smoothing. Apl. Mat ) Zbl [3] T. Cipra: Robust exponential smoothing. Journal of Forecasting 992) [4] T. Cipra: Dynamic credibility with outliers and missing observations. Appl. Math ) Zbl [5] T. Cipra J. Trujillo A. Rubio: Holt-Winter s method with missing observations. Manage. Sci ) Zbl [6] E. S. Gardner: Exponential smoothing: the state of the art. Journal of Forecasting 4 985) 28. [7] D. C. Montgomery L. A. Johnson: Forecasting and Time Series Analysis. McGraw-Hill New York 976. Zbl [8] T. Ratinger: Seasonal time series with missing observations. Appl. Math ) Zbl [9] D. J. Wright: Forecasting data published at irregular time intervals using an extension of Holt s method. Manage. Sci ) Author s address: T. Cipra Dept. of Statistics Faculty of Mathematics and Physics Charles University Sokolovská Praha 8 cipra@karlin.mff.cuni.cz. 604