Dear Nuno, dear All.
Reference [7] contains a system of two equations marked as (1). Just
after explaining the notation there's another equation linking alpha
to an algebraic structure. To obtain the golden alpha just put equal
values to V and W.
Reference [8] contains a similar system on page 711. The equation for
alpha follows immediately.
Reference [11] contains equations (5) and (6), which form the system
identical to system (1) from reference [7]. The equation for alpha
follows immediately, just put c=1 to get the golden mean.
Reference [1]: pages 302-303, equations (3.1), (3.2) and (3.9).
All the references listed (except for [6]) include both the
system of interest and an equation for alpha. None of the references below
directs one to golden numbers explicitly or implicitly.
The system of equations (5) and (6) in [11] involves two random
variables and presents a sort of convolution of each of their
distributions. Iff the variances of these RV are equal to each other,
one gets the golden alpha as a result of the equation for a smoothing
constant alpha. The equation for alpha links a statistical model (the
system of (5) and (6)) to the method known as simple exponential smoothing
in a simple way as shown in [6] and explained in [1].
---
Best wishes,
Andrey Kostenko
Some relevant sources:
>>
>> N.B.: Ignore equation (3) in [10], which is relevant to the subject
>> but incorrect.
>>
>> 1. Muth JF (1960) Optimal properties of exponentially weighted
>> forecasts.
>> J. Amer. Statist. Assoc., 55, 299-306
>> 2. Harrison PJ (1967) Exponential smoothing and short-term sales
>> forecasting.
>> Management Science, 13, 821-842
>> 3. Harvey A (1984) A unified view of statistical forecasting
>> procedures.
>> Journal of Forecasting, 3, 245-283
>> 4. Johnston FR and Harrison PJ (1986) The variance of the lead-time
>> demand.
>> Journal of the Operational Research Society, 37, 303-308
>> 5. Harvey A and Snyder R (1990) Structural time series models in
>> inventory control.
>> International Journal of Forecasting, 6, 187-198
>> 6. Kekre S, Morton Th and Smunt L (1990) Forecasting using partially
>> known demands.
>> International Journal of Forecasting, 6, 115-125
>> 7. Johnston FR (1993) Exponentially Weighted Moved Average (EWMA)
>> with irregular updating periods.
>> Journal of the Operational Research Society, 44, 711-716
>> 8. Johnston F and Boylan J (1994) How far ahead can EWMA model be
>> extrapolated.
>> Journal of the Operational Research Society, 45, 710-713
>> 9. Johnston F, Boylan J, Shale, E. and Meadows M. (1999)A robust
>> forecasting system, based on
>> the combination of two simple moving averages. Journal of the Operational
>> Research Society, 50, 1199-1204
>> [10]. Johnston F, Boylan J, Meadows M and Shale, E (1999) Some
>> properties of a simple moving
>> average when applied to forecasting a time series. Journal of the
>> Operational Research Society, 50, 1267-1271
>> 11. Chatfield C, Koehler A, Ord. J and Snyder R (2001) A new look at
>> models for exponential smoothing.
>> The Statistician, 50, 147-159
>> 12. Boylan J and Johnston F (2003) Optimality and robustness of
>> combinations of moving averages.
>> Journal of the Operational Research Society, 54, 109-115
>>
Monday, June 27, 2005, 8:19:46 AM, you wrote:
NC> Dear Andrey,
NC> Could you please direct me for a direct reference and page where this
NC> optimal parameter appears with this value?
NC> Regards,
NC> NC
NC> _______________________
NC> Nuno Crato
NC> Dept of Mathematics, ISEG
NC> R. Miguel Lupi 20,
NC> 1200-871 Lisboa, Portugal
NC> Fax/Phone: +351 213925846
NC> [log in to unmask]
NC> http://www.iseg.utl.pt/~ncrato
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