Dear mathematicians, dear statisticians, dear econometricians.
Dear TIMESERIES and ORNET listers.
I am not aware of any simple explanation of why one of the well-known Golden numbers,
namely 0.61803398…, occurs as the smoothing constant alpha in SES in case the ratio of
the two error variances in the state-space model known as ‘random work plus noise’ model
is unity. It is well known that SES is optimal for such a model and the model is considered
to be a model underlying simple exponential smoothing.
In other words, why the steady-state updating equations with equal error variances
correspond to a simple exponential smoothing with a smoothing constant being equal
precisely to the limit of the ratio of two successive terms of the Fibonacci sequence?
Can the community of scientists help a curious amateur find a simple reason WHY? There is a
great deal of scientific theory and practical observation that can be linked to the Golden
numbers to make me not think of the above observation of mine as a mere coincidence. With the
relationship 0.618033…* 1.618033…= 1, the Golden means are as much important in mathematics
as 3.141592… and 2.718281… This raises another question of whether the Muth’s discovery of 1960
has been sufficiently evaluated and appreciated by modern mathematicians, statisticians and
econometricians. Thanks.
Regards,
Andrey Kostenko,
21.06.05
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
1. See glossaries/dictionaries/Internet for entries such as Golden ratio/section/number/mean;
Fibonacci numbers/sequence
2. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
3. http://www.goldenmuseum.com
4. http://www.engineering.sdstate.edu/~fib/ - an academic journal devoted to the study of
Fibonacci numbers
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