General correcting formula of forecasting?

Abstract

The article raises the question of possible existence of gaps, ruptures in the probability scale and of possible values of these ruptures. The calculations give 1/3 of the standard deviation as the minimal value of these ruptures for a number of standard distributions.

Full text

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General correcting formula of forecasting?

Alexander Harin

Moscow Institute of Physics and Technology
Modern University for the Humanities



A general correcting formula of forecasting (as a
framework for long-use and standardized forecasts) is
proposed. The formula provides new forecasting resources
and areas of application including economic forecasting.



Contents

Introduction …………………………………………………… 1

1. Approaches …………………………………………………….
1.1. The principle of uncertain future
1.2. Optimal frames of reference
1.3. Formula of forecasting as a framework for forecasts
1.4. New forecasting resources and areas of application
2. Development of the formula. Errors …………………………
2.1. Ideal initial circumstances
2.1.1. Example. Hiroshima 1945
2.2. Non-ideal initial circumstances
3. Development of the formula. Functions ……………………..
3.1. Additive and multiplicative functions
3.2. Other functions
3.3. Versions of the formula. Transformations
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7

Conclusions ……………………………………………………
References ……………………………………………………..
10
11





Introduction
This paper presents in English, develops and generalizes the results of Harin
(2008) and a part of the results of Harin (2009). From other works of the item see,
e.g., Tsay (2008), Kasa (2000).

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1. Approaches
1.1. The principle of uncertain future
General principle of uncertain future:
“A future event contains an uncertainty”
Specific principle of uncertain future:
“The estimation of the probability of a future event should (manifestly)
contain an uncertainty”
or,
Pestimated ≈ Pestimated mean ± ΔP
Development and applications of the principle see in (Harin 2005 – Harin
2009).

1.1.1. The first consequence of the principle
1.1.1.1. A far analogy. Vibrations near a rigid wall
Suppose an electro-drill or any similar device, e.g., sewing-machine,
vibrosieve, machine-gun, electric hammer etc. which (when working) can vibrate
quickly. Presume the device has rigid flank sides and vibrates with the amplitude
of, say, 1 mm.
Can we approach a flank side of the non-working drill (or of the device) to a
rigid wall or ledge:
A) as close as at the distance, say, 0.1 mm;
B) tightly?
Certainly. Both A) and B).
And now turn the drill (the device) on. What will be the distance from the
rigid wall to the working drill? Vibrations will repulse, shift the drill from the wall.
Due to the vibrations:
A) the distance from the drill to the wall will be more than 0.1 mm;
B) the gap, rupture will arise between the drill and the wall.

1.1.1.2. An example. Aiming firing at a target
General conditions
Suppose a hypothetic transportable testing stand, arrangement for testing the
quality of rifles, guns, cartridges etc. To avoid human errors, the arrangement is
made in the form of a standing man, a rifle is fasten onto the arrangement and the
aiming is performed automatically. Suppose firing errors are minimized and are
much less than one point of the target.
Suppose the arrangement is placed near a railway or Metro. The vibrations of
the ground increase firing errors up to, say, 2 points. For the sake of simplicity,
assume the target is strongly elongated in one of directions. So, the consideration is
reduced to one-dimensional and uniform (without effects of curvature) case.
Suppose the points are located in the scale from “0” to “10”: “9”, “8”, “7” etc. are
located after “10”. Before “0” there is the blank space which is equivalent to “0”.
Suppose following dispersion takes place: one shot =exact; one shot =+2
points; one shot =-2 points.
If the aiming is performed at, say, “7”, the mean result is the same as the
aiming value. The result is (7+9+5)/3=7.

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A) The shift from the bounds to the middle of the target scale
If the aiming is performed at “9”, one bullet should hit beyond the bound “10”
at “11”, but really hits at “9”. The result is (9+9+7)/3=25/3=8⅓. One bullet,
instead of “11”, hits “9”, i.e. 2 less than the aiming value. The mean result is
shifted from the bound (from “10”) to the middle (to ~“5”) of the scale by 2/3
points.
If the aiming is performed at “1”, one bullet should hit beyond the bound “0”
at “-1”, but really hits at the blank space which is equivalent to “0”. The result is
(1+3+0)/3=1⅓. One bullet, instead of “-1”, hits “0”, i.e. 1 more than the aiming
value. The mean result is shifted from the bound (from “0”) to the middle (to ~“5”)
of the scale by 1/3 points.
A) The dispersion causes the shifts of the mean results from the bounds
to the middle of the target scale.

B) The ruptures in the target scale
If the aiming is performed at the bound of the target scale “10”, one bullet
should hit beyond the bound “10” at “12”, but really hits at “8”. The result is
(10+8+8)/3=26/3=8⅔. One bullet, instead of “12”, hits “8”, i.e. 4 less than the
aiming value. The rupture between the mean result and the bound “10” of the scale
is 1⅓ points.
If the aiming is performed at the bound of the scale “0”, one bullet should hit
beyond the bound “0” at “-2”, but really hits at the blank space which is equivalent
to “0”. The result is (0+2+0)/3=2/3. One bullet, instead of “-2”, hits “0”, i.e. 2
more than the aiming value. The rupture between the mean result and the bound
“0” of the scale is 2/3 points.
B) The dispersion causes the ruptures near the bounds of the target
scale.

1.1.1.3. The first consequence of the principle
Suppose we wish to test a probability value P, which is very close (but not
equal) to the bound Pbound of the probability scale, i.e. to 0% or 100%. The mean-
square error, the uncertainty value of the estimation of P is ΔP (taking into
account Novosyolov 2009). Let us examine two cases. Name them conditionally:
certain (P=Pcertain and ΔP=ΔPcertain) and uncertain (P=Puncertain and
ΔP=ΔPuncertain). Suppose the certain case is an initial one and the uncertain case is a
final one. Suppose for both cases the number of trials, tests, outcomes etc. is the
same. So, the difference, change ΔPuncertain-ΔPcertain is defined only by the
difference, change of noises, disturbances etc.
Suppose in the certain case the uncertainty value of the probability estimation
is equal to 0 or is much less than the difference between the initial probability
Pcertain and the bound of the probability scale Pbound
ΔPcertain << |Pbound - Pcertain|
Suppose in the uncertain case the uncertainty value of the probability
estimation is more than the difference between the initial probability Pcertain and
the bound of the probability scale Pbound
ΔPuncertain > |Pbound - Pcertain|

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A) Shifts in the probability scale
If, due to increasing of noises, the uncertainty ΔP of the probability
estimation increases from initial ΔPcertain<<|Pbound-Pcertain| to final
ΔPuncertain>|Pbound-Pcertain|, then the probability will be shifted, “pushed away” by
the noises from the bound to the middle of the probability scale.
Indeed, if, e.g., for initial Pcertain=99%, the uncertainty of the probability
estimation increases from initial ΔPcertain<<1% to final ΔPuncertain=5%, then,
evidently, the probability will be shifted from initial Pcertain=99% to final
Puncertain<99%. Similarly, if for initial Pcertain=1%, the uncertainty of the
probability estimation increases from initial ΔPcertain<<1% to final ΔPuncertain=5%,
then, evidently, the probability will be shifted from initial Pcertain=1% to final
Puncertain>1%.
A) The increasing of the probability estimation uncertainty (caused by
the increasing of the noises) shifts the probability from the bounds to the middle of
the probability scale. At high probabilities, the final probability Puncertain will be
lower than the initial Pcertain. At low probabilities, the final probability Puncertain
will be higher (*without the influence of the second consequence of the principle)
than the initial Pcertain.
Phigh uncertain < Phigh certain
*Plow uncertain > Plow certain

B) Ruptures in the probability scale
If the mean-square error, the uncertainty value ΔPcertain of the estimation of
the probability P is equal to zero, then the probability Pcertain can be arbitrarily
close to the bound Pbound of the probability scale. If, due to noises, the uncertainty
value ΔPuncertain of the estimation of the probability P is finite, then the
probability Puncertain can not be closer to the bound Pbound of the probability scale
than the finite quantity δPuncertain (see Harin 2009-2).
Indeed, if the uncertainty of the probability estimation increases from initial
value ΔPcertain<<1% to final value, e.g., ΔPuncertain=5%, then, evidently, the
probability Puncertain can not be closer to the bound Pbound of the probability scale
than δPuncertain=0.5%. So, the probability can not be more than 99.5%. It can not
be (*see the second consequence below) less than 0.5% also.
B) Due to noises, there will exist ruptures, gaps, forbidden bands in the
probability scale. Or
*|Pbound - Puncertain| ≥ δPuncertain
where
ΔPuncertain ≥ δPuncertain ≥ O(ΔPuncertain) ≥ const > 0.

Evidently, the statements A) and B) of the first consequence of the principle
are true both for the present and future.

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1.1.2. The second consequence of the principle.
Incompleteness of the present probability system of future events
The probability of an event, which is not forbidden by objective laws, is more
than zero (in the microcosm even virtual events can occur that infringe the laws of
conservation). Hence, at any real number of foreseen events, an unforeseen event
with the probability more than zero will occur in any forecast or plan. Or
∑ Pforeseen + ∑ Punforeseen = 100%
∑ Punforeseen > 0%
Hence
∑ Pforeseen < 100%
where and further
∑ Pforeseen - the sum of estimates of probabilities of all foreseen events;
∑ Punforeseen - the sum of estimates of probabilities of all unforeseen events;

1.1.3. Examples of applications of the principle
In the probability theory the principle provides the statement of existence of
ruptures in the probability scale near 0% and 100% due to noises and uncertainties
(Harin 2009-2).
In economics for the Allais paradox (Allais 1953), the “fourfold pattern”
paradox, risk aversion, loss aversion, overweighting of low probabilities, uniform
explanation of choices for both gains and losses, the equity premium puzzle, etc
(see, e.g., Di Mauro and Maffioletti 2004) the principle (Formation of ruptures in
the probability scale) provides an uniform solution (Harin 2007). For the problems
of the incompleteness of systems of preferences, ambiguity aversion, the Ellsberg
paradox (Ellsberg 1961), etc the principle (Incompleteness of the probability
system) provides uniform solution also (Harin 2007).
In the theory of complex systems the principle provides a possibility of
infringement of division into groups of inconsistent events for future events
(Karassev 2007).

1.2. Optimal frames of reference
From physics it is well known an event may be described in various frames of
reference. Optimal choice of frame of reference is well known to be valuable.
When one use various frames of reference, the expressions of transmission between
various frames of reference are necessary.

1.3. Formula of forecasting as a framework for forecasts
An unforeseen event can modify an ideally forecasted phenomenon. Hence if
a forecast is used after such unforeseen event the forecast should be corrected.
The correcting formula of forecasting represents the correction (in a sense, a
framework for forecasts) which should be done for the forecast to be true after
unforeseen events have been occurred. In general, corrections may involve
corrections of errors and functions.
The correcting formula of forecasting may be also used as an adapting tool in
addition to unified and standardized forecasts to take into account distinctive
features of specific situations.