Stochastic prediction procedure

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In probability theory and statistics, a stochastic prediction procedure is based on a Bernoulli space and may be used to make predictions under specific conditions. In contrast to a prediction obtained in traditional science, predictions obtained by means of a stochastic predíction procedure^[1] meet a given reliability requirement and are optimal with respect to accuracy. A prediction procedure refers to a random variable X and predicts future events for X depending on the initial conditions or more precisely said on what is known about the initial conditions.

Mathematical formulation

A stochastic prediction procedure is a mathematical function^[2] denoted ${\displaystyle A_{X}^{(\beta )}}$ defined on sets representing the possible initial conditions and having images that are the predictions. The function ${\displaystyle A_{X}^{(\beta )}}$ is derived meeting the following two requirements:

• The stochastic prediction procedure ${\displaystyle A_{X}^{(\beta )}}$ shall yield predictions which will occur with a probability of at least $\beta$ where $\beta$ is called the reliability level of ${\displaystyle A_{X}^{(\beta )}}$.
• The stochastic prediction procedure ${\displaystyle A_{X}^{(\beta )}}$ shall yield predictions with optimal accuracy, where accuracy is defined by the size of the prediction.

The mathematical task consists of deriving a function ${\displaystyle A_{X}^{(\beta )}}$ which meets the above formulated two requirements.

Stochastic predictions

The predictions made by means of ${\displaystyle A_{X}^{(\beta )}}$ refer to the random variable X. The random variable X has a certain range of variability denoted by ${\mathfrak {X}}$ implying that any meaningful prediction is a subset of ${\mathfrak {X}}$. The future development is uncertain and uncertainty is generated by ignorance and randomness. In order to meet the reliability requirement specified by the reliability level $\beta$ it is necessary to consider both. The two sources of uncertainty are explicitly contained in the corresponding Bernoulli Space.

The derivation of a stochastic prediction procedure is therefore based on a Bernoulli Space which represents a stochastic model of the change from past to future taking into account the characteristic human ignorance about the initial conditions and the inherent randomness of the evolution of universe. Thus, it becomes possible to derive predictions procedures which yield reliable predictions with optimal accuracy. Note that because of randomness it is in principle impossible to predict the indeterminate future outcome and at the same time meet a specified reliability requirement. This is impossible, even in the rare case that the initial conditions are known.

Predictions in science

The characteristic issues of a stochastic prediction procedures shall be illustrated by a comparison with a prediction made and published by the University Corporation for Atmospheric Research (UCAR) which is described in^[3] as follows:

NASA Sunspot Number Predictions in March 2004 for Solar cycle 23 and 24, which did not come true. Even in 2006 a new scientific model predicted with 98% accuracy that cycle 24 would start at the end of 2007, and that its peak would be higher than cycle 23. Nevertheless, the cycle started two years later and the current prediction says the peak will be lower than the previous one.

The involved scientists of the University Corporation for Atmospheric Research (UCAR) reported on March 6, 2006, about their new prediction method:^[4]

The scientists have confidence in the forecast because, in a series of test runs, the newly developed model simulated the strength of the past eight solar cycles with more than 98% accuracy. The forecasts are generated, in part, by tracking the subsurface movements of the sunspot remnants of the previous two solar cycles. The team is publishing its forecast in the current issue of Geophysical Research Letters.

"Our model has demonstrated the necessary skill to be used as a forecasting tool," says NCAR scientist Mausumi Dikpati, the leader of the forecast team at NCAR's High Altitude Observatory that also includes Peter Gilman and Giuliana de Toma.

The first striking difference refers to the use of the word "accuracy". In case of a stochastic prediction procedure it refers to the size of the predicted event. The accuracy is largest, if the predicted event is a singleton and thus has the size 1. The larger the size is, the worse the accuracy becomes. If the size of the predicted event is too large, then the prediction is useless.

In the case of the UCAR predictions, an accuracy of more than 98% is claimed for the comparison with the past eight solar cycles. The meaning of this statement for the future cycles remains unclear. It resembles in some sense a statement about the reliability, i.e., the probability that the predicted event will actually occur. But this interpretation is wrong, since the statement refers to the past and not to the future. Maybe, it constitutes a relative frequency, however, in this case the corresponding event is unclear. It is always possible to develop a model that matches the past, but this does not mean that the probability of predicted future events can be assessed. Moreover, it appears that the UCAR predictions are given by singletons. If this is really the case then the accuracy would be highest, however the reliability would be equal to zero or at least close to zero. This is by the way the usual case in physics, which yields predictions with highest accuracy which, however, almost never occur.

References

1. Elart von Collani (ed.), Defining the Science of Stochastics, Heldermann Verlag, Lemgo, 2004.
2. Elart von Collani, The Need for a Standard for Making Predictions, Economic Quality Control, Vol. 23, 287–299, 2008.
3. Wikipedia, Prediction.
4. Scientists Issue Unprecedented Forecast of Next Sunspot Cycle .

External links

• Stochastikon Ecyclopedia,
• E-Learning Programme Stochastikon Magister,
• Homepage of Stochastikon GmbH,
• Economic Quality Control,
• Journal of Uncertain Systems,