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In many practical situations like weather prediction, we are interested in large-scale (averaged) value of the predicted quantities. For example, it is impossible to predict the exact future temperature at different spatial locations, but we can reasonably well predict average temperature over a region. Traditionally, to obtain such large-scale predictions, we first perform a detailed integration of the corresponding differential equation and then average the resulting detailed solution. This procedure is often very time-consuming, since we need to process all the details of the original data. In our previous papers, we have shown that similar quality large-scale prediction results can be obtained if, instead, we apply a much faster procedure—first average the inputs (by applying an appropriate fuzzy transform) and then use these averaged inputs to solve the corresponding (discretization of the) differential equation. In this paper, we provide a general theoretical explanation of why our semiheuristic method works, that is, why fuzzy transforms are efficient in large-scale predictions.

One of the main objectives of science is to predict the future values of the physical quantities. For example, it is desirable to predict tomorrow's weather, the weather for several days ahead, and so forth. For a spreading flu epidemic, it is desirable to predict how this epidemic will spread if we do not introduce any restrictions on travel-and how this spread will change if such restrictions are introduced.

Of course, ideally, it is desirable to have predictions which are as detailed as possible. For example, ideally, we would like to know the exact value of tomorrow's temperature and wind speed at all possible spatial locations within a given region—or to predict exactly where the epidemics will spread and exactly how many people will fall ill if we do not introduce any travel restrictions.

However, in many practical situations, such a detailed prediction is impossible. In some of these situations, prediction is potentially possible, but it requires such a large amount of computations that even on the fastest modern computers, the computations finish long after the future event (that we are trying to predict) has already occurred.

In many practical situations in which we cannot predict the

For example, from the practical viewpoint, even though we cannot predict the exact value of tomorrow's temperature at all possible spatial locations, it would be beneficial to predict the

For predicting time series, for example, financial time series formed by the prices of different stocks at different moments of time, though it is impossible to predict the exact values of the future prices, it is desirable to at least be able to predict the

For clarity and simplicity, in the following text, we will describe the case when both the input

Instead of predicting the values

It is reasonable to assume that for different moments

A natural example of such averaging is a

A similar representation is often useful for other weight functions as well. In general, once we know this new weight function

Thus, from the mathematical viewpoint, the weighted averages are simply the values of the fuzzy transform.

Most relations in physics are described by differential equations. In particular, the relation between the observed signals

Since prediction usually means solving a known differential equation, a usual procedure for large-scale predictions is as follows:

first, we use the known values

then, we apply the weighted average procedure (

The main drawback of the traditional procedure is that we spend a lot of computation time to get a detailed solution

For example, in weather prediction, we spend hours of computer time on high-performance supercomputers to solve a complex system of differential equations with thousand of variables and then only use the large-scale weighted average of this solution.

We are only interested in

In other words,

traditionally, we first

what we propose is that we first

For several differential equations, we implemented the above idea of how to speed up computations. Specifically,

instead of the original input

then we use the values

we use the results

Surprisingly, we got a very good approximation to the values

In this paper, we provide a theoretical explanation for the empirical success of the fuzzy-transform-based methods of speeding up computations.

This explanation makes us confident that this fuzzy transform technique can be successfully used in other large-scale prediction problems as well.

Usually, the effect of each input value

extend the dependence of

ignore quadratic and higher order terms, and thus

keep only linear terms in this dependence.

In this case, we get the following dependence:

We are interested in systematic predictions, predictions that need to be repeated again and again. In these predictions, there is no fixed moment of time: if we start with the same input repeated later (i.e., shifted in time, from

For the formula (

first, we must have

second, we must have

Thus, we arrive at the following dependence:

In the traditional approach, we first find the detailed output (

An alternative approach is to first apply the same averaging to the original signal

In terms of the normalized weight function (

Similarly, in terms of the normalized weight function

In view of formulas (

To prove that these expressions coincide, let us try to transform them into each other. In expression (

The equality is proven.

In the ideal case, when quadratic terms can be completely ignored and there is no dependence on absolute time, the new method leads to

the quadratic terms are small but non-zero, and that

there may be an underlying trend-like dependence on absolute time (like global warming in weather prediction),

Since large-scale predictions are approximate anyway, this approximate equality means that, in terms of accuracy, the new predictions are, in effect, as good as the traditional ones. Since the new predictions are much faster to compute, they have a clear practical advantage.

This work was supported in part by the National Science Foundation Grant HRD-0734825, by Grant 1 T36 GM078000-01 from the National Institutes of Health, and by Grant MSM 6198898701 from MŠMT of Czech Republic. The authors are thankful to the anonymous referees for valuable suggestions.