### Dangers of Extrapolation

Extrapolating is something we do very frequently, even at a rather subconscious level - it is correlated to inductive logic too. But there are dangers within extrapolation that one should be aware of.

In mathemathical terms, extrapolation can be defined as follows. We have a function

The obvious risk is that the expression of

For example, if we did experimental measurements of some property in certain conditions and found a mathematical correlation, there is no guarantee that in different conditions said correlation will still be valid. In fact, in the scientific and engineering literature there are many examples of equations valid only in a given range of variables, or which require different coefficients for different ranges.

This means that results obtained through extrapolation should be taken with a grain of salt, because they can be amply different from reality. To make a pedestrian example, many saving bank accounts offer different interest rates for different amounts of money. Extrapolating the interest calculated for 1000€ to 10 000€ would thus give the wrong result.

Sure, not all extrapolations are the same: some can be taken with more confidence than others. However, the recommendation is not to extrapolate unless there is no other way.

An interesting twist is when the variable x is time. Extrapolating here means assuming that the time-trend of some quantity will remain the same. If the trend changes, the results of extrapolation will be wrong. Hence, infamous fiascos such as

Another complication is that, in the case of natural phenomena, often the function

Extrapolation is a useful technique, but saddled with considerable inherent uncertainty.

In mathemathical terms, extrapolation can be defined as follows. We have a function

*y*=*f(x)*which expression is known for a given interval of*x*, from*x1*to*x2*. Extrapolation measn assuming that the expression of*f(x)*will be the same also for*x*out of the considered interval. In this way, for*x3*we will simply have*y3*=*f(x3)*.The obvious risk is that the expression of

*f(x)*may not be the same out of the interval where it is known.For example, if we did experimental measurements of some property in certain conditions and found a mathematical correlation, there is no guarantee that in different conditions said correlation will still be valid. In fact, in the scientific and engineering literature there are many examples of equations valid only in a given range of variables, or which require different coefficients for different ranges.

This means that results obtained through extrapolation should be taken with a grain of salt, because they can be amply different from reality. To make a pedestrian example, many saving bank accounts offer different interest rates for different amounts of money. Extrapolating the interest calculated for 1000€ to 10 000€ would thus give the wrong result.

Sure, not all extrapolations are the same: some can be taken with more confidence than others. However, the recommendation is not to extrapolate unless there is no other way.

An interesting twist is when the variable x is time. Extrapolating here means assuming that the time-trend of some quantity will remain the same. If the trend changes, the results of extrapolation will be wrong. Hence, infamous fiascos such as

*The Population Bomb*.Another complication is that, in the case of natural phenomena, often the function

*f(x)*is obtained using regression or fitting techniques, which introduce errors by themselves. Using these functions for extrapolation then adds doubts to errors.Extrapolation is a useful technique, but saddled with considerable inherent uncertainty.

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