Linearity in the mathematical sense is more than a straight line graph. In the jargon, it means additive and homogeneous. These abstruse terms can be illustrated most simply by diagrams. We assume that a system produces output y for an input x:
In words, this means that a linear system produces the same output for two added signals, or for a signal multiplied by a constant, whether those operations are carried out before or after the signals pass through the system.
In algebraic notation it is written:
f(x1+x2) = f(x1) +f(x2)
f(kx) = kf(x)
Why it is important?
If a system is linear, it is amenable to analysis by linear algebra, which is a vast mathematical structure of immense power. Methods of linear algebra include algebraic geometry, matrices, vector spaces and transforms (such as Fourier).
There is no equivalent non-linear algebra and non –linear systems have in general to be treated on an ad hoc basis.
In the real world, apart from the electromagnetic properties of a vacuum, true linear behaviour is virtually unknown. Fortunately, however, many systems are effectively linear over a restricted range of variables.
Examples in mathematics
Functions and transforms that are linear include integration, differentiation and Fourier transform. Non-linear examples include squaring, exponential and logarithm.
An anomalous case is the process of adding a constant, which does not obey the equations above. This process is incrementally linear, which means that linear algebra is still available for changes in the variables.
A simple physical example
The ideal hi-fi system is perfectly linear. Real hi-fi systems have non-linearities and their minimisation is a measure of the quality of the system. The approximate linearity is, however, only applicable over a restricted range of inputs. If the maximum input is exceeded, then part of the system will “saturate”, becoming highly non-linear.
Linearity is not the same as frequency response. A non-linear amplifier will generate harmonics that are not present in the original signal. A linear filter will treat different frequencies differently, but it will not generate new ones. It is the presence of harmonics that make poor systems uncomfortable to listen to.
What are the consequences of incorrectly assuming linearity?
A good example of a serious error arising from a mistaken assumption of linearity is the so-called “Hockey stick” curve. This was adopted by the UN IPCC, resulting in potentially devastating economic consequences. The mathematical method employed by the authors was “principal component analysis”, which is a form of linear algebra applied to statistical data.
One of the main sources of data for this exercise was plant growth (tree rings).
It is easy to demonstrate that plant growth is a non-linear process. Plants require for growth nutriment, light, warmth and moisture. Consider just the last two of these. In the middle range of variables, increases in warmth and moisture both increase growth rates. However, at the extremes, this is not true. If it is very cold, then more moisture will impede growth, while if it is very dry, more heat will also reduce it. Thus plant growth is not only non-linear, it is not even monotonic, which implies a gross non-linearity and excludes the use of linear algebra.
The results of this analysis were used by the IPCC for the basis of a claim that phenomena such as the Little Ice Age and the Mediaeval Warm period never actually happened, despite the copious evidence to the contrary from history, art, archaeology, entomology etc. The error was compounded by arrogant dismissal of criticisms, attempts to prevent their publication and refusal to make public the computer programmes involved, but that is another story. It is curious that a prolonged and intricate argument has followed, when all that needs to be said is that the method used was not valid.
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