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Just a note

Taylor's series is not too difficult to express - but is less easy to explain. Provided you are not immediately concerned about exactly how this formula is evaluated, or used, this brief description may be of interest.

Essentially, a well behaved function ( f ) can be approximated about a point x, and moment μ, by expanding it into this infinitely-long series of terms,

f(x-μ) = [ f(x)/0! ]μ 0 + [ f'(x)/1! ]μ 1 + [ f''(x)/2! ]μ 2 + ... [ f (x)/∞! ]μ

If we replace the expressions in square brackets by 'constants' ( c1 to ci ), and collect all the bits to the right of the ith-order term into a remainder ( Ri ), this rather frightening series simplifies to (for example)

f(x-μ) ≅ c0μ 0 + c1μ 1 + c2μ 2 + R2

Which is recognisably a 2-order polynomial ( i = 2 ), plus a remainder term.

Now for the twiddly bits.

  • μ 0 = 1, μ 1 = μ, μ 2 = μ×μ, so μ = ∞, 1, or 0
  • f'(x) is, effectively a probability density, or the rate of change (or slope) of f(x) when plotted against x. For example the normal density function ( Z[x] or φ[x] ) is the rate of change (or slope) of is the slope of the cumulative normal distribution function ( Φ[x] ) at point x - so Z[x] could also be written as Φ'[x] ).
  • Similarly, f''(x) is the rate of change of slope at point x. These rates of change may be evaluated using differential Calculus.
  • 0! = 1, 1! = 1×1, 2! = 2×1
    So anything divided by ∞! is, almost certainly, zero.
  • In other words, unless you are dealing with a sincerely weird function, the higher the order of each term, the smaller it is. So, as a rule, if R2 is fairly small, R5 would be much smaller.

Confusingly, although the above series can be written more concisely as f(x-μ) = Σ( [fi(x)/i!]μ i ), it may be rearranged so that f(x) = Σ( [fi(μ)/i!][x-μ] i ).