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The log series, and the log series distribution

 

The log series distribution is named because of its similarity to the logarithmic series:

  • x1/1 + x2/2 x3/3 + x4/4 x5/5 + x6/6... = Σ[(-1)(i-1)xi/i]
    If −1<x<1 this series is infinitely long and convergent. Its sum is loge(1+x) - hence its name.

  • If all the terms in the series are positive, then x1/1 + x2/2 + x3/3 + x4/4 + ax5/5 + x6/6... = Σ[xi/i]
    Provided that 0<x<1 the sum of this series is loge(1x).

This distribution can be conveniently rescaled by multiplying each term by a constant, α.
Its sum is Σ[αxi/i] = αΣ[xi/i] = αloge(1x).

If α = 1/−loge(1x) then the sum of this series is one. This is the mass probability function of the log series distribution - as shown graphically below.

{Fig. 1}
lsdists.gif

Notice that this relationship is true irrespective of the value of x.

Provided this model describes the number of organisms (i) observed within each of s species, to work out the number of species represented by i individuals we set α to s/−loge(1x). In other words we multiply the height of each bar in the graph above by s.

  • If we multiply each term in the above series by i, we obtain a simpler series: x1 + x2 + x3 + x4 + x5 + x6... = Σ[xi] and, provided that 0<x<1 then its sum (n) is x/[1 - x].

    Once again, if we rescale these terms by multiplying each by a constant, α, their sum is αx/[1 - x].

So if each bar on the graph above corresponds to a set of species, each represented by i organisms, the total number of organisms in each bar must be si. The total number of organisms, n, is αx/[1 - x] - and therefore α = n[1 - x]/x.