3.2. Random distribution

   Random spatial distribution is simulated using poisson distribution.

   Simplest example: 100 people are fishing in the same lake for the same time (e.g.
   3 h); they have equal probability to catch a fish per unit time.
   Question: How many fishers catch 0, 1, 2, 3 etc. fish?

                                      No. of fish
                                       captured,
                                           i

                                        No. of
                                        fishers
                                         n(i)

                                      Proportion
                                      of fishers
                                         p(i)

                                        Poisson
                                     distribution
                                     n'(i)=Np'(i)

   0          11    0.11 10
   1          25    0.25 23
   2          21    0.21 27
   3          25    0.25 20
   4          9     0.09 12
   5          7     0.07 5
   6          2     0.02 2
   7          0     0.00 1
   Total      N=100 1.00 100

   Mean number of fish captured by 1 fisher, M = 2.30, and standard deviation, SD =
   1.41.

   Poisson distribution is described by equation:

                                     [eqpoiss.gif]

   where m is the mean and i!= 123 ... i, 0!=1; 1!=1.

   Theorem: In poisson distribution, mean = variance: [eqpoiss1.gif]

   Two main methods of parameter estimation
    1. Method of moments (m = M)
    2. Non-linear regression (iterative approximation)

   In the table above we used the method of moments: m = M = 2.3.

   Chi-square test is used to test if sample distribution is different from
   theoretical distribution. The equation is:

                                     [eqchisq.gif]

   where n(i) is the sample distribution (e.g., the number of fishers that captured
   i fish), and n'(i) is theoretical distribution (e.g., expected number of fishers
   that captured i fish according to poisson distribution). In our example,

                                    [eqchisq1.gif]

   The number of degrees of freedom is equal to the number of distribution classes
   (7 classes in our example) minus the number of parameters that were used to
   adjust the theoretical distribution to the sample distribution. We used 2
   parameters: m=2.3 and N = 100. Thus,

                                    df = 7 - 2 = 5

   [1]Critical value for chi-square for df = 5 and P = 0.05 is 11.07.
   Conclusion: Sample distribution does not differ significantly from the poisson
   distribution.

   Note. Chi-square test cannot prove that sample distribution is the same as the
   theoretical distribution! If there is no significant differences, it may mean two
   things: sample distribution is really very close to the theoretical distribution,
   or there may be just not enough data to distinguish these distributions.
   Suggestion: Use multiple hypothesis, i.e., compare sample distribution with
   several theoretical distributions.

   Relationship between fishers (our example) and spatial distribution. Imagine, we
   sample a population by counting organisms in sample areas (e.g., 1 sq.m.). Then,
   each sample area is equivalent to a fisher and the number of organisms found is
   equivalent to the number of fish captured. The notion "random distribution" can
   be defined using the model of random deposition of individual organisms. We start
   from empty space and put the first organism by random selection of its
   coordinates. The organism may end in a sample area (=fish captured). Then we add
   the second organism, and so on.

   Thus, the proportion of samples in which i organisms were found will correspond
   to the poisson distribution.

   Quick test for the type of spatial pattern:
   Coefficient of dispersion: [eqcd.gif]
   if CD > 1 then aggregated distribution

   Poisson distributions are asymmetric at low mean values, and almost symmetric at
   higher mean values:

                                     [poisson.gif]

   When mean increases to infinity, poisson distribution coincides with a normal
   distribution.

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     ____________________________________________________________________________

   [5]Alexei Sharov 12/28/95

References

   1. http://alexei.nfshost.com/PopEcol/tables/chisq.html
   2. http://alexei.nfshost.com/PopEcol/lec3/3distrib.html
   3. http://alexei.nfshost.com/PopEcol/lec3/spatdist.html
   4. http://alexei.nfshost.com/PopEcol/lec3/agregat.html
   5. http://alexei.nfshost.com/~sharov/alexei.html