 | Spatial distribution of organisms |
Three types of spatial distribution are usually considered |
The Poisson distribution, where m is the mean and i!= 1 ... 譱, 0!=1; 1!=1. |
Theorem: In poisson distribution, mean = variance |
Chi-square test equation, 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, |
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. |
Quick test for the type of spatial pattern:Coefficient of dispersion CD |
Poisson distributions are asymmetric at low mean values, and almost symmetric at higher mean values. When mean increases to infinity, poisson distribution coincides with a normal distribution. |
Aggregated spatial distributions, the equation3, where m is mean, k is "coefficient of aggregation" (aggregation increases with decreasing k). The zero term (the proportion of empty samples) equals to: |
Other terms can be estimated by iteration: |
Estimating NBD parameters using the method of moments: m=M |
For a random distribution, CD=1 and k is infinite. |
Coefficient of dispersion CD |
Indexes of aggregation. Consider probability distribution p(i) = the proportion of samples with i individuals. |
Mean crowding |
Density-Invariant Indexes of Aggregation. Eq4 - the "power law" |
Density-Invariant Indexes of Aggregation.Mean crowding regression |
Density-Invariant Indexes of Aggregation. Regression coefficients can be used to distinguish between different patterns of spatial distribution |
Effect of Quad Size on Aggregation Indexes. Eq5, where subscripts 1,2,...i stand for successively increasing sizes of quads. |
Density-Invariant Indexes of Aggregation.Ro index is used to determine characteristic distances in a spatial distribution |
Geostatistical Analysis of Population Distribution. Directional correlogram |
Example 1 - pair samples in south-north direction |
the set of all possible lag vectors in ex1.Vectors that end in the same cell are grouped into one class and correlogram value is estimated separately for each class. The number of directions may be different (4, 8, 16, etc.) |
Correlogram is estimated using equation 6, where indexes -h and +h refer to sample points located at the tail and head of vector h; z-h and z+h are organism counts in samples separated by lag vector h; summation is performed over all pairs of samples separated by vector h; Nh is the number of pairs of samples separated by vector h; M-h and M+h are mean values for samples located at the tail and head of vector h; s-h and s+h are standard deviations of samples located at the tail and head of vector h. |
Other measures of spatial dependence are covariance function |
and variogram (=semivariogram). The correlogram, covariance function, and variogram are all related. If the population mean and variance are constant over the sampling area (there is no trend) then: |
where C(0) is the covariance at zero lag = variance = squared standard deviation. |
Typical variogram has the following shape. |
|
Examples of one-dimensional spatial distributions with different nugget effects and corresponding variograms |
Anisotropy: different spatial relationships in different directions. Corresponding spatial pattern |
A variogram with anisotropy |
Log-transformation of data may be necessary before variogram estimation |
Some times indicator transformation is used eq7, where c is a threshold value ("a cutoff") |
Fractal is what will appear after infinite number of steps |
Hausdorf suggested to count the minimum number of equal spheres (circles in the picture) that cover the entire figure. The number of spheres, n, depends on their radius, r, and dimension was defined as: |
dimension. |
dimension of a line equals to 1 for that figure |
Normal" geometric figures have integer dimensions: 1 for a line, 2 for a square, 3 for a cube. However, fractals have FRACTIONAL dimensions, as in the example below: Here we use rather large circles, and thus, the precision is not high. For example, we got D=2.01 for a square instead of D=2. |
Dimension of a square |
Fractal |
the Mandelbrot set which is also a fractal |
Fractal dimension, D, is related to the slope of the variogram plotted in log-log scale, b. D = 2 - b/2 for a 1-dimensional space. D = 3 - b/2 for a 2-dimensional space. In the figure above, b=1, and thus, D=1.5 for a 1-dimensional space |
Estimate the fractal dimension of the "carpet of Serpinski":What is the area of this "carpet"? |
Tree Types of Spatial Distribution
3.2 Random Distribution
3.3 Aggregated Spatial Distribution
3.4 Indexes of Aggregation
3.5 Density-Invariant Indexes of Aggregation
3.6 Geostatistical Analysis of Population Distribution
3.7 Fractal Dimension of Population Distribution
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 the first equation.
Two main methods of parameter estimation
Method of moments (m = M)
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. See the equation.
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
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 CD:
if CD << 1 then regular distribution
if CD 1 then random distribution
if CD >> 1 then aggregated distribution.
3.3. Aggregated spatial distributions
There is no universal theoretical model for aggregated spatial distributions. However, there are empirical models, e.g., negative binomial distribution (NBD) which work very well in the majority of cases. NBD is described by the equation 3.
Estimating NBD parameters using the method of moments:
m = M
For a random distribution, CD=1 and k is infinite. When k increases to infinity then NBD distribution coincides with poisson distribution.
3.4. Indexes of aggregation
1. Coefficient of dispersion:
2. Mean crowding (Lloyd 1967) is equal to the mean number of "neighbors" in the same quad: Sample No. No. of individuals
(N) No. of neighbors
(N-1) N(N-1)
1 5 4 20
2 3 2 6
3 0 -1 0
4 1 0 0
5 7 6 42
Total 16 - 68
It means that the mean number of "neighbors" is = 4.25.
Note: mean crowding has biological sense only if the size of each quad corresponds to "interaction distance" among individuals.
Consider probability distribution p(i) = the proportion of samples with i individuals.
For a random (poisson) distribution, CD=1, and the mean number of "neighbors" is m.
3. Lloyd (1967) suggested a "patchiness" index: "mean crowding /m", which is 1 for a random distribution, > 1 for aggregated distributions, and <1 for regular distributions. However it does not make more sense than CD.
3.5. Density-Invariant Indexes of Aggregation
Simple indexes of aggregation are specific to a particular population sampled at particular time. They cannot be extrapolated neither in space nor in time, and this is their major limitation. In order to overcome this limitation, several density-invariant indexes has been developed.
1. The most frequently used is the "power law" (Taylor 1961) (see eq.4)
Coefficient b is considered as species-specific. This equation was shown to work well in the wide range of species density. Of course, it is hard to expect that b will be constant in any kind of environment, but for populations in similar environments it is usually stable.
2. Mean crowding regression (Iwao 1968):
Regression coefficients can be used to distinguish between different patterns of spatial distribution.
3. Negative binomial k. This is not a good index because usually it is not density-invariant.
Effect of Quad Size on Aggregation Indexes
Different scales of spatial distribution should be considered. The distribution may be random at small scales and aggregated at larger scales. Thus, it is important to examine the distribution using different quad sizes (quad size = spatial resolution).
For example, the distribution is random, if and only if the coefficient of dispersion (CD) is equal to 1 for all the range of quad sizes. Thus, if CD=1 for a specific quad size, then we cannot conclude that the distribution is random, we need to test other quad sizes.
Any index of aggregation can be plotted against quad size. However, there are specialized indexes designed for multiple quad sizes. For example, ro index (Iwao 1972) was defined as eq5.
3.6. Geostatistical Analysis of Population Distribution
The main idea of geostatistical methods is to relate the spatial variation among population densities to the distance lag. We already used one of geostatistical tools: an omnidirectional correlogram. Here we are going to explore a set of related tools: covariance functions and variograms. Also, we will discuss the phenomenon of anisotropy.
Directional correlogram is defined as correlation among population counts at points separated by space lag h. The difference from the omnidirectional correlogram is that h is a vector rather than a scalar (that is why it is in bold face). For example, if h={20,10}, then each pair of compared samples should be separated by 20 m in west-east direction and by 10 m in south-north direction (ex1).
In practice, it is difficult to find enough sample points which are separated by exactly the same lag vector h. Thus, the set of all possible lag vectors is usually partitioned into classes (ex1).
Vectors that end in the same cell are grouped into one class and correlogram value is estimated separately for each class. The number of directions may be different (4, 8, 16, etc.)
Correlogram is estimated using equation 6.
Interpretation of the nugget effect: It shows the pure random variation in population density (white noise) or it may be associated with sampling error.
Transformation is necessary to make the distribution more symmetrical and to remove the trend in variance. In a log-normal distribution, variance is proportional to the mean squared. Thus, in high-density areas, the variance is higher than in low-density areas. After log-transformation the variance becomes uniform.
Note: If zero values are present, use transformation: log(N+1).
Some times indicator transformation is used (eq7).
Usually, a series of thresholds is used, and variograms are estimated for all of them. If one threshold has to be selected, then the best is to take the median threshold m which corresponds to the 50% cumulative probability distribution.
3.7. Fractal Dimension of Population Distribution
Old definition of a fractal: a figure with self-similarity at all spatial scales.
Fractal is what will appear after infinite number of steps.
Examples of fractals were known to mathematicians for a long time, but the notion was formulated by Mandelbrot (1977).
New definition of a fractal: Fractal is a geometric figure with fractional dimension
It is not trivial to count the number of dimensions for a geometric figure. Geometric figure can be defined as an infinite set of points with distances specified for each pair of points. The question is how to count dimensions of such a figure. Hausdorf suggested to count the minimum number of equal spheres (circles in the picture below) that cover the entire figure
Main text - http://home.comcast.net/~sharov/PopEcol/lec3/random.html
http://home.comcast.net/~sharov/PopEcol/lec3/agindex.html
http://home.comcast.net/~sharov/PopEcol/lec3/geostat.html
http://home.comcast.net/~sharov/PopEcol/lec3/fracdim.html
