Spatial Issues - Problem Set 7.5

Note: I'm not sure I like the way I did the statistics.  It probably would have been better to convert the proportions to actual number of observations (dots).   This leaves the problem of some zero expected values though (which aren't allowed in chi-squared tests).

1. Look at Figure 1.  How are these points distributed spatially?  Prove whether it is aggregated, even, or randomly distributed.
   
   (Figure 1)
   
   Well to look at this, we first find out what the mean number of points in each quadrat is.   This is 1.5 points/quadrat.  The variance is 1.686.  Therefore the variance : mean ratio is 1.124 (slightly clumped).  The next question is whether this is statistically different from 1.  To do this we see what proportion of squares have 0 points in them, 1 points, etc.  The observed values and expected values are shown below (expected were generated by a poisson distribution).  A chi-squared test was then used to determine if there was a difference between the observed and expected values.  This chi-squared test showed a near perfect match (see a difference like this by chance alone 99% of the time).
   
   

   # of Points	Observed	Expected
   0	9	8.032686
   1	11	12.04903
   2	9	9.036771
   3	4	4.518386
   4	2	1.694395

   
   

2. Calculate Lloyd's mean crowding index for Figure 1.  Does this support your results from question 1 (why)?
   
   Lloyd's mean crowding index is: , where N = total # of individuals, and x_i represents the number of individuals in quadrat i.
   For our example, Q = 36, N = 54, and the mean crowding index is 1.59.  This is pretty much what we might expect - it's a moderate crowding index, but that will become more evident as we look at other examples.
   

3. Repeat question 1 for Figure 2.
   
   
   (Figure 2)
   

   Well to look at this, we first find out what the mean number of points in each quadrat is.  This is 1.361 points/quadrat.  The variance is 0.752.  Therefore the variance : mean ratio is 0.552 (even).  The next question is whether this is statistically different from 1.  To do this we see what proportion of squares have 0 points in them, 1 points, etc.  The observed values and expected values are shown below (expected were generated by a poisson distribution).  A chi-squared test was then used to determine if there was a difference between the observed and expected values.  This chi-squared test did show a significant difference (see a difference like this by chance alone 1.6 % of the time).
   
   

   # of Points	Observed	Expected
   0	3	9.229527
   1	22	12.56241
   2	7	8.549419
   3	3	3.878903
   4	2	1.319905

4. Repeat question 2 for Figure 2.
   
   For our example, Q = 36, N = 49, and the mean crowding index is .897.  This is pretty much what we might expect - it's much smaller indicating less crowding.
   

5. The distributions for Figures 1 and 2 correspond to two of the three possible spatial distributions (aggregate, even, or random).  Compose a quadrat map similar to these figures that corresponds with the third distribution.  Back this up empirically using the methods you used in questions 1 & 2.
   
   (aggregated)
   
   Now the mean, variance, and variance : mean ratio are .972, 1.51, and 1.56 respectively.   The mean crowding index is 1.49.  I'm not going to report the statistics here, but it is aggregated.
   
   

6. Look at Figure 3. Break this illustration up into quadrats. Repeat this process for a different (try quite a bit different) spatial scale. Using variance : mean ratios, does the spatial distribution depend on scale?
   
   (Figure 3)
   
   Here are my two scales:
   
   For the first one, there was a mean of 2.97, variance of 2.77 and variance : mean of .932 (slightly even).  The second scale has a mean of .305, a variance of 1.86, and a variance : mean of  1.086 (slightly aggregated). While there wasn't a huge difference between the two results, there was some and sometimes it can make a really big difference.
   
   
   

7. The data in Table 1 represents the positions of individuals of a crop-eating chrysomelid beetle. The positions are scaled to meters on two axes (north/south and east/west) of a crop field. How would you describe the spatial distribution of this site if you knew that these species regularly interacted with individuals within 7 meters of each other? What would your description be if the individuals only interacted much with others when less than 2.5 meters apart?
   
   Here's a graph of the beetles in space (with gridlines 7 meters apart).
   
   
   
   From this you can grid it up (you could generate this with a scatter plot in Excel with gridlines - there are alternatives too).  The scale of your grid is dependent on what measure you were doing.  For grids of 7 meters wide (if they were only interacting at this spatial scale, the appropriate grid size might be larger - do you see why this would be so?).  This one has a mean of .430 dots/grid, a variance of .470, and variance : mean of 1.09.  You should have classified this a nearly random and repeated the analysis at a different spatial scale..
   

8. Oftentimes it is useful to consider a frequency function, f(x), which describes the frequency of x individuals present in subsets of the sample (by area). Draw this function for random, aggregated, and even distributions (use the same average density for all three). How do they contrast?
   
   These are essentially the same as our even and expected distributions from problem 3.   With an aggregated curve added.  Here they are together graphed.
   
   Here the mean is 1.  It is useful to note that even distributions typically have a frequency peak around the mean.  The others peak at or near 0.  Aggregated typically has a very strong peak at zero, but also has a longer tail than the other distributions.
   
   

9. What are some reasons why you might want to examine the spatial distributions of populations (or other levels eg. communities)?
   
   This is one for you to thing about!

Table 1