summary(lansing) #returns summary stats for the dataset
## Marked planar point pattern:  2251 points
## Average intensity 2251 points per square unit (one unit = 924 feet)
## 
## *Pattern contains duplicated points*
## 
## Coordinates are given to 3 decimal places
## i.e. rounded to the nearest multiple of 0.001 units (one unit = 924 feet)
## 
## Multitype:
##          frequency proportion intensity
## blackoak       135 0.05997335       135
## hickory        703 0.31230560       703
## maple          514 0.22834300       514
## misc           105 0.04664594       105
## redoak         346 0.15370950       346
## whiteoak       448 0.19902270       448
## 
## Window: rectangle = [0, 1] x [0, 1] units
## Window area = 1 square unit
## Unit of length: 924 feet
str(lansing) #returns the structure of the dataset
## List of 6
##  $ window    :List of 4
##   ..$ type  : chr "rectangle"
##   ..$ xrange: num [1:2] 0 1
##   ..$ yrange: num [1:2] 0 1
##   ..$ units :List of 3
##   .. ..$ singular  : chr "foot"
##   .. ..$ plural    : chr "feet"
##   .. ..$ multiplier: num 924
##   .. ..- attr(*, "class")= chr "unitname"
##   ..- attr(*, "class")= chr "owin"
##  $ n         : int 2251
##  $ x         : num [1:2251] 0.078 0.076 0.051 0.015 0.03 0.102 0.135 0.121 0.04 0.065 ...
##  $ y         : num [1:2251] 0.091 0.266 0.225 0.366 0.426 0.474 0.498 0.489 0.596 0.608 ...
##  $ markformat: chr "vector"
##  $ marks     : Factor w/ 6 levels "blackoak","hickory",..: 1 1 1 1 1 1 1 1 1 1 ...
##  - attr(*, "class")= chr "ppp"
names(lansing) #returns names of callable elements of the dataset
## [1] "window"     "n"          "x"          "y"          "markformat"
## [6] "marks"
mytest <- quadrat.test(ants)
## Warning: Some expected counts are small; chi^2 approximation may be inaccurate
plot(mytest)

Question 1: It has 97 points in three columns: x, y, Marked.

ants
## Marked planar point pattern: 97 points
## Multitype, with levels = Cataglyphis, Messor 
## window: polygonal boundary
## enclosing rectangle: [-25, 803] x [-49, 717] units (one unit = 0.5 feet)

Question 2: They are marked, and the marks represent the species/type of ant.

Question 3: Because the p-value is much greater than 0.05, I would accept it.

ques3res <- quadrat.test(ants, nx=3, ny=3)
## Warning: Some expected counts are small; chi^2 approximation may be inaccurate
ques3res
## 
##  Chi-squared test of CSR using quadrat counts
## 
## data:  ants
## X2 = 7.5508, df = 8, p-value = 0.9571
## alternative hypothesis: two.sided
## 
## Quadrats: 9 tiles (irregular windows)

Question 4:

Question 5: for cataglyphis ants I got a p-value of 0.6071 and for messor ants I got a p-value of 0.7553, and would accept both of these.
cataglyphis_ants <- subset(ants, marks == "Cataglyphis")
messor_ants <- subset(ants, marks == "Messor")

cataglyphis_test <- quadrat.test(cataglyphis_ants, nx = 3, ny = 3)
## Warning: Some expected counts are small; chi^2 approximation may be inaccurate
messor_test <- quadrat.test(messor_ants, nx = 3, ny = 3)
## Warning: Some expected counts are small; chi^2 approximation may be inaccurate
cataglyphis_test
## 
##  Chi-squared test of CSR using quadrat counts
## 
## data:  cataglyphis_ants
## X2 = 5.5594, df = 8, p-value = 0.6071
## alternative hypothesis: two.sided
## 
## Quadrats: 9 tiles (irregular windows)
messor_test
## 
##  Chi-squared test of CSR using quadrat counts
## 
## data:  messor_ants
## X2 = 6.2221, df = 8, p-value = 0.7553
## alternative hypothesis: two.sided
## 
## Quadrats: 9 tiles (irregular windows)

Question 6: the observation falls in between the envelope, suggestion there it is random.

## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
## 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
## 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
## 99.
## 
## Done.

envelope(lansing, Gcross, nsim = 99, i = 'maple', j = 'hickory') |> plot()
## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
## 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
## 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
## 99.
## 
## Done.

envelope(lansing, Kcross, nsim = 99, i = 'maple', j = 'hickory') |> plot()
## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
## 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
## 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
## 99.
## 
## Done.

envelope(lansing, pcfcross, nsim = 99, i = 'maple', j = 'hickory') |> plot()
## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
## 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
## 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
## 99.
## 
## Done.

Question 7: There is a common trend with all of their observations falling below the envelope, suggesting a high distance between these groups.

Question 8: Most of the observations for all plots fall within the envelope or just above it - which could mean the species have some clustering together.

g_env_ants <- envelope(ants, Gcross, nsim = 99, i = "Cataglyphis", j = "Messor")
## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
## 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
## 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
## 99.
## 
## Done.
k_env_ants <- envelope(ants, Kcross, nsim = 99, i = "Cataglyphis", j = "Messor")
## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
## 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
## 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
## 99.
## 
## Done.
pcf_env_ants <- envelope(ants, pcfcross, nsim = 99, i = "Cataglyphis", j = "Messor")
## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
## 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
## 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
## 99.
## 
## Done.
plot(g_env_ants, main = "Gcross: Cataglyphis to Messor")

plot(k_env_ants, main = "Kcross: Cataglyphis to Messor")

plot(pcf_env_ants, main = "pcfcross: Cataglyphis to Messor")

E <- envelope(longleaf, markcorr, nsim=99)
## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
## 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
## 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
## 99.
## 
## Done.
plot(E)

Question 9: There is some deviation from the null from 0-20 meters, and around evey 10 meters after you pass 20 there is a spike where there is little/no deviation.

Question 10: It suggests a negative association between the two.

Question 11: This is showing that trees of different sizes are in clusters. Likely that between larger trees smaller ones are using the space inbetween them.

## Generating 99 simulations of CSR  ...
## 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
## 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
## 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
## 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
## 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 
## 99.
## 
## Done.

Question 12: From 0-15 meters the line is <1, and is a negative correlation suggestion the anemones are of different sizes in that area, and from meters 15+ the line remains near 1, if anything still just slightly below it, meaning there is little to no correlation if size and place of anemones. This could be happening because of the avalibility of resources, sunlight, etc. causing larger/older anemones having a random distribution but with clusters of saller/younger ones nearby because of avalibilty.