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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: