An unbiased estimator of the mean density per cell in the population is given by:

\(\hat{\mu} = \frac{1}{n}\sum\limits_{i=1}^{n}{\frac{y_i}{m_i}} = \frac{1}{n}\sum\limits_{i=1}^{n}{\bar{y_i}}\)

Which is the means of the network means. The estimated variance of this estimator is what you would expect from simple random sampling.

\(\hat{V}(\hat{\mu}) = (1 - \frac{n}{N}) \frac{s_\bar{y}^2}{n}\)

Where: \(\frac{s_\bar{y}^2}{n}\)

is the sample variance of the network means

ie: \(s_\bar{y}^2 = var(\frac{y_i}{m_i})\)

\(n = 5\)

\(N = 100\)

\(\hat{\mu} = \frac{1}{5}\sum{(\frac{0}{1} + \frac{12}{4} + \frac{1}{1} + \frac{15}{5} + \frac{7}{2})} = 2.1\)

\(s_\bar{y}^2 = var(\frac{y}{m}) = 2.3\)

\(\hat{V}(\hat{\mu}) = (1 - \frac{5}{100}) \frac{2.3}{5} = 0.437\)

example.world <- matrix(c( 1, 0, 3, 3, 0, 0,-1,
                           0, 0, 3, 0, 3, 0,-1,
                           3, 0, 0, 0, 0, 0, 0,
                           0, 4, 0, 0, 0, 3, 3,
                           0, 0, 0, 3, 3, 3, 0,
                          -1,-1, 0, 0, 0, 0, 0, 
                          -1,-1,-1,-1,-1,-1,-1),
                        nrow = 7, ncol = 7,byrow=TRUE)
example.world

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7]
## [1,]    1    0    3    3    0    0   -1
## [2,]    0    0    3    0    3    0   -1
## [3,]    3    0    0    0    0    0    0
## [4,]    0    4    0    0    0    3    3
## [5,]    0    0    0    3    3    3    0
## [6,]   -1   -1    0    0    0    0    0
## [7,]   -1   -1   -1   -1   -1   -1   -1

If you want to rebuild a world given some adaptive samples:

Think of a world like a game of minesweeper.

Since we know the y and m for a few samples we know a couple of things. Anything around these samples has to have a value of 0, as defined by what it means to take an adaptive sample.

Any square in the representation of the world that doesnt touch a known sample has some unknown value “-1”

First lets look at another example world:

figure10.2 <- matrix(rep(rep(0,10),10),
  nrow = 10, ncol = 10)

figure10.2[,10] <- c(4,0,0,0,0,0,0,0,0,7)
figure10.2[,9]  <- c(3,0,2,1,0,0,0,0,4,1)
figure10.2[,8]  <- c(0,0,0,4,5,0,0,3,2,0)
figure10.2[,7]  <- c(5,4,0,0,0,0,0,0,0,0)
figure10.2[,6]  <- c(0,0,0,2,0,1,0,0,7,0)
figure10.2[,5]  <- c(0,5,0,4,0,0,0,0,0,0)
figure10.2[,4]  <- c(0,0,3,0,0,4,0,3,0,0)
figure10.2[,3]  <- c(0,1,0,0,0,3,0,0,4,0)
figure10.2[,2]  <- c(0,0,0,2,6,0,0,0,0,0)
figure10.2[,1]  <- c(0,4,0,4,0,0,2,3,0,0)

This world is represented as a Matrix in memory.

In order to navigate this world and take an adpative sample this data structure will have to be reshaped in order to analyze its contents.

To understand the analysis some definitions of functions and parts of the algorithm used to take the adaptive sample need to be known.

for each index "i" of the matrix:
    - Is "i" in the Top Row?
    - Is "i" in the Bottom Row?
    - Is "i" in the Leftmost column?
    - Is "i" in the Rightmost Column?

    if (Not in Leftmost column)
      left <- i - 1
    
    if (Not in the Leftmost column or Topmost row)
      upleft <- i - 1 + ncol
    
    if (Not in the Topmost Row)
      up <- i + ncol
    
    if (Not in the Topmost row or Rightmost column)
      upright <- i + 1 + ncol
    
    if (Not in the Rightmost column)
      right <- i + 1
    
    if (Not in the Bottommost row or the Rightmost column)
      downright <- i + 1 - ncol
    
    if (Not in the Bottommost row)
      down <- i - ncol

A matrix where each cell is the vertex of a graph and all vertecies of this graph are fully connected to each of their neighbors

ie: Any vertex to the NE, N, NW, W, SW, S, SE, E is connected

Full adjaceny list:

adjlist(figure10.2)

## [[1]]
##  N NE  E 
## 11 12  2 
## 
## [[2]]
##  W NW  N NE  E 
##  1 11 12 13  3 
## 
## [[3]]
##  W NW  N NE  E 
##  2 12 13 14  4 
## 
## [[4]]
##  W NW  N NE  E 
##  3 13 14 15  5 
## 
## [[5]]
##  W NW  N NE  E 
##  4 14 15 16  6 
## 
## [[6]]
##  W NW  N NE  E 
##  5 15 16 17  7 
## 
## [[7]]
##  W NW  N NE  E 
##  6 16 17 18  8 
## 
## [[8]]
##  W NW  N NE  E 
##  7 17 18 19  9 
## 
## [[9]]
##  W NW  N NE  E 
##  8 18 19 20 10 
## 
## [[10]]
##  W NW  N 
##  9 19 20 
## 
## [[11]]
##  N NE  E  S SW 
## 21 22 12  1  2 
## 
## [[12]]
##  W NW  N NE  E SE  S SW 
## 11 21 22 23 13  1  2  3 
## 
## [[13]]
##  W NW  N NE  E SE  S SW 
## 12 22 23 24 14  2  3  4 
## 
## [[14]]
##  W NW  N NE  E SE  S SW 
## 13 23 24 25 15  3  4  5 
## 
## [[15]]
##  W NW  N NE  E SE  S SW 
## 14 24 25 26 16  4  5  6 
## 
## [[16]]
##  W NW  N NE  E SE  S SW 
## 15 25 26 27 17  5  6  7 
## 
## [[17]]
##  W NW  N NE  E SE  S SW 
## 16 26 27 28 18  6  7  8 
## 
## [[18]]
##  W NW  N NE  E SE  S SW 
## 17 27 28 29 19  7  8  9 
## 
## [[19]]
##  W NW  N NE  E SE  S SW 
## 18 28 29 30 20  8  9 10 
## 
## [[20]]
##  W NW  N SE  S 
## 19 29 30  9 10 
## 
## [[21]]
##  N NE  E  S SW 
## 31 32 22 11 12 
## 
## [[22]]
##  W NW  N NE  E SE  S SW 
## 21 31 32 33 23 11 12 13 
## 
## [[23]]
##  W NW  N NE  E SE  S SW 
## 22 32 33 34 24 12 13 14 
## 
## [[24]]
##  W NW  N NE  E SE  S SW 
## 23 33 34 35 25 13 14 15 
## 
## [[25]]
##  W NW  N NE  E SE  S SW 
## 24 34 35 36 26 14 15 16 
## 
## [[26]]
##  W NW  N NE  E SE  S SW 
## 25 35 36 37 27 15 16 17 
## 
## [[27]]
##  W NW  N NE  E SE  S SW 
## 26 36 37 38 28 16 17 18 
## 
## [[28]]
##  W NW  N NE  E SE  S SW 
## 27 37 38 39 29 17 18 19 
## 
## [[29]]
##  W NW  N NE  E SE  S SW 
## 28 38 39 40 30 18 19 20 
## 
## [[30]]
##  W NW  N SE  S 
## 29 39 40 19 20 
## 
## [[31]]
##  N NE  E  S SW 
## 41 42 32 21 22 
## 
## [[32]]
##  W NW  N NE  E SE  S SW 
## 31 41 42 43 33 21 22 23 
## 
## [[33]]
##  W NW  N NE  E SE  S SW 
## 32 42 43 44 34 22 23 24 
## 
## [[34]]
##  W NW  N NE  E SE  S SW 
## 33 43 44 45 35 23 24 25 
## 
## [[35]]
##  W NW  N NE  E SE  S SW 
## 34 44 45 46 36 24 25 26 
## 
## [[36]]
##  W NW  N NE  E SE  S SW 
## 35 45 46 47 37 25 26 27 
## 
## [[37]]
##  W NW  N NE  E SE  S SW 
## 36 46 47 48 38 26 27 28 
## 
## [[38]]
##  W NW  N NE  E SE  S SW 
## 37 47 48 49 39 27 28 29 
## 
## [[39]]
##  W NW  N NE  E SE  S SW 
## 38 48 49 50 40 28 29 30 
## 
## [[40]]
##  W NW  N SE  S 
## 39 49 50 29 30 
## 
## [[41]]
##  N NE  E  S SW 
## 51 52 42 31 32 
## 
## [[42]]
##  W NW  N NE  E SE  S SW 
## 41 51 52 53 43 31 32 33 
## 
## [[43]]
##  W NW  N NE  E SE  S SW 
## 42 52 53 54 44 32 33 34 
## 
## [[44]]
##  W NW  N NE  E SE  S SW 
## 43 53 54 55 45 33 34 35 
## 
## [[45]]
##  W NW  N NE  E SE  S SW 
## 44 54 55 56 46 34 35 36 
## 
## [[46]]
##  W NW  N NE  E SE  S SW 
## 45 55 56 57 47 35 36 37 
## 
## [[47]]
##  W NW  N NE  E SE  S SW 
## 46 56 57 58 48 36 37 38 
## 
## [[48]]
##  W NW  N NE  E SE  S SW 
## 47 57 58 59 49 37 38 39 
## 
## [[49]]
##  W NW  N NE  E SE  S SW 
## 48 58 59 60 50 38 39 40 
## 
## [[50]]
##  W NW  N SE  S 
## 49 59 60 39 40 
## 
## [[51]]
##  N NE  E  S SW 
## 61 62 52 41 42 
## 
## [[52]]
##  W NW  N NE  E SE  S SW 
## 51 61 62 63 53 41 42 43 
## 
## [[53]]
##  W NW  N NE  E SE  S SW 
## 52 62 63 64 54 42 43 44 
## 
## [[54]]
##  W NW  N NE  E SE  S SW 
## 53 63 64 65 55 43 44 45 
## 
## [[55]]
##  W NW  N NE  E SE  S SW 
## 54 64 65 66 56 44 45 46 
## 
## [[56]]
##  W NW  N NE  E SE  S SW 
## 55 65 66 67 57 45 46 47 
## 
## [[57]]
##  W NW  N NE  E SE  S SW 
## 56 66 67 68 58 46 47 48 
## 
## [[58]]
##  W NW  N NE  E SE  S SW 
## 57 67 68 69 59 47 48 49 
## 
## [[59]]
##  W NW  N NE  E SE  S SW 
## 58 68 69 70 60 48 49 50 
## 
## [[60]]
##  W NW  N SE  S 
## 59 69 70 49 50 
## 
## [[61]]
##  N NE  E  S SW 
## 71 72 62 51 52 
## 
## [[62]]
##  W NW  N NE  E SE  S SW 
## 61 71 72 73 63 51 52 53 
## 
## [[63]]
##  W NW  N NE  E SE  S SW 
## 62 72 73 74 64 52 53 54 
## 
## [[64]]
##  W NW  N NE  E SE  S SW 
## 63 73 74 75 65 53 54 55 
## 
## [[65]]
##  W NW  N NE  E SE  S SW 
## 64 74 75 76 66 54 55 56 
## 
## [[66]]
##  W NW  N NE  E SE  S SW 
## 65 75 76 77 67 55 56 57 
## 
## [[67]]
##  W NW  N NE  E SE  S SW 
## 66 76 77 78 68 56 57 58 
## 
## [[68]]
##  W NW  N NE  E SE  S SW 
## 67 77 78 79 69 57 58 59 
## 
## [[69]]
##  W NW  N NE  E SE  S SW 
## 68 78 79 80 70 58 59 60 
## 
## [[70]]
##  W NW  N SE  S 
## 69 79 80 59 60 
## 
## [[71]]
##  N NE  E  S SW 
## 81 82 72 61 62 
## 
## [[72]]
##  W NW  N NE  E SE  S SW 
## 71 81 82 83 73 61 62 63 
## 
## [[73]]
##  W NW  N NE  E SE  S SW 
## 72 82 83 84 74 62 63 64 
## 
## [[74]]
##  W NW  N NE  E SE  S SW 
## 73 83 84 85 75 63 64 65 
## 
## [[75]]
##  W NW  N NE  E SE  S SW 
## 74 84 85 86 76 64 65 66 
## 
## [[76]]
##  W NW  N NE  E SE  S SW 
## 75 85 86 87 77 65 66 67 
## 
## [[77]]
##  W NW  N NE  E SE  S SW 
## 76 86 87 88 78 66 67 68 
## 
## [[78]]
##  W NW  N NE  E SE  S SW 
## 77 87 88 89 79 67 68 69 
## 
## [[79]]
##  W NW  N NE  E SE  S SW 
## 78 88 89 90 80 68 69 70 
## 
## [[80]]
##  W NW  N SE  S 
## 79 89 90 69 70 
## 
## [[81]]
##  N NE  E  S SW 
## 91 92 82 71 72 
## 
## [[82]]
##  W NW  N NE  E SE  S SW 
## 81 91 92 93 83 71 72 73 
## 
## [[83]]
##  W NW  N NE  E SE  S SW 
## 82 92 93 94 84 72 73 74 
## 
## [[84]]
##  W NW  N NE  E SE  S SW 
## 83 93 94 95 85 73 74 75 
## 
## [[85]]
##  W NW  N NE  E SE  S SW 
## 84 94 95 96 86 74 75 76 
## 
## [[86]]
##  W NW  N NE  E SE  S SW 
## 85 95 96 97 87 75 76 77 
## 
## [[87]]
##  W NW  N NE  E SE  S SW 
## 86 96 97 98 88 76 77 78 
## 
## [[88]]
##  W NW  N NE  E SE  S SW 
## 87 97 98 99 89 77 78 79 
## 
## [[89]]
##   W  NW   N  NE   E  SE   S  SW 
##  88  98  99 100  90  78  79  80 
## 
## [[90]]
##   W  NW   N  SE   S 
##  89  99 100  79  80 
## 
## [[91]]
##  E  S SW 
## 92 81 82 
## 
## [[92]]
##  W  E SE  S SW 
## 91 93 81 82 83 
## 
## [[93]]
##  W  E SE  S SW 
## 92 94 82 83 84 
## 
## [[94]]
##  W  E SE  S SW 
## 93 95 83 84 85 
## 
## [[95]]
##  W  E SE  S SW 
## 94 96 84 85 86 
## 
## [[96]]
##  W  E SE  S SW 
## 95 97 85 86 87 
## 
## [[97]]
##  W  E SE  S SW 
## 96 98 86 87 88 
## 
## [[98]]
##  W  E SE  S SW 
## 97 99 87 88 89 
## 
## [[99]]
##   W   E  SE   S  SW 
##  98 100  88  89  90 
## 
## [[100]]
##  W SE  S 
## 99 89 90

Breadth First Search (BFS) on “s” using using the adjacency list generated from the adjlist.adaptive function:

Algorithm: Breadth First Search (BFS)
- BFS (V,E,s) {
- for each u within V - {s}
    - u.distance <- inf
    - u.parent <- u
- s.d <- 0
- Q <- 0
- Enqueue(Q,s)
- while Q is not empty
    - u <- Dequeue(Q)
    - for each v within G.adj[u]
        - if v.d == inf
        - v.d = u.d + 1
        - v.parent = u
        - Enqueue(Q,v)
- }

All of the visited squares have a value != -1 aka infinity

The value of \(-1\) means that there is an “infinite distance” between the two vertices, aka we cannot get there from where we are so these were not sampled for the adaptive sample on “s”

The adaptive sample Function:

adaptive.sample <- function(world, vertecies){
  world.adj <- adjlist.adaptive(world)
  samples <- data.frame("y" = numeric(0),"m" = numeric(0))
  for(i in vertecies){
    samples <- rbind(samples,BFS(world,world.adj,i)$adaptive.sample)
  }
  return(samples)
}

adaptive.sample.figure10.2 <- adaptive.sample(figure10.2, 
                                              c(97,83,56,44,39))
adaptive.sample.figure10.2

##    y m
## 1  0 1
## 2 12 4
## 3  1 1
## 4 15 5
## 5  0 1