DATA 609: Final Presentation
Dan Brooks, Keith Folsom
May 18, 2017
1989: The Midge Classification Problem (Keith)
Two species of midges, Af and Apf, have been identified by biologists Grogan and Wirth (1981) on the basis of antenna and wing length (Figure A.8). Each of nine Af midges is denoted by a circle, and each of six Apf midges is denoted by a square. It is important to be able to classify a specimen as Af or Apf, given the antenna and wing length
- The Apf midge is a carrier of a debilitating disease that causes swelling of the brain when a human is bitten by an infected midge
- The Af midge is quite harmless and a valuable pollinator
Midge Data
The midge data provided by species appears as shown below:
| Species | Ant.Length | Wing.Length |
|---|---|---|
| Af | 1.38 | 1.64 |
| Af | 1.40 | 1.70 |
| Af | 1.24 | 1.72 |
| Af | 1.36 | 1.74 |
| Af | 1.38 | 1.82 |
| Af | 1.48 | 1.82 |
| Af | 1.54 | 1.82 |
| Af | 1.38 | 1.90 |
| Af | 1.56 | 2.08 |
| Apf | 1.14 | 1.78 |
| Apf | 1.20 | 1.86 |
| Apf | 1.18 | 1.96 |
| Apf | 1.30 | 1.96 |
| Apf | 1.26 | 2.00 |
| Apf | 1.28 | 2.00 |
Apf & Af Midge Scatterplot
Given a midge that you know is species Af or Apf, how would you go about classifying it?
- Leverage the existence of the pointless region between the species to establish boundaries.
Problem Statement
Apply your method to three specimens with (antenna, wing) lengths (1.24, 1.80), (1.28, 1.84), (1.40, 2.04)
Problem : The three specimens fall within the pointless region between the two midge borders.
Assume that species Af is a valuable pollinator and that species Apf is a carrier of a debilitating disease. Would you modify your classification scheme and if so, how?
- Yes, the classification scheme needs to be modified to classifiy the species falling in between the existing Apf and Af boundaries.
Classification Method 1
- Consider the ratio of midge antenna to wing length as a means of classification
Classification Method 1 - Continued
Using the mean ratio for each species, we arrive at the following equations:
\[ \text{Af Midge: }Antenna.Length = 0.7846034 * Wing.Length \]
\[ \text{Apf Midge:} Antenna.Length = 0.6368195 * Wing.Length \]
Classification Method 1 - Continued
Superimposing on top of the midge graph:
Using the mean of the ratios between the two species, we establish a border between the two groups:
Classification Method 1 - Classification Scheme:
\[ \text{if Antenna.Length > 0.7107115 * Wing.Length then Af Midge} \]
\[ \text{if Antenna.Length < 0.7107115 * Wing.Length then Apf Midge} \]
Classification Method 2 - Linear Regression
- Create linear regression models for the Af and Apf species.
Applying predicted values based on the model with an 85% confidence level:
The predicted lower and upper bands for each species do not cover the unclassified midge at (1.8, 1.24).
Classification Method 2 - Linear Regression Continued
Given this missed midge, use a similar approach using the mid-way points between the two boundaries established by the linear models:
\[ \text{Af Midge:} Antenna.Length = 0.4789 * Wing.Length + 0.5491 \] \[ \text{Apf Midge:} Antenna.Length = 0.5582 * Wing.Length + 0.1512 \]
Classification Method 2 - Classification Scheme:
\[ \text{if Antenna.Length > 0.5185763 * Wing.Length + 0.3501444 then Af Midge} \]
\[ \text{if Antenna.Length < 0.5185763 * Wing.Length + 0.3501444 then Apf Midge} \]
1986: The Emergency-Facilities Location Problem (Dan)
The township of Rio Rancho has hitherto not had its own emergency facilities. It has secured funds to erect two emergency facilities in 1986, each of which will combine ambulance, fire, and police services. Figure A.4 indicates the demand, or number of emergencies per square block, for 1985. The L region in th north is an obstacle, whereas the rectangle in the south is a park with a shallow pond. It takes an emergency vehicle an average of 15 seconds to go one block in the N-S direction and 20 seconds in the E-W direction. Your task is to locate the two facilities so as to minimize the total response time.
Assumptions
Assume that the demand is concentrated at the center of the block and that the facilities will be located on corners.
Assume that the demand is uniformly distributed on the streets bordering each block and that the facilities may be located anywhere on the streets.
Approach One
Euclidean Distance
Treat all points as those they can be reached by all of the other points
Treat the barries as blocks with no incidencies
Number of incidences gets multiplied by the N/S weight and E/W weight to get the total distance for traveling to that location.
All distances and added up acr0ss the board and the two smallest distances are where the stations should be placed.
Below we can see the matrix of incidencies
Below that we see the distances. Block (5,3) and (5,4) have the smallest distances, so the stations should be placed there.
Ranchero
[,1] [,2] [,3] [,4] [,5]
[1,] 3 1 4 2 5
[2,] 3 2 3 3 2
[3,] 2 0 3 3 2
[4,] 3 0 0 3 1
[5,] 3 4 3 3 5
[6,] 2 3 4 4 0
[7,] 1 2 0 1 3
[8,] 0 2 0 3 2
[9,] 3 0 0 0 4
[10,] 3 1 0 4 2
print("/n")
[1] "/n"
raw.distance
[,1] [,2] [,3] [,4] [,5]
[1,] 11040 9820 9200 9260 10360
[2,] 9885 8665 8045 8105 9205
[3,] 9120 7900 7280 7340 8440
[4,] 8655 7435 6815 6875 7975
[5,] 8400 7180 6560 6620 7720
[6,] 8685 7465 6845 6905 8005
[7,] 9360 8140 7520 7580 8680
[8,] 10245 9025 8405 8465 9565
[9,] 11340 10120 9500 9560 10660
[10,] 12645 11425 10805 10865 11965
Approach Two
Treat the matrix as a transition matrix
Use graphical based algorithms instead of euclidean distance, because not all points are reached by other points
Obstacles are left outside of the transition matrix, meaning they are not able to be reached
Create a digraph of all the points in the grid
Use Djikstras shorest path algorithm to find the shorest distances between all of the points
Add up all of those distance to find the total sitances between all the possible points in the model
The N/S and E/W distances are divided by the number of incidencies at that location. Since there are 0 incidencies on some locations, 1 was added to everything to make the math possible.
Transition Matrix
Rio.Rancho
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[44,] 0.00 0 0 0 4 0.00 0.0 0 3 0.000000
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[50,] 0.00 0 0 0 4 0.00 0.0 0 4 0.000000
Plot
Digraph, showing the connection between all of the nodes
The nodes off to the sides are the barriers, they cannot be reached by anything
graph <- graph.adjacency(Rio.Rancho, mode = "directed", weight = TRUE)
plot.igraph(graph, layout = layout.fruchterman.reingold, vertex.label.color = "black", edge.color = "black")
Shortest Path
- Below we see the distance matrix, shows the distacne from the nodes on the rows, to each node across the columns.
- Below that is the two smallest distances. The translate to the points (5,4) and (6,4)
- The stations should now be stacked, compared to being next to each other. That could be due to the boundaries getting in the way.
s.path
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[27,] 12.000000 10.000000 3.750000 7.000000 11.000000 15.750000
[28,] 8.000000 14.000000 7.750000 3.000000 7.000000 11.750000
[29,] 4.000000 18.000000 11.750000 7.000000 3.000000 7.750000
[30,] 0.000000 22.000000 15.750000 11.000000 7.000000 3.750000
[31,] 22.000000 0.000000 6.666667 13.333333 21.000000 25.750000
[32,] 15.750000 6.666667 0.000000 6.666667 14.750000 19.500000
[33,] 11.000000 13.333333 6.666667 0.000000 10.000000 14.750000
[34,] 7.000000 21.000000 14.750000 10.000000 0.000000 5.000000
[35,] 3.750000 25.750000 19.500000 14.750000 5.000000 0.000000
[36,] 27.416667 7.500000 11.666667 18.333333 26.416667 31.166667
[37,] 20.750000 11.666667 5.000000 11.666667 19.750000 24.500000
[38,] NA NA NA NA NA NA
[39,] 10.750000 24.750000 18.500000 13.750000 3.750000 8.750000
[40,] 7.500000 29.500000 23.250000 18.500000 8.750000 3.750000
[41,] 30.750000 11.250000 15.000000 21.666667 29.750000 34.500000
[42,] 25.750000 16.250000 10.000000 16.666667 24.750000 29.500000
[43,] NA NA NA NA NA NA
[44,] 14.500000 28.500000 22.250000 17.500000 7.500000 10.750000
[45,] 10.500000 32.500000 26.250000 21.500000 11.500000 6.750000
[46,] 34.500000 15.000000 18.750000 25.416667 33.500000 38.250000
[47,] 33.250000 20.000000 17.500000 24.166667 32.250000 37.000000
[48,] 21.500000 33.750000 29.250000 24.500000 14.500000 17.750000
[49,] 17.500000 31.500000 25.250000 20.500000 10.500000 13.750000
[50,] 14.500000 35.500000 29.250000 24.500000 14.500000 10.750000
[,36] [,37] [,38] [,39] [,40] [,41] [,42] [,43]
[1,] 31.250000 30.750000 NA 20.75000 24.00000 35.00000 35.75000 NA
[2,] 36.250000 35.750000 NA 25.75000 29.00000 40.00000 40.75000 NA
[3,] 40.250000 38.750000 NA 28.75000 32.33333 44.00000 43.75000 NA
[4,] 41.416667 34.750000 NA 24.75000 28.33333 44.75000 39.75000 NA
[5,] 41.416667 34.750000 NA 24.75000 25.00000 44.75000 39.75000 NA
[6,] 27.500000 27.000000 NA 24.50000 27.75000 31.25000 32.00000 NA
[7,] 32.500000 32.000000 NA 29.50000 32.75000 36.25000 37.00000 NA
[8,] 37.500000 36.000000 NA 26.00000 29.58333 41.25000 41.00000 NA
[9,] 37.666667 31.000000 NA 21.00000 24.58333 41.00000 36.00000 NA
[10,] 38.916667 32.250000 NA 22.25000 22.50000 42.25000 37.25000 NA
[11,] 23.750000 23.250000 NA 28.25000 31.50000 27.50000 28.25000 NA
[12,] NA NA NA NA NA NA NA NA
[13,] 38.916667 32.250000 NA 22.25000 25.83333 42.25000 37.25000 NA
[14,] 33.916667 27.250000 NA 17.25000 20.83333 37.25000 32.25000 NA
[15,] 33.916667 27.250000 NA 17.25000 17.50000 37.25000 32.25000 NA
[16,] 20.000000 19.500000 NA 25.50000 30.25000 23.75000 24.50000 NA
[17,] NA NA NA NA NA NA NA NA
[18,] NA NA NA NA NA NA NA NA
[19,] 30.166667 23.500000 NA 13.50000 17.08333 33.50000 28.50000 NA
[20,] 32.250000 25.583333 NA 15.58333 12.50000 35.58333 30.58333 NA
[21,] 16.250000 15.750000 NA 21.75000 26.50000 20.00000 20.75000 NA
[22,] 18.416667 11.750000 NA 17.75000 22.50000 21.75000 16.75000 NA
[23,] 22.416667 15.750000 NA 13.75000 18.33333 25.75000 20.75000 NA
[24,] 26.416667 19.750000 NA 9.75000 13.33333 29.75000 24.75000 NA
[25,] 29.750000 23.083333 NA 13.08333 10.00000 33.08333 28.08333 NA
[26,] 12.500000 13.750000 NA 19.75000 24.50000 16.25000 18.75000 NA
[27,] 15.416667 8.750000 NA 14.75000 19.50000 18.75000 13.75000 NA
[28,] 19.416667 12.750000 NA 10.75000 15.50000 22.75000 17.75000 NA
[29,] 23.416667 16.750000 NA 6.75000 11.50000 26.75000 21.75000 NA
[30,] 27.416667 20.750000 NA 10.75000 7.50000 30.75000 25.75000 NA
[31,] 7.500000 11.666667 NA 24.75000 29.50000 11.25000 16.25000 NA
[32,] 11.666667 5.000000 NA 18.50000 23.25000 15.00000 10.00000 NA
[33,] 18.333333 11.666667 NA 13.75000 18.50000 21.66667 16.66667 NA
[34,] 26.416667 19.750000 NA 3.75000 8.75000 29.75000 24.75000 NA
[35,] 31.166667 24.500000 NA 8.75000 3.75000 34.50000 29.50000 NA
[36,] 0.000000 6.666667 NA 30.16667 34.91667 3.75000 8.75000 NA
[37,] 6.666667 0.000000 NA 23.50000 28.25000 10.00000 5.00000 NA
[38,] NA NA 0 NA NA NA NA NA
[39,] 30.166667 23.500000 NA 0.00000 5.00000 33.50000 28.50000 NA
[40,] 34.916667 28.250000 NA 5.00000 0.00000 38.25000 33.25000 NA
[41,] 3.750000 10.000000 NA 33.50000 38.25000 0.00000 5.00000 NA
[42,] 8.750000 5.000000 NA 28.50000 33.25000 5.00000 0.00000 NA
[43,] NA NA NA NA NA NA NA 0
[44,] 33.916667 27.250000 NA 3.75000 7.00000 35.75000 32.25000 NA
[45,] 37.916667 31.250000 NA 7.75000 3.00000 39.75000 36.25000 NA
[46,] 7.500000 13.750000 NA 35.75000 39.00000 3.75000 8.75000 NA
[47,] 12.500000 12.500000 NA 30.75000 34.00000 8.75000 7.50000 NA
[48,] 32.500000 32.500000 NA 10.75000 14.00000 28.75000 27.50000 NA
[49,] 36.500000 30.250000 NA 6.75000 10.00000 32.75000 31.50000 NA
[50,] 40.500000 34.250000 NA 10.75000 7.00000 36.75000 35.50000 NA
[,44] [,45] [,46] [,47] [,48] [,49] [,50]
[1,] 17.00000 21.00000 35.00000 30.00000 10.00000 14.00000 18.00000
[2,] 22.00000 26.00000 40.00000 35.00000 15.00000 19.00000 23.00000
[3,] 26.00000 30.00000 44.00000 39.00000 19.00000 23.00000 27.00000
[4,] 28.50000 31.33333 48.00000 43.00000 23.00000 27.00000 31.00000
[5,] 28.50000 28.00000 48.50000 46.33333 26.33333 30.33333 32.00000
[6,] 20.75000 24.75000 35.00000 33.75000 13.75000 17.75000 21.75000
[7,] 25.75000 29.75000 40.00000 38.75000 18.75000 22.75000 26.75000
[8,] 29.00000 32.58333 45.00000 42.00000 22.00000 26.00000 30.00000
[9,] 24.75000 27.58333 44.75000 43.50000 26.75000 27.75000 31.58333
[10,] 26.00000 25.50000 46.00000 44.75000 28.83333 29.00000 29.50000
[11,] 24.50000 28.50000 31.25000 35.75000 17.50000 21.50000 25.50000
[12,] NA NA NA NA NA NA NA
[13,] 26.00000 28.83333 46.00000 44.75000 25.75000 29.00000 32.83333
[14,] 21.00000 23.83333 41.00000 39.75000 28.00000 24.00000 27.83333
[15,] 21.00000 20.50000 41.00000 39.75000 28.00000 24.00000 24.50000
[16,] 28.25000 32.25000 27.50000 32.00000 21.25000 25.25000 29.25000
[17,] NA NA NA NA NA NA NA
[18,] NA NA NA NA NA NA NA
[19,] 17.25000 20.08333 37.25000 36.00000 24.25000 20.25000 24.08333
[20,] 19.33333 15.50000 39.33333 38.08333 26.33333 22.33333 19.50000
[21,] 25.50000 29.50000 23.75000 28.25000 25.00000 28.50000 32.50000
[22,] 21.50000 25.50000 25.50000 24.25000 28.50000 24.50000 28.50000
[23,] 17.50000 21.33333 29.50000 28.25000 24.50000 20.50000 24.50000
[24,] 13.50000 16.33333 33.50000 32.25000 20.50000 16.50000 20.33333
[25,] 16.83333 13.00000 36.83333 35.58333 23.83333 19.83333 17.00000
[26,] 23.50000 27.50000 20.00000 25.00000 28.75000 26.50000 30.50000
[27,] 18.50000 22.50000 22.50000 21.25000 25.50000 21.50000 25.50000
[28,] 14.50000 18.50000 26.50000 25.25000 21.50000 17.50000 21.50000
[29,] 10.50000 14.50000 30.50000 29.25000 17.50000 13.50000 17.50000
[30,] 14.50000 10.50000 34.50000 33.25000 21.50000 17.50000 14.50000
[31,] 28.50000 32.50000 15.00000 20.00000 33.75000 31.50000 35.50000
[32,] 22.25000 26.25000 18.75000 17.50000 29.25000 25.25000 29.25000
[33,] 17.50000 21.50000 25.41667 24.16667 24.50000 20.50000 24.50000
[34,] 7.50000 11.50000 33.50000 32.25000 14.50000 10.50000 14.50000
[35,] 10.75000 6.75000 38.25000 37.00000 17.75000 13.75000 10.75000
[36,] 33.91667 37.91667 7.50000 12.50000 32.50000 36.50000 40.50000
[37,] 27.25000 31.25000 13.75000 12.50000 32.50000 30.25000 34.25000
[38,] NA NA NA NA NA NA NA
[39,] 3.75000 7.75000 35.75000 30.75000 10.75000 6.75000 10.75000
[40,] 7.00000 3.00000 39.00000 34.00000 14.00000 10.00000 7.00000
[41,] 35.75000 39.75000 3.75000 8.75000 28.75000 32.75000 36.75000
[42,] 32.25000 36.25000 8.75000 7.50000 27.50000 31.50000 35.50000
[43,] NA NA NA NA NA NA NA
[44,] 0.00000 4.00000 32.00000 27.00000 7.00000 3.00000 7.00000
[45,] 4.00000 0.00000 36.00000 31.00000 11.00000 7.00000 4.00000
[46,] 32.00000 36.00000 0.00000 5.00000 25.00000 29.00000 33.00000
[47,] 27.00000 31.00000 5.00000 0.00000 20.00000 24.00000 28.00000
[48,] 7.00000 11.00000 25.00000 20.00000 0.00000 4.00000 8.00000
[49,] 3.00000 7.00000 29.00000 24.00000 4.00000 0.00000 4.00000
[50,] 7.00000 4.00000 33.00000 28.00000 8.00000 4.00000 0.00000
print("/n")
[1] "/n"
mins
[1] 663.3333 683.6667