From the previous projects, we have cleaned up abnormal data. Since aircraft and aircraft_num are the same, I am eliminating the aircraft variable as well, with knowing airbus = 1, and boeing = 0. I also converted the aircraft_num and distance_Y to factor variable.
library(dplyr)
library(nnet)
library(ggplot2)
FAA <- read.csv("FAA_wrangled.csv", header = TRUE)
#create a multinomial variable and attach it to dataset
FAA <- FAA %>%
mutate(distance_Y = case_when(
distance < 1000 ~ 1,
distance >= 1000 & distance < 2500 ~ 2,
TRUE ~ 3
))
#discard variable distance
FAA <- FAA[, c(-1,-7)]
FAA$distance_Y <- as.factor(FAA$distance_Y)
FAA$aircraft_num <- as.factor(FAA$aircraft_num)
attach(FAA)
str(FAA)'data.frame': 195 obs. of 8 variables:
$ no_pasg : int 41 44 47 48 48 48 49 49 49 51 ...
$ speed_ground: num 97.6 99.6 92.9 101.8 109.3 ...
$ speed_air : num 97 99.2 95.8 103.6 109.6 ...
$ height : num 38.4 35.2 23.8 23 33.1 ...
$ pitch : num 3.53 3.84 3.91 4.94 4.04 ...
$ duration : num 123 139 169 157 200 ...
$ aircraft_num: Factor w/ 2 levels "0","1": 2 2 2 2 2 2 2 2 2 2 ...
$ distance_Y : Factor w/ 2 levels "2","3": 1 1 1 2 2 2 2 2 2 2 ...summary(FAA) no_pasg speed_ground speed_air height pitch duration
Min. :41.00 Min. : 88.69 Min. : 90.00 Min. : 9.697 Min. :2.702 Min. : 45.5
1st Qu.:56.00 1st Qu.: 95.28 1st Qu.: 96.15 1st Qu.:23.365 1st Qu.:3.636 1st Qu.:115.9
Median :60.00 Median :100.75 Median :100.89 Median :29.837 Median :4.070 Median :149.3
Mean :59.83 Mean :103.43 Mean :103.50 Mean :30.359 Mean :4.043 Mean :150.9
3rd Qu.:65.00 3rd Qu.:109.57 3rd Qu.:109.42 3rd Qu.:36.590 3rd Qu.:4.442 3rd Qu.:185.4
Max. :80.00 Max. :132.78 Max. :132.91 Max. :58.228 Max. :5.311 Max. :287.0
aircraft_num distance_Y
0:118 2: 95
1: 77 3:100
Number of passenger is a count variable, so I am using Poisson distribution to model this variable.
#fit a GLM
mod <- glm(no_pasg ~ ., family = poisson, FAA)
summary(mod)
Call:
glm(formula = no_pasg ~ ., family = poisson, data = FAA)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 4.139e+00 1.782e-01 23.228 <2e-16 ***
speed_ground 7.798e-04 6.178e-03 0.126 0.900
speed_air -8.324e-04 6.310e-03 -0.132 0.895
height -5.875e-05 1.028e-03 -0.057 0.954
pitch -4.671e-03 1.800e-02 -0.260 0.795
duration -1.742e-04 1.961e-04 -0.889 0.374
aircraft_num1 1.244e-02 2.111e-02 0.589 0.556
distance_Y3 5.709e-04 2.971e-02 0.019 0.985
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 162.74 on 194 degrees of freedom
Residual deviance: 161.31 on 187 degrees of freedom
AIC: 1332.7
Number of Fisher Scoring iterations: 4step(mod)Start: AIC=1332.69
no_pasg ~ speed_ground + speed_air + height + pitch + duration +
aircraft_num + distance_Y
Df Deviance AIC
- distance_Y 1 161.31 1330.7
- height 1 161.31 1330.7
- speed_ground 1 161.32 1330.7
- speed_air 1 161.32 1330.7
- pitch 1 161.38 1330.8
- aircraft_num 1 161.65 1331.0
- duration 1 162.10 1331.5
<none> 161.31 1332.7
Step: AIC=1330.69
no_pasg ~ speed_ground + speed_air + height + pitch + duration +
aircraft_num
Df Deviance AIC
- height 1 161.31 1328.7
- speed_ground 1 161.32 1328.7
- speed_air 1 161.32 1328.7
- pitch 1 161.38 1328.8
- aircraft_num 1 161.67 1329.0
- duration 1 162.10 1329.5
<none> 161.31 1330.7
Step: AIC=1328.69
no_pasg ~ speed_ground + speed_air + pitch + duration + aircraft_num
Df Deviance AIC
- speed_ground 1 161.33 1326.7
- speed_air 1 161.33 1326.7
- pitch 1 161.38 1326.8
- aircraft_num 1 161.67 1327.0
- duration 1 162.11 1327.5
<none> 161.31 1328.7
Step: AIC=1326.71
no_pasg ~ speed_air + pitch + duration + aircraft_num
Df Deviance AIC
- speed_air 1 161.33 1324.7
- pitch 1 161.40 1324.8
- aircraft_num 1 161.69 1325.1
- duration 1 162.18 1325.6
<none> 161.33 1326.7
Step: AIC=1324.71
no_pasg ~ pitch + duration + aircraft_num
Df Deviance AIC
- pitch 1 161.40 1322.8
- aircraft_num 1 161.69 1323.1
- duration 1 162.18 1323.6
<none> 161.33 1324.7
Step: AIC=1322.78
no_pasg ~ duration + aircraft_num
Df Deviance AIC
- aircraft_num 1 161.97 1321.3
- duration 1 162.24 1321.6
<none> 161.40 1322.8
Step: AIC=1321.35
no_pasg ~ duration
Df Deviance AIC
- duration 1 162.74 1320.1
<none> 161.97 1321.3
Step: AIC=1320.12
no_pasg ~ 1
Call: glm(formula = no_pasg ~ 1, family = poisson, data = FAA)
Coefficients:
(Intercept)
4.091
Degrees of Freedom: 194 Total (i.e. Null); 194 Residual
Null Deviance: 162.7
Residual Deviance: 162.7 AIC: 1320drop1(mod, test = "LRT")Single term deletions
Model:
no_pasg ~ speed_ground + speed_air + height + pitch + duration +
aircraft_num + distance_Y
Df Deviance AIC LRT Pr(>Chi)
<none> 161.31 1332.7
speed_ground 1 161.32 1330.7 0.01593 0.8996
speed_air 1 161.32 1330.7 0.01740 0.8951
height 1 161.31 1330.7 0.00327 0.9544
pitch 1 161.38 1330.8 0.06734 0.7953
duration 1 162.10 1331.5 0.79052 0.3739
aircraft_num 1 161.65 1331.0 0.34706 0.5558
distance_Y 1 161.31 1330.7 0.00037 0.9847#check the correlation
round(cor(FAA[,1:6]), 2) no_pasg speed_ground speed_air height pitch duration
no_pasg 1.00 0.00 0.00 -0.01 -0.04 -0.07
speed_ground 0.00 1.00 0.99 -0.10 -0.06 0.02
speed_air 0.00 0.99 1.00 -0.09 -0.05 0.04
height -0.01 -0.10 -0.09 1.00 -0.03 0.07
pitch -0.04 -0.06 -0.05 -0.03 1.00 -0.06
duration -0.07 0.02 0.04 0.07 -0.06 1.00I built a model called “mod” and from the summary statistics, we can see that there is no variable that’s significant enough to impact the number of passengers variable. From the step method, I got the null model.
I also decided to check the correlation, and saw that only speed_ground and speed_air are highly correlated. We have learned to remove the speed_ground from the previous projects. This is not much of a help, as in the summary statistics, non of the variables showed significance.
In the next code, I tried to add the dispersion parameter and see if that could provide better result.
#take a look at the observed data and predicted data using the model
mod$y 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
41 44 47 48 48 48 49 49 49 51 52 52 53 53 54 54 54 55 55 55 56 56 56 56 57 58 58
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
58 58 59 60 60 60 60 60 61 61 61 62 62 62 62 62 62 62 63 63 63 63 63 64 64 65 65
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
65 65 65 65 66 66 66 66 66 66 67 67 67 68 68 68 69 70 71 72 73 75 80 43 44 46 46
82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
46 46 46 47 48 48 49 49 52 52 52 52 52 52 53 53 53 53 54 54 54 55 55 55 56 56 56
109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135
56 57 57 57 57 57 57 58 58 58 58 58 58 58 59 59 59 59 59 59 59 59 59 59 59 60 60
136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162
60 60 60 60 60 60 60 60 60 61 61 61 61 61 61 61 61 61 61 61 62 62 62 63 63 63 63
163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189
63 63 64 64 64 64 64 64 65 65 65 65 66 66 66 66 67 67 67 68 68 68 69 69 69 69 70
190 191 192 193 194 195
70 70 72 72 75 79 round(predict(mod, type = "response"),1) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22
60.7 60.5 60.0 59.9 59.7 60.3 60.3 60.8 59.9 60.5 60.7 60.5 60.0 60.4 60.5 59.7 60.3 59.1 60.0 59.8 61.3 59.7
23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
60.6 60.5 59.5 60.0 60.8 61.7 60.5 59.8 60.5 60.2 60.8 60.2 60.5 60.5 59.9 59.7 60.3 60.8 60.3 60.9 60.5 60.8
45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
60.4 61.0 60.6 60.3 59.9 60.8 60.7 60.5 60.2 59.8 61.3 60.7 60.8 59.7 59.5 59.7 59.5 61.3 60.7 60.6 60.0 60.4
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88
60.4 60.3 60.1 59.4 60.0 59.1 61.3 60.2 59.8 60.8 60.3 59.5 60.5 58.9 58.4 59.2 59.4 59.5 59.6 59.4 59.6 60.0
89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
59.7 60.1 59.7 58.9 59.7 58.0 60.4 57.9 59.9 59.1 60.0 58.8 60.5 58.4 60.3 60.2 59.4 60.1 59.2 59.9 58.7 59.7
111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132
60.1 59.7 59.7 59.9 59.8 58.9 59.9 58.7 59.8 59.6 59.0 58.6 58.7 59.8 58.7 59.2 58.9 60.0 58.7 59.6 59.6 59.5
133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154
58.2 58.9 60.3 60.1 59.5 59.1 60.2 59.7 59.7 59.8 59.6 60.0 60.0 59.0 59.7 58.9 59.3 59.3 59.9 59.7 59.5 60.1
155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176
59.4 59.2 60.4 59.1 59.1 59.3 59.1 59.8 59.0 59.5 60.0 60.0 59.9 58.6 60.0 59.9 59.2 58.7 58.9 59.4 59.9 59.4
177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195
59.4 60.2 60.1 60.0 59.2 59.7 59.9 59.2 59.3 60.1 59.7 58.9 60.1 60.1 60.1 59.9 59.4 60.1 59.5 gof<-sum(residuals(mod,type="pearson")^2)
#add dispersion parameter
dp <- gof/mod$df.res
summary(mod, dispersion = dp)
Call:
glm(formula = no_pasg ~ ., family = poisson, data = FAA)
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 4.139e+00 1.643e-01 25.191 <2e-16 ***
speed_ground 7.798e-04 5.696e-03 0.137 0.891
speed_air -8.324e-04 5.818e-03 -0.143 0.886
height -5.875e-05 9.480e-04 -0.062 0.951
pitch -4.671e-03 1.660e-02 -0.281 0.778
duration -1.742e-04 1.808e-04 -0.964 0.335
aircraft_num1 1.244e-02 1.946e-02 0.639 0.523
distance_Y3 5.709e-04 2.740e-02 0.021 0.983
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for poisson family taken to be 0.8502401)
Null deviance: 162.74 on 194 degrees of freedom
Residual deviance: 161.31 on 187 degrees of freedom
AIC: 1332.7
Number of Fisher Scoring iterations: 4From the new summary statistics with dispersion parameter, the P value for all variables other than the intercept is still showing non significant. Therefore, my conclusion is that there is no useful variables in the cleaned FAA dataset to predict the number of passengers on board.