Factors affecting Flight Landing
Goal: To study what factors and how they impact the landing distance of a commercial flight
Summary:
The first step is to import and clean the data so that it is ready for the next step i.e. statistical analysis. So, I imported the files, explored the data, identified extreme values (abnormal), merged the datasets and removed duplicate observations. Along with this, I also looked at the statistical summary and explored missing data. The number of observations left after merging, removing the duplicate rows and removing the abnormal values was 840.
Next step was to analyze the data using data visualization, correlation and regression techniques. The goal of this step was to find the relationship between the response variable (distance) and the predictors like speed_ground, speed_air, etc. (if any) using correlation and regression analysis. Also, I visualized the dataset using the scatter plot. Visualization is not enough as we need to see if the relationship is statistically significant. So, I moved on to Correlation analysis. Correlation helps to identify if there is any relationship between the predictor variable and the response. One thing where correlation fails is that it doesn’t explain the relationship between the response variable and all the predictor variables together. So, the next step I did was multiple regression. After performing multiple regression and considering the issue of multicollinearity, I found that speed_ground was not significant.
Finally, I applied StepAIC to perform forward variable selection based. Speed_air, aircraft, height and no_pasg predictors were selected based on lowest AIC value.
# Loading the relevant libraries
library(readxl)
library(dplyr)
library(ggplot2)
library(tidyr)
library(purrr)
library(MASS)# Importing the Excel files -----------------------------------------------
faa1 <- read_excel("C:/Users/ashis/Downloads/Stat Modelling/Week 1/FAA1.xls")
faa2 <- read_excel("C:/Users/ashis/Downloads/Stat Modelling/Week 1/FAA2.xls")
# Exploring the data ------------------------------------------------------
str(faa1)## Classes 'tbl_df', 'tbl' and 'data.frame': 800 obs. of 8 variables:
## $ aircraft : chr "boeing" "boeing" "boeing" "boeing" ...
## $ duration : num 98.5 125.7 112 196.8 90.1 ...
## $ no_pasg : num 53 69 61 56 70 55 54 57 61 56 ...
## $ speed_ground: num 107.9 101.7 71.1 85.8 59.9 ...
## $ speed_air : num 109 103 NA NA NA ...
## $ height : num 27.4 27.8 18.6 30.7 32.4 ...
## $ pitch : num 4.04 4.12 4.43 3.88 4.03 ...
## $ distance : num 3370 2988 1145 1664 1050 ...str(faa2)## Classes 'tbl_df', 'tbl' and 'data.frame': 150 obs. of 7 variables:
## $ aircraft : chr "boeing" "boeing" "boeing" "boeing" ...
## $ no_pasg : num 53 69 61 56 70 55 54 57 61 56 ...
## $ speed_ground: num 107.9 101.7 71.1 85.8 59.9 ...
## $ speed_air : num 109 103 NA NA NA ...
## $ height : num 27.4 27.8 18.6 30.7 32.4 ...
## $ pitch : num 4.04 4.12 4.43 3.88 4.03 ...
## $ distance : num 3370 2988 1145 1664 1050 ...head(faa1)## # A tibble: 6 x 8
## aircraft duration no_pasg speed_ground speed_air height pitch distance
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 boeing 98.5 53 108. 109. 27.4 4.04 3370.
## 2 boeing 126. 69 102. 103. 27.8 4.12 2988.
## 3 boeing 112. 61 71.1 NA 18.6 4.43 1145.
## 4 boeing 197. 56 85.8 NA 30.7 3.88 1664.
## 5 boeing 90.1 70 59.9 NA 32.4 4.03 1050.
## 6 boeing 138. 55 75.0 NA 41.2 4.20 1627.head(faa2)## # A tibble: 6 x 7
## aircraft no_pasg speed_ground speed_air height pitch distance
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 boeing 53 108. 109. 27.4 4.04 3370.
## 2 boeing 69 102. 103. 27.8 4.12 2988.
## 3 boeing 61 71.1 NA 18.6 4.43 1145.
## 4 boeing 56 85.8 NA 30.7 3.88 1664.
## 5 boeing 70 59.9 NA 32.4 4.03 1050.
## 6 boeing 55 75.0 NA 41.2 4.20 1627.dim(faa1)## [1] 800 8dim(faa2)## [1] 150 7identical(faa1,faa2)## [1] FALSE# Creating an additional column 'duration' in FAA2 --------------------------
faa2 <- faa2 %>% mutate(duration = NA)
# Converting the data type of 'duration' column to numeric --------------
faa2$duration <- as.numeric(faa2$duration)
# Arranging the columns of faa2 in the same order as faa1 -----------------
faa2 <- faa2 %>% dplyr::select(aircraft,duration,no_pasg,speed_ground,speed_air,
height,pitch,distance)
# Merging the datasets faa1 and faa2 --------------------------------------
faa_merged <- rbind(faa1,faa2)
# Fetching the index of duplicate rows after removing the duration column. We can see that 100 rows are duplicate
which(duplicated(faa_merged[,-2]))## [1] 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817
## [18] 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834
## [35] 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851
## [52] 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868
## [69] 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885
## [86] 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900faa_duplicate <- faa_merged[which(duplicated(faa_merged[,-2])),]
faa_duplicate## # A tibble: 100 x 8
## aircraft duration no_pasg speed_ground speed_air height pitch distance
## <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 boeing NA 53 108. 109. 27.4 4.04 3370.
## 2 boeing NA 69 102. 103. 27.8 4.12 2988.
## 3 boeing NA 61 71.1 NA 18.6 4.43 1145.
## 4 boeing NA 56 85.8 NA 30.7 3.88 1664.
## 5 boeing NA 70 59.9 NA 32.4 4.03 1050.
## 6 boeing NA 55 75.0 NA 41.2 4.20 1627.
## 7 boeing NA 54 54.4 NA 24.0 3.84 805.
## 8 boeing NA 57 57.1 NA 19.4 4.64 574.
## 9 boeing NA 61 85.4 NA 35.4 4.23 1699.
## 10 boeing NA 56 61.8 NA 36.7 4.18 1138.
## # ... with 90 more rows# Removing the duplicate rows from the merged dataset ---------------------
faa_merged <- faa_merged[-801:-900,]
# Summary statistic of the merged data. We are left with 850 rows
str(faa_merged)## Classes 'tbl_df', 'tbl' and 'data.frame': 850 obs. of 8 variables:
## $ aircraft : chr "boeing" "boeing" "boeing" "boeing" ...
## $ duration : num 98.5 125.7 112 196.8 90.1 ...
## $ no_pasg : num 53 69 61 56 70 55 54 57 61 56 ...
## $ speed_ground: num 107.9 101.7 71.1 85.8 59.9 ...
## $ speed_air : num 109 103 NA NA NA ...
## $ height : num 27.4 27.8 18.6 30.7 32.4 ...
## $ pitch : num 4.04 4.12 4.43 3.88 4.03 ...
## $ distance : num 3370 2988 1145 1664 1050 ...summary(faa_merged)## aircraft duration no_pasg speed_ground
## Length:850 Min. : 14.76 Min. :29.0 Min. : 27.74
## Class :character 1st Qu.:119.49 1st Qu.:55.0 1st Qu.: 65.90
## Mode :character Median :153.95 Median :60.0 Median : 79.64
## Mean :154.01 Mean :60.1 Mean : 79.45
## 3rd Qu.:188.91 3rd Qu.:65.0 3rd Qu.: 92.06
## Max. :305.62 Max. :87.0 Max. :141.22
## NA's :50
## speed_air height pitch distance
## Min. : 90.00 Min. :-3.546 Min. :2.284 Min. : 34.08
## 1st Qu.: 96.25 1st Qu.:23.314 1st Qu.:3.642 1st Qu.: 883.79
## Median :101.15 Median :30.093 Median :4.008 Median :1258.09
## Mean :103.80 Mean :30.144 Mean :4.009 Mean :1526.02
## 3rd Qu.:109.40 3rd Qu.:36.993 3rd Qu.:4.377 3rd Qu.:1936.95
## Max. :141.72 Max. :59.946 Max. :5.927 Max. :6533.05
## NA's :642faa_merged$aircraft <- as.factor(faa_merged$aircraft)
# Removing the abnormal values in height and duration columns -------------
faa_merged <- faa_merged %>% filter((is.na(duration) | duration > 40) & height > 0)
# Plotting density plot for all the variables except aircraft variable ---------
faa_merged[,-1] %>%
gather() %>%
ggplot(aes(value)) +
facet_wrap(~ key, scales = "free") +
geom_density() +
theme_minimal()
# Computing the correlation with distance for all the variables but aircraft
correlation_vector <- vector("numeric", 6)
for (i in 2:(ncol(faa_merged) - 1)) {
correlation_vector[i - 1] <- as.numeric(cor(faa_merged[,i], faa_merged[ncol(faa_merged)],
use = "complete.obs"))
}
names(correlation_vector) <- c("duration","no_pasg","speed_ground","speed_air",
"height", "pitch")
correlation_vector## duration no_pasg speed_ground speed_air height
## -0.05287377 -0.02457070 0.86166762 0.94786540 0.11543232
## pitch
## 0.09726243# Plotting the scatter matrix ---------------------------------------------
GGally::ggpairs(faa_merged[, ])
# Applying the linear regression model to all the variables one by one -----
list_model <- list()
summary_model <- list()
for (i in names(faa_merged[,1:7])) {
list_model[[i]] <- lm(paste("distance ~", i), data = faa_merged)
summary_model[[i]] <- summary( list_model[[i]])
}
merged_coeff <- map(summary_model,coefficients)
pvalue_merged <- sort(unlist(map(merged_coeff,8)))
# Scaling the predictors and applying lm model ----------------------------
faa_scaled <- data.frame(scale(faa_merged[,2:7]))
class(faa_scaled)## [1] "data.frame"faa_scaled[,c(7,8)] <- faa_merged[,c(1,8)]
scaled_model <- list()
scaled_summary <- list()
for (i in names(faa_scaled[,1:7])) {
scaled_model[[i]] <- lm(paste("distance ~", i), data = faa_scaled)
scaled_summary[[i]] <- summary( scaled_model[[i]])
}
scaled_coeff <- map(scaled_summary,coefficients)
scaled_beta <- sort(unlist(map(scaled_coeff,2)), decreasing = TRUE)
# Printing correlation, p-value and slope coefficient. In terms of correlation, speed_air is highly correlated with distance, then speed_ground and so on. Looking at p values, all the p values are less than 5% but speed_ground is the lowest, then speed_air, etc. In terms of coefficient, speed_air is the highest, then speed_ground, etc.
sort(correlation_vector, decreasing = TRUE)## speed_air speed_ground height pitch no_pasg
## 0.94786540 0.86166762 0.11543232 0.09726243 -0.02457070
## duration
## -0.05287377pvalue_merged## speed_ground speed_air aircraft height pitch
## 4.372864e-249 8.896065e-103 4.011232e-12 8.029532e-04 4.781048e-03
## duration no_pasg
## 1.375954e-01 4.769768e-01scaled_beta## speed_air speed_ground aircraft height pitch
## 846.84021 798.14333 438.26834 106.92236 90.09200
## no_pasg duration
## -22.75928 -49.48541# Applying lm model (speed_air, speed_ground and both simultaneously) to check multicollinearity.
model_colli1 <- lm(distance~speed_ground, data = faa_merged)
model_colli2 <- lm(distance~speed_air, data = faa_merged)
model_colli3 <- lm(distance~speed_air + speed_ground, data = faa_merged)
summ_colli1 <- summary(model_colli1)
summ_colli2 <- summary(model_colli2)
summ_colli3 <- summary(model_colli3)
# Printing p values for speed_ground, speed_air & speed_ground+speed_air
summ_colli1$coefficients[8]## [1] 4.372864e-249summ_colli2$coefficients[8]## [1] 8.896065e-103summ_colli3$coefficients[11]## [1] 6.321636e-12summ_colli3$coefficients[12]## [1] 0.2605788# Applying forward selection based on variables selected after step 16. Considered speed_air as the best variable based on p-value, correlation and slope coefficient, then speed_ground, etc. Adding the variable one at time and looking at R2, Adjusted R2 and AIC value.
model_forward_sel <- list()
model_forward_sel[[1]] <- lm(distance~speed_air, data = faa_merged)
model_forward_sel[[2]] <- lm(distance~speed_air+speed_ground, data = faa_merged)
model_forward_sel[[3]] <- lm(distance~speed_air+speed_ground+aircraft, data = faa_merged)
model_forward_sel[[4]] <- lm(distance~speed_air+speed_ground+aircraft+height,
data = faa_merged)
model_forward_sel[[5]] <- lm(distance~speed_air+speed_ground+aircraft+height+pitch,
data = faa_merged)
model_forward_sel[[6]] <- lm(distance~speed_air+speed_ground+aircraft+height+pitch+no_pasg,
data = faa_merged)
model_forward <- list()
for (i in 1:length(model_forward_sel)) {
model_forward[[i]] <- summary(model_forward_sel[[i]])
}
r_square <- map(model_forward,"r.squared")
adjusted_rsquare <- map(model_forward,"adj.r.squared")
# Printing R square, Adjusted R Square & AIC. We can see that R2 increases as the number of variables are increasing. Adjusted R2 is also increasing even though it is applying penalty for adding the variable. AIC is lowest for speed_air+speed_ground+aircraft+height
r_square## [[1]]
## [1] 0.8984488
##
## [[2]]
## [1] 0.8990847
##
## [[3]]
## [1] 0.9540371
##
## [[4]]
## [1] 0.9754762
##
## [[5]]
## [1] 0.975483
##
## [[6]]
## [1] 0.9758141adjusted_rsquare## [[1]]
## [1] 0.8979486
##
## [[2]]
## [1] 0.8980855
##
## [[3]]
## [1] 0.9533511
##
## [[4]]
## [1] 0.9749857
##
## [[5]]
## [1] 0.974867
##
## [[6]]
## [1] 0.9750812map(model_forward_sel, AIC)## [[1]]
## [1] 2903.86
##
## [[2]]
## [1] 2904.573
##
## [[3]]
## [1] 2745.351
##
## [[4]]
## [1] 2618.572
##
## [[5]]
## [1] 2620.515
##
## [[6]]
## [1] 2619.728# Performing forward variable selection using stepAIC. We found that AIC is lowest for speed_air + aircraft + height + no_pasg i.e. 1955.48
entire_model <- lm(distance~., data = na.omit(faa_merged))
null_model <- lm(distance~1, data = na.omit(faa_merged))
stepAIC(null_model, scope = list(lower = null_model,upper = entire_model),
direction = "forward")## Start: AIC=2682.54
## distance ~ 1
##
## Df Sum of Sq RSS AIC
## + speed_air 1 143944957 15942768 2230.4
## + speed_ground 1 140048296 19839429 2273.4
## + aircraft 1 5775391 154112334 2677.3
## <none> 159887725 2682.5
## + pitch 1 991324 158896401 2683.3
## + height 1 707080 159180645 2683.7
## + duration 1 370042 159517683 2684.1
## + no_pasg 1 200305 159687420 2684.3
##
## Step: AIC=2230.36
## distance ~ speed_air
##
## Df Sum of Sq RSS AIC
## + aircraft 1 8642406 7300363 2078.5
## + height 1 2868393 13074375 2193.3
## + pitch 1 1062502 14880266 2218.8
## <none> 15942768 2230.4
## + no_pasg 1 142329 15800439 2230.6
## + speed_ground 1 75985 15866783 2231.4
## + duration 1 8648 15934120 2232.2
##
## Step: AIC=2078.49
## distance ~ speed_air + aircraft
##
## Df Sum of Sq RSS AIC
## + height 1 3426505 3873858 1955.7
## <none> 7300363 2078.5
## + duration 1 52208 7248155 2079.1
## + no_pasg 1 44258 7256104 2079.3
## + speed_ground 1 42073 7258290 2079.3
## + pitch 1 3705 7296658 2080.4
##
## Step: AIC=1955.65
## distance ~ speed_air + aircraft + height
##
## Df Sum of Sq RSS AIC
## + no_pasg 1 42386 3831472 1955.5
## <none> 3873858 1955.7
## + duration 1 10970 3862888 1957.1
## + speed_ground 1 7715 3866143 1957.3
## + pitch 1 488 3873370 1957.6
##
## Step: AIC=1955.48
## distance ~ speed_air + aircraft + height + no_pasg
##
## Df Sum of Sq RSS AIC
## <none> 3831472 1955.5
## + duration 1 7980.7 3823491 1957.1
## + speed_ground 1 6881.7 3824590 1957.1
## + pitch 1 755.5 3830716 1957.5##
## Call:
## lm(formula = distance ~ speed_air + aircraft + height + no_pasg,
## data = na.omit(faa_merged))
##
## Coefficients:
## (Intercept) speed_air aircraftboeing height
## -6446.140 83.713 442.235 14.105
## no_pasg
## -2.098