NYCFlightsDataModel

Summarize Data

daily <- flights %>%
  mutate(date = make_date(year, month, day)) %>%
  group_by(date) %>%
  summarize(n = n())

ggplot(daily, aes(date, n)) +
  geom_line()

Investigate Daily-Weekly Pattern

daily <- daily %>%
  mutate(wday = wday(date, label = TRUE))
ggplot(daily, aes(wday,n)) +
  geom_boxplot()

mod = lm(n ~ wday, data = daily)

grid <- daily %>%
  data_grid(wday) %>%
  add_predictions(mod, "n")

ggplot(daily, aes(wday, n)) +
  geom_boxplot() +
  geom_point(data = grid, color = "orange", size = 4)

Investigate residuals

daily <- daily %>%
  add_residuals(mod)

daily %>%
  ggplot(aes(date, resid)) +
  geom_ref_line(h = 0) +
  geom_line()

ggplot(daily, aes(date, resid, color = wday)) +
  geom_ref_line(h = 0, colour = "red") +
  geom_line()

daily %>%
  filter(resid < -100)

## # A tibble: 11 x 4
##    date           n wday  resid
##    <date>     <int> <ord> <dbl>
##  1 2013-01-01   842 Tue   -109.
##  2 2013-01-20   786 Sun   -105.
##  3 2013-05-26   729 Sun   -162.
##  4 2013-07-04   737 Thu   -229.
##  5 2013-07-05   822 Fri   -145.
##  6 2013-09-01   718 Sun   -173.
##  7 2013-11-28   634 Thu   -332.
##  8 2013-11-29   661 Fri   -306.
##  9 2013-12-24   761 Tue   -190.
## 10 2013-12-25   719 Wed   -244.
## 11 2013-12-31   776 Tue   -175.

daily %>%
  ggplot(aes(date, resid)) +
  geom_ref_line(h = 0, colour = "red", size = 1) +
  geom_line(color = "grey50") +
  geom_smooth(se = FALSE, span = 0.20)

## `geom_smooth()` using method = 'loess' and formula 'y ~ x'

Seasonal Saturday effect

daily %>%
  filter(wday == "Sat") %>%
  ggplot(aes(date, n)) +
  geom_point()+
  geom_line() +
  scale_x_date(
    NULL,
    date_breaks = "1 month",
    date_labels = "%b"
  )

Add Seasonal Variable

term <- function(date) {
  cut(date,
      breaks = ymd(20130101, 20130605, 20130825, 20140101),
      labels = c("spring", "summer", "fall")
      )
}

daily <- daily %>%
  mutate(term = term(date))

daily %>%
  filter(wday == "Sat") %>%
  ggplot(aes(date, n, color = term)) +
  geom_point(alpha = 1/3)+
  geom_line() +
  scale_x_date(
    NULL,
    date_breaks = "1 month",
    date_labels = "%b"
  )

daily %>%
  ggplot(aes(wday, n, color = term)) +
  geom_boxplot()

mod1 <- lm(n ~ wday, data = daily)
mod2 <- lm(n ~ wday * term, data = daily)

daily %>%
  gather_residuals(without_term = mod1, with_term = mod2) %>%
  ggplot(aes(date, resid, color = model)) +
  geom_line(alpha = 0.75)

grid <- daily %>%
  data_grid(wday, term) %>%
  add_predictions(mod2, "n")

ggplot(daily, aes(wday, n)) +
  geom_boxplot() +
  geom_point(data = grid, color = "red") +
  facet_wrap(~ term)

Better model for outliers (Robust regression)

mod3 <- MASS::rlm(n ~ wday * term, data = daily)

daily %>%
  add_residuals(mod3, "resid") %>%
  ggplot(aes(date, resid)) +
  geom_hline(yintercept = 0, size = 2, color = "red") +
  geom_line()

Computed Variables

# If you are creating variables it might be a good idea to bundle the creation of the variables up into a function
compute_vars <- function(data) {
  data %>%
    mutate(term = term(date),
           wday = wday(date, label = TRUE)
           )
}

# Another option would be to put the transformations directly in the model formula:

wday2 <- function(x) wday(x, label = TRUE)
mod3 <- lm(n ~ wday2(date) * term(date), data = daily)

Time of Year: An Alternative Approach

# We could use a more flexible model to capture the pattern of school term in the data
library(splines)
mod <- MASS::rlm(n ~ wday * ns(date, 5), data = daily)

daily %>% 
  data_grid(wday, date = seq_range(date, n = 13)) %>% 
  add_predictions(mod) %>% 
  ggplot(aes(date, pred, color = wday)) +
  geom_line() +
  geom_point()

# We see a strong pattern in the numbers of Sat flights.  This is reassuring, because we also saw that pattern in the raw data.  It's a good sign when you get the same signal from different approaches.

Question #1

Why are there fewer than expected flights on January 20, May 26 and September 1? (Hint: they all have the same explanation.) How would these days generalize into another year?

### January 20, May 26 and September 1 are closed holidays. The actual dates change each year so they will not be generalized into another year.

Question #2

What do the three days with high positive residuals represent? How would these days generalize to another year?

daily %>%
  top_n(3, resid)

## # A tibble: 3 x 5
##   date           n wday  resid term 
##   <date>     <int> <ord> <dbl> <fct>
## 1 2013-11-30   857 Sat   112.  fall 
## 2 2013-12-01   987 Sun    95.5 fall 
## 3 2013-12-28   814 Sat    69.4 fall

### They are thankgiving and Christmas holidays.

Question #3

Create a new variable that splits the “wday” variable into terms, but only for Saturdays, i.e., it should have Thurs, Fri, but Sat-summer, Sat-spring, Sat-fall. How does this model compare with the model with every combination of “wday” and “term”?

daily_q3 <- daily %>%
  mutate(wday2 = 
         case_when(.$wday == "Sat" & .$term == "summer" ~ "Sat-summer",
         .$wday == "Sat" & .$term == "fall" ~ "Sat-fall",
         .$wday == "Sat" & .$term == "spring" ~ "Sat-spring",
         TRUE ~ as.character(.$wday)))

mod2 <- lm(n ~ wday * term, data = daily_q3)

mod_q3 <- lm(n ~ wday2, data = daily_q3)

daily_q3 %>% 
  gather_residuals(Q3 = mod_q3, All_combine = mod2) %>% 
  ggplot(aes(date, resid, colour = model)) +
    geom_line(alpha = 0.75)

AIC(mod2)

## [1] 3855.73

AIC(mod_q3)

## [1] 3862.885

### This model has smaller AIC.

Question #4

Create a new “wday” variable that combines the day of week, term(for Saturdays), and public holidays. What do the residuals of the model look like?

mod2 <- lm(n ~ wday * term, data = daily)

mod_q5 <- lm(n ~ wday * month(date), data = daily)
daily %>% 
  gather_residuals(Q4 = mod_q5, All_combine = mod2) %>% 
  ggplot(aes(date, resid, colour = model)) +
    geom_line(alpha = 0.75)

Question #5

What happens if you fit a day-of-week effect that varies by month (i.e.m n ~ wday*month)? Why is this not very helpful?

mod2 <- lm(n ~ wday * term, data = daily)

mod_q5 <- lm(n ~ wday * month(date), data = daily)
daily %>% 
  gather_residuals(Q4 = mod_q5, All_combine = mod2) %>% 
  ggplot(aes(date, resid, colour = model)) +
    geom_line(alpha = 0.75)

### The fewer samples and bigger variabilities during summer time make the residuals during that time bigger than the usual.

Question #6

What would you expect the model n ~ wday + ns(date,5) to look like? Knowing what you know about the data, why would you expect it not to be particularly effective?

### No trend can be observed across weekdays, and the weekday conditions. Therefore, the model wouldn’t be effective.

Question #7

We hypothesized that people leaving on Sundays are more likely to be business travelers who need to be somewhere on Monday. Explore the hypothesis by seeing how if breaks down based on distance and time: if it’s true, you’d expect to see more Sunday evening flights to places that are far away.

q7 <- flights[flights$sched_dep_time >= 2000, ]

q7 %>% 
  mutate(date = make_date(year, month, day)) %>%
  mutate(wday = wday(date, label = TRUE)) %>%
  group_by(wday) %>%
  summarize(n = n()) %>%
  ggplot(aes(as.factor(wday), n)) +
  geom_bar(stat = "identity")

q7 %>% 
  mutate(date = make_date(year, month, day)) %>%
  mutate(wday = wday(date, label = TRUE)) %>%
  ggplot(aes(as.factor(wday), distance)) +
  geom_boxplot() +
  geom_point()

### More flights departed on Sunday evening but the distances of these flights were not longer than flights departing on rest of the weekdays.