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?

blockeddays <- c("0121", "0526", "0902")
yearbounds <- 2013:2019
map(yearbounds, ~ wday(ymd(paste0(.x, blockeddays, sep = "")), label = TRUE))

## [[1]]
## [1] Mon Sun Mon
## Levels: Sun < Mon < Tue < Wed < Thu < Fri < Sat
## 
## [[2]]
## [1] Tue Mon Tue
## Levels: Sun < Mon < Tue < Wed < Thu < Fri < Sat
## 
## [[3]]
## [1] Wed Tue Wed
## Levels: Sun < Mon < Tue < Wed < Thu < Fri < Sat
## 
## [[4]]
## [1] Thu Thu Fri
## Levels: Sun < Mon < Tue < Wed < Thu < Fri < Sat
## 
## [[5]]
## [1] Sat Fri Sat
## Levels: Sun < Mon < Tue < Wed < Thu < Fri < Sat
## 
## [[6]]
## [1] Sun Sat Sun
## Levels: Sun < Mon < Tue < Wed < Thu < Fri < Sat
## 
## [[7]]
## [1] Mon Sun Mon
## Levels: Sun < Mon < Tue < Wed < Thu < Fri < Sat

Looking at the days provided for analysis we figured that these days come either before or after the holidays. Looking on the above we see variation of days hence it comes to be different for every 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

The days with high positive residuals represents that Sat,Sun model underpredicts the no. of flights. Although another year might be different. So we need to be sure that this imprecision is date or weekend effect accordingly.

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”?

dailyFlights <-
  flights %>% 
  mutate(date = make_date(year, month, day)) %>%
  group_by(date) %>% 
  summarize(n = n()) %>% 
  mutate(wday = wday(date, label = TRUE))
modFlights <- lm(n ~ wday, data = dailyFlights)
dailyFlights <- add_residuals(dailyFlights, modFlights)
term <- function(date) {
  cut(date,
      breaks = ymd(20130101, 20130605, 20130825, 20140101),
      labels = c("spring", "summer", "fall")
      )
}
dailyFlights <-
  dailyFlights %>% 
  mutate(term = term(date))

dailyFlights_new <-
  dailyFlights %>% 
  mutate(wday = as.character(wday),
         term_sat = ifelse(wday == "Sat", paste0(wday, "-", term), wday))
modFlights_new <- MASS::rlm(n ~ term_sat, data = dailyFlights_new)
dailyFlights_new %>% 
  add_residuals(modFlights_new) %>% 
  ggplot(aes(date, resid)) +
  geom_line()

Based on the above analysis we see that outliers from winter,summer are visible. The January to March fits in our prediction model.

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?

holi_dailyFlight <-
  dailyFlights_new %>% 
  mutate(holidays = case_when(date %in% ymd(c(20130101, 20130121, 20130218, 20130527, 
 20130704,  20130902,  20131028,   20131111,  20131128,  20131225)) ~ "holidays", TRUE ~ "None")) %>% 
  unite(newTerm, term_sat, holidays)
modFlights2 <- lm(n ~ newTerm, data = holi_dailyFlight)
holi_dailyFlight %>% 
  add_residuals(modFlights2) %>% 
  ggplot(aes(date, resid)) +
  geom_line()

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?

modFlight2 <- lm(n ~ wday * month(date), data = holi_dailyFlight)
holi_dailyFlight %>% 
  add_residuals(modFlight2) %>% 
  ggplot(aes(date, resid)) +
  geom_line()

Looks like we see more outliers in the graph. Less observations in each cell makes it hard for us to take care of uncertain predictions.

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?

The above model that we have in place works for this specific year.However we know the various days in each year changes so it cannot be genralized into other respective years.

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.

weekHypothesis <- function(x) {
  fct_relevel(x, "Sun", after = 7)
}
dailyFlights %>% 
  mutate(wday = weekHypothesis(wday)) %>% 
  ggplot(aes(wday, n)) +
  geom_boxplot()