ANLY 505 - NYCFlightsDataModel
Week 3
your name
2020-04-13
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, na.action = na.warn)
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, na.action = na.warn)
mod2 <- lm(n ~ wday * term, data = daily, na.action = na.warn)
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, na.action = na.warn)
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, na.action = na.warn)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, na.action = na.warn)
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?
The daily-weekly pattern shows that the number of flight on weekends are always fewer than weekdays. January 20, May 26 and September 1 are all Sunday, thus, there are less fights on these days. I think it can be applied to other years if there are fewer flights on the same days.
Question #2
What do the three days with high positive residuals represent? How would these days generalize to another year?
# Use this chunk to answer question 2
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 fallThese days represent thanks giving day, the day after of that and the day after of Chrismas holiday, so it is reseanable that they can not represents average level and more extremely high. In general, the same days in another year also have high positive residual.
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”?
# Use this chunk to answer question 3
daily <- daily %>%
mutate(wday = as.character(wday),
sat = ifelse(wday=="Sat", paste0(wday, "-", term), wday))
fit1 <- lm(n~sat, data = daily)
summary(fit1)##
## Call:
## lm(formula = n ~ sat, data = daily)
##
## Residuals:
## Min 1Q Median 3Q Max
## -331.75 -8.81 11.31 23.64 141.00
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 967.462 6.568 147.308 < 2e-16 ***
## satMon 7.346 9.288 0.791 0.4295
## satSat-fall -251.462 12.951 -19.416 < 2e-16 ***
## satSat-spring -230.143 12.045 -19.107 < 2e-16 ***
## satSat-summer -166.545 15.167 -10.981 < 2e-16 ***
## satSun -75.981 9.288 -8.181 5.06e-15 ***
## satThu -1.712 9.288 -0.184 0.8539
## satTue -16.103 9.244 -1.742 0.0824 .
## satWed -4.769 9.288 -0.513 0.6079
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 47.36 on 356 degrees of freedom
## Multiple R-squared: 0.7357, Adjusted R-squared: 0.7297
## F-statistic: 123.8 on 8 and 356 DF, p-value: < 2.2e-16summary(mod2)##
## Call:
## lm(formula = n ~ wday * term, data = daily, na.action = na.warn)
##
## Residuals:
## Min 1Q Median 3Q Max
## -326.33 -8.00 9.82 20.67 141.00
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 912.8684 3.7086 246.151 < 2e-16 ***
## wday.L -71.4674 9.8349 -7.267 2.49e-12 ***
## wday.Q -161.0112 9.8196 -16.397 < 2e-16 ***
## wday.C -65.6215 9.8068 -6.691 8.98e-11 ***
## wday^4 -82.0997 9.8412 -8.342 1.78e-15 ***
## wday^5 -1.3425 9.7787 -0.137 0.8909
## wday^6 -12.4501 9.7903 -1.272 0.2043
## termsummer 37.6922 6.3336 5.951 6.56e-09 ***
## termfall 3.7598 5.5033 0.683 0.4949
## wday.L:termsummer -6.2215 16.8049 -0.370 0.7114
## wday.Q:termsummer 26.4121 16.7501 1.577 0.1158
## wday.C:termsummer 26.7451 16.7885 1.593 0.1121
## wday^4:termsummer 23.5933 16.7523 1.408 0.1599
## wday^5:termsummer -13.1343 16.7720 -0.783 0.4341
## wday^6:termsummer -5.0026 16.6744 -0.300 0.7643
## wday.L:termfall -31.9048 14.5607 -2.191 0.0291 *
## wday.Q:termfall -0.6959 14.5707 -0.048 0.9619
## wday.C:termfall -8.8349 14.5417 -0.608 0.5439
## wday^4:termfall -9.8570 14.5899 -0.676 0.4997
## wday^5:termfall -3.2107 14.5228 -0.221 0.8252
## wday^6:termfall -8.2061 14.5769 -0.563 0.5738
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 46.17 on 344 degrees of freedom
## Multiple R-squared: 0.7573, Adjusted R-squared: 0.7432
## F-statistic: 53.67 on 20 and 344 DF, p-value: < 2.2e-16For the new model, the power of prediction is lower than original model, but it estimate fewer number of parametrics which can reduce extreme residuals.
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?
# Use this chunk to answer question 4
daily <- daily %>%
mutate(hol = case_when(date %in% ymd(c(20130101, 20130704, 20130902, 20131128, 20131225))~"holiday",
T ~ as.character(sat)))
fit2 <- lm(n~hol, data = daily)
daily <- daily%>%
add_residuals(fit2, "residual2")
ggplot(daily, aes(date, residual2)) +
geom_line() +
labs(x="date", y="residual")
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?
# Use this chunk to answer question 5
daily <- daily%>%
mutate(month=month(date, label = T))
fit3 <- lm(n~wday*month, data = daily)
daily <- daily%>%
add_residuals(fit3, "residual3")
ggplot(daily, aes(date, residual3)) +
geom_line() +
labs(x="date", y="residual")
The fluaction in the right of plot shows that there are extreme residuals in the regression, which means we can not solve the problem.
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?
# Use this chunk to answer question 6
fit4 <- lm(n~wday+splines::ns(date, 5), data = daily)
daily <- daily%>%
add_residuals(fit4, "residual4")
ggplot(daily, aes(date, residual4)) +
geom_line() +
labs(x="date", y="residual")
This model has different trend in diverse year, so we can not use it to predict value of another year.
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.
# Use this chunk to answer question 7
flights <- flights%>%
mutate(date = make_date(year, month, day),
wday = wday(date, label = T)) %>%
filter(hour>20)
ggplot(flights, aes(y=distance, x=wday)) +
geom_boxplot() +
labs(x="weekday", y="distance")
From the result, we can see there is no significant difference between Sunday and other days.