In a post last week I examined how often Fairbanks gets more than two inches of snow in late spring. We only got 1.1 inches on April 24th, so that event didn’t qualify, but another snowstorm hit Fairbanks this week. Enough that I skied to work a couple days ago (April 30th) and could have skied this morning too.

Another, probably more relevant statistic would be to look at storm totals rather than the amount of snow that fell within a single, somewhat arbitrary 24-hour period (midnight to midnight for the Fairbanks Airport station, 6 AM to 6 AM for my COOP station). With SQL window functions we can examine the totals over a moving window, in this case five days and see what the largest late season snowfall totals were in the historical record.

Here’s a list of the late spring (after April 21st) snowfall totals for Fairbanks where the five day snowfall was greater than three inches:

Anyone who was here in 1992 remembers that “summer,” with more than a foot of snow in mid May, and two feet of snow in a pair of storms starting on September 11th, 1992. I don’t expect that all the late spring cold weather and snow we’re experiencing this year will necessarily translate into a short summer like 1992, but we should keep the possibility in mind.

I’m writing this blog post on May 1st, looking outside as the snow continues to fall. We’ve gotten three inches in the last day and a half, and I even skied to work yesterday. It’s still winter here in Fairbanks.

The image shows the normalized temperature anomaly calendar heatmap for April. The bluer the squares are, the colder that day was compared with the 30-year climate normal daily temperature for Fairbanks. There were several days where the temperature was more than three standard deviations colder than the mean anomaly (zero), something that happens very infrequently.

Here are the top ten coldest average April temperatures for the Fairbanks Airport Station.

The averages come from the Global Historical Climate Network - Daily data set, with some fairly dubious additions to extend the Fairbanks record back before the 1956 start of the current station. Here’s the query to get the historical data:

SELECT rank() OVER (ORDER BY tavg) AS rank,
       year, round(c_to_f(tavg), 1) AS tavg
FROM (
    SELECT year, avg(tavg) AS tavg
    FROM (
        SELECT extract(year from dte) AS year,
               dte, (tmin + tmax) / 2.0 AS tavg
        FROM (
            SELECT dte,
                   sum(CASE WHEN variable = 'TMIN'
                            THEN raw_value * 0.1
                            ELSE 0 END) AS tmin,
                   sum(CASE WHEN variable = 'TMAX'
                            THEN raw_value * 0.1
                            ELSE 0 END) AS tmax
            FROM ghcnd_obs
            WHERE variable IN ('TMIN', 'TMAX')
                  AND station_id = 'USW00026411'
                  AND extract(month from dte) = 4 GROUP BY dte
        ) AS foo
    ) AS bar GROUP BY year
) AS foobie
ORDER BY rank;

And the way I calculated the average temperature for this April. pafg is a text file that includes the data from each day’s National Weather Service Daily Climate Summary. Average daily temperature is in column 9.

$ tail -n 30 pafg | \
  awk 'BEGIN {sum = 0; n = 0}; {n = n + 1; sum += $9} END { print sum / n; }'
18.1667

This morning’s weather forecast includes this section:

.WEDNESDAY...CLOUDY. A CHANCE OF SNOW IN THE MORNING...THEN SNOW LIKELY IN THE AFTERNOON. SNOW ACCUMULATION OF 1 TO 2 INCHES. HIGHS AROUND 40. WEST WINDS INCREASING TO 15 TO 20 MPH.

.WEDNESDAY NIGHT...CLOUDY. SNOW LIKELY IN THE EVENING...THEN A CHANCE OF SNOW AFTER MIDNIGHT. LOWS IN THE 20S. WEST WINDS TO 20 MPH DIMINISHING.

Here’s a look at how often Fairbanks gets two or more inches of snow later than April 23rd:

It’s not all that frequent, with only 18 occurrences in the last 98 years, and two of those 18 coming two days in a row. The pattern is also curious, with several in the early 1900s, one or two in each decade until the 2000s when there were several events.

In any case, I’m not looking forward to it. We’ve still got a lot of hardpack on the road from the 5+ inches we got a couple weeks ago and I’ve just started riding my bicycle to work every day. If we do get 2 inches of snow, that’ll slow breakup even more, and mess up the shoulders of the road for a few days.

Next month I’ll be attending a game at Wrigley Field and my brother and I had some discussion about the best strategy for us to meet up somewhere in Chicago after the game. Knowing how long a game could be, and how many people are likely to be crowding the train platforms is useful information that can be inferred from the game data that http://www.retrosheet.org/ collects and distributes.

It’s also a good excuse to fiddle around the the IPython Notebook, pandas and the rest of the Python scientific computing stack.

I downloaded the game log data from http://www.retrosheet.org/gamelogs/index.html using this bash one-liner:

for i in `seq 1871 2012`; \
do  wget http://www.retrosheet.org/gamelogs/gl${i}.zip ; \
    unzip gl${i}.zip; \
    rm gl${i}.zip; \
done

Game Length

Game length in minutes is in column 19. Let’s read it and analyze it with Pandas.

import pandas as pd
import matplotlib.pyplot as plt
import datetime

def fix_df(input_df, cols):
    """ Pulls out the columns from cols, labels them and returns
        a new DataFrame """
    df = pd.DataFrame(index=range(len(input_df)))
    for k, v in cols.items():
        df[v] = input_df[k]

    return df

cols = {0:'dt_str', 2:'day_of_week', 3:'visiting_team',
        6:'home_team', 9:'visitor_score', 10:'home_score',
       11:'game_outs', 12:'day_night', 13:'complete',
       16:'park_id', 17:'attendance', 18:'game_time_min'}
raw_gamelogs = []
for i in range(1871, 2013):
    fn = "GL{0}.TXT".format(i)
    raw_gamelogs.append(pd.read_csv(fn, header=None, index_col=None))
raw_gamelog = pd.concat(raw_gamelogs, ignore_index=True)
gamelog = fix_df(raw_gamelog, cols)
gamelog['dte'] = gamelog.apply(
    lambda x: datetime.datetime.strptime(str(x['dt_str']), '%Y%m%d').date(),
    axis=1)
gamelog.ix[0:5, 1:] # .head() but without the dt_str col

	dow	vis	hme	vis	hme	outs	day	park	att	time	dte
0	Thu	CL1	FW1	0	2	54	D	FOR01	200	120	1871-05-04
1	Fri	BS1	WS3	20	18	54	D	WAS01	5000	145	1871-05-05
2	Sat	CL1	RC1	12	4	54	D	RCK01	1000	140	1871-05-06
3	Mon	CL1	CH1	12	14	54	D	CHI01	5000	150	1871-05-08
4	Tue	BS1	TRO	9	5	54	D	TRO01	3250	145	1871-05-09
5	Thu	CH1	CL1	18	10	48	D	CLE01	2500	120	1871-05-11

(Note that I’ve abbreviated the column names so they fit)

The data looks reasonable, although I don’t know quite what to make of the team names from back in 1871. Now I’ll take a look at the game time data for the whole data set:

times = gamelog['game_time_min']
times.describe()

count    162259.000000
mean        153.886145
std          74.850459
min          21.000000
25%         131.000000
50%         152.000000
75%         173.000000
max       16000.000000

The statistics look reasonable, except that it’s unlikley that there was a completed game in 21 minutes, or that a game took 11 days, so we have some outliers in the data. Let’s see if something else in the data might help us filter out these records.

First the games longer than 24 hours:

print(gamelog[gamelog.game_time_min > 60 * 24][['visiting_team',
                    'home_team', 'game_outs', 'game_time_min']])
print("Removing all NaN game_outs: {0}".format(
        len(gamelog[np.isnan(gamelog.game_outs)])))
print("Max date removed: {0}".format(
        max(gamelog[np.isnan(gamelog.game_outs)].dte)))

      visiting_team home_team  game_outs  game_time_min
26664           PHA       WS1        NaN          12963
26679           PHA       WS1        NaN           4137
26685           WS1       PHA        NaN          16000
26707           NYA       PHA        NaN          15115
26716           CLE       CHA        NaN           3478
26717           DET       SLA        NaN           1800
26787           PHA       NYA        NaN           3000
26801           PHA       NYA        NaN           6000
26880           CHA       WS1        NaN           2528
26914           SLA       WS1        NaN           2245
26921           SLA       WS1        NaN           1845
26929           CLE       WS1        NaN           3215
Removing all NaN game_outs: 37890
Max date removed: 1915-10-03

There’s no value for game_outs, so there isn’t data for how long the game actually was. We remove 37,890 records by eliminating this data, but these are all games from prior to the 1916 season, so it seems reasonable:

gamelog = gamelog[pd.notnull(gamelog.game_outs)]
gamelog.game_time_min.describe()

count    156639.000000
mean        154.808975
std          32.534916
min          21.000000
25%         133.000000
50%         154.000000
75%         174.000000
max        1150.000000

What about the really short games?

gamelog[gamelog.game_time_min < 60][[
        'dte', 'game_time_min', 'game_outs']].tail()

               dte  game_time_min  game_outs
79976   1948-07-02             24         54
80138   1948-07-22             59         30
80982   1949-05-29             21         42
113455  1971-07-30             48         27
123502  1976-09-10             57         30

Many of these aren’t nine inning games because there are less than 51 outs (8 innings for a home team, 9 for the visitor in a home team win ✕ 3 innings = 51). At the moment, I’m interested in looking at how long a game is likely to be, rather than the pattern of game length over time, so we can leave these records in the data.

Now we filter the data to just the games played since 2000.

twenty_first = gamelog[gamelog['dte'] > datetime.date(1999, 12, 31)]
times_21 = twenty_first['game_time_min']
times_21.describe()

count    31580.000000
mean       175.084421
std         26.621124
min         79.000000
25%        157.000000
50%        172.000000
75%        189.000000
max        413.000000

The average game length between 2000 and 2012 is 175 minutes, or just under three hours. And three quarters of all the games played are under three hours and ten minutes.

Here’s the code to look at the distribution of these times. The top plot shows a histogram (the count of the games in each bin) and a density estimation of the same data.

The second plot is a cumulative density plot, which makes it easier to see what percentage of games are played within a certain time.

from scipy import stats
# Calculate a kernel density function
density = stats.kde.gaussian_kde(times_21)
x = range(90, 300)
rc('grid', linestyle='-', alpha=0.25)
fig, axes = plt.subplots(ncols=1, nrows=2, figsize=(8, 6))

# First plot (histogram / density)
ax = axes[0]
ax2 = ax.twinx()
h = ax.hist(times_21, bins=50, facecolor="lightgrey", histtype="stepfilled")
d = ax2.plot(x, density(x))
ax2.set_ylabel("Density")
ax.set_ylabel("Count")
plt.title("Distribution of MLB game times, 2000-2012")
ax.grid(True)
ax.set_xlim(90, 300)
ax.set_xticks(range(90, 301, 15))

# Second plot (cumulative histogram)
ax1 = axes[1]
n, bins, patches = ax1.hist(times_21, bins=50, normed=True, cumulative=True,
        facecolor="lightgrey", histtype="stepfilled")
y = n / float(n[-1]) # Convert counts to percentage of total
y = np.concatenate(([0], y))
bins = bins - ((bins[1] - bins[0]) / 2.0) # Center curve on bars
ax1.plot(bins, y, color="blue")
ax1.set_ylabel("Cumulative percentage")
ax1.set_xlabel("Game time (minutes)")
ax1.grid(True)
ax1.set_xlim(90, 300)
ax1.set_xticks(range(90, 301, 15))
ax1.set_yticks(np.array(range(0, 101, 10)) / 100.0)
plt.savefig('game_time.png', dpi=87.5)
plt.savefig('game_time.svg')
plt.savefig('game_time.pdf')

The two plots show what the descriptive statistics did: 70% of games are under three hours but it’s not uncommon for games to last three hours and fifteen minutes. Beyond that, it’s pretty uncommon.

Change in game times in the last 100 years

Now let’s look at how game times have changed over the years. First we eliminate all the games that weren’t complete in 51 or 54 innings to standardize the “game” were’re evaluating.

nine_innings = gamelog[gamelog.game_outs.isin([51, 54])]
nine_innings['year'] = nine_innings.apply(lambda x: x['dte'].year, axis=1)
nine_groupby_year = nine_innings.groupby('year')
mean_time_by_year = nine_groupby_year['game_time_min'].aggregate(np.mean)
fig = plt.figure()
p = mean_time_by_year.plot(figsize=(8, 4))
p.set_xlim(1915, 2015)
p.set_xlabel('Year')
p.set_ylabel('Game Time (minutes)')
p.set_title('Average MLB nine-inning game time since 1915')
p.set_xticks(range(1915, 2016, 10))
plt.savefig('game_time_by_year.png', dpi=87.5)
plt.savefig('game_time_by_year.svg')
plt.savefig('game_time_by_year.pdf')

That shows a pretty clear upward trend interrupted by a slight decline between 1960 and 1975 (when offense was down across baseball). Since the mid-90s, game times have hovered around 2:50, so maybe MLB’s efforts at increasing the pace of the game have at least stopped what had been an almost continual rise in game times.

Attendance

I’ll be seeing a game at Wrigley Field, so let’s examine attendance at Wrigley.

Attendance is field 17, “Park ID” is field 16, but we can probably use Home team (field 6) = “CHN” for seasons after 1914 when it opened.

twenty_first[twenty_first['home_team'] == 'CHN']['attendance'].describe()

count     1053.000000
mean     37326.561254
std       5544.121773
min          0.000000
25%      36797.000000
50%      38938.000000
75%      40163.000000
max      55000.000000

We see the minimum is zero, which may indicate bad or missing data. Let’s look at all the games at Wrigley with less than 10,000 fans:

twenty_first[(twenty_first.home_team == 'CHN') &
             (twenty_first.attendance < 10000)][[
            'dte', 'visiting_team', 'home_team', 'visitor_score',
            'home_score', 'game_outs', 'day_night', 'attendance',
            'game_time_min']]

	dte	vis	hme	vis	hme	outs	day	att	game_time
172776	2000-06-01	ATL	CHN	3	5	51	D	5267	160
173897	2000-08-25	LAN	CHN	5	3	54	D	0	189
174645	2001-04-18	PHI	CHN	3	4	51	D	0	159
176290	2001-08-20	MIL	CHN	4	7	51	D	0	180
177218	2002-04-28	LAN	CHN	5	4	54	D	0	196
177521	2002-05-21	PIT	CHN	12	1	54	N	0	158
178910	2002-09-02	MIL	CHN	4	2	54	D	0	193
181695	2003-09-27	PIT	CHN	2	4	51	D	0	169
183806	2004-09-10	FLO	CHN	7	0	54	D	0	146
184265	2005-04-13	SDN	CHN	8	3	54	D	0	148
188186	2006-08-03	ARI	CHN	10	2	54	D	0	197

Looks like the zeros are just missing data because these games have relevant data for the other fields and it’s impossible to believe that not a single person was in the stands. We’ll get rid of them for the attendance analysis.

Now we’ll filter the games so we’re only looking at day games played at Wrigley where the attendance value is above zero, and group the data by day of the week.

groupby_dow = twenty_first[(twenty_first['home_team'] == 'CHN') &
           (twenty_first['day_night'] == 'D') &
           (twenty_first['attendance'] > 0)].groupby('day_of_week')

groupby_dow['attendance'].aggregate(np.mean).order()

day_of_week
Tue            34492.428571
Wed            35312.265957
Thu            35684.737864
Mon            37938.757576
Fri            38066.060976
Sun            38583.833333
Sat            39737.428571
Name: attendance

And plot it:

filtered = twenty_first[(twenty_first['home_team'] == 'CHN') &
                           (twenty_first['day_night'] == 'D') &
                           (twenty_first['attendance'] > 0)]
dows = {'Sun':0, 'Mon':1, 'Tue':2, 'Wed':3, 'Thu':4, 'Fri':5, 'Sat':6}
filtered['dow'] = filtered.apply(lambda x: dows[x['day_of_week']], axis=1)
filtered['attendance'] = filtered['attendance'] / 1000.0
fig = plt.figure()
d = filtered.boxplot(column='attendance', by='dow', figsize=(8, 4))
plt.suptitle('')
d.set_xlabel('Day of the week')
d.set_ylabel('Attendance (thousands)')
d.set_title('Wrigley Field attendance by day of the week, 2000-2012')
labels = ['Sun', 'Mon', 'Tue', 'Wed', 'Thu', 'Fri', 'Sat']
l = d.set_xticklabels(labels)
d.set_ylim(15, 45)
plt.savefig('attendance.png', dpi=87.5)
plt.savefig('attendance.svg') # lines don't show up?
plt.savefig('attendance.pdf')

There’s a pretty clear pattern here, with increasing attendance from Tuesday through Monday, and larger variances in attendance during the week.

It’s a bit surprising that Monday games are as popular as Saturday games, especially since we’re only looking at day games. On any given Monday when the Cubs are at home, there’s a good chance that there will be fourty thousand people not showing up to work!

Cold November

Several years ago I showed some R code to make a heatmap showing the rank of the Oakland A’s players for various hitting and pitching statistics.

Last week I used this same style of plot to make a new weather visualization on my web site: a calendar heatmap of the difference between daily average temperature and the “climate normal” daily temperature for all dates in the last ten years. “Climate normals” are generated every ten years and are the averages for a variety of statistics for the previous 30-year period, currently 1981—2010.

A calendar heatmap looks like a normal calendar, except that each date box is colored according to the statistic of interest, in this case the difference in temperature between the temperature on that date and the climate normal temperature for that date. I also created a normalized version based on the standard deviations of temperature on each date.

Here’s the temperature anomaly plot showing all the temperature differences for the last ten years:

It’s a pretty incredible way to look at a lot of data at the same time, and it makes it really easy to pick out anomalous events such as the cold November and December of 2012. One thing you can see in this plot is that the more dramatic temperature differences are always in the winter; summer anomalies are generally smaller. This is because the range of likely temperatures is much larger in winter, and in order to equalize that difference, we need to normalize the anomalies by this range.

One way to do that is to divide the actual temperature difference by the standard deviation of the 30-year climate normal mean temperature. Because of the nature of the distribution standard deviations are based on, approximately 66% of the variation occurrs within -1 and 1 standard deviation, 95% between -2 and 2, and 99% between -3 and 3 standard deviations. That means that deep red or blue dates, those outside of -3 and 3, in the normalized calendar plot are fairly rare occurrances.

Here’s the normalized anomalies for the last twelve months:

The tricky part in generating either of these plots is getting the temperature data into the right format. The plots are faceted by month and year (or YYYYY-MM in the twelve month plot), so each record needs to have month and year. That part is easy. Each individual plot is a single calendar month, and is organized by day of the week along the x-axis, and the inverse of week number along the y-axis (the first week in a month is at the top of the plot, the last at the bottom).

Here’s how to get the data formatted properly:

library(lubridate)
cal <- function(dt) {
    # Reads a date object and returns a tuple (weekrow, daycol)
    # where weekrow starts at 1 and daycol starts at 1 for Sunday
    year <- year(dt)
    month <- month(dt)
    day <- day(dt)
    wday_first <- wday(ymd(paste(year, month, 1, sep = '-'), quiet = TRUE))
    offset <- 7 + (wday_first - 2)
    weekrow <- ((day + offset) %/% 7) - 1
    daycol <- (day + offset) %% 7

    c(weekrow, daycol)
}
weekrow <- function(dt) {
    cal(dt)[1]
}
daycol <- function(dt) {
    cal(dt)[2]
}
vweekrow <- function(dts) {
    sapply(dts, weekrow)
}
vdaycol <- function(dts) {
    sapply(dts, daycol)
}
pafg$temp_anomaly <- pafg$mean_temp - pafg$average_mean_temp
pafg$month <- month(pafg$dt, label = TRUE, abbr = TRUE)
pafg$year <- year(pafg$dt)
pafg$weekrow <- factor(vweekrow(pafg$dt),
   levels = c(5, 4, 3, 2, 1, 0),
   labels = c('6', '5', '4', '3', '2', '1'))
pafg$daycol <- factor(vdaycol(pafg$dt),
   labels = c('u', 'm', 't', 'w', 'r', 'f', 's'))

And the plotting code:

library(ggplot2)
library(scales)
library(grid)
svg('temp_anomaly_heatmap.svg', width = 11, height = 10)
q <- ggplot(data = subset(pafg, year > max(pafg$year) - 11),
            aes(x = daycol, y = weekrow, fill = temp_anomaly)) +
    theme_bw() +
    theme(axis.text.x = element_blank(),
          axis.text.y = element_blank(),
          panel.grid.major = element_blank(),
          panel.grid.minor = element_blank(),
          axis.ticks.x = element_blank(),
          axis.ticks.y = element_blank(),
          axis.title.x = element_blank(),
          axis.title.y = element_blank(),
          legend.position = "bottom",
          legend.key.width = unit(1, "in"),
          legend.margin = unit(0, "in")) +
    geom_tile(colour = "white") +
    facet_grid(year ~ month) +
    scale_fill_gradient2(name = "Temperature anomaly (°F)",
          low = 'blue', mid = 'lightyellow', high = 'red',
          breaks = pretty_breaks(n = 10)) +
    ggtitle("Difference between daily mean temperature\
             and 30-year average mean temperature")
print(q)
dev.off()

You can find the current versions of the temperature and normalized anomaly plots at:

• Heatmap of temperature anomaly, last ten years
• Heatmap of temperature anomlay, last twelve months
• Normalized heatmap of temperature anomaly, last ten years
• Normalized heatmap of temperature anomaly, last twelve month