mon, 07-sep-2015, 14:12

Thirty yards of wood chips

Every couple years we cover our dog yard with a fresh layer of wood chips from the local sawmill, Northland Wood. This year I decided to keep closer track of how much effort it takes to move all 30 yards of wood chips by counting each wheelbarrow load, recording how much time I spent, and by using a heart rate monitor to keep track of effort.

The image below show the tally board. Tick marks indicate wheelbarrow-loads, the numbers under each set of five were the number of minutes since the start of each bout of work, and the numbers on the right are total loads and total minutes. I didn’t keep track of time, or heart rate, for the first set of 36 loads.

It’s not on the chalkboard, but my heart rate averaged 96 beats per minute for the first effort on Saturday morning, and 104, 96, 103, and 103 bpm for the rest. That averages out to 100.9 beats per minute.

For the loads where I was keeping track of time, I averaged 3 minutes and 12 seconds per load. Using that average for the 36 loads on Friday afternoon, that means I spent around 795 minutes, or 13 hours and 15 minutes moving and spreading 248 wheelbarrow-loads of chips.

Using a formula found in [Keytel LR, et al. 2005. Prediction of energy expenditure from heart rate monitoring during submaximal exercise. J Sports Sci. 23(3):289-97], I calculate that I burned 4,903 calories above the amount I would have if I’d been sitting around all weekend. To put that in perspective, I burned 3,935 calories running the Equinox Marathon in September, 2013.

As I was loading the wheelbarrow, I was mentally keeping track of how many pitchfork-loads it took to fill the wheelbarrow, and the number hovered right around 17. That means there are about 4,216 pitchfork loads in 30 yards of wood chips.

To summarize: 30 yards of wood chips is equivalent to 248 wheelbarrow loads. Each wheelbarrow-load is 0.1209 yards, or 3.26 cubic feet. Thirty yards of wood chips is also equivalent to 4,216 pitchfork loads, each of which is 0.19 cubic feet. It took me 13.25 hours to move and spread it all, or 3.2 minutes per wheelbarrow-load, or 11 seconds per pitchfork-load.

One final note: this amount completely covered all but a few square feet of the dog yard. In some places the chips were at least six inches deep, and in others there’s just a light covering of new over old. I don’t have a good measure of the yard, but if I did, I’d be able to calculate the average depth of the chips. My guess is that it is around 2,500 square feet, which is what 30 yards would cover to an average depth of 4 inches.

sat, 30-may-2015, 09:21

# Introduction

Often when I’m watching Major League Baseball games a player will come up to bat or pitch and I’ll comment “former Oakland Athletic” and the player’s name. It seems like there’s always one or two players on the roster of every team that used to be an Athletic.

Let’s find out. We’ll use the Retrosheet database again, this time using the roster lists from 1990 through 2014 and comparing it against the 40-man rosters of current teams. That data will have to be scraped off the web, since Retrosheet data doesn’t exist for the current season and rosters change frequently during the season.

# Methods

As usual, I’ll use R for the analysis, and rmarkdown to produce this post.

```library(plyr)
library(dplyr)
library(rvest)
options(stringsAsFactors=FALSE)
```

We’re using plyr and dplyr for most of the data manipulation and rvest to grab the 40-man rosters for each team from the MLB website. Setting stringsAsFactors to false prevents various base R packages from converting everything to factors. We're not doing any statistics with this data, so factors aren't necessary and make comparisons and joins between data frames more difficult.

## Players by team for previous years

```retrosheet_db <- src_postgres(host="localhost", port=5438,
dbname="retrosheet", user="cswingley")

rosters <- tbl(retrosheet_db, "rosters")

all_recent_players <-
rosters %>%
filter(year>1989) %>%
collect() %>%
mutate(player=paste(first_name, last_name),
team=team_id) %>%
select(player, team, year)

save(all_recent_players, file="all_recent_players.rdata", compress="gzip")
```

The Retrosheet database lives in PostgreSQL on my computer, but one of the advantages of using dplyr for retrieval is it would be easy to change the source statement to connect to another sort of database (SQLite, MySQL, etc.) and the rest of the commands would be the same.

We only grab data since 1990 and we combine the first and last names into a single field because that’s how player names are listed on the 40-man roster pages on the web.

Now we filter the list down to Oakland Athletic players, combine the rows for each Oakland player, summarizing the years they played for the A’s into a single column.

```oakland_players <- all_recent_players %>%
filter(team=='OAK') %>%
group_by(player) %>%
summarise(years=paste(year, collapse=', '))
```

Here’s what that looks like:

```kable(head(oakland_players))
```
player years
A.J. Griffin 2012, 2013
A.J. Hinch 1998, 1999, 2000
Aaron Cunningham 2008, 2009
Aaron Harang 2002, 2003
Aaron Small 1996, 1997, 1998
... ...

## Current 40-man rosters

Major League Baseball has the 40-man rosters for each team on their site. In order to extract them, we create a list of the team identifiers (oak, sf, etc.), then loop over this list, grabbing the team name and all the player names. We also set up lists for the team names (“Athletics”, “Giants”, etc.) so we can replace the short identifiers with real names later.

```teams=c("ana", "ari", "atl", "bal", "bos", "cws", "chc", "cin", "cle", "col",
"det", "mia", "hou", "kc", "la", "mil", "min", "nyy", "nym", "oak",
"phi", "pit", "sd", "sea", "sf", "stl", "tb", "tex", "tor", "was")

team_names = c("Angels", "Diamondbacks", "Braves", "Orioles", "Red Sox",
"White Sox", "Cubs", "Reds", "Indians", "Rockies", "Tigers",
"Marlins", "Astros", "Royals", "Dodgers", "Brewers", "Twins",
"Yankees", "Mets", "Athletics", "Phillies", "Pirates",
"Padres", "Mariners", "Giants", "Cardinals", "Rays", "Rangers",
"Blue Jays", "Nationals")

get_players <- function(team) {
# reads the 40-man roster data for a team, returns a data frame

roster_html <- html(paste("http://www.mlb.com/team/roster_40man.jsp?c_id=",
team,
sep=''))

players <- roster_html %>%
html_nodes("#roster_40_man a") %>%
html_text()

data.frame(team=team, player=players)
}

current_rosters <- ldply(teams, get_players)

save(current_rosters, file="current_rosters.rdata", compress="gzip")
```

Here’s what that data looks like:

```kable(head(current_rosters))
```
team player
ana Jose Alvarez
ana Cam Bedrosian
ana Andrew Heaney
ana Jeremy McBryde
ana Mike Morin
ana Vinnie Pestano
... ...

# Combine the data

To find out how many players on each Major League team used to play for the A’s we combine the former A’s players with the current rosters using player name. This may not be perfect due to differences in spelling (accented characters being the most likely issue), but the results look pretty good.

```roster_with_oakland_time <- current_rosters %>%
left_join(oakland_players, by="player") %>%
filter(!is.na(years))

```
team player years
ana Huston Street 2005, 2006, 2007, 2008
ana Grant Green 2013
ana Collin Cowgill 2012
ari Brad Ziegler 2008, 2009, 2010, 2011
ari Cliff Pennington 2008, 2009, 2010, 2011, 2012
atl Trevor Cahill 2009, 2010, 2011
... ... ...

You can see from this table (just the first six rows of the results) that the Angels have three players that were Athletics.

Let’s do the math and find out how many former A’s are on each team’s roster.

```n_former_players_by_team <-
roster_with_oakland_time %>%
group_by(team) %>%
arrange(player) %>%
summarise(number_of_players=n(),
players=paste(player, collapse=", ")) %>%
arrange(desc(number_of_players)) %>%
inner_join(data.frame(team=teams, team_name=team_names),
by="team") %>%
select(team_name, number_of_players, players)

names(n_former_players_by_team) <- c('Team', 'Number',
'Former Oakland Athletics')
kable(n_former_players_by_team,
align=c('l', 'r', 'r'))
```
Team Number Former Oakland Athletics
Athletics 22 A.J. Griffin, Andy Parrino, Billy Burns, Coco Crisp, Craig Gentry, Dan Otero, Drew Pomeranz, Eric O'Flaherty, Eric Sogard, Evan Scribner, Fernando Abad, Fernando Rodriguez, Jarrod Parker, Jesse Chavez, Josh Reddick, Nate Freiman, Ryan Cook, Sam Fuld, Scott Kazmir, Sean Doolittle, Sonny Gray, Stephen Vogt
Astros 5 Chris Carter, Dan Straily, Jed Lowrie, Luke Gregerson, Pat Neshek
Braves 4 Jim Johnson, Jonny Gomes, Josh Outman, Trevor Cahill
Rangers 4 Adam Rosales, Colby Lewis, Kyle Blanks, Michael Choice
Angels 3 Collin Cowgill, Grant Green, Huston Street
Cubs 3 Chris Denorfia, Jason Hammel, Jon Lester
Dodgers 3 Alberto Callaspo, Brandon McCarthy, Brett Anderson
Mets 3 Anthony Recker, Bartolo Colon, Jerry Blevins
Yankees 3 Chris Young, Gregorio Petit, Stephen Drew
Rays 3 David DeJesus, Erasmo Ramirez, John Jaso
Diamondbacks 2 Brad Ziegler, Cliff Pennington
Indians 2 Brandon Moss, Nick Swisher
White Sox 2 Geovany Soto, Jeff Samardzija
Tigers 2 Rajai Davis, Yoenis Cespedes
Royals 2 Chris Young, Joe Blanton
Marlins 2 Dan Haren, Vin Mazzaro
Padres 2 Derek Norris, Tyson Ross
Giants 2 Santiago Casilla, Tim Hudson
Nationals 2 Gio Gonzalez, Michael Taylor
Red Sox 1 Craig Breslow
Rockies 1 Carlos Gonzalez
Brewers 1 Shane Peterson
Twins 1 Kurt Suzuki
Phillies 1 Aaron Harang
Mariners 1 Seth Smith
Cardinals 1 Matt Holliday
Blue Jays 1 Josh Donaldson

Pretty cool. I do notice one problem: there are actually two Chris Young’s playing in baseball today. Chris Young the outfielder played for the A’s in 2013 and now plays for the Yankees. There’s also a pitcher named Chris Young who shows up on our list as a former A’s player who now plays for the Royals. This Chris Young never actually played for the A’s. The Retrosheet roster data includes which hand (left and/or right) a player bats and throws with, and it’s possible this could be used with the MLB 40-man roster data to eliminate incorrect joins like this, but even with that enhancement, we still have the problem that we’re joining on things that aren’t guaranteed to uniquely identify a player. That’s the nature of attempting to combine data from different sources.

One other interesting thing. I kept the A’s in the list because the number of former A’s currently playing for the A’s is a measure of how much turnover there is within an organization. Of the 40 players on the current A’s roster, only 22 of them have ever played for the A’s. That means that 18 came from other teams or are promotions from the minors that haven’t played for any Major League teams yet.

# All teams

## Teams with players on other teams

Now that we’ve looked at how many A’s players have played for other teams, let’s see how the number of players playing for other teams is related to team. My gut feeling is that the A’s will be at the top of this list as a small market, low budget team who is forced to turn players over regularly in order to try and stay competitive.

We already have the data for this, but need to manipulate it in a different way to get the result.

```teams <- c("ANA", "ARI", "ATL", "BAL", "BOS", "CAL", "CHA", "CHN", "CIN",
"CLE", "COL", "DET", "FLO", "HOU", "KCA", "LAN", "MIA", "MIL",
"MIN", "MON", "NYA", "NYN", "OAK", "PHI", "PIT", "SDN", "SEA",
"SFN", "SLN", "TBA", "TEX", "TOR", "WAS")

team_names <- c("Angels", "Diamondbacks", "Braves", "Orioles", "Red Sox",
"Angels", "White Sox", "Cubs", "Reds", "Indians", "Rockies",
"Tigers", "Marlins", "Astros", "Royals", "Dodgers", "Marlins",
"Brewers", "Twins", "Expos", "Yankees", "Mets", "Athletics",
"Cardinals", "Rays", "Rangers", "Blue Jays", "Nationals")

players_on_other_teams <- all_recent_players %>%
group_by(player, team) %>%
summarise(years=paste(year, collapse=", ")) %>%
inner_join(current_rosters, by="player") %>%
mutate(current_team=team.y, former_team=team.x) %>%
select(player, current_team, former_team, years) %>%
inner_join(data.frame(former_team=teams, former_team_name=team_names),
by="former_team") %>%
group_by(former_team_name, current_team) %>%
summarise(n=n()) %>%
group_by(former_team_name) %>%
arrange(desc(n)) %>%
mutate(rank=row_number()) %>%
filter(rank!=1) %>%
summarise(n=sum(n)) %>%
arrange(desc(n))
```

This is a pretty complicated set of operations. The main trick (and possible flaw in the analysis) is to get a list similar to the one we got for the A’s earlier, and eliminate the first row (the number of players on a team who played for that same team in the past) before counting the total players who have played for other teams. It would probably be better to eliminate that case using team name, but the team codes vary between Retrosheet and the MLB roster data.

Here are the results:

```names(players_on_other_teams) <- c('Former Team', 'Number of players')

kable(players_on_other_teams)
```
Former Team Number of players
Athletics 57
Marlins 56
Rangers 55
Diamondbacks 51
Braves 50
Yankees 50
Angels 49
Red Sox 47
Pirates 46
Royals 44
Dodgers 43
Mariners 42
Rockies 42
Cubs 40
Tigers 40
Astros 38
Blue Jays 38
Rays 38
White Sox 38
Indians 35
Mets 35
Twins 33
Cardinals 31
Nationals 31
Orioles 31
Reds 28
Brewers 26
Phillies 25
Giants 24
Expos 4

The A’s are indeed on the top of the list, but surprisingly, the Padres are also at the top. I had no idea the Padres had so much turnover. At the bottom of the list are teams like the Giants and Phillies that have been on recent winning streaks and aren’t trading their players to other teams.

## Current players on the same team

We can look at the reverse situation: how many players on the current roster played for that same team in past years. Instead of removing the current × former team combination with the highest number, we include only that combination, which is almost certainly the combination where the former and current team is the same.

```players_on_same_team <- all_recent_players %>%
group_by(player, team) %>%
summarise(years=paste(year, collapse=", ")) %>%
inner_join(current_rosters, by="player") %>%
mutate(current_team=team.y, former_team=team.x) %>%
select(player, current_team, former_team, years) %>%
inner_join(data.frame(former_team=teams, former_team_name=team_names),
by="former_team") %>%
group_by(former_team_name, current_team) %>%
summarise(n=n()) %>%
group_by(former_team_name) %>%
arrange(desc(n)) %>%
mutate(rank=row_number()) %>%
filter(rank==1,
former_team_name!="Expos") %>%
summarise(n=sum(n)) %>%
arrange(desc(n))

names(players_on_same_team) <- c('Team', 'Number of players')

kable(players_on_same_team)
```
Team Number of players
Rangers 31
Rockies 31
Twins 30
Giants 29
Indians 29
Cardinals 28
Mets 28
Orioles 28
Tigers 28
Brewers 27
Diamondbacks 27
Mariners 27
Phillies 27
Reds 26
Royals 26
Angels 25
Astros 25
Cubs 25
Nationals 25
Pirates 25
Rays 25
Red Sox 24
Blue Jays 23
Athletics 22
Marlins 22
Yankees 22
Dodgers 21
White Sox 20
Braves 13

The A’s are near the bottom of this list, along with other teams that have been retooling because of a lack of recent success such as the Yankees and Dodgers. You would think there would be an inverse relationship between this table and the previous one (if a lot of your former players are currently playing on other teams they’re not playing on your team), but this isn’t always the case. The White Sox, for example, only have 20 players on their roster that were Sox in the past, and there aren’t very many of them playing on other teams either. Their current roster must have been developed from their own farm system or international signings, rather than by exchanging players with other teams.

sun, 03-may-2015, 09:22

Yesterday I saw something I’ve never seen in a baseball game before: a runner getting hit by a batted ball, which according to Rule 7.08(f) means the runner is out and the ball is dead. It turns out that this isn’t as unusual an event as I’d thought (see below), but what was unusal is that this ended the game between the Angels and Giants. Even stranger, this is also how the game between the Diamondbacks and Dodgers ended.

Let’s use Retrosheet data to see how often this happens. Retrosheet data is organized into game data, roster data and event data. Event files contain a record of every event in a game and include the code BR for when a runner is hit by a batted ball. Here’s a SQL query to find all the matching events, who got hit and whether it was the last out in the game.

```SELECT sub.game_id, teams, date, inn_ct, outs_ct, bat_team, event_tx,
first_name || ' ' || last_name AS runner,
CASE WHEN event_id = max_event_id THEN 'last out' ELSE '' END AS last_out
FROM (
SELECT year, game_id, away_team_id || ' @ ' || home_team_id AS teams,
date, inn_ct,
CASE WHEN bat_home_id = 1
THEN home_team_id
ELSE away_team_id END AS bat_team, outs_ct, event_tx,
CASE regexp_replace(event_tx, '.*([1-3])X[1-3H].*', E'\\1')
WHEN '1' THEN base1_run_id
WHEN '2' THEN base2_run_id
WHEN '3' THEN base3_run_id END AS runner_id,
event_id
FROM events
WHERE event_tx ~ 'BR'
) AS sub
INNER JOIN rosters
ON sub.year=rosters.year
AND runner_id=player_id
AND rosters.team_id = bat_team
INNER JOIN (
SELECT game_id, max(event_id) AS max_event_id
FROM events
GROUP BY game_id
) AS max_events
ON sub.game_id = max_events.game_id
ORDER BY date;
```

Here's what the query does. The first sub-query sub finds all the events with the BR code, determines which team was batting and finds the id for the player who was running. This is joined with the roster table so we can assign a name to the runner. Finally, it’s joined with a subquery, max_events, which finds the last event in each game. Once we’ve got all that, the SELECT statement at the very top retrieves the columns of interest, and records whether the event was the last out of the game.

Retrosheet has event data going back to 1922, but the event files don’t include every game played in a season until the mid-50s. Starting in 1955 a runner being hit by a batted ball has a game twelve times, most recently in 2010. On average, runners get hit (and are called out) about fourteen times a season.

Here are the twelve times a runner got hit to end the game, since 1955. Until yesterday, when it happened twice in one day:

Runners hit by a batted ball to end a game, since 1955
Date Teams Batting Event Runner
1956-09-30 NY1 @ PHI PHI S4/BR.1X2(4)# Granny Hamner
1961-09-16 PHI @ CIN PHI S/BR/G4.1X2(4) Clarence Coleman
1971-08-07 SDN @ HOU SDN S/BR.1X2(3) Ed Spiezio
1979-04-07 CAL @ SEA SEA S/BR.1X2(4) Larry Milbourne
1979-08-15 TOR @ OAK TOR S/BR.3-3;2X3(4)# Alfredo Griffin
1980-09-22 CLE @ NYA CLE S/BR.3XH(5) Toby Harrah
1984-04-06 MIL @ SEA MIL S/BR.1X2(4) Robin Yount
1987-06-25 ATL @ LAN ATL S/L3/BR.1X2(3) Glenn Hubbard
1994-06-13 HOU @ SFN HOU S/BR.1X2(4) James Mouton
2001-08-04 NYN @ ARI ARI S/BR.2X3(6) David Dellucci
2003-04-09 KCA @ DET DET S/BR.1X2(4) Bobby Higginson
2010-06-27 PIT @ OAK PIT S/BR/G.1X2(3) Pedro Alvarez

And all runners hit last season:

Runners hit by a batted ball in the 2014 MLB season
Date Teams Batting Event Runner
2014-05-07 SEA @ OAK OAK S/BR/G.1X2(3) Derek Norris
2014-05-11 MIN @ DET DET S/BR/G.3-3;2X3(6);1-2 Austin Jackson
2014-05-23 CLE @ BAL BAL S/BR/G.2X3(6);1-2 Chris Davis
2014-05-27 NYA @ SLN SLN S/BR/G.1X2(3) Matt Holliday
2014-06-14 CHN @ PHI CHN S/BR/G.1X2(4) Justin Ruggiano
2014-07-13 OAK @ SEA SEA S/BR/G.1X2(4) Kyle Seager
2014-07-18 PHI @ ATL PHI S/BR/G.1X2(4) Grady Sizemore
2014-07-25 BAL @ SEA SEA S/BR/G.1X2(4) Brad Miller
2014-08-05 NYN @ WAS WAS S/BR/G.2X3(6);3-3 Asdrubal Cabrera
2014-09-04 SLN @ MIL SLN S/BR/G.2X3(6);1-2 Matt Carpenter
2014-09-09 SDN @ LAN LAN S/BR/G.2X3(6) Matt Kemp
2014-09-18 BOS @ PIT BOS S/BR/G.3XH(5);1-2;B-1 Jemile Weeks
thu, 30-apr-2015, 18:14

photo by Tom Vivian

Yesterday, the Baltimore Orioles and Chicago White Sox played a game at Camden Yards in downtown Baltimore. The game was “closed to fans” due to the riots that broke out in the city after the funeral for a man who died in police custody. It’s the first time a Major League Baseball game has been played without any fans in the stands, but unfortunately it’s not the first time there have been riots in Baltimore.

After Martin Luther King, Jr. was murdered in April 1968, Baltimore rioted for six days, with local police, and more than eleven thousand National Guard, Army troops, and Marines brought in to restore order. According to wikipedia six people died, more than 700 were injured, 1,000 businesses were damaged and close to six thousand people were arrested.

At that time, the Orioles played in Memorial Stadium, about 4 miles north-northwest of where they play now. I don’t know much about that area of Baltimore, but I was curious to know whether the Orioles played any baseball games during those riots.

Retrosheet has one game, on April 10, 1968, with a reported attendance of 22,050. The Orioles defeated the Oakland Athletics by a score of 3–1. Thomas Phoebus got the win over future Hall of Famer Catfish Hunter. Other popular players in the game included Reggie Jackson, Sal Bando, Rick Mondy and Bert Campaneris for the A’s and Brooks Robinson, Frank Robinson, Davey Johnson, and Boog Powell for the Orioles.

The box score and play-by-play can be viewed here.

sat, 25-apr-2015, 10:21

# Introduction

One of the best sources of weather data in the United States comes from the National Weather Service's Cooperative Observer Network (COOP), which is available from NCDC. It's daily data, collected by volunteers at more than 10,000 locations. We participate in this program at our house (station id DW1454 / GHCND:USC00503368), collecting daily minimum and maximum temperature, liquid precipitation, snowfall and snow depth. We also collect river heights for Goldstream Creek as part of the Alaska Pacific River Forecast Center (station GSCA2). Traditionally, daily temperature measurements were collecting using a minimum maximum thermometer, which meant that the only way to calculate average daily temperature was by averaging the minimum and maximum temperature. Even though COOP observers typically have an electronic instrument that could calculate average daily temperature from continuous observations, the daily minimum and maximum data is still what is reported.

In an earlier post we looked at methods used to calculate average daily temperature, and if there are any biases present in the way the National Weather Service calculates this using the average of the minimum and maximum daily temperature. We looked at five years of data collected at my house every five minutes, comparing the average of these temperatures against the average of the daily minimum and maximum. Here, we will be repeating this analysis using data from the Climate Reference Network stations in the United States.

The US Climate Reference Network is a collection of 132 weather stations that are properly sited, maintained, and include multiple redundant measures of temperature and precipitation. Data is available from http://www1.ncdc.noaa.gov/pub/data/uscrn/products/ and includes monthly, daily, and hourly statistics, and sub-hourly (5-minute) observations. We’ll be focusing on the sub-hourly data, since it closely matches the data collected at my weather station.

A similar analysis using daily and hourly CRN data appears here.

# Getting the raw data

```\$ wget http://www1.ncdc.noaa.gov/pub/data/uscrn/products/stations.tsv
\$ wget -np -m http://www1.ncdc.noaa.gov/pub/data/uscrn/products/subhourly01/
\$ find www1.ncdc.noaa.gov/ -type f -name 'CRN*.txt' -exec gzip {} \;
```

The code to insert all of this data into a database can be found here. Once inserted, I have a table named crn_stations that has the station data, and one named crn_subhourly with the five minute observation data.

# Methods

Once again, we’ll use R to read the data, process it, and produce plots.

## Libraries

```library(dplyr)
library(lubridate)
library(ggplot2)
library(scales)
library(grid)
```

Connect to the database and load the data tables.

```noaa_db <- src_postgres(dbname="noaa", host="mason")

crn_stations <- tbl(noaa_db, "crn_stations") %>%
collect()

crn_subhourly <- tbl(noaa_db, "crn_subhourly")
```

Remove observations without temperature data, group by station and date, calculate average daily temperature using the two methods, remove any daily data without a full set of data, and collect the results into an R data frame. This looks very similar to the code used to analyze the data from my weather station.

```crn_daily <-
crn_subhourly %>%
filter(!is.na(air_temperature)) %>%
mutate(date=date(timestamp)) %>%
group_by(wbanno, date) %>%
summarize(t_mean=mean(air_temperature),
t_minmax_avg=(min(air_temperature)+
max(air_temperature))/2.0,
n=n()) %>%
filter(n==24*12) %>%
mutate(anomaly=t_minmax_avg-t_mean) %>%
select(wbanno, date, t_mean, t_minmax_avg, anomaly) %>%
collect()
```

The two types of daily average temperatures are calculated in this step:

```summarize(t_mean=mean(air_temperature),
t_minmax_avg=(min(air_temperature)+
max(air_temperature))/2.0)
```

Where t_mean is the value calculated from all 288 five minute observations, and t_minmax_avg is the value from the daily minimum and maximum.

Now we join the observation data with the station data. This attaches station information such as the name and latitude of the station to each record.

```crn_daily_stations <-
crn_daily %>%
inner_join(crn_stations, by="wbanno") %>%
select(wbanno, date, state, location, latitude, longitude,
t_mean, t_minmax_avg, anomaly)
```

Finally, save the data so we don’t have to do these steps again.

```save(crn_daily_stations, file="crn_daily_averages.rdata")
```

# Results

Here are the overall results of the analysis.

```summary(crn_daily_stations\$anomaly)
```
```##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
## -11.9000  -0.1028   0.4441   0.4641   1.0190  10.7900
```

The average anomaly across all stations and all dates is 0.44 degrees Celsius (0.79 degrees Farenheit). That’s a pretty significant error. Half the data is between −0.1 and 1.0°C (−0.23 and +1.8°F) and the full range is −11.9 to +10.8°C (−21.4 to +19.4°F).

# Plots

Let’s look at some plots.

## Raw data by latitude

To start, we’ll look at all the anomalies by station latitude. The plot only shows one percent of the actual anomalies because plotting 512,460 points would take a long time and the general pattern is clear from the reduced data set.

```set.seed(43)
p <- ggplot(data=crn_daily_stations %>% sample_frac(0.01),
aes(x=latitude, y=anomaly)) +
geom_point(position="jitter", alpha="0.2") +
geom_smooth(method="lm", se=FALSE) +
theme_bw() +
scale_x_continuous(name="Station latitude", breaks=pretty_breaks(n=10)) +
scale_y_continuous(name="Temperature anomaly (degrees C)",
breaks=pretty_breaks(n=10))

print(p)
```

The clouds of points show the differences between the min/max daily average and the actual daily average temperature, where numbers above zero represent cases where the min/max calculation overestimates daily average temperature. The blue line is the fit of a linear model relating latitude with temperature anomaly. We can see that the anomaly is always positive, averaging around half a degree at lower latitudes and drops somewhat as we proceed northward. You also get a sense from the actual data of how variable the anomaly is, and at what latitudes most of the stations are found.

Here are the regression results:

```summary(lm(anomaly ~ latitude, data=crn_daily_stations))
```
```##
## Call:
## lm(formula = anomaly ~ latitude, data = crn_daily_stations)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -12.3738  -0.5625  -0.0199   0.5499  10.3485
##
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.7403021  0.0070381  105.19   <2e-16 ***
## latitude    -0.0071276  0.0001783  -39.98   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.9632 on 512458 degrees of freedom
## Multiple R-squared:  0.00311,    Adjusted R-squared:  0.003108
## F-statistic:  1599 on 1 and 512458 DF,  p-value: < 2.2e-16
```

The overall model and coefficients are highly significant, and show a slight decrease in the positive anomaly as we move farther north. Perhaps this is part of the reason why the analysis of my station (at a latitude of 64.89) showed an average anomaly close to zero (−0.07°C / −0.13°F).

## Anomalies by month and latitude

One of the results of our earlier analysis was a seasonal pattern in the anomalies at our station. Since we also know there is a latitudinal pattern, in the data, let’s combine the two, plotting anomaly by month, and faceting by latitude.

Station latitude are binned into groups for plotting, and the plots themselves show the range that cover half of all anomalies for that latitude category × month. Including the full range of anomalies in each group tends to obscure the overall pattern, and the plot of the raw data didn’t show an obvious skew to the rarer anomalies.

Here’s how we set up the data frames for the plot.

```crn_daily_by_month <-
crn_daily_stations %>%
mutate(month=month(date),
lat_bin=factor(ifelse(latitude<30, '<30',
ifelse(latitude>60, '>60',
paste(floor(latitude/10)*10,
(floor(latitude/10)+1)*10,
sep='-'))),
levels=c('<30', '30-40', '40-50',
'50-60', '>60')))

summary_stats <- function(l) {
s <- summary(l)
data.frame(min=s['Min.'],
first=s['1st Qu.'],
median=s['Median'],
mean=s['Mean'],
third=s['3rd Qu.'],
max=s['Max.'])
}

crn_by_month_lat_bin <-
crn_daily_by_month %>%
group_by(month, lat_bin) %>%
do(summary_stats(.\$anomaly)) %>%
ungroup()

station_years <-
crn_daily_by_month %>%
mutate(year=year(date)) %>%
group_by(wbanno, lat_bin) %>%
summarize() %>%
group_by(lat_bin) %>%
summarize(station_years=n())
```

And the plot itself. At the end, we’re using a function called facet_adjust, which adds x-axis tick labels to the facet on the right that wouldn't ordinarily have them. The code comes from this stack overflow post.

```p <- ggplot(data=crn_by_month_lat_bin,
aes(x=month, ymin=first, ymax=third, y=mean)) +
geom_hline(yintercept=0, alpha=0.2) +
geom_hline(data=crn_by_month_lat_bin %>%
group_by(lat_bin) %>%
summarize(mean=mean(mean)),
aes(yintercept=mean), colour="darkorange", alpha=0.5) +
geom_pointrange() +
facet_wrap(~ lat_bin, ncol=3) +
geom_text(data=station_years, size=4,
aes(x=2.25, y=-0.5, ymin=0, ymax=0,
label=paste('n =', station_years))) +
scale_y_continuous(name="Range including 50% of temperature anomalies") +
scale_x_discrete(breaks=1:12,
labels=c('Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun',
'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec')) +
theme_bw() +
theme(axis.text.x=element_text(angle=45, hjust=1, vjust=1.25),
axis.title.x=element_blank())
```

Each plot shows the range of anomalies from the first to the third quartile (50% of the observed anomalies) by month, with the dot near the middle of the line at the mean anomaly. The orange horizontal line shows the overall mean anomaly for that latitude category, and the count at the bottom of the plot indicates the number of “station years” for that latitude category.

It’s clear that there are seasonal patterns in the differences between the mean daily temperature and the min/max estimate. But each plot looks so different from the next that it’s not clear if the patterns we are seeing in each latitude category are real or artificial. It is also problematic that three of our latitude categories have very little data compared with the other two. It may be worth performing this analysis in a few years when the lower and higher latitude stations have a bit more data.

# Conclusion

This analysis shows that there is a clear bias in using the average of minimum and maximum daily temperature to estimate average daily temperature. Across all of the CRN stations, the min/max estimator overestimates daily average temperature by almost a half a degree Celsius (0.8°F).

We also found that this error is larger at lower latitudes, and that there are seasonal patterns to the anomalies, although the seasonal patterns don’t seem to have clear transitions moving from lower to higher latitudes.

The current length of the CRN record is quite short, especially for the sub-hourly data used here, so the patterns may not be representative of the true situation.

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