mon, 09-jan-2017, 09:49

# Introduction

The latest forecast discussions for Northern Alaska have included warnings that we are likely to experience an extended period of below normal temperatures starting at the end of this week, and yesterday’s Deep Cold blog post discusses the similarity of model forecast patterns to patterns seen in the 1989 and 1999 extreme cold events.

Our dogs spend most of their time in the house when we’re home, but if both of us are at work they’re outside in the dog yard. They have insulated dog houses, but when it’s colder than −15° F, we put them into a heated dog barn. That means one of us has to come home in the middle of the day to let them out to go to the bathroom.

Since we’re past the Winter Solstice, and day length is now increasing, I was curious to see if that has an effect on daily temperature, hopeful that the frequency of days when we need to put the dogs in the barn is decreasing.

# Methods

We’ll use daily minimum and maximum temperature data from the Fairbanks International Airport station, keeping track of how many years the temperatures are below −15° F and dividing by the total to get a frequency. We live in a cold valley on Goldstream Creek, so our temperatures are typically several degrees colder than the Fairbanks Airport, and we often don’t warm up as much during the day as in other places, but minimum airport temperature is a reasonable proxy for the overall winter temperature at our house.

# Results

The following plot shows the frequency of minimum (the top of each line) and maximum (the bottom) temperature colder than −15° F at the airport over the period of record, 1904−2016. The curved blue line represents a best fit line through the minimum temperature frequency, and the vertical blue line is drawn at the date when the frequency is the highest.

The maximum frequency is January 12th, so we have a few more days before the likelihood of needing to put the dogs in the barn starts to decline. The plot also shows that we could still reach that threshold all the way into April.

For fun, here’s the same plot using −40° as the threshold:

The date when the frequency starts to decline is shifted slightly to January 15th, and you can see the frequencies are lower. In mid-January, we can expect minimum temperature to be colder than −15° F more than half the time, but temperatures colder than −40° are just under 15%. There’s also an interesting anomaly in mid to late December where the frequency of very cold temperatures appears to drop.

# Appendix: R code

```library(tidyverse)
library(lubridate)
library(scales)

noaa <- src_postgres(host="localhost", dbname="noaa")

fairbanks <- tbl(noaa, build_sql("SELECT * FROM ghcnd_pivot
WHERE station_name='FAIRBANKS INTL AP'")) %>%
collect()

save(fairbanks, file="fairbanks_ghcnd.rdat")

for_plot <- fairbanks %>%
mutate(doy=yday(dte),
dte_str=format(dte, "%d %b"),
min_below=ifelse(tmin_c < -26.11,1,0),
max_below=ifelse(tmax_c < -26.11,1,0)) %>%
filter(dte_str!="29 Feb") %>%
mutate(doy=ifelse(leap_year(dte) & doy>60, doy-1, doy),
doy=(doy+31+28+31+30)%%365) %>%
group_by(doy, dte_str) %>%
mutate(n_min=sum(ifelse(!is.na(min_below), 1, 0)),
n_max=sum(ifelse(!is.na(max_below), 1, 0))) %>%
summarize(min_freq=sum(min_below, na.rm=TRUE)/max(n_min, na.rm=TRUE),
max_freq=sum(max_below, na.rm=TRUE)/max(n_max, na.rm=TRUE))

x_breaks <- for_plot %>%
filter(doy %in% seq(49, 224, 7))

stats <- tibble(doy=seq(49, 224),
pred=predict(loess(min_freq ~ doy,
for_plot %>%
filter(doy >= 49, doy <= 224))))

max_stats <- stats %>%

p <- ggplot(data=for_plot,
aes(x=doy, ymin=min_freq, ymax=max_freq)) +
geom_linerange() +
geom_smooth(aes(y=min_freq), se=FALSE, size=0.5) +
geom_segment(aes(x=max_stats\$doy, xend=max_stats\$doy,
y=-Inf, yend=max_stats\$pred),
colour="blue", size=0.5) +
scale_x_continuous(name=NULL,
limits=c(49, 224),
breaks=x_breaks\$doy,
labels=x_breaks\$dte_str) +
scale_y_continuous(name="Frequency of days colder than −15° F",
breaks=pretty_breaks(n=10)) +
theme_bw() +
theme(axis.text.x=element_text(angle=30, hjust=1))

# Minus 40
for_plot <- fairbanks %>%
mutate(doy=yday(dte),
dte_str=format(dte, "%d %b"),
min_below=ifelse(tmin_c < -40,1,0),
max_below=ifelse(tmax_c < -40,1,0)) %>%
filter(dte_str!="29 Feb") %>%
mutate(doy=ifelse(leap_year(dte) & doy>60, doy-1, doy),
doy=(doy+31+28+31+30)%%365) %>%
group_by(doy, dte_str) %>%
mutate(n_min=sum(ifelse(!is.na(min_below), 1, 0)),
n_max=sum(ifelse(!is.na(max_below), 1, 0))) %>%
summarize(min_freq=sum(min_below, na.rm=TRUE)/max(n_min, na.rm=TRUE),
max_freq=sum(max_below, na.rm=TRUE)/max(n_max, na.rm=TRUE))

x_breaks <- for_plot %>%
filter(doy %in% seq(63, 203, 7))

stats <- tibble(doy=seq(63, 203),
pred=predict(loess(min_freq ~ doy,
for_plot %>%
filter(doy >= 63, doy <= 203))))

max_stats <- stats %>%

q <- ggplot(data=for_plot,
aes(x=doy, ymin=min_freq, ymax=max_freq)) +
geom_linerange() +
geom_smooth(aes(y=min_freq), se=FALSE, size=0.5) +
geom_segment(aes(x=max_stats\$doy, xend=max_stats\$doy,
y=-Inf, yend=max_stats\$pred),
colour="blue", size=0.5) +
scale_x_continuous(name=NULL,
limits=c(63, 203),
breaks=x_breaks\$doy,
labels=x_breaks\$dte_str) +
scale_y_continuous(name="Frequency of days colder than −40°",
breaks=pretty_breaks(n=10)) +
theme_bw() +
theme(axis.text.x=element_text(angle=30, hjust=1))
```
fri, 13-may-2016, 06:02

This morning’s weather forecast:

```SUNNY. HIGHS IN THE UPPER 70S TO LOWER 80S. LIGHT WINDS.
```

May 13th seems very early in the year to hit 80 degrees in Fairbanks, so I decided to check it out. What I’m doing here is selecting all the dates where the temperature is above 80°F, then ranking those dates by year and date, and extracting the “winner” for each year (where rank is 1).

```WITH warm AS (
SELECT extract(year from dte) AS year, dte,
c_to_f(tmax_c) AS tmax_f
FROM ghcnd_pivot
WHERE station_name = 'FAIRBANKS INTL AP'
AND c_to_f(tmax_c) >= 80.0),
ranked AS (
SELECT year, dte, tmax_f,
row_number() OVER (PARTITION BY year
ORDER BY dte) AS rank
FROM warm)
SELECT dte,
extract(doy from dte) AS doy,
round(tmax_f, 1) as tmax_f
FROM ranked
WHERE rank = 1
ORDER BY doy;
```

And the results:

Earliest 80 degree dates, Fairbanks Airport
Date Day of year High temperature (°F)
1995-05-09 129 80.1
1975-05-11 131 80.1
1942-05-12 132 81.0
1915-05-14 134 80.1
1993-05-16 136 82.0
2002-05-20 140 80.1
2015-05-22 142 80.1
1963-05-22 142 84.0
1960-05-23 144 80.1
2009-05-24 144 80.1

If we hit 80°F today, it’ll be the fourth earliest day of year to hit that temperature since records started being kept in 1904.

Update: We didn’t reach 80°F on the 13th, but got to 82°F on May 14th, tied with that date in 1915 for the fourth earliest 80 degree temperature.

mon, 21-dec-2015, 16:58

# Introduction

While riding to work this morning I figured out a way to disentangle the effects of trail quality and physical conditioning (both of which improve over the season) from temperature, which also tends to increase throughout the season. As you recall in my previous post, I found that days into the season (winter day of year) and minimum temperature were both negatively related with fat bike energy consumption. But because those variables are also related to each other, we can’t make statements about them individually.

But what if we look at pairs of trips that are within two days of each other and look at the difference in temperature between those trips and the difference in energy consumption? We’ll only pair trips going the same direction (to or from work), and we’ll restrict the pairings to two days or less. That eliminates seasonality from the data because we’re always comparing two trips from the same few days.

# Data

For this analysis, I’m using SQL to filter the data because I’m better at window functions and filtering in SQL than R. Here’s the code to grab the data from the database. (The CSV file and RMarkdown script is on my GitHub repo for this analysis). The trick here is to categorize trips as being to work (“north”) or from work (“south”) and then include this field in the partition statement of the window function so I’m only getting the next trip that matches direction.

```library(dplyr)
library(ggplot2)
library(scales)

exercise_db <- src_postgres(host="example.com", dbname="exercise_data")

diffs <- tbl(exercise_db,
build_sql(
"WITH all_to_work AS (
SELECT *,
CASE WHEN extract(hour from start_time) < 11
THEN 'north' ELSE 'south' END AS direction
FROM track_stats
WHERE type = 'Fat Biking'
AND miles between 4 and 4.3
), with_next AS (
SELECT track_id, start_time, direction, kcal, miles, min_temp,
FROM all_to_work
WINDOW w AS (PARTITION BY direction ORDER BY start_time)
)
SELECT start_time, next_start_time, direction,
min_temp, next_min_temp,
kcal / miles AS kcal_per_mile,
next_kcal / next_miles as next_kcal_per_mile,
next_min_temp - min_temp AS temp_diff,
(next_kcal / next_miles) - (kcal / miles) AS kcal_per_mile_diff
FROM with_next
WHERE next_start_time - start_time < '60 hours'
ORDER BY start_time")) %>% collect()

write.csv(diffs, file="fat_biking_trip_diffs.csv", quote=TRUE,
row.names=FALSE)
```
start time next start time temp diff kcal / mile diff
2013-12-03 06:21:49 2013-12-05 06:31:54 3.0 -13.843866
2013-12-03 15:41:48 2013-12-05 15:24:10 3.7 -8.823329
2013-12-05 06:31:54 2013-12-06 06:39:04 23.4 -22.510564
2013-12-05 15:24:10 2013-12-06 16:38:31 13.6 -5.505662
2013-12-09 06:41:07 2013-12-11 06:15:32 -27.7 -10.227048
2013-12-09 13:44:59 2013-12-11 16:00:11 -25.4 -1.034789

Out of a total of 123 trips, 70 took place within 2 days of each other. We still don’t have a measure of trail quality, so pairs where the trail is smooth and hard one day and covered with fresh snow the next won’t be particularly good data points.

Let’s look at a plot of the data.

```s = ggplot(data=diffs,
aes(x=temp_diff, y=kcal_per_mile_diff)) +
geom_point() +
geom_smooth(method="lm", se=FALSE) +
scale_x_continuous(name="Temperature difference between paired trips (degrees F)",
breaks=pretty_breaks(n=10)) +
scale_y_continuous(name="Energy consumption difference (kcal / mile)",
breaks=pretty_breaks(n=10)) +
theme_bw() +
ggtitle("Paired fat bike trips to and from work within 2 days of each other")

print(s)
```

This shows that when the temperature difference between two paired trips is negative (the second trip is colder than the first), additional energy is required for the second (colder) trip. This matches the pattern we saw in my earlier post where minimum temperature and winter day of year were negatively associated with energy consumption. But because we’ve used differences to remove seasonal effects, we can actually determine how large of an effect temperature has.

There are quite a few outliers here. Those that are in the region with very little difference in temperature are likey due to snowfall changing the trail conditions from one trip to the next. I’m not sure why there is so much scatter among the points on the left side of the graph, but I don’t see any particular pattern among those points that might explain the higher than normal variation, and we don’t see the same variation in the points with a large positive difference in temperature, so I think this is just normal variation in the data not explained by temperature.

## Results

Here’s the linear regression results for this data.

```summary(lm(data=diffs, kcal_per_mile_diff ~ temp_diff))
```
```##
## Call:
## lm(formula = kcal_per_mile_diff ~ temp_diff, data = diffs)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -40.839  -4.584  -0.169   3.740  47.063
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  -2.1696     1.5253  -1.422    0.159
## temp_diff    -0.7778     0.1434  -5.424 8.37e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.76 on 68 degrees of freedom
## Multiple R-squared:  0.302,  Adjusted R-squared:  0.2917
## F-statistic: 29.42 on 1 and 68 DF,  p-value: 8.367e-07
```

The model and coefficient are both highly signficant, and as we might expect, the intercept in the model is not significantly different from zero (if there wasn’t a difference in temperature between two trips there shouldn’t be a difference in energy consumption either, on average). Temperature alone explains 30% of the variation in energy consumption, and the coefficient tells us the scale of the effect: each degree drop in temperature results in an increase in energy consumption of 0.78 kcalories per mile. So for a 4 mile commute like mine, the difference between a trip at 10°F vs −20°F is an additional 93 kilocalories (30 × 0.7778 × 4 = 93.34) on the colder trip. That might not sound like much in the context of the calories in food (93 kilocalories is about the energy in a large orange or a light beer), but my average energy consumption across all fat bike trips to and from work is 377 kilocalories so 93 represents a large portion of the total.

sun, 20-dec-2015, 15:16

# Introduction

I’ve had a fat bike since late November 2013, mostly using it to commute the 4.1 miles to and from work on the Goldstream Valley trail system. I used to classic ski exclusively, but that’s not particularly pleasant once the temperatures are below 0°F because I can’t keep my hands and feet warm enough, and the amount of glide you get on skis declines as the temperature goes down.

However, it’s also true that fat biking gets much harder the colder it gets. I think this is partly due to biking while wearing lots of extra layers, but also because of increased friction between the large tires and tubes in a fat bike. In this post I will look at how temperature and other variables affect the performance of a fat bike (and it’s rider).

The code and data for this post is available on GitHub.

# Data

I log all my commutes (and other exercise) using the RunKeeper app, which uses the phone’s GPS to keep track of distance and speed, and connects to my heart rate monitor to track heart rate. I had been using a Polar HR chest strap, but after about a year it became flaky and I replaced it with a Scosche Rhythm+ arm band monitor. The data from RunKeeper is exported into GPX files, which I process and insert into a PostgreSQL database.

From the heart rate data, I estimate energy consumption (in kilocalories, or what appears on food labels as calories) using a formula from Keytel LR, et al. 2005, which I talk about in this blog post.

Let’s take a look at the data:

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

tbl_df() %>%

```
start_time miles time hours mph hr kcal min_temp max_temp
2013-11-27 06:22:13 4.17 0:35:11 0.59 7.12 157.8 518.4 -1.1 1.0
2013-11-27 15:27:01 4.11 0:35:49 0.60 6.89 156.0 513.6 1.1 2.2
2013-12-01 12:29:27 4.79 0:55:08 0.92 5.21 172.6 951.5 -25.9 -23.9
2013-12-03 06:21:49 4.19 0:39:16 0.65 6.40 148.4 526.8 -4.6 -2.1
2013-12-03 15:41:48 4.22 0:30:56 0.52 8.19 154.6 434.5 6.0 7.9
2013-12-05 06:31:54 4.14 0:32:14 0.54 7.71 155.8 463.2 -1.6 2.9

There are a few things we need to do to the raw data before analyzing it. First, we want to restrict the data to just my commutes to and from work, and we want to categorize them as being one or the other. That way we can analyze trips to ABR and home separately, and we’ll reduce the variation within each analysis. If we were to analyze all fat biking trips together, we’d be lumping short and long trips, as well as those with a different proportion of hills or more challenging conditions. To get just trips to and from work, I’m restricting the distance to trips between 4.0 and 4.3 miles, and only those activities where there were two of them in a single day (to work and home from work). To categorize them into commutes to work and home, I filter based on the time of day.

I’m also calculating energy per mile, and adding a “winter day of year” variable (wdoy), which is a measure of how far into the winter season the trip took place. We can’t just use day of year because that starts over on January 1st, so we subtract the number of days between January and May from the date and get day of year from that. Finally, we split the data into trips to work and home.

I’m also excluding the really early season data from 2015 because the trail was in really poor condition.

```fat_bike_commute <- fat_bike %>%
filter(miles>4, miles<4.3) %>%
mutate(direction=ifelse(hour(start_time)<10, 'north', 'south'),
wdoy=yday(date-days(120)),
kcal_per_mile=kcal/miles) %>%
group_by(date) %>%
mutate(n=n()) %>%
ungroup() %>%
filter(n>1)

to_abr <- fat_bike_commute %>% filter(direction=='north',
wdoy>210)
to_home <- fat_bike_commute %>% filter(direction=='south',
wdoy>210)
```
start_time miles time hours mph hr min_temp max_temp direction wdoy kcal_per_mile
2013-11-27 15:27:01 4.11 0:35:49 0.60 6.89 156.0 1.1 2.2 south 211 124.96350
2013-12-03 15:41:48 4.22 0:30:56 0.52 8.19 154.6 6.0 7.9 south 217 102.96209
2013-12-05 15:24:10 4.18 0:29:07 0.49 8.60 150.7 9.7 12.0 south 219 94.13876
2013-12-06 16:38:31 4.17 0:26:04 0.43 9.60 154.3 23.3 24.7 south 220 88.63309
2013-12-09 13:44:59 4.11 0:32:06 0.54 7.69 161.3 27.5 28.5 south 223 119.19708
2013-12-11 16:00:11 4.19 0:33:48 0.56 7.44 157.6 2.1 4.5 south 225 118.16229

# Analysis

Here a plot of the data. We’re plotting all trips with winter day of year on the x-axis and energy per mile on the y-axis. The color of the points indicates the minimum temperature and the straight line shows the trend of the relationship.

```s <- ggplot(data=fat_bike_commute %>% filter(wdoy>210), aes(x=wdoy, y=kcal_per_mile, colour=min_temp)) +
geom_smooth(method="lm", se=FALSE, colour=mnsl("10B 7/10", fix=TRUE)) +
geom_point(size=3) +
scale_x_continuous(name=NULL,
breaks=c(215, 246, 277, 305, 336),
labels=c('1-Dec', '1-Jan', '1-Feb', '1-Mar', '1-Apr')) +
scale_y_continuous(name="Energy (kcal)", breaks=pretty_breaks(n=10)) +
scale_colour_continuous(low=mnsl("7.5B 5/12", fix=TRUE), high=mnsl("7.5R 5/12", fix=TRUE),
breaks=pretty_breaks(n=5),
guide=guide_colourbar(title="Min temp (°F)", reverse=FALSE, barheight=8)) +
ggtitle("All fat bike trips") +
theme_bw()
print(s)
```

Across all trips, we can see that as the winter progresses, I consume less energy per mile. This is hopefully because my physical condition improves the more I ride, and also because the trail conditions also improve as the snow pack develops and the trail gets harder with use. You can also see a pattern in the color of the dots, with the bluer (and colder) points near the top and the warmer temperature trips near the bottom.

Let’s look at the temperature relationship:

```s <- ggplot(data=fat_bike_commute %>% filter(wdoy>210), aes(x=min_temp, y=kcal_per_mile, colour=wdoy)) +
geom_smooth(method="lm", se=FALSE, colour=mnsl("10B 7/10", fix=TRUE)) +
geom_point(size=3) +
scale_x_continuous(name="Minimum temperature (degrees F)", breaks=pretty_breaks(n=10)) +
scale_y_continuous(name="Energy (kcal)", breaks=pretty_breaks(n=10)) +
scale_colour_continuous(low=mnsl("7.5PB 2/12", fix=TRUE), high=mnsl("7.5PB 8/12", fix=TRUE),
breaks=c(215, 246, 277, 305, 336),
labels=c('1-Dec', '1-Jan', '1-Feb', '1-Mar', '1-Apr'),
guide=guide_colourbar(title=NULL, reverse=TRUE, barheight=8)) +
ggtitle("All fat bike trips") +
theme_bw()
print(s)
```

A similar pattern. As the temperature drops, it takes more energy to go the same distance. And the color of the points also shows the relationship from the earlier plot where trips taken later in the season require less energy.

There is also be a correlation between winter day of year and temperature. Since the winter fat biking season essentially begins in December, it tends to warm up throughout.

# Results

The relationship between winter day of year and temperature means that we’ve got multicollinearity in any model that includes both of them. This doesn’t mean we shouldn’t include them, nor that the significance or predictive power of the model is reduced. All it means is that we can’t use the individual regression coefficients to make predictions.

Here are the linear models for trips to work, and home:

```to_abr_lm <- lm(data=to_abr, kcal_per_mile ~ min_temp + wdoy)
print(summary(to_abr_lm))
```
```##
## Call:
## lm(formula = kcal_per_mile ~ min_temp + wdoy, data = to_abr)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -27.845  -6.964  -3.186   3.609  53.697
##
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 170.81359   15.54834  10.986 1.07e-14 ***
## min_temp     -0.45694    0.18368  -2.488   0.0164 *
## wdoy         -0.29974    0.05913  -5.069 6.36e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15.9 on 48 degrees of freedom
## Multiple R-squared:  0.4069, Adjusted R-squared:  0.3822
## F-statistic: 16.46 on 2 and 48 DF,  p-value: 3.595e-06
```
```to_home_lm <- lm(data=to_home, kcal_per_mile ~ min_temp + wdoy)
print(summary(to_home_lm))
```
```##
## Call:
## lm(formula = kcal_per_mile ~ min_temp + wdoy, data = to_home)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -21.615 -10.200  -1.068   3.741  39.005
##
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept) 144.16615   18.55826   7.768 4.94e-10 ***
## min_temp     -0.47659    0.16466  -2.894  0.00570 **
## wdoy         -0.20581    0.07502  -2.743  0.00852 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.49 on 48 degrees of freedom
## Multiple R-squared:  0.5637, Adjusted R-squared:  0.5455
## F-statistic: 31.01 on 2 and 48 DF,  p-value: 2.261e-09
```

The models confirm what we saw in the plots. Both regression coefficients are negative, which means that as the temperature rises (and as the winter goes on) I consume less energy per mile. The models themselves are significant as are the coefficients, although less so in the trips to work. The amount of variation in kcal/mile explained by minimum temperature and winter day of year is 41% for trips to work and 56% for trips home.

What accounts for the rest of the variation? My guess is that trail conditions are the missing factor here; specifically fresh snow, or a trail churned up by snowmachiners. I think that’s also why the results are better on trips home than to work. On days when we get snow overnight, I am almost certainly riding on an pristine snow-covered trail, but by the time I leave work, the trail will be smoother and harder due to all the traffic it’s seen over the course of the day.

# Conclusions

We didn’t really find anything surprising here: it is significantly harder to ride a fat bike when it’s colder. Because of conditioning, improved trail conditions, as well as the tendency for warmer weather later in the season, it also gets easier to ride as the winter goes on.

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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