wed, 14-nov-2012, 05:29

Early-season ski from work

Yesterday a co-worker and I were talking about how we weren’t able to enjoy the new snow because the weather had turned cold as soon as the snow stopped falling. Along the way, she mentioned that it seemed to her that the really cold winter weather was coming later and later each year. She mentioned years past when it was bitter cold by Halloween.

The first question to ask before trying to determine if there has been a change in the date of the first cold snap is what qualifies as “cold.” My officemate said that she and her friends had a contest to guess the first date when the temperature didn’t rise above -20°F. So I started there, looking for the month and day of the winter when the maximum daily temperature was below -20°F.

I’m using the GHCN-Daily dataset from NCDC, which includes daily minimum and maximum temperatures, along with other variables collected at each station in the database.

When I brought in the data for the Fairbanks Airport, which has data available from 1948 to the present, there was absolutely no relationship between the first -20°F or colder daily maximum and year.

However, when I changed the definition of “cold” to the first date when the daily minimum temperature is below -40, I got a weak (but not statistically significant) positive trend between date and year.

The SQL query looks like this:

```SELECT year, water_year, water_doy, mmdd, temp
FROM (
SELECT year, water_year, water_doy, mmdd, temp,
row_number() OVER (PARTITION BY water_year ORDER BY water_doy) AS rank
FROM (
SELECT extract(year from dte) AS year,
extract(year from dte + interval '92 days') AS water_year,
extract(doy from dte + interval '92 days') AS water_doy,
to_char(dte, 'mm-dd') AS mmdd,
sum(CASE WHEN variable = 'TMIN'
THEN raw_value * raw_multiplier
ELSE NULL END
) AS temp
FROM ghcnd_obs
INNER JOIN ghcnd_variables USING(variable)
WHERE station_id = 'USW00026411'
GROUP BY extract(year from dte),
extract(year from dte + interval '92 days'),
extract(doy from dte + interval '92 days'),
to_char(dte, 'mm-dd')
ORDER BY water_year, water_doy
) AS foo
WHERE temp < -40 AND temp > -80
) AS bar
WHERE rank = 1
ORDER BY water_year;
```

I used “water year” instead of the actual year because the winter is split between two years. The water year starts on October 1st (we’re in the 2013 water year right now, for example), which converts a split winter (winter of 2012/2013) into a single year (2013, in this case). To get the water year, you add 92 days (the sum of the days in October, November and December) to the date and use that as the year.

Here’s what it looks like (click on the image to view a PDF version):

The dots are the observed date of first -40° daily minimum temperature for each water year, and the blue line shows a linear regression model fitted to the data (with 95% confidence intervals in grey). Despite the scatter, you can see a slightly positive slope, which would indicate that colder temperatures in Fairbanks are coming later now, than they were in the past.

As mentioned, however, our eyes often deceive us, so we need to look at the regression model to see if the visible relationship is significant. Here’s the R lm results:

```Call:
lm(formula = water_doy ~ water_year, data = first_cold)

Residuals:
Min      1Q  Median      3Q     Max
-45.264 -15.147  -1.409  13.387  70.282

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -365.3713   330.4598  -1.106    0.274
water_year     0.2270     0.1669   1.360    0.180

Residual standard error: 23.7 on 54 degrees of freedom
Multiple R-squared: 0.0331,     Adjusted R-squared: 0.01519
F-statistic: 1.848 on 1 and 54 DF,  p-value: 0.1796
```

The first thing to check in the model summary is the p-value for the entire model on the last line of the results. It’s only 0.1796, which means that there’s an 18% chance of getting these results simply by chance. Typically, we’d like this to be below 5% before we’d consider the model to be valid.

You’ll also notice that the coefficient of the independent variable (water_year) is positive (0.2270), which means the model predicts that the earliest cold snap is 0.2 days later every year, but that this value is not significantly different from zero (a p-value of 0.180).

Still, this seems like a relationship worth watching and investigating further. It might be interesting to look at other definitions of “cold,” such as requiring three (or more) consecutive days of -40° temperatures before including that period as the earliest cold snap. I have a sense that this might reduce the year to year variation in the date seen with the definition used here.

sun, 04-mar-2012, 12:15

I re-ran the analysis of my ski speeds discussed in an earlier post. The model looks like this:

```lm(formula = mph ~ season_days + temp, data = ski)

Residuals:
Min       1Q   Median       3Q      Max
-1.76466 -0.20838  0.02245  0.15600  0.90117

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.414677   0.199258  22.156  < 2e-16 ***
season_days 0.008510   0.001723   4.938 5.66e-06 ***
temp        0.027334   0.003571   7.655 1.10e-10 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.428 on 66 degrees of freedom
Multiple R-squared: 0.5321, Adjusted R-squared: 0.5179
F-statistic: 37.52 on 2 and 66 DF,  p-value: 1.307e-11
```

What this is saying is that about half the variation in my ski speeds can be explained by the temperature when I start skiing and how far along in the season we are (season_days). Temperature certainly makes sense—I was reminded of how little glide you get at cold temperatures skiing to work this week at -25°F. And it’s gratifying that my speeds are increasing as the season goes on. It’s not my imagination that my legs are getting stronger and my technique better.

The following figure shows the relationship of each of these two variables (season_days and temp) to the average speed of the trip. I used the melt function from the reshape package to make the plot:

```melted <- melt(data = ski,
measure.vars = c('season_days', 'temp'),
id.vars = 'mph')
q <- ggplot(data = melted, aes(x = value, y = mph))
q + geom_point()
+ facet_wrap(~ variable, scales = 'free_x')
+ stat_smooth(method = 'lm')
+ theme_bw()
```

Last week I replaced by eighteen-year-old ski boots with a new pair, and they’re hurting my ankles a little. Worse, the first four trips with my new boots were so slow and frustrating that I thought maybe I’d made a mistake in the pair I’d bought. My trip home on Friday afternoon was another frustrating ski until I stopped and applied warmer kick wax and had a much more enjoyable mile and a half home. There are a lot of other unmeasured factors including the sort of snow on the ground (fresh snow vs. smooth trail vs. a trail ripped up by snowmachines), whether I applied the proper kick wax or not, whether my boots are hurting me, how many times I stopped to let dog teams by, and many other things I can’t think of. Explaining half of the variation in speed is pretty impressive.

tags: skiing  R  statistics
wed, 01-feb-2012, 18:41

January 2012 was a historically cold month in Fairbanks, the fifth-coldest in more than 100 years of records. According to the National Weather Service office in Fairbanks:

January 2012 was the coldest month in more than 40 years in Fairbanks. Not since January 1971 has the Fairbanks area endured a month as cold as this.

The average high temperature for January was 18.2 below and the average low was 35 below. The monthly average temperature of 26.9 below was 19 degrees below normal and made this the fifth coldest January of record. The coldest January of record was 1971, when the average temperature was 31.7 below. The highest temperature at the airport was 21 degrees on the 10th, one of only three days when the temperature rose above zero. This ties with 1966 as the most days in January with highs of zero or lower. There were 16 days with a low temperature of 40 below or lower. Only four months in Fairbanks in more than a century of weather records have had more 40 below days. The lowest temperature at the airport was 51 below on the 29th.

Here’s a figure showing some of the relevant information:

The vertical bars show how much colder (or warmer for the red bars) the average daily temperature at the airport was compared with the 30-year average. You can see from these bars that we had only four days where the temperature was slightly above normal. The blue horizontal line shows the average anomaly for the period, and the orange (Fairbanks airport) and dark cyan (Goldstream Creek) horizontal lines show the actual average temperatures over the period. The average temperature at our house was -27.7°F for the month of January.

Finally, the circles and + symbols represent the minimum daily temperatures recorded at the airport (orange) and our house (dark cyan). You can see the two days late in the month where we got down to -54 and -55°F; cold enough that the propane in our tank remained a liquid and we couldn’t use our stove without heating up the tank.

No matter how you slice it, it was a very cold month.

Here’s some of the R code used to make the plot:

```library(lubridate)
library(ggplot2)
library(RPostgreSQL)
# (Read in dw1454 data here)
dw_1454\$date <- force_tz(
ymd(as.character(dw_1454\$date)),
tzone = "America/Anchorage")
dw_1454\$label <- 'dw1454 average'
plot_data\$line_color <- as.factor(
as.numeric(plot_data\$avg_temp_anomaly > 0))
plot_data\$anomaly <- as.factor(
ifelse(plot_data\$line_color == 0,
"degrees colder",
"degrees warmer"))
plot_data\$daily <- 'FAI average'

q <- ggplot(data = plot_data,
aes(x = date + hours(9))) # TZ?
q + geom_hline(y = avg_mean_anomaly,
colour = "blue", size = 0.25) +
geom_hline(y = avg_mean_pafg,
colour = "orange", size = 0.25) +
geom_hline(y = avg_mean_dw1454,
colour = "darkcyan", size = 0.25) +
geom_linerange(aes(ymin = avg_temp_anomaly,
ymax = 0, colour = anomaly)) +
theme_bw() +
scale_y_continuous(name = "Temperature (degrees F)") +
scale_color_manual(name = "Daily temperature",
c("degrees colder" = "blue",
"degrees warmer" = "red",
"FAI average" = "orange",
"dw1454 average" = "darkcyan")) +
scale_x_datetime(name = "Date") +
geom_point(aes(y = min_temp,
colour = daily), shape = 1, size = 1) +
geom_point(data = dw_1454,
aes(x = date, y = dw1454_min,
colour = label), shape = 3, size = 1) +
opts(title = "Average Daily Temperature Anomaly") +
geom_text(aes(x = ymd('2012-01-31'),
y = avg_mean_dw1454 - 1.5),
label = round(avg_mean_dw1454, 1),
colour = "darkcyan", size = 4)
```
tags: weather  temperature  R
tue, 31-jan-2012, 19:05

Skiing at -34

This morning I skied to work at the coldest temperatures I’ve ever attempted (-31°F when I left). We also got more than an inch of snow yesterday, so not only was it cold, but I was skiing in fresh snow. It was the slowest 4.1 miles I’d ever skied to work (57+ minutes!) and as I was going, I thought about what factors might explain how fast I ski to and from work.

Time to fire up R and run some PostgreSQL queries. The first query grabs the skiing data for this winter:

```SELECT start_time,
(extract(epoch from start_time) - extract(epoch from '2011-10-01':date))
/ (24 * 60 * 60) AS season_days,
mph,
dense_rank() OVER (
PARTITION BY
extract(year from start_time)
|| '-' || extract(week from start_time)
ORDER BY date(start_time)
) AS week_count,
CASE WHEN extract(hour from start_time) < 12 THEN 'morning'
ELSE 'afternoon'
END AS time_of_day
FROM track_stats
WHERE type = 'Skiing'
AND start_time > '2011-07-03' AND miles > 3.9;
```

This yields data that looks like this:

start_time season_days miles mph week_count time_of_day
2011-11-30 06:04:21 60.29469 4.11 4.70 1 morning
2011-11-30 15:15:43 60.67758 4.16 4.65 1 afternoon
2011-12-02 06:01:05 62.29242 4.07 4.75 2 morning
2011-12-02 15:19:59 62.68054 4.11 4.62 2 afternoon

Most of these are what you’d expect. The unconventional ones are season_days, the number of days (and fraction of a day) since October 1st 2011; week_count, the count of the number of days in that week that I skied. What I really wanted week_count to be was the number of days in a row I’d skied, but I couldn’t come up with a quick SQL query to get that, and I think this one is pretty close.

I got this into R using the following code:

```library(lubridate)
library(ggplot2)
library(RPostgreSQL)
drv <- dbDriver("PostgreSQL")
con <- dbConnect(drv, dbname=...)
ski <- dbGetQuery(con, query)
ski\$start_time <- ymd_hms(as.character(ski\$start_time))
ski\$time_of_day <- factor(ski\$time_of_day, levels = c('morning', 'afternoon'))
```

Next, I wanted to add the temperature at the start time, so I wrote a function in R that grabs this for any date passed in:

```get_temp <- function(dt) {
query <- paste("SELECT ... FROM arduino WHERE obs_dt > '",
dt,
"' ORDER BY obs_dt LIMIT 1;", sep = "")
temp <- dbGetQuery(con, query)
temp[[1]]
}
```

The query is simplified, but the basic idea is to build a query that finds the next temperature observation after I started skiing. To add this to the existing data:

```temps <- sapply(ski[,'start_time'], FUN = get_temp)
ski\$temp <- temps
```

Now to do some statistics:

```model <- lm(data = ski, mph ~ season_days + week_count + time_of_day + temp)
```

Here’s what I would expect. I’d think that season_days would be positively related to speed because I should be getting faster as I build up strength and improve my skill level. week_count should be negatively related to speed because the more I ski during the week, the more tired I will be. I’m not sure if time_of_day is relevant, but I always get the sense that I’m faster on the way home so afternoon should be positively associated with speed. Finally, temp should be positively associated with speed because the glide you can get from a properly waxed pair of skis decreases as the temperature drops.

Here's the results:

```summary(model)
Coefficients:
Estimate  Std. Error t value Pr(>|t|)
(Intercept)          4.143760   0.549018   7.548 1.66e-08 ***
season_days          0.006687   0.006097   1.097  0.28119
week_count           0.201717   0.087426   2.307  0.02788 *
time_of_dayafternoon 0.137982   0.143660   0.960  0.34425
temp                 0.021539   0.007694   2.799  0.00873 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.4302 on 31 degrees of freedom
Multiple R-squared: 0.4393,    Adjusted R-squared: 0.367
F-statistic: 6.072 on 4 and 31 DF,  p-value: 0.000995
```

The model is significant, and explains about 37% of the variation in speed. The only variables that are significant are week_count and temp, but oddly, week_count is positively associated with speed, meaning the more I ski during the week, the faster I get by the end of the week. That doesn’t make any sense, but it may be because the variable isn’t a good proxy for the “consecutive days” variable I was hoping for. Temperature is positively associated with speed, which means that I ski faster when it’s warmer.

The other refinement to this model that might have a big impact would be to add a variable for how much snow fell the night before I skied. I am fairly certain that the reason this morning’s -31°F ski was much slower than my return home at -34°F was because I was skiing on an inch of fresh snow in the morning and had tracks to ski in on the way home.

sun, 30-jan-2011, 09:52

Location map

A couple years ago we got iPhones, and one of my favorite apps is the RunKeeper app, which tracks your outdoor activities using the phone’s built-in GPS. When I first started using it I compared the results of the tracks from the phone to a Garmin eTrex, and they were so close that I’ve given up carrying the Garmin. The fact that the phone is always with me, makes keeping track of all my walks with Nika, and trips to work on my bicycle or skis pretty easy. Just like having a camera with you all the time means you capture a lot more images of daily life, having a GPS with you means you have the opportunity to keep much better track of where you go.

RunKeeper records locations on your phone and transfers the data to the RunKeeper web site when you get home (or during your trip if you’ve got a good enough cell signal). Once on the web site, you can look at the tracks on a Google map, and RunKeeper generates all kinds of statistics on your travels. You can also download the data as GPX files, which is what I’m working with here.

The GPX files are processed by a Python script that inserts each point into a spatially-enabled PostgreSQL database (PostGIS), and ties it to a track.

Summary views allow me to generate statistics like this, a summary of all my travels in 2010:

 Type Miles Hours Speed Bicycling 538.68 39.17 13.74 Hiking 211.81 92.84 2.29 Skiing 3.17 0.95 3.34

Another cool thing I can do is use R to generate a map showing where I’ve spent the most time. That’s what’s shown in the image on the right. If you’re familiar at all with the west side of the Goldstream Valley, you’ll be able to identify the roads, Creek, and trails I’ve been on in the last two years. The scale bar is the number of GPS coordinates fell within that grid, and you can get a sense of where I’ve travelled most. I’m just starting to learn what R can do with spatial data, so this is a pretty crude “analysis,” but here’s how I did it (in R):

```library(RPostgreSQL)
library(spatstat)
drv <- dbDriver("PostgreSQL")
con <- dbConnect(drv, dbname="new_gps", host="nsyn")
points <- dbGetQuery(con,
"SELECT type,
ST_X(ST_Transform(the_geom, 32606)) AS x,
ST_Y(ST_Transform(the_geom, 32606)) AS y
FROM points
INNER JOIN tracks USING (track_id)
INNER JOIN types USING (type_id)
WHERE ST_Y(the_geom) > 60 AND ST_X(the_geom) > -148;"
)
points_ppp <- ppp(points\$x, points\$y, c(min(points\$x), max(points\$x)), c(min(points\$y), max(points\$y)))
Lab.palette <- colorRampPalette(c("blue", "magenta", "red", "yellow", "white"), bias=2, space="Lab")
spatstat.options(npixel = c(500, 500))
map <- pixellate(points_ppp)
png("loc_map.png", width = 700, height = 600)
image(map, col = Lab.palette(256), main = "Gridded location counts")
dev.off()
```

Here’s a similar map showing just my walks with Nika and Piper:

Walks with Nika and Piper

And here's something similar using ggplot2:

```library(ggplot2)
m <- ggplot(data = points, aes(x = x, y = y)) + stat_density2d(geom = "tile", aes(fill = ..density..), contour = FALSE)
m + scale_fill_gradient2(low = "white", mid = "blue", high = "red", midpoint = 5e-07)
```

I trimmed off the legend and axis labels:

ggplot2, geom_density2d

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