Yesterday I looked at how wind might be affecting my bicycling to and from work. Today I’ll examine the idea that Miller Hill is confounding the effect of wind on average speed by excluding this portion of the trip from the analysis. To do this, I include a bounding box comparison in the SQL statement that extracts the wind factors for track points. The additional `WHERE` condition looks like this:

ST_Within(point_utm, ST_SetSRID(ST_MakeBox2D(ST_Point(454861,7193973), ST_Point(458232,7199159)), 32606))

The same `ST_Within` test is used in the calculation of average speed for each of the trips from work to home. After compiling the wind factors and average speeds, we compare the two using R. Here are the updated results:

lm(formula = mph ~ wind, data = data) Residuals: Min 1Q Median 3Q Max -1.87808 -0.55299 0.04038 0.62790 1.19076 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 16.8544 0.2176 77.442 <2e-16 *** wind 0.3896 0.2002 1.946 0.0683 . --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.9445 on 17 degrees of freedom Multiple R-squared: 0.1822, Adjusted R-squared: 0.1341 F-statistic: 3.788 on 1 and 17 DF, p-value: 0.06834

This time around the model and both coefficients are statistically significant (finally!), and “wind factor” is positively correlated with my average speed over the part of the route that doesn’t include Miller Hill and Railroad drive. It’s not a major contributor, but it does explain approximately 18% of the variation in average speed.

Cycling with the wind

Photo by ~BostonBill~

I decided to look at wind a little more deeply after yesterday’s bike ride home. It seemed clear to me that the wind was strongly at my back for much of the route. It wasn’t my fastest ride home, but it was close, and it didn’t feel like I was working all that hard.

Here’s the process. First, examine all my bicycling tracks individually, using PostGIS’s `ST_Azimuth` function to calculate the direction I was traveling at each point. The query uses another of the new window functions (`lead`) in PostgreSQL 8.4.

SELECT point_id, dt_local, ST_Azimuth( point_utm, lead(point_utm) OVER (PARTITION BY tid ORDER BY dt_local) ) / (2 * pi()) *360 FROM points WHERE tid = TID ORDER BY dt_local;

Then, for each point, find the direction the wind was blowing. This is a pretty slow query, but I haven’t found a better way to compare timestamps in the database to find the closest record. This technique, based on converting both timestamps to “epoch,” which is the number of seconds since January 1st, 1970, is faster than using an `interval` type of operation (like: `WHERE obs_dt - POINT_DT BETWEEN interval '-3 minutes' AND interval '3 minutes'`).

SELECT obs_dt, wdir, wspd FROM observations WHERE abs(extract(epoch from obs_dt) - extract(epoch from POINT_DT)) < 5 * 60 AND wspd IS NOT NULL AND wdir IS NOT NULL ORDER BY abs(extract(epoch from obs_dt) - extract(epoch from POINT_DT)) LIMIT 1;

Now I’ve got the direction I was traveling and the direction the wind is coming from. I wrote a Python function that returns a value from –1 (wind is in my face) to 1 (wind is at my back). The procedure is to convert the wind directions to unit *u* and *v* vectors and get the distance between the endpoints of each vector. The distances are then scaled such that wind behind the direction traveled range from 0 – 1, and from –1 – 0 for wind blowing against the direction traveled.

def wind_effect(mydir, winddir): """ Returns a number from 1 (wind at my back) to -1 (wind in my face) based on the directions passed in. Remember that wind direction is where the wind is *from*, so a wind direction of 0 and a mydir of 0 means the wind is in my face. """ try: mydir = float(mydir) winddir = float(winddir) except: return(None) my_spd = 1.0 wind_spd = 1.0 u_mydir = -1 * my_spd * math.sin(math.radians(mydir)) v_mydir = -1 * my_spd * math.cos(math.radians(mydir)) u_winddir = -1 * wind_spd * math.sin(math.radians(winddir)) v_winddir = -1 * wind_spd * math.cos(math.radians(winddir)) distance = math.sqrt((u_mydir - u_winddir)**2 + (v_mydir - v_winddir)**2) factor = (1.41421356 - distance) if factor < 0.0: factor = factor / -0.58578644 else: factor = factor / -1.41421356 return(factor)

Finally, multiply this value by the wind speed at that time, and sum all these values for an entire bicycling track. The result is a “wind factor.” A positive wind factor means the wind was generally at my back during the ride, negative means it was blowing in my face. Yesterday’s ride home had the highest wind factor (1.07) among trips since June. So the wind really was at my back!

Can “wind factor” help predict average speed? Here’s the R and results:

$ R --save < wind_from_abr.R > data<-read.table('wind_factor_from_abr',header=TRUE) > model<-lm(speed ~ wind, data) > summary(model) Call: lm(formula = speed ~ wind, data = data) Residuals: Min 1Q Median 3Q Max -0.90395 -0.46782 -0.04334 0.40286 0.85918 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 14.7796 0.1471 100.48 <2e-16 *** wind 0.4369 0.2875 1.52 0.147 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.5522 on 17 degrees of freedom Multiple R-squared: 0.1196, Adjusted R-squared: 0.06784 F-statistic: 2.31 on 1 and 17 DF, p-value: 0.1469

Hmm. Not a whole lot of help here. The model is close to being statistically significant (although it’s not…), and it’s not very predictive (only 12% of the variation in average speed is explained by wind factor). However, the directionality of the (not quite statistically significant) `wind` coefficient is correct. A positive wind factor is (weakly) correlated with a higher average speed.

Thinking more about my route from work, I suspect that the route is actually two trips: the trip from ABR to the bottom of Miller Hill (4.8 miles) and the two mile trip over Miller Hill to our house. I’ll bet that wind becomes statistically significant if I only consider the first part of the trip: wind doesn’t have as much effect on a hill climb, and after making it over the top, the rest is a bumpy, gravel road where speed is determined more by safety than wind or how hard I’m pedalling. I think this might also resolve the question of why the ride home is so much easier than to work. It’s not because I’m glad to be out of work or because I’m carrying a lunchbox full of food to work, it’s because it’s downhill from ABR to the bottom of Miller Hill.