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Peter Norvig authored and 
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@@ -8,7 +8,12 @@
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"\n",
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"Like most people, I bike slower when I'm going up a steep hill than on a flat road. But how much slower?\n",
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"To answer that, I downloaded data on my past rides \n",
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"from [Strava](https://www.strava.com/athletes/575579) and manipulated the `data` to create two lists, `X` holding the number of feet of climbing per mile for each ride, and `Y` holding the speed in miles per hour for each corresponding ride. (Notice that I only keep rides longer than 30 miles, so these should all be at a similar level of seriousness.)"
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"from [Strava](https://www.strava.com/athletes/575579) and manipulated the `data` to create two lists:\n",
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"\n",
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"- `X`: the *grade* of each ride, in feet of ascent per mile. \n",
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"- `Y`: the *speed* of each (corresponding) ride, in miles per hour.\n",
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"\n",
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"I omit rides shorter than 30 miles, because on some short rides I was sprinting at an unsustainable speed, and on others I was slowed by city traffic. Here is the code to collect and plot the X:Y data:"
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]
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},
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{
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@@ -17,8 +22,20 @@
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"outputs": [
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{
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"data": {
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"image/png": 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truncated
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"text/plain": [
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"<matplotlib.figure.Figure at 0x115deb278>"
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]
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},
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"metadata": {},
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"output_type": "display_data"
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}
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],
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"source": [
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"%matplotlib inline\n",
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"import csv\n",
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@@ -26,21 +43,24 @@
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"def hours(sec): return float(sec) / 60 / 60\n",
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"def feet(meters): return float(meters) * 100 / 2.54 / 12\n",
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"def miles(meters): return feet(meters) / 5280\n",
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"def getcsv(fname): return [r for r in csv.reader(open(fname)) if miles(r[0]) > 30]\n",
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"\n",
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"# A file with three fields per line: distance (m), climb (m), elapsed time (sec)\n",
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"data = getcsv('dist-climb-time.csv') \n",
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"X = [feet(climb) / miles(dist) for (dist, climb, time) in data]\n",
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"Y = [miles(dist) / hours(time) for (dist, climb, time) in data]"
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"# Read data file with three fields: distance (m), climb (m), elapsed time (sec)\n",
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"X, Y = [], []\n",
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"for (dist, climb, time) in csv.reader(open('dist-climb-time.csv')):\n",
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" if miles(dist) > 30:\n",
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" X.append(feet(climb) / miles(dist))\n",
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" Y.append(miles(dist) / hours(time))\n",
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"\n",
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"plt.plot(X, Y, 'o');"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Now I'll plot `X` versus `Y`, along with a best-fit polynomial to the points.\n",
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"(I use a polynomial of degree two because the raw data looks like a curve, not a straight line,\n",
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"and because a cubic polynomial seems like too many degress of freedom for y noisy data.)"
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"As expected, the speeds get slower as the grade gets steeper. The data has a lot of variance, and I can say that it \n",
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"looks more like a curve than a straight line, so I'll fit a cubic (degree two) polynomial to the data,\n",
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"and make the plot prettier:"
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]
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},
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{
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@@ -52,45 +72,34 @@
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"outputs": [
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{
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"data": {
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"image/png": 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truncated
|
||||
"image/png": 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truncated
|
||||
"text/plain": [
|
||||
"<matplotlib.figure.Figure at 0x116d67eb8>"
|
||||
"<matplotlib.figure.Figure at 0x119d56550>"
|
||||
]
|
||||
},
|
||||
"metadata": {},
|
||||
"output_type": "display_data"
|
||||
},
|
||||
{
|
||||
"name": "stdout",
|
||||
"output_type": "stream",
|
||||
"text": [
|
||||
" 0 ft/mile: 13.8 mph\n",
|
||||
" 20 ft/mile: 13.8 mph\n",
|
||||
" 40 ft/mile: 13.5 mph\n",
|
||||
" 60 ft/mile: 13.0 mph\n",
|
||||
" 80 ft/mile: 12.1 mph\n",
|
||||
"100 ft/mile: 11.0 mph\n",
|
||||
"120 ft/mile: 9.6 mph\n",
|
||||
"140 ft/mile: 7.8 mph\n"
|
||||
]
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"def poly(X, Y, n):\n",
|
||||
" \"Return best degree-n polynomial to fit X, Y data.\"\n",
|
||||
" \"Best-fit degree-n polynomial for X, Y data.\"\n",
|
||||
" C = np.polyfit(X, Y, n)[::-1] # Array of coefficients, reversed\n",
|
||||
" return lambda x: sum(C[i] * x ** i for i in range(n + 1)) \n",
|
||||
"\n",
|
||||
"def show(X, Y, grades=range(0, 141, 20), degree=2): \n",
|
||||
" plt.plot(X, Y, '+')\n",
|
||||
" F = poly(X, Y, degree)\n",
|
||||
" plt.plot(grades, [F(x) for x in grades], '-')\n",
|
||||
" plt.ylabel('speed in miles per hour')\n",
|
||||
" plt.xlabel('grade in feet per mile')\n",
|
||||
" plt.grid()\n",
|
||||
" plt.show()\n",
|
||||
" for g in grades:\n",
|
||||
" print('{:3} ft/mile: {:4.1f} mph'.format(g, F(g)))\n",
|
||||
"F = poly(X, Y, 2) # defines y = F(x); x in ft/mile, y in mph\n",
|
||||
"\n",
|
||||
"def show(X, Y, F=F): \n",
|
||||
" plt.rcParams[\"figure.figsize\"] = (10, 6)\n",
|
||||
" plt.style.use('fivethirtyeight')\n",
|
||||
" plt.plot(X, Y, 'o')\n",
|
||||
" X1 = list(range(int(max(X))))\n",
|
||||
" plt.plot(X1, [F(x) for x in X1], 'o')\n",
|
||||
" plt.ylabel('Speed (mph)')\n",
|
||||
" plt.xlabel('Grade (feet/mile)')\n",
|
||||
" plt.minorticks_on()\n",
|
||||
" plt.grid(True, which='major')\n",
|
||||
" plt.grid(True, which='minor', alpha=0.2)\n",
|
||||
" \n",
|
||||
"show(X, Y) "
|
||||
]
|
||||
@@ -99,7 +108,61 @@
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"So, it looks like I go about 14 mph when the road is fairly flat, with a lot of variability from 12 to 16 mph, depending more on my level of effort than on the grade of the road. But from 60 ft/mile and up, speed falls off quickly at 1 mph for every 20 ft/mile, and by 140 ft/mile, I'm down to 7.8 mph. Note that 140 ft/mile is only 2.7% grade, but if you figure a typical route is 1/3 up, 1/3 down, and 1/3 flatish, then that's 8% grade on the up part."
|
||||
"So, I average about 14 mph when the road is fairly flat, with a lot of variability from 12 to 16 mph, depending more on my level of effort than on the grade of the road. But from 60 ft/mile and up, speed falls off quickly at 1 mph for every 20 ft/mile, and by 140 ft/mile, I'm down around 8 mph. Note that 140 ft/mile is only 2.7% grade, but if you figure a typical route is 1/3 up, 1/3 down, and 1/3 flatish, then that's 8% grade on the up part.\n",
|
||||
"\n",
|
||||
"I can use the polynomial `F` to estimate the duration of a route:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 3,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
"def duration(dist, climb, F=F):\n",
|
||||
" \"Given a distance in miles and total climb in feet, return estimated time in minutes.\"\n",
|
||||
" return dist / F(climb / dist) * 60"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"For example, to get to Pescadero, I can take the flatter coast route, or follow the creek, saving 4 miles of distance, but adding 300 ft of climbing:"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 4,
|
||||
"metadata": {
|
||||
"collapsed": false
|
||||
},
|
||||
"outputs": [
|
||||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"(157.39103492491944, 149.35787723449531, 8.0331576904241331)"
|
||||
]
|
||||
},
|
||||
"execution_count": 4,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"coast = duration(33.4, 2201)\n",
|
||||
"creek = duration(29.4, 2520)\n",
|
||||
"\n",
|
||||
"coast, creek, coast - creek"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "markdown",
|
||||
"metadata": {},
|
||||
"source": [
|
||||
"The coast route takes about 8 minutes longer. Good to know, but other factors are probably more important in making the choice."
|
||||
]
|
||||
}
|
||||
],
|
||||
|
||||
Reference in new issue
Block a user