10_linear_regression


Linear Regression

Agenda

  1. Introducing the bikeshare dataset
    • Reading in the data
    • Visualizing the data
  2. Linear regression basics
    • Form of linear regression
    • Building a linear regression model
    • Using the model for prediction
    • Does the scale of the features matter?
  3. Working with multiple features
    • Visualizing the data (part 2)
    • Adding more features to the model
  4. Choosing between models
    • Feature selection
    • Evaluation metrics for regression problems
    • Comparing models with train/test split and RMSE
    • Comparing testing RMSE with null RMSE
  5. Creating features
    • Handling categorical features
    • Feature engineering
  6. Comparing linear regression with other models

Reading in the data

We'll be working with a dataset from Capital Bikeshare that was used in a Kaggle competition (data dictionary).


In [223]:
# read the data and set the datetime as the index
import seaborn as sns
import matplotlib.pyplot as plt
%matplotlib inline
plt.rcParams['figure.figsize'] = (8, 6)
plt.rcParams['font.size'] = 14
import pandas as pd
urls = ['../data/KDCA-201601.csv', '../data/KDCA-201602.csv', '../data/KDCA-201603.csv']
frames = [pd.read_csv(url) for url in urls]
weather = pd.concat(frames)
cols = 'WBAN	Date	Time	StationType	SkyCondition	Visibility	WeatherType	DryBulbFarenheit	DryBulbCelsius	WetBulbFarenheit	WetBulbCelsius	DewPointFarenheit	DewPointCelsius	RelativeHumidity	WindSpeed	WindDirection	ValueForWindCharacter	StationPressure	PressureTendency	PressureChange	SeaLevelPressure	RecordType	HourlyPrecip	Altimeter'
cols = cols.split()
weather = weather[cols]
weather.rename(columns={'DryBulbFarenheit':'temp',
                       'RelativeHumidity': 'humidity'}, inplace=True)
# weather['humidity'] = pd.to_numeric(weather.humidity, errors='coerce')

weather['datetime'] = pd.to_datetime(weather.Date.astype(str) + weather.Time.apply('{0:0>4}'.format))
weather['datetime_hour'] = weather.datetime.dt.floor(freq='h')
weather['month'] = weather.datetime.dt.month

bikes = pd.read_csv('../data/2016-Q1-Trips-History-Data.csv')
bikes['start'] = pd.to_datetime(bikes['Start date'], infer_datetime_format=True)
bikes['end'] = pd.to_datetime(bikes['End date'], infer_datetime_format=True)
bikes['datetime_hour'] = bikes.start.dt.floor(freq='h')
weather[['datetime', 'temp']].hist(bins=30)
print(weather.columns)
weather.head()


Index(['WBAN', 'Date', 'Time', 'StationType', 'SkyCondition', 'Visibility',
       'WeatherType', 'temp', 'DryBulbCelsius', 'WetBulbFarenheit',
       'WetBulbCelsius', 'DewPointFarenheit', 'DewPointCelsius', 'humidity',
       'WindSpeed', 'WindDirection', 'ValueForWindCharacter',
       'StationPressure', 'PressureTendency', 'PressureChange',
       'SeaLevelPressure', 'RecordType', 'HourlyPrecip', 'Altimeter',
       'datetime', 'datetime_hour', 'month'],
      dtype='object')
Out[223]:
WBAN Date Time StationType SkyCondition Visibility WeatherType temp DryBulbCelsius WetBulbFarenheit ... StationPressure PressureTendency PressureChange SeaLevelPressure RecordType HourlyPrecip Altimeter datetime datetime_hour month
0 13743 20160101 52 11 BKN046 OVC250 10.0 44 6.7 39 ... 30.09 30.15 AA 30.16 2016-01-01 00:52:00 2016-01-01 00:00:00 1
1 13743 20160101 152 11 FEW045 BKN230 10.0 42 5.6 38 ... 30.08 30.14 AA 30.15 2016-01-01 01:52:00 2016-01-01 01:00:00 1
2 13743 20160101 252 11 FEW045 BKN220 10.0 42 5.6 38 ... 30.09 30.15 AA 30.16 2016-01-01 02:52:00 2016-01-01 02:00:00 1
3 13743 20160101 352 11 BKN220 10.0 42 5.6 37 ... 30.07 30.14 AA 30.14 2016-01-01 03:52:00 2016-01-01 03:00:00 1
4 13743 20160101 452 11 BKN180 OVC230 10.0 42 5.6 37 ... 30.07 30.14 AA 30.14 2016-01-01 04:52:00 2016-01-01 04:00:00 1

5 rows × 27 columns


In [224]:
bikes.merge(weather[['temp', 'datetime_hour', 'datetime']], on='datetime_hour')


Out[224]:
Duration (ms) Start date End date Start station number Start station End station number End station Bike number Member Type start end datetime_hour temp datetime
0 301295 3/31/2016 23:59 4/1/2016 0:04 31280 11th & S St NW 31506 1st & Rhode Island Ave NW W00022 Registered 2016-03-31 23:59:00 2016-04-01 00:04:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
1 557887 3/31/2016 23:59 4/1/2016 0:08 31275 New Hampshire Ave & 24th St NW 31114 18th St & Wyoming Ave NW W01294 Registered 2016-03-31 23:59:00 2016-04-01 00:08:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
2 555944 3/31/2016 23:59 4/1/2016 0:08 31101 14th & V St NW 31221 18th & M St NW W01416 Registered 2016-03-31 23:59:00 2016-04-01 00:08:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
3 766916 3/31/2016 23:57 4/1/2016 0:09 31226 34th St & Wisconsin Ave NW 31214 17th & Corcoran St NW W01090 Registered 2016-03-31 23:57:00 2016-04-01 00:09:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
4 139656 3/31/2016 23:57 3/31/2016 23:59 31011 23rd & Crystal Dr 31009 27th & Crystal Dr W21934 Registered 2016-03-31 23:57:00 2016-03-31 23:59:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
5 967713 3/31/2016 23:57 4/1/2016 0:13 31266 11th & M St NW 31600 5th & K St NW W20562 Casual 2016-03-31 23:57:00 2016-04-01 00:13:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
6 534836 3/31/2016 23:57 4/1/2016 0:06 31222 New York Ave & 15th St NW 31278 18th & R St NW W20222 Registered 2016-03-31 23:57:00 2016-04-01 00:06:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
7 243864 3/31/2016 23:56 4/1/2016 0:00 31228 8th & H St NW 31600 5th & K St NW W20291 Registered 2016-03-31 23:56:00 2016-04-01 00:00:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
8 372524 3/31/2016 23:55 4/1/2016 0:01 31113 Columbia Rd & Belmont St NW 31234 20th & O St NW / Dupont South W20590 Registered 2016-03-31 23:55:00 2016-04-01 00:01:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
9 215194 3/31/2016 23:55 3/31/2016 23:59 31263 10th & K St NW 31265 5th St & Massachusetts Ave NW W21876 Registered 2016-03-31 23:55:00 2016-03-31 23:59:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
10 498903 3/31/2016 23:55 4/1/2016 0:03 31243 Maryland & Independence Ave SW 31209 1st & N St SE W20973 Registered 2016-03-31 23:55:00 2016-04-01 00:03:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
11 389082 3/31/2016 23:54 4/1/2016 0:01 31079 Lee Hwy & N Cleveland St 31093 21st St N & N Pierce St W01197 Registered 2016-03-31 23:54:00 2016-04-01 00:01:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
12 1680745 3/31/2016 23:54 4/1/2016 0:22 31258 Lincoln Memorial 31269 3rd St & Pennsylvania Ave SE W01191 Casual 2016-03-31 23:54:00 2016-04-01 00:22:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
13 1687026 3/31/2016 23:54 4/1/2016 0:23 31258 Lincoln Memorial 31269 3rd St & Pennsylvania Ave SE W20449 Casual 2016-03-31 23:54:00 2016-04-01 00:23:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
14 544541 3/31/2016 23:53 4/1/2016 0:02 31245 7th & R St NW / Shaw Library 31505 Eckington Pl & Q St NE W20888 Registered 2016-03-31 23:53:00 2016-04-01 00:02:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
15 1001144 3/31/2016 23:51 4/1/2016 0:08 31106 Calvert & Biltmore St NW 31226 34th St & Wisconsin Ave NW W22196 Casual 2016-03-31 23:51:00 2016-04-01 00:08:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
16 1262663 3/31/2016 23:51 4/1/2016 0:12 31111 10th & U St NW 31226 34th St & Wisconsin Ave NW W21553 Casual 2016-03-31 23:51:00 2016-04-01 00:12:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
17 451821 3/31/2016 23:51 3/31/2016 23:59 31613 Eastern Market Metro / Pennsylvania Ave & 7th ... 31617 Bladensburg Rd & Benning Rd NE W20614 Registered 2016-03-31 23:51:00 2016-03-31 23:59:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
18 305172 3/31/2016 23:50 3/31/2016 23:55 31269 3rd St & Pennsylvania Ave SE 31639 2nd & G St NE W22068 Registered 2016-03-31 23:50:00 2016-03-31 23:55:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
19 5230964 3/31/2016 23:50 4/1/2016 1:17 31248 Smithsonian / Jefferson Dr & 12th St SW 31248 Smithsonian / Jefferson Dr & 12th St SW W01458 Casual 2016-03-31 23:50:00 2016-04-01 01:17:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
20 5226179 3/31/2016 23:50 4/1/2016 1:17 31248 Smithsonian / Jefferson Dr & 12th St SW 31248 Smithsonian / Jefferson Dr & 12th St SW W22256 Casual 2016-03-31 23:50:00 2016-04-01 01:17:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
21 534398 3/31/2016 23:51 3/31/2016 23:59 31106 Calvert & Biltmore St NW 31278 18th & R St NW W21441 Casual 2016-03-31 23:51:00 2016-03-31 23:59:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
22 611293 3/31/2016 23:49 3/31/2016 23:59 31230 Metro Center / 12th & G St NW 31212 21st & M St NW W00928 Registered 2016-03-31 23:49:00 2016-03-31 23:59:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
23 1782313 3/31/2016 23:48 4/1/2016 0:18 31602 Park Rd & Holmead Pl NW 31258 Lincoln Memorial W20734 Registered 2016-03-31 23:48:00 2016-04-01 00:18:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
24 1772821 3/31/2016 23:48 4/1/2016 0:18 31602 Park Rd & Holmead Pl NW 31258 Lincoln Memorial W00353 Registered 2016-03-31 23:48:00 2016-04-01 00:18:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
25 173074 3/31/2016 23:45 3/31/2016 23:48 31019 Wilson Blvd & N Edgewood St 31079 Lee Hwy & N Cleveland St W00719 Registered 2016-03-31 23:45:00 2016-03-31 23:48:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
26 412372 3/31/2016 23:45 3/31/2016 23:52 31202 14th & R St NW 31114 18th St & Wyoming Ave NW W20924 Registered 2016-03-31 23:45:00 2016-03-31 23:52:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
27 401291 3/31/2016 23:45 3/31/2016 23:52 31202 14th & R St NW 31114 18th St & Wyoming Ave NW W20768 Registered 2016-03-31 23:45:00 2016-03-31 23:52:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
28 617906 3/31/2016 23:45 3/31/2016 23:56 31234 20th & O St NW / Dupont South 31117 15th & Euclid St NW W00613 Registered 2016-03-31 23:45:00 2016-03-31 23:56:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
29 527203 3/31/2016 23:44 3/31/2016 23:53 31613 Eastern Market Metro / Pennsylvania Ave & 7th ... 31208 M St & New Jersey Ave SE W01039 Casual 2016-03-31 23:44:00 2016-03-31 23:53:00 2016-03-31 23:00:00 67 2016-03-31 23:52:00
... ... ... ... ... ... ... ... ... ... ... ... ... ... ...
674708 919064 1/1/2016 0:38 1/1/2016 0:53 31107 Lamont & Mt Pleasant NW 31503 Florida Ave & R St NW W21252 Registered 2016-01-01 00:38:00 2016-01-01 00:53:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674709 453855 1/1/2016 0:36 1/1/2016 0:43 31032 Washington Blvd & 7th St N 31067 Columbia Pike & S Walter Reed Dr W20276 Registered 2016-01-01 00:36:00 2016-01-01 00:43:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674710 598544 1/1/2016 0:35 1/1/2016 0:45 31509 New Jersey Ave & R St NW 31203 14th & Rhode Island Ave NW W20356 Registered 2016-01-01 00:35:00 2016-01-01 00:45:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674711 300498 1/1/2016 0:35 1/1/2016 0:40 31513 Rhode Island Ave & V St NE 31516 Rhode Island Ave Metro W20812 Registered 2016-01-01 00:35:00 2016-01-01 00:40:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674712 303506 1/1/2016 0:35 1/1/2016 0:40 31513 Rhode Island Ave & V St NE 31516 Rhode Island Ave Metro W21375 Registered 2016-01-01 00:35:00 2016-01-01 00:40:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674713 228193 1/1/2016 0:33 1/1/2016 0:37 31611 13th & H St NE 31622 13th & D St NE W20851 Registered 2016-01-01 00:33:00 2016-01-01 00:37:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674714 304174 1/1/2016 0:32 1/1/2016 0:37 31613 Eastern Market Metro / Pennsylvania Ave & 7th ... 31626 15th St & Massachusetts Ave SE W21227 Registered 2016-01-01 00:32:00 2016-01-01 00:37:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674715 2018681 1/1/2016 0:32 1/1/2016 1:06 31246 M St & Pennsylvania Ave NW 31513 Rhode Island Ave & V St NE W22120 Casual 2016-01-01 00:32:00 2016-01-01 01:06:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674716 1119945 1/1/2016 0:31 1/1/2016 0:49 31258 Lincoln Memorial 31051 Arlington Blvd & N Queen St W20374 Registered 2016-01-01 00:31:00 2016-01-01 00:49:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674717 1125841 1/1/2016 0:31 1/1/2016 0:50 31258 Lincoln Memorial 31051 Arlington Blvd & N Queen St W20756 Registered 2016-01-01 00:31:00 2016-01-01 00:50:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674718 638476 1/1/2016 0:32 1/1/2016 0:42 31611 13th & H St NE 31613 Eastern Market Metro / Pennsylvania Ave & 7th ... W21985 Registered 2016-01-01 00:32:00 2016-01-01 00:42:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674719 1107609 1/1/2016 0:31 1/1/2016 0:49 31262 11th & F St NW 31505 Eckington Pl & Q St NE W00630 Registered 2016-01-01 00:31:00 2016-01-01 00:49:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674720 313542 1/1/2016 0:31 1/1/2016 0:36 31623 Columbus Circle / Union Station 31614 11th & H St NE W20643 Registered 2016-01-01 00:31:00 2016-01-01 00:36:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674721 873789 1/1/2016 0:29 1/1/2016 0:43 31616 3rd & H St NE 31281 8th & O St NW W20910 Registered 2016-01-01 00:29:00 2016-01-01 00:43:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674722 862067 1/1/2016 0:28 1/1/2016 0:42 31640 Maryland Ave & E St NE 31500 4th & W St NE W21281 Registered 2016-01-01 00:28:00 2016-01-01 00:42:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674723 336250 1/1/2016 0:28 1/1/2016 0:34 31116 California St & Florida Ave NW 31239 17th & Rhode Island Ave NW W01296 Registered 2016-01-01 00:28:00 2016-01-01 00:34:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674724 398655 1/1/2016 0:28 1/1/2016 0:35 31101 14th & V St NW 31109 7th & T St NW W20747 Registered 2016-01-01 00:28:00 2016-01-01 00:35:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674725 278474 1/1/2016 0:27 1/1/2016 0:32 31231 14th & D St NW / Ronald Reagan Building 31262 11th & F St NW W00922 Casual 2016-01-01 00:27:00 2016-01-01 00:32:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674726 379090 1/1/2016 0:27 1/1/2016 0:34 31509 New Jersey Ave & R St NW 31202 14th & R St NW W20020 Registered 2016-01-01 00:27:00 2016-01-01 00:34:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674727 882262 1/1/2016 0:26 1/1/2016 0:40 31245 7th & R St NW / Shaw Library 31244 4th & E St SW W21360 Registered 2016-01-01 00:26:00 2016-01-01 00:40:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674728 325824 1/1/2016 0:26 1/1/2016 0:31 31245 7th & R St NW / Shaw Library 31201 15th & P St NW W00295 Registered 2016-01-01 00:26:00 2016-01-01 00:31:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674729 813328 1/1/2016 0:20 1/1/2016 0:33 31041 Prince St & Union St 31084 Ballenger Ave & Dulaney St W22235 Registered 2016-01-01 00:20:00 2016-01-01 00:33:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674730 1102053 1/1/2016 0:18 1/1/2016 0:36 31506 1st & Rhode Island Ave NW 31116 California St & Florida Ave NW W00568 Registered 2016-01-01 00:18:00 2016-01-01 00:36:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674731 872980 1/1/2016 0:16 1/1/2016 0:30 31041 Prince St & Union St 31048 King St Metro South W22058 Registered 2016-01-01 00:16:00 2016-01-01 00:30:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674732 859170 1/1/2016 0:16 1/1/2016 0:30 31041 Prince St & Union St 31048 King St Metro South W01122 Casual 2016-01-01 00:16:00 2016-01-01 00:30:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674733 782042 1/1/2016 0:16 1/1/2016 0:29 31266 11th & M St NW 31278 18th & R St NW W22090 Registered 2016-01-01 00:16:00 2016-01-01 00:29:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674734 213976 1/1/2016 0:15 1/1/2016 0:19 31506 1st & Rhode Island Ave NW 31509 New Jersey Ave & R St NW W01294 Registered 2016-01-01 00:15:00 2016-01-01 00:19:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674735 715013 1/1/2016 0:13 1/1/2016 0:25 31222 New York Ave & 15th St NW 31214 17th & Corcoran St NW W21427 Registered 2016-01-01 00:13:00 2016-01-01 00:25:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674736 448007 1/1/2016 0:10 1/1/2016 0:17 32039 Old Georgetown Rd & Southwick St 32002 Bethesda Ave & Arlington Rd W22202 Registered 2016-01-01 00:10:00 2016-01-01 00:17:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00
674737 166066 1/1/2016 0:06 1/1/2016 0:09 31102 11th & Kenyon St NW 31105 14th & Harvard St NW W01346 Registered 2016-01-01 00:06:00 2016-01-01 00:09:00 2016-01-01 00:00:00 44 2016-01-01 00:52:00

674738 rows × 14 columns


In [225]:
hours = bikes.groupby('datetime_hour').agg('count')
hours['datetime_hour'] = hours.index
hours.head()
hours['total'] = hours.start
hours = hours[['total', 'datetime_hour']]
hours.total.plot()
hours_weather = hours.merge(weather, on='datetime_hour')
hours_weather.plot(kind='scatter', x='temp', y='total')


Out[225]:
<matplotlib.axes._subplots.AxesSubplot at 0x130b294e0>

In [226]:
weekday = hours_weather[(hours_weather.datetime.dt.hour==11) & (hours_weather.datetime.dt.dayofweek<5) ]
weekday.plot(kind='scatter', x='temp', y='total')


Out[226]:
<matplotlib.axes._subplots.AxesSubplot at 0x127d0bef0>

In [227]:
# import seaborn as sns

sns.lmplot(x='temp', y='total', data=weekday, aspect=1.5, scatter_kws={'alpha':0.8})


Out[227]:
<seaborn.axisgrid.FacetGrid at 0x127fd3400>

Questions:

  • What does each observation represent?
  • What is the response variable (as defined by Kaggle)?
  • How many features are there?

Form of linear regression

$y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_nx_n$

  • $y$ is the response
  • $\beta_0$ is the intercept
  • $\beta_1$ is the coefficient for $x_1$ (the first feature)
  • $\beta_n$ is the coefficient for $x_n$ (the nth feature)

The $\beta$ values are called the model coefficients:

  • These values are estimated (or "learned") during the model fitting process using the least squares criterion.
  • Specifically, we are find the line (mathematically) which minimizes the sum of squared residuals (or "sum of squared errors").
  • And once we've learned these coefficients, we can use the model to predict the response.

In the diagram above:

  • The black dots are the observed values of x and y.
  • The blue line is our least squares line.
  • The red lines are the residuals, which are the vertical distances between the observed values and the least squares line.

Building a linear regression model


In [228]:
# create X and y
feature_cols = ['temp']
X = hours_weather[feature_cols]
y = hours_weather.total

In [229]:
# import, instantiate, fit
from sklearn.linear_model import LinearRegression
linreg = LinearRegression()
linreg.fit(X, y)


Out[229]:
LinearRegression(copy_X=True, fit_intercept=True, n_jobs=1, normalize=False)

In [230]:
# print the coefficients
print(linreg.intercept_)
print(linreg.coef_)


-176.632263746
[ 9.88748864]

Interpreting the intercept ($\beta_0$):

  • It is the value of $y$ when $x$=0.
  • Thus, it is the estimated number of rentals when the temperature is 0 degrees Celsius.
  • Note: It does not always make sense to interpret the intercept. (Why?)

Interpreting the "temp" coefficient ($\beta_1$):

  • It is the change in $y$ divided by change in $x$, or the "slope".
  • Thus, a temperature increase of 1 degree F is associated with a rental increase of 9.17 bikes.
  • This is not a statement of causation.
  • $\beta_1$ would be negative if an increase in temperature was associated with a decrease in rentals.

Using the model for prediction

How many bike rentals would we predict if the temperature was 77 degrees F?


In [231]:
# manually calculate the prediction
linreg.intercept_ + linreg.coef_ * 77


Out[231]:
array([ 584.70436123])

In [232]:
# use the predict method
linreg.predict(77)


Out[232]:
array([ 584.70436123])

Does the scale of the features matter?

Let's say that temperature was measured in Fahrenheit, rather than Celsius. How would that affect the model?


In [233]:
# create a new column for Fahrenheit temperature
hours_weather['temp_C'] = (hours_weather.temp - 32) * 5/9
hours_weather.head()


Out[233]:
total datetime_hour WBAN Date Time StationType SkyCondition Visibility WeatherType temp ... StationPressure PressureTendency PressureChange SeaLevelPressure RecordType HourlyPrecip Altimeter datetime month temp_C
0 56 2016-01-01 00:00:00 13743 20160101 52 11 BKN046 OVC250 10.0 44 ... 30.09 30.15 AA 30.16 2016-01-01 00:52:00 1 6.666667
1 105 2016-01-01 01:00:00 13743 20160101 152 11 FEW045 BKN230 10.0 42 ... 30.08 30.14 AA 30.15 2016-01-01 01:52:00 1 5.555556
2 74 2016-01-01 02:00:00 13743 20160101 252 11 FEW045 BKN220 10.0 42 ... 30.09 30.15 AA 30.16 2016-01-01 02:52:00 1 5.555556
3 32 2016-01-01 03:00:00 13743 20160101 352 11 BKN220 10.0 42 ... 30.07 30.14 AA 30.14 2016-01-01 03:52:00 1 5.555556
4 13 2016-01-01 04:00:00 13743 20160101 452 11 BKN180 OVC230 10.0 42 ... 30.07 30.14 AA 30.14 2016-01-01 04:52:00 1 5.555556

5 rows × 29 columns


In [234]:
# Seaborn scatter plot with regression line
sns.lmplot(x='temp_C', y='total', data=hours_weather, aspect=1.5, scatter_kws={'alpha':0.2})
sns.lmplot(x='temp', y='total', data=hours_weather, aspect=1.5, scatter_kws={'alpha':0.2})


Out[234]:
<seaborn.axisgrid.FacetGrid at 0x12665c2b0>

In [235]:
# create X and y
feature_cols = ['temp_C']
X = hours_weather[feature_cols]
y = hours_weather.total

# instantiate and fit
linreg = LinearRegression()
linreg.fit(X, y)

# print the coefficients
print(linreg.intercept_, linreg.coef_)


139.767372607 [ 17.79747954]

In [236]:
# convert 77 degrees Fahrenheit to Celsius
(77 - 32)* 5/9


Out[236]:
25.0

In [237]:
# predict rentals for 25 degrees Celsius
linreg.predict(25)


Out[237]:
array([ 584.70436123])

Conclusion: The scale of the features is irrelevant for linear regression models. When changing the scale, we simply change our interpretation of the coefficients.


In [239]:
# remove the temp_F column
# bikes.drop('temp_C', axis=1, inplace=True)

Visualizing the data (part 2)


In [240]:
# explore more features
feature_cols = ['temp', 'month', 'humidity']

In [241]:
# multiple scatter plots in Seaborn
# print(hours_weather.humidity != 'M')
hours_weather.humidity = hours_weather.humidity.apply(lambda x: -1 if isinstance(x, str) else x)
# hours_weather.loc[hours_weather.humidity.dtype != int].humidity = 100
sns.pairplot(hours_weather, x_vars=feature_cols, y_vars='total', kind='reg')


Out[241]:
<seaborn.axisgrid.PairGrid at 0x12734bcf8>

In [242]:
# multiple scatter plots in Pandas
fig, axs = plt.subplots(1, len(feature_cols), sharey=True)
for index, feature in enumerate(feature_cols):
    hours_weather.plot(kind='scatter', x=feature, y='total', ax=axs[index], figsize=(16, 3))


Are you seeing anything that you did not expect?


In [244]:
# cross-tabulation of season and month
pd.crosstab(hours_weather.month, hours_weather.datetime.dt.dayofweek)


Out[244]:
datetime 0 1 2 3 4 5 6
month
1 72 77 98 103 170 151 129
2 203 182 183 115 102 114 125
3 129 139 124 134 123 115 134

In [245]:
# box plot of rentals, grouped by season
hours_weather.boxplot(column='total', by='month')


Out[245]:
<matplotlib.axes._subplots.AxesSubplot at 0x125982588>

In [246]:
# line plot of rentals
hours_weather.total.plot()


Out[246]:
<matplotlib.axes._subplots.AxesSubplot at 0x125a1cb00>

What does this tell us?

There are more rentals in the winter than the spring, but only because the system is experiencing overall growth and the winter months happen to come after the spring months.


In [247]:
# correlation matrix (ranges from 1 to -1)
hours_weather.corr()


Out[247]:
total WBAN Date Time StationType Visibility temp DryBulbCelsius DewPointFarenheit DewPointCelsius humidity Altimeter month temp_C
total 1.000000 NaN 0.240069 0.283064 NaN 0.224845 0.437242 0.437254 0.156470 0.156632 -0.109927 -0.038122 0.236981 0.437242
WBAN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
Date 0.240069 NaN 1.000000 -0.004699 NaN 0.168070 0.506872 0.506715 0.371529 0.371576 -0.791303 -0.037263 0.994602 0.506872
Time 0.283064 NaN -0.004699 1.000000 NaN 0.026204 0.120424 0.120322 0.008574 0.008571 -0.025222 -0.032535 -0.007578 0.120424
StationType NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN
Visibility 0.224845 NaN 0.168070 0.026204 NaN 1.000000 0.119073 0.118922 -0.257975 -0.257868 -0.187697 0.075111 0.167172 0.119073
temp 0.437242 NaN 0.506872 0.120424 NaN 0.119073 1.000000 0.999982 0.789360 0.789495 -0.269786 -0.465325 0.499250 1.000000
DryBulbCelsius 0.437254 NaN 0.506715 0.120322 NaN 0.118922 0.999982 1.000000 0.789386 0.789520 -0.269803 -0.465342 0.499113 0.999982
DewPointFarenheit 0.156470 NaN 0.371529 0.008574 NaN -0.257975 0.789360 0.789386 1.000000 0.999987 -0.121292 -0.533671 0.370406 0.789360
DewPointCelsius 0.156632 NaN 0.371576 0.008571 NaN -0.257868 0.789495 0.789520 0.999987 1.000000 -0.121358 -0.533625 0.370438 0.789495
humidity -0.109927 NaN -0.791303 -0.025222 NaN -0.187697 -0.269786 -0.269803 -0.121292 -0.121358 1.000000 -0.016122 -0.793716 -0.269786
Altimeter -0.038122 NaN -0.037263 -0.032535 NaN 0.075111 -0.465325 -0.465342 -0.533671 -0.533625 -0.016122 1.000000 -0.020321 -0.465325
month 0.236981 NaN 0.994602 -0.007578 NaN 0.167172 0.499250 0.499113 0.370406 0.370438 -0.793716 -0.020321 1.000000 0.499250
temp_C 0.437242 NaN 0.506872 0.120424 NaN 0.119073 1.000000 0.999982 0.789360 0.789495 -0.269786 -0.465325 0.499250 1.000000

In [248]:
# visualize correlation matrix in Seaborn using a heatmap
sns.heatmap(hours_weather.corr())


Out[248]:
<matplotlib.axes._subplots.AxesSubplot at 0x1265a3ef0>

What relationships do you notice?

Adding more features to the model


In [251]:
# create a list of features
feature_cols = ['temp', 'month', 'humidity']

In [252]:
# create X and y
X = hours_weather[feature_cols]
y = hours_weather.total

# instantiate and fit
linreg = LinearRegression()
linreg.fit(X, y)

# print the coefficients
print(linreg.intercept_, linreg.coef_)


-226.700163985 [  9.33945424  30.62665914   0.60483008]

In [253]:
# pair the feature names with the coefficients
list(zip(feature_cols, linreg.coef_))


Out[253]:
[('temp', 9.3394542400038798),
 ('month', 30.626659142039028),
 ('humidity', 0.60483007751778484)]

Interpreting the coefficients:

  • Holding all other features fixed, a 1 unit increase in temperature is associated with a rental increase of 9.3 bikes.
  • Holding all other features fixed, a 1 unit increase in month is associated with a rental increase of 30.6 bikes.
  • Holding all other features fixed, a 1 unit increase in humidity is associated with a rental decrease of .60 bikes.

Does anything look incorrect?

Feature selection

How do we choose which features to include in the model? We're going to use train/test split (and eventually cross-validation).

Why not use of p-values or R-squared for feature selection?

  • Linear models rely upon a lot of assumptions (such as the features being independent), and if those assumptions are violated, p-values and R-squared are less reliable. Train/test split relies on fewer assumptions.
  • Features that are unrelated to the response can still have significant p-values.
  • Adding features to your model that are unrelated to the response will always increase the R-squared value, and adjusted R-squared does not sufficiently account for this.
  • p-values and R-squared are proxies for our goal of generalization, whereas train/test split and cross-validation attempt to directly estimate how well the model will generalize to out-of-sample data.

More generally:

  • There are different methodologies that can be used for solving any given data science problem, and this course follows a machine learning methodology.
  • This course focuses on general purpose approaches that can be applied to any model, rather than model-specific approaches.

Evaluation metrics for regression problems

Evaluation metrics for classification problems, such as accuracy, are not useful for regression problems. We need evaluation metrics designed for comparing continuous values.

Here are three common evaluation metrics for regression problems:

Mean Absolute Error (MAE) is the mean of the absolute value of the errors:

$$\frac 1n\sum_{i=1}^n|y_i-\hat{y}_i|$$

Mean Squared Error (MSE) is the mean of the squared errors:

$$\frac 1n\sum_{i=1}^n(y_i-\hat{y}_i)^2$$

Root Mean Squared Error (RMSE) is the square root of the mean of the squared errors:

$$\sqrt{\frac 1n\sum_{i=1}^n(y_i-\hat{y}_i)^2}$$

In [254]:
# example true and predicted response values
true = [10, 7, 5, 5]
pred = [8, 6, 5, 10]

In [256]:
# calculate these metrics by hand!
from sklearn import metrics
import numpy as np
print('MAE:', metrics.mean_absolute_error(true, pred))
print('MSE:', metrics.mean_squared_error(true, pred))
print('RMSE:', np.sqrt(metrics.mean_squared_error(true, pred)))


MAE: 2.0
MSE: 7.5
RMSE: 2.73861278753

Comparing these metrics:

  • MAE is the easiest to understand, because it's the average error.
  • MSE is more popular than MAE, because MSE "punishes" larger errors, which tends to be useful in the real world.
  • RMSE is even more popular than MSE, because RMSE is interpretable in the "y" units.

All of these are loss functions, because we want to minimize them.

Here's an additional example, to demonstrate how MSE/RMSE punish larger errors:


In [258]:
# same true values as above
true = [10, 7, 5, 5]

# new set of predicted values
pred = [10, 7, 5, 13]

# MAE is the same as before
print('MAE:', metrics.mean_absolute_error(true, pred))

# MSE and RMSE are larger than before
print('MSE:', metrics.mean_squared_error(true, pred))
print('RMSE:', np.sqrt(metrics.mean_squared_error(true, pred)))


MAE: 2.0
MSE: 16.0
RMSE: 4.0

Comparing models with train/test split and RMSE


In [259]:
from sklearn.cross_validation import train_test_split
import sklearn.metrics as metrics
import numpy as np

# define a function that accepts a list of features and returns testing RMSE
def train_test_rmse(feature_cols, data):
    X = data[feature_cols]
    y = data.total
    X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=123)
    linreg = LinearRegression()
    linreg.fit(X_train, y_train)
    y_pred = linreg.predict(X_test)
    return np.sqrt(metrics.mean_squared_error(y_test, y_pred))

In [260]:
# compare different sets of features
print(train_test_rmse(['temp', 'month', 'humidity'], hours_weather))
print(train_test_rmse(['temp', 'month'], hours_weather))
print(train_test_rmse(['temp', 'humidity'], hours_weather))


246.451640105
246.783327002
246.572708806

Comparing testing RMSE with null RMSE

Null RMSE is the RMSE that could be achieved by always predicting the mean response value. It is a benchmark against which you may want to measure your regression model.


In [261]:
# split X and y into training and testing sets
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=123)

# create a NumPy array with the same shape as y_test
y_null = np.zeros_like(y_test, dtype=float)

# fill the array with the mean value of y_test
y_null.fill(y_test.mean())
y_null


Out[261]:
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        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,
        238.6989721,  238.6989721,  238.6989721,  238.6989721,  238.6989721])

In [262]:
# compute null RMSE
np.sqrt(metrics.mean_squared_error(y_test, y_null))


Out[262]:
274.56092645120714

Handling categorical features

scikit-learn expects all features to be numeric. So how do we include a categorical feature in our model?

  • Ordered categories: transform them to sensible numeric values (example: small=1, medium=2, large=3)
  • Unordered categories: use dummy encoding (0/1)

What are the categorical features in our dataset?

  • Ordered categories: weather (already encoded with sensible numeric values)
  • Unordered categories: season (needs dummy encoding), holiday (already dummy encoded), workingday (already dummy encoded)

For season, we can't simply leave the encoding as 1 = spring, 2 = summer, 3 = fall, and 4 = winter, because that would imply an ordered relationship. Instead, we create multiple dummy variables:


In [263]:
# create dummy variables
season_dummies = pd.get_dummies(hours_weather.month, prefix='month')

# print 5 random rows
season_dummies.sample(n=5, random_state=1)


Out[263]:
month_1 month_2 month_3
1443 0.0 1.0 0.0
1152 0.0 1.0 0.0
1348 0.0 1.0 0.0
1334 0.0 1.0 0.0
2700 0.0 0.0 1.0

In general, if you have a categorical feature with k possible values, you create k-1 dummy variables.

If that's confusing, think about why we only need one dummy variable for holiday, not two dummy variables (holiday_yes and holiday_no).


In [264]:
# concatenate the original DataFrame and the dummy DataFrame (axis=0 means rows, axis=1 means columns)
hw_dum = pd.concat([hours_weather, season_dummies], axis=1)

# print 5 random rows
hw_dum.sample(n=5, random_state=1)


Out[264]:
total datetime_hour WBAN Date Time StationType SkyCondition Visibility WeatherType temp ... SeaLevelPressure RecordType HourlyPrecip Altimeter datetime month temp_C month_1 month_2 month_3
1443 2 2016-02-19 02:00:00 13743 20160219 252 11 CLR 10.00 30 ... 30.57 AA 30.57 2016-02-19 02:52:00 2 -1.111111 0.0 1.0 0.0
1152 40 2016-02-12 05:00:00 13743 20160212 552 11 BKN180 10.00 20 ... 30.32 AA 30.32 2016-02-12 05:52:00 2 -6.666667 0.0 1.0 0.0
1348 3 2016-02-16 04:00:00 13743 20160216 405 11 VV002 0.12 FZFG 31 ... M SP 29.83 2016-02-16 04:05:00 2 -0.555556 0.0 1.0 0.0
1334 8 2016-02-15 23:00:00 13743 20160215 2328 11 OVC006 7.00 -FZDZ 30 ... M SP 30.06 2016-02-15 23:28:00 2 -1.111111 0.0 1.0 0.0
2700 2 2016-03-31 03:00:00 13743 20160331 352 11 SCT250 10.00 55 ... 30.03 AA 30.04 2016-03-31 03:52:00 3 12.777778 0.0 0.0 1.0

5 rows × 32 columns


In [265]:
# include dummy variables for season in the model
feature_cols = ['temp', 'month_2', 'month_3', 'humidity']
X = hw_dum[feature_cols]
y = hw_dum.total
linreg = LinearRegression()
linreg.fit(X, y)
list(zip(feature_cols, linreg.coef_))


Out[265]:
[('temp', 9.4424422534193315),
 ('month_2', -246.40126166629125),
 ('month_3', -163.90969444931002),
 ('humidity', -2.7144997426796031)]

In [266]:
# compare original season variable with dummy variables
print(train_test_rmse(['temp', 'month', 'humidity'], hw_dum))
print(train_test_rmse(['temp', 'month_2', 'month', 'humidity'], hw_dum))


246.451640105
240.628565747

Feature engineering

See if you can create the following features:

  • hour: as a single numeric feature (0 through 23)
  • hour: as a categorical feature (use 23 dummy variables)
  • daytime: as a single categorical feature (daytime=1 from 7am to 8pm, and daytime=0 otherwise)

Then, try using each of the three features (on its own) with train_test_rmse to see which one performs the best!


In [267]:
# hour as a numeric feature
hw_dum['hour'] = hw_dum.datetime.dt.hour

In [268]:
# hour as a categorical feature
hour_dummies = pd.get_dummies(hw_dum.hour, prefix='hour')
# hour_dummies.drop(hour_dummies.columns[0], axis=1, inplace=True)
hw_dum = pd.concat([hw_dum, hour_dummies], axis=1)

In [269]:
# daytime as a categorical feature
hw_dum['daytime'] = ((hw_dum.hour > 6) & (hw_dum.hour < 21)).astype(int)

In [270]:
print(train_test_rmse(['hour'], hw_dum),
    train_test_rmse(hw_dum.columns[hw_dum.columns.str.startswith('hour_')], hw_dum)
    ,train_test_rmse(['daytime'], hw_dum))


261.836828231 212.27200657 229.412966447

Comparing linear regression with other models

Advantages of linear regression:

  • Simple to explain
  • Highly interpretable
  • Model training and prediction are fast
  • No tuning is required (excluding regularization)
  • Features don't need scaling
  • Can perform well with a small number of observations
  • Well-understood

Disadvantages of linear regression:

  • Presumes a linear relationship between the features and the response
  • Performance is (generally) not competitive with the best supervised learning methods due to high bias
  • Can't automatically learn feature interactions