Key Word(s): Regression, kNN, Scikit-Learn

# Lecture 4: Introduction to Regression¶

### Data Science 1: CS 109A/STAT 121A/AC 209A/ E 109A Instructors: Pavlos Protopapas, Kevin Rader, Rahul Dave, Margo Levine

#### Harvard University Fall 2017

In [1]:
import pandas as pd
import sys
import numpy as np
import scipy as sp
import matplotlib
import matplotlib.pyplot as plt
import seaborn as sns
import sklearn as sk
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn.neighbors import KNeighborsRegressor
sns.set(style="ticks")
%matplotlib inline

import seaborn as sns
pd.set_option('display.width', 1500)
pd.set_option('display.max_columns', 100)

sns.set_context('poster')
# import matplotlib.pylab as pylab
# params = {'legend.fontsize': 'large',
#           'figure.figsize': (15, 5),
#          'axes.labelsize': 'large',
#          'axes.titlesize':'large',
#          'xtick.labelsize':'large',
#          'ytick.labelsize':'large'}
# pylab.rcParams.update(params)}


## Step #1: Load and Explore Data¶

In [2]:
# GET THE FULL DATA SET FROM
nyc_cab_df = pd.read_csv('data/nyc_car_hire_data.csv', low_memory=False) # DO WE NEED THE LOW MEMORY

In [3]:
nyc_cab_df.shape

Out[3]:
(1873671, 30)
In [4]:
nyc_cab_df.head()

Out[4]:
AWND Base Day Dropoff_latitude Dropoff_longitude Ehail_fee Extra Fare_amount Lpep_dropoff_datetime MTA_tax PRCP Passenger_count Payment_type Pickup_latitude Pickup_longitude RateCodeID SNOW SNWD Store_and_fwd_flag TMAX TMIN Tip_amount Tolls_amount Total_amount Trip_distance Trip_type Type VendorID lpep_pickup_datetime Trip Length (min)
0 4.7 B02512 1 NaN NaN NaN NaN 33.863498 2014-04-01 00:24:00 NaN 0.0 NaN NaN 40.7690 -73.9549 NaN 0.0 0.0 NaN 60 39 NaN NaN NaN 4.083561 NaN 1 NaN 2014-04-01 00:11:00 13.0
1 4.7 B02512 1 NaN NaN NaN NaN 19.022892 2014-04-01 00:29:00 NaN 0.0 NaN NaN 40.7267 -74.0345 NaN 0.0 0.0 NaN 60 39 NaN NaN NaN 3.605694 NaN 1 NaN 2014-04-01 00:17:00 12.0
2 4.7 B02512 1 NaN NaN NaN NaN 25.498981 2014-04-01 00:34:00 NaN 0.0 NaN NaN 40.7316 -73.9873 NaN 0.0 0.0 NaN 60 39 NaN NaN NaN 4.221763 NaN 1 NaN 2014-04-01 00:21:00 13.0
3 4.7 B02512 1 NaN NaN NaN NaN 28.024628 2014-04-01 00:39:00 NaN 0.0 NaN NaN 40.7588 -73.9776 NaN 0.0 0.0 NaN 60 39 NaN NaN NaN 2.955510 NaN 1 NaN 2014-04-01 00:28:00 11.0
4 4.7 B02512 1 NaN NaN NaN NaN 12.083589 2014-04-01 00:40:00 NaN 0.0 NaN NaN 40.7594 -73.9722 NaN 0.0 0.0 NaN 60 39 NaN NaN NaN 1.922292 NaN 1 NaN 2014-04-01 00:33:00 7.0
In [5]:
nyc_cab_df.describe()

Out[5]:
AWND Day Dropoff_latitude Dropoff_longitude Ehail_fee Extra Fare_amount MTA_tax PRCP Passenger_count Payment_type Pickup_latitude Pickup_longitude RateCodeID SNOW SNWD TMAX TMIN Tip_amount Tolls_amount Total_amount Trip_distance Trip_type Type VendorID Trip Length (min)
count 1.873671e+06 1.873671e+06 1.309155e+06 1.309155e+06 0.0 1.309155e+06 1.873671e+06 1.309155e+06 1.873671e+06 1.309155e+06 1.309155e+06 1.873671e+06 1.873671e+06 1.309155e+06 1873671.0 1873671.0 1.873671e+06 1.873671e+06 1.309155e+06 1.309155e+06 1.309155e+06 1.873671e+06 1.225061e+06 1.873671e+06 1.309155e+06 1.873671e+06
mean 6.349472e+00 1.573272e+01 4.069035e+01 -7.380999e+01 NaN 3.718944e-01 1.480038e+01 4.885869e-01 3.277113e-01 1.424879e+00 1.613783e+00 4.070361e+01 -7.385354e+01 1.096059e+00 0.0 0.0 6.094607e+01 4.334624e+01 1.013024e+00 1.069806e-01 1.432070e+01 2.917737e+00 1.020388e+00 3.012888e-01 1.781165e+00 1.132344e+01
std 1.782350e+00 8.786914e+00 1.653301e+00 2.997829e+00 NaN 3.705219e-01 9.899925e+00 7.478954e-02 1.033659e+00 1.129275e+00 5.014897e-01 1.435388e+00 2.603334e+00 6.177816e-01 0.0 0.0 8.434375e+00 6.049009e+00 2.219229e+00 8.479594e-01 1.138924e+01 2.529476e+00 1.413220e-01 4.588180e-01 4.134567e-01 7.875921e+00
min 2.000000e+00 1.000000e+00 0.000000e+00 -1.219265e+02 NaN 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00 1.000000e+00 -4.075827e+01 -1.219263e+02 1.000000e+00 0.0 0.0 4.700000e+01 3.100000e+01 0.000000e+00 0.000000e+00 0.000000e+00 -3.483963e-01 1.000000e+00 0.000000e+00 1.000000e+00 0.000000e+00
25% 4.900000e+00 8.000000e+00 4.071491e+01 -7.396558e+01 NaN 0.000000e+00 7.500000e+00 5.000000e-01 0.000000e+00 1.000000e+00 1.000000e+00 4.071750e+01 -7.397948e+01 1.000000e+00 0.0 0.0 5.400000e+01 4.000000e+01 0.000000e+00 0.000000e+00 7.500000e+00 1.340000e+00 1.000000e+00 0.000000e+00 2.000000e+00 6.000000e+00
50% 6.000000e+00 1.600000e+01 4.075613e+01 -7.394200e+01 NaN 5.000000e-01 1.300000e+01 5.000000e-01 0.000000e+00 1.000000e+00 2.000000e+00 4.074918e+01 -7.395300e+01 1.000000e+00 0.0 0.0 6.100000e+01 4.300000e+01 0.000000e+00 0.000000e+00 1.100000e+01 2.330000e+00 1.000000e+00 0.000000e+00 2.000000e+00 1.000000e+01
75% 8.300000e+00 2.400000e+01 4.080378e+01 -7.390412e+01 NaN 5.000000e-01 2.037272e+01 5.000000e-01 7.000000e-02 1.000000e+00 2.000000e+00 4.079704e+01 -7.392220e+01 1.000000e+00 0.0 0.0 6.700000e+01 4.600000e+01 1.500000e+00 0.000000e+00 1.750000e+01 3.640000e+00 1.000000e+00 1.000000e+00 2.000000e+00 1.400000e+01
max 9.400000e+00 3.000000e+01 4.289498e+01 0.000000e+00 NaN 3.000000e+01 3.351000e+03 1.500000e+00 4.970000e+00 9.000000e+00 5.000000e+00 4.211660e+01 7.393764e+01 9.900000e+01 0.0 0.0 7.700000e+01 5.900000e+01 5.000000e+02 1.500000e+02 3.352500e+03 2.313600e+02 2.000000e+00 1.000000e+00 2.000000e+00 5.900000e+01
In [6]:
nyc_cab_df.Passenger_count.hist()

Out[6]:
In [7]:
nyc_cab_df.dtypes

Out[7]:
AWND                     float64
Base                      object
Day                        int64
Dropoff_latitude         float64
Dropoff_longitude        float64
Ehail_fee                float64
Extra                    float64
Fare_amount              float64
Lpep_dropoff_datetime     object
MTA_tax                  float64
PRCP                     float64
Passenger_count          float64
Payment_type             float64
Pickup_latitude          float64
Pickup_longitude         float64
RateCodeID               float64
SNOW                     float64
SNWD                     float64
Store_and_fwd_flag        object
TMAX                       int64
TMIN                       int64
Tip_amount               float64
Tolls_amount             float64
Total_amount             float64
Trip_distance            float64
Trip_type                float64
Type                       int64
VendorID                 float64
lpep_pickup_datetime      object
Trip Length (min)        float64
dtype: object
In [8]:
nyc_cab_df.columns

Out[8]:
Index(['AWND', 'Base', 'Day', 'Dropoff_latitude', 'Dropoff_longitude', 'Ehail_fee', 'Extra', 'Fare_amount', 'Lpep_dropoff_datetime', 'MTA_tax', 'PRCP', 'Passenger_count', 'Payment_type', 'Pickup_latitude', 'Pickup_longitude', 'RateCodeID', 'SNOW', 'SNWD', 'Store_and_fwd_flag', 'TMAX', 'TMIN', 'Tip_amount', 'Tolls_amount', 'Total_amount', 'Trip_distance', 'Trip_type', 'Type', 'VendorID', 'lpep_pickup_datetime', 'Trip Length (min)'], dtype='object')

## Clean data¶

How would you clean the data?

• Extreme/outliers
• Impute missing values
• Drop some columns?
• Drop rows with problematic values
In [9]:
nyc_cab_sample = nyc_cab_df.sample(n=1000, random_state=6)
fares = nyc_cab_sample['Fare_amount'].values
trip_lengths = nyc_cab_sample[['Trip Length (min)']].values

In [10]:
fig, ax = plt.subplots(1, 1, figsize=(10, 7))

ax.scatter(trip_lengths, fares, color='gray', alpha=0.1, s=20)

ax.set_xlabel('Trip Length (min)')
ax.set_ylabel('Trip Fare ($)') ax.set_title('NYC Care Hire Data:\n Trip Duration vs Fare Scatter Plot')  Out[10]: ## Step #2: Modeling the Data¶ In [11]: #nyc_cab_sample = nyc_cab_df.sample(n=1000, random_state=1) y = nyc_cab_sample['Fare_amount'].values X = nyc_cab_sample[['Trip Length (min)']].values X.shape  Out[11]: (1000, 1) In [26]: regression = LinearRegression(fit_intercept=True) regression.fit(X, y) # this is the estimated statistical model (f_hat) using the predicted coefficients regression_line = lambda x: regression.intercept_ + regression.coef_ * x print('The estimated equation of the regression line is: {} + {} * x'.format(regression.intercept_, regression.coef_[0]))  The estimated equation of the regression line is: 6.429017573516136 + 0.7339670486476195 * x  In [14]: fig, ax = plt.subplots(1, 1, figsize=(10, 10)) x_vals = np.linspace(0, 70, 100) ax.plot(x_vals, regression_line(x_vals), color='red', linewidth=1.0, label='regression line') ax.scatter(X, y, color='gray', alpha=0.2, label='data') ax.set_xlabel('Trip Length (min)') ax.set_ylabel('Trip Fare ($)')
ax.set_title('NYC Care Hire Data:\n Trip Duration vs Fare Scatter Plot')
ax.legend(loc='best')

Out[14]:

## Step #3: Evaluate and Interpret the Model¶

### 1. Train vs Test Error¶

In [15]:
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33, random_state=6)

train_MSE= np.mean((y_train - regression.predict(X_train))**2)
test_MSE= np.mean((y_test - regression.predict(X_test))**2)
print('The train MSE is {}, the test MSE is {}'.format(train_MSE, test_MSE))

The train MSE is 48.59886786981504, the test MSE is 52.199666372395114


Why is the MSE for test higher?

### 2. Uncertainty in the Model Parameter Estimates¶

In [16]:
def find_regression_params(regression_model, samples):
# randomly select number of samples from the big data
nyc_cab_sample = nyc_cab_df.sample(n=samples)

y = nyc_cab_sample['Fare_amount'].values
X = nyc_cab_sample[['Trip Length (min)']].values

regression_model.fit(X, y)

return regression_model.intercept_, regression_model.coef_[0]

In [17]:
regression_model = LinearRegression(fit_intercept=True)

total_draws = 500
samples = 1000
regression_params = []

for i in range(total_draws):
if i % 10 == 0:
out = i * 1. / total_draws * 100
sys.stdout.write("\r%d%%" % out)
sys.stdout.flush()

regression_params.append(find_regression_params(regression_model, samples))

sys.stdout.write("\r%d%%" % 100)
regression_params = np.array(regression_params)

100%
In [18]:
fig, ax = plt.subplots(1, 2, figsize=(10, 5))

ax[0].hist(regression_params[:, 0], bins=50, color='blue', edgecolor='white', linewidth=1, alpha=0.2)
ax[0].axvline(x=regression_params[:, 0].mean(), color='red', label='mean = {0:.2f}'.format(regression_params[:, 0].mean()))
ax[0].axvline(x=regression_params[:, 0].mean() - 2 * regression_params[:, 0].std(), color='green', linestyle='--', label='std = {0:.2f}'.format(regression_params[:, 0].std()))
ax[0].axvline(x=regression_params[:, 0].mean() + 2 * regression_params[:, 0].std(), color='green', linestyle='--')

ax[0].set_xlabel('Intercept')
ax[0].set_ylabel('Frequency')
ax[0].set_title('Histogram of Estimates of Intercept')
ax[0].legend(loc='best')

ax[1].hist(regression_params[:, 1], bins=50, color='gray', edgecolor='white', linewidth=1, alpha=0.2)
ax[1].axvline(x=regression_params[:, 1].mean(), color='red', label='mean = {0:.2f}'.format(regression_params[:, 1].mean()))
ax[1].axvline(x=regression_params[:, 1].mean() - 2 * regression_params[:, 1].std(), color='green', linestyle='--', label='std = {0:.2f}'.format(regression_params[:, 1].std()))
ax[1].axvline(x=regression_params[:, 1].mean() + 2 * regression_params[:, 1].std(), color='green', linestyle='--')

ax[1].set_xlabel('Slope')
ax[1].set_ylabel('Frequency')
ax[1].set_title('Histogram of Estimates of Slope')
ax[1].legend(loc='best')

plt.tight_layout()


### 3. The Effect of Sample Size on Uncertainty¶

In [19]:
regression_model = LinearRegression(fit_intercept=True)

def compute_SE(total_draws, samples, regression_model):

regression_params = []

for i in range(total_draws):
regression_params.append(find_regression_params(regression_model, samples))

regression_params = np.array(regression_params)
return np.std(regression_params[:, 0]), np.std(regression_params[:, 1])

total_draws = 100
samples = range(100, 10001, 900)
ses = []

for i in range(len(samples)):
out = i * 1. / len(samples) * 100
sys.stdout.write("\r%d%%" % out)
sys.stdout.flush()
ses.append(compute_SE(total_draws, samples[i], regression_model))

sys.stdout.write("\r%d%%" % 100)

100%
In [20]:
ses = np.array(ses)

fig, ax = plt.subplots(1, 1, figsize=(10, 5))

ax.plot(samples, ses[:, 0], color='teal', label='SE of intercept')
ax.plot(samples, ses[:, 1], color='brown', label='SE of slope')

ax.set_xlabel('Number of Samples')
ax.set_ylabel('SE')
ax.set_title('Uncertainty in Parameter Estimates vs Sample Size')
ax.legend(loc='best')

Out[20]:

## Step 2: Consider a Different Model for the Data - kNN¶

In [21]:
nyc_cab_sample = nyc_cab_df.sample(n=1000, random_state=1)

y = nyc_cab_sample['Fare_amount'].values
X = nyc_cab_sample[['Trip Length (min)']].values

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33, random_state=0)

In [22]:
knn_model = KNeighborsRegressor(n_neighbors=2)
knn_model.fit(X_train, y_train)

Out[22]:
KNeighborsRegressor(algorithm='auto', leaf_size=30, metric='minkowski',
metric_params=None, n_jobs=1, n_neighbors=2, p=2,
weights='uniform')
In [23]:
fig, ax = plt.subplots(1, 2, figsize=(15, 8))
ax[0].scatter(X_train, knn_model.predict(X_train), color='red', alpha=0.1, label='knn model')
ax[0].scatter(X_train, y_train, color='gray', alpha=0.1, label='training data')

ax[0].set_xlabel('Trip Length (min)')
ax[0].set_ylabel('Trip Fare ($)') ax[0].set_title('NYC Care Hire Data:\n Trip Duration vs Fare Scatter Plot') ax[0].legend(loc='best') ax[1].scatter(X_test, knn_model.predict(X_test), color='blue', alpha=0.1, label='knn model') ax[1].scatter(X_test, y_test, color='gray', alpha=0.1, label='testing data') ax[1].set_xlabel('Trip Length (min)') ax[1].set_ylabel('Trip Fare ($)')
ax[1].set_title('NYC Care Hire Data:\n Trip Duration vs Fare Scatter Plot')
ax[1].legend(loc='best')

Out[23]:
In [24]:
train_MSE= np.mean((y_train - knn_model.predict(X_train))**2)
test_MSE= np.mean((y_test - knn_model.predict(X_test))**2)

print('The train MSE is {}, the test MSE is {}'.format(train_MSE, test_MSE))

The train MSE is 70.9224876379618, the test MSE is 77.62695077723178


## Step #3: Evaluate and Interpret the kNN Model¶

In [25]:
train_MSEs = []
test_MSEs = []

for k in range(1, 100):
sys.stdout.write("\r%d%%" % k)
sys.stdout.flush()

knn_model = KNeighborsRegressor(n_neighbors=k)
knn_model.fit(X_train, y_train)
train_MSEs.append(np.mean((y_train - knn_model.predict(X_train))**2))
test_MSEs.append(np.mean((y_test - knn_model.predict(X_test))**2))

sys.stdout.write("\r%d%%" % 100)

100%
In [ ]:
fig, ax = plt.subplots(1, 1, figsize=(10, 10))

ax.plot(range(1, 100), train_MSEs, color='red', linewidth=1.0, label='train MSE')
ax.plot(range(1, 100), test_MSEs, color='blue', linewidth=1.0, label='test MSE')

ax.set_xlabel('k number of neighbors')
ax.set_ylabel('MSE')
ax.set_title('Train and Test MSE of the kNN model vs k')
ax.legend(loc='best')