What if your data doesn't look linear at all? Let's look at some more realistic-looking page speed / purchase data:
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%matplotlib inline
from pylab import *
import numpy as np
np.random.seed(2)
pageSpeeds = np.random.normal(3.0, 1.0, 1000)
purchaseAmount = np.random.normal(50.0, 10.0, 1000) / pageSpeeds
scatter(pageSpeeds, purchaseAmount)
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numpy has a handy polyfit function we can use, to let us construct an nth-degree polynomial model of our data that minimizes squared error. Let's try it with a 4th degree polynomial:
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x = np.array(pageSpeeds)
y = np.array(purchaseAmount)
p4 = np.poly1d(np.polyfit(x, y, 4))
We'll visualize our original scatter plot, together with a plot of our predicted values using the polynomial for page speed times ranging from 0-7 seconds:
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import matplotlib.pyplot as plt
xp = np.linspace(0, 7, 100)
plt.scatter(x, y)
plt.plot(xp, p4(xp), c='r')
plt.show()
Looks pretty good! Let's measure the r-squared error:
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from sklearn.metrics import r2_score
r2 = r2_score(y, p4(x))
print(r2)
Try different polynomial orders. Can you get a better fit with higher orders? Do you start to see overfitting, even though the r-squared score looks good for this particular data set?
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