Python_Data_Science_第七课

polynomial regression

Not all relationships are linear.
Linear formula: y=mx+b
  This is a "first order" or "first degree" polynomial , as the power of x is 1.
Second order polynomial: y=ax**2 + bx + c
Third order polynomial: y = ax**3 + bx**2 + cx + d
Higher orders polynomial produce more complex curves.

#beware overfitting
Don't use more degrees than you need.
Visualize your data first to see how complex of a curve there might really be.
Visualize the fit - is your curve going out of its way to accomodate outliers?
A high r-squared simply means your curve fits your training data well, but it may not be a good predictor.

code:

#fabricate data
np.random.seed(2)
pageSpeeds = np.random.normal(3.0, 1.0, 1000)
purchaseAmount = np.random.normal(50.0, 10.0, 1000) / pageSpeeds
plt.scatter(pageSpeeds, purchaseAmount)
plt.show()
#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.
x=np.array(pageSpeeds)
y=np.array(purchaseAmount)
p4=np.poly1d(np.polyfit(x,y, 4))

#visualize
xp=np.linspace(0, 7, 100)
plt.scatter(x, y)
plt.plot(xp, p4(xp), c='r')
plt.show()

#measure the r-squared error, 0 is bad, and 1 is good.
from sklearn.metrics import r2_score
r2=r2_score(y, p4(x))
print(r2)
#output will be ,pretty good
0.82937663963

#change the order to 8
In [14]: p4=np.poly1d(np.polyfit(x,y, 8))
    ...:
In [15]: xp=np.linspace(0, 7, 100)
    ...: plt.scatter(x, y)
    ...: plt.plot(xp, p4(xp), c='r')
    ...: plt.show()
    ...:

In [16]: from sklearn.metrics import r2_score
    ...: r2=r2_score(y, p4(x))
    ...: print(r2)
    ...:
#more accurate than order of 4
0.881439566368

#change the order to 1 , this will be linear regression.
p4=np.poly1d(np.polyfit(x,y, 1))
xp=np.linspace(0, 7, 100)
plt.scatter(x, y)
plt.plot(xp, p4(xp), c='r')
plt.show()
from sklearn.metrics import r2_score
r2=r2_score(y, p4(x))
print(r2)
#r-squared is only 0.50
0.502494130455