Python_data_science_第七课

polynomial regression

1. 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.

2. 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.

3. 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