%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt

xs = np.linspace(0.0, 3.0, 31)
ys = np.sin(xs) + np.random.normal(0.0, 0.1, len(xs))

xp = np.linspace(-0.1, 3.1, 100)
plt.plot(xs,ys,"o", xp, np.sin(xp), "-")
plt.ylim(-0.2, 1.5)

xp = np.linspace(-0.1, 3.1, 1000)
plt.plot(xs,ys, "o", xp, np.poly1d(np.polyfit(xs,ys,1))(xp), "-", xp, np.poly1d(np.polyfit(xs,ys,2))(xp), "--")
plt.ylim(-0.2, 1.2)

plt.plot(xs,ys, "o", xp, np.poly1d(np.polyfit(xs,ys,30))(xp), "-")
plt.xlim(-0.2, 3.1)
plt.ylim(-1, 2)

def fit(i):
    return np.poly1d(np.polyfit(xs,ys,i))
def s2(i):
    return ((ys - fit(i)(xs))**2).sum()/len(xs)
degs = np.array([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15])
plt.plot(degs, np.vectorize(s2)(degs))
plt.yscale('log')

def aic(i):
    return len(xs)*np.log(s2(i))+2*(i+1)
aics = np.vectorize(aic)(degs)
plt.plot(degs, aics)



X2 = np.polynomial.polynomial.polyvander(xs, 2)
from sklearn.linear_model import LinearRegression
lr = LinearRegression(normalize=True)
lr.fit(X2, ys)
predicted = lr.predict(X2)
plt.plot(xs, ys, "o", xs, predicted, "-")