def test_LeastSquares_R2() :
"""Tests the R2 score method of the Least Squares class
The test is done with a known beta array, comparing results to a known
MSE value.
"""
N = 5
P = 3
x = np.linspace(0,1,N)
random.seed(10)
y = 3*x**2 - 9*x - 2.4*x**5 + 3.1
X = np.zeros(shape=(N,P))
X[:,0] = 1.0
for j in range(1,P) :
X[:,j] = x**j
OLS = LeastSquares(backend='skl')
beta_skl = OLS.fit(X,y)
R2_skl = OLS.R2()
OLS = LeastSquares(backend='manual')
beta_manual = OLS.fit(X,y)
# Ensure the manual and the skl fit both use the exact same beta
# values.
OLS.beta = beta_skl
R2_manual = OLS.R2()
yHat = np.dot(X, beta_skl)
R2_true = 1.0 - np.sum((y - yHat)**2) / np.sum((y - np.mean(y))**2)
# beta:
# 2.98147321428571299151
# -6.48616071428570872826
# -1.66071428571428914012
#
# yHat = beta0 + beta1 x + beta2 x^2
# 2.98147321428571299151
# 1.25613839285714279370
# -0.67678571428571365765
# -2.81729910714285569640
# -5.16540178571428487686
#
# y = 3x^2 - 9x -2.4x^5 + 3.1
# 3.10000000000000008882
# 1.03515625000000000000
# -0.72500000000000008882
# -2.53203124999999973355
# -5.30000000000000071054
#
# R2 = 1.0 - sum(yHat - y)**2 / sum(yHat - mean(y))**2
# 0.99605957942938250227
assert R2_skl == pytest.approx(R2_manual, abs=1e-15)
assert R2_skl == pytest.approx(R2_true, abs=1e-15)
assert R2_manual == pytest.approx(R2_true, abs=1e-15)
def plot_MSE_R2() :
leastSquares = LeastSquares(backend='manual')
N = int(1e4)
x = np.random.rand(N)
y = np.random.rand(N)
x_data = np.zeros(shape=(N,2))
x_data[:,0] = x
x_data[:,1] = y
y_data = np.zeros(shape=(N))
@jit(nopython=True, cache=True)
def computeFrankeValues(x_data, y) :
N = x_data.shape[0]
for i in range(N) :
y[i] = franke(x_data[i,0], x_data[i,1])
computeFrankeValues(x_data, y_data)
p_max = 10
p = [i for i in range(2, p_max+1)]
R2 = [None for i in range(2, p_max+1)]
MSE = [None for i in range(2, p_max+1)]
for degree in p :
designMatrix = DesignMatrix('polynomial2D', degree)
X = designMatrix.getMatrix(x_data)
leastSquares.fit(X, y_data)
_ = leastSquares.predict()
R2[degree-2] = leastSquares.R2()
MSE[degree-2] = leastSquares.MSE()
print(R2[degree-2])
print(MSE[degree-2])
p = np.array(p)
plt.semilogy(p, 1-np.array(R2),'b-o', markersize=3)
plt.rc('text', usetex=True)
plt.xlabel(r"$p$", fontsize=10)
plt.ylabel(r"$1-(R^2$ score$)$", fontsize=10)
plt.subplots_adjust(left=0.2,bottom=0.2)
#plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', 'OLS_R2.png'), transparent=True, bbox_inches='tight')
plt.show()
plt.figure()
ax = plt.gca()
plt.semilogy(p, MSE, 'r-o', markersize=3)
plt.rc('text', usetex=True)
plt.xlabel(r"$p$", fontsize=10)
plt.ylabel(r"MSE", fontsize=10)
plt.subplots_adjust(left=0.2,bottom=0.2)
#plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', 'OLS_MSE.png'), transparent=True, bbox_inches='tight')
plt.show()
def part_a(plotting=False) :
MSE_degree = []
R2_degree = []
betaVariance_degree = []
for degree in [2,3,4,5]: #,6,7,8,9] :
designMatrix = DesignMatrix('polynomial2D', degree)
leastSquares = LeastSquares(backend='manual')
bootstrap = Bootstrap(leastSquares, designMatrix)
N = int(1e4)
x = np.random.rand(N)
y = np.random.rand(N)
x_data = np.zeros(shape=(N,2))
x_data[:,0] = x
x_data[:,1] = y
y_data = np.zeros(shape=(N))
@jit(nopython=True, cache=True)
def computeFrankeValues(x_data, y) :
N = x_data.shape[0]
for i in range(N) :
y[i] = franke(x_data[i,0], x_data[i,1])
computeFrankeValues(x_data, y_data)
bootstrap.resample(x_data, y_data, 1000)
MSE_degree. append(leastSquares.MSE())
R2_degree. append(leastSquares.R2())
betaVariance_degree.append(bootstrap.betaVariance)
if plotting :
print("MSE: ", MSE_degree[-1])
print("R2: ", R2_degree[-1])
print("Beta Variance: ")
for b in betaVariance_degree[-1] : print(b)
print("Beta: ")
for b in leastSquares.beta : print(b)
print(" ")
M = 100
fig = plt.figure()
ax = fig.gca(projection='3d')
x = np.linspace(0, 1, M)
y = np.linspace(0, 1, M)
X, Y = np.meshgrid(x,y)
x_data = np.vstack([X.ravel(), Y.ravel()]).T
# When plotting the Franke function itself, we use these lines.
yy_data = np.zeros(shape=(x_data.data.shape[0]))
computeFrankeValues(x_data, yy_data)
# When plotting the linear regression model:
XX = designMatrix.getMatrix(x_data)
leastSquares.X = XX
y_data = leastSquares.predict()
Z = np.reshape(y_data.T, X.shape)
ZF = np.reshape(yy_data.T, X.shape)
#ax.plot_surface(X,Y,Z,cmap=cm.coolwarm,linewidth=0, antialiased=False)
ax.plot_surface(X,Y,abs(Z-ZF),cmap=cm.coolwarm,linewidth=0, antialiased=False)
ax.set_zlim(-0.10, 1.40)
ax.zaxis.set_major_locator(LinearLocator(5))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))
ax.view_init(30, 45)
#plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', 'franke.png'), transparent=True, bbox_inches='tight')
#plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', 'OLS'+str(degree)+'.png'), transparent=True, bbox_inches='tight')
#plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', 'OLS'+str(degree)+'_diff.png'), transparent=True, bbox_inches='tight')
plt.show()
print("\nMSE :")
print(MSE_degree)
print("\nR2 :")
print(R2_degree)
print("\nσ²(β) :")
print(betaVariance_degree)
return MSE_degree, R2_degree, betaVariance_degree
def fit_franke_noise() :
R2 = []
MSE = []
R2_noise = []
MSE_noise = []
beta_noise = []
betaVariance_noise = []
noise = np.logspace(-4,0,50)
k = 1
@jit(nopython=True, cache=True)
def computeFrankeValues(x_data, y) :
N = x_data.shape[0]
for i in range(N) :
y[i] = franke(x_data[i,0], x_data[i,1])
for eta in noise :
designMatrix = DesignMatrix('polynomial2D', 10)
leastSquares = LeastSquares(backend='manual')
bootstrap = Bootstrap(leastSquares, designMatrix)
N = int(1e5)
x = np.random.rand(N)
y = np.random.rand(N)
x_data = np.zeros(shape=(N,2))
x_data[:,0] = x
x_data[:,1] = y
y_data = np.zeros(shape=(N))
computeFrankeValues(x_data, y_data)
y_data_noise = y_data + eta * np.random.standard_normal(size=N)
bootstrap.resample(x_data, y_data_noise, k)
MSE_noise. append(leastSquares.MSE())
R2_noise. append(leastSquares.R2())
beta_noise. append(bootstrap.beta)
betaVariance_noise.append(bootstrap.betaVariance)
leastSquares.y = y_data
MSE.append(leastSquares.MSE())
R2. append(leastSquares.R2())
"""
betaVariance_noise = np.array(betaVariance_noise)
for beta in betaVariance_noise :
print(beta)
colors = ['b', 'g', 'r', 'c', 'm', 'y', 'k','#1f77b4', '#ff7f0e', '#2ca02c', '#d62728',
'#9467bd', '#8c564b', '#e377c2', '#7f7f7f',
'#bcbd22', '#17becf']
for i in range(6) :
plt.loglog(noise, betaVariance_noise[:,i], colors[i]+'-o', markersize=2)
plt.rc('text', usetex=True)
#plt.rc('font',**{'family':'sans-serif','sans-serif':['Helvetica']})
## for Palatino and other serif fonts use:
#plt.rc('font',**{'family':'serif','serif':['Palatino']}) plt.xlabel(r"$p$", fontsize=16)
plt.xlabel(r"noise scale $\eta$", fontsize=10)
plt.ylabel(r"$ \sigma^2(\beta_j)$", fontsize=10)
plt.legend([r"intercept",
r"$\beta_{x}$",
r"$\beta_{y}$",
r"$\beta_{x^2}$",
r"$\beta_{xy}$",
r"$\beta_{y^2}$"], fontsize=10)
plt.subplots_adjust(left=0.2,bottom=0.2)
#plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', 'beta_variance_OLS_noise.png'), transparent=True, bbox_inches='tight')
#plt.show()
"""
print(R2_noise)
print(1-np.array(R2_noise))
fig, ax1 = plt.subplots()
ax1.loglog(noise, 1-np.array(R2_noise),'k-o',markersize=2)
ax1.loglog(noise, 1-np.array(R2),'k--',markersize=2)
plt.xlabel(r"noise scale $\eta$", fontsize=10)
plt.ylabel(r"$1-R^2$", color='k', fontsize=10)
ax2 = ax1.twinx()
ax2.loglog(noise, np.array(MSE_noise), 'b-o',markersize=2)
ax2.loglog(noise, np.array(MSE), 'b--',markersize=2)
plt.ylabel(r"MSE", color='b', fontsize=10)
plt.subplots_adjust(left=0.2,bottom=0.2,right=0.9)
ax1.set_ylim([0.95*min(min(MSE_noise), min(R2_noise)), 1.05*(max(max(MSE_noise), max(R2_noise)))])
ax2.set_ylim([0.95*min(min(MSE_noise), min(R2_noise)), 1.05*(max(max(MSE_noise), max(R2_noise)))])
ax2.get_yaxis().set_ticks([])
plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', 'R2MSE_OLS_noise.png'), transparent=True, bbox_inches='tight')
plt.show()
def part_b():
R2 = []
MSE = []
R2_noise = []
MSE_noise = []
beta_noise = []
betaVariance_noise = []
noise = np.linspace(0, 1.0, 50)
k = 1
fig, ax1 = plt.subplots()
plt.rc('text', usetex=True)
@jit(nopython=True, cache=True)
def computeFrankeValues(x_data, y):
N = x_data.shape[0]
for i in range(N):
y[i] = franke(x_data[i, 0], x_data[i, 1])
ind = -1
for lambda_ in np.logspace(-2, 0, 3):
ind += 1
MSE_noise = []
for eta in noise:
designMatrix = DesignMatrix('polynomial2D', 10)
if ind == 0:
leastSquares = LeastSquares(backend='manual', method='ols')
else:
leastSquares = LeastSquares(backend='manual', method='ridge')
leastSquares.setLambda(lambda_)
bootstrap = Bootstrap(leastSquares, designMatrix)
N = int(1000)
x = np.random.rand(N)
y = np.random.rand(N)
x_data = np.zeros(shape=(N, 2))
x_data[:, 0] = x
x_data[:, 1] = y
y_data = np.zeros(shape=(N))
computeFrankeValues(x_data, y_data)
y_data_noise = y_data + eta * np.random.standard_normal(size=N)
bootstrap.resample(x_data, y_data_noise, k)
MSE_noise.append(leastSquares.MSE())
R2_noise.append(leastSquares.R2())
beta_noise.append(bootstrap.beta)
betaVariance_noise.append(bootstrap.betaVariance)
# Different noise, test data
N = int(1000)
x = np.random.rand(N)
y = np.random.rand(N)
x_data = np.zeros(shape=(N, 2))
x_data[:, 0] = x
x_data[:, 1] = y
y_data = np.zeros(shape=(N))
computeFrankeValues(x_data, y_data)
y_data_noise = y_data + eta * np.random.standard_normal(size=N)
X = designMatrix.getMatrix(x_data)
leastSquares.X = X
leastSquares.predict()
leastSquares.y = y_data_noise
MSE.append(leastSquares.MSE())
R2.append(leastSquares.R2())
colors = ['b', 'g', 'r', 'c', 'm', 'y', 'k']
if ind == 0:
ax1.loglog(noise,
np.array(MSE_noise),
colors[ind] + '--',
markersize=1,
label=r"OLS")
else:
ax1.loglog(noise,
np.array(MSE_noise),
colors[ind] + '-',
markersize=1,
label=r"$\lambda=10^{%d}$" % (int(np.log10(lambda_))))
plt.ylabel(r"MSE", fontsize=10)
plt.xlabel(r"noise scale $\eta$", fontsize=10)
plt.subplots_adjust(left=0.2, bottom=0.2)
#ax1.set_ylim([0.95*min(min(MSE_noise), min(R2_noise)), 1.05*(max(max(MSE_noise), max(R2_noise)))])
ax1.legend()
#plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', 'MSE_ridge_noise.png'), transparent=True, bbox_inches='tight')
plt.show()
def plot_beta_ridge():
beta = []
betaVariance = []
MSE = []
R2 = []
k = 10000
fig, ax1 = plt.subplots()
plt.rc('text', usetex=True)
@jit(nopython=True, cache=True)
def computeFrankeValues(x_data, y):
N = x_data.shape[0]
for i in range(N):
y[i] = franke(x_data[i, 0], x_data[i, 1])
ind = -1
lam = np.logspace(-3, 5, 20)
for lambda_ in lam:
if ind == 0:
leastSquares = LeastSquares(backend='manual', method='ols')
else:
leastSquares = LeastSquares(backend='manual', method='ridge')
designMatrix = DesignMatrix('polynomial2D', 3)
bootstrap = Bootstrap(leastSquares, designMatrix)
leastSquares.setLambda(lambda_)
ind += 1
N = int(1e4)
x = np.random.rand(N)
y = np.random.rand(N)
x_data = np.zeros(shape=(N, 2))
x_data[:, 0] = x
x_data[:, 1] = y
y_data = np.zeros(shape=(N))
computeFrankeValues(x_data, y_data)
eta = 1.0
y_data_noise = y_data + eta * np.random.standard_normal(size=N)
bootstrap.resample(x_data, y_data_noise, k)
MSE.append(leastSquares.MSE())
R2.append(leastSquares.R2())
beta.append(bootstrap.beta)
betaVariance.append(bootstrap.betaVariance)
leastSquares.y = y_data
MSE.append(leastSquares.MSE())
R2.append(leastSquares.R2())
beta = np.array(beta)
betaVariance = np.array(betaVariance)
monomial = [
'1', 'x', 'y', 'x^2', 'xy', 'y^2', 'x^3', 'x^2y', 'xy^2', 'y^3'
]
colors = [
'#1f77b4', '#ff7f0e', '#2ca02c', '#d62728', '#9467bd', '#8c564b',
'#e377c2', '#7f7f7f', '#bcbd22', '#17becf'
]
for i in range(10):
plt.errorbar(lam[1:],
beta[1:, i],
yerr=2 * betaVariance[1:, i],
fmt='-o',
markersize=2,
linewidth=1,
color=colors[i],
elinewidth=0.5,
capsize=2,
capthick=0.5,
label=r"$\beta_{%s}$" % (monomial[i]))
plt.rc('text', usetex=True)
plt.ylabel(r"$\beta_j$", fontsize=10)
plt.xlabel(r"shrinkage parameter $\lambda$", fontsize=10)
plt.subplots_adjust(left=0.2, bottom=0.2)
plt.legend(fontsize=8)
for i in range(10):
plt.errorbar(1e-3,
beta[0, i],
yerr=2 * betaVariance[0, i],
fmt='-o',
markersize=2,
linewidth=1,
color=colors[i],
elinewidth=0.5,
capsize=2,
capthick=0.5)
fig.gca().set_xscale('log')
#plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', 'beta_ridge.png'), transparent=True, bbox_inches='tight')
plt.show()