def visualize_beta():
L = 10
nn = pickle.load(open('nn5_final.p', 'rb'))
#for w in nn.weights[:1] :
# with np.printoptions(precision=2, suppress=True) :
# print(w.reshape((-1,L)))
#plt.imshow(nn.weights[0].reshape((L,-1)))
#plt.show()
N = 5000
training_fraction = 0.4
ising = Ising(L, N)
#X, y = ising.generateTrainingData1D()
D, ry = ising.generateDesignMatrix1D()
#y /= L
ols = LeastSquares(method='ols', backend='manual')
ols.setLambda(0.1)
ols.fit(D, ry)
print(ising.states.shape)
#W = nn.weights[0].reshape((-1,L))*nn.weights[1]
W = ols.beta.reshape((-1, L))
J = ising.J
for i in range(10):
row = ising.states[i, :]
des = D[i, :]
E = ry[i]
print(row.shape)
row = np.expand_dims(row, 1)
print("s W s:", row.T @ W @ row)
print("s (W+W')/2 s:", row.T @ (W + W.T) / 2 @ row)
print("s J s:", row.T @ J @ row)
#print("D w: ", W.T @ des * nn.weights[1])
#print("pred: ", nn.predict(np.expand_dims(des.T,1)))
print("E: ", E)
print("")
for i in range(N):
row = ising.states[i, :]
des = D[i, :]
E = ry[i]
atol = 1e-14
rtol = 1e-14
#assert np.allclose(row.T @ W @ row, row.T @ (W+W.T)/2 @ row, atol=atol, rtol=rtol)
#assert np.allclose(row.T @ (W+W.T)/2 @ row, row.T @ J @ row, atol=atol, rtol=rtol)
with np.printoptions(precision=2, suppress=True):
for a in np.linalg.eig(W):
print(a)
with np.printoptions(precision=2, suppress=True):
print("det=", np.linalg.det(W))
print("cond=", np.linalg.cond(W))
print("")
print("J:\n:", J)
print("W+W'/2\n", (W + W.T) / 2)
print("J+J'/2\n", (J + J.T) / 2)
print("Tr D:", np.sum(np.diag((W + W.T) / 2)))
def part_e(plotting=False):
x_train, y_train, x_test, y_test = real_data(file_number=2, plotting=False)
for method in ['ols', 'ridge', 'lasso']:
designMatrix = DesignMatrix('polynomial2D', 10)
leastSquares = LeastSquares(backend='manual', method=method)
leastSquares.setLambda(1e-3)
if method == 'lasso':
leastSquares.setLambda(1e-4)
X = designMatrix.getMatrix(x_train)
leastSquares.fit(X, y_train)
X_test = designMatrix.getMatrix(x_test)
leastSquares.predict()
leastSquares.y = y_test
print(leastSquares.MSE())
if plotting:
x = np.linspace(0, 1, 60)
y = np.copy(x)
XX, YY = np.meshgrid(x, y)
ZZ = np.reshape(leastSquares.yHat, XX.shape)
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(XX,
YY,
ZZ,
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 + 90)
#plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', method+'terrain.png'), transparent=True, bbox_inches='tight')
plt.show()
if plotting:
x = np.linspace(0, 1, 60)
y = np.copy(x)
XX, YY = np.meshgrid(x, y)
ZZ = np.reshape(y_test, XX.shape)
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(XX,
YY,
ZZ,
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 + 90)
#plt.savefig(os.path.join(os.path.dirname(__file__), 'figures', 'test_terrain.png'), transparent=True, bbox_inches='tight')
plt.show()
def test_LeastSquares_fit_lasso() :
"""Tests the fit method of the Least Squares class with method='lasso'
"""
N = 500
P = 5
x = np.linspace(0,1,N)
y = 2.0 + x + x**2
X = np.zeros(shape=(N,P))
X[:,0] = 1.0
for j in range(1,P) :
X[:,j] = x**j
lasso = LeastSquares(method='lasso', backend='skl')
lasso.setLambda(0.01)
beta_lasso = lasso.fit(X,y)
# Make sure lasso regression zeroes out x*3 and x**4 beta terms.
assert beta_lasso[-2:] == pytest.approx(np.zeros(2), abs=1e-15)
def test_LeastSquares_fit_ridge() :
"""Tests the fit method of the Least Squares class with method='ridge'
"""
N = 5
P = 5
x = np.linspace(0,1,N)
y = x + x**2 - (1.0 - x)**5
X = np.zeros(shape=(N,P))
X[:,0] = 1.0
for j in range(1,P) :
X[:,j] = x**j
OLS = LeastSquares(method='ols', backend='manual')
beta_ols = OLS.fit(X,y)
OLS = LeastSquares(method='ridge', backend='manual')
OLS.setLambda(0.0)
beta_lambda0 = OLS.fit(X,y)
assert beta_lambda0 == pytest.approx(beta_ols, abs=1e-15)
# Make sure the skl and the manual backends give the same result
SKL = LeastSquares(method='ridge', backend='skl')
SKL.setLambda(0.0)
beta_skl = SKL.fit(X,y)
assert beta_lambda0 == pytest.approx(beta_skl, abs=1e-10)
SKL.setLambda = 0.5
OLS.setLambda = 0.5
beta_skl = SKL.fit(X,y)
beta_lambda0 = OLS.fit(X,y)
print(beta_lambda0)
print(beta_skl)
assert beta_lambda0 == pytest.approx(beta_skl, abs=1e-10)
if __name__ == '__main__':
Degree = 1 + np.arange(20)
train_MSE = []
test_MSE = []
print("#########################################################")
for degree in Degree:
degree = int(degree)
designMatrix = DesignMatrix('polynomial2D', degree)
leastSquares = LeastSquares(method="ridge", backend='manual')
Lambda = 4
leastSquares.setLambda(Lambda)
crossvalidation = CrossValidation(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))
X = designMatrix.getMatrix(x_data)
XT_X = np.dot(X.T, X)
print("det(X^T*X): %g" %
(np.linalg.det(XT_X + Lambda * np.eye(XT_X.shape[0]))))
def part_e_2():
x_train, y_train, x_test, y_test = real_data(file_number=2, plotting=False)
degree = 10
designMatrix = DesignMatrix('polynomial2D', degree)
leastSquares = LeastSquares(backend='manual', method='ols')
X = designMatrix.getMatrix(x_train)
leastSquares.fit(X, y_train)
X_test = designMatrix.getMatrix(x_test)
leastSquares.predict()
leastSquares.y = y_test
ols_MSE = leastSquares.MSE()
print(ols_MSE)
ols_MSE = np.array([ols_MSE, ols_MSE])
ols_lambda = np.array([1e-5, 1])
ridge_lambda = np.logspace(-5, 0, 20)
ridge_MSE = []
for lambda_ in ridge_lambda:
print("ridge " + str(lambda_))
designMatrix = DesignMatrix('polynomial2D', degree)
leastSquares = LeastSquares(backend='manual', method='ridge')
leastSquares.setLambda(lambda_)
X = designMatrix.getMatrix(x_train)
leastSquares.fit(X, y_train)
X_test = designMatrix.getMatrix(x_test)
leastSquares.predict()
leastSquares.y = y_test
ridge_MSE.append(leastSquares.MSE())
print(leastSquares.MSE())
ridge_MSE = np.array(ridge_MSE)
lasso_lambda = np.logspace(-4, 0, 20)
lasso_MSE = []
for lambda_ in lasso_lambda:
print("lasso " + str(lambda_))
designMatrix = DesignMatrix('polynomial2D', degree)
leastSquares = LeastSquares(backend='manual', method='lasso')
leastSquares.setLambda(lambda_)
X = designMatrix.getMatrix(x_train)
leastSquares.fit(X, y_train)
X_test = designMatrix.getMatrix(x_test)
leastSquares.predict()
leastSquares.y = y_test
lasso_MSE.append(leastSquares.MSE())
lasso_MSE = np.array(ridge_MSE)
########################################################
plt.rc('text', usetex=True)
plt.loglog(ols_lambda,
ols_MSE,
'k--o',
markersize=1,
linewidth=1,
label=r'OLS')
plt.loglog(ridge_lambda,
ridge_MSE,
'r-o',
markersize=1,
linewidth=1,
label=r'Ridge')
plt.loglog(lasso_lambda,
lasso_MSE,
'b-o',
markersize=1,
linewidth=1,
label=r'Lasso')
plt.xlabel(r"$\lambda$", fontsize=10)
plt.ylabel(r"MSE", fontsize=10)
#plt.subplots_adjust(left=0.2,bottom=0.2)
plt.legend(fontsize=10)
plt.savefig(os.path.join(os.path.dirname(__file__), 'figures',
'lambda_terrain.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()
def R2_versus_lasso():
L = 3
N = 10000
training_fraction = 0.4
ising = Ising(L, N)
D, ry = ising.generateDesignMatrix1D()
X, y = ising.generateTrainingData1D()
y /= L
D_train = D[int(training_fraction * N):, :]
ry_train = ry[int(training_fraction * N):]
D_validation = D[:int(training_fraction * N), :]
ry_validation = ry[:int(training_fraction * N)]
lasso = LeastSquares(method='lasso', backend='skl')
lasso.setLambda(1e-2)
lasso.fit(D_train, ry_train)
lasso.y = ry_validation
lasso_R2 = sklearn.metrics.mean_squared_error(
ry_validation / L,
lasso.predict(D_validation) / L)
n_samples, n_features = X.shape
nn = NeuralNetwork(inputs=L * L,
neurons=L,
outputs=1,
activations='identity',
cost='mse',
silent=False)
nn.addLayer(neurons=1)
nn.addOutputLayer(activations='identity')
validation_skip = 100
epochs = 50000
nn.fit(D.T,
ry,
shuffle=True,
batch_size=2000,
validation_fraction=1 - training_fraction,
learning_rate=0.0001,
verbose=False,
silent=False,
epochs=epochs,
validation_skip=validation_skip,
optimizer='adam')
plt.rc('text', usetex=True)
validation_loss = nn.validation_loss_improving
validation_ep = np.linspace(0, epochs, len(nn.validation_loss_improving))
plt.semilogy(validation_ep, validation_loss, 'r-', label=r'NN')
plt.semilogy([0, epochs],
np.array([lasso_R2, lasso_R2]),
'k--',
label=r'Lasso')
plt.xlabel(r'Epoch', fontsize=10)
plt.ylabel(r'Mean squared error', fontsize=10)
plt.legend(fontsize=10)
plt.xlim((0, epochs))
ax = plt.gca()
ymin, ymax = ax.get_ylim()
if ymin > pow(10, -5):
ymin = pow(10, -5)
#plt.ylim((ymin,ymax))
plt.savefig(os.path.join(os.path.dirname(__file__), 'figures',
'NN_compare_lasso.png'),
transparent=True,
bbox_inches='tight')