update
This commit is contained in:
Morten Hjorth-Jensen
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from mpl_toolkits.mplot3d import Axes3D
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from matplotlib import cm
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from matplotlib.ticker import LinearLocator, FormatStrFormatter
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import numpy as np
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import pandas as pd
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import matplotlib.pyplot as plt
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from sklearn.model_selection import train_test_split
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from sklearn import linear_model
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def MSE(y_data,y_model):
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n = np.size(y_model)
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return np.sum((y_data-y_model)**2)/n
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def FrankeFunction(x,y):
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term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
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term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
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term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
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term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
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return term1 + term2 + term3 + term4
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def create_X(x, y, n ):
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if len(x.shape) > 1:
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x = np.ravel(x)
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y = np.ravel(y)
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N = len(x)
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l = int((n+1)*(n+2)/2) # Number of elements in beta
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X = np.ones((N,l))
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for i in range(1,n+1):
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q = int((i)*(i+1)/2)
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for k in range(i+1):
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X[:,q+k] = (x**(i-k))*(y**k)
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return X
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# Making meshgrid of datapoints and compute Franke's function
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n = 10
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N = 1000
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x = np.sort(np.random.uniform(0, 1, N))
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y = np.sort(np.random.uniform(0, 1, N))
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z = FrankeFunction(x, y)
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X = create_X(x, y, n=n)
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# We split the data in test and training data
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X_train, X_test, y_train, y_test = train_test_split(X, z, test_size=0.2)
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# matrix inversion to find beta
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OLSbeta = np.linalg.pinv(X_train.T @ X_train) @ X_train.T @ y_train
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print(OLSbeta)
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# and then make the prediction
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ytildeOLS = X_train @ OLSbeta
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print("Training MSE for OLS")
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print(MSE(y_train,ytildeOLS))
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ypredictOLS = X_test @ OLSbeta
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print("Test MSE OLS")
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print(MSE(y_test,ypredictOLS))
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p = len(OLSbeta)
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I = np.eye(p,p)
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# Decide which values of lambda to use
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nlambdas = 100
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MSEOwnRidgePredict = np.zeros(nlambdas)
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MSEOwnRidgeTrain = np.zeros(nlambdas)
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MSERidgePredict = np.zeros(nlambdas)
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MSERidgeTrain = np.zeros(nlambdas)
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lambdas = np.logspace(-4, 4, nlambdas)
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for i in range(nlambdas):
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lmb = lambdas[i]
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OwnRidgeBeta = np.linalg.pinv(X_train.T @ X_train+lmb*I) @ X_train.T @ y_train
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# include lasso using Scikit-Learn
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# Note: we include the intercept
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RegRidge = linear_model.Ridge(lmb,fit_intercept=False)
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RegRidge.fit(X_train,y_train)
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# and then make the prediction
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ytildeOwnRidge = X_train @ OwnRidgeBeta
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ypredictOwnRidge = X_test @ OwnRidgeBeta
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ytildeRidge = RegRidge.predict(X_train)
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ypredictRidge = RegRidge.predict(X_test)
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MSEOwnRidgePredict[i] = MSE(y_test,ypredictOwnRidge)
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MSEOwnRidgeTrain[i] = MSE(y_train,ytildeOwnRidge)
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MSERidgePredict[i] = MSE(y_test,ypredictRidge)
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MSERidgeTrain[i] = MSE(y_train,ytildeRidge)
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print("Beta values for own Ridge implementation")
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print(OwnRidgeBeta)
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print("Beta values for Scikit-Learn Ridge implementation")
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print(RegRidge.coef_)
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# Now plot the results
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plt.figure()
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plt.plot(np.log10(lambdas), MSEOwnRidgeTrain, 'r', label = 'MSE own Ridge train')
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plt.plot(np.log10(lambdas), MSEOwnRidgePredict, 'b--', label = 'MSE own Ridge Test')
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plt.plot(np.log10(lambdas), MSERidgeTrain, 'y', label = 'MSE SL Ridge train')
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plt.plot(np.log10(lambdas), MSERidgePredict, 'g--', label = 'MSE SL Ridge Test')
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plt.xlabel('log10(lambda)')
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plt.ylabel('MSE')
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plt.legend()
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plt.show()
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# Now plot the function
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fig = plt.figure()
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ax=fig.add_subplot(projection='3d')
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# Make data.
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x, y = np.meshgrid(x,y)
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z = FrankeFunction(x, y)
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znew = X @ OLSbeta
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# Plot the surface.
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surf = ax.plot_surface(x, y, z, cmap=cm.coolwarm,
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linewidth=0, antialiased=False)
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# Customize the z axis.
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ax.set_zlim(-0.10, 1.40)
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ax.zaxis.set_major_locator(LinearLocator(10))
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ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))
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# Add a color bar which maps values to colors.
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fig.colorbar(surf, shrink=0.5, aspect=5)
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plt.show()
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@@ -1,72 +0,0 @@
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import matplotlib.pyplot as plt
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import numpy as np
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from sklearn.linear_model import LinearRegression
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from sklearn.preprocessing import PolynomialFeatures
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from sklearn.model_selection import train_test_split
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from sklearn.preprocessing import StandardScaler
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def MSE(y_data,y_model):
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n = np.size(y_model)
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return np.sum((y_data-y_model)**2)/n
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def OLS_fit_beta(X, y):
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return np.linalg.pinv(X.T @ X) @ X.T @ y
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def Ridge_fit_beta(X, y,L,d):
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I = np.eye(d,d)
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return np.linalg.pinv(X.T @ X + L*I) @ X.T @ y
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np.random.seed(2018)
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n = 100
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d = 3
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L = 0.001
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true_beta = [2, 0.5, 3.7]
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# Make data set.
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x = np.linspace(-3, 3, n)
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y_real = 2 + 0.5*x + 3.7*x**2
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y = np.sum(
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np.asarray([x ** p * b for p, b in enumerate(true_beta)]),
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axis=0) + 0.1 * np.random.normal(size=len(x))
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#Design matrix X including the intercept
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X = np.zeros((len(x), d))
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for p in range(d): # (d-1)
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X[:, p] = x ** (p) # (p+1 if not intercept included)
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#Split datamatrix
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X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
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#Calculate beta, own code
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beta_OLS = OLS_fit_beta(X_train, y_train)
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beta_Ridge = Ridge_fit_beta(X_train, y_train,L,d)
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print(beta_OLS)
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print(beta_Ridge)
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#predict value
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ytilde_test_OLS = X_test @ beta_OLS
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ytilde_test_Ridge = X_test @ beta_Ridge
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#Calculate MSE
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print(" ")
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print("test MSE of OLS:")
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print(MSE(y_test,ytilde_test_OLS))
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print(" ")
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print("test MSE of Ridge")
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print(MSE(y_test,ytilde_test_Ridge))
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plt.scatter(x,y,label='Data')
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#plt.plot(x,y_real,label='no noise')
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plt.plot(x, X @ beta_OLS,'*', label="OLS_Fit")
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plt.plot(x, X @ beta_Ridge, label="Ridge_Fit")
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plt.grid()
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plt.legend()
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plt.show()
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from mpl_toolkits.mplot3d import Axes3D
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import matplotlib.pyplot as plt
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from matplotlib import cm
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from matplotlib.ticker import LinearLocator, FormatStrFormatter
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import numpy as np
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from random import random, seed
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fig = plt.figure()
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ax=fig.add_subplot(projection='3d')
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# Make data.
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x = np.arange(0, 1, 0.05)
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y = np.arange(0, 1, 0.05)
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x, y = np.meshgrid(x,y)
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def FrankeFunction(x,y):
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term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
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term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
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term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
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term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
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return term1 + term2 + term3 + term4
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z = FrankeFunction(x, y)
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# Plot the surface.
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surf = ax.plot_surface(x, y, z, cmap=cm.coolwarm,
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linewidth=0, antialiased=False)
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# Customize the z axis.
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ax.set_zlim(-0.10, 1.40)
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ax.zaxis.set_major_locator(LinearLocator(10))
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ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))
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# Add a color bar which maps values to colors.
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fig.colorbar(surf, shrink=0.5, aspect=5)
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plt.show()
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@@ -1,88 +0,0 @@
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import matplotlib.pyplot as plt
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import numpy as np
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from sklearn.linear_model import LinearRegression
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from sklearn.preprocessing import PolynomialFeatures
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from sklearn.model_selection import train_test_split
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from sklearn.preprocessing import StandardScaler
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def MSE(y_data,y_model):
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n = np.size(y_model)
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return np.sum((y_data-y_model)**2)/n
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def OLS_fit_beta(X, y):
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return np.linalg.pinv(X.T @ X) @ X.T @ y
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def Ridge_fit_beta(X, y,L,d):
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I = np.eye(d,d)
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return np.linalg.pinv(X.T @ X + L*I) @ X.T @ y
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np.random.seed(2018)
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n = 1000
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d = 3
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L = 0.001
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true_beta = [2, 0.5, 3.7]
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# Make data set.
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x = np.linspace(-3, 3, n)
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y_real = 2 + 0.5*x + 3.7*x**2
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y = np.sum(
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np.asarray([x ** p * b for p, b in enumerate(true_beta)]),
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axis=0) + 0.1 * np.random.normal(size=len(x))
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#Design matrix X does include the intercept
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X = np.zeros((len(x), d))
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for p in range(d): # (d-1)
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X[:, p] = x ** (p+1) # (p+1 if not intercept included)
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#Split datamatrix
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X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2)
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scaler = StandardScaler()
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yscaler = StandardScaler()
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scaler.fit(X_train)
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yscaler.fit(y_train)
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X_train_scaled = scaler.transform(X_train)
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X_test_scaled = scaler.transform(X_test)
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y_train_scaled = yscaler.transform(y_train)
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y_test_scaled = yscaler.transform(y_test)
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#Calculate beta
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beta_OLS = OLS_fit_beta(X_train_scaled, y_train_scaled)
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beta_Ridge = Ridge_fit_beta(X_train_scaled, y_train_scaled,L,d)
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print(beta_OLS)
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print(beta_Ridge)
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"""
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interceptOLS = y_scaler - X_train_mean @ beta_OLS
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interceptRidge = y_scaler - X_train_mean @ beta_Ridge
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print(interceptOLS)
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print(interceptRidge)
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"""
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#predict value
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ytilde_test_OLS = X_test_scaled @ beta_OLS
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ytilde_test_Ridge = X_test_scaled @ beta_Ridge
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#Calculate MSE
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print(" ")
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print("test MSE of OLS:")
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print(MSE(y_test,ytilde_test_OLS))
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print(" ")
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print("test MSE of Ridge")
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print(MSE(y_test,ytilde_test_Ridge))
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plt.scatter(x,y,label='Data')
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#plt.plot(x,y_real,label='no noise')
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plt.plot(x, X @ beta_OLS,'*', label="OLS_Fit")
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plt.plot(x, X @ beta_Ridge, label="Ridge_Fit")
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plt.grid()
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plt.legend()
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plt.show()
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