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%%javascript
$.getScript('https://kmahelona.github.io/ipython_notebook_goodies/ipython_notebook_toc.js')
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# import the necessary package at the very beginning
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
import sklearn
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## Type Your Answer Below ##
np.random.seed(1)
X = np.random.random([100, 1]).ravel()*10 # generate a set of 100 random float in range [0, 10]
X[:5]
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random_error = np.random.randn(100) # genrate a set of 100 random error from a standard normal distribution
random_error[:5]
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Y = np.sin(X) + random_error # y = sin(x)+a where a is a small random error
Y[:5]
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## Type Your Answer Below ##
plt.scatter(x=X, y=Y, marker='o', alpha=0.4, color='b')
plt.xlabel('X')
plt.ylabel('Y')
plt.title('Y=sin(X) + random_error')
print('X: ', X.shape, ' ', 'Y: ', Y.shape )
You need to split the data into training and testing data before you build the model. This is the same procedure you need to do in the following questions.
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## Type Your Answer Below ##
# reshape X from row vector in shape(100, ) to column vector in shape (100, 1)
X_re = X.reshape(X.shape[0], 1)
X_re.shape
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# initiate a linear regression model
from sklearn.linear_model import LinearRegression
lr = LinearRegression()
lr
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# Use train_test_split to train and test lr
from sklearn import model_selection
Xtrain, Xtest, Ytrain, Ytest = model_selection.train_test_split(X_re, Y, train_size=70, random_state=1)
print(Xtrain.shape, Xtest.shape, Ytrain.shape, Ytest.shape)
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lr.fit(Xtrain, Ytrain)
Ypred = lr.predict(Xtest)
print('The mathematical formula of linear regression model: ', 'Y = ' + str(lr.coef_) + '*' + 'X + ' + str(lr.intercept_), '\n')
print('The coefficient of determination R^2 of the training set: ', lr.score(Xtrain, Ytrain), '\n')
print('The coefficient of determination R^2 of the testing set: ', lr.score(Xtest, Ytest), '\n')
plt.scatter(Ytest, Ypred, marker='o', alpha=0.5)
plt.xlabel('Ytest')
plt.ylabel('Ypred')
plt.title('Linear regression model performance')
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# Get the training and validation score of your model
# training and validation score具体指的什么？
from sklearn.model_selection import cross_val_score
cv_scores = cross_val_score(lr, X_re, Y, cv=3) # 3-fold cross validation
print('cv_scores: ', cv_scores)
print('mean of cv_scores: ', cv_scores.mean())
#The mean score and the 95% confidence interval of the score estimate are hence given by:
print("Accuracy: %0.2f (+/- %0.2f)" % (cv_scores.mean(), cv_scores.std() * 2))
怎么理解cv_scores是负数？
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## Type Your Answer Below ##
# show predicted y in red color
Ypred = lr.predict(X_re)
plt.plot(X, Ypred, label='Predicted Y', color='r')
# show real y in blue color
plt.scatter(X, Y, label='Real Y', color='b')
# show ground truth - sin(X) in green color
Yground = np.sin(X)
plt.scatter(X, Yground, label='Ground truth Y', color='g')
plt.xlabel('X')
plt.ylabel('Y')
plt.title('Three types of Y in a plot')
plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)
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## Type Your Answer Below ##
X2 = X_re**2
X2 = np.hstack([X_re, X2])
print(X2.shape)
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lr2 = LinearRegression()
lr2.fit(X2, Y)
cv_scores2 = cross_val_score(lr2, X2, Y, cv=3)
print('cv_scores for model using x and x^2: ', cv_scores2)
print('mean of cv_scores for model using x and x^2: ', cv_scores2.mean())
#The mean score and the 95% confidence interval of the score estimate are hence given by:
print("Accuracy: %0.2f (+/- %0.2f)" % (cv_scores2.mean(), cv_scores2.std() * 2))
print('The mathematical formula of linear regression model: ', 'Y = ' + str(lr2.coef_[0]) + '*X ' + str(lr2.coef_[1]) + "*X^2 + " + str(lr.intercept_), '\n')
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# visualize new set of Ypred, Y, Yground_truth
Ypred2 = lr2.predict(X2)
Yground = np.sin(X)
plt.scatter(X, Ypred2, label='predicted y using x and x**2', color='r')
plt.scatter(X, Y, label='real y', color='b')
plt.scatter(X, Yground, label='ground truth - sin(x)', color='g')
plt.xlabel('X')
plt.ylabel('Y')
plt.title('Three types of Y in a plot')
plt.legend(bbox_to_anchor=(1.05, 1), loc=2, borderaxespad=0.)
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In this question, you need to build 15 models, with features of [x],[x,x^2],[x,x^2,x^3],...,[x,x^2,...,x^15]. For each model you need to calculate the training score and validation score then make the plot as we required.
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from sklearn.model_selection import validation_curve
from sklearn.linear_model import Ridge
from sklearn.model_selection import cross_val_score
index =[] # generate an array with number 1 to 15
for i in range(1, 16):
index.append(i)
df = pd.DataFrame(columns = index) # create a new dataframe with 15 columns
df.iloc[:, 0] = X # the 1st column is X**1
mean_cv_scores = []
mean_train_scores = []
mean_valid_scores= []
for i in index:
print("################ Adding " + "x**" + str(i) + " ######################")
df.loc[:, i] = X**i # Add a new column of values
lr = LinearRegression() # start a new linear regression model with the new column taking into consideration
#lr.fit(df.iloc[:, :i], Y)
#Ypredict = lr.predict(df.iloc[:, :i])
cv_scores = cross_val_score(lr, df.iloc[:, :i], Y, cv=3)
print("mean cv score for the model is:", np.mean(cv_scores))
mean_cv_scores.append(np.mean(cv_scores))
train_score, valid_score = validation_curve(Ridge(), df.iloc[:, :i], Y, "alpha", np.logspace(-7, 3, 3))
print("mean train score is: ", np.mean(train_score))
print("mean valid score is: ", np.mean(valid_score))
mean_train_scores.append(np.mean(train_score))
mean_valid_scores.append(np.mean(valid_score))
print()
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plt.plot(df.columns, mean_train_scores, c='b', label='mean train scores') #plot the training score and validation score showing what happens when feature set gets larger
plt.plot(df.columns, mean_valid_scores, c='r', label = 'mean valid scores')
plt.xlabel('feature')
plt.ylabel('mean of evaluation scores')
plt.legend(loc=0)
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plt.plot(df.columns, mean_cv_scores, label='mean cv scores') #plot the training score and validation score showing what happens when feature set gets larger
plt.xlabel('feature')
plt.ylabel('mean of cross validation score')
plt.legend(loc=0)
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