In this notebook, we will implement ridge regression via gradient descent. You will:
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import graphlab
import numpy as np
import pandas as pd
from sklearn import linear_model
import matplotlib as mpl
import matplotlib.pyplot as plt
import seaborn as sns
sns.set_style('darkgrid')
%matplotlib inline
If we want to do any "feature engineering" like creating new features or adjusting existing ones we should do this directly using the SFrames as seen in the first notebook of Week 2. For this notebook, however, we will work with the existing features.
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sales = graphlab.SFrame('kc_house_data.gl/')
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plt.figure(figsize=(8,6))
plt.plot(sales['sqft_living'], sales['price'],'.')
plt.xlabel('Living Area (ft^2)', fontsize=16)
plt.ylabel('House Price ($)', fontsize=16)
plt.title('King County, Seattle House Price Data', fontsize=18)
plt.axis([0.0, 14000.0, 0.0, 8000000.0])
plt.show()
As in Week 2, we convert the SFrame into a 2D Numpy array. Copy and paste get_num_data() from the second notebook of Week 2.
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def get_numpy_data(input_sframe, features, output):
input_sframe['constant'] = 1 # Adding column 'constant' to input SFrame with all values = 1.0
features = ['constant'] + features # Adding 'constant' to List of features
# Selecting the columns for the feature_matrux and output_array
features_sframe = input_sframe[features]
output_sarray = input_sframe[output]
# Converting sframes to numpy.ndarrays
feature_matrix = features_sframe.to_numpy()
output_array = output_sarray.to_numpy()
return(feature_matrix, output_array)
Also, copy and paste the predict_output() function to compute the predictions for an entire matrix of features given the matrix and the weights:
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def predict_output(feature_matrix, weights):
predictions = np.dot(feature_matrix, weights)
return predictions
We are now going to move to computing the derivative of the regression cost function. Recall that the cost function is the sum over the data points of the squared difference between an observed output and a predicted output, plus the L2 penalty term.
Cost(w)
= SUM[ (prediction - output)^2 ]
+ l2_penalty*(w[0]^2 + w[1]^2 + ... + w[k]^2).Since the derivative of a sum is the sum of the derivatives, we can take the derivative of the first part (the RSS) as we did in the notebook for the unregularized case in Week 2 and add the derivative of the regularization part. As we saw, the derivative of the RSS with respect to w[i] can be written as:
2*SUM[ error*[feature_i] ].The derivative of the regularization term with respect to w[i] is:
2*l2_penalty*w[i].Summing both, we get
2*SUM[ error*[feature_i] ] + 2*l2_penalty*w[i].That is, the derivative for the weight for feature i is the sum (over data points) of 2 times the product of the error and the feature itself, plus 2*l2_penalty*w[i].
We will not regularize the constant. Thus, in the case of the constant, the derivative is just twice the sum of the errors (without the 2*l2_penalty*w[0] term).
Recall that twice the sum of the product of two vectors is just twice the dot product of the two vectors. Therefore the derivative for the weight for feature_i is just two times the dot product between the values of feature_i and the current errors, plus 2*l2_penalty*w[i].
With this in mind complete the following derivative function which computes the derivative of the weight given the value of the feature (over all data points) and the errors (over all data points). To decide when to we are dealing with the constant (so we don't regularize it) we added the extra parameter to the call feature_is_constant which you should set to True when computing the derivative of the constant and False otherwise.
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def feature_derivative_ridge(errors, feature, weight, l2_penalty, feature_is_constant):
# If feature_is_constant is True, derivative is twice the dot product of errors and feature
if feature_is_constant==True:
derivative = 2.0*np.dot(errors, feature)
# Otherwise, derivative is twice the dot product plus 2*l2_penalty*weight
else:
derivative = 2.0*np.dot(errors, feature) + 2.0*l2_penalty*weight
return derivative
To test your feature derivartive run the following:
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(example_features, example_output) = get_numpy_data(sales, ['sqft_living'], 'price')
my_weights = np.array([1., 10.])
test_predictions = predict_output(example_features, my_weights)
errors = test_predictions - example_output # prediction errors
# next two lines should print the same values
print feature_derivative_ridge(errors, example_features[:,1], my_weights[1], 1, False)
print np.sum(errors*example_features[:,1])*2+20.
print ''
# next two lines should print the same values
print feature_derivative_ridge(errors, example_features[:,0], my_weights[0], 1, True)
print np.sum(errors)*2.
Now we will write a function that performs a gradient descent. The basic premise is simple. Given a starting point we update the current weights by moving in the negative gradient direction. Recall that the gradient is the direction of increase and therefore the negative gradient is the direction of decrease and we're trying to minimize a cost function.
The amount by which we move in the negative gradient direction is called the 'step size'. We stop when we are 'sufficiently close' to the optimum. Unlike in Week 2, this time we will set a maximum number of iterations and take gradient steps until we reach this maximum number. If no maximum number is supplied, the maximum should be set 100 by default. (Use default parameter values in Python.)
With this in mind, complete the following gradient descent function below using your derivative function above. For each step in the gradient descent, we update the weight for each feature before computing our stopping criteria.
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def ridge_regression_gradient_descent(feature_matrix, output, initial_weights, step_size, l2_penalty, max_iterations):
weights = np.array(initial_weights) # make sure it's a numpy array
iteration_count = 0
#while not reached maximum number of iterations:
while iteration_count < max_iterations:
predictions = predict_output(feature_matrix, weights) # computing predictions w/ feature_matrix and weights
errors = predictions - output # compute the errors as predictions - output
# loop over each weight
for i in xrange(len(weights)):
# Recall that feature_matrix[:,i] is the feature column associated with weights[i]
# compute the derivative for weight[i].
#(Remember: when i=0, you are computing the derivative of the constant!)
if i == 0:
derivative = feature_derivative_ridge(errors, feature_matrix[:,0], weights[0], l2_penalty, True)
else:
derivative = feature_derivative_ridge(errors, feature_matrix[:,i], weights[i], l2_penalty, False)
weights[i] = weights[i] - step_size*derivative
# Incrementing the iteration count
iteration_count += 1
return weights
The L2 penalty gets its name because it causes weights to have small L2 norms than otherwise. Let's see how large weights get penalized. Let us consider a simple model with 1 feature:
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simple_features = ['sqft_living']
my_output = 'price'
Load the training set and test set.
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train_data,test_data = sales.random_split(.8,seed=0)
In this part, we will only use 'sqft_living' to predict 'price'. Use the get_numpy_data function to get a Numpy versions of your data with only this feature, for both the train_data and the test_data.
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(simple_feature_matrix, output) = get_numpy_data(train_data, simple_features, my_output)
(simple_test_feature_matrix, test_output) = get_numpy_data(test_data, simple_features, my_output)
Let's set the parameters for our optimization:
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initial_weights = np.array([0.0, 0.0])
step_size = 1e-12
max_iterations=1000
First, let's consider no regularization. Set the l2_penalty to 0.0 and run your ridge regression algorithm to learn the weights of your model. Call your weights:
simple_weights_0_penalty
we'll use them later.
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l2_penalty = 0.0
simple_weights_0_penalty = ridge_regression_gradient_descent(simple_feature_matrix, output, initial_weights, step_size, l2_penalty, max_iterations)
Next, let's consider high regularization. Set the l2_penalty to 1e11 and run your ridge regression algorithm to learn the weights of your model. Call your weights:
simple_weights_high_penalty
we'll use them later.
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l2_penalty = 1.0e11
simple_weights_high_penalty = ridge_regression_gradient_descent(simple_feature_matrix, output, initial_weights, step_size, l2_penalty, max_iterations)
This code will plot the two learned models. (The green line is for the model with no regularization and the red line is for the one with high regularization.)
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plt.figure(figsize=(8,6))
plt.plot(simple_feature_matrix[:,1],output,'.', label= 'House Price Data')
plt.hold(True)
plt.plot(simple_feature_matrix[:,1], predict_output(simple_feature_matrix, simple_weights_0_penalty),'-', label= 'No L2 Penalty')
plt.plot(simple_feature_matrix[:,1], predict_output(simple_feature_matrix, simple_weights_high_penalty),'-', label= 'Large L2 Penalty')
plt.hold(False)
plt.legend(loc='upper left', fontsize=16)
plt.xlabel('Living Area (ft^2)', fontsize=16)
plt.ylabel('House Price ($)', fontsize=16)
plt.title('King County, Seattle House Price Data', fontsize=18)
plt.axis([0.0, 14000.0, 0.0, 8000000.0])
plt.show()
Compute the RSS on the TEST data for the following three sets of weights:
Which weights perform best?
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test_pred_weights_0 = predict_output(simple_test_feature_matrix, initial_weights)
RSS_test_weights_0 = sum( (test_output - test_pred_weights_0)**2.0 )
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test_pred_no_reg = predict_output(simple_test_feature_matrix, simple_weights_0_penalty)
RSS_test_no_reg = sum( (test_output - test_pred_no_reg)**2.0 )
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test_pred_high_reg = predict_output(simple_test_feature_matrix, simple_weights_high_penalty)
RSS_test_high_reg = sum( (test_output - test_pred_high_reg)**2.0 )
QUIZ QUESTIONS
Q1: What is the value of the coefficient for sqft_living that you learned with no regularization, rounded to 1 decimal place? What about the one with high regularization?
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print 'No Regulatization sqft_living weight: %.1f' %(simple_weights_0_penalty[1])
print 'High Regulatization sqft_living weight: %.1f' %(simple_weights_high_penalty[1])
Q2: Comparing the lines you fit with the with no regularization versus high regularization, which one is steeper?
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print 'Line with No Regularization is steeper'
Q3: What are the RSS on the test data for each of the set of weights above (initial, no regularization, high regularization)?
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print 'Test set RSS with initial weights all set to 0.0: %.1e' %(RSS_test_weights_0)
print 'Test set RSS with initial weights set to weights learned with no regularization: %.1e' %(RSS_test_no_reg)
print 'Test set RSS with initial weights set to weights learned with high regularization: %.1e' %(RSS_test_high_reg)
Initial weights learned with no regularization performed best on the Test Set (lowest RSS value)
Let us now consider a model with 2 features: ['sqft_living', 'sqft_living15'].
First, create Numpy versions of your training and test data with these two features.
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model_features = ['sqft_living', 'sqft_living15'] # sqft_living15 is the average squarefeet for the nearest 15 neighbors.
my_output = 'price'
(feature_matrix, output) = get_numpy_data(train_data, model_features, my_output)
(test_feature_matrix, test_output) = get_numpy_data(test_data, model_features, my_output)
We need to re-inialize the weights, since we have one extra parameter. Let us also set the step size and maximum number of iterations.
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initial_weights = np.array([0.0,0.0,0.0])
step_size = 1e-12
max_iterations = 1000
First, let's consider no regularization. Set the l2_penalty to 0.0 and run your ridge regression algorithm to learn the weights of your model. Call your weights:
multiple_weights_0_penalty
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l2_penalty = 0.0
multiple_weights_0_penalty = ridge_regression_gradient_descent(feature_matrix, output, initial_weights, step_size, l2_penalty, max_iterations)
Next, let's consider high regularization. Set the l2_penalty to 1e11 and run your ridge regression algorithm to learn the weights of your model. Call your weights:
multiple_weights_high_penalty
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l2_penalty = 1.0e11
multiple_weights_high_penalty = ridge_regression_gradient_descent(feature_matrix, output, initial_weights, step_size, l2_penalty, max_iterations)
Compute the RSS on the TEST data for the following three sets of weights:
Which weights perform best?
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test_pred_mul_feat_weights_0 = predict_output(test_feature_matrix, initial_weights)
RSS_test_mul_feat_weights_0 = sum( (test_output - test_pred_mul_feat_weights_0)**2.0 )
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test_pred_mul_feat_no_reg = predict_output(test_feature_matrix, multiple_weights_0_penalty)
RSS_test_mul_feat_no_reg = sum( (test_output - test_pred_mul_feat_no_reg)**2.0 )
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test_pred_mul_feat_high_reg = predict_output(test_feature_matrix, multiple_weights_high_penalty)
RSS_test_mul_feat_high_reg = sum( (test_output - test_pred_mul_feat_high_reg)**2.0 )
Predict the house price for the 1st house in the test set using the no regularization and high regularization models. (Remember that python starts indexing from 0.) How far is the prediction from the actual price? Which weights perform best for the 1st house?
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print 'Pred. price of 1st house in Test Set with weights learned with no reg.: %.2f' %(test_pred_mul_feat_no_reg[0])
print 'Pred. price of 1st house in Test Set with weights learned with high reg.: %.2f' %(test_pred_mul_feat_high_reg[0])
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print 'Pred. price - actual prize of 1st house in Test Set, using weights w/ no reg.: %.2f' %(abs(test_output[0] - test_pred_mul_feat_no_reg[0]))
print 'Pred. price - actual prize of 1st house in Test Set, using weights w/ high reg.: %.2f' %(abs(test_output[0] - test_pred_mul_feat_high_reg[0]))
Weights with high regularization perform best on 1st house in Test Set
QUIZ QUESTIONS
Q1: What is the value of the coefficient for sqft_living that you learned with no regularization, rounded to 1 decimal place? What about the one with high regularization?
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print 'No Regulatization sqft_living weight: %.1f' %(multiple_weights_0_penalty[1])
print 'High Regulatization sqft_living weight: %.1f' %(multiple_weights_high_penalty[1])
Q2: What are the RSS on the test data for each of the set of weights above (initial, no regularization, high regularization)?
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print 'Test set RSS with initial weights all set to 0.0: %.1e' %(RSS_test_mul_feat_weights_0)
print 'Test set RSS with initial weights set to weights learned with no regularization: %.1e' %(RSS_test_mul_feat_no_reg)
print 'Test set RSS with initial weights set to weights learned with high regularization: %.1e' %(RSS_test_mul_feat_high_reg)
Q3: We make prediction for the first house in the test set using two sets of weights (no regularization vs high regularization). Which weights make better prediction for that particular house?
Weights with high regularization perform best on 1st house in Test Set
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