In this notebook, you will use LASSO to select features, building on a pre-implemented solver for LASSO (using GraphLab Create, though you can use other solvers). You will:
In the second notebook, you will implement your own LASSO solver, using coordinate descent.
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import graphlab
In [2]:
sales = graphlab.SFrame('kc_house_data.gl/')
As in Week 2, we consider features that are some transformations of inputs.
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from math import log, sqrt
sales['sqft_living_sqrt'] = sales['sqft_living'].apply(sqrt)
sales['sqft_lot_sqrt'] = sales['sqft_lot'].apply(sqrt)
sales['bedrooms_square'] = sales['bedrooms']*sales['bedrooms']
# In the dataset, 'floors' was defined with type string,
# so we'll convert them to float, before creating a new feature.
sales['floors'] = sales['floors'].astype(float)
sales['floors_square'] = sales['floors']*sales['floors']
Let us fit a model with all the features available, plus the features we just created above.
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all_features = ['bedrooms', 'bedrooms_square',
'bathrooms',
'sqft_living', 'sqft_living_sqrt',
'sqft_lot', 'sqft_lot_sqrt',
'floors', 'floors_square',
'waterfront', 'view', 'condition', 'grade',
'sqft_above',
'sqft_basement',
'yr_built', 'yr_renovated']
Applying L1 penalty requires adding an extra parameter (l1_penalty) to the linear regression call in GraphLab Create. (Other tools may have separate implementations of LASSO.) Note that it's important to set l2_penalty=0 to ensure we don't introduce an additional L2 penalty.
In [5]:
model_all = graphlab.linear_regression.create(sales, target='price', features=all_features,
validation_set=None,
l2_penalty=0., l1_penalty=1e10)
Find what features had non-zero weight.
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model_all.coefficients.sort('value', ascending = False).print_rows(num_rows = 50)
Note that a majority of the weights have been set to zero. So by setting an L1 penalty that's large enough, we are performing a subset selection.
QUIZ QUESTION: According to this list of weights, which of the features have been chosen?
To find a good L1 penalty, we will explore multiple values using a validation set. Let us do three way split into train, validation, and test sets:
Be very careful that you use seed = 1 to ensure you get the same answer!
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(training_and_validation, testing) = sales.random_split(.9,seed=1) # initial train/test split
(training, validation) = training_and_validation.random_split(0.5, seed=1) # split training into train and validate
Next, we write a loop that does the following:
l1_penalty in [10^1, 10^1.5, 10^2, 10^2.5, ..., 10^7] (to get this in Python, type np.logspace(1, 7, num=13).)l1_penalty on TRAIN data. Specify l1_penalty=l1_penalty and l2_penalty=0. in the parameter list..predict()) for that l1_penaltyl1_penalty produced the lowest RSS on validation data.When you call linear_regression.create() make sure you set validation_set = None.
Note: you can turn off the print out of linear_regression.create() with verbose = False
In [43]:
import numpy as np
import sys
def choose_l1(l1s, nnz_threshold = -1):
best_model = 0
best_l1 = sys.maxint
min_nnz = sys.maxint
for l1 in l1s:
model = graphlab.linear_regression.create(training, target='price', features=all_features,
validation_set=None,
l2_penalty=0., l1_penalty=l1, verbose = False)
validation['pred'] = model.predict(validation)
testing['pred'] = model.predict(testing)
this_rss = validation.apply(lambda x : (x['price'] - x['pred']) ** 2).sum()
test_rss = testing.apply(lambda x : (x['price'] - x['pred']) ** 2).sum()
nnz_num = model['coefficients']['value'].nnz()
if (this_rss < best_l1 or rss < 0) and (nnz_threshold <= 0 or nnz_num == nnz_threshold):
best_l1 = this_rss
best_model = model
best_l1 =
print best_l1, test_rss, l1
model.coefficients.sort('value', ascending = False).print_rows(num_rows = 50)
if nnz_num < min_nnz:
min_nnz = nnz_num
return best_model, best_l1, min_nnz
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In [ ]:
model, l1, mnn = choose_l1(np.logspace(1, 7, num=13))
mnn
QUIZ QUESTIONS
l1_penalty?l1_penalty?
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QUIZ QUESTION Also, using this value of L1 penalty, how many nonzero weights do you have?
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In this section, you are going to implement a simple, two phase procedure to achive this goal:
l1_penalty values to find a narrow region of l1_penalty values where models are likely to have the desired number of non-zero weights.l1_penalty that achieves the desired sparsity. Here, we will again use a validation set to choose the best value for l1_penalty.
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max_nonzeros = 7
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l1_penalty_values = np.logspace(8, 10, num=20)
Now, implement a loop that search through this space of possible l1_penalty values:
l1_penalty in np.logspace(8, 10, num=20):l1_penalty on TRAIN data. Specify l1_penalty=l1_penalty and l2_penalty=0. in the parameter list. When you call linear_regression.create() make sure you set validation_set = Nonemodel['coefficients']['value'] gives you an SArray with the parameters you learned. If you call the method .nnz() on it, you will find the number of non-zero parameters!
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this_model, this_l1, this_mnn = choose_l1(l1_penalty_values)
this_mnn
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In [35]:
this_model['coefficients']['value'].nnz()
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Out of this large range, we want to find the two ends of our desired narrow range of l1_penalty. At one end, we will have l1_penalty values that have too few non-zeros, and at the other end, we will have an l1_penalty that has too many non-zeros.
More formally, find:
l1_penalty that has more non-zeros than max_nonzero (if we pick a penalty smaller than this value, we will definitely have too many non-zero weights)l1_penalty_min (we will use it later)l1_penalty that has fewer non-zeros than max_nonzero (if we pick a penalty larger than this value, we will definitely have too few non-zero weights)l1_penalty_max (we will use it later)Hint: there are many ways to do this, e.g.:
l1_penalty and inspecting it to find the appropriate boundaries.
In [40]:
def choose_largest_l1(l1s):
best_model = 0
best_l1 = sys.maxint
rss = sys.maxint
min_nnz = sys.maxint
l1_penalty_min = -sys.maxint
l1_penalty_max = sys.maxint
for l1 in l1s:
model = graphlab.linear_regression.create(training, target='price', features=all_features,
validation_set=None,
l2_penalty=0., l1_penalty=l1, verbose = False)
validation['pred'] = model.predict(validation)
testing['pred'] = model.predict(testing)
this_rss = validation.apply(lambda x : (x['price'] - x['pred']) ** 2).sum()
test_rss = testing.apply(lambda x : (x['price'] - x['pred']) ** 2).sum()
nnz_num = model['coefficients']['value'].nnz()
if this_rss < rss or rss < 0:
rss = this_rss
best_model = model
print rss, test_rss, l1
model.coefficients.sort('value', ascending = False).print_rows(num_rows = 50)
if nnz_num < max_nonzeros and l1 < l1_penalty_max:
l1_penalty_max = l1
if nnz_num > max_nonzeros and l1 > l1_penalty_min:
l1_penalty_min = l1
return l1_penalty_min, l1_penalty_max
l1_penalty_min, l1_penalty_max = choose_largest_l1(l1_penalty_values)
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l1_penalty_min, l1_penalty_max
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QUIZ QUESTIONS
What values did you find for l1_penalty_min andl1_penalty_max?
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l1_penalty_values = np.linspace(l1_penalty_min,l1_penalty_max,20)
l1_penalty in np.linspace(l1_penalty_min,l1_penalty_max,20):l1_penalty on TRAIN data. Specify l1_penalty=l1_penalty and l2_penalty=0. in the parameter list. When you call linear_regression.create() make sure you set validation_set = NoneFind the model that the lowest RSS on the VALIDATION set and has sparsity equal to max_nonzero.
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model_new, l1_new, nnz_new = choose_l1(l1_penalty_values, max_nonzeros)
l1_new, nnz_new
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QUIZ QUESTIONS
l1_penalty in our narrow range has the lowest RSS on the VALIDATION set and has sparsity equal to max_nonzeros?
In [47]:
l1_penalty_values
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