Welcome to the first project of the Machine Learning Engineer Nanodegree! In this notebook, some template code has already been written. You will need to implement additional functionality to successfully answer all of the questions for this project. Unless it is requested, do not modify any of the code that has already been included. In this template code, there are four sections which you must complete to successfully produce a prediction with your model. Each section where you will write code is preceded by a STEP X header with comments describing what must be done. Please read the instructions carefully!
In addition to implementing code, there will be questions that you must answer that relate to the project and your implementation. Each section where you will answer a question is preceded by a QUESTION X header. Be sure that you have carefully read each question and provide thorough answers in the text boxes that begin with "Answer:". Your project submission will be evaluated based on your answers to each of the questions.
A description of the dataset can be found here, which is provided by the UCI Machine Learning Repository.
To familiarize yourself with an iPython Notebook, try double clicking on this cell. You will notice that the text changes so that all the formatting is removed. This allows you to make edits to the block of text you see here. This block of text (and mostly anything that's not code) is written using Markdown, which is a way to format text using headers, links, italics, and many other options! Whether you're editing a Markdown text block or a code block (like the one below), you can use the keyboard shortcut Shift + Enter or Shift + Return to execute the code or text block. In this case, it will show the formatted text.
Let's start by setting up some code we will need to get the rest of the project up and running. Use the keyboard shortcut mentioned above on the following code block to execute it. Alternatively, depending on your iPython Notebook program, you can press the Play button in the hotbar. You'll know the code block executes successfully if the message "Boston Housing dataset loaded successfully!" is printed.
In [1]:
# Importing a few necessary libraries
import numpy as np
import matplotlib.pyplot as pl
from sklearn import datasets
from sklearn.tree import DecisionTreeRegressor
# Make matplotlib show our plots inline (nicely formatted in the notebook)
%matplotlib inline
# Create our client's feature set for which we will be predicting a selling price
CLIENT_FEATURES = [[11.95, 0.00, 18.100, 0, 0.6590, 5.6090, 90.00, 1.385, 24, 680.0, 20.20, 332.09, 12.13]]
# Load the Boston Housing dataset into the city_data variable
city_data = datasets.load_boston()
# Initialize the housing prices and housing features
housing_prices = city_data.target
housing_features = city_data.data
print "Boston Housing dataset loaded successfully!"
In this first section of the project, you will quickly investigate a few basic statistics about the dataset you are working with. In addition, you'll look at the client's feature set in CLIENT_FEATURES and see how this particular sample relates to the features of the dataset. Familiarizing yourself with the data through an explorative process is a fundamental practice to help you better understand your results.
In the code block below, use the imported numpy library to calculate the requested statistics. You will need to replace each None you find with the appropriate numpy coding for the proper statistic to be printed. Be sure to execute the code block each time to test if your implementation is working successfully. The print statements will show the statistics you calculate!
In [2]:
import numpy
# Number of houses in the dataset
total_houses = housing_features.shape[0]
# Number of features in the dataset
total_features = housing_features.shape[1]
# Minimum housing value in the dataset
minimum_price = housing_prices.min()
# Maximum housing value in the dataset
maximum_price = housing_prices.min()
# Mean house value of the dataset
mean_price = housing_prices.mean()
# Median house value of the dataset
median_price = numpy.median(housing_prices)
# Standard deviation of housing values of the dataset
std_dev = numpy.std(housing_prices)
# Show the calculated statistics
print "Boston Housing dataset statistics (in $1000's):\n"
print "Total number of houses:", total_houses
print "Total number of features:", total_features
print "Minimum house price:", minimum_price
print "Maximum house price:", maximum_price
print "Mean house price: {0:.3f}".format(mean_price)
print "Median house price:", median_price
print "Standard deviation of house price: {0:.3f}".format(std_dev)
As a reminder, you can view a description of the Boston Housing dataset here, where you can find the different features under Attribute Information. The MEDV attribute relates to the values stored in our housing_prices variable, so we do not consider that a feature of the data.
Of the features available for each data point, choose three that you feel are significant and give a brief description for each of what they measure.
Remember, you can double click the text box below to add your answer!
In [3]:
import pandas
import seaborn
columns = ('crime big_lots industrial charles_river nox rooms old distance'
' highway_access tax_rate pupil_teacher_ratio blacks lower_status'.split())
housing_data = pandas.DataFrame(housing_features, columns=columns)
housing_data['median_value'] = housing_prices
client_data = pandas.DataFrame(CLIENT_FEATURES, columns=columns)
seaborn.set_style('whitegrid')
for column in housing_data.columns:
grid = seaborn.lmplot(column, 'median_value', data=housing_data, size=8)
axe = grid.fig.gca()
title = axe.set_title('{0} vs price'.format(column))
CRIM - the per-capita crime rate. INDUS - the proportion of non-retail business acres per town. LSTAT - the percentage of the population that is of lower status.
In [4]:
print CLIENT_FEATURES
In [5]:
print(client_data.crime)
print(client_data.industrial)
print(client_data.lower_status)
CRIM : 11.95, INDUS: 18.1, LSTAT: 12.13
In the code block below, you will need to implement code so that the shuffle_split_data function does the following:
X and target labels (housing values) y.If you use any functions not already acessible from the imported libraries above, remember to include your import statement below as well!
Ensure that you have executed the code block once you are done. You'll know if the shuffle_split_data function is working if the statement "Successfully shuffled and split the data!" is printed.
In [6]:
# Put any import statements you need for this code block here
from sklearn import cross_validation
def shuffle_split_data(X, y):
""" Shuffles and splits data into 70% training and 30% testing subsets,
then returns the training and testing subsets. """
# Shuffle and split the data
X_train, X_test, y_train, y_test = cross_validation.train_test_split(X,
y,
test_size=.3,
random_state=0)
# Return the training and testing data subsets
return X_train, y_train, X_test, y_test
# Test shuffle_split_data
try:
X_train, y_train, X_test, y_test = shuffle_split_data(housing_features, housing_prices)
print "Successfully shuffled and split the data!"
except:
print "Something went wrong with shuffling and splitting the data."
So that we can assess the model using a different data-set than what it was trained on, thus reducing the likelihood of overfitting the model to the training data and increasing the likelihood that it will generalize to other data.
In the code block below, you will need to implement code so that the performance_metric function does the following:
y labels y_true and the predicted values of the y labels y_predict.You will need to first choose an appropriate performance metric for this problem. See the sklearn metrics documentation to view a list of available metric functions. Hint: Look at the question below to see a list of the metrics that were covered in the supporting course for this project.
Once you have determined which metric you will use, remember to include the necessary import statement as well!
Ensure that you have executed the code block once you are done. You'll know if the performance_metric function is working if the statement "Successfully performed a metric calculation!" is printed.
In [7]:
# Put any import statements you need for this code block here
from sklearn.metrics import mean_squared_error
def performance_metric(y_true, y_predict):
""" Calculates and returns the total error between true and predicted values
based on a performance metric chosen by the student. """
error = mean_squared_error(y_true, y_predict)
return error
# Test performance_metric
try:
total_error = performance_metric(y_train, y_train)
print "Successfully performed a metric calculation!"
except:
print "Something went wrong with performing a metric calculation."
Mean Squared Error was the most appropriate performance metric for predicting housing prices because we are predicting a numeric value (this is a regression problem) and while Mean Absolute Error could also be used, the MSE emphasizes larger errors more (due to the squaring) and so is preferable.
In the code block below, you will need to implement code so that the fit_model function does the following:
make_scorer documentation.regressor, parameters, and scoring_function. See the sklearn documentation on GridSearchCV.When building the scoring function and GridSearchCV object, be sure that you read the parameters documentation thoroughly. It is not always the case that a default parameter for a function is the appropriate setting for the problem you are working on.
Since you are using sklearn functions, remember to include the necessary import statements below as well!
Ensure that you have executed the code block once you are done. You'll know if the fit_model function is working if the statement "Successfully fit a model to the data!" is printed.
In [8]:
# Put any import statements you need for this code block
from sklearn.metrics import make_scorer
from sklearn.grid_search import GridSearchCV
def fit_model(X, y):
""" Tunes a decision tree regressor model using GridSearchCV on the input data X
and target labels y and returns this optimal model. """
# Create a decision tree regressor object
regressor = DecisionTreeRegressor()
# Set up the parameters we wish to tune
parameters = {'max_depth':(1,2,3,4,5,6,7,8,9,10)}
# Make an appropriate scoring function
scoring_function = make_scorer(mean_squared_error, greater_is_better=False)
# Make the GridSearchCV object
reg = GridSearchCV(regressor, param_grid=parameters,
scoring=scoring_function, cv=10)
# Fit the learner to the data to obtain the optimal model with tuned parameters
reg.fit(X, y)
# Return the optimal model
return reg
# Test fit_model on entire dataset
try:
reg = fit_model(housing_features, housing_prices)
print "Successfully fit a model!"
except:
print "Something went wrong with fitting a model."
The GridSearchCV algorithm exhaustively works through the parameters it is given to tune the model. Because it is exhaustive it is appropriate when the parameters are relatively limited and the model-creation is not computationally intensive, otherwise its run-time might be infeasible.
Cross-validation is a method of testing a model by partitioning the data into subsets, with each subset taking a turn as the test set while the data not being used as a test-set is used as the training set. This allows the model to be tested against all the data-points, rather than having some data reserved exclusively as training data and the remainder exclusively as testing data.
Because grid-search attempts to find the optimal parameters for a model, it's advantageous to use the same training and testing data in each case (case meaning a particular permutation of the parameters) so that the comparisons are equitable. One could simply perform an initial train-validation-test split and use this throughout the grid search, but this then risks the possibility that there was something in the initial split that will bias the outcome. By using all the partitions of the data as both test and training data, as cross-validation does, the chance of a bias in the splitting is reduced and at the same time all the parameter permutations are given the same data to be tested against.
You have now successfully completed your last code implementation section. Pat yourself on the back! All of your functions written above will be executed in the remaining sections below, and questions will be asked about various results for you to analyze. To prepare the Analysis and Prediction sections, you will need to intialize the two functions below. Remember, there's no need to implement any more code, so sit back and execute the code blocks! Some code comments are provided if you find yourself interested in the functionality.
In [9]:
def learning_curves(X_train, y_train, X_test, y_test):
""" Calculates the performance of several models with varying sizes of training data.
The learning and testing error rates for each model are then plotted. """
print "Creating learning curve graphs for max_depths of 1, 3, 6, and 10. . ."
# Create the figure window
fig = pl.figure(figsize=(10,8))
# We will vary the training set size so that we have 50 different sizes
sizes = np.round(np.linspace(1, len(X_train), 50))
train_err = np.zeros(len(sizes))
test_err = np.zeros(len(sizes))
# Create four different models based on max_depth
for k, depth in enumerate([1,3,6,10]):
for i, s in enumerate(sizes):
# Setup a decision tree regressor so that it learns a tree with max_depth = depth
regressor = DecisionTreeRegressor(max_depth = depth)
# Fit the learner to the training data
regressor.fit(X_train[:s], y_train[:s])
# Find the performance on the training set
train_err[i] = performance_metric(y_train[:s], regressor.predict(X_train[:s]))
# Find the performance on the testing set
test_err[i] = performance_metric(y_test, regressor.predict(X_test))
# Subplot the learning curve graph
ax = fig.add_subplot(2, 2, k+1)
ax.plot(sizes, test_err, lw = 2, label = 'Testing Error')
ax.plot(sizes, train_err, lw = 2, label = 'Training Error')
ax.legend()
ax.set_title('max_depth = %s'%(depth))
ax.set_xlabel('Number of Data Points in Training Set')
ax.set_ylabel('Total Error')
ax.set_xlim([0, len(X_train)])
# Visual aesthetics
fig.suptitle('Decision Tree Regressor Learning Performances', fontsize=18, y=1.03)
fig.tight_layout()
fig.show()
In [10]:
def model_complexity(X_train, y_train, X_test, y_test):
""" Calculates the performance of the model as model complexity increases.
The learning and testing errors rates are then plotted. """
print "Creating a model complexity graph. . . "
# We will vary the max_depth of a decision tree model from 1 to 14
max_depth = np.arange(1, 14)
train_err = np.zeros(len(max_depth))
test_err = np.zeros(len(max_depth))
for i, d in enumerate(max_depth):
# Setup a Decision Tree Regressor so that it learns a tree with depth d
regressor = DecisionTreeRegressor(max_depth = d)
# Fit the learner to the training data
regressor.fit(X_train, y_train)
# Find the performance on the training set
train_err[i] = performance_metric(y_train, regressor.predict(X_train))
# Find the performance on the testing set
test_err[i] = performance_metric(y_test, regressor.predict(X_test))
# Plot the model complexity graph
pl.figure(figsize=(7, 5))
pl.title('Decision Tree Regressor Complexity Performance')
pl.plot(max_depth, test_err, lw=2, label = 'Testing Error')
pl.plot(max_depth, train_err, lw=2, label = 'Training Error')
pl.legend()
pl.xlabel('Maximum Depth')
pl.ylabel('Total Error')
pl.show()
In this third section of the project, you'll take a look at several models' learning and testing error rates on various subsets of training data. Additionally, you'll investigate one particular algorithm with an increasing max_depth parameter on the full training set to observe how model complexity affects learning and testing errors. Graphing your model's performance based on varying criteria can be beneficial in the analysis process, such as visualizing behavior that may not have been apparent from the results alone.
In [11]:
learning_curves(X_train, y_train, X_test, y_test)
Looking at the model with max-depth of 3, as the size of the training set increases, the training error gradually increases. The testing error initially decreases, the seems to more or less stabilize.
The training and testing plots for the model with max-depth 1 move toward convergence with an error near 50, indicating a high bias (the model is too simple, and the additional data isn't improving the generalization of the model). For the model with max-depth 10, the curves haven't converged, and the training error remains near 0, indicating that it suffers from high variance, and should be improved with more data.
In [12]:
model_complexity(X_train, y_train, X_test, y_test)
As max-depth increases the training error improves, while the testing error decreases up until a depth of 5 and then begins a slight increase as the depth is increased. Based on this I would say that the max-depth of 5 created the model that best generalized the dataset, as it minimized the testing error.
In [13]:
print "Final model optimal parameters:", reg.best_params_
In [ ]:
repetitions = 1000
models = [fit_model(housing_features, housing_prices) for model in range(repetitions)]
params_scores = [(model.best_params_, model.best_score_) for model in models]
parameters = numpy.array([param_score[0]['max_depth'] for param_score in params_scores])
scores = numpy.array([param_score[1] for param_score in params_scores])
In [ ]:
best_models = pandas.DataFrame.from_dict({'parameter':parameters, 'score': scores})
x_labels = sorted(best_models.parameter.unique())
figure = pl.figure()
axe = figure.gca()
grid = seaborn.boxplot('parameter', 'score', data = best_models,
order=x_labels, ax=axe)
title = axe.set_title("Best Parameters vs Best Scores")
In [ ]:
best_index = np.where(scores==np.max(scores))
print(scores[best_index])
print(parameters[best_index])
In [ ]:
bin_range = best_models.parameter.max() - best_models.parameter.min()
bins = pandas.cut(best_models.parameter,
bin_range)
In [ ]:
parameter_group = pandas.groupby(best_models, 'parameter')
parameter_group.median()
In [ ]:
parameter_group.max()
While a max-depth of 4 was the most common best-parameter, the max-depth of 5 was the median max-depth, had the highest median score, and had the highest overall score, so I will say that the optimal max_depth parameter is 5. This is in line with what I had guessed, based on the Complexity Performance plot.
In [ ]:
best_model = models[best_index[0][0]]
sale_price = best_model.predict(CLIENT_FEATURES)
predicted = sale_price[0] * 1000
actual_median = housing_data.median_value.median() * 1000
print ("Predicted value of client's home: ${0:,.2f}".format(predicted))
print("Median Value - predicted: ${0:,.2f}".format(actual_median - predicted))
The predicted value of the client's home is $20,968.
I think that this model seems to make a reasonable prediction for the given data (Boston Suburbs in 1970), but I'm not sure that I agree that the data for an entire suburb is necessarily generalizable for a specific unit within a suburb. What this model predicts is that a suburb with attributes similar to the client's would have our predicted median value, but within each suburb there would likely be a bit of variance. I would also think that separating out the upper-class houses would give a better model for this particular client, given the sub-median values for his or her features, and the right-skew of the data. If the goal was to predict median prices for suburbs, then I would use this model, but not for individual sales.