Welcome to the first project of the Machine Learning Engineer Nanodegree! In this notebook, some template code has already been provided for you, and you will need to implement additional functionality to successfully complete this project. You will not need to modify the included code beyond what is requested. Sections that begin with 'Implementation' in the header indicate that the following block of code will require additional functionality which you must provide. Instructions will be provided for each section and the specifics of the implementation are marked in the code block with a 'TODO' statement. Please be sure to read the instructions carefully!
In addition to implementing code, there will be questions that you must answer which relate to the project and your implementation. Each section where you will answer a question is preceded by a 'Question X' header. Carefully read each question and provide thorough answers in the following text boxes that begin with 'Answer:'. Your project submission will be evaluated based on your answers to each of the questions and the implementation you provide.
Note: Code and Markdown cells can be executed using the Shift + Enter keyboard shortcut. In addition, Markdown cells can be edited by typically double-clicking the cell to enter edit mode.
In this project, you will evaluate the performance and predictive power of a model that has been trained and tested on data collected from homes in suburbs of Boston, Massachusetts. A model trained on this data that is seen as a good fit could then be used to make certain predictions about a home — in particular, its monetary value. This model would prove to be invaluable for someone like a real estate agent who could make use of such information on a daily basis.
The dataset for this project originates from the UCI Machine Learning Repository. The Boston housing data was collected in 1978 and each of the 506 entries represent aggregated data about 14 features for homes from various suburbs in Boston, Massachusetts. For the purposes of this project, the following preprocessing steps have been made to the dataset:
'MEDV' value of 50.0. These data points likely contain missing or censored values and have been removed.'RM' value of 8.78. This data point can be considered an outlier and has been removed.'RM', 'LSTAT', 'PTRATIO', and 'MEDV' are essential. The remaining non-relevant features have been excluded.'MEDV' has been multiplicatively scaled to account for 35 years of market inflation.Run the code cell below to load the Boston housing dataset, along with a few of the necessary Python libraries required for this project. You will know the dataset loaded successfully if the size of the dataset is reported.
In [1]:
# Import libraries necessary for this project
import numpy as np
import pandas as pd
import visuals as vs # Supplementary code
from sklearn.cross_validation import ShuffleSplit
# Pretty display for notebooks
%matplotlib inline
# Load the Boston housing dataset
data = pd.read_csv('housing.csv')
prices = data['MEDV']
features = data.drop('MEDV', axis = 1)
# Success
print "Boston housing dataset has {} data points with {} variables each.".format(*data.shape)
In this first section of this project, you will make a cursory investigation about the Boston housing data and provide your observations. Familiarizing yourself with the data through an explorative process is a fundamental practice to help you better understand and justify your results.
Since the main goal of this project is to construct a working model which has the capability of predicting the value of houses, we will need to separate the dataset into features and the target variable. The features, 'RM', 'LSTAT', and 'PTRATIO', give us quantitative information about each data point. The target variable, 'MEDV', will be the variable we seek to predict. These are stored in features and prices, respectively.
For your very first coding implementation, you will calculate descriptive statistics about the Boston housing prices. Since numpy has already been imported for you, use this library to perform the necessary calculations. These statistics will be extremely important later on to analyze various prediction results from the constructed model.
In the code cell below, you will need to implement the following:
'MEDV', which is stored in prices.
In [2]:
# TODO: Minimum price of the data
minimum_price = np.min(prices)
# TODO: Maximum price of the data
maximum_price = np.max(prices)
# TODO: Mean price of the data
mean_price = np.mean(prices)
# TODO: Median price of the data
median_price = np.median(prices)
# TODO: Standard deviation of prices of the data
std_price = np.std(prices)
# Show the calculated statistics
print "Statistics for Boston housing dataset:\n"
print "Minimum price: ${:,.2f}".format(minimum_price)
print "Maximum price: ${:,.2f}".format(maximum_price)
print "Mean price: ${:,.2f}".format(mean_price)
print "Median price ${:,.2f}".format(median_price)
print "Standard deviation of prices: ${:,.2f}".format(std_price)
As a reminder, we are using three features from the Boston housing dataset: 'RM', 'LSTAT', and 'PTRATIO'. For each data point (neighborhood):
'RM' is the average number of rooms among homes in the neighborhood.'LSTAT' is the percentage of homeowners in the neighborhood considered "lower class" (working poor).'PTRATIO' is the ratio of students to teachers in primary and secondary schools in the neighborhood.Using your intuition, for each of the three features above, do you think that an increase in the value of that feature would lead to an increase in the value of 'MEDV' or a decrease in the value of 'MEDV'? Justify your answer for each.
Hint: Would you expect a home that has an 'RM' value of 6 be worth more or less than a home that has an 'RM' value of 7?
Answer:
'RM': An increase in 'RM' would lead to an increase in the value of 'MEDV', since houses with more rooms are generally larger and require more land and material.'LSTAT': An increase in 'LTSTAT' would lead to a decrease in the value of 'MEDV' because housing in an area considered more "lower-class" would tend to be affordable to members of the lower-class - meaning the prices would tend downward.'PTRATIO': I would expect an increase in 'PTRATIO' would lead to a decrease in 'MEDV'. Think inversely, an increase in student-to-teacher ratio means a decrease in teacher-to-student ratio. One might consider an area with a lower teacher-to-student ratio to have lower academic standards. Areas with lower academic standards are less attractive to homebuyers who might be looking to start a family.In this second section of the project, you will develop the tools and techniques necessary for a model to make a prediction. Being able to make accurate evaluations of each model's performance through the use of these tools and techniques helps to greatly reinforce the confidence in your predictions.
It is difficult to measure the quality of a given model without quantifying its performance over training and testing. This is typically done using some type of performance metric, whether it is through calculating some type of error, the goodness of fit, or some other useful measurement. For this project, you will be calculating the coefficient of determination, R2, to quantify your model's performance. The coefficient of determination for a model is a useful statistic in regression analysis, as it often describes how "good" that model is at making predictions.
The values for R2 range from 0 to 1, which captures the percentage of squared correlation between the predicted and actual values of the target variable. A model with an R2 of 0 always fails to predict the target variable, whereas a model with an R2 of 1 perfectly predicts the target variable. Any value between 0 and 1 indicates what percentage of the target variable, using this model, can be explained by the features. A model can be given a negative R2 as well, which indicates that the model is no better than one that naively predicts the mean of the target variable.
For the performance_metric function in the code cell below, you will need to implement the following:
r2_score from sklearn.metrics to perform a performance calculation between y_true and y_predict.score variable.
In [4]:
# TODO: Import 'r2_score'
from sklearn.metrics import r2_score
def performance_metric(y_true, y_predict):
""" Calculates and returns the performance score between
true and predicted values based on the metric chosen. """
# TODO: Calculate the performance score between 'y_true' and 'y_predict'
score = r2_score(y_true, y_predict)
# Return the score
return score
Assume that a dataset contains five data points and a model made the following predictions for the target variable:
| True Value | Prediction |
|---|---|
| 3.0 | 2.5 |
| -0.5 | 0.0 |
| 2.0 | 2.1 |
| 7.0 | 7.8 |
| 4.2 | 5.3 |
Would you consider this model to have successfully captured the variation of the target variable? Why or why not?
Run the code cell below to use the performance_metric function and calculate this model's coefficient of determination.
In [5]:
# Calculate the performance of this model
score = performance_metric([3, -0.5, 2, 7, 4.2], [2.5, 0.0, 2.1, 7.8, 5.3])
print "Model has a coefficient of determination, R^2, of {:.3f}.".format(score)
Answer:
It would depend on what the data represents, and how you measure success. The coeficient of determination is 0.923, meaning that only 7.7% of the variance is unexplained by the model. In most cases, I would think this would meet success criteria.
Your next implementation requires that you take the Boston housing dataset and split the data into training and testing subsets. Typically, the data is also shuffled into a random order when creating the training and testing subsets to remove any bias in the ordering of the dataset.
For the code cell below, you will need to implement the following:
train_test_split from sklearn.cross_validation to shuffle and split the features and prices data into training and testing sets.random_state for train_test_split to a value of your choice. This ensures results are consistent.X_train, X_test, y_train, and y_test.
In [6]:
# TODO: Import 'train_test_split'
from sklearn.cross_validation import train_test_split
# TODO: Shuffle and split the data into training and testing subsets
X_train, X_test, y_train, y_test = train_test_split(features, prices, random_state=14, train_size=0.8)
# Success
print "Training and testing split was successful."
Answer:
Having both training and testing subsets allows you to both train your model, and measure how well your model performs. If you didn't have a way to test your model, but rather used all of the data available to train it, then you wouldn't have a way to check if your model is over-fitting the data, and you wouldn't have a true estimate of its performance.
In this third section of the project, you'll take a look at several models' learning and testing performances 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 performance. 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.
The following code cell produces four graphs for a decision tree model with different maximum depths. Each graph visualizes the learning curves of the model for both training and testing as the size of the training set is increased. Note that the shaded region of a learning curve denotes the uncertainty of that curve (measured as the standard deviation). The model is scored on both the training and testing sets using R2, the coefficient of determination.
Run the code cell below and use these graphs to answer the following question.
In [8]:
# Produce learning curves for varying training set sizes and maximum depths
vs.ModelLearning(features, prices)
Choose one of the graphs above and state the maximum depth for the model. What happens to the score of the training curve as more training points are added? What about the testing curve? Would having more training points benefit the model?
Hint: Are the learning curves converging to particular scores?
Answer:
I have chosen the graph of the model with a maximum depth of 10. The training score decreases as more points are added. For the testing curve, it sharply increases until about 50 points, but as more points are added, the the magnitude of this increase is lower. Adding more training points would not benefit this model, as it appears to have learned all that it could by around 250 training points.
The following code cell produces a graph for a decision tree model that has been trained and validated on the training data using different maximum depths. The graph produces two complexity curves — one for training and one for validation. Similar to the learning curves, the shaded regions of both the complexity curves denote the uncertainty in those curves, and the model is scored on both the training and validation sets using the performance_metric function.
Run the code cell below and use this graph to answer the following two questions.
In [9]:
vs.ModelComplexity(X_train, y_train)
When the model is trained with a maximum depth of 1, does the model suffer from high bias or from high variance? How about when the model is trained with a maximum depth of 10? What visual cues in the graph justify your conclusions?
Hint: How do you know when a model is suffering from high bias or high variance?
Answer: When the model is trained with a maximum depth of 1, the model suffers from high bias, since both the training and test scores are very low. This means that the model is not complex enough to learn general trends. With a maximum depth of 10, the model suffers from high variance. This is evident by the high accuracy of the training data, but the low accuracy of the test data. It appears that the model is too complex, over-fitting the data. The trends it finds are too specific. It is not able to generalize the trends, so when new information is presented, it is not able to predict the target.
Answer: It looks like a maximum depth of 3 provides the best generalization of the unseen data. I say this because of three observations:
Answer: The grid search technique involves sweeping over a defined set of parameters, each with a set of values to try. The algorithm trains the model with each combination of parameters and outputs the parameter set with the best cross-validation score.
Answer:
The K-fold cross-validation training technique involves splitting a data set into K bins. The algorithm trains and tests the data K times, using each bin as the test data exactly once, and the remaining K-1 bins as the training data. The algorithm then averages the results for each train/test set. For grid search, using K-fold cross-validation ensures that you train over every aspect of the data, so that your search is not sensative to any given parameter. By this, I mean that there may be some inherent trend in the training set that will favor a certain value for some parameter. This trend may not generalize to the entire data set. By using K-fold cross validation, you test every parameter on every piece of data, which minimizes this effect.
Your final implementation requires that you bring everything together and train a model using the decision tree algorithm. To ensure that you are producing an optimized model, you will train the model using the grid search technique to optimize the 'max_depth' parameter for the decision tree. The 'max_depth' parameter can be thought of as how many questions the decision tree algorithm is allowed to ask about the data before making a prediction. Decision trees are part of a class of algorithms called supervised learning algorithms.
For the fit_model function in the code cell below, you will need to implement the following:
DecisionTreeRegressor from sklearn.tree to create a decision tree regressor object.'regressor' variable.'max_depth' with the values from 1 to 10, and assign this to the 'params' variable.make_scorer from sklearn.metrics to create a scoring function object.performance_metric function as a parameter to the object.'scoring_fnc' variable.GridSearchCV from sklearn.grid_search to create a grid search object.'regressor', 'params', 'scoring_fnc', and 'cv_sets' as parameters to the object. GridSearchCV object to the 'grid' variable.
In [16]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.tree import DecisionTreeRegressor
from sklearn.metrics import make_scorer
from sklearn.grid_search import GridSearchCV
def fit_model(X, y):
""" Performs grid search over the 'max_depth' parameter for a
decision tree regressor trained on the input data [X, y]. """
# Create cross-validation sets from the training data
cv_sets = ShuffleSplit(X.shape[0], n_iter = 10, test_size = 0.20, random_state = 0)
# TODO: Create a decision tree regressor object
regressor = DecisionTreeRegressor()
# TODO: Create a dictionary for the parameter 'max_depth' with a range from 1 to 10
params = {'max_depth': [1,2,3,4,5,6,7,8,9,10]}
# TODO: Transform 'performance_metric' into a scoring function using 'make_scorer'
scoring_fnc = make_scorer(performance_metric)
# TODO: Create the grid search object
grid = GridSearchCV(regressor, params, scoring=scoring_fnc, cv=cv_sets)
# Fit the grid search object to the data to compute the optimal model
grid = grid.fit(X, y)
# Return the optimal model after fitting the data
return grid.best_estimator_
Once a model has been trained on a given set of data, it can now be used to make predictions on new sets of input data. In the case of a decision tree regressor, the model has learned what the best questions to ask about the input data are, and can respond with a prediction for the target variable. You can use these predictions to gain information about data where the value of the target variable is unknown — such as data the model was not trained on.
In [17]:
# Fit the training data to the model using grid search
reg = fit_model(X_train, y_train)
# Produce the value for 'max_depth'
print "Parameter 'max_depth' is {} for the optimal model.".format(reg.get_params()['max_depth'])
Answer:
The optimal maximum depth for the decision tree model is 4. I considered this as an option in question 6, but ultimately went with 3. However, the learning curve was not derived for a maximum depth of 4, which might have swayed my decision. In the end, this value agrees with the complexity curve calculated for the maximum depth parameter.
Imagine that you were a real estate agent in the Boston area looking to use this model to help price homes owned by your clients that they wish to sell. You have collected the following information from three of your clients:
| Feature | Client 1 | Client 2 | Client 3 |
|---|---|---|---|
| Total number of rooms in home | 5 rooms | 4 rooms | 8 rooms |
| Neighborhood poverty level (as %) | 17% | 32% | 3% |
| Student-teacher ratio of nearby schools | 15-to-1 | 22-to-1 | 12-to-1 |
What price would you recommend each client sell his/her home at? Do these prices seem reasonable given the values for the respective features?
Hint: Use the statistics you calculated in the Data Exploration section to help justify your response.
Run the code block below to have your optimized model make predictions for each client's home.
In [23]:
# Produce a matrix for client data
client_data = [[5, 17, 15], # Client 1
[4, 32, 22], # Client 2
[8, 3, 12]] # Client 3
# Show predictions
for i, price in enumerate(reg.predict(client_data)):
print "Predicted selling price for Client {}'s home: ${:,.2f}".format(i+1, price)
Answer:
These prices do fall within the range given by the data set. Client 1 and 2's price fall within 2 standard deviations of the mean, but Client 3's price lies toward the outer edge of 3 standard deviations from the mean (i.e., is close to the top 1% of home prices in the data set). However, it can be seen that Client 3's house has "good" values for all three metrics that we are looking at. The trends for all clients agree with what was predicted to be the using 'LSTAT', 'RM', and 'PTRATIO'.
An optimal model is not necessarily a robust model. Sometimes, a model is either too complex or too simple to sufficiently generalize to new data. Sometimes, a model could use a learning algorithm that is not appropriate for the structure of the data given. Other times, the data itself could be too noisy or contain too few samples to allow a model to adequately capture the target variable — i.e., the model is underfitted. Run the code cell below to run the fit_model function ten times with different training and testing sets to see how the prediction for a specific client changes with the data it's trained on.
In [25]:
vs.PredictTrials(features, prices, fit_model, client_data)
In a few sentences, discuss whether the constructed model should or should not be used in a real-world setting.
Hint: Some questions to answering:
Answer:
In short, no, this model should not be used in a real world setting. There are less than 500 samples for an entire city, and as shown above, the model varies widely with different train/test splits. The model is not very complex - it is based on three metrics, but there are many others that might play a role, such as crime rate, house condition, school ratings, and proximity to points of interest. The 'PTRATIO' is a questionable metric, as it doesn't really tell you about how good education is. The model is also extrapolated linearly from data taken almost 40 years prior, without taking into consideration changes such as city layout or industry/availability of work, or any other factors that could drastically impact the model.