Machine Learning Engineer Nanodegree¶

Model Evaluation & Validation¶

Project: Predicting Boston Housing Prices¶

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.

Getting Started¶

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:

  • 16 data points have an 'MEDV' value of 50.0. These data points likely contain missing or censored values and have been removed.
  • 1 data point has an 'RM' value of 8.78. This data point can be considered an outlier and has been removed.
  • The features 'RM', 'LSTAT', 'PTRATIO', and 'MEDV' are essential. The remaining non-relevant features have been excluded.
  • The feature '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 [2]:
# Import libraries necessary for this project
import numpy as np
import pandas as pd
from sklearn.cross_validation import ShuffleSplit

# Import supplementary visualizations code
import visuals as vs

# 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)
Boston housing dataset has 489 data points with 4 variables each.

Data Exploration¶

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.

Implementation: Calculate Statistics¶

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:

  • Calculate the minimum, maximum, mean, median, and standard deviation of 'MEDV', which is stored in prices.
    • Store each calculation in their respective variable.
In [3]:
prices_numpy = prices.as_matrix()
# TODO: Minimum price of the data
minimum_price = prices_numpy.min()

# TODO: Maximum price of the data
maximum_price = prices_numpy.max()

# TODO: Mean price of the data
mean_price = np.mean(prices_numpy)

# TODO: Median price of the data
median_price = np.median(prices_numpy)

# TODO: Standard deviation of prices of the data
std_price = prices_numpy.std()

# 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)
Statistics for Boston housing dataset:

Minimum price: $105,000.00
Maximum price: $1,024,800.00
Mean price: $454,342.94
Median price $438,900.00
Standard deviation of prices: $165,171.13

Question 1 - Feature Observation¶

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: For each of the three given feature RM, LSTAT and PTRATIO, change in these values affects increase and decrease in MEDV value. RM which is average number of rooms among homes in the neighbourhood increases then the value of MEDV also increases because more rooms means more bigger homes. LSTAT which is percentage of homeowners in the neighbourhood considered lower class when increases will decrease the value of MEDV because more number of lower class homeowners means they would spend less or no money on maintaining their homes and the neighbourhood would not look good considering homes that are not maintained and clean. PTRATIO which is ratio of students to teacher in primary and secondary schools in neighbourhood when increases may decrease the value of MEDV because families might want to live closer to schools where their children can get more attension from their teacher and quality of education improves if student to teacher ratio is optimum. Also time to commute to these schools will reduce.

Developing a Model¶

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.

Implementation: Define a Performance Metric¶

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 is no better than a model that always predicts the mean of 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 arbitrarily worse than one that always predicts the mean of the target variable.

For the performance_metric function in the code cell below, you will need to implement the following:

  • Use r2_score from sklearn.metrics to perform a performance calculation between y_true and y_predict.
  • Assign the performance score to the 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

Question 2 - Goodness of Fit¶

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)
Model has a coefficient of determination, R^2, of 0.923.

Answer: This model has coefficient of determination, R^2 score of 0.923 which mean 92.3% of the variation of the dependent variable can be explained by the model. Therefore, yes this model has successfully captured the variation the variation of the target variable.

Implementation: Shuffle and Split Data¶

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:

  • Use train_test_split from sklearn.cross_validation to shuffle and split the features and prices data into training and testing sets.
    • Split the data into 80% training and 20% testing.
    • Set the random_state for train_test_split to a value of your choice. This ensures results are consistent.
  • Assign the train and testing splits to X_train, X_test, y_train, and y_test.
In [6]:
# TODO: Import 'train_test_split'
from sklearn import cross_validation
# TODO: Shuffle and split the data into training and testing subsets
X_train, X_test, y_train, y_test = cross_validation.train_test_split(features, prices, test_size = 0.2, random_state = 2)

# Success
print "Training and testing split was successful."
Training and testing split was successful.

Question 3 - Training and Testing¶

What is the benefit to splitting a dataset into some ratio of training and testing subsets for a learning algorithm?
Hint: What could go wrong with not having a way to test your model?

Answer: Splitting the dataset into some ratio of training and testing subset is important because if we dont, then we would be training and testing our model on same set of values which may lead to overfitting of our model. Any dataset other than the given dataset will not yeild the desired result. The benefit of splitting the dataset is that from the given dataset we can train our model from training subset of the dataset and then we will be able to test our model from training subset. In this way our model is free from any overfitting and underfitting issues and we can get correct predictions for other datasets.

Analyzing Model 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.

Learning Curves¶

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 [7]:
# Produce learning curves for varying training set sizes and maximum depths
vs.ModelLearning(features, prices)

Question 4 - Learning the Data¶

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: Second graph, maximum depth is 3. When there are zero training points the score of the training curve is maximum that is 1. As more training points are added score of the training curve decreases to approximately 0.8, the decreases is noticeable for first 300 training points. Beyond that the score remains same even if more training points are added. When training points are added the score of the testing curve increases until 300 training points. From zero to 50 score increases rapidly and from 50 to 300 score increased by a very small amount. Beyond 300, adding more training points does not affect the score of the testing curve. Adding more training points benefits the model but after a certain limit it will not matter. For example: Just having 50 training points is not sufficient for the given model. Training model with atleast 300 training points is considered sufficient in this case. But training points beyond 300 training points does not affect the model. So it is always better to have more training points than less. One more thing to notice is that this is the ideal curve without high bias or variation compared to other graphs.

Complexity Curves¶

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 [8]:
vs.ModelComplexity(X_train, y_train)

Question 5 - Bias-Variance Tradeoff¶

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 model is trained with a maximum depth of 1 then model suffers from high bias. The validation score and training score does not have much difference. Model doesnot generalizes well in this situation. The model trained with maximum depth of 10 has high variation. There is a significant difference between training score and validated score. At maximum depth 10 model fits well but does not generalizes well because training score is higher than validated score

Question 6 - Best-Guess Optimal Model¶

Which maximum depth do you think results in a model that best generalizes to unseen data? What intuition lead you to this answer?

Answer: Model at maximum depth 4 generalizes to unseen data well because the gap between training curve and validated score curve is not much therefore no high variance scenario.

Evaluating Model Performance¶

In this final section of the project, you will construct a model and make a prediction on the client's feature set using an optimized model from fit_model.

What is the grid search technique and how it can be applied to optimize a learning algorithm?

Answer: In the grid search technique it allows one to define a grid of parameters that will be searched using K-fold cross-validation. This technique tries every combination of the provided hyper-parameter values in order to find the best model. Then the parameter with highest cross-validation accuracy will optimize the learning algorithm.

Question 8 - Cross-Validation¶

What is the k-fold cross-validation training technique? What benefit does this technique provide for grid search when optimizing a model?
Hint: Much like the reasoning behind having a testing set, what could go wrong with using grid search without a cross-validated set?

Answer: In K-fold cross-validation, dataset is split into k folds of equal size. Each fold then acts like a testing set 1 time, and then act like training set K-1 times. Average testing performance which is also known as cross-validated performance is used. Benefits of k-fold cross-validation is that it gives more reliable estimate compared to train/test split. Also a primary advantage of this method is that it matters less how the data gets divided. Every data point gets to be in a test set exactly once, and gets to be in a training set k-1 times. It is recommended when we dont have enough samples.

Implementation: Fitting a Model¶

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.

In addition, you will find your implementation is using ShuffleSplit() for an alternative form of cross-validation (see the 'cv_sets' variable). While it is not the K-Fold cross-validation technique you describe in Question 8, this type of cross-validation technique is just as useful!. The ShuffleSplit() implementation below will create 10 ('n_splits') shuffled sets, and for each shuffle, 20% ('test_size') of the data will be used as the validation set. While you're working on your implementation, think about the contrasts and similarities it has to the K-fold cross-validation technique.

Please note that ShuffleSplit has different parameters in scikit-learn versions 0.17 and 0.18. For the fit_model function in the code cell below, you will need to implement the following:

  • Use DecisionTreeRegressor from sklearn.tree to create a decision tree regressor object.
    • Assign this object to the 'regressor' variable.
  • Create a dictionary for 'max_depth' with the values from 1 to 10, and assign this to the 'params' variable.
  • Use make_scorer from sklearn.metrics to create a scoring function object.
    • Pass the performance_metric function as a parameter to the object.
    • Assign this scoring function to the 'scoring_fnc' variable.
  • Use GridSearchCV from sklearn.grid_search to create a grid search object.
    • Pass the variables 'regressor', 'params', 'scoring_fnc', and 'cv_sets' as parameters to the object.
    • Assign the GridSearchCV object to the 'grid' variable.
In [9]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.metrics import make_scorer
from sklearn.grid_search import GridSearchCV
from sklearn.tree import DecisionTreeRegressor
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
    # sklearn version 0.18: ShuffleSplit(n_splits=10, test_size=0.1, train_size=None, random_state=None)
    # sklearn versiin 0.17: ShuffleSplit(n, n_iter=10, test_size=0.1, train_size=None, random_state=None)
    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(random_state=0)

    # TODO: Create a dictionary for the parameter 'max_depth' with a range from 1 to 10
    drange = range(1, 11)
    params = dict(max_depth=drange)

    # 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, cv=cv_sets, scoring=scoring_fnc)

    # Fit the grid search object to the data to compute the optimal model
    grid =, y)

    # Return the optimal model after fitting the data
    return grid.best_estimator_

Making Predictions¶

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.

Question 9 - Optimal Model¶

What maximum depth does the optimal model have? How does this result compare to your guess in Question 6?

Run the code block below to fit the decision tree regressor to the training data and produce an optimal model.

In [10]:
# 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'])
Parameter 'max_depth' is 4 for the optimal model.

Answer: Parameter 'max_depth' is 4 for the optimal model. Result is same as question 6

Question 10 - Predicting Selling Prices¶

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 [11]:
# 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)
Predicted selling price for Client 1's home: $415,800.00
Predicted selling price for Client 2's home: $236,478.26
Predicted selling price for Client 3's home: $888,720.00

Answer: *
Predicted selling price for Client 1 home: 415,800.00
Predicted selling price for Client 2 home: 236,478.26
Predicted selling price for Client 3 home: 888,720.00

Result from Data Exploration:
Minimum price: 105,000.00
Maximum price: 1,024,800.00
Mean price: 454,342.94
Median price 438,900.00
Standard deviation of prices: 165,171.13

Price for Client 1 and Client 2 is resonable is it is below mean and median price. Also poverty level is increased compared to Client 3 hence price seems resonable. Price for Client 3 is way above mean and median price. But price seems resonable because of low poverty level and more number of rooms compared to other Clients. *


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 [12]:
vs.PredictTrials(features, prices, fit_model, client_data)
Trial 1: $391,183.33
Trial 2: $424,935.00
Trial 3: $415,800.00
Trial 4: $420,622.22
Trial 5: $418,377.27
Trial 6: $411,931.58
Trial 7: $399,663.16
Trial 8: $407,232.00
Trial 9: $351,577.61
Trial 10: $413,700.00

Range in prices: $73,357.39

Question 11 - Applicability¶

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:

  • How relevant today is data that was collected from 1978?
  • Are the features present in the data sufficient to describe a home?
  • Is the model robust enough to make consistent predictions?
  • Would data collected in an urban city like Boston be applicable in a rural city?

Answer: Data collected from 1978 is not relevant since a lot has changed since 1978. More relevant features can used because there are lot of changes since 1978. Features that are applicable to neighbourhoods of today in boston should be considered such as access to public transportation, crime rate, development, etc. Data collected from urban city like Boston may not be applicable to rural city because demographics are not the same for both, more relevant features for rurla city must be used to fit data.

Note: Once you have completed all of the code implementations and successfully answered each question above, you may finalize your work by exporting the iPython Notebook as an HTML document. You can do this by using the menu above and navigating to
File -> Download as -> HTML (.html). Include the finished document along with this notebook as your submission.