Machine Learning Engineer Nanodegree

Model Evaluation & Validation

Project 1: 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 preoprocessing steps have been made to the dataset:

  • 16 data points have an 'MDEV' 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 'MDEV' are essential. The remaining non-relevant features have been excluded.
  • The feature 'MDEV' 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 [12]:
# 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
import matplotlib.pyplot as pl

# Pretty display for notebooks
%matplotlib inline

# Load the Boston housing dataset
data = pd.read_csv('housing.csv')
prices = data['MDEV']
features = data.drop('MDEV', 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, 'MDEV', 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 'MDEV', which is stored in prices.
    • Store each calculation in their respective variable.
In [14]:
# 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)

# Extra Practice:
first_quartile = np.percentile(prices, 25)
third_quartile = np.percentile(prices, 75)

# Histogram of house prices
pl.hist(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)

# Add the 1st and 3nd quartile 
print ""
print "1st quartile: ${:,.2f}".format(first_quartile)
print "3nd quartile: ${:,.2f}".format(third_quartile)

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

1st quartile: $350,700.00
3nd quartile: $518,700.00

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 all Boston homeowners who have a greater net worth than homeowners in the neighborhood.
  • '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 'MDEV' or a decrease in the value of 'MDEV'? 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' (increase) is positively correlated to 'MDEV' (increase), because more rooms may mean bigger spaces and higher price. The 'RM' slope is positive in the Graph-3.
  • 'LSTAT' (increase) is negatively correlated to 'MDEV' (decrease), because the higher percentage of owners may mean the lower demand for purchasing houses. The 'LSTAT' slope is negative in the Grape-1.
  • 'PTRATIO' is not a real effective feature for 'MDEV'. For instance, a higher ratio students to teachers may caused by that the more citizens living in one place are seeking for a top-school for supporting well educations to their kids , or may caused by that the citizens prefer to have more kids and lack wealth for education. The Graph-2 also show that there is no fitted pattern or clear relative slope.
In [4]:
# Analyst
import matplotlib.pyplot as plt
plt.figure(figsize = (20, 5))
for i, col in enumerate(features.columns):
    plt.subplot(1, 3, i)
    plt.plot(data[col], prices, 'o')
    plt.title(col)
    plt.xlabel(col)
    plt.ylabel('prices')

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 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:

  • 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 [5]:
# 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 [6]:
# 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: For evaluating models, the 0.923 R^2 score (coefficient of determination) means the model explain 92.30% information of 'True Value'. If the model has real economic meaning or making sense for practical relation, it seemed to have successfully captured 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 [7]:
# 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, test_size = 0.2, random_state = 42)

# 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: To evaluate a model should consider the significant of predication. The testing subsets fitted process help to avoid overfitting and omitting variables. To improve the accuracy of models prediction is the target of modeling. So, to splitting a datasets into training and testing subsets is important process in modeling.

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 reigon 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 [9]:
# 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:

I prefer to choose the max_depth = 3.

In the training curve the performance decrease as the number of training points increases, but after about 300 points the curve turn flat, adding more points don't benefit the training.

In the testing curve, the performance improves as the number of training points increases, the score increase fast in the first 50 points, and increase slow between 50 and 300, but after 300 points there is no more benefit. Both curves converge at score 0.8.

The balance of spliting training scores and testing scores should be considered for avoiding overfitting and decreasing the bias.

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 [10]:
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:

The maximum depth of 1: the model may suffer from high bias or converged with errors, when the training score and validation score are low.

The maximum depth of 10: The large gap between the training and testing scores may mean the model suffer from high variance. Unlike a biased model, the training socres of model are imporved with sacrificing the validation scores.

Based on the visual hints in the graph, the maximum depth of 3 or 4 may be a good choice for balancing bias and variances.

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: The maximum depth of 4 may be the best generalized to unseen data. Basing on the plot, the max_depth of 4 model reaches a well balance between the training score and validation score, which may avoid overfitting and decrese bias.

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.

Question 7 - Grid Search

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

Answer:

The traditional way of performing hyperparameter optimization has been grid search, or a parameter sweep, which is simply an exhaustive searching through a manually specified subset of the hyperparameter space of a learning algorithm.

A grid search algorithm must be guided by some performance metric, typically measured by cross-validation on the training set or evaluation on a held-out validation set.

It uses cross-validation for assessing how the results of a statistical analysis will generalize to an independent data set. In the process, features can be leveraged to perform a more efficient cross-validation used for model selection of this parameter, it can try all the parameter combinations and get a fitted classifier that is automatically tuned to the optimal parameter combination.

The goal of grid search is tuning the model, so the model will learn well and don't overfit. And the beauty part is that it can work through many combinations in only a couple extra lines of code.

Question 8 - Cross-Validation

What is the k-fold cross-validation training technique and how is it performed on a learning algorithm?

Answer: In the k-fold Cross-Valiation apporach, the training set is split into k smaller sets. The performance measure reported by k-fold cross-validation is then the average of the values computed in loop. The process is :

  • split the data in k bins of data potins
  • run k separate learning experiments
  • in each of the experiments, pick one of the subsets as the testing data
  • put the rest of the subsets are together into the training set
  • average the performance result of the k experiments

This approach can be computationally expensive, but does not waste too much data (as it is the case when fixing an arbitrary test set), which is a major advantage in problem such as inverse inference where the number of samples is very small. The k-fold CV results in higher accuracy.

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.

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 [14]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.metrics import make_scorer
from sklearn.tree import DecisionTreeRegressor
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': range(1, 11)}

    # 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_

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 [15]:
# 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: The optimal model has a maximum depth of 4. The results match with my guess in Q6.

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
Household net worth (income) Top 34th percent Bottom 45th percent Top 7th percent
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 [16]:
# Produce a matrix for client data
client_data = [[5, 34, 15], # Client 1
               [4, 55, 22], # Client 2
               [8, 7, 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: $344,400.00
Predicted selling price for Client 2's home: $237,478.72
Predicted selling price for Client 3's home: $931,636.36

Answer: I would recommend each client the specific price for each house, basing on the statistic stylization and logical sense.

The price prediction of client 1's home is $344,400, which lower than the mean price $454,342 and in one standard deviation of price $165,171. The three variances are close the means for each one.

The price prediction of client 2's home is $237,478, which higher than the minimum price $105,000. The price based on the less rooms, higher percent of household rate, and the higher ratio of students to teachers.

The price prediction of client 3's home is $931,636, closing to the maximum price $1,024,800. The house has 8 rooms as the biggest home in the three houses, the lowest student-teacher ratio, and the lowest household ratio.

Sensitivity

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 [17]:
vs.PredictTrials(features, prices, fit_model, client_data)

Trial 1: $324,240.00
Trial 2: $324,450.00
Trial 3: $346,500.00
Trial 4: $420,622.22
Trial 5: $302,400.00
Trial 6: $411,931.58
Trial 7: $344,750.00
Trial 8: $407,232.00
Trial 9: $352,315.38
Trial 10: $316,890.00

Range in prices: $118,222.22

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:

  • The data collected from 1978 is not suitable for analyzing current events, because the relevent environment and relevant situcations have changed a lot.
  • No, the features present in the dataset are qualify, but not sufficient. For instance, the qualify of medical security, or the age of house, may also be a effective variance to describe a home.
  • No, the model is not robust enough to make consistent prediction. The standard deviation should be considered in prediction of model.
  • No, the data collected in an urban may different with the data collected in a rural city, the other endogenous and exogenous variances will change the relevant features. It's important to use suitable data to train model. For instance, the farm area in a rural may be a effective feature like the infrastructures in an urban do.