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 [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 visuals.py
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))
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 [3]:
prices_a = prices.as_matrix()
# TODO: Minimum price of the data
minimum_price = np.amin(prices_a)
# TODO: Maximum price of the data
maximum_price = np.amax(prices_a)
# TODO: Mean price of the data
mean_price = np.mean(prices_a)
# TODO: Median price of the data
median_price = np.median(prices_a)
# TODO: Standard deviation of prices of the data
std_price = np.std(prices_a)
# 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 should increase the MEDV as it increases its value (directly proportional) because it impacts with the house size (more rooms). LSTAT should be reversely proportional to MEDV, because as we have more "lower class" in the neighborhood, we probably will have a simpler house (not well finished) and face more security problems, so lower price. PTRATIO, it is hard to guess, because more students could be a family area that is good, but we need to analyze the LSTAT in conjunction, also if we have more teachers, could be a more quite region (good) or a not residential area that will be (not good). I think there's a correlation with the other classes that gives us a better hint.
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 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:
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: The estmator got an excellent R2 value, 0.923 is very close to 1, so it can predicts very well the model. But I think it could not captures the variation (of a real example) because it has only 5 data points, and that is too few. However, it has a good prediction to the model for the given data, and for that it captures the variation.
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.model_selection import train_test_split
prices_a = prices.as_matrix()
features_a = features.as_matrix()
# TODO: Shuffle and split the data into training and testing subsets
X_train, X_test, y_train, y_test = train_test_split(features_a, prices_a, test_size=0.2, random_state=42)
# Success
print("Training and testing split was successful.")
Answer: If you don't have a test subset, you can not estimate if your predictor is good enought for the "new data", or data that is not in the traning set, so you will probably overfit the predicted model and you can not verify it until new data arives. So, probably, the predictor will work great with the training set, but could not be a good estimator for the real data. Also, it will be hard to analyze the predictor efficiency.
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 [7]:
# 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: Analysing max_depth = 10, with more training points we improve our model, but we can notice that at some time it do not increase the score. As we add more training points we have better test results because we start to generalize our model, but it also has a convergence, so more training points do not affect the test at some point. Until it reaches the stagnation point, the added points were good, but after that, it could not improve the model.
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)
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: If the model uses a maximum depth of 1, it will be oversimplified, and it suffers from high bias. At the figure above, it is possible to say that the performance is low (r2), even for the training score, what means that it is not capturing the features correlation. If it uses a maximum depth of 10, it, probably, is overfitting (not generalized well), so it suffers from the high variance. It is noticed because the error on the test set is much higher (lower r2) that in the training set. So it a start to get not generalized behavior, but some specific behavior from the training set.
Answer: We can use 3 or 4 as de maximum depth (I go with 3). The validation score has an inflection at 4, so it starts to decrease the validation score (changes the tangential direction).
Answer: It is a systematically way to optimize an algorithm parameter (or parameters). GridSearch tries to analyze the parameters variation we want to test and select the optimal values for them. It outputs parameters that achieve the best performance (as a combination). The parameters can be manually specified and which range or values we want to use. Also, it depends on the training set.
Answer: The k-fold cross-validation creates k equal sized subsamples. k times, one subsample is selected for validation and the remaining k - 1 are used for training. After all tries, k-fold computes the average error. It could generalize the model and gives more accuracy because the model will be trained and test aginst all data, and also " matters less how the data gets divided". Each data value will be used k -1 times for training and one time for validation. We can also use the multiple training for optimization, as we have more sets to test. A simple train/test split does not give a good way to evalute the best parameter, because the goal rely on get a good result with train and test set, so there no much changes to do, but, with k-fold we need to get a good result with much more train/test variation, so we force the estimator to be more generalized.
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.
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 [9]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.tree import DecisionTreeRegressor
from sklearn.metrics import make_scorer
from sklearn.grid_search import GridSearchCV
#from sklearn.model_selection import ShuffleSplit
from sklearn.cross_validation import ShuffleSplit
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)
#cv_sets = ShuffleSplit(n_splits = 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
#params = {'max_depth': range(1,11)}
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, greater_is_better = True)
# TODO: Create the grid search object
#grid = GridSearchCV(regressor, param_grid=params, scoring=scoring_fnc, cv=cv_sets.get_n_splits())
grid = GridSearchCV(regressor, param_grid=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 [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']))
Answer: The answer is 4, and it is the range we "guess" on question 6. But it is logical because we are optimizing one parameter (so it is not so difficult) and we have the graphs to know the exact estimator behavior against the parameter change.
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))
features.describe()
Out[11]:
Answer:
Client 1: 403025.00 The value looks good, the value is close to the medium, and if we analyse the features, we can note that this client want the features close to the medium of each feature. So the predictor will go to something to close to it.
Client 2: 237478.72 Also it is good, the client chooses values close to the minimum RM (which decreases the price), close to the maximum LSTAT (which also decreases the price) and PTRATIO at the maximum. So all those features forces the estimator to a lower price.
Client 3: 931636.36 This client is the oposite of client 2, the RM is too close to the maximum value, LSTAT is close to the minimum and the PTRATIO is close to the minimum. Thus, it is a set that the estimator will try to reach a higher value.
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)
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: This model is simplified because there are some other items more important to describe the house price. I think that data from 1978 is not so relevant because one of the most important features we need to know about this market is the location, and the location could change a lot from 39 years, it could be a better place now or a dangerous place. As I said, location is an important feature, and it is not present. I think the model could not give us an accurate prediction for real-world data, but for the exercise, it does a good job. Also, this data is for urban houses, rural predictions need other features (like size), so the predictor needs rural data to predict correctly, but we can use isUrban as a feature.
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.