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
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 [3]:
# TODO: Minimum price of the data
minimum_price = np.amin(prices)
# TODO: Maximum price of the data
maximum_price = np.amax(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' will increase value of 'MEDV'. Having more rooms in a house implies that the size of the house increases as well, due to the extra space requiered. Having more square feet in the property increases the value. 'LSTAT' will decrease the value of 'MEDV'. Prices of the houses are generally also driven by the quality and exclusivity of the area. A high number of (working poor) means that the area is not exlusive, so there is no price value added to it.'PTRATIO' will decrease value of 'MEDV'. (Assuming we calculate the ratio by students/teachers) Ex, a ratio of 5 is more valuable than one of 14. Better education is provided when classes have a lower ratio of student to teacher, since the teacher is capable of providing a more detailed and intimate service.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'
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'
from sklearn.metrics import r2_score as r2
score = r2(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:
Since the obtained score of R^2 is 0.923, and considering that 1 is the optimal value then we can asses that the model was predicted correctly.
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, test_size=.20, train_size=.80, random_state=11)
# Success
print "Training and testing split was successful."
Answer:
The main benefit of dividing our model into a training and testing portion, is that it will allow us to analyse how our model behaves under new data. Without a testing portion it is not posible for us to analyze how our model is doing in an actual scenario, since we can only use the data we used to train it. This could lead to a model sensitive to overfitting, which will give us unaccurate predictions along the way.
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:
Based on the performance metric used for testing in training, which is R2, we know that the best scenario should be the one in which the gap between graphs is minimun and where both graphs are closer to 1. With this information in mind we can get the following observations:
So in conclusion, the graph with the _maxdepth=3 will be selected as the maximun depth. When analyzing all the graphs, as more points are added the assymptote growth of the training curve gets closer to one, by comparing the limit in depth 1 (.45) and depth 10 (.95). In case of the testing score we can observe how the limit grows from depth 1 to 3, reaching a maxima point close to (.8) and then it is decreasing it's value in while more depth points are added. So having even more points will not benefit the model cause it will make the model sensitive to variance creating overfitting.
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:
Answer:
I'll personally pick a max_depth of 3, it is the best estimation that provides the results closer to each other and also the closest to one as we can obtain (roughly around .75). Balancing in this case in between bias and variance to an optimun level.
Answer:
Grid Search is a way to systematically train a specific set of models, which each of them differ from the others based on the given parameters, inside a specific grid. The evaluation of this models will be obtained by using cross-validation against other areas of the grid. The best performing model will be used.
In order to apply it to a lerning algoritms there is a couple of things that you need to define. First, you need to declare the parameters and the accepted values for those. Then, create a classifier and search over those variables in an estimator. Finally, use a fit functions that will try all parameters and return the optimal one.
Answer:
The K-fold cross-validation technique works under the premise of partitioning the data set in k-bins of equal size. Then k separate experiments will be run, using one bin as testing and the rest as training, ensuring in this way that all data in the data set is used to train our model. The average of all scenarios is returned.
A grid search technique will ensure us to return an optimal model. However with out the use of cross-validation this model might be a subject to high variance. By using K-fold cross-validation we also ensure that our model will not suffer from variance in new datasets (If the features and dataset are well constructed).
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 [9]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.metrics import make_scorer
from sklearn.tree import DecisionTreeRegressor as DT
from sklearn.grid_search import GridSearchCV as GS
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 = DT()
# 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 = GS(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 optimal _maxdepth is 4. I predicted to be 3 using a safer bet, knowing that 4 had a better score in terms of the evaluation and training set but the gap in between both being bigger.
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)
print '\nMean of Average Room number {:,.2f}'.format(np.mean(data['RM']))
print 'Mean of Neighborhood poverty level {:,.2f}'.format(np.mean(data['LSTAT']))
print 'Mean of Student-Teacher ratio {:,.2f}'.format(np.mean(data['PTRATIO']))
import matplotlib.pyplot as plt
plt.hist(prices, bins = 30)
for price in reg.predict(client_data):
plt.axvline(price, c = 'r', lw = 3)
Answer:
First, lets remeber the stats:
From here we can observe that all the predictions are in the range described by the minimun and the maximun price. In terms of the features, the combination that should generate the more expensive house should have a greater number of rooms, a smaller ratio of student-teacher and the low poverty level in the Neighborhood. On contrast, the opposite will give you a cheaper house, with some laying in the average.
More specifically:
Overall I'll recommend our clients to go a portion higher than the calculated prices, maybe a 5-10% to have some room for negotiation. The prices are consistent with the parameters features in the dataset.
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:
The collected data, although the prices where normalized to macth nowadays inflation, is not particularly relevant in the actual Boston. In a 38 years span neighborhoods can be affected by the development of the city, like new buldings and/ or new areas becoming more revelant. The price of the same areas we have in the dataset are not the same as nowadays.
The features present in the dataset are also not sufficent to rate an actual value of a house, although relevant houses with the same characteristics might not have the same price based on several factors like terrain size, construction size, materials used, when was the house erected, etc.
The model, as seen by the section in sensitivity, is not robust enough to make consistent predictions. A range in the prices of almost $70,000 is really big in relation with the house princing, since it represents a 10-20% of the final price of the house.
Data collected in an urban city should not be applied to a rural setting for different reasons. First of all, the price points of a rural city tend to be lower due to the the lesser amount of services available and the amount of people looking for housing in a specific area. Which just invalidades all of our model.
So in conclusion the model should not be used in a real life scenario, just as a learning expirience.