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 [37]:
# 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.
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# TODO: Minimum price of the data
minimum_price = np.min(prices)
# TODO: Maximum price of the data
maximum_price = np.max(prices)
# TODO: Mean price of the data
mean_price = np.mean(prices)
# TODO: Median price of the data
median_price = np.median(prices)
# TODO: Standard deviation of prices of the data
std_price = np.std(prices)
# Show the calculated statistics
print "Statistics for Boston housing dataset:\n"
print "Minimum price: ${:,.2f}".format(minimum_price)
print "Maximum price: ${:,.2f}".format(maximum_price)
print "Mean price: ${:,.2f}".format(mean_price)
print "Median price ${:,.2f}".format(median_price)
print "Standard deviation of prices: ${:,.2f}".format(std_price)
As a reminder, we are using three features from the Boston housing dataset: 'RM', 'LSTAT', and 'PTRATIO'. For each data point (neighborhood):
'RM' is the average number of rooms among homes in the neighborhood.'LSTAT' is the percentage of homeowners in the neighborhood considered "lower class" (working poor).'PTRATIO' is the ratio of students to teachers in primary and secondary schools in the neighborhood.Using your intuition, for each of the three features above, do you think that an increase in the value of that feature would lead to an increase in the value of 'MEDV' or a decrease in the value of 'MEDV'? Justify your answer for each.
Hint: Would you expect a home that has an 'RM' value of 6 be worth more or less than a home that has an 'RM' value of 7?
| Variable | Expected relation of house price with respect to variable |
|---|---|
RM |
positive |
LSTAT |
negative |
PTRATO |
positive |
The relations I would expect to exist in the data are summarized in the table above. I will explain the reasoning behind each conclusion next. I expect that the average number of rooms in a neighborhood will have a positive relation on MEDV. I base this on the observation that an increase in number of rooms will most likely mean a higher surface of the house. Since the price per square foot is always positive, a higher surface will lead to a higher house price.
I expect the LSTAT to have a negative impact on MEDV. I base this on the assumption that a higher percentage of "lower class" home owners will likely mean that they have less income to spend on purchasing a house. As a result the house prices in these neighbourhoods will likely be lower than a neighborhood where homeowners are richer.
I expect that PTRATIO, the ratio of students to teachers, will have a negative impact on the house prices. I base this assumption on the premise that home owners would be willing to pay more for houses close to a school were less students are assigned per teacher than for a house close to a school were there are more students assigned per teacher. Since the number will likely be within the range of [1/40, 1/10] or [0.025, 0.1] we may need to take the $x_{\text{PTRATO}}\tick = \left x_{\text{PTRATO}} \right^{-1}$ transformation. Applying this transformation would scale our variable from [10, 40]. In that case the sign would flip.
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.
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# 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. """
from sklearn.metrics import r2_score
# 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.
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# 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: Based on the $R^2$ I would consider this model as being successfull at capturing the variation of the target variable because 0.923 is close to 1. As such we can conclude that there is a high correlation between the prediction and the true value, which leads me to conclude that the model is of acceptable quality.
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 [41]:
# 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,
random_state = 0,
train_size = 0.80)
# Success
print "Training and testing split was successful."
The benefit of splitting the data in a training and test set is that the test set can be used as an independent test of the quality of our model. We can use different models on our training set and ultimately compare the predictive power of different models on our test set.
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.
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# 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: If we focus on the performance plot for trees with a maximum depth of three levels deep, we see that the performance on the test data converges to 75 percent. We see that with fifty data points the model has a in sample performance of 85 percent and out of sample performance of around 60 percent. As fifty points are added we see the in-sample performance dropping and the out-of-sample performance increasing. The reason for this is that more variance needs to be explained in the in-sample data, and not all the variance can be captured by the tree since the depth is too shallow. However, it does lead to an improvement in the out-of-sample dataset because the model is able to forecast the outcome in more generic terms. We see that the data still improves up to two hundred data points for the out-of-sample data set. It then decreases and increases again, however, at a much slower rate than the first fifty data points. This is because the decision tree algorithm cannot capture more variance in the model.
We see from the graph that adding additional data after two hundred does increase the model quality marginally. Adding more data would be good up to 350 data points but drop after that. It is however a very marginal increase which is hard to identify graphically. Maybe a log transform of the last end of the graph would permit us to decide whether we want to add more data.
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.
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vs.ModelComplexity(X_train, y_train)
When the model is trained with a maximum depth of 1, does the model suffer from high bias or from high variance? How about when the model is trained with a maximum depth of 10? What visual cues in the graph justify your conclusions?
Hint: How do you know when a model is suffering from high bias or high variance?
Answer: When the model has a maximum depth of one, the model has a strong bias. This is due to the fact that the model is over-simplistic for the complexity in the data. As the restriction on the maximum depth is relaxed, we see that the bias is reduced. We can see this from the graph because the model goes to a performance of 1 on the training set. The performance on the validation set behaves differently. It increases up to a maximum depth of four and then decreases again. This shows that the model has a high variance. It is unable to generalize the results that it learned ont he training set on the validation set which leads to a lower performance.
Another hint that shows us that a tree with max depth one has high bias is that the score is low and therefore the error is high. Lastly, if we stand still to the structure of a one level deep tree we quickly realize that the model is likely an oversimplification of the data. It is very unlikely that a single attributes exists in the data that is able to predict the price accurately.
Answer: I believe that a maximum depth of four is ideal for the current model given the split we created. I base this premise on the fact that the performance on the validation score increases until four and then decreases again. If we would define the problem as $\arg\max_{d \in [1, 10]}(r(tree(d))$, where $r$ calculates the out-of-sample performance on the validation data and $tree(d)$ is a function that yields a tree with depth $d$, the solution would be four.
Answer: A grid search technique in general terms is a method that allows to tune a parameter in a learning algorithm that leads to the best predicitive model. In the case above the grid search algorithm would compute the $\arg\max$ problem that we defined. The term grid search refers to the fact that typical numerical optimization such as the Levenberg-Marquandt algorithms cannot typically be used to find the optimum since it is often not possible to define a smooth first and/or second order derivative for the function that is to be optimised. As a result a grid is defined for the parameters of the machine learning algorithm and the algorithm is computed for each possible combination of values in the grid. Finally the global optimum in the grid is selected.
Answer: A $k$-fold cross validation will create $k$ sized samples without replacement and train a model on the sample and measure the performance on left out data. It will do this $k$ times. At each iteration the method selects one sample and combines the remaining $k-1$ samples to create the training data. It will fit the model on that training data and measure the performance on the left out data. It will repeat this on all the samples. The final error of a model is measured as the average error on all the left out training sets.
The benefit of using this in combination with the grid search is that it will permit grid search to find a more optimal solution. If one were to use grid search on itself without $k$ fold cross validation grid search would likely overfit the model on the data which in turn would lead to poor results on the test data set.
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 [60]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.metrics import make_scorer
from sklearn import grid_search
from sklearn import tree
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 = tree.DecisionTreeRegressor()
# TODO: Create a dictionary for the parameter 'max_depth' with a range from 1 to 10
params = {'max_depth': range(1, 10)}
# TODO: Transform 'performance_metric' into a scoring function using 'make_scorer'
scoring_fnc = make_scorer(performance_metric)
# TODO: Create the grid search object
grid = grid_search.GridSearchCV(estimator = regressor, param_grid = params,
scoring = scoring_fnc,
n_jobs = 1, 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.
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# 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: By applying the grid search with $k$ fold cross validation we find that the optimal max_depth is four, which aligns with our observation in Question 6.
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 [61]:
# Produce a matrix for client data
client_data = [[5, 17, 15], # Client 1
[4, 32, 22], # Client 2
[8, 3, 12]] # Client 3
# Show predictions
for i, price in enumerate(reg.predict(client_data)):
print "Predicted selling price for Client {}'s home: ${:,.2f}".format(i+1, price)
Answer:
Predicted selling price for Client 1's home: \$391,183.33
Predicted selling price for Client 2's home: \$189,123.53
Predicted selling price for Client 3's home: \$942,666.67
We see that the house in the poorest neighborhood and highest student-teacher ratio has the lowest price. This aligns with our hypothesis that we had put forward in Question 1. The last house, with 8 rooms, least poverty and low student to teacher ratio has the highest price. This again aligns with the hypothesis put forward in Question 1.
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 [59]:
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: I would not recommend to use the model to predict current day prices either in Boston or in any other US city. The model we found does not fluctuate too much. We should however test the model on a sample that we leave out in our $k$-fold cross validation and our grid search. This would show us whether the model would have performed well in 1978.
Whether this model can be utilized on current data is very unlikely. Data was collected in 1978, the US has seen a very significant drop in crime since than and a strong move from suburbia back to cities. In some cities this push is so strong that even proximity to public transportation is a strong driver of house prices. Since this data is not part of the data set it is likely that the house prices would be wrongly predicted. Moreover, since 1978 there has been an inflation. This has lead to overall house price increase. The model would not be able to capture this change.
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