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Sajal Sharma

Predicting Boston Housing Prices

06.01.2017 -
Python, Decision Tree, Regression, Model Complexity Analysis


In this project, we 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's monetary value.

The dataset for this project originates from the UCI Machine Learning Repository. The Boston housing data was collected in 1978 and each of the 506 entries represent aggregated data about 14 features for homes from various suburbs in Boston, Massachusetts.

For the purposes of this project, the following preprocessing steps have been made to the dataset:

  • 16 data points have an 'MEDV' value of 50.0. These data points likely contain missing or censored values and have been removed.
  • 1 data point has an 'RM' value of 8.78. This data point can be considered an outlier and has been removed.
  • The features 'RM', 'LSTAT', 'PTRATIO', and 'MEDV' are essential. The remaining non-relevant features have been excluded.
  • The feature 'MEDV' has been multiplicatively scaled to account for 35 years of market inflation.

We'll start with the reading in the data, and separating the features and prices for homes into different pandas dataframes.

1# Import libraries necessary for this project
2import numpy as np
3import pandas as pd
4from sklearn.cross_validation import ShuffleSplit
6# Import supplementary visualizations code
7import visuals as vs
9# Pretty display for notebooks
10%matplotlib inline
12# Load the Boston housing dataset
13data = pd.read_csv('housing.csv')
14prices = data['MEDV']
15features = data.drop('MEDV', axis = 1)
17# Success
18print "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 section, we will make a cursory investigation about the Boston housing data.

Since the main goal of this project is to construct a working model which has the capability of predicting the value of houses, we have separated 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.

Calculating Statistics

We'll start with calculating some descriptive statistics about the Boston housing prices.

1# Minimum price of the data
2minimum_price = np.min(prices)
4# Maximum price of the data
5maximum_price = np.max(prices)
7# Mean price of the data
8mean_price = np.mean(prices)
10# Median price of the data
11median_price = np.median(prices)
13# Standard deviation of prices of the data
14std_price = np.std(prices)
16# Show the calculated statistics
17print "Statistics for Boston housing dataset:\n"
18print "Minimum price: ${:,.2f}".format(minimum_price)
19print "Maximum price: ${:,.2f}".format(maximum_price)
20print "Mean price: ${:,.2f}".format(mean_price)
21print "Median price ${:,.2f}".format(median_price)
22print "Standard deviation of prices: ${:,.2f}".format(std_price)
Statistics for Boston housing dataset:
Minimum price: $105,000.00
Maximum price: $1,024,800.00
Mean price: $454,342.94
Median price $438,900.00
Standard deviation of prices: $165,171.13

Feature Observation

To dive a bit deeper int our data, 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.

Without building a model, let's try to figure out if an increase in the value of a feature would lead to an increase in the value of 'MEDV' or a decrease in the value of 'MEDV'.

  • 'RM': An increase in the value of this feature will lead to an increase in the value of 'MEDV'. This is because for you'd expect a home with a higher number of rooms to be more expensive that a home with lower number of rooms.
  • 'LSTAT': An increase in the value of this feature will lead to a decrease in the value of 'MEDV'. A lower class homeowner might not be able to afford expensive houses, so you'd expect them to leave in a cheaper home. A higher percentage of such people could correlate to cheaper homes in an area, and thus, a lower 'MEDV' value.
  • 'PTRATIO': An increase in the value of this feature will lead to an decrease in the value of 'MEDV'. A low student to teacher ration is typically associated with better education level of a school, as a teacher is able to focus on individual students better (than if there were more students). So, due to the presence of better quality schools, people might be willing to pay more to live in those areas, to provide their children with better education, and the prices might be higher.

We can build scatterplots to see if our intuition is correct.

1import matplotlib.pyplot as plt
2import seaborn as sns
4for var in features.columns:
5 sns.regplot(data[var],prices)




All three scatterplots above confirm our intuition.

Developing a Model

In this section of the project, we 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

For this project, we will calculate the coefficient of determination, R2, to quantify our 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.

Let's define a function that returns the r2 score for given true and predicted data.

1# Import 'r2_score'
2from sklearn.metrics import r2_score
4def performance_metric(y_true, y_predict):
5 """ Calculates and returns the performance score between
6 true and predicted values based on the metric chosen. """
8 # Calculate the performance score between 'y_true' and 'y_predict'
9 score = r2_score(y_true,y_predict)
11 # Return the score
12 return score

Goodness of Fit

As an aside, assume that a dataset contains five data points and a model made the following predictions for the target variable:

True ValuePrediction

Let's use the performance_metric function and calculate this model's coefficient of determination.

1# Calculate the performance of this model
2score = performance_metric([3, -0.5, 2, 7, 4.2], [2.5, 0.0, 2.1, 7.8, 5.3])
3print "Model has a coefficient of determination, R^2, of {:.3f}.".format(score)
1Model has a coefficient of determination, R^2, of 0.923.

The above model has a coefficient of determination, R2, of 0.923. Considering that the worse score for R2 is 0, and the best score is 1, a score of 0.923. This means that the model successfully captures more than 90 percent of the variation in the target variable. Though it is important to note that how good the score is, depends on the domain of the problem.

Shuffle and Split Data

Now we'll 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.

1# TODO: Import 'train_test_split'
2from sklearn.cross_validation import train_test_split
4# TODO: Shuffle and split the data into training and testing subsets
5X_train, X_test, y_train, y_test = train_test_split(features,prices,test_size=0.2,random_state=100)
7# Success
8print "Training and testing split was successful."
1Training and testing split was successful.

By spitting a dataset into training and testing subsets, we can train our model on the training subset, and then feed it with unseen data from the test subset to evaluate the performance of our model.

Training and testing on the same data doesn't give us a genuine evaluation of the model, at it has already seen testing the data when training, and thus might not perform well in real-world scenarios where we often deal with unseen data. Related to this is the problem of "overfitting", i.e. the model can be really accurate on the training data, but perform poorly on the training data.

Analyzing Model Performance

In this third section of the project, we'll take a look at several models' learning and testing performances on various subsets of training data. Additionally, we'll investigate the Decision Tree algorithm with an increasing 'max_depth' parameter on the full training set to observe how model complexity affects performance.

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. 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.

_Note: The section uses helper functions supplied for the purposes of this project and available in the 'visuals' module.

1# Produce learning curves for varying training set sizes and maximum depths
2vs.ModelLearning(features, prices)


Looking at the learning curve for the model with max_depth = 3,

  • The score of the training curve decreases as more training points are added. This happens because with fewer training points, the model can modify its paramters to better approximate the targets; but as the number of training points increases, perfectly fitting them becomes more difficult, and the training score goes down.
  • The score of the training curve increases as more training points are added, but there's a slight dip after adding more than 350 training points. The score starts lower because the model has not yet learned enough to predict test points. As the model receives more training points, and hence, more information, it is better suited to predict unseen data.
  • The training and testing curve seem to be converging to a score of 0.8. This usually happens when the model has stretched its limits of extracting information from the training data even though more training points are being added. So the score stabilizes.

Therefore, having more training points might not benefit the model (with max_depth of 3).

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.

1vs.ModelComplexity(X_train, y_train)


Bias-Variance Tradeoff

  • Model trained with max_depth of 1: The model suffers from high bias at this depth. Looking at the graph, we can see that both training and validation scores are low, and similar. We can say that the model is over-simplified and is not capturing the underlying relationships present in the data for both training and validation datasets.

  • Model trained with max_depth of 10: The model suffers from high variance at this depth, and is overfitting on the training data. In the graph, we can see that the training score at this depth is almost equal to 1.0, while the validation score is lower, at around 0.7. The curves also seem to be diverging away from each other at this point.

In my opinion, a maximum depth of 3 results in a model that best generalizes to unseen data. That depth is the sweet spot for model complexity, as our model performs similar on training and validation data, while the overall score for both is still relatively high at between 0.7 to 0.8. A depth lower than that gives us poor training and validation score, while a higher depth overfits on the testing data, leading to a lower validation score.

Evaluating Model Performance

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

We'll be using Grid Search and Cross Validation techniques in this section.

The grid search technique is a systematic way of going through different combinations of parameter values while cross validating the results to determine the parameter combination which gives the best performance based on a scoring technique.

In order to optimize a learning algorithm, we can apply grid search by specifying the parameters, and the possible values of those parameters. The grid search then returns the best parameter values for our model, after fitting the supplied data. This takes out the guess-work involved in seeking out the opitimal paramter values for a classifier.


The k-fold crossvalidation training technique is a way of splitting the dataset into k partitions of equal size, and then running k separate learning experiments on the training data. In each of the experiments, we chose a training set of the size of k-1 partitions, train our model on that partition, and evaluate the results on the remaining test data. The results/scores for the k experiments are then averaged out.

This technique is benefitial when using grid search to optimize a model because it allows us to look for parameter settings that perform well for different test sets. If we had a single testing set, it's easy to tune a model to perform well for that specific test set (and result in overfitting on the test set in this case), while cross validation allows us to generalize the results.

Fitting a Model

Now we'll bring everything together and train a model using the decision tree algorithm. To ensure that we are producing an optimized model, we 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.

1# Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
2from sklearn.tree import DecisionTreeRegressor
3from sklearn.metrics import make_scorer
4from sklearn.grid_search import GridSearchCV
6def fit_model(X, y):
7 """ Performs grid search over the 'max_depth' parameter for a
8 decision tree regressor trained on the input data [X, y]. """
10 # Create cross-validation sets from the training data
11 cv_sets = ShuffleSplit(X.shape[0], n_iter = 10, test_size = 0.20, random_state = 0)
13 # Create a decision tree regressor object
14 regressor = DecisionTreeRegressor()
16 # Create a dictionary for the parameter 'max_depth' with a range from 1 to 10
17 params = {'max_depth': range(1,11)}
19 # Transform 'performance_metric' into a scoring function using 'make_scorer'
20 scoring_fnc = make_scorer(performance_metric)
22 # Create the grid search object
23 grid = GridSearchCV(regressor,params,scoring_fnc,cv=cv_sets)
25 # Fit the grid search object to the data to compute the optimal model
26 grid =, y)
28 # Return the optimal model after fitting the data
29 return grid.best_estimator_

Making Predictions

Once a model has been trained on a given set of data, it can 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.

Optimal Model

1# Fit the training data to the model using grid search
2reg = fit_model(X_train, y_train)
4# Produce the value for 'max_depth'
5print "Parameter 'max_depth' is {} for the optimal model.".format(reg.get_params()['max_depth'])
1Parameter 'max_depth' is 4 for the optimal model.

Answer: The optimal model has a max depth of 4, which is close enough to the previous guess of 3 earlier. Going back to the complexity graph, the model might perform better on the test set with a max depth of 4, but since the difference is minimal (which can also be due to noise), it might make sense to chose the simpler model.

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:

FeatureClient 1Client 2Client 3
Total number of rooms in home5 rooms4 rooms8 rooms
Neighborhood poverty level (as %)17%32%3%
Student-teacher ratio of nearby schools15-to-122-to-112-to-1

Let's see what prices our model will predict for these clients, and if they seem reasonable given the features.

1# Produce a matrix for client data
2client_data = [[5, 17, 15], # Client 1
3 [4, 32, 22], # Client 2
4 [8, 3, 12]] # Client 3
6# Show predictions
7for i, price in enumerate(reg.predict(client_data)):
8 print "Predicted selling price for Client {}'s home: ${:,.2f}".format(i+1, price)
1Predicted selling price for Client 1's home: $401,333.33
2Predicted selling price for Client 2's home: $240,947.37
3Predicted selling price for Client 3's home: $893,700.00

The prices seem reasonable given the features of the homes. We can see that the highest priced home has the most rooms, lowest neighbourhood poverty level, and the lowest student-teacher ratio, all of which make intuitive sense as discussed in the Question 1. On the contrary, the home with the lowest number of rooms, highest neighbourhood poverty level, and highest student-teacher ratio is priced the lowest in our predictions.

This finishes our project. We have built a decision tree regressor that performs reasonably well given the 3 features. But should the model should be used in a real-world setting? It probably shouldn't because:

  • The data collected in 1978 is not really relevant today due to rising population levels and changing population density of different areas.
  • The features present in the data that we built our model on are not likely be sufficient to describe a home. Examples of interesting features to look at may be proximity to city center, or neighbourhood crime rate.
  • The data collected in an urban city will not be applicable in a rural city, because the people might value different aspects of a home depending on whether they live in an urban city or a rural area. For example, a person living in a rural city might value number of rooms in a home over the proximity to the city.
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