Machine Learning Engineer Nanodegree

Model Evaluation & Validation

Project: 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 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.

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 [1]:
# Import libraries and check the versions
import pandas as pd
import sys
import numpy as np
import sklearn

print('Python version: {}'.format(sys.version))
print('Numpy version {}'.format(np.__version__))
print('Pandas version {}'.format(pd.__version__))
print('Sklearn version: {}').format(sklearn.__version__)

Python version: 2.7.13 |Anaconda, Inc.| (default, Sep 21 2017, 17:38:20) 
[GCC 4.2.1 Compatible Clang 4.0.1 (tags/RELEASE_401/final)]
Numpy version 1.13.1
Pandas version 0.20.3
Sklearn version: 0.19.0

In [2]:
# Import libraries necessary for this project
import numpy as np
import pandas as pd
from sklearn.cross_validation import ShuffleSplit
import matplotlib.pyplot as plt 

# Import supplementary visualizations code
import visuals as vs

# Pretty display for notebooks
%matplotlib inline

# for more clear plots
from IPython.display import set_matplotlib_formats

# 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)

Boston housing dataset has 489 data points with 4 variables each.
/anaconda/envs/python2/lib/python2.7/site-packages/sklearn/ DeprecationWarning: This module was deprecated in version 0.18 in favor of the model_selection module into which all the refactored classes and functions are moved. Also note that the interface of the new CV iterators are different from that of this module. This module will be removed in 0.20.
  "This module will be removed in 0.20.", DeprecationWarning)
/anaconda/envs/python2/lib/python2.7/site-packages/sklearn/ DeprecationWarning: This module was deprecated in version 0.18 in favor of the model_selection module into which all the functions are moved. This module will be removed in 0.20

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, 'MEDV', 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 'MEDV', which is stored in prices.
    • Store each calculation in their respective variable.

In [3]:
# look at first five value of features

0 6.575 4.98 15.3
1 6.421 9.14 17.8
2 7.185 4.03 17.8
3 6.998 2.94 18.7
4 7.147 5.33 18.7

In [4]:
# look at first five value of target 

0    504000.0
1    453600.0
2    728700.0
3    701400.0
4    760200.0
Name: MEDV, dtype: float64

In [5]:
# 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)

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

In [6]:
#pandas describe method gives us the summary statistics

count    4.890000e+02
mean     4.543429e+05
std      1.653403e+05
min      1.050000e+05
25%      3.507000e+05
50%      4.389000e+05
75%      5.187000e+05
max      1.024800e+06
Name: MEDV, dtype: float64

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 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: This problem can phrased using examples like below.

  • Would you expect a home that has an 'RM' value(number of rooms) of 6 be worth more or less than a home that has an 'RM' value of 7?
  • Would you expect a neighborhood that has an 'LSTAT' value(percent of lower class workers) of 15 have home prices be worth more or less than a neighborhood that has an 'LSTAT' value of 20?
  • Would you expect a neighborhood that has an 'PTRATIO' value(ratio of students to teachers) of 10 have home prices be worth more or less than a neighborhood that has an 'PTRATIO' value of 15?


Let's draw correlation matrix to see how features are correlated. We will use Seaborn library to do that. +1 is a strong positive correlation which means if a value increases, the other should increase. -1 is a strong negative correlation which means if a value increase, the other should decrease.

In [7]:
import seaborn as sns

# calculate the correlation
corr = data.corr()

# plot the heatmap

RM and MEDV has positive correlation. If 'RM' value increases, I expect 'MEDV' to go up. Therefore, a house with 5 rooms should be more expensive than a house with 2 rooms.

LSTAT and MEDV has strong negative correlation. If 'LSTAT' value(percent of lower class workers) increases, 'MEDV' should decrease. So, if there are more lower class, then house prices is expected to be lower.

PTRATIO and MEDV has negative correlation. If 'PTRATIO' value increase, I expect 'MEDV' to decrease. That's why a teacher with 15 students in a school should be a more expensive negihbourhood than a teacher with 35 students.

In [8]:
for i, col in enumerate(features.columns):
    plt.subplot(1,3, i+1)
    sns.regplot(x=data[col], y=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 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:

  • 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 [9]:
# 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

Run the code cell below to use the performance_metric function and calculate this model's coefficient of determination.

In [10]:
# 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.
  • Would you consider this model to have successfully captured the variation of the target variable?
  • Why or why not?

Hint: The R2 score is the proportion of the variance in the dependent variable that is predictable from the independent variable. In other words:

  • R2 score of 0 means that the dependent variable cannot be predicted from the independent variable.
  • R2 score of 1 means the dependent variable can be predicted from the independent variable.
  • R2 score between 0 and 1 indicates the extent to which the dependent variable is predictable. An
  • R2 score of 0.40 means that 40 percent of the variance in Y is predictable from X.


This model succesfully generalize the data and predicts very well based on the R2 score being 92%. R2, the coefficient of determination measures how well a model predicts. R2 is always between 0 and 1. It could be less than 0 and that is a terrible model.

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 [11]:
# TODO: Import 'train_test_split'
from sklearn.model_selection 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=.2,random_state=0)

# 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: Think about how overfitting or underfitting is contingent upon how splits on data is done.


If we want to test a model, it should be tested on a dataset that the model has never seen. On the contrary, if we test the model on the data it has seen before, the model memorize it. Therefore it does a bad job on the unseen data.

Splitting data into test and training allow us to measure our model in different ways. It is common to split data 70% -80% traninig set.

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 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 [12]:
# 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? Generally speaking, the more data you have, the better. But if your training and testing curves are converging with a score above your benchmark threshold, would this be necessary? Think about the pros and cons of adding more training points based on if the training and testing curves are converging.


I choose the fourth graph with the max depth of 10. In this graph training score is really high but testing score is low. The model learned, even memorized the training data but it is not doing a good job on the test data. It is overfitting in other words, high variance.

Increasing number of points would not make a difference in this model since it is reached it is optimal score and and has converged.

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 Q5 and Q6.

In [13]:
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: High bias is a sign of underfitting(model is not complex enough to pick up the nuances in the data) and high variance is a sign of overfitting(model is by-hearting the data and cannot generalize well). Think about which model(depth 1 or 10) aligns with which part of the tradeoff.


When the model is trained with a maximum depth of 1, it suffers from high bias(underfitting). Both training and testing scores are low. It is not a useful model.

When the model is trained with a maximum depth of 10, this time it suffers from high variance(overfitting). Training score increases because model is learning the training data, however, it is terrible on the data it has never seen beeen, for instance, testing data.

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?

Hint: Look at the graph above Question 5 and see where the validation scores lie for the various depths that have been assigned to the model. Does it get better with increased depth? At what point do we get our best validation score without overcomplicating our model? And remember, Occams Razor states "Among competing hypotheses, the one with the fewest assumptions should be selected."


For this particular case, max depth of 3 or 4 would be the sweet spot. The model is doing great on both testing and training score. After max depth of 4, gap between training and testing score is increasing, this is the sign of high variance (overfitting).

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.

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

Hint: When explaining the Grid Search technique, be sure to touch upon why it is used, what the 'grid' entails and what the end goal of this method is. To solidify your answer, you can also give an example of a parameter in a model that can be optimized using this approach.


Grid search technique is for finding the best hyperparameters of a model. The goal of the method is to tune hyperparameters of the model to fit the data best without overfitting. At the end of the grid search we find the best values for our hyperparameters.

Below is a demonstration of grid search. Random Forest is a great algorithm for both classification and regression tasks. Since this is a regression problem, we are going to use Random Forest Regressor. Here, 3 hyperparameters are being tested. n_estimators, criterion, and max_depth with the scoring type R_squared. GridSearchCV tests exhaustively all values given in the param_grid parameter.

In [14]:
from sklearn.ensemble import RandomForestRegressor
from sklearn.grid_search import GridSearchCV

#define parameters and values for the grid
params={'n_estimators':[5,10,15, 20],
        'criterion':['mse', 'mae'],

grid = GridSearchCV(RandomForestRegressor(), param_grid=params, cv=5, scoring='r2')

/anaconda/envs/python2/lib/python2.7/site-packages/sklearn/ DeprecationWarning: This module was deprecated in version 0.18 in favor of the model_selection module into which all the refactored classes and functions are moved. This module will be removed in 0.20.

In [15]:, prices)

GridSearchCV(cv=5, error_score='raise',
       estimator=RandomForestRegressor(bootstrap=True, criterion='mse', max_depth=None,
           max_features='auto', max_leaf_nodes=None,
           min_impurity_decrease=0.0, min_impurity_split=None,
           min_samples_leaf=1, min_samples_split=2,
           min_weight_fraction_leaf=0.0, n_estimators=10, n_jobs=1,
           oob_score=False, random_state=None, verbose=0, warm_start=False),
       fit_params={}, iid=True, n_jobs=1,
       param_grid={'n_estimators': [5, 10, 15, 20], 'criterion': ['mse', 'mae'], 'max_depth': [1, 5, 10]},
       pre_dispatch='2*n_jobs', refit=True, scoring='r2', verbose=0)

In [16]:
#best parameters

{'criterion': 'mae', 'max_depth': 5, 'n_estimators': 20}

In [17]:
# best estimator

RandomForestRegressor(bootstrap=True, criterion='mae', max_depth=5,
           max_features='auto', max_leaf_nodes=None,
           min_impurity_decrease=0.0, min_impurity_split=None,
           min_samples_leaf=1, min_samples_split=2,
           min_weight_fraction_leaf=0.0, n_estimators=20, n_jobs=1,
           oob_score=False, random_state=None, verbose=0, warm_start=False)

Question 8 - Cross-Validation

  • What is the k-fold cross-validation training technique?

  • What benefit does this technique provide for grid search when optimizing a model?

Hint: When explaining the k-fold cross validation technique, be sure to touch upon what 'k' is, how the dataset is split into different parts for training and testing and the number of times it is run based on the 'k' value.

When thinking about how k-fold cross validation helps grid search, think about the main drawbacks of grid search which are hinged upon using a particular subset of data for training or testing and how k-fold cv could help alleviate that. You can refer to the docs for your answer.


K-fold cv randomly divides data into k subsets and each time, it uses one block for validation and k-1 block for training. Each iteration, scoring method (accuracy, error etc) is being saved and then be averaged to produce a single value.

In [18]:
from IPython.display import Image

# credits of image:
# other source:


In grid search, we are finding the best hyperparameters for our model. K-Fold cv allows us to use every single data point for validation at least once. This gives us a better estimate of how the model is doing. However, this is still biased. Once we complete cv, we do test our model on testing data which our model has never seen.

For example, if we use a single train-validation split in the grid search, we get the best parameters for that particular split. However, those parameters won't do a good job on the testing data. Using K-Fold cv on grid search, we use various validation set, so we select the best parameters for generalize case which will perform better on testing data.

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.

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.

Please note that ShuffleSplit has different parameters in scikit-learn versions 0.17 and 0.18. 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 [19]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.tree import DecisionTreeRegressor
from sklearn.metrics import make_scorer
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
    # sklearn version 0.18: ShuffleSplit(n_splits=10, test_size=0.1, train_size=None, random_state=None)
    # sklearn versiin 0.17: ShuffleSplit(n, n_iter=10, test_size=0.1, train_size=None, random_state=None)
    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(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)}

    # TODO: Transform 'performance_metric' into a scoring function using 'make_scorer' 
    scoring_fnc = make_scorer(performance_metric)

    # TODO: Create the grid search cv object --> GridSearchCV()
    # Make sure to include the right parameters in the object:
    # (estimator, param_grid, scoring, cv) which have values 'regressor', 'params', 'scoring_fnc', and 'cv_sets' respectively.
    grid = GridSearchCV(estimator=regressor, param_grid=params, scoring=scoring_fnc, cv=cv_sets )

    # Fit the grid search object to the data to compute the optimal model
    grid =, 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 [20]:
# 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.

Hint: The answer comes from the output of the code snipped above.


Parameter 'max_depth' is 4 for the optimal model. In Question 6, I was guessing that 3 or 4 would be the best max depth values for this model.

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
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. Of the three clients, client 3 has has the biggest house, in the best public school neighborhood with the lowest poverty level; while client 2 has the smallest house, in a neighborhood with a relatively high poverty rate and not the best public schools.

Run the code block below to have your optimized model make predictions for each client's home.

In [21]:
# 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)

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


Client1 ($391,183.33): RM seems reasonable. LSTATS value is also in the price range. PTRATIO is below the average. But RM and LSTAT are the most important features in the model and both are in the price range. Overall, the price seems good.

Client2 ($189,123.53): 4 bedroom house sells range about 300K-500K. The predicted price is low. Predicted price based on LSTAT value is lower than average. Predicted price for PTRATIO seems good. House prices for 22 ratio is around 200K. Overall, I would recommend to increase to price and not to sell it less than 200K.

Client3 ($942,666.67): 8 bedroom houses are around 1M. Price seems good. Houses are selling about 700K-800K for LSTAT value is 3. Predicted price is really high in that case. For PTRATIO=12, houses are about 600K-700K. Predicted price is high again. I would recommend to lower the price. 800K is reasonable price.

In [22]:
for feature in features:
    g = sns.JointGrid(x=features[feature], y=prices, data=data)  
    g.plot_joint(sns.regplot, order=2)  

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