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 [45]:
# Import libraries necessary for this project
import numpy as np
import pandas as pd
from sklearn.cross_validation import ShuffleSplit

# Import supplementary visualizations code visuals.py
import visuals as vs

# Pretty display for notebooks
%matplotlib inline

# Load the Boston housing dataset
data = pd.read_csv('housing.csv')
prices = data['MEDV']
print (prices.head())
features = data.drop('MEDV', axis = 1)
print (features.head())
    
# Success
print ("Boston housing dataset has {} data points with {} variables each.".format(*data.shape))

0    504000.0
1    453600.0
2    728700.0
3    701400.0
4    760200.0
Name: MEDV, dtype: float64
      RM  LSTAT  PTRATIO
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
Boston housing dataset has 489 data points with 4 variables each.

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 [46]:
# 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))

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

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?

Answer: The feature RM is often correlated to the livable surface of a house, so a home with more rooms can be an indicator of higher prices. Therefore we can expect higher RM value to mean higher prices.

The feature LSTAT indicates the percent of lower class workers in the neighborhood. This can give an insight of the trend value of houses in the neighborhood as it could give a threshold to what people can afford in the area. A higher LSTAT value can indicate lower house prices.

The feature PTRATIO indicates the ratio of students/teachers and can be an indicator of education quality. Generally, the less student per teacher, the better the education as the teacher can give more care to each student. This can also be correlated to the parents wealth, as good schools can be expensive. Therefore a higher PTRATIO can indicate lower house 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 [47]:
# 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 [48]:
# 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.

Answer:The obtained R2 score is equal to 0.923 meaning that 92.3% of the variance in the target variable can be explained by the feature. Appreciation of what is a good R2 score depends on the problem itself and how thorough we want to be considering the prediction. For this example however, as no further precision are mentioned about the data and the end application, I would arbitrarily consider that the model successfully captures the variation of the target variable as 92.3% can be considered (in general) as a satisfactory score.

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 [49]:
# 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 = 0.2, random_state = 42)

# 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.

Answer: How can we determine if a model is good ? We need to be able to test it. Testing implies spliting the data in two sets, the training set and the testing set. The training set will allow us to train the model. The set needs to be big enough to allow the model to find the best general rules underlying the data. The testing set is use at the very end of the process (we never use them for training), once the model is trained, to see if the predictions made by the model fit the labels. The testing process allows us to detect underfitting and overfitting behaviors.

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 [50]:
# 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.

Answer: I choose the graph on top right with a mwximum depth of 3.

The score of the training curve decreases as more training points are added. Inversely, the score of the testing curve increases as more training points are added. We see that the training and testing curves converge quickly to a score around 0.8. Already with only 50 training points the testing score reaches a little bit more than 0.6 and increasing the number of training points to 200 leads us close enough to the testing curve 0.8 score plateau.

Adding more training points to that would not be beneficial to the model as it wouldn't increase the testing score by much considering the standard deviation and would only slower the execution time.

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 [51]:
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.

Answer: When trained with a maximum depth of 1, the model suffers high bias because both the training and the validation score are rather low around 0.4. This means that the model isn't doing a good job at fitting the data in the training set and therefore the testing points are not precisely predicted, it is a case of underfitting the data.

When trained with a maximum depth of 10, the model suffers high variance because the training score is close to 1 and the validation score is rather low, around 0.7 only. This means that the model is being too specific at memorizing the points on the training set leading to an almost perfect training score, but when it comes to the validation set, the model is unable to generalize well indicating overfitting.

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."

Answer: A maximum depth of 4 would give a model that best generalizes the data.

The intuition that lead me to think so is that the validation score is at its highest around a maximum depth of 4 meaning that it is where the model is best. The validation score then decreases as we add more depth to the decision tree whereas the testing score inscreases up to 1 meaning that after this point, the model gets overcomplicated and leads to 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.

Question 7 - Grid Search

  • 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.

Answer: Grid search technique is used to search for the optimal combination of tuning parameter. Tuning parameters can be a very tedious work to do manually especially when we want to look at several different parameters. Grid search automates this process, we just have to specify the parameters we want to look at and the different values it should be tested for. For each possible combination, a model is trained and evaluated. All scores are compared and the best one gives the best model with the best combination of parameter values. We can then use this optimized model to predict new points.

For example, we want to train a decision tree to make predictions. We can picture the grid search where each grid is a different model with different combinations of parameter values. In the case of our decision tree we want to tune the parameter depth and test it for all values ranging from 1 to 20. We specify the parameter and the estimator when using the grid search method and for each value of depth, a new model will be trained and evaluated, meaning 20 models will be trained. The best score then indicates which model with which value of depth was best.

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.

Answer: When we split our data into a training and a testing set, we technically 'loose' the information of the points in the testing set for training. The k-fold cross-validation allows us to recycle information.

For that, the data is split into k smaller sets of equal size that I will call buckets. Each bucket will be used 1 time as a testing set and k - 1 time as a training set. The operation is repeated k times as the bucket for testing set changes each iteration. Testing performances are then averaged to get our model. In doing so, all the points are used for training at some point (when not shuffled) and this is more reliable then doing the training-test slip only once, especially when we have few data points. However this technique can be computationally expensive as it need to train the model k times.

In the case of grid search, the number of model tested is as big as all the different combination of parameters to test and values. Cross validation allows to mitigate the bias brought when splitting the data into a training and a testing set only once. Also each model in the grid search has to be evaluated but we can't use the testing set for that as it would break the golden rule of only using the testing set at the very end. So cross validation implies using a third type of data set called validation set. This is the set used for model validation and the testing set is only used once the best model is picked to test it.

The whole process makes the result more reliable but in the same time, the work load is multipled k times for each combination of parameter. The computational expense of cross validation can be reduced by shuffling the data points in each bucket and reducing the number of iteration.

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 [52]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.tree import DecisionTreeRegressor
from sklearn.metrics import make_scorer
from sklearn.model_selection 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()

    # TODO: Create a dictionary for the parameter 'max_depth' with a range from 1 to 10
    params = {'max_depth': list(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(regressor, 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_

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 [53]:
# 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.

Answer: The optimal model has a max_depth of 4 which is corresponds to my guess in question 6 so everything works out well =)

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 [54]:
# 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: $403,025.00
Predicted selling price for Client 2's home: $237,478.72
Predicted selling price for Client 3's home: $931,636.36

Answer: Predicted prices are \$403,025.00 for client 1, \$237,478.72 for client 2 and \$931,636.36 for client 3. When looking back at the statistics calculated previously in Data Exploration, we can see that the mean price is \$454,342.94 and that the standard deviation (std) is \$165,171. Assuming house prices in Boston is normally distributed, we can discuss the predictions considering the mean and std calculated above.

The predicted selling price for Client 1's home is 0.31 std away from the mean and therefore is in the normal range of prices. It seems like a reasonable assumption regarding the assets of the house.

The predicted selling price for Client 2's home is 1.31 std away from the mean. Being more than 1 std away from the mean makes this predicted price rather untypical towards lower prices. This can be explained by the fact that the house is the smallest of the sample and that the neighborhood is relatively poor with not the best public schools. This is a plausible predicted price which might need some checks on site to be confirmed.

Finally the predicted selling price for Client 3's home is 2.89 std away from the mean. Being more than 2 std away from the mean makes this predicted price very unlikely towards higher prices (less than 0.001 actually). However it is noteworthy that the house must be fairly big with 8 rooms, and has a very nice neighborhood with the best public schools and the lowest poverty level of the sample. The house has to be beyond exceptional if we want to keep the predicted price as it is and would need thorough checks on site before being confirmed.

Sensitivity

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 respect to the data it's trained on.

In [55]:
vs.PredictTrials(features, prices, fit_model, client_data)

Trial 1: $391,183.33
Trial 2: $424,935.00
Trial 3: $415,800.00
Trial 4: $420,622.22
Trial 5: $413,334.78
Trial 6: $411,931.58
Trial 7: $399,663.16
Trial 8: $407,232.00
Trial 9: $351,577.61
Trial 10: $413,700.00

Range in prices: $73,357.39

Question 11 - Applicability

  • In a few sentences, discuss whether the constructed model should or should not be used in a real-world setting.

Hint: Take a look at the range in prices as calculated in the code snippet above. Some questions to answering:

  • How relevant today is data that was collected from 1978? How important is inflation?
  • Are the features present in the data sufficient to describe a home? Do you think factors like quality of apppliances in the home, square feet of the plot area, presence of pool or not etc should factor in?
  • Is the model robust enough to make consistent predictions?
  • Would data collected in an urban city like Boston be applicable in a rural city?
  • Is it fair to judge the price of an individual home based on the characteristics of the entire neighborhood?

Answer: I don't think this model should be used in a real-world setting for the following reasons:

Obsolete data : Dating from 1978, the reality of house pricing then and now might have drastically change. Even though the feature 'MEDV' has been multiplicatively scaled to account for 35 years of market inflation, other factors about the neighborhood might have changed. One neighborhood that was nice in 1978 might not be the same today. The other features like 'RM', 'LSTAT', and 'PTRATIO' might not be reliable anymore as new schools might have open when other closed for example.

Absence of other features : The features present in the data is not sufficient. 2 out of 3 of them are features about the neighborhood and not the house itself. As mentioned above, other features like livable square feet surface, quality of insulation, presence of a pool, the surface of a garden if any, should be taken into account like. Concerning the neighborhood, easy access to transportation, commercial areas, schools, crime levels, pollution are among other important features. As described in the UCI Machine Learning Repository (https://archive.ics.uci.edu/ml/machine-learning-databases/housing/housing.names) other features were present in the dataset that we did not use :

1. CRIM      per capita crime rate by town
2. ZN        proportion of residential land zoned for lots over 
             25,000 sq.ft.
3. INDUS     proportion of non-retail business acres per town
4. CHAS      Charles River dummy variable (= 1 if tract bounds 
             river; 0 otherwise)
5. NOX       nitric oxides concentration (parts per 10 million)
6. RM        average number of rooms per dwelling
7. AGE       proportion of owner-occupied units built prior to 1940
8. DIS       weighted distances to five Boston employment centres
9. RAD       index of accessibility to radial highways
10. TAX      full-value property-tax rate per $10,000
11. PTRATIO  pupil-teacher ratio by town
12. B        1000(Bk - 0.63)^2 where Bk is the proportion of blacks 
             by town
13. LSTAT    % lower status of the population
14. MEDV     Median value of owner-occupied homes in $1000's

Robustness : As we can see above predictions can vary from \$351,577.61 to \$424,935.00 representing almost a 20% inscrease of the price. \$73,357.39, can make a big difference for a buyer or a seller, showing that the model is not robust enough to make consistent predictions.

Geographical zone: While some features are applicable anywhere, like square feet surface, some of the data collected in an urban city are hardly applicable in a rural city. For instance easy access to transportation, commercial areas or even schools might be inexistant in some rural cities.

According to me, the characteristics of the neighborhood alone is not sufficient to judge the price of a home although it is inherent.

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