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 necessary for this project
import numpy as np
import pandas as pd
from sklearn.model_selection 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']
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.

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 [2]:
# 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

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:

  1. 'RM' or average number of rooms have a direct relation to the area of the house (in general, may not be always true). This means a home with more number of rooms will cover more area than a home with comparatively less number of rooms. This may lead to a conclusion that in most cases (Keeping other factors constant) the relation between price of the house and 'RM' will be directly proportional, that is, the increase in 'RM' means increase in house's median value 'MEDV'(bigger house larger the price).
  2. Locality defines the area surrouding the house which inturn defines how well-off a person living in that locality is and how much can he/she pay to buy a house. 'LSTAT' is the percentage of homeowners in the neighborhood considered "lower class". A person of lower class have a far less chance of living in a well-off place and generally signifies the houses in the area to cost less. This means the correlation can be inverse. That is more the value of 'LSTAT' lesser the 'MDEV' value.
  3. Education has become a necessary requirement of our daily life. This means there is a requirement of more number of teachers with the ever increasing population. An area which can supply an ample ratio of teachers to students is generally considered a nice to place to live because of the opportunities available. This means increase in teacher student ratio will increase the price of the houses in a locality. So the inverse of Teacher student ratio, i.e., 'PTRATIO', will be inversely proportional to the 'MEDV'. That is more the value of 'PTRATIO' lesser the 'MDEV' value.

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 [3]:
# 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 [4]:
# 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:

R2 score also knowns as coeeficient of determination is a number in range of [0, 1] where 0 represents that there is no correlation between the dependent variable and the independent vairbale. 1 represents that the dependent variable is correlated to the independent vairbale and dependent vairbale can be predicted with a very high accuracy using the independent vairbale.

A score of 0.923 represents that 92.3% of variability in the dependent variable is accountable using the independent variable (high correlation). In the data displayed, it is clear that there is general correlation, as the independent vairbale(true value increases) so does the dependent variable (predicted value). So we can clearly state that the the model successfully captured (much part of) the trend in the target variable

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 [5]:
# TODO: Import 'train_test_split'
from sklearn.model_selection import train_test_split
import time

# 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=10)

# 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:

Any Machine Learning approach must take into account the trade-off between variance and bias (in other words overfitting and underfitting). A model always learn from a set of data. This data may or may not be a general display of the trend in the population from which it is taken. In the world of software development, testing is always a crucial part. This train and test set split gives us a way to train our model over the majority of data and to see whether it is performing aptly on unseen (or alien) values or not. There arrives two cases where the system might produce bad results:

  1. Overfitting (high variance): The model might generalizes over the dataset it has trained over and in doing so it precisely sticks to each data point in the training set. This might look good but in a general environment this might cause a new point to be classified wrongly.
  2. Underfitting (high bias): The model in this case gets so little of the data for training (or less time with data; as in number of iterations) that it fails to capture the trend in the data and as a result it really isn't able to perform well on any real world data point and/or even on the training dataset.

This problem is captured by performing test over the trained model to check its real world implications over alien data points.

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 [6]:
# 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:

Taking into account the graph with max_depth = 3

As more and more training points are added:

  1. Training Curve tends to level down and come closer to the testing curve displaying the increase in variance (r2 score) of the model over training set. In other words the uncertainity of the system is increasing over training set.
  2. Testing Curve tends to level up and come closer to the score of 0.8 almost near to the training curve. The increase in score also displays an increase in certainity over the new data points in the training set as the number of data points in training set increases. Although the shaded region displys uncertainity (standard deviation in the values of prediction)

Both the curves clearly seems to get saturated at an r2 score of approximately 0.8 for 300 data points or more in training set. Adding more points will clearly add an overhead of running the model over the data even though there is no need as the model is at peak score for max_depth=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.

Run the code cell below and use this graph to answer the following two questions Q5 and Q6.

In [7]:
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:

Looking at the curve it is clear that the score over both the training and the validation set was low at depth 1, depicting that the model was unable to get a better sense of the trend in the data through training. In other words the system went into underfitting. Thus at depth 1 the system suffered high bias problem.

As the depth increased we see a general trend of increase in the performance over the training set. However the performance began to drop over the testing set (cross validations) at the depths greater than 4. At the depth of 10 there is a huge gap in the training score and the validation score. This clearly depicts that model was performing exceptionally well over the training set but performed very poorly over the cross validations. This gap in the learning curves is in itself the case of overfitting. Thus at depth 10 the system suffered high variance problem.

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:

Looking at the complexity curves it is clear that the x coordinate of the peak of the validation curve best genralizes over the validation set while at the same time giving a near good performance on the training set. The peak value at which we get this best generalization is depth=4.

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

A grid search technique is generally a mechanism of finding the best possible combination of values (from set of values provided) for the hyperparameters of a learning algorithm. Going into it in lay man terms we can say that grid search is an algorithm to find a way and get another algorithm to work more efficiently.

We can explain this by using the example of a simple Support Vector Classifier(SVC) from the sklearn library. Let us say we want to make it work more efficiently over a set of data. The way to do this is to choose the hyperparameters aptly. If we want to do this using the values of C and gamma only. We can select a set of values for C and gamma and generate a grid. For example,

C gamma
1 0.1
10 0.2
100 0.5
1000 1.0

The grid search algorithm will take in the scoring function (to check the best possible combination) the dataset as well as the object of the SVC and will result the values of gamma and c which will give us the best possible value of the SVC.

NOTE: The values might be like 10, 1.0 that is it may or may not return a row as result. Also C column might be having only 3 possible values while gamma might be having 4 or more.

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:

k-fold cross Validation

K-fold cross validation is a technique for validating the working of a model over the alien data. In this technique we divide the available dataset into k equal parts and run the algorithm k times over the training and test dataset. Here training dataset equal to k-1 parts of the k parts we got from division of data and the test dataset is the kth part of the original data. This means the algorithm is ran for testing over each part of the original dataset atleast once. The overall score of the model is then computed as the average score of the model over the k sets.

k-fold can help in techniques like grid search to get to the results of the best values for hyperparamters by providing different training and test sets everytime. But the shuffling of the data points in the set before slicing it in k parts is necessary. Reason for shuffling lies in the order in which we recieve or gather our dataset. If the dataset, for example, is in order it might be possible that the training set gets all the values of one type of data (In the case of our dataset it might be possible that data is ordered in terms of LSTAT). This will result in model to generalize well over some kind of data while missing over other data points.

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 [8]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.metrics import make_scorer
from sklearn.tree import DecisionTreeRegressor
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 = ShuffleSplit(n_splits = 10, test_size = 0.2, random_state = 10)

    # 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': 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, param_grid=params, scoring=scoring_fnc, cv=cv)

    # 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 [9]:
# 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:

Optimal Depth of model equals 4. My intuition hit the point with the optimal depth for the model. The complexity curve gives a clear insight of how well the model can work with increasing complexities and can saturate and then get a dip in performance after a while.

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 [10]:
# 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: $406,933.33
Predicted selling price for Client 2's home: $232,200.00
Predicted selling price for Client 3's home: $938,053.85

Answer:

The recommended prices:

  1. Client 1 - \$ 419,045 : The price of \$419k has a value of z score as 0.213 (not taking into account negative sign). It is clear that the point belongs to the 64% dataset and is well placed (or predicted). Thus the prediction is good enough and can be recommended to the client.
  2. Client 2 - \$ 229,303: The price of \$229k might seem a much while looking at the constitution of a low level neighborhood and the scarcity of teachers; but the number of rooms increase the value of the house giving it a value that is much more than the minumum value (\$100k) of the price.
  3. Client 3 - \$ 949,200: The Values of all the three parameters RM, LSTAT and PTRATIO have values that are pretty obviously in favor of the home. Highest value of \$1024k displays that the predicted value is an optimal price for the house.

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 [11]:
vs.PredictTrials(features, prices, fit_model, client_data)

Trial 1: $411,000.00
Trial 2: $411,417.39
Trial 3: $415,800.00
Trial 4: $420,622.22
Trial 5: $413,334.78
Trial 6: $411,931.58
Trial 7: $400,350.00
Trial 8: $407,232.00
Trial 9: $402,531.82
Trial 10: $379,125.00

Range in prices: $41,497.22

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:

The model should not be used as it is in a real-world setting. There are more than one reasons to believe this:

Relevancy (in 2017):

According to Zillow
The median home value in Boston is \$558,300. Boston home values have gone up 11.5% over the past year and Zillow predicts they will rise 4.5% within the next year. This can be found here.

This clearly displays that a price from 39 years old dataset is totally out of comparison and is irrelevant to today's price range. Also there might be variation in values of 'LSTAT', 'RM' and 'PTRATIO' of locality over 39 years.

Features:

Many features like price per square feet, crime rate in the locality and the age of the building along with the 3 features we used are some more possible values that can give model a way to be more precise in predicting house prices.

Robustness

As seen in the sensitivity section, the model produces values in range of \$41k for same values of input. This clearly depicts lack of robustness of model over the data and the lack of its generalization over the dataset.

Applicability

A model trained on dataset of a well-off place have certain standards which far bests the standards of a rural area.

Fairness

It is clearly fair to judge price of a house based on the entire neighborhood as a person will take into account the facilities available to him/her and how well all his needs will be fulfilled after spending loads of money.

Note: Once you have completed all of the code implementations and successfully answered each question above, you may finalize your work by exporting the iPython Notebook as an HTML document. You can do this by using the menu above and navigating to
File -> Download as -> HTML (.html). Include the finished document along with this notebook as your submission.