Machine Learning Foundations 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.

Before start

Make sure you have the same scikit-learn version used in this notebool Run the following cell to check if it's right.

In [66]:
import sklearn
print("Your scikit-learn version is ", sklearn.__version__)
if sklearn.__version__ >= '0.18':
    print("Good!!")
else:
    print("You need to upgrade your scikit-learn version or change the code to make it work.")
    print("You may run:\n")
    print("pip install scikit-learn==0.18.1")

Your scikit-learn version is  0.18.1
Good!!

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 [67]:
# 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 [68]:
# TODO: Minimum price of the data
minimum_price = prices.min()

# TODO: Maximum price of the data
maximum_price = prices.max()

# TODO: Mean price of the data
mean_price = prices.mean()

# TODO: Median price of the data
median_price = prices.median()

# TODO: Standard deviation of prices of the data
std_price = prices.std()

# 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: $165,340.28

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:

An increase in RM would lead to an increase in MEDV, because the number of rooms in a house is directly proportional to its size; therefore, a house containing more rooms would be larger and cost more.

On the other hand, a neighborhood that has an increased LSTAT value tends to be associated with higher crime rates, hence its MEDV would be considerably lower.

Lastly, a higher PTRATIO means a neighborhood has more teachers available, which could potentially indicate that there are more schools nearby, an inference that brings more appeal to a neighborhood and raises real estate prices. Thus, MEDV and PTRATIO are directly proportional as well.

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 [69]:
# 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 [70]:
# 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.
  • R2 score of 0.40 means that 40 percent of the variance in Y is predictable from X.

Answer:

Yes, because a R2 score of 0.923 means that 92.3 percent of the variance in Y is predictable from X.

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 [71]:
# 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=27)

# 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:

Before elaborating on an answer constrained by the realms of Machine Learning, an appropriate analogy to explore is how students are taught a new subject at school. A student who can only solve the exercises that the teacher previously solved in class has not fully mastered the subject; it is essential for them to acquire a sufficiently general understanding from those examples so that they can also solve new questions about the same topic.

Machines are taught similarly. The purpose of a model is to predict something based on previously collected data. When the model is trained, it tries its best to find out a pattern in the training dataset while minimizing the error rate. However, if the model goes beyond finding patterns, it may memorize the training data. Such behaviour is called overfitting. Additionally, if the model fails to find patterns, it underfits the data and will not predict well.

The benefit to splitting a dataset into training and testing subsets lies in mainly mitigating data overfitting issues. By testing a model with a subset that was not used to train it ― and this is of paramount importance, we can evaluate how well a model performs.

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 [72]:
# 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:

Chosen graph has max_depth=6. The Training Score slightly decreases and converges to around 0.8 as the number of points increases. As for the Testing Score we see a sudden increase for the 0-50 number of points range, another slight increase ranging 50-150 and then stabilizes and converges to score 0.6. Therefore, the sweet spot for max_depth=6 lies at 150 Training Points, where we have the maximum Testing Score without compromising the Training Score so much.

Based on the analysis of the curves, it is safe to conclude that the model has slightly overfitted the data (High Variance scenario) ― but not as much as max_depth=10 though, meaning the gap between the curves as they converge is larger than aimed for.

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 [73]:
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:

With a maximum depth of 1, the model suffers from high bias, meaning it is underfitting and oversimplified. The visual cue is a low score of both Training and Validation sets.

As for the maximum depth of 10, the model suffers from high variance or overfitting. The visual cues are the great gap between Training and Validation scores, as well as a high score for the Training set in contrast to a much lower score for the Validation set. This means that the model memorized the data and overfit it, but performs badly with a testing/validation sets.

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 3 best generalizes the unseen data because the Training and Validation score curves are tightly close together, locked at a score ranging from 0.7 to 0.8, give or take. The curves for maximum depth of 2 are even closer, but at 3 the scores are slightly higher, which is a fair trade-off.

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:

In a nutshell, Grid Search is a technique to find the optimal combination of parameters for a given model. This optimization usually aims to minimize the training error of a model ― let's call it $M$ ― that takes the training sample $S_{train}$ and additional parameters, say $\alpha$ and $\beta$ (also known as hyperparemeters).

The end goal is to find values for $\alpha$ and $\beta$ that tune the model for a minimal training error. As for most optimization problems, finding the perfect fit is notably expensive.

Therefore we resort to Grid Search, which is achieved by preselecting a finite set of values of $\alpha$ $(\alpha_1, \alpha_2, ...)$ and $\beta$ $(\beta_1, \beta_2, ...)$ and solving the optimization problem for each permutation pair of values $(\alpha_1, \beta_1), (\alpha_2, \beta_1), ..., (\alpha_2, \beta_1), (\alpha_2, \beta_2), ...$ that, when plotted in space, look like a grid ― hence the name.

For a Decision Tree Classifier model, max_depth is an example of parameter that can be optmized using Grid Search.

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:

The golden rule of Machine Learning dictates that testing data should never be used for training a model. Yet, deciding on a befitting model implicates in evaluating each candidate model before making a final judgment. That's where Cross-Validation comes in to play.

In short, Cross-Validation is a resampling procedure used to evaluate a machine learning model on a limited data sample.

K-fold technique is one variant of Cross Validation which involves randomly dividing the original sample into $k$ groups, or folds, of approximately equal size. The first fold is treated as a validation set, and the method is fit on the remaining $k − 1$ folds. This process is then repeated $k$ times, with each of the $k$ subsamples used exactly once as the validation data. The $k$ results can then be averaged to produce a single estimation.

It is a fairly popular method because it generally results in a less biased estimate of the model skill than other methods, such as a simple train/test split; this helps preventing overfitting and is certainly beneficial to Grid Search when optimizing a model.

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.

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 [74]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.metrics import make_scorer
from sklearn.tree import DecisionTreeRegressor

# Using GridSearchCV from model_selection instead of grid_search to avoid deprecation warning
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)
    cv_sets = ShuffleSplit(n_splits=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': np.arange(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_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 [75]:
# 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 maximum depth of 4, which is very close to my guess of max_depth=3 in Question 6.

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 [76]:
# 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: $417,320.69
Predicted selling price for Client 2's home: $229,693.33
Predicted selling price for Client 3's home: $879,900.00

Answer:

Follows some computations to assist my reasoning.

In [77]:
# Features for cheapest and most expensive house respectively
f_min = features.iloc[prices.idxmin]
f_max = features.iloc[prices.idxmax]
#'RM', 'LSTAT', and 'PTRATIO'
msg_templ = "RM={:.2f}\t LSTAT={:.2f}\t PTRATIO={:.2f}\t for {} house."
print(msg_templ.format(f_min['RM'], f_min['LSTAT'], f_min['PTRATIO'], "cheapest"))
print(msg_templ.format(f_max['RM'], f_max['LSTAT'], f_max['PTRATIO'], "most expensive"))

RM=5.45	 LSTAT=30.59	 PTRATIO=20.20	 for cheapest house.
RM=8.40	 LSTAT=5.91	 PTRATIO=13.00	 for most expensive house.
Feature Minimum Price Maximum Price
Total number of rooms in home 5 rooms 8 rooms
Neighborhood poverty level (as %) 30.6% 5.9%
Student-teacher ratio of nearby schools 20-to-1 13-to-1

Prices recommended, rounded to the nearest hundred, are:

  • Client 1: $417,300.00
  • Client 2: $229,700.00
  • Client 3: $879,900.00

Houses' prices for Client 1 and Client 2 are below the average price of $454,342.94 found on the Data Exploration section. Client 3's house is priced over 90% higher than the average and only nearly 15% lower than the maximum house price of $1,024,800.00.

The house price for Client 1 seems reasonable due to average poverty level of 17% and student-teacher ratio 15-to-1 nearing that of high priced neighborhoods (as seen on the table above, the student-teacher ratio for the most expensive house is 13-to-1).

The house price for Client 2 is also justifiable due to high poverty levels and higher student-teacher ratio, though slightly overpriced, considering that the lowest-priced house in the dataset has 1 more room and lower poverty level and student-teacher ratio. However, this could also either mean that the latter is underpriced or that other factors are responsible for the price discrepancy.

Lastly, the house price for Client 3 seems reasonable due to very low poverty level and student-teacher ratio. However, as forementioned, the model predicted a price 15% lower than the maximum price, a significant deviation considering the first ranks higher for the same three given features. Similar to the previous analysis, this could either mean that the maximum price house is overpriced or that the model was not able to capture the sensibility of the house market basing solely on number of rooms, poverty level and student-teacher ratio of nearby schools.

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 [78]:
vs.PredictTrials(features, prices, fit_model, client_data)

Trial 1: $391,183.33
Trial 2: $419,700.00
Trial 3: $415,800.00
Trial 4: $420,622.22
Trial 5: $418,377.27
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: $69,044.61

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 appliances 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?

Answers:

How relevant today is data that was collected from 1978? How important is inflation?
Inflation measures the buying power of a particular currency over time. This Inflation Calculator shows that an item costing 1 dollar in 1978 would cost roughly 3.85 dollars in 2018. That is a 285.1% increase! It doesn't necessarily mean, however, that every asset priced in 1978 should cost 285.1% more in 2018. It simply serves as a takeaway that prices do increase over time and inflation must be accounted for.

Demographics also changes a lot over the course of 40 years. For example, population growth and government policies may certainly have an effect on poverty level as well as student-teacher ratio; but at the same time new schools could have been built as well. All of these, combined with inflation and other pertinent factors, exert profound influence on housing market prices. Therefore, data collected from 1978 might not be so relevant today.

Are the features present in the data sufficient to describe a home? Do you think factors like quality of appliances in the home, square feet of the plot area, presence of pool or not etc should factor in?
The features present in the data are unquestionably relevant, but not sufficient. Factors like property maintenance, local amenities nearby ― such as hospitals, parks, grocery stores, shopping malls and public transport access; distance to downtown district, crime rates, air and noise pollution levels etc should all factor in.

Is the model robust enough to make consistent predictions?
Yes, to an extent. The model is constrained by the lack of additional relevant factors that should weigh in when estimating the selling price of houses accurately. These additional factors could help explain the deviations that section Question 10 - Predicting Selling Prices answer touched upon.

Would data collected in an urban city like Boston be applicable in a rural city?
No, it may not be applicable due to demographics differences between urban and rural cities. A model for a rural city may take into consideration factors that wouldn't otherwise be so relevant to an urban model. For an instance, the distance to nearest urban center or metropole city would probably be deemed pertinent for predicting rural house prices.

Is it fair to judge the price of an individual home based on the characteristics of the entire neighborhood?
Yes, it definitely is. Let's consider two identical houses built on two distinct neighborhoods A and B. Neighborhood A holds the lowest crime rates in the city and is situated near parks, health centers, police and fire departments. On the other hand, Neighborhood B is situated on the outskirts of the city and holds not alarming but slightly higher crime rates. Even though the houses look exactly the same down to every detail, their vicinity may be extremely different, which may lead to a huge impact in an individual's quality of life. Law of Supply and Demand endorses that there would be more potential buyers (i.e demand) willing to buy houses on Neighborhood A. That alone is sufficient to increase house prices on Neighborhood A in contrast to Neighborhood B.

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.