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.
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:
'MEDV' value of 50.0. These data points likely contain missing or censored values and have been removed.'RM' value of 8.78. This data point can be considered an outlier and has been removed.'RM', 'LSTAT', 'PTRATIO', and 'MEDV' are essential. The remaining non-relevant features have been excluded.'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 [10]:
# 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']
features = data.drop('MEDV', axis = 1)
# Success
print "Boston housing dataset has {} data points with {} variables each.".format(*data.shape)
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.
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:
'MEDV', which is stored in prices.
In [11]:
# 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)
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.
'RM' value(number of rooms) of 6 be worth more or less than a home that has an 'RM' value of 7?'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?'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:
We should expect home with higher 'RM' value to be worth more than home with lower 'RM' value, this implies that 'MEDV' would be higher. Since a house is value by total number of square feet and price per square feet. Therefore, all else being equal, a home with more rooms should be larger in terms of total number of square feet and would warrant for a higher price.
On the other hand, we would expect an increase in 'LSTAT' and 'PTRATIO' to caused a decrease in 'MEDV'. We can safely assume that the lower class would only be able to afford cheaper homes rather than expensive ones and private schools which are less affordable, would have a lower ratio of students to teacher. Hence, the intuition would be an increase in 'LSTAT' and 'PTRATIO' would meant a decrease in 'MEDV'.
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.
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:
r2_score from sklearn.metrics to perform a performance calculation between y_true and y_predict.score variable.
In [19]:
# 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
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 [20]:
# 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)
Hint: The R2 score is the proportion of the variance in the dependent variable that is predictable from the independent variable. In other words:
Answer:
R^2 score of 0.923 implies that 92.3% of the variance in the target variable can be explained by the independent variables. A model with R^2 score of 1 would be ideal. Since the R^2 score is close to 1.0, we can safely say that the variation of the target variable is being captured by the model above.
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:
train_test_split from sklearn.cross_validation to shuffle and split the features and prices data into training and testing sets.random_state for train_test_split to a value of your choice. This ensures results are consistent.X_train, X_test, y_train, and y_test.
In [21]:
# TODO: Import 'train_test_split'
from sklearn.model_selection import train_test_split
# TODO: Shuffle and split the data into training and testing subsets
X_train, X_test, y_train, y_test = train_test_split(features, prices, test_size=0.2, random_state=17)
# Success
print "Training and testing split was successful."
Answer:
If the model is underfitting the training data, it has not captured the information in the examples you already have. Underfitting is easy to detect given a good performance metric. We can rectify this by trying alternate learning algorithms. On the other hand, overfitting happens when a model learns the detail with the noise in the training data to the extent that it negatively impacts the performance of the model when shown new data. This means that the noise is picked up and learned by the model. The problem is that these noise do not apply to new data and negatively impact the models ability to generalize.
Ideally, we want a model in between underfitting and overfitting. To achieve this, we can look at the performance of the learning algorithm over time as it is traning on the training data. By splitting a dataset into training and testing subsets, we can evaluate the model performance based on data that was not used for training. By doing so, we can use the testing set to evaluate if our model is underfitting or overfitting. We can plot both the skill on the training data and the skill on a test dataset we have held back from the training process.
Over time, as the algorithm learns, the error for the model on the training data goes down and so does the error on the test dataset. If we train for too long, the performance on the training dataset may continue to decrease because the model is overfitting and learning the irrelevant detail and noise in the training dataset. At the same time the error for the test set starts to rise again as the model’s ability to generalize decreases.
The stopping point would be before the error on the test dataset starts to increase where the model has good skill on both the training dataset and the unseen test dataset.
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.
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 [22]:
# Produce learning curves for varying training set sizes and maximum depths
vs.ModelLearning(features, prices)
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:
For this question, we will analyse the graph with max_depth equal 10. Initially, as the number of training points increases, the gap between training score and testing score narrowed. This effect persists up to approximately 200 trianing points. Thereafter, increasing number of training points does not help to narrow the gap between the training score and testing score. Both the training score and testing score appears to plateaued at approximately 0.98 and 0.7 respectively.
Generally speaking, the more data we have, the better. However, in this case adding more data would not benefit the model but increase the training time unnecessarily.
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 [23]:
vs.ModelComplexity(X_train, y_train)
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:
The model trained with a maximum depth of 1 suffers from high bias, where it has training and validation score which are very small and the difference between them is small.
On the other hand, the model trained with a maximum depth of 10 suffers from high variance. The training score is extremely close to 1, but the validation score is approximately 0.65 which is significantly lower.
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 graph above, we can conclude that the model with maximum depth of 4 would generalizes on unseen data best. As can be observed from the graph above, training score is slightly above 0.8, and validation score is slightly below 0.8, which are reasonable scores. In addition to that, the gap between both scores is reasonable, which indicates that the variance is not too high or too low. As can be seen that, any maximum depth before 4, has variance which is too low and for maximum depth after 4 the training score and valdiation score begins to diverge.
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 is a hyperparameter optimization technique. In other words, it is a technique use for choosing a best performing set of hyperparameter values from all the possible combinations of values provided. In the case of a decision tree algorithm, we often face with the problem of how to select the optimal hyperparameters (e.g. maximum depth, number of samples in leaf nodes and etc) of the model. Instead of varying the hyperparameters to search for the optimal hyperparameters manually, we can leverage on grid search. We simply have to define a range of possible values for each hyperparameter (e.g. maximum depth from 1 to 5). Using grid search, each of the model is then fit for each combination of the parameters on the training subset. Then with a suitable error metric, we can validate the model with a validation set. Finally, a combination of parameters having the best performance is chosen. We can be sure that this combination of parameters would be optimal.
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:
In k-fold cross validation, the training set is split into k subsets and the learning algorithm would be train using k - 1 folds while the remaining would be used for validation (each folds will be part of the test data once).
When using grid search without k-fold cross validation, we usually split our dataset into training set, validation set and testing set. By doing so, we reduced the size of the dataset available for fitting the model. When using grid search With k-fold cross-validation, we can use the whole training set for fitting and validating the model and obtain more accurate validation scores as the model is fit and validated on different subsets each of the k times. This technique makes validation less dependent on random splits of the training set.
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:
DecisionTreeRegressor from sklearn.tree to create a decision tree regressor object.'regressor' variable.'max_depth' with the values from 1 to 10, and assign this to the 'params' variable.make_scorer from sklearn.metrics to create a scoring function object.performance_metric function as a parameter to the object.'scoring_fnc' variable.GridSearchCV from sklearn.grid_search to create a grid search object.'regressor', 'params', 'scoring_fnc', and 'cv_sets' as parameters to the object. GridSearchCV object to the 'grid' variable.
In [25]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.metrics import make_scorer
from sklearn.tree import DecisionTreeRegressor
from sklearn.grid_search import GridSearchCV
def fit_model(X, y):
""" Performs grid search over the 'max_depth' parameter for a
decision tree regressor trained on the input data [X, y]. """
# Create cross-validation sets from the training data
# sklearn version 0.18: ShuffleSplit(n_splits=10, test_size=0.1, train_size=None, random_state=None)
# sklearn versiin 0.17: ShuffleSplit(n, n_iter=10, test_size=0.1, train_size=None, random_state=None)
cv_sets = ShuffleSplit(X.shape[0], n_iter = 10, test_size = 0.20, random_state = 0)
# TODO: Create a decision tree regressor object
regressor = DecisionTreeRegressor()
# 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, cv=cv_sets, scoring=scoring_fnc)
# 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_
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.
In [26]:
# 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'])
Hint: The answer comes from the output of the code snipped above.
Answer:
The optimal model have a max_depth of 4 which is in agreement with my guess in Question 6.
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 |
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 [27]:
# 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)
Answer:
Based on the model, the price recommended for each client to sell their home are $412,950.00, $234,529.79 and $896,962.50 for client 1, 2 and 3 respectively. If we explore the dataset, with the given the values of the features the predicted prices seem reasonable. The predicted price for client 1's home is only slightly lower than mean and homes with prices close to that have similar feature values. For client 2, we can notice that the range of prices are wide, but the predicted price of client 2's home lies within that range. Similarly, if we compare features of client 3's home to features of homes with prices greater than \$900000. It Turns out that these homes have similar feature values and that the predicted price of client 3's home is very close.
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 [29]:
vs.PredictTrials(features, prices, fit_model, client_data)
Hint: Take a look at the range in prices as calculated in the code snippet above. Some questions to answering:
Answer:
The housing price dataset from 1978 would have become obsolete. The dataset would not be able to reflect the social and economic conditions of Boston today. In my opinion, the features presented in the dataset are insufficient to represent how valuation of housing is done in real world. Housing valuation is more complex than that and there are many contribution factors such as location of the house in Boston, the total number of square feet, the age of the property, and etc. Since the size of the dataset is not large, the model will become sensitive to a shuffle of train-test split. Intuitively, we expect a larger dataset would make the model more robust. The data collected in an urban city like Boston would not be applicable in a rural city. Simply because of the difference in socio-economic conditions and also valuation of housing might be done differently in a rural city. This warrant a datasets that consists of features which represents housing valuation in rural city. I think we should not judge the price of an individual home based on the characteristics of the entire neighborhood solely. Although, we could capture some patterns in pricing within the neighborhood, but within the neighborhood, we should still expect characteristics of home to vary. Hence, the price of houses within the same neighborhood would still differ.
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.