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 [18]:
# 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.
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# TODO: Minimum price of the data
minimum_price = np.amin(prices)
# TODO: Maximum price of the data
maximum_price = np.amax(prices)
# TODO: Mean price of the data
mean_price = np.mean(prices)
# TODO: Median price of the data
median_price = np.median(prices)
# TODO: Standard deviation of prices of the data
std_price = np.std(prices)
# Show the calculated statistics
print ("Statistics for Boston housing dataset:\n")
print ("Minimum price: ${:,.2f}".format(minimum_price))
print ("Maximum price: ${:,.2f}".format(maximum_price))
print ("Mean price: ${:,.2f}".format(mean_price))
print ("Median price ${:,.2f}".format(median_price))
print ("Standard deviation of prices: ${:,.2f}".format(std_price))
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
RM Given the dataset and the statistics above, I notice there can be a higher MEDV. One reason is that a room in a given home can have a larger area space therefore making the cost of that house higher.
LSTAT If there is a lower MEDV, then we can assume that there may be a higher LSTAT. In some cases, LSTAT may not be ideal for family with teenagers because the lower class is usually prone to crime and violence. there would be a lower teacher-to-student ratio resulting in less attention dedicated to each student that may impair their performance in school.
PTRATIO prices of houses around public schools are generally lower than those around private schools. Hence one would expect a lower price given a high student-to-teacher ratio due to a lower demand for houses in such areas.
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.
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# 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.
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# 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 I would suppose this model has successfully captured the variation of the target variable because we have a perfect score (of 0.923) that fall between 0 and 1, signifying that the extent to which the dependent variable is predictable; this result is exactly what we want.
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.
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# 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=33)
# Success
print ("Training and testing split was successful.")
print(features.shape[0])
print(float(X_train.shape[0]) / float(features.shape[0]))
print(float(X_test.shape[0]) / float(features.shape[0]))
Answer: The best answer would be to avoid "overfitting" and "underfitting". So in theory, you want to your test result to be as precise as possible, you want to make sure you trim your dataset to meet some requirements. For this, best to split the dataset into two separate categories, a "training set" and a "testing set", in which case you would utilize the training set throughout most of the model. Benefits: Cross-validation: So here your final model is the one that gives the minimum average validation error. Using a fixed validation set: And here you train on the first part and do model selection using the second part.
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.
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# 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:
I've chosen graph 1 (the one of the upper left corner) which have the max_depth of 1. This can be classified as High Bias Scenario. As observations increases the testing score also increase. Also we can take notice that the training score decreases according to the number of observations. In addition to this, I notice the model does not fit the data well.
max_depth = 3 would be an Ideal Scenario. Evidently the testing score increases according with the training points. Also, there is not high bias neither high variance; I can point that it fits the data well. As more training points are add, thus it becomes an ideal model.
When max_depth is 6, testing score increases with training points, while training score decreases slightly with training points, and there seems to be high variance scenario. Also, we notice a high score, and an overfitting of dataset as the model nigh ideal scenario.
When max_depth is 10, we get almost a perfect score, we notice the dataset overfitting even though there is a high variance. The testing score increases with training points; the training score seems to slightly decrease with training points at -1.0; and the model generalize well.
If more training points are added, the graph will most likely drop to lesser depth while keeping it's linear path.
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.
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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: Maximum depth of 1: High Bias In this case, since it is a maximum depth of 1, it is safe to note High Bias. Both training and testing scores are low. It is clear that there is barely a gap between the training and testing scores. Therefore, it indicates the model is not fitting the dataset well and not generalizing well hence the model is suffering from high bias.
Maximum depth of 10: High Variance While the maximum depth of 10, we can say High Variance Training score is high. And testing score is low. Therefore there is a substantial gap between the training and testing scores. This indicates the model is fitting the dataset well but not generalizing well hence the model is suffering from high variance.
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: Acording to the learning curves, evidence points to the maximum depth of 3. The graph seem as though the training score seem to plateau, which means the highest possible score for the model's ability to generalize to unseen data; And also it's conclusive that the model may not be suffering from a high variance scenario.
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: It is said that the grid search technique allows one to define a grid of parameters that will be searched using K-fold cross-validation; it exhaustively tries every combination of the provided hyper-parameter values in order to find the best model. Using this technique can help find the highest cross-validation accuracy that matches with the corresponding parameters that optimizes the learning algorithm.
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 data analysis the "k" in K-fold is when a dataset is split into a folds of equal size; by which either fold represent a testing set as a single instance and also represent the training set on basis of k-1 instance. K-fold is also known as cross-validated performance where average testing performance is used as the estimate of out-of-sample performance. One advantage can be that it gives more reliable estimate of out-of-sample performance than train/test split, and also reduces the variance of a single trial of a train/test. When using k-fold cross-validation, we are using the best/accurate average on testing as a benchmark to decide which is the most optimal set of parameters for the learning algorithm. Usually different sets of optimal parameters are outputted if k-fold cross-validation set is not use while running grid-search.
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.
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# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.grid_search import GridSearchCV as gscv
from sklearn.metrics import make_scorer
from sklearn.tree import DecisionTreeRegressor as dtr
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 = dtr(random_state=0)
# TODO: Create a dictionary for the parameter 'max_depth' with a range from 1 to 10
depth_range = range(1, 10)
#params = dict(max_depth=depth_range)
params = {'max_depth': (1, 2, 3, 4, 5, 6, 7, 8, 9, 10)}
# TODO: Transform 'performance_metric' into a scoring function using 'make_scorer'
scoring_fnc = ms(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 = gscv(estimator=regressor,param_grid=params,scoring=scoring_fnc,cv=cv_sets)
# Fit the grid search object to the data to compute the optimal model
grid = 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.
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# 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 has a maximum depth of 4. My guess was slightly higher from 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 [78]:
# 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))
# TODO: Minimum price of the data
minimum_price = np.amin(price)
# TODO: Maximum price of the data
maximum_price = np.amax(price)
# TODO: Mean price of the data
mean_price = np.mean(price)
# TODO: Median price of the data
median_price = np.median(price)
# TODO: Standard deviation of prices of the data
std_price = np.std(price)
# Show the calculated statistics
print("\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))
Answer: According to the dataset, these are the prices I recommend:
Client 3: \$938,053.85.
Data Exploration Findings:
As for Client 1, according to the Data Exploration:
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 [76]:
import matplotlib.pyplot as plt
vs.PredictTrials(features, prices, fit_model, client_data)
plt.hist(prices, bins = 20)
for price in reg.predict(client_data):
plt.axvline(price, lw = 5, c = 'r')
Hint: Take a look at the range in prices as calculated in the code snippet above. Some questions to answering:
Answer: Given a data that was collected from 1978 and the fact that demographic have since changed, it may not be wise to use data collected from a rural city to be aplicable. There are some aspect to apply when concern about demographics, there may be changes in features (Appliances and etc). Comparing rural and big cities, amenities do vary. In regards to the algorithm used above to conclude the this finding, it may not be wise to apply (in whole) the algorithm used. Resources have changed, features have changed, demographics have change. And also, some of the features included above may be insufficient in present days. We can consider features like crime rates, demographics of shopping centers, procximity to city and hospitals, and law enforcement offices and etc.
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.
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