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 [5]:
# Import libraries necessary for this project
import warnings
warnings.filterwarnings("ignore", category = UserWarning)
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 [3]:
data.describe()
Out[3]:
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 [8]:
# TODO: Minimum price of the data
minimum_price = np.min(prices) #making these changes based on first review - Alternative using pandas prices.min()
# TODO: Maximum price of the data
maximum_price = np.max(prices) #making these changes based on first review - Alternative using pandas prices.max()
# TODO: Mean price of the data
mean_price = np.mean(prices) #making these changes based on first review - Alternative using pandas prices.mean()
# TODO: Median price of the data
median_price = np.median(prices) #making these changes based on first review - Alternative using pandas prices.mean()
# TODO: Standard deviation of prices of the data
std_price = np.std(prices, ddof=0) #My intuition is that I should not use prices.std() because it treats the dataset as a sample and divides bu N-1
# 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: Would you expect a home that has an 'RM' value of 6 be worth more or less than a home that has an 'RM' value of 7?
In [9]:
import matplotlib.pyplot
import seaborn
xLab = {'RM':'No of Rooms', 'LSTAT': 'Population Percentage with Lower Status', 'PTRATIO': 'Students-to-Teacher ratio'}
for my_Features in ['LSTAT', 'PTRATIO', 'RM']:
plot = seaborn.regplot(data[my_Features], prices)
plot.set(xlabel=xLab[my_Features], ylabel='Median Values')
matplotlib.pyplot.show()
Looking at plots above a "trend" in behavior prices vis-a-vis these three features is obvious. Hence:
Expect an increase in Median Prices "MEDV" for increase in Number of Rooms "RM" (Larger the houses bigger the price) Expect a decrease in Median Prices "MEDV" for increase in Population Percentage with Lower Income Status "LSTAT" (Two factors that home prices... Customers say: Why should I pay big price for a house that is in a poor neighborhood? Housing builders / promoters say: If this neighborhood is generally poor why would anybody build an expensive house there? Who will buy it? As a result the houe prices get depressed in predominantly) Expect a decrease in Median Prices "MEDV" for increase in Students-to-Teacher ratio "PTRATIO" (If a buyers have school-going kids, they are willing to pay higher price for a house that is served by a school with better Student-to-Teacher ratio. I have seen even empty-nesters would buy a house in a neighborhood served by schools with better Student-to-Teacher ratio because the selling price is better if they decide to move out)
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 [6]:
# TODO: Import 'r2_score'
from sklearn.metrics import r2_score #done !
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 |
Would you consider this model to have successfully captured the variation of the target variable? Why or why not?
Run the code cell below to use the performance_metric function and calculate this model's coefficient of determination.
In [7]:
# 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)
In [8]:
score = performance_metric([3, -0.5], [2.5, 0.0])
print "This two point Model has a coefficient of determination, R^2, of {:.3f}.".format(score)
Mathematically the score R^2 of 0.923 is great, but I am not happy about the size of the data. Maths can lie ;o) For example a perfect straight line can be drawn between two points. Now a R^2 value for the model with just two observations is 0.918 ! Why not 1.0? A larger sample size would be better to comment about the variability of the target variable
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 [9]:
# TODO: Import 'train_test_split'
from sklearn.cross_validation import train_test_split #Done!
# 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=7)
# Success
print "Training and testing split was successful."
If we are forced to use up all the dataset for the model, then we will be relying on accuracy as indicated by the residuals (RMSE etc.) and not know how well the model will do when it is asked to make new predictions for data it has not already seen.
It's like having a student driver who has driven the same backroad several times and has now mastered driving on the same route but has the student learnt enough to go on the highway on his / her own without the instructor along?
That is the risk of not having a model trained and not tested
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 [10]:
# Produce learning curves for varying training set sizes and maximum depths
vs.ModelLearning(features, prices)
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?
I like results of max_depth = 3 As more training points are added, the training score drops a little until it does not drop terribly which shows in the convergence of the training curve at a score of 0.8 in this case. The testing scores are also better with more training points. However, since that curve (testing scores) is also converging to 0.8 in this case, having more training points does not benefit the model.
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.
In [11]:
vs.ModelComplexity(X_train, y_train)
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: How do you know when a model is suffering from high bias or high variance?
Here is my answer after my first review and re-read of the subject area A model is is a learner. It could have preconceived notion / assumption of the outcome or it may assume nothing and fit itself to the data. When it has some preconceived idea it's behavior to training data and validation data is same. (It goes equally right or equally wrong). This bias. When it has no such bias, it will wrap itself around the training data to an extent that it's score with training data is high and with unseen data its score is low. In other words it goes all over the place when it comes to prediction but is well behaved with training data. This is example of variance.
The above plot shows that the model with maximum depth 10 suffers from high variance (i.e. poor prediction with unseen data compared to good score with training data) Conversely the model with maximum depth of 1 has equally good (or bad) score on training data as well as testing data. (i.e equal amount of error with both) It suffers from a high bias (or low variance).
Below is my orginal answer which shows I have clearly not understood this part well. I will leave the answer as a reminder for myslef for future When model is trained to a lower level of complexity say 1, it has a lower testing score which means it does poorly with known data points (training data). It also has a low validation score which means it does poorly with data it had not seen during training.
I would say at Maximum Depth of 1, the Validation score of 0.4 suggests that the model suffers from high variance.
With a higher level of complexity both scores improve. Beyond a certain level of complexity (as from Maximum Depth =3 onwards), only the training scores improve but validation scores in fact drop meaning the model does well with known data points (training data) and does poorly with unseen data. Then the model is biased towards what its knows best and not sensitive to responding to data unseen.
I would say at Maximum Depth of 10, the model suffers from high bias.
*** You will notice that I am calling it "the model" out of my own precoceived notion or bias ;0). Algorithm with Maximum Depth 1 is one model and with Maximum depth 10 is another model (not the same) and the two behave differently. Algorithms are inherently high bias or high variance. Examples of high-bias machine learning algorithms include: Linear Regression, Linear Discriminant Analysis and Logistic Regression and Examples of high-variance machine learning algorithms include: Decision Trees, k-Nearest Neighbors and Support Vector Machines
I would pick between Maximum depth of 3 or 4. Maximum depth of 3 results in a model that is best genalized to unseen data. It may not give as good a score as Maximum Depth 4 does but is less complex. Maximum depth of 4 gives a better score but it's relying on a more complex model and it could be sluggish (not responsive comparatively) It's a toss up between the two models
Its a technique that uses a number of models with like algorithms where-in parameters have been changed in their plausible range thus forming a grid (say similar to having algorithms along x and parameters along the y).
Testing these models in this grid and obtaining scores will suggest selection of a learning algorithm that is not heavily biased and has good sensitivity to unseen data
The k-fold cross validation method splits the training data into k equal subsets with random data points from the dataset. It then uses sequentially one of the k subsets as testing data and the other k-1 subsets as training data.
The model is then validated against each set and the average of all validation scores is used to rate the model. I should add based on my first review and advice from the reviewer:
Now imagive we have data that is limited to a number of observations however large. If I do a single split of traing dataset and validation dataset I will get a validation score. Will the validation score be the same if some unseen data that the model saw during validation was in fact in training dataset? Maybe the model should be exposed to different data points to evaluated / cross-validate its performance. The k-fold method is wonderful. The advantage of this method is that it does not matter how the data gets divided. It ensures that every data point gets to be in a test set exactly once, and gets to be in a training set k-1 times. Thus the model gets exposed to every data-point and its prformace is validated. (cross-validated if right term is to be used) *Now a note to myself The disadvantage of k-fold method is that the training algorithm has to be rerun from scratch k times, which means it takes k times as much computation to make an evaluation. There is a variant of this method which is to randomly divide the data into a test and training set k different times. In that you independently choose how large each test set is and how many trials you average over.
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 [15]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
import warnings
warnings.filterwarnings("ignore", category = UserWarning)
from sklearn.metrics import make_scorer
from sklearn.grid_search import GridSearchCV
from sklearn.tree import DecisionTreeRegressor
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
# ShuffleSplit(X.shape[0], n_iter = 10, test_size = 0.20, random_state = 0)
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(random_state = 42 * 7) # I am playing with the value for parameter random_state.
#Is the data scientist expected to feed a random number to the parameter or just the seed?
#If I run the code multiple times...should I not use a different seed?
# TODO: Create a dictionary for the parameter 'max_depth' with a range from 1 to 10
params = [{"max_depth":range(1,11)}]
# TODO: Transform 'performance_metric' into a scoring function using 'make_scorer'
scoring_fnc = make_scorer(performance_metric)
# TODO: Create the grid search object
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_
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 [14]:
# 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'])
Wow I completely left this unanswered ! Here is my answer The optimal model has maximum depth of 4. It matches 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 |
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.
Run the code block below to have your optimized model make predictions for each client's home.
In [15]:
# 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)
My recommendations to the three clients would be:
Client 1: Expect your house to sell around 407,000 Client 2: Expect your house to sell around 229,000 Client 3: Expect your house to sell around 979,000
We know that in Boston Housing market: Minimum price: $105,000.00, Maximum price: $1,024,800.00, Mean price: $454,342.94, Median price $438,900.00 and Standard deviation of prices: $165,171.13
Thus Client 1's house price is very close to the mean price of $454,342.94.
It's a 5 room house (which is below the mean of 6.24 for this market and all things being the same should have been below the mean price of the market that is 454,342.94). The neighborhood poverty level of 17% is also is worse than the market's mean poverty level of 12.94% but it is shored up by the Student-to-Teacher ratio of 15-to-1 which is better than the Market's mean of 18.5. Client 1 should be happy at the location of this house.
The Client 2's house is unfortunately in the lowest 18 percent points of the Boston Market (Mean Price minus 1.363 times the standard deviation). It's a 4 room house that is below the market's mean of 6.24 rooms (thus all other things being the same the price will be below the market's mean price of 454,342.94). The Povert Level in the neighborhood is 32% which is worse than the market's mean poverty level of 12.94% (thus the price for the house is further depressed) The Student-to-Teacher ratio is 22-to-1 which compares unfavorably with the Market's mean of 18.5 (Yet another reason for the price to be depressed compared to the mean) If this client is a buyer, I would say he is getting a good deal...he can live in Boston !
The Client 3's house price at $979,300.00 is a dream ! It has everything going for it. It's a 8 room house (much bigger than the market's mean of 6.24 rooms), It's in a neighborhood where poverty level is only 3 percentage points (compared to Boston market mean of 12.94 percentage points and Student-to-Teacher ratio of 12-to-1 is as good as it gets in Boston area wher the best ratio is 12.6-to-1 (I would be suspicious of this 12-to-1 number because it is better than the population mean)
The prices generally look reasonable to me. Looks like Students-to-Teacher ratio of Client 3's house listing looks suspicious.
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 the data it's trained on.
In [16]:
vs.PredictTrials(features, prices, fit_model, client_data)
In a few sentences, discuss whether the constructed model should or should not be used in a real-world setting.
Hint: Some questions to answering:
Data from 1978 may not be relevant today since housing markets are also affected by interest rates, availability of financing and housing markets have been affected by other factors such as economic conditions which are of a cyclical nature.
This analysis is based on only 3 of the 14 features present the original dataset at https://archive.ics.uci.edu/ml/datasets/Housing It may be worthwhile exploring these other 11 features.
The model is as good as the data it is built on. If the data is irrelevant today, the model is also outdated hence may not be robust.
The model works on samples that are representative of the population. It is unlikely that model based on Boston Housing Market will apply equally well to a rural data sample
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.