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:
'MDEV' 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 'MDEV' are essential. The remaining non-relevant features have been excluded.'MDEV' has been multiplicatively scaled to account for 35 years of market inflation.Run the code cell below to load the Boston housing dataset, along with a few of the necessary Python libraries required for this project. You will know the dataset loaded successfully if the size of the dataset is reported.
In [1]:
# Import libraries necessary for this project
import numpy as np
import pandas as pd
import seaborn as sns
import visuals as vs # Supplementary code
from sklearn.cross_validation import ShuffleSplit
# Pretty display for notebooks
%matplotlib inline
# Load the Boston housing dataset
data = pd.read_csv('housing.csv')
prices = data['MDEV']
features = data.drop('MDEV', 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, 'MDEV', 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:
'MDEV', which is stored in prices.
In [2]:
# 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(ddof=0)
# 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 all Boston homeowners who have a greater net worth than homeowners in the neighborhood.'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 'MDEV' or a decrease in the value of 'MDEV'? 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?
Answer:
RM is the average number of rooms among homes in the neighborhood. The price of a three room house is likely to be less than the price of a five/six room house. It's likely that the increase in room number will lead to an increase in the house price.
LSTAT is the percentage of the Boston homeowners who had a greater networth than the homeowners in the neighborhood. If the neighborhood is poor, then it's likely the LSTAT will be high as more Boston homeowners will have a higher networth than the homeowners of the respective neighborhood. In a poor neighborhood, the house price is low. It's likely that increase in LSTAT will lead to a decrease in the house price MDEV.
PTRATIO is the ratio of students to teachers in the primary and secondary school in the neighborhood. If PTRATIO is 12/1 it means 1 teacher has to attend to at least 10 students. In poor neighborhoods there's often not enough fund in schools to hire more teachers. Thus I'd assume higher PTRATIO will correspond to a poor neighborhood where the house prices will be low.
To confirm my intuition I created a pairplot of MDEV against the three values, RM, LSTAT and PTRATIO. RM shows a positive relationship, LSTAT shows a negative one and PTRATIO also shows a strict negative relationship after 20 and moderative negative relationship before 20 with some occassional upper end prices.
In [3]:
sns.pairplot(data,x_vars = ["RM","LSTAT","PTRATIO"], y_vars = ["MDEV"],kind = "scatter")
Out[3]:
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 always fails to predict 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 no better than one that naively 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 [4]:
# 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 |
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 [5]:
# 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)
Answer:
Given R^2 captures the percentage of squared correlation between the predicted and actual values of the target variable and it ranges from 0 to 1 where close to 1 means better performance, and given this model had a R^2 score of 0.923, it appears that the model has been able to successfully capture the variation 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 [6]:
# 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 = 0)
# Success
print "Training and testing split was successful."
Answer:
If we train and test our model on the same data set, the model will most likely to be good at predicting as it has already seen the data. But it would give a false sense of high performance(with respect to the evaluation metric we choose). The end goal of a machine learning model is to make accurate predictions on the data it has not seen before.
If we split the dataset into training and testing sets, then we can train the model on the training data and test it on the independent testing dataset. After assessing the model's performance on the testing data set that the model has not seen before, we can be more confident about it's performance.
So we split the data set into training and testing subsets to avoid overfitting and to test the model's performance against a data set it has not seen before.
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 [7]:
# 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?
Answer:
It appears that both training and testing curve is showing lower scores compared to other graphs.
As more training points are being added, the score of the training curve goes down significantly. At first it went down to around 0.6 when there was about 50 data points, but after adding more data points the score went down even furture and levelled off near 0.4 after 100 data points.
In the graph with max depth 1, the testing accuracy is near 0.4 at maximum and then mostly stays there. It achieves the maximum accuracy with 50 data points, but after that as more data gets added it does nothing to improve the performance of the testing curve.
It seems like a classic case of underfitting as the model is too simple to capture the complexities inherent in the data set. This model has high bias. Adding more training points will not benefit the model as it will consistently under fit the data set because the model is just too simple to capture complexities of the process that developed the data.
In this graph the scores seem to be converging to 0.4. In models with high bias, training and testing scores tend to converge to same score because the model fails to capture complexities above that level systematically. This is yet another evidence to claim that this model has high bias even adding lots of data points will not improve the performance of the model significantly.
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 [8]:
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?
Answer:
When the model is trained with a maximum depth of one, the model suffers from high bias because the accuracy for both training and testing score is relatively low,around 0.4 for the testing score and 0.5 for the training score.
This suggests the model failed to capture the complex relationships inherent in the data set and suffers from underfitting. This is a classic case of high bias.
When the model is trained with a maximum depth of 10 the difference between training score and testing score is pretty high. Training score is showing as 'near accurate'/very close to one while testing score is around 0.7.
This suggests that the model suffers from high variance consdering it has very good performance on training score because it has learnt the quirks of the training dataset and overfitted to it.
However, it fails to generalize it's predictions to the testing dataset because of overfitting. This is a classic case of suffering from high variance.
In max depth one the training score and testing score is pretty close and both of them are low scores. This was the visual cue for high bias because models with high bias converge to similar training and testing scores and underfits the data.
In max depth 10 the visual cue was the big gap between training and testing score and the training score being close to one. This suggested the model has overfitted the training data and scored very high on it, while it failed to generalize to the testing data.
Answer:
I think the model with max depth 4 best generalizes to the unseen data because it has the highest testing score which is very close to 0.8 and that's also the depth where the difference between the training score and the testing score is really small. Until the depth of four, the testing score keeps rising.
Above the depth of four, the training score of the model keeps increasing and goes to one but the testing score keeps decreasing while the gap between the training and testing scores also increases.This signals overfitting and a high variance model. Those models will not be able to generalize over independent data sets.
Under the depth of four the gap between training and testing score is low, but the score it self is really low which signals underfitting and high bias. Those models will not be able to capture the complexities of the data sets for both training and testing sets.
Answer:
Many Learning algorithms tend to have parameters that can be tuned for optimal model performance over a particular data set. However, figuring out the best combination of the parameters manually can be a lengthy process.
Grid Search automates this process by trying out multiple combinations of the given parameters over an estimator/model to find out the best combination for those parameters under the chosen evaluation metric to determine the best model. This way we can optimize a learning algorithm by choosing the model with the best parameter combination returned by the grid search. Note that Grid Search will exhaustively try all combinations of the given parameters instead of a random sample of the parameters.
Answer:
When we train a model we want to use as much data as possible, but we also want to test our model using as much data as possible. As we split the data into training and testing sets, we have to make a trade off when we pick the amount of data that will go into the training set vs the amount of the data that will go to the testing set.
K-Ford cross validation overcomes this problem by randomly splitting the data into k subsets. For k iterations, each time one subset of the data set is hold out for the testing and the rest goes into training the model. This way we can evaluate the model's performance k times and the average score over the k-iterations is likely to be a good predictor of the model's actual performance over new data sets when it will have to generalize it's predictions compared to the situation where we only split the data set into two subsets, training and testing and evaluate performance only once. This way the model learns from the whole data set and performs testing on the entire data sets leading to better performance and more accurate evaluations.
When Grid Search technique is trying out different combinations of the given parameters for the model, if it evaluates the model only by splitting the data into two subsets, training and testing it can easily return an overfitted model which has learnt only the quirks of the training subset. Not to mention in this case we will not be using the whole data set for training and testing either, k-fold cross validation can be combined the grid search technique to ensure that the grid-search is choosing the model that best generalizes instead of the model that overfits.
In this case we will provide the grid search with some parameters and a given model and the data set. The k-fold cross validation will evaluate the grid search's chosen model with some combination of the parameters in each iteration and return the average score. Grid Search will choose the model that performed best on k-fold cross validation and return that model with the chosen combination of the parameters.
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.
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 [9]:
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.tree import DecisionTreeRegressor
from sklearn.metrics import make_scorer
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
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":(1,2,3,4,5,6,7,8,9,10)}
# TODO: Transform 'performance_metric' into a scoring function using 'make_scorer'
scoring_fnc = make_scorer(performance_metric) # as performance metric is R^2 which is a scoring function, not a loss function
# greater_is_better defaults to true
# TODO: Create the grid search object
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 [10]:
# 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'])
Answer:
It appears that the optimal model has the max depth of four. My initial guess was that choosing the model with max depth of four would be the optimal choice because the difference between training and testing score was small and the testing score was the highest near 0.8 at the max depth of four in the model complexity graph.
However, since the grid search has choosen the model with max depth of four it verifies my assumption.
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 |
| Household net worth (income) | Top 34th percent | Bottom 45th percent | Top 7th percent |
| 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 [11]:
# Produce a matrix for client data
client_data = [[5, 34, 15], # Client 1
[4, 55, 22], # Client 2
[8, 7, 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:
The statistics from the data exploration section looks like(all in USD) :
The first client's house has 5 rooms, the household networth is top 34 percent and the student teacher ratio of nearby schools is 15 to 1. The model predicted his house should be sold for $324,240,00. The house has average number of rooms and similar to average PTRATIO, I believe the LSTAT number 34th percentile got this houses price down. The predicted price is pretty close to the median price though ( 438,900.00) so it's more or less reasonable.
The second client's house has four rooms, the household networth is around bottom 45 percent and the student teacher ratio of nearby schools is 22 to 1. From the data exploratory section we have noticed that the low household networth and high student teacher ratio is negatively correlated with the house price. The model has predicted the house price around 189,123.53 USD which is near the minimum. The maximum PTRATIO in this data set is also 22 and the maximum LSTAT is around 37. Given the house does seem to be owned by poor owners(with top 55% household networth greater than this household owners) and the PTRATIO is around the maximum(22), the model's predictions that this house will sale below the average house price( mean house price = 438,900.00 USD) and near the minimum price( 105,000.00) seems reasonable to me.
The third client's house has eight rooms, the house hold networth is around top 7 percent and PTRATIO is at the data set's minimum which is 12 to one. The maximum number of rooms in this data set also happens to be eight. Since the number of rooms is positively correlated with the price, the number of household networth and PTRATIO is negatively correlated, it appears that the models prediction this house will sale at the price of 942,666.57, which is near the maximum price (1,024,800.00) is a reasonable guess.
From the nearest neighbors algorithm(below) we can see that the numbers for the predictions made by the decision-tree-regressor are pretty close, at least between one standard deviation above/below the predictions made by nearest neighbors. Note that the nearest neighbors part was added from the pro-tips part of the last review.
In [12]:
from sklearn.neighbors import NearestNeighbors
num_neighbors=5
def nearest_neighbor_price(x):
def find_nearest_neighbor_indexes(x, X): # x is your vector and X is the data set.
neigh = NearestNeighbors( num_neighbors )
neigh.fit(X)
distance, indexes = neigh.kneighbors( x )
return indexes
indexes = find_nearest_neighbor_indexes(x, features)
sum_prices = []
for i in indexes:
sum_prices.append(prices[i])
neighbor_avg = np.mean(sum_prices)
return neighbor_avg
print nearest_neighbor_price( [4, 55, 22])
index = 0
for i in client_data:
val=nearest_neighbor_price(i)
index += 1
print "The predicted {} nearest neighbors price for home {} is: ${:,.2f}".format(num_neighbors,index, val)
In [13]:
data[["RM","LSTAT","PTRATIO"]].describe()
Out[13]:
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 [14]:
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:
Answer:
I believe this model should not be used in a real world setting.
Some reasons :
The model, decision tree regressor is a reasonable selection to predict housing prices, but the data set we have trained the model on, is 20 years old and is not a good refletion of the current trends. Perhaps we should collect better data and train another model before making any prediction in this situation.
In [ ]: