Machine Learning Engineer Nanodegree¶
Model Evaluation & Validation¶
Project: Predicting Boston Housing Prices¶
Welcome to the first project of the Machine Learning Engineer Nanodegree! In this notebook, some template code has already been provided for you, and you will need to implement additional functionality to successfully complete this project. You will not need to modify the included code beyond what is requested. Sections that begin with 'Implementation' in the header indicate that the following block of code will require additional functionality which you must provide. Instructions will be provided for each section and the specifics of the implementation are marked in the code block with a 'TODO' statement. Please be sure to read the instructions carefully!
+In addition to implementing code, there will be questions that you must answer which relate to the project and your implementation. Each section where you will answer a question is preceded by a 'Question X' header. Carefully read each question and provide thorough answers in the following text boxes that begin with 'Answer:'. Your project submission will be evaluated based on your answers to each of the questions and the implementation you provide.
++ +Note: Code and Markdown cells can be executed using the Shift + Enter keyboard shortcut. In addition, Markdown cells can be edited by typically double-clicking the cell to enter edit mode.
+
Getting Started¶
In this project, you will evaluate the performance and predictive power of a model that has been trained and tested on data collected from homes in suburbs of Boston, Massachusetts. A model trained on this data that is seen as a good fit could then be used to make certain predictions about a home — in particular, its monetary value. This model would prove to be invaluable for someone like a real estate agent who could make use of such information on a daily basis.
+The dataset for this project originates from the UCI Machine Learning Repository. The Boston housing data was collected in 1978 and each of the 506 entries represent aggregated data about 14 features for homes from various suburbs in Boston, Massachusetts. For the purposes of this project, the following preprocessing steps have been made to the dataset:
+-
+
- 16 data points have an
'MEDV'value of 50.0. These data points likely contain missing or censored values and have been removed.
+ - 1 data point has an
'RM'value of 8.78. This data point can be considered an outlier and has been removed.
+ - The features
'RM','LSTAT','PTRATIO', and'MEDV'are essential. The remaining non-relevant features have been excluded.
+ - The feature
'MEDV'has been multiplicatively scaled to account for 35 years of market inflation.
+
Run the code cell below to load the Boston housing dataset, along with a few of the necessary Python libraries required for this project. You will know the dataset loaded successfully if the size of the dataset is reported.
+ +# 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)
+Data Exploration¶
In this first section of this project, you will make a cursory investigation about the Boston housing data and provide your observations. Familiarizing yourself with the data through an explorative process is a fundamental practice to help you better understand and justify your results.
+Since the main goal of this project is to construct a working model which has the capability of predicting the value of houses, we will need to separate the dataset into features and the target variable. The features, 'RM', 'LSTAT', and 'PTRATIO', give us quantitative information about each data point. The target variable, 'MEDV', will be the variable we seek to predict. These are stored in features and prices, respectively.
Implementation: Calculate Statistics¶
For your very first coding implementation, you will calculate descriptive statistics about the Boston housing prices. Since numpy has already been imported for you, use this library to perform the necessary calculations. These statistics will be extremely important later on to analyze various prediction results from the constructed model.
In the code cell below, you will need to implement the following:
+-
+
- Calculate the minimum, maximum, mean, median, and standard deviation of
'MEDV', which is stored inprices.-
+
- Store each calculation in their respective variable. +
+
# TODO: Minimum price of the data
+minimum_price = np.min(prices)
+
+# TODO: Maximum price of the data
+maximum_price = np.max(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)
+Question 1 - Feature Observation¶
As a reminder, we are using three features from the Boston housing dataset: 'RM', 'LSTAT', and 'PTRATIO'. For each data point (neighborhood):
-
+
'RM'is the average number of rooms among homes in the neighborhood.
+'LSTAT'is the percentage of homeowners in the neighborhood considered "lower class" (working poor).
+'PTRATIO'is the ratio of students to teachers in primary and secondary schools in the neighborhood.
+
Using your intuition, for each of the three features above, do you think that an increase in the value of that feature would lead to an increase in the value of 'MEDV' or a decrease in the value of 'MEDV'? Justify your answer for each.
+Hint: 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: higher RM should lead to an increase in MEDV, as more rooms means higher price. Higher LSTAT should lead to a decrease in MEDV, as more lower class people means less purchasing power so lower home prices. Higher PTRATIO should lead to a decrease in home prices, as richer people may go to neighbourhoods with more advantageous, lower student to teacher ratios.
+ ++
Developing a Model¶
In this second section of the project, you will develop the tools and techniques necessary for a model to make a prediction. Being able to make accurate evaluations of each model's performance through the use of these tools and techniques helps to greatly reinforce the confidence in your predictions.
+ +Implementation: Define a Performance Metric¶
It is difficult to measure the quality of a given model without quantifying its performance over training and testing. This is typically done using some type of performance metric, whether it is through calculating some type of error, the goodness of fit, or some other useful measurement. For this project, you will be calculating the coefficient of determination, R2, to quantify your model's performance. The coefficient of determination for a model is a useful statistic in regression analysis, as it often describes how "good" that model is at making predictions.
+The values for R2 range from 0 to 1, which captures the percentage of squared correlation between the predicted and actual values of the target variable. A model with an R2 of 0 is no better than a model that always predicts the mean of the target variable, whereas a model with an R2 of 1 perfectly predicts the target variable. Any value between 0 and 1 indicates what percentage of the target variable, using this model, can be explained by the features. A model can be given a negative R2 as well, which indicates that the model is arbitrarily worse than one that always predicts the mean of the target variable.
+For the performance_metric function in the code cell below, you will need to implement the following:
-
+
- Use
r2_scorefromsklearn.metricsto perform a performance calculation betweeny_trueandy_predict.
+ - Assign the performance score to the
scorevariable.
+
# TODO: 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. """
+
+ from sklearn.metrics import r2_score
+ # TODO: Calculate the performance score between 'y_true' and 'y_predict'
+ score = r2_score(y_true, y_predict)
+
+ # Return the score
+ return score
+Question 2 - Goodness of Fit¶
Assume that a dataset contains five data points and a model made the following predictions for the target variable:
+| True Value | +Prediction | +
|---|---|
| 3.0 | +2.5 | +
| -0.5 | +0.0 | +
| 2.0 | +2.1 | +
| 7.0 | +7.8 | +
| 4.2 | +5.3 | +
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.
# 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:yes, because r squared is high and thus 92.3 % of the dependent variable is explained by the independent variable.
+ +Implementation: Shuffle and Split Data¶
Your next implementation requires that you take the Boston housing dataset and split the data into training and testing subsets. Typically, the data is also shuffled into a random order when creating the training and testing subsets to remove any bias in the ordering of the dataset.
+For the code cell below, you will need to implement the following:
+-
+
- Use
train_test_splitfromsklearn.cross_validationto shuffle and split thefeaturesandpricesdata into training and testing sets.-
+
- Split the data into 80% training and 20% testing. +
- Set the
random_statefortrain_test_splitto a value of your choice. This ensures results are consistent.
+
+ - Assign the train and testing splits to
X_train,X_test,y_train, andy_test.
+
# 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=5)
+
+# Success
+print "Training and testing split was successful."
+Question 3 - Training and Testing¶
What is the benefit to splitting a dataset into some ratio of training and testing subsets for a learning algorithm?
+Hint: What could go wrong with not having a way to test your model?
Answer: It helps to prevent overfitting ensuring the trained model generalizes well to out of sample data.
+ ++
Analyzing Model Performance¶
In this third section of the project, you'll take a look at several models' learning and testing performances on various subsets of training data. Additionally, you'll investigate one particular algorithm with an increasing 'max_depth' parameter on the full training set to observe how model complexity affects performance. Graphing your model's performance based on varying criteria can be beneficial in the analysis process, such as visualizing behavior that may not have been apparent from the results alone.
Learning Curves¶
The following code cell produces four graphs for a decision tree model with different maximum depths. Each graph visualizes the learning curves of the model for both training and testing as the size of the training set is increased. Note that the shaded region of a learning curve denotes the uncertainty of that curve (measured as the standard deviation). The model is scored on both the training and testing sets using R2, the coefficient of determination.
+Run the code cell below and use these graphs to answer the following question.
+ +# Produce learning curves for varying training set sizes and maximum depths
+vs.ModelLearning(features, prices)
+
+Question 4 - Learning the Data¶
Choose one of the graphs above and state the maximum depth for the model. What happens to the score of the training curve as more training points are added? What about the testing curve? Would having more training points benefit the model?
+Hint: Are the learning curves converging to particular scores?
Answer: The max depth is 3, after that overfitting occurs. As more examples are added, the training score will decline a bit as it gets harder to fit all examples. But testing also gets better, until the two almost converge. Adding more training point would benefit the model a bit, as training and testing curves would converge even more. However, additional points are not always helpful, as at some point no new information becomes available, so the model predictions on the test set will not improve in accuracy further, so the testing score doesnt improve more. The reason why the training score is high initially is that it is easy to fit the data as there are few examples. As more examples becomes available, it gets more difficult to fit so testing score drops.
+ +Complexity Curves¶
The following code cell produces a graph for a decision tree model that has been trained and validated on the training data using different maximum depths. The graph produces two complexity curves — one for training and one for validation. Similar to the learning curves, the shaded regions of both the complexity curves denote the uncertainty in those curves, and the model is scored on both the training and validation sets using the performance_metric function.
Run the code cell below and use this graph to answer the following two questions.
+ +vs.ModelComplexity(X_train, y_train)
+
+Question 5 - Bias-Variance Tradeoff¶
When the model is trained with a maximum depth of 1, does the model suffer from high bias or from high variance? How about when the model is trained with a maximum depth of 10? What visual cues in the graph justify your conclusions?
+Hint: How do you know when a model is suffering from high bias or high variance?
Answer: At max depth 1, it suffers from high bias - r^2 is generally low for both training and validation, so it has low predictive power. At max depth 10, it suffers from high variance, as r^2 for training goes to 1, but validation remains low, meaning model is overfitting and does not generalise well. These conclusions are based on the fact that the training and validation scores diverge at higher max depth, with training improving, whereas validation is dropping off and even declining.
+ +Question 6 - Best-Guess Optimal Model¶
Which maximum depth do you think results in a model that best generalizes to unseen data? What intuition lead you to this answer?
+ +Answer: It seems to be max depth of 3, as afterwards validation declines whereas training continues to rise, showing the model cannot generalise well anymore, so there is no benefit to increasing max depth.
+ ++
Evaluating Model Performance¶
In this final section of the project, you will construct a model and make a prediction on the client's feature set using an optimized model from fit_model.
Question 7 - Grid Search¶
What is the grid search technique and how it can be applied to optimize a learning algorithm?
+ +Answer: The grid search space is a space occupied by parameters to a model. The grid search technique will try all possible points on this space (combination of parameters) The output of grid search are many models, each using different parameters. This is a selective search, as only points on the parameter grid will be used. The search is guided by chosing a range of values for each parameter dimension picking a step size large enough to expect the output to vary in some interesting way. In case of enum parameters, we could earch across several different classifiers for example, to get an idea which works best.
+ +Question 8 - Cross-Validation¶
What is the k-fold cross-validation training technique? What benefit does this technique provide for grid search when optimizing a model?
+Hint: Much like the reasoning behind having a testing set, what could go wrong with using grid search without a cross-validated set?
Answer: The K Fold cross validation technique splits the dataset into training and cross validation sets. More specifically, it splits the data into K subsets, using one of the subsets as validation and the k-1 others combined as the training set. It then trains and evaluates the ML algorithm. In the next iteration it will pick a different training and cross validation set and repeat the previous step. Once all K combinations have been explored, an average performance estimate is computed across the outputs of each of the k steps run. The benefit of this technique to grid search is that you get a more reliable, stable estimate for each point on the parameter grid, as it will be run using several splits of training / testing data, eliminating potential bias in the dataset and given stable values no matter how the dataset is divided into training and testing. This K fold validation is useful for grid search as each point on the grid will get a more reliable estimate of model performance than not using K folder validation. If we limit grid search to a single dataset (no K fold validation) overfitting would be an issue, as the specific choice of model function might lead to lack of generalisation to out of sample data. Applying this generalisation inside the dataset, many times over, make the estimate more stable.
+ +Implementation: Fitting a Model¶
Your final implementation requires that you bring everything together and train a model using the decision tree algorithm. To ensure that you are producing an optimized model, you will train the model using the grid search technique to optimize the 'max_depth' parameter for the decision tree. The 'max_depth' parameter can be thought of as how many questions the decision tree algorithm is allowed to ask about the data before making a prediction. Decision trees are part of a class of algorithms called supervised learning algorithms.
In addition, you will find your implementation is using ShuffleSplit() for an alternative form of cross-validation (see the 'cv_sets' variable). While it is not the K-Fold cross-validation technique you describe in Question 8, this type of cross-validation technique is just as useful!. The ShuffleSplit() implementation below will create 10 ('n_splits') shuffled sets, and for each shuffle, 20% ('test_size') of the data will be used as the validation set. While you're working on your implementation, think about the contrasts and similarities it has to the K-fold cross-validation technique.
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:
-
+
- Use
DecisionTreeRegressorfromsklearn.treeto create a decision tree regressor object.-
+
- Assign this object to the
'regressor'variable.
+
+ - Assign this object to the
- Create a dictionary for
'max_depth'with the values from 1 to 10, and assign this to the'params'variable.
+ - Use
make_scorerfromsklearn.metricsto create a scoring function object.-
+
- Pass the
performance_metricfunction as a parameter to the object.
+ - Assign this scoring function to the
'scoring_fnc'variable.
+
+ - Pass the
- Use
GridSearchCVfromsklearn.grid_searchto create a grid search object.-
+
- Pass the variables
'regressor','params','scoring_fnc', and'cv_sets'as parameters to the object.
+ - Assign the
GridSearchCVobject to the'grid'variable.
+
+ - Pass the variables
# TODO: Import 'make_scorer', 'DecisionTreeRegressor', and '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
+ from sklearn.tree import DecisionTreeRegressor
+ regressor = DecisionTreeRegressor(random_state =1)
+
+ # TODO: Create a dictionary for the parameter 'max_depth' with a range from 1 to 10
+ import numpy as np
+ params = {"max_depth": np.arange(1,11)}
+ print("params", params)
+
+ # TODO: Transform 'performance_metric' into a scoring function using 'make_scorer'
+ from sklearn.metrics import make_scorer
+ scoring_fnc = make_scorer(performance_metric)
+
+ # TODO: Create the grid search object
+ from sklearn.grid_search import GridSearchCV
+ grid = GridSearchCV(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_
+Making Predictions¶
Once a model has been trained on a given set of data, it can now be used to make predictions on new sets of input data. In the case of a decision tree regressor, the model has learned what the best questions to ask about the input data are, and can respond with a prediction for the target variable. You can use these predictions to gain information about data where the value of the target variable is unknown — such as data the model was not trained on.
+ +Question 9 - Optimal Model¶
What maximum depth does the optimal model have? How does this result compare to your guess in Question 6?
+Run the code block below to fit the decision tree regressor to the training data and produce an optimal model.
+ +# 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: The max depth of the optimal model is 4. My guess would have been 3, as it seems with 4, training accuracy goes up, but much more so than test accuracy, which almost flatlines.
+ +Question 10 - Predicting Selling Prices¶
Imagine that you were a real estate agent in the Boston area looking to use this model to help price homes owned by your clients that they wish to sell. You have collected the following information from three of your clients:
+| Feature | +Client 1 | +Client 2 | +Client 3 | +
|---|---|---|---|
| Total number of rooms in home | +5 rooms | +4 rooms | +8 rooms | +
| Neighborhood poverty level (as %) | +17% | +32% | +3% | +
| Student-teacher ratio of nearby schools | +15-to-1 | +22-to-1 | +12-to-1 | +
What price would you recommend each client sell his/her home at? Do these prices seem reasonable given the values for the respective features?
+Hint: Use the statistics you calculated in the Data Exploration section to help justify your response.
Run the code block below to have your optimized model make predictions for each client's home.
+ +# 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: Client 1's home: around \$410k, Client 2's home: around\$235k, Client 3's home: around \$920k. Prices seem reasonable, as Client 3's home has low neigbourhood poverty, low student teacher ratio and a lot of bedrooms. C2 home should be the lowest, as least rooms, worst student teacher ratio, and highest neighbourhood poverty. C1 works too and is in between C3 and C2, as better student teacher ratio, half the poverty level, and 1 extra room compared to C2. More generally, prices are in the range of the dataset and a large house in a good neighborhood is close to the maximum prices which is expected
+ +Sensitivity¶
An optimal model is not necessarily a robust model. Sometimes, a model is either too complex or too simple to sufficiently generalize to new data. Sometimes, a model could use a learning algorithm that is not appropriate for the structure of the data given. Other times, the data itself could be too noisy or contain too few samples to allow a model to adequately capture the target variable — i.e., the model is underfitted. Run the code cell below to run the fit_model function ten times with different training and testing sets to see how the prediction for a specific client changes with the data it's trained on.
vs.PredictTrials(features, prices, fit_model, client_data) # predicts client 1
+Question 11 - Applicability¶
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:
-
+
- How relevant today is data that was collected from 1978? +
- Are the features present in the data sufficient to describe a home? +
- Is the model robust enough to make consistent predictions? +
- Would data collected in an urban city like Boston be applicable in a rural city? +
Answer: Data from 1978 may not be that relevant, as underlying factors may have changed, such as the distribution of wealth across neighbourhoods, or the number of teachers and students in schools. However, if there were no such trends, the dataset could be a good proxy, as price was corrected for inflation. The number of features are probably not sufficient to describe a home - condition, size of land, age and other features may have additional or more important predictive power. The model does not seem very robust - a range of around 70k in predictions on just 10 iterations would be unacceptable to a prospective seller. Finally, urban data may apply to rural settings as there are less neigbourhoods, prices are generally lower, and properties illiquid so not many transactions so price discovery may not follow the same pattern as in a city. Thus the model's use should be restricted to an urban setting, additional features should be added to control the variance of predictions, and ideally updated data would give a better price estimate that may come closer to a real transaction as of today.
+ ++ +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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