Machine Learning Engineer Nanodegree¶

Model Evaluation & Validation¶

Project: Predicting Boston Housing Prices¶

Getting Started¶

In this project, we 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.

Code cell below load the Boston housing dataset, along with a few of the necessary Python libraries required for this project. We 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.model_selection 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))

Boston housing dataset has 489 data points with 4 variables each.

Data Exploration¶

In this first section of this project, we will make a cursory investigation about the Boston housing data and provide our observations. Familiarizing ourself with the data through an explorative process is a fundamental practice to help us better understand and justify our 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¶

We will calculate descriptive statistics about the Boston housing prices. These statistics will be extremely important later on to analyze various prediction results from the constructed model.

# Minimum price of the data
minimum_price = np.min(prices)

# Maximum price of the data
maximum_price = np.max(prices)

# Mean price of the data
mean_price = np.mean(prices)

# Median price of the data
median_price = np.median(prices)

# 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: ${}".format(minimum_price)) 
print("Maximum price: ${}".format(maximum_price))
print("Mean price: ${}".format(mean_price))
print("Median price ${}".format(median_price))
print("Standard deviation of prices: ${}".format(std_price))

Statistics for Boston housing dataset:

Minimum price: $105000.0
Maximum price: $1024800.0
Mean price: $454342.9447852761
Median price $438900.0
Standard deviation of prices: $165171.13154429477

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.

Question: What do we think that an increase in the value of these feature would lead to an increase in the value of 'MEDV' or a decrease in the value of 'MEDV'?

Hint: This problem can phrased using examples like below.

Would we expect a home that has an 'RM' value(number of rooms) of 6 be worth more or less than a home that has an 'RM' value of 7?
Would we expect a neighborhood that has an '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?
Would we expect a neighborhood that has an '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: So, For each of the three features above, let us calculate 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'

#### How will 'RM' (Number of room) value affect 'MEDV'? So, if the 'RM' (Number of room) value increases the value of 'MEDV' will definitely increase. Since, More rooms means more people can stay and if there are more numbers of room per dwelling the cost of that dwelling will also increase.
#### How will 'LSTAT' (Percent of lower class workers) value affect 'MEDV'? So, if the 'LSTAT' (Percent of lower class workers) value increases the value of 'MEDV' will definitely decrease. Since, More the percent of lower class workers staying in the area the less will be the percent of rich people staying in the area, Because the people who are rich will like to leave in higher value houses, which will lead to decrease in the value of house as poor people wont be able to pay more.
#### How will 'PTRATIO' (Ratio of students to teachers) value affect 'MEDV'? So, if the 'PTRATIO' (Ratio of students to teachers) value increases the value of 'MEDV' will definitely decrease. Since, More the ratio means more number of students would like to stay in the area since they can directly interact with teachers, which will lead to students paying more for the houses.

Developing a Model¶

In this second section of the project, we 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 our 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, we will be calculating the coefficient of determination, R², to quantify our 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, WE have implementeed the following:

Used r2_score from sklearn.metrics to perform a performance calculation between y_true and y_predict.
Assigned the performance score to the score variable.

# 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. """
    
    # 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

Below we used 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))

Model has a coefficient of determination, R^2, of 0.923.

Question:

Would we consider this model to have successfully captured the variation of the target variable?
Why or why not?

Hint: The R2 score is the proportion of the variance in the dependent variable that is predictable from the independent variable. In other words:

R2 score of 0 means that the dependent variable cannot be predicted from the independent variable.
R2 score of 1 means the dependent variable can be predicted from the independent variable.
R2 score between 0 and 1 indicates the extent to which the dependent variable is predictable. An
R2 score of 0.40 means that 40 percent of the variance in Y is predictable from X.

Answer: Yes, we will definitely consider this model to have successfully captured the variation of the target variable. Because, the r2 score lies between 0 to 1 and more it is closer to 1, more is the probablity that a models will find a predicted value with less errors. Also the varriance of difference is 0.472 and that of true value is 6.118, so 1 - 0.472/6.118 = 0.923

Implementation: Shuffle and Split Data¶

Our next implementation required that we 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, we implemented the following:

Used train_test_split from sklearn.model_selection to shuffle and split the features and prices data into training and testing sets.
- Splited the data into 80% training and 20% testing.
- Set the random_state for train_test_split to a value of our choice. This ensures results are consistent.
Assigned the train and testing splits to X_train, X_test, y_train, and y_test.

# Import 'train_test_split'
from sklearn.model_selection import train_test_split

# Shuffle and split the data into training and testing subsets
X_train, X_test, y_train, y_test = train_test_split(features, prices, train_size=0.8, random_state=42)

# Success
print("Training and testing split was successful.")

Training and testing split was successful.

C:\Users\paras.jain\AppData\Local\Continuum\anaconda3\lib\site-packages\sklearn\model_selection\_split.py:2069: FutureWarning: From version 0.21, test_size will always complement train_size unless both are specified.
  FutureWarning)

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: Think about how overfitting or underfitting is contingent upon how splits on data is done.

Answer: The benefit to splitting a dataset into some ratio of training and testing subsets for a learning algorithm is very important. Beacuse, if we split the data, we can then run performance metric on the predicted value of the test data with the original data, and can find out if our model is making correct predictions or not with the help of performance metrics like r2_score. Also, This helps us identify whether the model is underfitted or overfitted and if it needs some tweaking.

Analyzing Model Performance¶

In this third section of the project, we took a look at several models' learning and testing performances on various subsets of training data. Additionally, we have investigated one particular algorithm with an increasing 'max_depth' parameter on the full training set to observe how model complexity affects performance. Graphing our 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 produced 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.

# 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? Generally speaking, the more data we have, the better. But if our training and testing curves are converging with a score above our 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: So, if we see the graph with maximum depth = 3, We can see that the training data has a score of 0.7 to 0.8 and also the testing data has a score of 0.7 to 0.8, and difference is also very less between the training and testing scores. Also if we increase the number of training point the model wont be much affected as we can see a constant uncertainity after 300 training points.

Complexity Curves¶

The following code cell produced 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.

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 our conclusions?

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: When the model is trained with a maximum depth of 1, the model suffer from high bias. Because the training and validation score are very less, which means the model is not sufficient enough to capture complex problem. And when the model is trained with a maximum depth of 10, the model suffer from high variance. Beacuse the training score is good but the validation score is less with high uncertaininty.

Question 6 - Best-Guess Optimal Model¶

Which maximum depth do we think results in a model that best generalizes to unseen data?
What intuition lead us to this answer?

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: The maximum depth I think results in a model that best generalizes to unseen data is 3. Since the training and validation scores are at peak over depth 3 and less unceratainity. Also they started diverging from depth 4 where the training data is doing good but not the testing data, that to with high uncertainity.

Evaluating Model Performance¶

In this final section of the project, we will constructed a model and made prediction on the client's feature set using an optimized model from fit_model.

Question 7 - Grid Search¶

What is the grid search technique?
How it can be applied to optimize a learning algorithm?

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 our answer, we can also give an example of a parameter in a model that can be optimized using this approach.

Answer: A Machine learning algorithm has various parameters which are configured to improve the performance of an algorithm on the dataset. Grid search is a technique that can search accross a parameter space to determine which values of the parameters give the best performance during the learning stage of the algorithm. E.g For decesion tree we have parameters for depth as we have seen above. Grid Search allows us to provide various models to apply in our data in order to see which one provides the most useful insight. Similarly the individual hyperparameter of each model can be adjusted and tested to see which hyperparameter settings provide the optimal results. This optimal result can be measured as a score such as R-Squared, F1 etc. depending what we want to use as the score measure, note here that not only F1 score is a metric for Grid Search. Without Grid-Search, we have to set the hyperparameter to produce the best outcome running every combination time at time. Nevertheless, with a grid search we run the one function without the "manual effort".

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: 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. We can refer to the docs for our answer.

Answer: A k-fold cross validation training technique takes a dataset and splits it into k subsets. During the training stage the algorithm is trained k times using each subset as a testing set once, giving back k different learned algorithms. If the grid search algorithm is applied to only one training, testing split the parameters are overfitted for that single set, but if the grid search algorithm is applied to k training and testing split, we will be able to decide the best parameter by looking all the learning models.

Implementation: Fitting a Model¶

Our final implementation requires that we bring everything together and train a model using the decision tree algorithm. To ensure that we are producing an optimized model, we 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, we will find our 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 we 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 we're working on our implementation, think about the contrasts and similarities it has to the K-fold cross-validation technique.

For the fit_model function below, we implemented the following:

Used DecisionTreeRegressor from sklearn.tree to create a decision tree regressor object.
- Assign this object to the 'regressor' variable.
Created a dictionary for 'max_depth' with the values from 1 to 10, and assigned this to the 'params' variable.
Used make_scorer from sklearn.metrics to create a scoring function object.
- Passed the performance_metric function as a parameter to the object.
- Assigned this scoring function to the 'scoring_fnc' variable.
Used GridSearchCV from sklearn.model_selection to create a grid search object.
- Pass the variables 'regressor', 'params', 'scoring_fnc', and 'cv_sets' as parameters to the object.
- Assigned the GridSearchCV object to the 'grid' variable.

# Import 'make_scorer', 'DecisionTreeRegressor', and 'GridSearchCV'
from sklearn.tree import DecisionTreeRegressor
from sklearn.metrics import make_scorer
from sklearn.model_selection 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(n_splits = 10, test_size = 0.20, random_state = 0)

    # Create a decision tree regressor object
    regressor = DecisionTreeRegressor()

    # Create a dictionary for the parameter 'max_depth' with a range from 1 to 10
    params = {'max_depth': range(1,11)}

    # Transform 'performance_metric' into a scoring function using 'make_scorer' 
    scoring_fnc = make_scorer(performance_metric)

    # Create the grid search cv object --> GridSearchCV()
    # Make sure to include the right parameters in the object:
    # (estimator, param_grid, scoring, cv) which have values 'regressor', 'params', 'scoring_fnc', and 'cv_sets' respectively.
    grid = GridSearchCV(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. we 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 our guess in Question 6?

Runed the code block below to fit the decision tree regressor to the training data and produced 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']))

Parameter 'max_depth' is 4 for the optimal model.

Hint: The answer comes from the output of the code snipped above.

Answer: Maximum depth the optimal model have is 4. We choosed maxed depth as 3 in Question 6, it may be because we had learned the model only on one training set.

Question 10 - Predicting Selling Prices¶

Imagine that we were a real estate agent in the Boston area looking to use this model to help price homes owned by our clients that they wish to sell. we have collected the following information from three of our 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 we recommend each client sell his/her home at?
Do these prices seem reasonable given the values for the respective features?

Hint: Use the statistics we calculated in the Data Exploration section to help justify our 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.

Runed the code block below to have our 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))

Predicted selling price for Client 1's home: $403,025.00
Predicted selling price for Client 2's home: $237,478.72
Predicted selling price for Client 3's home: $931,636.36

data.describe()

Answer: Predicted selling price for Client 1's home is $403,025.00 having 5 Rooms, 17% as Neighborhood poverty level and Student-teacher ratio of nearby schools is 15. So we can see that selling price lies approx to the 50% data in the MEDV collumn which is 1 RM std below the mean, 1 LSTAT std above the mean and 2 PTRATIO std below the mean. The prediction seems consistent with stats, since all the features are mostly approx to mean.

Predicted selling price for Client 2's home is $237,478.72 having 4 Rooms, 32% as Neighborhood poverty level and Student-teacher ratio of nearby schools is 22. So we can see that selling price lies between the low 25% data in the MEDV collumn which is 3 RM std below the mean, 2 LSTAT std above the mean and 2 PTRATIO std above the mean. The prediction seems consistent with stats since the poverty value is very high.

Predicted selling price for Client 3's home is $931,636.36 having 8 Rooms, 3% as Neighborhood poverty level and Student-teacher ratio of nearby schools is 12. So we can see that selling price lies between the top 25% data in the MEDV collumn which is 2 RM std above the mean, 1.5 LSTAT std below the mean and 4 PTRATIO std below the mean. The prediction seems consistent with stats since the number of room is much greater and the poverty level is much less though the student -teacher ration is less then mean.

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.

Runed 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.

vs.PredictTrials(features, prices, fit_model, client_data)

Trial 1: $391,183.33
Trial 2: $419,700.00
Trial 3: $415,800.00
Trial 4: $420,622.22
Trial 5: $413,334.78
Trial 6: $411,931.58
Trial 7: $399,663.16
Trial 8: $407,232.00
Trial 9: $351,577.61
Trial 10: $413,700.00

Range in prices: $69,044.61

Question 11 - Applicability¶

In a few sentences, discuss whether the constructed model should or should not be used in a real-world setting.

Hint: Take a look at the range in prices as calculated in the code snippet above. Some questions to answering:

How relevant today is data that was collected from 1978? How important is inflation?
Are the features present in the data sufficient to describe a home? Do we think factors like quality of apppliances in the home, square feet of the plot area, presence of pool or not etc should factor in?
Is the model robust enough to make consistent predictions?
Would data collected in an urban city like Boston be applicable in a rural city?
Is it fair to judge the price of an individual home based on the characteristics of the entire neighborhood?

Answer: The constructed model should not be used in a real world situation as the model seems to be giving a very high range in prices depending on the data it was trained on, A price change of $69,044.61 could mean anything approx 20% change in the value of the clients house, which doesnt seems good.

The market is likely to have changed quite a bit since 1978, the current model doesnt account for inflation over time. Also it does not account for now and then good neighbourhoods and the rise of people income because of inflation.

The features present in the data is not sufficient to describe a home. I think factors like quality of apppliances in the home, square feet of the plot area, presence of pool or not etc should have a effect on value.

Data collected in an urban city like Boston are not applicable in a rural city because there are people with different characteristic and monthly incomes acrros rural city which cannot be same as urban cities.

Also it is not fair to judge the price of an individual home based on the characteristics of the entire neighborhood.

	RM	LSTAT	PTRATIO	MEDV
count	489.000000	489.000000	489.000000	4.890000e+02
mean	6.240288	12.939632	18.516564	4.543429e+05
std	0.643650	7.081990	2.111268	1.653403e+05
min	3.561000	1.980000	12.600000	1.050000e+05
25%	5.880000	7.370000	17.400000	3.507000e+05
50%	6.185000	11.690000	19.100000	4.389000e+05
75%	6.575000	17.120000	20.200000	5.187000e+05
max	8.398000	37.970000	22.000000	1.024800e+06