Welcome to the third project of the Machine Learning Engineer Nanodegree! In this notebook, some template code has already been provided for you, and it will be your job to implement the additional functionality necessary to successfully complete this project. 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 analyze a dataset containing data on various customers' annual spending amounts (reported in monetary units) of diverse product categories for internal structure. One goal of this project is to best describe the variation in the different types of customers that a wholesale distributor interacts with. Doing so would equip the distributor with insight into how to best structure their delivery service to meet the needs of each customer.
The dataset for this project can be found on the UCI Machine Learning Repository. For the purposes of this project, the features 'Channel' and 'Region' will be excluded in the analysis — with focus instead on the six product categories recorded for customers.
Run the code block below to load the wholesale customers 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 renders as rs
from IPython.display import display # Allows the use of display() for DataFrames
# Show matplotlib plots inline (nicely formatted in the notebook)
%matplotlib inline
# Load the wholesale customers dataset
try:
data = pd.read_csv("customers.csv")
data.drop(['Region', 'Channel'], axis = 1, inplace = True)
print "Wholesale customers dataset has {} samples with {} features each.".format(*data.shape)
except:
print "Dataset could not be loaded. Is the dataset missing?"
In this section, you will begin exploring the data through visualizations and code to understand how each feature is related to the others. You will observe a statistical description of the dataset, consider the relevance of each feature, and select a few sample data points from the dataset which you will track through the course of this project.
Run the code block below to observe a statistical description of the dataset. Note that the dataset is composed of six important product categories: 'Fresh', 'Milk', 'Grocery', 'Frozen', 'Detergents_Paper', and 'Delicatessen'. Consider what each category represents in terms of products you could purchase.
In [2]:
# Display a description of the dataset
display(data.describe())
To get a better understanding of the customers and how their data will transform through the analysis, it would be best to select a few sample data points and explore them in more detail. In the code block below, add three indices of your choice to the indices list which will represent the customers to track. It is suggested to try different sets of samples until you obtain customers that vary significantly from one another.
In [3]:
# TODO: Select three indices of your choice you wish to sample from the dataset
indices = [10, 47, 53]
# Create a DataFrame of the chosen samples
samples = pd.DataFrame(data.loc[indices], columns = data.keys())#.reset_index(drop = True)
print "Chosen samples of wholesale customers dataset:"
display(samples)
print "Deviation from the mean:"
display(samples - data.mean().round())
print "Deviation from the median:"
display(samples - data.median().round())
Consider the total purchase cost of each product category and the statistical description of the dataset above for your sample customers.
What kind of establishment (customer) could each of the three samples you've chosen represent?
Hint: Examples of establishments include places like markets, cafes, and retailers, among many others. Avoid using names for establishments, such as saying "McDonalds" when describing a sample customer as a restaurant.
Answer:
As suggested in a review, I make use of the deviations of these points in respect to the mean and median values esplicitly.
One interesting thought to consider is if one (or more) of the six product categories is actually relevant for understanding customer purchasing. That is to say, is it possible to determine whether customers purchasing some amount of one category of products will necessarily purchase some proportional amount of another category of products? We can make this determination quite easily by training a supervised regression learner on a subset of the data with one feature removed, and then score how well that model can predict the removed feature.
In the code block below, you will need to implement the following:
new_data a copy of the data by removing a feature of your choice using the DataFrame.drop function.sklearn.cross_validation.train_test_split to split the dataset into training and testing sets.test_size of 0.25 and set a random_state.random_state, and fit the learner to the training data.score function.
In [4]:
# TODO: Make a copy of the DataFrame, using the 'drop' function to drop the given feature
feature = 'Grocery'
new_data = data.drop([feature], axis=1)
# TODO: Split the data into training and testing sets using the given feature as the target
from sklearn.cross_validation import train_test_split
X_train, X_test, y_train, y_test = train_test_split(new_data, data[feature], test_size = 0.2, random_state=0)
# TODO: Create a decision tree regressor and fit it to the training set
from sklearn.tree import DecisionTreeRegressor
regressor = DecisionTreeRegressor()
regressor.fit(X_train, y_train)
# TODO: Report the score of the prediction using the testing set
score = regressor.score(X_test, y_test)
Which feature did you attempt to predict? What was the reported prediction score? Is this feature is necessary for identifying customers' spending habits?
Hint: The coefficient of determination, R^2, is scored between 0 and 1, with 1 being a perfect fit. A negative R^2 implies the model fails to fit the data.
Answer: I chose Grocery and, without tuning the model parameters, I get R^2 = 0.623, which shows that a certain degree of correlations with the other variables is present. This feature may not be therefore necessary for identifying the customer habits.
To get a better understanding of the dataset, we can construct a scatter matrix of each of the six product features present in the data. If you found that the feature you attempted to predict above is relevant for identifying a specific customer, then the scatter matrix below may not show any correlation between that feature and the others. Conversely, if you believe that feature is not relevant for identifying a specific customer, the scatter matrix might show a correlation between that feature and another feature in the data. Run the code block below to produce a scatter matrix.
In [5]:
# Produce a scatter matrix for each pair of features in the data
pd.scatter_matrix(data, alpha = 0.3, figsize = (14,8), diagonal = 'kde');
Are there any pairs of features which exhibit some degree of correlation? Does this confirm or deny your suspicions about the relevance of the feature you attempted to predict? How is the data for those features distributed?
Hint: Is the data normally distributed? Where do most of the data points lie?
Answer: The Grocery feature and the Detergent_Paper features are clearly linearly correlated. This explaines the R^2 value found above. It therefore confirms the suspiciuos about the relevance of the Grocery feature. One can see a certain degree of correlation between Grocery and Milk, and, Milk and Detergent as well. These three variables seem to be connected (in fact, see PCA first component below). The distributions are long tailed but not Gaussian, since they are highly positively skewed. Most data points lie close to the origin, but the maximum appears at a finite non zero value.
In this section, you will preprocess the data to create a better representation of customers by performing a scaling on the data and detecting (and optionally removing) outliers. Preprocessing data is often times a critical step in assuring that results you obtain from your analysis are significant and meaningful.
If data is not normally distributed, especially if the mean and median vary significantly (indicating a large skew), it is most often appropriate to apply a non-linear scaling — particularly for financial data. One way to achieve this scaling is by using a Box-Cox test, which calculates the best power transformation of the data that reduces skewness. A simpler approach which can work in most cases would be applying the natural logarithm.
In the code block below, you will need to implement the following:
log_data after applying a logarithm scaling. Use the np.log function for this.log_samples after applying a logrithm scaling. Again, use np.log.
In [6]:
# TODO: Scale the data using the natural logarithm
log_data = data.apply(np.log)
# TODO: Scale the sample data using the natural logarithm
log_samples = samples.apply(np.log)
# Produce a scatter matrix for each pair of newly-transformed features
pd.scatter_matrix(log_data, alpha = 0.3, figsize = (14,8), diagonal = 'kde');
After applying a natural logarithm scaling to the data, the distribution of each feature should appear much more normal. For any pairs of features you may have identified earlier as being correlated, observe here whether that correlation is still present (and whether it is now stronger or weaker than before).
Run the code below to see how the sample data has changed after having the natural logarithm applied to it.
In [7]:
# Display the log-transformed sample data
display(log_samples)
Detecting outliers in the data is extremely important in the data preprocessing step of any analysis. The presence of outliers can often skew results which take into consideration these data points. There are many "rules of thumb" for what constitutes an outlier in a dataset. Here, we will use Tukey's Method for identfying outliers: An outlier step is calculated as 1.5 times the interquartile range (IQR). A data point with a feature that is beyond an outlier step outside of the IQR for that feature is considered abnormal.
In the code block below, you will need to implement the following:
Q1. Use np.percentile for this.Q3. Again, use np.percentile.step.outliers list.NOTE: If you choose to remove any outliers, ensure that the sample data does not contain any of these points!
Once you have performed this implementation, the dataset will be stored in the variable good_data.
In [8]:
opf = {} #outliers per feature
# For each feature find the data points with extreme high or low values
for feature in log_data.keys():
# TODO: Calculate Q1 (25th percentile of the data) for the given feature
Q1 = np.percentile(log_data[feature], 25)
# TODO: Calculate Q3 (75th percentile of the data) for the given feature
Q3 = np.percentile(log_data[feature], 75)
# TODO: Use the interquartile range to calculate an outlier step (1.5 times the interquartile range)
step = (Q3-Q1)*1.5
# Display the outliers
#print "Data points considered outliers for the feature '{}':".format(feature)
#display(log_data[~((log_data[feature] >= Q1 - step) & (log_data[feature] <= Q3 + step))])
opf[feature] = list(log_data[~((log_data[feature] >= Q1 - step)
& (log_data[feature] <= Q3 + step))].index)
# OPTIONAL: Select the indices for data points you wish to remove
method = 'pair'
def outliers_remover(out_dic, method='no'):
outliers = []
if method == 'no':
print 'No outlier removed'
elif method == 'all':
for f in out_dic.keys():
outliers += opf[f]
print 'Removed all outliers according to the interquartile method range'
elif method == 'pair':
for i in range(len(out_dic.keys())):
for j in range(i+1, len(out_dic.keys())):
#print data.keys()[i] , data.keys()[j], np.intersect1d(opf[data.keys()[i]], opf[data.keys()[j]])
outliers += list(np.intersect1d(out_dic[out_dic.keys()[i]], out_dic[out_dic.keys()[j]]))
print 'Removed all outliers common to at least two features'
return outliers
outliers = sorted(list(set(outliers_remover(opf, method))))
print len(outliers)
# Remove the outliers, if any were specified
good_data = log_data.drop(log_data.index[outliers]).reset_index(drop = True)
Answer: Yes there are 5 points [65, 66, 75, 128, 154] considered outliers by more than one feature. The set of points considered outliers by at least one feature is much bigger (42 points). The removal or not of these points may affect the clustering methods below, especially the Gaussian Mixture Model. I performed some test and see that the results obtained by removing the 5 points or the whole 42 points are very similar, meaning that the important outliers are actually those that are considered as such by more than one feature. If these are not removed, the GMM clustering scores much worse. For these reasons I decided to limit the removal only to the points [65, 66, 75, 128, 154].
In this section you will use principal component analysis (PCA) to draw conclusions about the underlying structure of the wholesale customer data. Since using PCA on a dataset calculates the dimensions which best maximize variance, we will find which compound combinations of features best describe customers.
Now that the data has been scaled to a more normal distribution and has had any necessary outliers removed, we can now apply PCA to the good_data to discover which dimensions about the data best maximize the variance of features involved. In addition to finding these dimensions, PCA will also report the explained variance ratio of each dimension — how much variance within the data is explained by that dimension alone. Note that a component (dimension) from PCA can be considered a new "feature" of the space, however it is a composition of the original features present in the data.
In the code block below, you will need to implement the following:
sklearn.decomposition.PCA and assign the results of fitting PCA in six dimensions with good_data to pca.log_samples using pca.transform, and assign the results to pca_samples.
In [9]:
# TODO: Apply PCA by fitting the good data with the same number of dimensions as features
from sklearn.decomposition import PCA
pca = PCA(n_components=6)
pca.fit(good_data)
# TODO: Transform the sample log-data using the PCA fit above
pca_samples = pca.transform(log_samples)
# Generate PCA results plot
pca_results = rs.pca_results(good_data, pca)
How much variance in the data is explained in total by the first and second principal component? What about the first four principal components? Using the visualization provided above, discuss what the first four dimensions best represent in terms of customer spending.
Hint: A positive increase in a specific dimension corresponds with an increase of the positive-weighted features and a decrease of the negative-weighted features. The rate of increase or decrease is based on the indivdual feature weights.
Answer:
The first four components exprain around 93% (0.9311) of the variance
The first component represents a combination of Detergent_papers, Grocery and Milk in terms of spending
Here we can comment that these two components specialize on the different 6 products independently, somehow hinting that two variables, given by the combination of those, are dominating.
Run the code below to see how the log-transformed sample data has changed after having a PCA transformation applied to it in six dimensions. Observe the numerical value for the first four dimensions of the sample points. Consider if this is consistent with your initial interpretation of the sample points.
In [10]:
# Display sample log-data after having a PCA transformation applied
display(pd.DataFrame(np.round(pca_samples, 4), columns = pca_results.index.values))
When using principal component analysis, one of the main goals is to reduce the dimensionality of the data — in effect, reducing the complexity of the problem. Dimensionality reduction comes at a cost: Fewer dimensions used implies less of the total variance in the data is being explained. Because of this, the cumulative explained variance ratio is extremely important for knowing how many dimensions are necessary for the problem. Additionally, if a signifiant amount of variance is explained by only two or three dimensions, the reduced data can be visualized afterwards.
In the code block below, you will need to implement the following:
good_data to pca.good_data using pca.transform, and assign the reuslts to reduced_data.log_samples using pca.transform, and assign the results to pca_samples.
In [11]:
# TODO: Apply PCA by fitting the good data with only two dimensions
pca = PCA(n_components=2)
pca.fit(good_data)
# TODO: Transform the good data using the PCA fit above
reduced_data = pca.transform(good_data)
# TODO: Transform the sample log-data using the PCA fit above
pca_samples = pca.transform(log_samples)
# Create a DataFrame for the reduced data
reduced_data = pd.DataFrame(reduced_data, columns = ['Dimension 1', 'Dimension 2'])
In [12]:
# Display sample log-data after applying PCA transformation in two dimensions
display(pd.DataFrame(np.round(pca_samples, 4), columns = ['Dimension 1', 'Dimension 2']))
In this section, you will choose to use either a K-Means clustering algorithm or a Gaussian Mixture Model clustering algorithm to identify the various customer segments hidden in the data. You will then recover specific data points from the clusters to understand their significance by transforming them back into their original dimension and scale.
Answer:
Gaussian Mixture Model, instead, allows for cluster into clusters. It contains K-means as a non probabilistic limit case, however it is usually much slower in convergence. K-means may be used before in order to find suitable initialization of the GMM.
Given that PCA shows that 2 components are dominant and observing the distribution of points in this 2D plane, I see that the data are quite uniformly distributed and no clear structure is present. Morevove the outliers have been removed. For this reason I chose to perform the analysis with K-Means. A-posteriori, I find that the two techniques reach very similar conclusions.
Depending on the problem, the number of clusters that you expect to be in the data may already be known. When the number of clusters is not known a priori, there is no guarantee that a given number of clusters best segments the data, since it is unclear what structure exists in the data — if any. However, we can quantify the "goodness" of a clustering by calculating each data point's silhouette coefficient. The silhouette coefficient for a data point measures how similar it is to its assigned cluster from -1 (dissimilar) to 1 (similar). Calculating the mean silhouette coefficient provides for a simple scoring method of a given clustering.
In the code block below, you will need to implement the following:
reduced_data and assign it to clusterer.reduced_data using clusterer.predict and assign them to preds.centers.pca_samples and assign them sample_preds.reduced_data against preds.score and print the result.
In [13]:
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score
# TODO: Apply your clustering algorithm of choice to the reduced data
kmeans = KMeans(n_clusters=2, random_state=0)
clusterer = kmeans.fit(reduced_data)
# TODO: Predict the cluster for each data point
preds = kmeans.predict(reduced_data)
# TODO: Find the cluster centers
centers = kmeans.cluster_centers_
# TODO: Predict the cluster for each transformed sample data point
sample_preds = kmeans.predict(pca_samples)
# TODO: Calculate the mean silhouette coefficient for the number of clusters chosen
score = silhouette_score(reduced_data, preds)
Answer: For the K-Means Algorithm
Once you've chosen the optimal number of clusters for your clustering algorithm using the scoring metric above, you can now visualize the results by executing the code block below. Note that, for experimentation purposes, you are welcome to adjust the number of clusters for your clustering algorithm to see various visualizations. The final visualization provided should, however, correspond with the optimal number of clusters.
In [14]:
# Display the results of the clustering from implementation
rs.cluster_results(reduced_data, preds, centers, pca_samples)
Each cluster present in the visualization above has a central point. These centers (or means) are not specifically data points from the data, but rather the averages of all the data points predicted in the respective clusters. For the problem of creating customer segments, a cluster's center point corresponds to the average customer of that segment. Since the data is currently reduced in dimension and scaled by a logarithm, we can recover the representative customer spending from these data points by applying the inverse transformations.
In the code block below, you will need to implement the following:
centers using pca.inverse_transform and assign the new centers to log_centers.np.log to log_centers using np.exp and assign the true centers to true_centers.
In [15]:
# TODO: Inverse transform the centers
log_centers = pca.inverse_transform(centers)
# TODO: Exponentiate the centers
true_centers = np.exp(log_centers)
# Display the true centers
segments = ['Segment {}'.format(i) for i in range(0,len(centers))]
true_centers = pd.DataFrame(np.round(true_centers), columns = data.keys())
true_centers.index = segments
display(true_centers)
print "Deviation from the mean:"
display(true_centers - data.mean().round())
print "Deviation from the median:"
display(true_centers - data.median().round())
Consider the total purchase cost of each product category for the representative data points above, and reference the statistical description of the dataset at the beginning of this project. What set of establishments could each of the customer segments represent?
Hint: A customer who is assigned to 'Cluster X' should best identify with the establishments represented by the feature set of 'Segment X'.
Answer:
As suggested in a review we make use of the deviation of the representative centroids in respect to the data means and medians.
In [16]:
# Display the predictions
for i, pred in enumerate(sample_preds):
print "Sample point", i, "predicted to be in Cluster", pred
print "\nWe diplay again the centroids coordinates in the original feature space"
display(true_centers)
print "and the sample points in the original feature space"
display(samples)
Answer:
The following analysis refers to the two tables above:
For Sample '0' (index 10), the values of all features, but the Frozen one, are very close to the cluster 1 centroid. This point is therefore well represented by the centroid itself and therefore by the correspondent cluster.
For Sample '1' (index 47): this is a high-seller customer. It did not appear in any of the outlier categories calculated, and therefore we cannot consider it as an outlier. On the other hand looking at the relative values of all the features in respect to the ones pertaining the centroids, we can safely state that it belongs fo the cluster 1, since apart for the Frozen and Fresh features, the centroid of cluster 1 seem to be represent more accurately the high selling instances, in respect to the centroid of cluster 0. Sample 1 therefore can be considered to be an extreme (boundary) point of cluster 1.
For Sample '2' (index 53), The values of Milk, Grocery, Detergents and Frozen are close to the cluster 1 centroid. A relative small deviation exist for the Delicatessen feature and a strong one for the Fresh feature. Overall this sample is relatively well represented by the cluster 1 centroid and therefore we can safely assign it to cluster 1.
To conclude: The clustering algorithm assigned all the points to the Cluster 1, so to the Retailer cluster. In two cases I was correct in the preliminary analysis (Sample 0 and 1 to belong to a retailer class). In the third case I opted for a small activity, a bar, but the analysis shows actually that also this example should belong to the retailer class.
In this final section, you will investigate ways that you can make use of the clustered data. First, you will consider how the different groups of customers, the customer segments, may be affected differently by a specific delivery scheme. Next, you will consider how giving a label to each customer (which segment that customer belongs to) can provide for additional features about the customer data. Finally, you will compare the customer segments to a hidden variable present in the data, to see whether the clustering identified certain relationships.
Companies will often run A/B tests when making small changes to their products or services to determine whether making that change will affect its customers positively or negatively. The wholesale distributor is considering changing its delivery service from currently 5 days a week to 3 days a week. However, the distributor will only make this change in delivery service for customers that react positively. How can the wholesale distributor use the customer segments to determine which customers, if any, would reach positively to the change in delivery service?
Hint: Can we assume the change affects all customers equally? How can we determine which group of customers it affects the most?
Answer: Assuming the clustering is correct, the two types of customers may have different needs and therefore react differently to the change. In order to test whether the decision to change the service is correct, the company will have to separate the two classes, and run the A/B test in each of them separately. The results in each class in fact may be different and a combined result, therefore, misleading. Once the test are conducted in the A/B form, the company, may draw conclusion whether the change is useful for customers belonging to cluster 0 and, if it is useful for those belonging to cluster 1.
Additional structure is derived from originally unlabeled data when using clustering techniques. Since each customer has a customer segment it best identifies with (depending on the clustering algorithm applied), we can consider 'customer segment' as an engineered feature for the data. Assume the wholesale distributor recently acquired ten new customers and each provided estimates for anticipated annual spending of each product category. Knowing these estimates, the wholesale distributor wants to classify each new customer to a customer segment to determine the most appropriate delivery service.
How can the wholesale distributor label the new customers using only their estimated product spending and the customer segment data?
Hint: A supervised learner could be used to train on the original customers. What would be the target variable?
Answer:
We could run a supervised classification algorithm using the customer segment data as training set. Within the training phase we can make use of cross-validation and all other methods to obtain a suitable model. Then the prediction phase will take as inputs the estimated product spending, and will output the classification, i.e. to which segment the new customers are more likely to belong to.
At the beginning of this project, it was discussed that the 'Channel' and 'Region' features would be excluded from the dataset so that the customer product categories were emphasized in the analysis. By reintroducing the 'Channel' feature to the dataset, an interesting structure emerges when considering the same PCA dimensionality reduction applied earlier to the original dataset.
Run the code block below to see how each data point is labeled either 'HoReCa' (Hotel/Restaurant/Cafe) or 'Retail' the reduced space. In addition, you will find the sample points are circled in the plot, which will identify their labeling.
In [17]:
# Display the clustering results based on 'Channel' data
rs.channel_results(reduced_data, outliers, pca_samples)
How well does the clustering algorithm and number of clusters you've chosen compare to this underlying distribution of Hotel/Restaurant/Cafe customers to Retailer customers? Are there customer segments that would be classified as purely 'Retailers' or 'Hotels/Restaurants/Cafes' by this distribution? Would you consider these classifications as consistent with your previous definition of the customer segments?
Answer:
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.