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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Machine Learning Engineer Nanodegree
Unsupervised Learning
Project
Step1:  Data Exploration
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
Step2:  Implementation
Step3:  Question 1
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
Step4:  Question 2
Which feature did you attempt to predict? What was the reported prediction score? Is this feature is necessary for identifying customers' spending habits?
Hint
Step5:  Question 3
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
Step6:  Observation
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.
Step7:  Implementation
Step8:  Question 4
Are there any data points considered outliers for more than one feature based on the definition above? Should these data points be removed from the dataset? If any data points were added to the outliers list to be removed, explain why. 
Answer
Step9:  Question 5
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
Step10:  Implementation
Step11:  Observation
Run the code below to see how the log-transformed sample data has changed after having a PCA transformation applied to it using only two dimensions. Observe how the values for the first two dimensions remains unchanged when compared to a PCA transformation in six dimensions.
Step12:  Visualizing a Biplot
A biplot is a scatterplot where each data point is represented by its scores along the principal components. The axes are the principal components (in this case Dimension 1 and Dimension 2). In addition, the biplot shows the projection of the original features along the components. A biplot can help us interpret the reduced dimensions of the data, and discover relationships between the principal components and original features.
Run the code cell below to produce a biplot of the reduced-dimension data.
Step13:  Observation
Once we have the original feature projections (in red), it is easier to interpret the relative position of each data point in the scatterplot. For instance, a point the lower right corner of the figure will likely correspond to a customer that spends a lot on 'Milk', 'Grocery' and 'Detergents_Paper', but not so much on the other product categories. 
From the biplot, which of the original features are most strongly correlated with the first component? What about those that are associated with the second component? Do these observations agree with the pca_results plot you obtained earlier?
Clustering
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. 
Question 6
What are the advantages to using a K-Means clustering algorithm? What are the advantages to using a Gaussian Mixture Model clustering algorithm? Given your observations about the wholesale customer data so far, which of the two algorithms will you use and why?
Answer
Step14:  Question 7
Report the silhouette score for several cluster numbers you tried. Of these, which number of clusters has the best silhouette score? 
Sillhouette score for n=2
Step15:  Implementation
Step16:  Question 8
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
Step17:  Answer
<ASSISTANT_TASK:>
Python Code:
# Import libraries necessary for this project
import numpy as np
import pandas as pd
from IPython.display import display # Allows the use of display() for DataFrames
# Import supplementary visualizations code visuals.py
import visuals as vs
# Pretty display for notebooks
%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?"
Explanation: Machine Learning Engineer Nanodegree
Unsupervised Learning
Project: Creating Customer Segments
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.
Getting Started
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.
End of explanation
# Display a description of the dataset
display(data.describe())
Explanation: Data Exploration
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.
End of explanation
# TODO: Select three indices of your choice you wish to sample from the dataset
import random
random.seed(14)
indices = [random.randint(0, data.shape[0]) for x in range(3)]
sampleIndices = indices
print("Indices: {}".format(indices))
# 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)
Explanation: Implementation: Selecting Samples
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.
End of explanation
from sklearn.cross_validation import train_test_split
from sklearn.tree import DecisionTreeRegressor
def find_relevance(data, target_label):
    # TODO: Make a copy of the DataFrame, using the 'drop' function to drop the given feature
    new_data = data.drop([target_label], axis=1, inplace=False)
    target = data[target_label]
    # TODO: Split the data into training and testing sets using the given feature as the target
    X_train, X_test, y_train, y_test = train_test_split(new_data, target, test_size=0.25, random_state=14)
    # TODO: Create a decision tree regressor and fit it to the training set
    regressor = DecisionTreeRegressor(random_state=14)
    regressor.fit(X_train, y_train)
    # TODO: Report the score of the prediction using the testing set
    score = regressor.score(X_test, y_test)
    return score
for target_label in data.columns:
    score = find_relevance(data, target_label)
    print("{:>20s}: {:+0.3f}".format(target_label, score))
Explanation: Question 1
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:
|         |   Index   |   Establishment           |   Reasoning 
|:-------:|:---------:|:-------------------------:|:-------------------
|  0  |     47    | Large supermarket         | Sales for Fresh, Milk, Grocery, and Detergents_Paper are well over the 75% quartile.
|  1  |     309   | Hotel                     | There are proportionally high sales in Milk, Grocery, and Detergents_Paper, all greater than 75% of the population.
|  2  |     287   | Restaurant                | Fresh, Frozen, and Delicatessen sales are all greater than the median.
Implementation: Feature Relevance
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:
 - Assign new_data a copy of the data by removing a feature of your choice using the DataFrame.drop function.
 - Use sklearn.cross_validation.train_test_split to split the dataset into training and testing sets.
   - Use the removed feature as your target label. Set a test_size of 0.25 and set a random_state.
 - Import a decision tree regressor, set a random_state, and fit the learner to the training data.
 - Report the prediction score of the testing set using the regressor's score function.
End of explanation
# Produce a scatter matrix for each pair of features in the data
pd.scatter_matrix(data, alpha = 0.3, figsize = (14,8), diagonal = 'kde');
Explanation: Question 2
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 attempted to predict the Milk feature. The reported score was 0.397. While there is some correlation, it is not strong -- 39.7% of the variation is explained using the other features. This feature may be useful in identifying customers' spending habits, though clearly not as important as some of the other features (Fresh, Frozen, and Delicatessen).
Visualize Feature Distributions
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.
End of explanation
# TODO: Scale the data using the natural logarithm
log_data = np.log(data)
# TODO: Scale the sample data using the natural logarithm
log_samples = np.log(samples)
# Produce a scatter matrix for each pair of newly-transformed features
pd.scatter_matrix(log_data, alpha = 0.3, figsize = (14,8), diagonal = 'kde');
Explanation: Question 3
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 following pairs show some correlation:
Milk & Grocery<br>
Milk & Detergents_Paper<br>
Detergents_Paper & Grocery
Milk appears to be correlated with both Grocery and Detergents_Paper, which agrees with the suspicion that Milk is not completely necessary. 
None of the data appears to be normally distributed, they are all skewed right. They are all centered around values < 10,000 with the exception of Fresh which is centered around 12,000. I see no distinction in terms of distibution between features that appear to be correlated with other features and those that don't.
Data Preprocessing
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.
Implementation: Feature Scaling
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:
 - Assign a copy of the data to log_data after applying logarithmic scaling. Use the np.log function for this.
 - Assign a copy of the sample data to log_samples after applying logarithmic scaling. Again, use np.log.
End of explanation
# Display the log-transformed sample data
display(log_samples)
Explanation: Observation
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.
End of explanation
features = log_data.columns
outlierLimitDict = {}
outlierDict = {}
# For each feature find the data points with extreme high or low values
for feature in features:
    # 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)
    iqr = Q3 - Q1
    step = 1.5 * iqr
    outlierLimitDict[feature] = (Q1 - step, Q3 + step)
    
    # Display the outliers
    outliers = log_data[~((log_data[feature] >= Q1 - step) & (log_data[feature] <= Q3 + step))]
    for index in outliers.index:
        originalCount = outlierDict.get(index, 0)
        outlierDict[index] = originalCount + 1
    print "Data points considered outliers for the feature '{}':".format(feature)
    display(outliers)
    
# Print indices of rows that are outliers for multiple features    
for index in sorted(outlierDict.keys()):
    if outlierDict[index] > 1:
        print("{:3}: {}".format(index, outlierDict[index]))
        
# OPTIONAL: Select the indices for data points you wish to remove
outliers  = []
# Remove the outliers, if any were specified
good_data = log_data.drop(log_data.index[outliers]).reset_index(drop = True)
# Make sure samples don't contain these indices
for index in sampleIndices:
    if index in outliers:
        raise Exception("The samples contain an outlier (index {})".format(index))
     
def color_point(row):
    if row.name in outliers:
        return "red"
    if row.name in outlierDict.keys():
        return "green"
    return "black"
    
pd.scatter_matrix(log_data, figsize = (14,8), diagonal = 'kde', alpha=1, lw=0, c=log_data.apply(color_point, axis=1));
Explanation: Implementation: Outlier Detection
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:
 - Assign the value of the 25th percentile for the given feature to Q1. Use np.percentile for this.
 - Assign the value of the 75th percentile for the given feature to Q3. Again, use np.percentile.
 - Assign the calculation of an outlier step for the given feature to step.
 - Optionally remove data points from the dataset by adding indices to the 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.
End of explanation
from sklearn.decomposition import PCA
# TODO: Apply PCA by fitting the good data with the same number of dimensions as features
n = min(good_data.shape)
pca = PCA(n_components=n)
pca.fit(good_data)
# TODO: Transform log_samples using the PCA fit above
pca_samples = pca.transform(log_samples)
# Generate PCA results plot
pca_results = vs.pca_results(good_data, pca)
for i in range(1,n+1):
    print("The total variance explained by the first {} principle component{} is {}.".format(
            i,
            " " if i == 1 else "s",
            sum(pca.explained_variance_ratio_[0:i])
    ))
Explanation: Question 4
Are there any data points considered outliers for more than one feature based on the definition above? Should these data points be removed from the dataset? If any data points were added to the outliers list to be removed, explain why. 
Answer:
There were several rows that were considered outliers for more than one feature
Row  | Outlier in N features
:---:|:----:
65   |  2
66   |  2
75   |  2
128  |  2
154  |  3
I've chosen to keep all outliers rather than discard them. All the data points are feasible and do not appear do be erronous inputs. The presence of the outliers affects both the results and assumptions about the data, so it is not legitimate to drop them, as it would be hiding a trend that may exist.
Feature Transformation
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.
Implementation: PCA
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:
 - Import sklearn.decomposition.PCA and assign the results of fitting PCA in six dimensions with good_data to pca.
 - Apply a PCA transformation of log_samples using pca.transform, and assign the results to pca_samples.
End of explanation
# Display sample log-data after having a PCA transformation applied
display(pd.DataFrame(np.round(pca_samples, 4), columns = pca_results.index.values))
Explanation: Question 5
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 total variance explained by the first 2 principle components is 0.718945231737.
The total variance explained by the first 4 principle components is 0.931295845055.
In each of the dimensions in the visualizations, the largest bar represents the most important or most meaningful feature. There can be multiple meaningful features in a single dimension. If I was to label the first four newly generated dimensions, they would be as follows:
Dimension |       Label      | Explanation
:---------:|:----------------:|:---------------------
     1     | Consumer retail spending  | There's a strong positive weight on Detergents_Paper and fairly strong positive weights on Milk and Grocery, which is in line with the type of spending that occurs at retail stores.
     2     | Commercial food service | There are similarly strong positive weights on Fresh, Frozen, and Delicatessen. This indicates that the dimension is driven by spending on food-related products. Since the weight is solely on food-related products, it indicates that the spending is done by customers in the food industry, rather than general consumers.
     3     | Health-conscious spending | There is a strong positive weight on Fresh and a strong negative weight on "Delicatessen". This suggests that the more fresh goods (healthy) that are purchased, the less dessert goods (unhealthy) are purchased.
     4     | "On-the-Go"-style food | A large positive weight on Frozen, a mild positive weight on Detergents_Paper, and mild negative weight on Fresh indicate spending on goods that are used by households, but don't have a dedicated cook. The large negative weight on Delicatessen bolsters that assumption in that consumers who rely on pre-made meals don't necessarily eat junk food.
Observation
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.
End of explanation
# TODO: Apply PCA by fitting the good data with only two dimensions
pca = PCA(n_components=2)
# TODO: Transform the good data using the PCA fit above
reduced_data = pca.fit_transform(good_data)
# TODO: Transform log_samples 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'])
Explanation: Implementation: Dimensionality Reduction
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:
 - Assign the results of fitting PCA in two dimensions with good_data to pca.
 - Apply a PCA transformation of good_data using pca.transform, and assign the results to reduced_data.
 - Apply a PCA transformation of log_samples using pca.transform, and assign the results to pca_samples.
End of explanation
# Display sample log-data after applying PCA transformation in two dimensions
display(pd.DataFrame(np.round(pca_samples, 4), columns = ['Dimension 1', 'Dimension 2']))
Explanation: Observation
Run the code below to see how the log-transformed sample data has changed after having a PCA transformation applied to it using only two dimensions. Observe how the values for the first two dimensions remains unchanged when compared to a PCA transformation in six dimensions.
End of explanation
# Create a biplot
vs.biplot(good_data, reduced_data, pca)
Explanation: Visualizing a Biplot
A biplot is a scatterplot where each data point is represented by its scores along the principal components. The axes are the principal components (in this case Dimension 1 and Dimension 2). In addition, the biplot shows the projection of the original features along the components. A biplot can help us interpret the reduced dimensions of the data, and discover relationships between the principal components and original features.
Run the code cell below to produce a biplot of the reduced-dimension data.
End of explanation
from sklearn.mixture import GMM
from sklearn.metrics import silhouette_score
# TODO: Apply your clustering algorithm of choice to the reduced data 
def getGmmSilhouetteScore(n, data, samples):
    clusterer = GMM(n_components=n, random_state=14)
    clusterer.fit(data)
    # TODO: Predict the cluster for each data point
    preds = clusterer.predict(data)
    # TODO: Find the cluster centers
    centers = clusterer.means_
    # TODO: Predict the cluster for each transformed sample data point
    sample_preds = clusterer.predict(samples)
    # TODO: Calculate the mean silhouette coefficient for the number of clusters chosen
    score = silhouette_score(data, preds)
    return score, centers, preds, sample_preds
bestSilhouetteScoreN = 0
bestSilhouetteScore = -1
maxN = 10
for n in range(2, maxN):
    score, _, _, _ = getGmmSilhouetteScore(n, reduced_data, pca_samples)
    if score > bestSilhouetteScore:
        bestSilhouetteScore = score
        bestSilhouetteScoreN = n
    print("Sillhouette score for n={}: {}".format(n, score))
score, centers, preds, sample_preds = getGmmSilhouetteScore(bestSilhouetteScoreN, reduced_data, pca_samples)
print("")
print("Best n is {} with a silhouette score of {}.".format(bestSilhouetteScoreN, score))
    
Explanation: Observation
Once we have the original feature projections (in red), it is easier to interpret the relative position of each data point in the scatterplot. For instance, a point the lower right corner of the figure will likely correspond to a customer that spends a lot on 'Milk', 'Grocery' and 'Detergents_Paper', but not so much on the other product categories. 
From the biplot, which of the original features are most strongly correlated with the first component? What about those that are associated with the second component? Do these observations agree with the pca_results plot you obtained earlier?
Clustering
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. 
Question 6
What are the advantages to using a K-Means clustering algorithm? What are the advantages to using a Gaussian Mixture Model clustering algorithm? Given your observations about the wholesale customer data so far, which of the two algorithms will you use and why?
Answer:
K-Means clustering requires less computational power than the Gaussian Mixture Model (GMM). K-Means also does not make assumptions about the distribution of the data. GMM is more robust in that it allows soft clustering, where one point may belong to multiple klusters to varying degrees. This helps find hidden relationships in data. It is also less prone to falling into local minima, which is a problem for K-Means.
The data that we have observed up until this point suggests that data points may belong to multiple clusters. For example, if we look at the 2<sup>nd</sup> and 3<sup>rd</sup> principle components, we see that Delicatessen is a strong indicator in both. If we use K-Means, we would force data points into one or the other, which will could negatively impact the analyze and assign steps of the next iteration and change the shape of the cluster. On the other hand, if we use GMM, we give the option of the data points assigning to either cluster and don't allow it to affect the shape of the clusters. Because the data is overlapping like this, I'm opting to use GMM.
Implementation: Creating Clusters
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:
 - Fit a clustering algorithm to the reduced_data and assign it to clusterer.
 - Predict the cluster for each data point in reduced_data using clusterer.predict and assign them to preds.
 - Find the cluster centers using the algorithm's respective attribute and assign them to centers.
 - Predict the cluster for each sample data point in pca_samples and assign them sample_preds.
 - Import sklearn.metrics.silhouette_score and calculate the silhouette score of reduced_data against preds.
   - Assign the silhouette score to score and print the result.
End of explanation
# Display the results of the clustering from implementation
vs.cluster_results(reduced_data, preds, centers, pca_samples)
Explanation: Question 7
Report the silhouette score for several cluster numbers you tried. Of these, which number of clusters has the best silhouette score? 
Sillhouette score for n=2: 0.316017379116
Sillhouette score for n=3: 0.375222595239
Sillhouette score for n=4: 0.333662047955
Sillhouette score for n=5: 0.257867358339
Sillhouette score for n=6: 0.262324563865
Sillhouette score for n=7: 0.313909530829
Sillhouette score for n=8: 0.295714674659
Sillhouette score for n=9: 0.32036621781
Best n is 3 with a silhouette score of 0.375222595239.
Cluster Visualization
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.
End of explanation
# 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)
Explanation: Implementation: Data Recovery
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:
 - Apply the inverse transform to centers using pca.inverse_transform and assign the new centers to log_centers.
 - Apply the inverse function of np.log to log_centers using np.exp and assign the true centers to true_centers.
End of explanation
# Display the predictions
for i, pred in enumerate(sample_preds):
    print "Sample point", i, "predicted to be in Cluster", pred
Explanation: Question 8
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:
Considering the median and quartiles for the statistical description (since the mean is convoluted due to one or more outliers), my observations are as follows:
In Segment 0, Milk, Grocery, Detergents_Paper all lie very close to the 75% quartile, indicating that customers segments in this cluster tend toward every-day-use products - something you would find in retail stores and department stores.
In Segment 1, Milk, Grocery, and Detergents_Paper lie close to the 25% quartile, and Frozen lies closer to the 75% quartile. The rest fall close to the median. Based on this, the customer segment appears to belong to establishments with emphasis on food and desserts, such as restaurants and cafes.
In Segement 2, Fresh, Frozen, and Delicatessen lie at or below the 25% quartile, and the rest follow below the median. This segment is characterized by low overall spending in all 6 categories. Milk and Grocery are relatively high, indicating that this may be a store which sells small amounts of everyday groceries, like small markets, or pharmacies.
|            |  Fresh         | Milk             | Grocery         |  Frozen         | Detergents_Paper   | Delicatessen
| ----------:|:--------------:|:----------------:|:---------------:|:---------------:|:------------------:|:-------------:
| count  | 440.000000     |  440.000000      | 440.000000      |  440.000000     | 440.000000         | 440.000000
| mean   | 12000.297727   |  5796.265909     | 7951.277273     |  3071.931818    | 2881.493182        | 1524.870455
| std    |  12647.328865  |  7380.377175     | 9503.162829     |  4854.673333    | 4767.854448        | 2820.105937
| min    |  3.000000      |  55.000000       | 3.000000        |  25.000000      | 3.000000           | 3.000000
| 25%    |  3127.750000   |  1533.000000     | 2153.000000     |  742.250000     | 256.750000         | 408.250000
| 50%    |  8504.000000   |  3627.000000     | 4755.500000     |  1526.000000    | 816.500000         | 965.500000
| 75%    |  16933.750000  |  7190.250000     | 10655.750000    |  3554.250000    | 3922.000000        | 1820.250000
| max    |  112151.000000 |  73498.000000    | 92780.000000    |  60869.000000   | 40827.000000       | 47943.000000
Question 9
For each sample point, which customer segment from Question 8 best represents it? Are the predictions for each sample point consistent with this?
Run the code block below to find which cluster each sample point is predicted to be.
End of explanation
# Display the clustering results based on 'Channel' data
vs.channel_results(reduced_data, outliers, pca_samples)
Explanation: Answer:
Yes, sample points 0 and 2 agree with my earlier predictions. Samples 0 and 1 have higher than median sales in Milk, Grocery, and Detergents_Paper which put them close to Cluster 0's centroid. I didn't mention hotel as an establishment for Cluster 0, but according do my description, it fits. Similarly, Sample 2's sales in Frozen are above the median, while Milk, Grocery, and Detergents_Paper are below the median, placing it closer to Cluster 1's median.
|         | Index | Establishment   | Fresh | Milk  | Grocery   |  Frozen  |  Detergents_Paper |  Delicatessen
|:-------:|:---------:|:-------------------:|:-----:|:-----:|:---------:|:--------:|:-----------------:|:------------:
|  0  |     47    | Large supermarket   | 44466 | 54259 |  55571    |   7782   |    24171          |   6465
|  1  |     309   | Hotel               | 918   | 20655 |  13567    |   1465   |    6846           |   806
|  2  |     287   | Restaurant          | 15354 | 2102  |  2828     |   8366   |    386            |   1027
Conclusion
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.
Question 10
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 react 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:
A change of delivery service from 5 days a week to 3 days a week would negatively affect customers who go through the products they buy quickly, such as fresh food which can't be stored in bulk for long periods of time. This might negatively impact restaurants, cafes, and fresh food markets. On the other hand, it would not affect large supermarkets, retail stores, and other similar customers who buy in bulk because the products are not time-sensative. I don't think this segment would react positively, but more likely would be neutral to the change.
This information can be used to randomly sample from each of the two segments to receive useful feedback as to whether or not each segment would react positively to the change.
Question 11
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 train a supervised learner using the 6 spending categories as the features and the customer segment as the label. Then we could input the new customers and have the learner classify them, assigning each a label.
Visualizing Underlying Distributions
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.
End of explanation
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