Profiling of credit card clients (abbreviated version).

Ricardo Ibarra-Gil

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This is my first project using unsupervised machine learning techniques in Python.

The objective is to analyze credit card usage data of a bank's customers to identify their different profiles.

Tools used include Principal Components Analysis, KMeans Clustering and Hierarchical Clustering.

Results of clients' profiles are intended to help the bank to better target its clients with tailored offers and products.

Content

1. About the data
2. Data cleaning
3. PCA reduction
4. Model. Hierarchical clustering, using 4 components
5. Conclusions and recommendations
6. Closing comments
7. References

1. About the data

• Data set from Kaggle (https://www.kaggle.com/arjunbhasin2013/ccdata)
• It contains information about usage of credit cards from approx. 9,000 clients of a bank, and 18 attributes, plus a unique customer identifier

Variables:

• CUSTID: Unique identifier
• BALANCE: Monthly average balance in credit card (CC) statement
• BALANCEFREQUENCY: Percentage of last 12 months with balance in CC statement
• PURCHASES: Total purchase amount in last 12 months
• ONEOFF_PURCHASES: Total amount of one-off purchases
• INSTALLMENTS_PURCHASES: Total amount of installment purchases
• CASH_ADVANCE: Total cash-advance amount
• PURCHASES_FREQUENCY: Frequency of purchases (% of months w/purchases ≥1)
• ONEOFF_PURCHASES_FREQUENCY: Frequency of one-off-purchases
• PURCHASES_INSTALLMENTS_FREQUENCY: Frequency of installment purchases
• CASH_ADVANCE_FREQUENCY: Cash-Advance frequency
• AVERAGE_PURCHASE_TRX: Average amount per purchase transaction
• CASH_ADVANCE_TRX: Average amount per cash-advance transaction
• PURCHASES_TRX: Average amount per purchase transaction
• CREDIT_LIMIT: Credit limit
• PAYMENTS: Total payments in the period
• MINIMUM_PAYMENTS: Total minimum payments due in the period
• PRC_FULL_PAYMENT: % of months with full payment of the due statement balance
• TENURE: Number of months as a customer

2. Data Cleaning

CC_DF = pd.read_csv('CC GENERAL.csv', sep = ',')
CC_DF.isnull().sum()

CUST_ID                               0
BALANCE                               0
BALANCE_FREQUENCY                     0
PURCHASES                             0
ONEOFF_PURCHASES                      0
INSTALLMENTS_PURCHASES                0
CASH_ADVANCE                          0
PURCHASES_FREQUENCY                   0
ONEOFF_PURCHASES_FREQUENCY            0
PURCHASES_INSTALLMENTS_FREQUENCY      0
CASH_ADVANCE_FREQUENCY                0
CASH_ADVANCE_TRX                      0
PURCHASES_TRX                         0
CREDIT_LIMIT                          1
PAYMENTS                              0
MINIMUM_PAYMENTS                    313
PRC_FULL_PAYMENT                      0
TENURE                                0
dtype: int64

Some initial considerations about the data:

• There are 313 missing values in the data for MINIMUM_PAYMENTS, and 1 missing value for CREDIT_LIMIT
• I set the MINIMUM_PAYMENTS missing values to the median of the variable and drop the missing value of CREDIT_LIMit
• I also drop unique identifier

CC_DF.describe()

	BALANCE	BALANCE_FREQUENCY	PURCHASES	ONEOFF_PURCHASES	INSTALLMENTS_PURCHASES	CASH_ADVANCE	PURCHASES_FREQUENCY	ONEOFF_PURCHASES_FREQUENCY	PURCHASES_INSTALLMENTS_FREQUENCY	CASH_ADVANCE_FREQUENCY	CASH_ADVANCE_TRX	PURCHASES_TRX	CREDIT_LIMIT	PAYMENTS	MINIMUM_PAYMENTS	PRC_FULL_PAYMENT	TENURE
count	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000	8950.000000
mean	1564.474828	0.877271	1003.204834	592.437371	411.067645	978.871112	0.490351	0.202458	0.364437	0.135144	3.248827	14.709832	4494.282473	1733.143852	844.906767	0.153715	11.517318
std	2081.531879	0.236904	2136.634782	1659.887917	904.338115	2097.163877	0.401371	0.298336	0.397448	0.200121	6.824647	24.857649	3638.646702	2895.063757	2332.792322	0.292499	1.338331
min	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	50.000000	0.000000	0.019163	0.000000	6.000000
25%	128.281915	0.888889	39.635000	0.000000	0.000000	0.000000	0.083333	0.000000	0.000000	0.000000	0.000000	1.000000	1600.000000	383.276166	170.857654	0.000000	12.000000
50%	873.385231	1.000000	361.280000	38.000000	89.000000	0.000000	0.500000	0.083333	0.166667	0.000000	0.000000	7.000000	3000.000000	856.901546	312.343947	0.000000	12.000000
75%	2054.140036	1.000000	1110.130000	577.405000	468.637500	1113.821139	0.916667	0.300000	0.750000	0.222222	4.000000	17.000000	6500.000000	1901.134317	788.713501	0.142857	12.000000
max	19043.138560	1.000000	49039.570000	40761.250000	22500.000000	47137.211760	1.000000	1.000000	1.000000	1.500000	123.000000	358.000000	30000.000000	50721.483360	76406.207520	1.000000	12.000000

Next, I should do something about the outliers

Standarization

In the next table, it can be seen that the maximum value for some of the variables are far from the mean. For instance, the max value for MINIMUM_PAYMENTS is located 32.4 standard deviations away from the mean. This clearly points to the presence of outliers in the data.

CC_DF_Z = []
for i in CC_DF.columns:
    CC_DF_Z.append(((CC_DF.describe()[i][7]) - (CC_DF.describe()[i][1])) / CC_DF.describe()[i][2])
CC_DF_Z = pd.DataFrame(CC_DF_Z, columns = ['Standard deviations from the mean of MAX'])
CC_DF_Z.insert(0,'Variable',CC_DF.columns)
CC_DF_Z.sort_values(by = ['Standard deviations from the mean of MAX'], ascending = False)

	Variable	Standard deviations from the mean of MAX
14	MINIMUM_PAYMENTS	32.390925
4	INSTALLMENTS_PURCHASES	24.425524
3	ONEOFF_PURCHASES	24.199714
2	PURCHASES	22.482254
5	CASH_ADVANCE	22.009887
10	CASH_ADVANCE_TRX	17.546868
13	PAYMENTS	16.921334
11	PURCHASES_TRX	13.810243
0	BALANCE	8.397019
12	CREDIT_LIMIT	7.009671
9	CASH_ADVANCE_FREQUENCY	6.820140
15	PRC_FULL_PAYMENT	2.893291
7	ONEOFF_PURCHASES_FREQUENCY	2.673302
8	PURCHASES_INSTALLMENTS_FREQUENCY	1.599110
6	PURCHASES_FREQUENCY	1.269772
1	BALANCE_FREQUENCY	0.518055
16	TENURE	0.360659

I considered two methods for standarization of the data and dealing with outliers:

• Robust Scaler
• Power Transformation

To have an idea of the results, let's take the variable MINIMUM_PAYMNETS as example:

table_1_CC_DF = [CC_DF_std, CC_DF_z]
table_1_CC_RS = [CC_RS_std, CC_RS_z]
table_1_CC_T = [CC_T_std, CC_T_z]
table_1 = pd.DataFrame([table_1_CC_DF, table_1_CC_RS, table_1_CC_T], columns = ['Standard deviation', 'Distance of Max to mean (stdv)']) 
table_1.insert(0,'Method', ['Original Data', 'Robust Scaler', 'Power Transformation'])
table_1

	Method	Standard deviation	Distance of Max to mean (stdv)
0	Original Data	2332.792322	32.390925
1	Robust Scaler	3.775626	32.390925
2	Power Transformation	1.000056	4.036337

The table above shows that, for MINIMUM_PAYMENTS variable:

• The original data has a substantially large standard deviation, and the maximum value is 32.4 standard deviarions from the mean.
• Robust Scaler decreases standard deviation (observations are closer to each oher), but the outliers are still far away.
• Power transformation compacts data and brings outliers closer: the maximum value is now 4 standar deviations from mean.

The following graph shows the effect of the power transofrmation on the distribution of the variable MINIMUM_PAYMENTS:

# Graph to compare original data woth transformed: one hist and one boxplot for each case
f, axs = plt.subplots(2,2, figsize = (10,6), sharex = False, gridspec_kw = dict(height_ratios = [1,3]))
sns.histplot(data = CC_DF, x = "MINIMUM_PAYMENTS", ax = axs[0,0], legend = False)
sns.boxplot(data = CC_DF, x = "MINIMUM_PAYMENTS", ax = axs[1,0])
sns.histplot(data = CC_T, x ="MINIMUM_PAYMENTS", ax = axs[0,1], legend = False)
sns.boxplot(data = CC_T, x = "MINIMUM_PAYMENTS", ax = axs[1,1])
f.tight_layout()

For the graph above, the left hand side represents the variable's original distribution, which ranges from 0 to 8,950, with a median of 312.3. The right hand side graph shows the distribution after the power transformation. Now, the variable ranges from -5.8 to 8.9, and its maximum value is 4.0 standard deviations away from its mean. We now have data more concentrated around its central values.

The following graph is the equivalent of the graph above, but for another variable: BALANCE. Similar conclusions can be drawn from it.

f, axs = plt.subplots(2,2, figsize = (10,6), sharex = False, gridspec_kw = dict(height_ratios = [1,3]))
# makde density plot along x-axis without legend
sns.histplot(data = CC_DF, x = "BALANCE", ax = axs[0,0],legend = False)
sns.boxplot(data = CC_DF, x = "BALANCE", ax = axs[1,0])
sns.histplot(data = CC_T, x ="BALANCE", ax = axs[0,1], legend = False)
sns.boxplot(data = CC_T, x = "BALANCE", ax = axs[1,1])
f.tight_layout()

As a final comment on the descriptive statitics of the data, following correlation heatmap shows some expected correlations, such as the ones between the variables levels, their frequencies (_FREQUENCY) and their average amount per transaction (TRX).
Interestingly enough, PRC_FULL_PAYMENT (how frequently the client pays its amount due totally) has negative correlations with some other variables such as:

• BALANCE (average monthly balance in the client's card)
• CASH_ADVANCE and CHAS_ADVANCE_TRX (how much the client asks for cash advance form its credit card)
• MINIMUM_PAYMENTS (total minimum payment due)

sns.heatmap(CC_T.corr(), cmap= 'YlGnBu')

<AxesSubplot:>

4. Principal Components Analysis (PCA)

Before PCA reduction, a Kmeans analysis on the transformed data was performed, but for the purposes of brevity, it is ommited in this presentation.
An alternative to improve the results of the analysis is to apply dimension reduction with Principal Component Analysis (PCA).
I run PCA setting the number of components to 8.

# Define optimal number of components
optimal_PC = 4
f, axs = plt.subplots(1,2, figsize = (8,4))
sns.lineplot(x = np.arange(1,9), y = PCA_1.explained_variance_ratio_, ax = axs[0])
sns.lineplot(x = np.arange(1,9), y = var_explained, ax = axs[1])
axs[0].set_title("Screetest")
axs[1].set_title("Accumulated variance explained")
axs[0].set_xlabel('Principal Components')
axs[1].set_xlabel('Principal Components')
axs[0].axvline(x = optimal_PC, color='r', linestyle=':')
axs[1].axhline(y = var_explained[(optimal_PC-1)], color='r', linestyle=':')
f.tight_layout()

For the analysis, I take only the first 4 components. Since:

• Variance of components explains 76.0% of variance of explained variables (right hand side graph)
• 5th component marginal contribution is only 6.7%, so we have a decreased marginal contribution to the explanation of data variance (left hand side graph)

The resulting loadings represent the relation between variables and components, and are useful to give meaning to our 4 components.

• I set a threshold, and display only loadings greater than |0.28|:

PCA_lv_1.iloc[:, : optimal_PC][(PCA_lv_1 > 0.28) | (PCA_lv_1 < -0.28)].fillna('-')

	V1	V2	V3	V4
BALANCE	-	0.433761	-	-
BALANCE_FREQUENCY	-	0.30332	-	-0.356114
PURCHASES	0.353849	-	-	-
ONEOFF_PURCHASES	-	-	0.50492	-
INSTALLMENTS_PURCHASES	0.317172	-	-0.389956	-
CASH_ADVANCE	-0.299421	-	-	0.292591
PURCHASES_FREQUENCY	0.355422	-	-	-
ONEOFF_PURCHASES_FREQUENCY	-	-	0.470402	-
PURCHASES_INSTALLMENTS_FREQUENCY	0.309445	-	-0.447063	-
CASH_ADVANCE_FREQUENCY	-0.291577	-	-	0.298498
CASH_ADVANCE_TRX	-0.294386	-	-	0.292329
PURCHASES_TRX	0.364443	-	-	-
CREDIT_LIMIT	-	-	-	-
PAYMENTS	-	0.321255	-	0.31056
MINIMUM_PAYMENTS	-	0.37198	-	-0.290518
PRC_FULL_PAYMENT	-	-	-	0.512515
TENURE	-	-	-	-0.296175

Definition of components (interpretation) from the information of our loadings

1. PC1: Intensiveness of credit card use, especially for big amounts in installments, and not for cash advance.
   Total purchase amount in last year is high. The total amount spent in the last year is high, and the amount per transaction is also high. Purchases amount in installments during last year is also high Frequency of purchases overall and in installments is high, meaning most months has at least one purchase

2. PC2: Indebtedness. The purchases are not high, but the balance is.
   High monthly average balance, they have accumulated debt. The number of months with balance is high, frequently having balance, so is having debt all moths High amount due and paid by the customer to reduce their balance total minimum payments due are also high, so payments are being made, but still has balance

3. PC3: Buying in one-off, no installments.
   One off purchases amount are high, and installments purchases amount are low Not using card for installments, only for one off purchases Exact same relation but with the frequency

4. PC4: Pay card in full, and use it especially for cash in advance: being a relatively new client.
   Paying card in total every month is a high feature in component Tenure is low, relatively new clients Also using credit card for asking cash advance. And not using card every month, negatively related to frequency

Having the four components interpreted, in the following graph we can see that maybe some concentrations of data (shadows) might become clusters in our clustering analysis

sns.pairplot(df_pca[['PC1', 'PC2', 'PC3', 'PC4']], kind = "scatter", plot_kws = dict(alpha = 0.03), height = 1.4)

<seaborn.axisgrid.PairGrid at 0x1679b2d60>

5. Model. Hierarchical clustering, using 4 components, ward linkage

After analyzing different specifications for the model, the conclusion was to use a ward linkage KMeans classification model. Complete, single and average were tried: average and single were suffering serious problems, and complete and ward led similar results.

DENDROGRAM: Selected three cluster, to cut the dendrogram around 160.

CLUSTER_X = df_pca.iloc[:,:4]
C_model_5 = AgglomerativeClustering(distance_threshold = 0, n_clusters = None, linkage = 'ward')
C_model_5.fit_predict(CLUSTER_X)
plt.title("Hierarchical Clustering Dendrogram: Ward linkage")
plot_dendrogram(C_model_5, truncate_mode="level", p = 12)
plt.axhline(y = 160, color='r', linestyle=':')
plt.xlabel("Number of points in node (or index of point if no parenthesis).")
plt.show()

I will use three clusters with ward linkage hierarchical clustering

classes = ['Cluster 0', 'Cluster 1', 'Cluster 2']
fig, ax = plt.subplots(1, 3, figsize=(14, 6))
scatter_0 = ax[0].scatter(CLUSTER_X['PC1'], CLUSTER_X['PC2'], c = C_model_10.labels_, alpha = .8, cmap = 'RdYlGn')
ax[0].set_xlabel("Intensiveness, use installments", fontsize = 16)
ax[0].set_ylabel("Indebtedness", fontsize = 16)
ax[0].legend(handles = scatter_0.legend_elements()[0], labels = classes)
scatter_1 = ax[1].scatter(CLUSTER_X['PC1'], CLUSTER_X['PC3'], c = C_model_10.labels_, alpha = 0.8, cmap = 'RdYlGn')
ax[1].set_xlabel("Intensiveness, use installments", fontsize = 16)
ax[1].set_ylabel("Purchases one-off", fontsize = 16)
ax[1].legend(handles = scatter_1.legend_elements()[0], labels = classes)
scatter_2 = ax[2].scatter(CLUSTER_X['PC1'], CLUSTER_X['PC4'], c = C_model_10.labels_, alpha = 0.8, cmap = 'RdYlGn')
ax[2].set_xlabel("Intensiveness, use installments", fontsize = 16)
ax[2].set_ylabel("Pay in full, use cash in advance & new clients", fontsize = 16)
ax[2].legend(handles = scatter_2.legend_elements()[0], labels = classes)
plt.tight_layout()
plt.show()

• Cluster 0 (red) is mostly in the mid-lower end of indebtedness, with few observations in the upper part. It also comprises observations in the medium to high range of intensiveness of use, especially in installments, all the range of purchases in one - off, and the middle and lower range of pay in full.
• Cluster 1 (yellow) is well defined as those who have lower intensiveness of use, especially in purchasing in installments. It is also distributed all along indebtedness and pay in full (especially using the card for cash in advance, and new clients).
• Cluster 2 (green) has most of its data points in the higher end of indebtedness and in the middle part of intensiveness of use.

fig, ax = plt.subplots(1, 2, figsize=(10, 6))
scatter_0 = ax[0].scatter(CLUSTER_X['PC2'], CLUSTER_X['PC3'], c = C_model_10.labels_, alpha = 0.8, cmap = 'RdYlGn')
ax[0].set_xlabel("Indebtedness", fontsize = 16)
ax[0].set_ylabel("Purchases one-off", fontsize = 16)
ax[0].legend(handles = scatter_0.legend_elements()[0], labels = classes)
scatter_1 = ax[1].scatter(CLUSTER_X['PC2'], CLUSTER_X['PC4'], c = C_model_10.labels_, alpha = 0.8, cmap = 'RdYlGn')
ax[1].set_xlabel("Indebtedness", fontsize = 16)
ax[1].set_ylabel("Pay in full, use cash in advance & new clients", fontsize = 16)
ax[1].legend(handles = scatter_1.legend_elements()[0], labels = classes)
plt.tight_layout()
plt.show()

From this second set of graphs that shows components 2, 3 and 4, conclusions are a bit more fuzzy or unclear, because clusters 0 and 1 (blue and green) show overlapping. But we can still see:

• Cluster 0 (red) is somewhat more concentrated towards the low part of pay in full compared esepcially to cluster 2 (green)
• Cluster 0 again appears covering all the range of purchases one-off.
• Cluster 1 (yellow) is concentrated in the middle part of purchases one-off
• Cluster 2 (green) is concentrated in the middle part of indebtedness

5. Conclusions and recommendations

Interpretation of clusters

From the information of the graphs, and remembering our definition of the Components, we can try to define our clients profile for very cluster:

Cluster 0

• People who use their credit card somewhat intensively: their frequency of purchases is high.
• They tend to use their credit cards for large amounts, and in installments.
• They are located in the middle part of indebtedness. They pay their balances but not completely.
• These clients have a middle tendency to pay their credit cards in full and do not use much their cards for cash in advance.

Cluster 1

• Clients who use their cards less frequently.
• Their level of indebtedness is average. Again, it could be due to their lower intesiveness of credit card use.
• They do not tend to buy in installments, and they are located right in the middle range of buying one-off purchases.
• Their data points are located all along the use of cash in advance and payment their card in full.

Cluster 2

• Clients in the middle to lower spectrum of intensiveness of use: these clients have an average frequency of credit card use.
• In the middle to high end of indebtedness: they have high balance in their cards.
• The data points of this cluster are located in the middle to high end of tendency to pay in full, which could be counterintuitive to the fact that they are also middle-high in indebtedness. So, the fact that cluster 2 data points are in the middle upper part of component 4 could be explained rather by other factors included in that component, such as: use of credit card for cash in advance, not using their card very frequently, and being relatively newer clients.

Having described the clusters to the extent that the results of the model allowed from the graphs, I go back to the original data set to try to corroborate or rectify insights about the clients of each cluster.

CC_DF['CLUSTER'] = C_model_10.labels_
# selected_vars = ['BALANCE', 'ONEOFF_PURCHASES', 'INSTALLMENTS_PURCHASES', 'CASH_ADVANCE', 'CREDIT_LIMIT', 'TENURE', 'CLUSTER']
# mean_complete = CC_DF[selected_vars].mean()
mean_complete = CC_DF.mean()
mean_complete = pd.DataFrame(mean_complete)
mean_complete.drop('CLUSTER', axis = 0, inplace = True)
# mean_grouped = CC_DF[selected_vars].groupby('CLUSTER').mean()
mean_grouped = CC_DF.groupby('CLUSTER').mean()
final_table = pd.concat([mean_complete.T, mean_grouped], ignore_index = True, axis = 0)
final_table.rename(index = {0 : 'Complete data', 1 : 'Cluster 0', 2 : 'Cluster 1', 3 : 'Cluster 2'}, inplace = True)
final_table 

	BALANCE	BALANCE_FREQUENCY	PURCHASES	ONEOFF_PURCHASES	INSTALLMENTS_PURCHASES	CASH_ADVANCE	PURCHASES_FREQUENCY	ONEOFF_PURCHASES_FREQUENCY	PURCHASES_INSTALLMENTS_FREQUENCY	CASH_ADVANCE_FREQUENCY	CASH_ADVANCE_TRX	PURCHASES_TRX	CREDIT_LIMIT	PAYMENTS	MINIMUM_PAYMENTS	PRC_FULL_PAYMENT	TENURE
Complete data	1564.474828	0.877271	1003.204834	592.437371	411.067645	978.871112	0.490351	0.202458	0.364437	0.135144	3.248827	14.709832	4494.282473	1733.143852	844.906767	0.153715	11.517318
Cluster 0	933.706163	0.849932	1249.809328	695.620828	554.629137	184.732268	0.666572	0.234626	0.517996	0.025218	0.588490	18.815924	4304.695666	1388.925592	671.797045	0.228381	11.605834
Cluster 1	2131.115470	0.882550	0.883766	0.753361	0.172329	1974.015044	0.001715	0.001300	0.000496	0.273109	6.288332	0.020251	3982.240437	1634.708858	992.030606	0.045101	11.305689
Cluster 2	2688.396865	0.948175	1462.440511	982.892297	479.750150	2069.621381	0.556545	0.343402	0.350929	0.285881	7.241398	20.054939	5617.444870	2815.673059	1163.011032	0.068479	11.511654

Comparing the conclusions drawn frome the graph with the numbers of the table, it can be stated that most of them hold.

There are, however, a couple of observations:

Cluster 1. Most data points are concentrated around the middle value of one-off purchases (component 3): this would lead to the conclusion that those clients buy in one off purchases, but this is not corroborated in the table. Nevertheless, for other variables that are also related to component 3, the values of the table are more in line to what we would have expected: clients in this cluster do not use their credit cards for purchases in installments, and have also a low frequency of purchases, which is a variable whose loading is negative (-.2) with respect to component 3.

About cluster 2. After reviewing the table we can confirm that those clients fall within the middle-upper part of component 4 (Pay in full, use cash in advance & new clients) because they use their credit cards intensively for cash in advance, and they have a relatively low frequency of use.

Examples of tailored strategies
Some examples of the kind of targeted promotions could be:

For clients that belong to cluster 0: frequently offer them the option to pay they purchases in interest-free installments if the amount of the purchase exceedes certain threshold.

For clients in cluster 1: motivate them to use their credit card more for purchases by offering them lower rate or lower fees in their cash in advance operations once they reach a certain amount of monthly purchases.

For clients in cluster 2: the bank could incentivice this customers to use their cards more intensively, either by offering them lower fees in their cash in aadvance operations, or by increasing their credit limit.

6. Closing comments

• After trying different model specifications, the best results were attained using hierarchical clustering with 3 clusters after dimensionality reduction using Principal Component Analysis and keeping the main 4 components.
• A satisfactory (by no means perfect) clustering of clients was achieved in this exercise, and come interesting recommendations for the bank can be drawn from the analysis.
• There is possible future work that could be made to strengtehn and extend the analysis. For instance, with information about the kind of commerces where clients mostly use their cards we could use hat as input and provide additional insights to further tailor promotions and offers for clients.
• An initial exploration of non Euclidian distances was made, but results did not improve from euclidian distance. Additional analysis with variations in the distance methods could be made.
• Sparse PCA is a optino that I would like to explore for this data set because some variables are concentrated around zero.

7. References

The original data set can be downloaded from: https://www.kaggle.com/arjunbhasin2013/ccdata

The following material online was consulted:

About Standarization with outliers and power transformation

• https://machinelearningmastery.com/robust-scaler-transforms-for-machine-learning/
• https://scikit-learn.org/stable/auto_examples/preprocessing/plot_all_scaling.html
• https://scikit-learn.org/stable/modules/preprocessing.html
• https://machinelearningmastery.com/power-transforms-with-scikit-learn/

Dendrograms:

• https://joernhees.de/blog/2015/08/26/scipy-hierarchical-clustering-and-dendrogram-tutorial/
• https://scikit-learn.org/stable/auto_examples/cluster/plot_agglomerative_dendrogram.html

About PCA

• https://www.datasklr.com/principal-component-analysis-and-factor-analysis/principal-component-analysis
• https://medium.com/more-python-less-problems/principal-component-analysis-and-k-means-clustering-to-visualize-a-high-dimensional-dataset-577b2a7a5fe2
• https://365datascience.com/tutorials/python-tutorials/pca-k-means/
• https://medium.com/more-python-less-problems/principal-component-analysis-and-k-means-clustering-to-visualize-a-high-dimensional-dataset-577b2a7a5fe2
• https://stats.stackexchange.com/questions/157621/how-would-pca-help-with-a-k-means-clustering-analysis

Some materials from labs and homeworks from class 95791 Data Mining (Heinz College, Carnegie Mellon University), taught by Dr. Gabriela Gongora Swartzman were also used.