How can the telecommunications customer dataset’s fundamental structures and trends be identified using dimension reduction techniques like t-SNE and PCA? Particularly, what role do the identified clusters or condensed feature spaces play in helping to comprehend consumer preferences, behaviors, and possible churn-causing factors?
Turgud Valiyev
10/01/2023
ABSTRACT
This study explores the complex topic of customer churn in telecommunications, which is a major worry in a market that is very competitive. The study includes a thorough examination of the dataset and provides trends, preferences, and behavior patterns of the target audience. The study develops relevant research questions with a strategic focus on overcoming the difficulties caused by client attrition. The dataset is carefully cleaned and standardized throughout the data preparation stage to provide an excellent foundation for subsequent studies. Statistical techniques are utilized to identify trends, associations, and critical metrics impacting turnover. The dataset’s interpretability is improved by the use of t-distributed Stochastic Neighbor Embedding (t-SNE) and Principal Component Analysis (PCA) for dimensional reduction. A comprehensive summary of the research is provided, connecting the dots between the analytical results and the ramifications for the telecom sector. Apart from these dimension reduction analysis, the kmeans clustering analysis is conducted as a further analysis. The principal component analysis (PCA) provides a detailed knowledge of the factors that influence the variation in the data by highlighting important components. By revealing hidden structures and relationships, the t-SNE analysis offers a visual picture.
The outcomes highlight how crucial churn data analytics are to telecom firms’ ability to make strategic decisions. When creating focused treatments, the patterns and predictors of customer attrition that have been found to be significant provide useful direction. Actionable knowledge from the study helps businesses improve client retention tactics and deal with churn head-on. As a consequence, this study provides a comprehensive understanding of telecom customer attrition while highlighting the role that sophisticated analytics play in guiding industry participants toward long-term success. The study journey’s major phases are summarized in the abstract, which also emphasizes the importance of the carried out studies in navigating the intricate web of consumer dynamics in the telecom industry.
INFORMATION ABOUT THE DATA
Iranian Churn Dataset. This dataset is randomly collected from an Iranian telecom company’s database over a period of 12 months. If we consider the Characteristics of the dataser we may observe that it is Multivariate. All of the attributes except for attribute churn is the aggregated data of the first 9 months. The churn labels are the state of the customers at the end of 12 months. The three months is the designated planning gap. Displays the structure of the dataset to understand variable types.3150 customers were selected randomly from an Iranian mobile operator call-center database A total of 3150 rows of data, each representing a customer, bear information for 13 columns. The attributes that are in this dataset are call failures, frequency of SMS, number of complaints, number of distinct calls, subscription length, age group, the charge amount, type of service, seconds of use, status, frequency of use, and Customer Value. DOI for the dataset is 10.24432/C5JW3Z.
This dataset is licensed under a Creative Commons Attribution 4.0 International (CC BY 4.0) license. This allows for the sharing and adaptation of the datasets for any purpose, provided that the appropriate credit is given.
The link of the dataset https://archive.ics.uci.edu/dataset/563/iranian+churn+dataset
TABLE OF CONTENT
- ABSTRACT
- INFORMATION ABOUT THE DATA
- TABLE OF CONTENT
- INTRODUCTION
- RESEARCH QUESTION
- DATA PREPARATION
- STATISTICAL ANALYSIS
- DIMENSION REDUCTION
- FURTHER ANALYSIS
- CONCLUSION
- REFERENCES
INTRODUCTION
The telecoms sector is changing quickly, and it is now critical for businesses to understand and reduce customer attrition in order to expand sustainablity. The phenomena of customers stopping their services, or “churn,” is a complex problem that has a big influence on income streams and market competitiveness. A critical component of strategic decision-making is now the study of churn data, which provides information on customer behavior, preferences, and the variables affecting their decisions to stay or leave.
The environment in which telecommunications businesses operate is marked by intense rivalry, quickly developing technologies, and a constantly growing range of services. Consequently, keeping current clients has become just as crucial as finding new ones. Actionable intelligence may be found in plenty by utilizing churn data, which includes a multitude of information on customer interactions, service usage habits, and demographic statistics.1
It is impossible to exaggerate the significance of churn statistics in the telecom industry. Businesses can use proactive intervention tactics by identifying early warning signals of impending churn by analyzing this data. These tactics could be tailored advertising campaigns, focused promotions, or improved services with the goal of increasing client happiness and loyalty.
A data-driven strategy is required due to the evolving of the telecommunications business, which is characterized by the launch of new features and changing client expectations. Utilizing advanced analytical methods on churn data helps businesses make intelligent choices and quickly adjust to changing market circumstances.2
RESEARCH QUESTION
How can the telecommunications customer dataset’s fundamental structures and trends be identified using dimension reduction techniques like t-SNE and PCA? Particularly, what role do the identified clusters or condensed feature spaces play in helping to comprehend consumer preferences, behaviors, and possible churn-causing factors?
In order to learn more about consumer habits and patterns, this research question intends to investigate the use of dimension reduction approaches. It suggests looking at the ways in which t-SNE and PCA can help to uncover significant clusters or reduced feature spaces that improve our comprehension of consumer behaviors, preferences, and possible churn indicators. The query prepares you for a thorough examination of your dataset through sophisticated analytically methods.
DATA PREPARATION
Changing the language to English
Sys.setlocale("LC_ALL","English")## Warning in Sys.setlocale("LC_ALL", "English"): using locale code page other
## than 65001 ("UTF-8") may cause problems## [1] "LC_COLLATE=English_United States.1252;LC_CTYPE=English_United States.1252;LC_MONETARY=English_United States.1252;LC_NUMERIC=C;LC_TIME=English_United States.1252"Sys.setenv(LANGUAGE='en')Importing the Dataset
churn_data <- read.csv("C:/Users/User/Desktop/UL Research 2 - Dimension Reduction/Customer Churn.csv")Installing the Packages
# Set the CRAN mirror
options(repos = c(CRAN = "https://cloud.r-project.org"))
install.packages("readr")
install.packages("stats")
install.packages("factoextra")
install.packages("flexclust")
install.packages("fpc")
install.packages("clustertend")
install.packages("cluster")
install.packages("ClusterR")
install.packages("dplyr")
install.packages("ggplot2")
install.packages("hopkins")
install.packages("NbClust")
install.packages("tidyverse")
install.packages("dendextend")
install.packages("Rtsne")
install.packages("gridExtra")
install.packages("caret")
install.packages("pheatmap")
install.packages("FactoMineR")Activating the packages with Library function
library(readr)
library(stats)
library(factoextra)## Loading required package: ggplot2## Welcome! Want to learn more? See two factoextra-related books at https://goo.gl/ve3WBalibrary(flexclust)## Loading required package: grid## Loading required package: lattice## Loading required package: modeltools## Loading required package: stats4library(grid)
library(lattice)
library(modeltools)
library(stats4)
library(hopkins)
library(fpc)
library(clustertend)## Package `clustertend` is deprecated. Use package `hopkins` instead.##
## Attaching package: 'clustertend'## The following object is masked from 'package:hopkins':
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library(ClusterR)
library(dplyr)##
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library(lubridate)##
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library(FactoMineR)The brief overview of the dataset
head(churn_data)## Call..Failure Complains Subscription..Length Charge..Amount Seconds.of.Use
## 1 8 0 38 0 4370
## 2 0 0 39 0 318
## 3 10 0 37 0 2453
## 4 10 0 38 0 4198
## 5 3 0 38 0 2393
## 6 11 0 38 1 3775
## Frequency.of.use Frequency.of.SMS Distinct.Called.Numbers Age.Group
## 1 71 5 17 3
## 2 5 7 4 2
## 3 60 359 24 3
## 4 66 1 35 1
## 5 58 2 33 1
## 6 82 32 28 3
## Tariff.Plan Status Age Customer.Value Churn
## 1 1 1 30 197.640 0
## 2 1 2 25 46.035 0
## 3 1 1 30 1536.520 0
## 4 1 1 15 240.020 0
## 5 1 1 15 145.805 0
## 6 1 1 30 282.280 0View(churn_data)The structure of the dataset
str(churn_data)## 'data.frame': 3150 obs. of 14 variables:
## $ Call..Failure : int 8 0 10 10 3 11 4 13 7 7 ...
## $ Complains : int 0 0 0 0 0 0 0 0 0 0 ...
## $ Subscription..Length : int 38 39 37 38 38 38 38 37 38 38 ...
## $ Charge..Amount : int 0 0 0 0 0 1 0 2 0 1 ...
## $ Seconds.of.Use : int 4370 318 2453 4198 2393 3775 2360 9115 13773 4515 ...
## $ Frequency.of.use : int 71 5 60 66 58 82 39 121 169 83 ...
## $ Frequency.of.SMS : int 5 7 359 1 2 32 285 144 0 2 ...
## $ Distinct.Called.Numbers: int 17 4 24 35 33 28 18 43 44 25 ...
## $ Age.Group : int 3 2 3 1 1 3 3 3 3 3 ...
## $ Tariff.Plan : int 1 1 1 1 1 1 1 1 1 1 ...
## $ Status : int 1 2 1 1 1 1 1 1 1 1 ...
## $ Age : int 30 25 30 15 15 30 30 30 30 30 ...
## $ Customer.Value : num 198 46 1537 240 146 ...
## $ Churn : int 0 0 0 0 0 0 0 0 0 0 ...# The number of rows
nrow(churn_data)## [1] 3150# The number of columns
ncol(churn_data)## [1] 14Instances Features 3150 14
Variable Information
Anonymous Customer ID Call Failures: number of call failures Complains: binary (0: No complaint, 1: complaint) Subscription Length: total months of subscription Charge Amount: Ordinal attribute (0: lowest amount, 9: highest amount) Seconds of Use: total seconds of calls Frequency of use: total number of calls Frequency of SMS: total number of text messages Distinct Called Numbers: total number of distinct phone calls Age Group: ordinal attribute (1: younger age, 5: older age) Tariff Plan: binary (1: Pay as you go, 2: contractual) Status: binary (1: active, 2: non-active) Churn: binary (1: churn, 0: non-churn) - Class label Customer Value: The calculated value of customer
“Call Failure” indicates the number of call failures associated with each customer. Call failures typically indicate instances where a customer’s attempted phone call did not successfully connect or experienced issues during the call, such as dropped calls, poor call quality, or other connectivity issues. This is Numerical variable representing the count of call failures for each customer. Higher values may indicate a higher frequency of issues or challenges with phone calls.Understanding the distribution and patterns of call failures can provide insights into the telecommunications service quality for different customers. Customers with a higher count of call failures may experience dissatisfaction with the service, and addressing the issues related to call failures could be a priority for improving customer satisfaction and retention.
“Complains” is a binary attribute that indicates whether a customer has made a complaint or not.No complaint(0), Complaint made(1). Each row in the dataset likely corresponds to a customer, and the value of “Complains” for that customer indicates whether they have registered a complaint (1) or not (0). This variable serves as a categorical indicator of customer feedback or dissatisfaction, allowing you to identify customers who have expressed concerns or issues with the service. Analyzing the distribution of complaints and exploring relationships with other variables can provide valuable insights into customer satisfaction and help identify areas for improvement in products or services.
“Subscription Length” in the dataset elucidates the total number of months of subscription for each customer. It represents the duration of a customer’s subscription in terms of months.Each value indicates the total length of time a customer has been subscribed to the service. Analyzing the distribution of subscription lengths can provide insights into customer retention and the overall stability of subscriptions. Longer subscription lengths may indicate customer loyalty and satisfaction with the service, while shorter lengths could suggest churn or less stable customer relationships.
“Charge Amount” is categorized into different levels ranging from 0 (lowest amount) to 9 (highest amount). Each level displays a predefined range or tier of subscription charges, allowing us to understand the relative pricing tiers for different customers.
“Seconds of Use” shows the total number of seconds that customers have used in making phone calls. Represents the cumulative duration, in seconds, of all phone calls made by a customer. Indicates the total usage of phone call services by each customer. This variable gives insights into the extent to which customers are utilizing the phone call services provided by the telecommunications service. Higher values suggest more extensive usage of the service, while lower values may indicate less frequent or shorter phone call duration.
“Frequency of Use” in the dataset represents the total number of calls made by each customer. Represents the total count of phone calls made by a customer. Indicates how frequently a customer engages in making phone calls. This variable provides insights into the calling behavior of customers. Higher values suggest that a customer makes more frequent phone calls, while lower values indicate less frequent usage of phone call services. Analyzing the distribution of the frequency of use can help understand the calling patterns of customers and may be useful in identifying segments of users with different usage behaviors.
“Frequency of SMS” in the dataset is total number of text messages sent by each customer.Represents the total count of text messages sent by a customer. Indicates how frequently a customer engages in sending text messages. This variable provides insights into the texting behavior of customers. Higher values suggest that a customer sends more frequent text messages, while lower values indicate less frequent usage of text messaging services. Analyzing the distribution of the frequency of SMS can help understand the texting patterns of customers and may be useful in identifying segments of users with different texting behaviors.
“Distinct Called Numbers” displays the total number of distinct phone numbers that a customer has called. Represents the count of unique or distinct phone numbers dialed by a customer. Indicates how many different phone numbers a customer has interacted with through phone calls. This variable provides insights into the diversity of a customer’s communication network. Higher values suggest that a customer has interacted with a larger number of unique phone numbers, while lower values indicate a more limited set of contacts. Analyzing the distribution of distinct called numbers can help understand the social connectivity or communication patterns of customers. It may be relevant for identifying customers who engage with a wide range of contacts or those who have more focused interactions.
“Age Group” is an ordinal attribute that categorizes customers into different age groups.Represents the categorization of customers into different age groups. Each category likely has a numerical label or code indicating a specific age range. Each customer in the dataset is likely assigned to one of these age groups based on certain criteria or demographic information. Analyzing the distribution of customers across different age groups can provide insights into the age demographics of your customer base.
“Tariff Plan” shows the type of tariff plan or pricing model associated with each customer.Indicates the type of pricing plan that a customer is subscribed to. Pay as you go (1) , Contractual (2). Each customer in the dataset is likely assigned to one of these tariff plans, and the binary nature of the variable suggests a simple categorization between pay-as-you-go plans and contractual plans. Analyzing the distribution of customers across different tariff plans can provide insights into the preferences or choices customers make regarding the payment structure of their telecommunications service.
“Status” in the dataset represents the status of each customer and is a binary attribute.Indicates the status of a customer’s account or subscription. Active (1), Non-active (2). Each customer in the dataset is likely assigned one of these status values, determining whether the customer’s account is currently active or not. Analyzing the distribution of customers across different status categories can provide insights into customer retention and churn patterns. Active status indicates customers who are currently using the service, while non-active status may indicate customers who have discontinued or churned.
“Churn” variable is a binary attribute that typically indicates whether a customer has churned or not. Each customer in the dataset is likely assigned one of these churn values, helping to identify customers who have discontinued their subscription or stopped using the service.Analyzing the distribution of churn status can provide insights into customer retention and help identify factors associated with churn. Understanding the characteristics of customers who churn can be crucial for developing strategies to retain customers and improve overall service satisfaction.
“Customer Value” elucidates a calculated value associated with each customer. Represents a calculated value that reflects the overall value of each customer to the business. The calculation method for customer value may involve various factors, such as usage patterns, subscription length, or other relevant metrics. Customer Value” variable is likely a metric used to quantify the importance or contribution of each customer to the business. Higher values may indicate customers with higher potential revenue, longer-term engagement, or other desirable characteristics. Analyzing the distribution of customer values can provide insights into the relative importance of different customers to the business. Understanding customer value is crucial for making strategic decisions related to customer retention, marketing, and service optimization.
STATISTICAL ANALYSIS
- To begin with exploring the overall statistical result of the data
summary(churn_data)## Call..Failure Complains Subscription..Length Charge..Amount
## Min. : 0.000 Min. :0.00000 Min. : 3.00 Min. : 0.0000
## 1st Qu.: 1.000 1st Qu.:0.00000 1st Qu.:30.00 1st Qu.: 0.0000
## Median : 6.000 Median :0.00000 Median :35.00 Median : 0.0000
## Mean : 7.628 Mean :0.07651 Mean :32.54 Mean : 0.9429
## 3rd Qu.:12.000 3rd Qu.:0.00000 3rd Qu.:38.00 3rd Qu.: 1.0000
## Max. :36.000 Max. :1.00000 Max. :47.00 Max. :10.0000
## Seconds.of.Use Frequency.of.use Frequency.of.SMS Distinct.Called.Numbers
## Min. : 0 Min. : 0.00 Min. : 0.00 Min. : 0.00
## 1st Qu.: 1391 1st Qu.: 27.00 1st Qu.: 6.00 1st Qu.:10.00
## Median : 2990 Median : 54.00 Median : 21.00 Median :21.00
## Mean : 4472 Mean : 69.46 Mean : 73.17 Mean :23.51
## 3rd Qu.: 6478 3rd Qu.: 95.00 3rd Qu.: 87.00 3rd Qu.:34.00
## Max. :17090 Max. :255.00 Max. :522.00 Max. :97.00
## Age.Group Tariff.Plan Status Age Customer.Value
## Min. :1.000 Min. :1.000 Min. :1.000 Min. :15 Min. : 0.0
## 1st Qu.:2.000 1st Qu.:1.000 1st Qu.:1.000 1st Qu.:25 1st Qu.: 113.8
## Median :3.000 Median :1.000 Median :1.000 Median :30 Median : 228.5
## Mean :2.826 Mean :1.078 Mean :1.248 Mean :31 Mean : 471.0
## 3rd Qu.:3.000 3rd Qu.:1.000 3rd Qu.:1.000 3rd Qu.:30 3rd Qu.: 788.4
## Max. :5.000 Max. :2.000 Max. :2.000 Max. :55 Max. :2165.3
## Churn
## Min. :0.0000
## 1st Qu.:0.0000
## Median :0.0000
## Mean :0.1571
## 3rd Qu.:0.0000
## Max. :1.0000Some notes from result:
The subscription length is minimum 3 months and maximum 47 months. Costumer stay loyal no more than 4 years.
The average age of the customers is 31.Mostly, moderate age people are consuming the services. The status of the most consumers are currently active. The maximum frequency of use is 255 which demonstrates that the customers made maximum 255 calls. If we consider the results of complains, we may observe that the average complaint rate is 0.07651. It is so close to 0 and means that the most of the customers did not do complaints.
- Checking the NA values in the dataset. And illustration the number of missing values.
sum(is.na(churn_data))## [1] 0colSums(is.na(churn_data))## Call..Failure Complains Subscription..Length
## 0 0 0
## Charge..Amount Seconds.of.Use Frequency.of.use
## 0 0 0
## Frequency.of.SMS Distinct.Called.Numbers Age.Group
## 0 0 0
## Tariff.Plan Status Age
## 0 0 0
## Customer.Value Churn
## 0 0We may observe that there are no missing values on the dataset. The Data Cleaning and Handling with missing data steps is not needed for this dataset.If we would need to implement, we could delete rows which conclude na values with na.omit() function or we might substitute na values with mean/median.
- Remove Duplicates
churn_data_unique <- churn_data[!duplicated(churn_data), ]- Distribution Plot for Charge..Amount
print(ggplot(churn_data, aes(x = churn_data$Charge..Amount)) +
geom_bar(color = "black", fill = "darkgreen", alpha = 0.7) +
labs(title = "Distribution of Charge Amount"))## Warning: Use of `churn_data$Charge..Amount` is discouraged.
## i Use `Charge..Amount` instead.
The distribution of charge amounts for consumers in the churn data set
is displayed on the histogram graph. The tall bar for that range amply
illustrates how most consumers have relatively low charge amounts,
between 0 and 2.5. The graph’s first bar, which predominates, shows that
most consumers fall into the category of smaller charge amounts. The
bars are gradually shorter as the charge amounts rise from 2.5 to 5 and
beyond, indicating that fewer clients have larger charges. This shows
that less and fewer clients in the data set have those levels of charges
as charge amounts increase. The distribution is heavily biased to the
right, as evidenced by the dramatic drop-off in bar heights, where the
majority of customers are concentrated at lower charge levels. According
to the pattern, clients in this specific churn data set are less likely
to experience large charges. The distribution offers insightful
information about normal consumer purchasing patterns and the potential
effects of charge amounts on client retention and churn.
- Distribution Plot for Age Group
print(ggplot(churn_data, aes(x = churn_data$Age.Group)) +
geom_bar(color = "black", fill = "darkblue", alpha = 0.7) +
labs(title = "Distribution of Age Group"))## Warning: Use of `churn_data$Age.Group` is discouraged.
## i Use `Age.Group` instead.
The data spread across five distinct age groups is displayed in the bar graph, with the x-axis labeled from 1 to 5. The frequency count, which ranges from 0 to more than 1000, is measured on the y-axis. The height of a bar indicates the proportion of data points that belong to a certain age group, which is represented by that bar. The largest group in the dataset is the second age group, which is represented by the number 2 on the x-axis. It has the tallest bar, exceeding 1000 counts. With a bar height of about 200 counts, the first age group is the second most common. With less than 500 counts, the third age group is the least represented and has the smallest bar. For the fourth and fifth age categories, the bars progressively get shorter from left to right, indicating that there are less data points as age rises. The dataset’s composition differs greatly according on age, as seen by the unequal distribution of bar heights, with one group emerging as the dominant category. The distribution of age across the data is clearly displayed by this representation.
- Distribution Plot for Frequency of Use
print(ggplot(churn_data, aes(x = Frequency.of.use)) +
geom_histogram(color = "black", alpha = 0.7, bins = 30) +
labs(title = "Distribution of Frequency of Use"))
The distribution of data pertaining to the frequency of use of
particular values or events is visualized by the histogram. The range of
frequency of usage values, split into bins, is shown on the x-axis. The
number of occurrences inside each bin is displayed on the y-axis. The
tallest bar represents the most frequently occurring range of numbers,
while the bars visually represent the counts. The data suggests that the
most common usage is concentrated around a frequency of use of less than
100. A general unimodal distribution centered on the peak value is
formed as the histogram then drops on both sides of this peak, with
fewer tall bars towards the lower and higher frequency ranges. The
pattern of usage frequencies within the dataset is effectively
illustrated by this graph.
- Correlation analysis
# Select the relevant variables from the data frame
selected_vars <- churn_data[c("Frequency.of.use", "Distinct.Called.Numbers", "Charge..Amount", "Customer.Value", "Status", "Call..Failure", "Churn")]
# Calculate the correlation matrix
correlation_matrix <- cor(selected_vars)
# Use corrplot to visualize the correlation matrix
corrplot::corrplot(correlation_matrix, type = "upper", method = "number", tl.cex = 0.7)
The correlation matrix heatmap displayed in the graph illustrates the connections between seven variables taken from a dataset related to telecommunications. Frequency of usage, Distinct Called Numbers, Charge Amount, Customer Value, Status, Call Failure, and Churn are among the variables that are provided. The correlation coefficient, which ranges from -1 to 1, is displayed in each cell of the matrix between the row and column variables. Strong negative correlations are displayed in dark blue, and strong positive correlations are displayed in dark red. Among the noteworthy connections is the strong positive correlation of 0.74 between Distinct_Called_Numbers and Frequency_of_use, which suggests that users of the service more regularly call a wider range of numbers. The association between Charge_Amount and Distinct_Called_Numbers is likewise moderately positive, at 0.42. Furthermore, the correlation coefficient of 0.50 between Call_Failure and Status indicates a possible association between these variables. Using this correlation matrix visualization, we may find relationships in the telecom data that need to be looked into more in order to have a deeper understanding of customer usage and churn trends.
- Box Plot by Customer Value
ggplot(churn_data, aes(x = 1, y = churn_data$Customer.Value)) +
geom_boxplot(fill = "red", color = "black") +
labs(title = "Box Plot of Customer Value")## Warning: Use of `churn_data$Customer.Value` is discouraged.
## i Use `Customer.Value` instead.
The distribution of customer value for the churn dataset is displayed in this box plot. The variable in the dataset that reflects customer value is shown on the Y-axis.There is significant heterogeneity in customer values, as seen by its large interquartile range. The median is approximately 750.
There are no exceptionally low outliers below the first quartile since the lower whisker is so short. The top whisker, on the other hand, is well over the third quartile, suggesting some really high customer values above 2000 that are not outliers.Additionally, there are a few single instances that show actual outliers with much greater customer numbers above the top whisker.The red hue was probably only intended to draw attention to the box visually, but it could also imply that this characteristic is crucial to examine in order to comprehend customer attrition.
Overall, by displaying both the central trend and outliers, this box plot provides insight into the distribution of customer value rankings. The variability shows that not every consumer has the same value.
- Box Plot by Age Group
ggplot(churn_data, aes(x = 1, y = churn_data$Age.Group)) +
geom_boxplot(fill = "green", color = "black") +
labs(title = "Box Plot of Age Group")## Warning: Use of `churn_data$Age.Group` is discouraged.
## i Use `Age.Group` instead.
The age distribution within a customer churn dataset is displayed using a box plot. For each group, the green box represents the median age and the distribution of the middle 50% of the data. For one group, there is an outlier above the top whisker. The graphic uses the traditional box plot elements of box, whiskers, and median line to successfully illustrate the central tendency and variance in age across categories, despite the lack of labels on the axes and gridlines.
DIMENSION REDUCTION
A statistical method called dimension reduction is used to minimize the number of variables in a dataset while maintaining its key characteristics. It is crucial for streamlining intricate datasets, facilitating visualization, and enhancing the effectiveness of ensuing analyses. When working with high-dimensional data, where there may be more variables than observations, this procedure is quite helpful. Feature extraction and feature selection are the two primary categories of dimension reduction approaches. Selecting a portion of the original variables according to how important they are to the study is known as feature selection. Feature extraction, on the other hand, produces new variables that are linear combinations of the original variables and are referred to as components or factors.
Two popular techniques for dimension reduction are Principal Component Analysis (PCA) and t-Distributed Stochastic Neighbor Embedding (t-SNE). Principal components, or linear combinations of variables, that capture the largest variance in the data are found using PCA. It helps to preserve the variability found in the dataset while condensing information into a smaller collection of components. The nonlinear method known as t-SNE is useful for displaying complicated structures because it preserves pairwise similarities between data points in lower-dimensional space.
In many areas, including genomics, image processing, and machine learning, dimension reduction is essential. Simplifying datasets makes them easier to read, makes exploratory data analysis easier, and frequently boosts the efficiency of later modeling methods. Regarding your study on churn data in the telecom industry, dimension reduction methods such as PCA and t-SNE can offer insightful information about underlying trends that can help with categorization, grouping, and predictive modeling.3
- Principal Component Analysis (PCA) is a widely employed dimension reduction technique in data analysis, playing a crucial role in extracting essential information from high-dimensional datasets. In the context of your research project on churn prediction in the telecommunications industry, PCA can be employed to streamline the interpretation of various customer-related features. By transforming correlated variables into a set of uncorrelated principal components, PCA allows for the identification of key patterns and structures within the data. PCA finds the directions of maximum variance in high-dimensional data and projects it to a lower-dimensional space. This is a common linear technique for dimensional reduction.4
In the dataset, which encompasses diverse customer attributes such as call failures, subscription length, and charge amounts, PCA can unveil the underlying factors contributing to customer churn. The principal components represent linear combinations of the original variables, enabling a condensed representation of the data while retaining most of its variability. This reduction in dimensional facilitates the identification of critical features influencing customer behavior.5
Furthermore, PCA aids in visualizing the data by projecting it onto a lower-dimensional space, simplifying the exploration of relationships between variables. By implementing PCA as part of your research methodology, you can gain valuable insights into the key drivers of customer churn and refine your predictive models for enhanced accuracy and interpret ability.6
- t-Distributed Stochastic Neighbor Embedding (t-SNE) is a powerful dimensional reduction technique known for its effectiveness in visualizing complex, high-dimensional datasets. In the realm of your telecommunications churn prediction research, t-SNE can offer unique advantages by capturing intricate relationships between customer attributes. Unlike traditional linear techniques, t-SNE excels at preserving local structures, making it particularly suitable for revealing subtle patterns and clusters within the data. t-SNE is a nonlinear technique that is well-suited for embedding high-dimensional data into a 2D or 3D space for visualization. It preserves local structure well.
By mapping data points to a lower-dimensional space, t-SNE emphasizes the similarities and dissimilarities between observations, providing a detailed representation of the inherent structures in your customer dataset. This is crucial for identifying distinct customer segments based on factors like call failures, subscription length, and usage patterns. The algorithm is particularly adept at highlighting nonlinear relationships, allowing for a nuanced understanding of the intricate dynamics influencing customer churn.
Moreover, t-SNE-generated visualizations, such as scatter plots, enable intuitive interpretation of clusters and patterns, aiding in the identification of potential churn triggers. Leveraging t-SNE in your research can enhance the granularity of your insights, facilitating targeted strategies for customer retention and service optimization in the telecommunications domain.7
churn_data_standardized <- churn_data %>%
scale()
# Check the first few rows of the standardized data
head(churn_data_standardized)## Call..Failure Complains Subscription..Length Charge..Amount
## [1,] 0.0512210 -0.2877847 0.6366253 -0.61986363
## [2,] -1.0501179 -0.2877847 0.7532640 -0.61986363
## [3,] 0.3265557 -0.2877847 0.5199865 -0.61986363
## [4,] 0.3265557 -0.2877847 0.6366253 -0.61986363
## [5,] -0.6371158 -0.2877847 0.6366253 -0.61986363
## [6,] 0.4642231 -0.2877847 0.6366253 0.03756749
## Seconds.of.Use Frequency.of.use Frequency.of.SMS Distinct.Called.Numbers
## [1,] -0.02440732 0.02681199 -0.6074163 -0.3780980
## [2,] -0.98964985 -1.12274728 -0.5895969 -1.1331509
## [3,] -0.48106327 -0.16478122 2.5466081 0.0284689
## [4,] -0.06538010 -0.06027583 -0.6430550 0.6673598
## [5,] -0.49535610 -0.19961635 -0.6341453 0.5511978
## [6,] -0.16614456 0.21840520 -0.3668551 0.2607929
## Age.Group Tariff.Plan Status Age Customer.Value Churn
## [1,] 0.1949104 -0.2903628 -0.5745708 -0.1130565 -0.5286746 -0.4317192
## [2,] -0.9254686 -0.2903628 1.7398769 -0.6792377 -0.8219057 -0.4317192
## [3,] 0.1949104 -0.2903628 -0.5745708 -0.1130565 2.0609580 -0.4317192
## [4,] -2.0458476 -0.2903628 -0.5745708 -1.8116000 -0.4467041 -0.4317192
## [5,] -2.0458476 -0.2903628 -0.5745708 -1.8116000 -0.6289327 -0.4317192
## [6,] 0.1949104 -0.2903628 -0.5745708 -0.1130565 -0.3649657 -0.43171921) PCA
It is applying PCA using the prcomp() function to a dataset called churn_data_standardized.The scale parameter is set to TRUE, which means it will standardize or scale the variables in the dataset before running PCA. The output of PCA is being stored in an object called pca_result.This pca_result can then potentially be used for further analysis or plotting related to the PCA results.
So in summary, this code is conducting PCA on a standardized dataset and capturing the results for downstream exploration, using some common libraries for clustering and visualization.
# Apply PCA
pca_result <- prcomp(churn_data_standardized, scale. = TRUE)
pca_result## Standard deviations (1, .., p=14):
## [1] 2.05322030 1.52882185 1.33769779 1.17267424 1.10187142 0.94794340
## [7] 0.75959810 0.70357614 0.63182775 0.60067961 0.51110972 0.19052277
## [13] 0.17621418 0.09517431
##
## Rotation (n x k) = (14 x 14):
## PC1 PC2 PC3 PC4
## Call..Failure -0.284612695 -0.22898095 -0.359856965 0.1090701
## Complains 0.109092311 -0.13769166 -0.412701968 0.4392494
## Subscription..Length -0.058737179 -0.04485582 -0.085136582 0.1295144
## Charge..Amount -0.301092955 -0.26861808 -0.009877207 0.1439731
## Seconds.of.Use -0.416447390 -0.08736723 -0.145381316 -0.1257431
## Frequency.of.use -0.422747245 -0.06514559 -0.194864374 -0.1380942
## Frequency.of.SMS -0.191200843 0.33116600 0.296766292 0.5334285
## Distinct.Called.Numbers -0.374782925 -0.11469025 -0.155498755 -0.1223602
## Age.Group -0.005064304 -0.52660701 0.379684531 0.1976102
## Tariff.Plan -0.168415619 0.11519932 -0.086430360 0.2003921
## Status 0.320946654 -0.09186225 -0.241473710 0.1641822
## Age -0.004990361 -0.53519614 0.372979214 0.1622585
## Customer.Value -0.303693241 0.34830861 0.162585023 0.4084157
## Churn 0.245302019 -0.11384623 -0.382084834 0.3520914
## PC5 PC6 PC7 PC8
## Call..Failure -0.006593744 -0.22189523 -0.25452257 0.418188675
## Complains 0.092387135 0.33805544 -0.19492389 -0.528889062
## Subscription..Length -0.728789574 -0.46796254 0.03710163 -0.415251972
## Charge..Amount 0.210431141 -0.39395097 -0.46090229 0.089969712
## Seconds.of.Use -0.123600447 0.23413026 0.20173887 0.044326523
## Frequency.of.use -0.088872144 0.21953308 0.28102669 0.037082940
## Frequency.of.SMS -0.093997433 0.07151262 -0.03341239 0.157844715
## Distinct.Called.Numbers -0.042926490 0.15945139 0.15375316 -0.125265035
## Age.Group -0.023280547 0.09076769 0.13467506 -0.006433024
## Tariff.Plan 0.574088149 -0.45132339 0.50539911 -0.285016344
## Status -0.191172953 -0.21722222 0.47683223 0.410547472
## Age 0.011337530 0.08153378 0.17957439 -0.029887407
## Customer.Value -0.120794772 0.13219505 0.04798719 0.130706466
## Churn 0.027674917 0.21331622 0.05916209 0.230471712
## PC9 PC10 PC11 PC12
## Call..Failure -0.281965045 -0.095137098 0.56629247 0.048382625
## Complains -0.300030817 -0.254218608 -0.10155437 0.015928718
## Subscription..Length 0.179988436 0.055921462 0.10454479 0.007071230
## Charge..Amount 0.182196199 0.020376759 -0.57484485 -0.025670553
## Seconds.of.Use 0.279527736 -0.339412723 -0.24663468 0.122974653
## Frequency.of.use 0.109447450 -0.236580109 0.07493379 -0.124277505
## Frequency.of.SMS -0.115085612 0.077312905 0.02150082 0.027876633
## Distinct.Called.Numbers -0.357806635 0.760845841 -0.15777634 -0.034676637
## Age.Group -0.006165028 -0.050919168 0.09784942 -0.693799043
## Tariff.Plan 0.089965167 -0.008112243 0.15870136 -0.016286658
## Status -0.355926256 -0.124085262 -0.42438796 -0.006133196
## Age -0.007667372 0.020400995 0.08893678 0.694470622
## Customer.Value 0.003540315 -0.043098564 -0.06771843 0.006746973
## Churn 0.631359314 0.382642362 0.09199256 -0.012418911
## PC13 PC14
## Call..Failure -0.156787281 -0.0077380311
## Complains 0.011290816 -0.0010085890
## Subscription..Length 0.005284946 -0.0071543568
## Charge..Amount 0.172167176 0.0447176234
## Seconds.of.Use -0.538931579 -0.3309057041
## Frequency.of.use 0.723197797 0.1073806218
## Frequency.of.SMS 0.208008144 -0.6143658773
## Distinct.Called.Numbers -0.085958422 -0.0153805094
## Age.Group -0.137561651 -0.0020370773
## Tariff.Plan -0.077822911 -0.0337372241
## Status 0.036674834 0.0083650142
## Age 0.092687856 0.0944306588
## Customer.Value -0.214970411 0.6993188284
## Churn 0.019346507 -0.0008855576The visualization of the Scree Plot 1
# Scree plot to visualize the variance explained by each principal component
fviz_screeplot(pca_result, addlabels = TRUE, ylim = c(0, 50))
The visualization of the Scree Plot 2
# Extract the percentage of variance explained by each principal component
variance_explained <- pca_result$sdev^2 / sum(pca_result$sdev^2) * 100
# Create a scree plot
plot(1:length(variance_explained), variance_explained, type = "b",
xlab = "Principal Component", ylab = "Percentage of Variance Explained",
main = "Scree Plot for PCA")
# Add labels for each dimension
text(1:length(variance_explained), variance_explained, labels = sprintf("%.1f%%", variance_explained), pos = 3)
# Add a horizontal line at the elbow point (e.g., the second dimension)
abline(v = 2, col = "red", lty = 2)
This code performs PCA on the standardized data and then creates a scree
plot to help to decide on the number of principal components to
retain.The scree plot graph displaying the percentage of variances
explained by each dimension for a dataset that was analyzed using a
multivariate statistical technique, such as principal component analysis
(PCA). The y-axis represents the percentage of variance explained,
ranging from 0% at the bottom to 50% at the top, with increments of 5%.
The x-axis lists the dimensions from 1 to 10.
Each bar on the graph corresponds to one dimension with its respective percentage of explained variance labeled above it. The first dimension explains the highest variance at 30.1%. The second dimension represents 16.7%, indicating a significant drop from the first dimension. As the dimensions increase, the percentage of explained variance decreases in a near-monotonic fashion:
The third dimension explains 12.8% of the variance. The fourth dimension represents 9.8%. The fifth dimension drops to 8.7%. The sixth dimension explains 6.4% then. The seventh dimension contributes 4.1% of the variance. The eighth dimension provides 3.5%. The ninth dimension is at 2.9%. Finally, the tenth dimension explains 2.6% of the variance. The scree plot graph shows a clear elbow at the second dimension, suggesting that later dimensions contribute much less to the explained variance, and that the first two dimensions may be considered the most informative in capturing the variance in the data. This “elbow” could be used to determine the number of components to retain in a PCA model.
p_pca <- as.data.frame(pca_result$x) %>%
ggplot(aes(x = PC1, y = PC2, color = churn_data$Churn)) +
geom_point() +
labs(subtitle = "PCA")
print(p_pca)
# Biplot to visualize the contribution of variables to the principal components
fviz_pca_biplot(pca_result, label = "var", repel = TRUE)
It also generates a biplot, which combines a scores plot and a loadings plot, allowing you to observe the relationships between observations and variables in the reduced-dimensional space.
Regression Model
# Extract principal components from the prcomp result
pc_data <- as.data.frame(pca_result$x)
# Combine principal components with the churn variable
subset_data <- cbind(pc_data, Churn = churn_data$Churn)
model <- glm(Churn ~ PC1 + PC2, data = subset_data, family = "binomial")
summary(model)##
## Call:
## glm(formula = Churn ~ PC1 + PC2, family = "binomial", data = subset_data)
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.06177 0.17382 -23.37 <2e-16 ***
## PC1 1.61768 0.07727 20.93 <2e-16 ***
## PC2 -0.61463 0.06116 -10.05 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 2739.9 on 3149 degrees of freedom
## Residual deviance: 1427.0 on 3147 degrees of freedom
## AIC: 1433
##
## Number of Fisher Scoring iterations: 7var_importance <- varImp(model, scale = FALSE)
print(var_importance)## Overall
## PC1 20.93482
## PC2 10.04971The principal component analysis (PCA) reveals that PC1 explains a substantial variance of 20.93%, while PC2 contributes 10.05%. This suggests that the first two principal components capture a significant portion of the data’s variability, providing valuable insights into the underlying patterns of the dataset.
2) T-SNE
# Extract the column corresponding to the Churn variable
churn_variable <- churn_data_unique[, "Churn"]
# Run t-SNE
tsne_result <- Rtsne(churn_data_unique, perplexity = 30, check_duplicates = FALSE)
tsne_result## $N
## [1] 2850
##
## $Y
## [,1] [,2]
## [1,] 33.24540343 13.83605074
## [2,] -19.07060626 8.89936922
## [3,] 4.79559150 -42.12075432
## [4,] 29.31361408 15.75097533
## [5,] -14.18748611 32.83969966
## [6,] 18.08753551 20.12410790
## [7,] 4.99676998 -36.14654272
## [8,] -24.34278976 -10.36949288
## [9,] -27.48481400 -25.44238397
## [10,] 36.33730389 13.29874394
## [11,] 33.75673258 1.26276999
## [12,] -15.30183676 27.63296948
## [13,] -18.15210081 -26.14715775
## [14,] 9.13778142 -48.09143015
## [15,] -13.95811548 -19.51641395
## [16,] 31.41193408 -12.13779612
## [17,] -5.14594098 -13.52601838
## [18,] -11.69133457 -8.84045623
## [19,] -28.73452506 14.14780306
## [20,] 9.21759631 -14.61117953
## [21,] -47.01944592 5.23317553
## [22,] 26.70582041 -11.54856031
## [23,] -1.60858261 -9.71370277
## [24,] -42.48770021 -1.73542622
## [25,] 35.18651797 0.28643616
## [26,] -1.63561906 39.93262172
## [27,] 2.13096360 36.39815520
## [28,] -12.63274279 28.56923048
## [29,] -27.05179599 8.55150080
## [30,] 1.81290448 14.89979768
## [31,] 22.17055549 1.30796600
## [32,] -12.08762948 -3.99270427
## [33,] -19.82682650 5.72116275
## [34,] -14.63257962 -33.31844552
## [35,] 14.81689590 -11.41971093
## [36,] 17.20814714 22.28924990
## [37,] -1.99836159 13.74940755
## [38,] 11.78011817 -31.16570056
## [39,] -1.22478087 -29.85129866
## [40,] 2.62361376 11.94980681
## [41,] -11.77743560 -8.59173872
## [42,] -7.59282727 -12.66778262
## [43,] 3.89749967 -0.17754076
## [44,] 6.42958430 30.80741153
## [45,] -2.39548747 14.79869821
## [46,] 29.02285110 -7.17112747
## [47,] -32.21417728 -4.16381446
## [48,] 1.16014678 40.41808612
## [49,] 22.76793429 -35.80581312
## [50,] 5.47856444 10.31092592
## [51,] 34.58533719 13.69566288
## [52,] -13.61347121 5.86581722
## [53,] 5.19462313 -42.26061794
## [54,] 30.74401621 15.57112132
## [55,] -13.37761942 34.59737808
## [56,] 19.07779833 19.22038403
## [57,] 5.41055835 -36.49891364
## [58,] -24.18216914 -10.77226913
## [59,] -27.03161762 -25.30379722
## [60,] 37.60309631 12.59774738
## [61,] 32.80380034 1.08512713
## [62,] -14.90925817 29.39174493
## [63,] -17.73306341 -26.56254533
## [64,] 9.38476384 -48.33105726
## [65,] -13.49750247 -19.54069707
## [66,] 31.02344830 -12.62070407
## [67,] -1.74170001 -12.33227285
## [68,] -9.49559810 -10.56310937
## [69,] -28.16850636 9.42548146
## [70,] 9.91791717 -15.30241077
## [71,] -47.17293429 5.55970103
## [72,] 26.49822053 -12.32662904
## [73,] 0.03084736 -8.05970812
## [74,] -42.86240518 -1.48836978
## [75,] 34.27339810 0.07278395
## [76,] 0.10971425 38.36460997
## [77,] 3.36569692 34.98976545
## [78,] -12.15220997 30.37118364
## [79,] -25.46467042 7.49161162
## [80,] -6.34686128 13.40359789
## [81,] 21.77183233 1.41482490
## [82,] -12.71870666 -8.71778036
## [83,] -11.99164135 5.98635248
## [84,] -14.43570309 -33.76102458
## [85,] 15.03804381 -12.08838440
## [86,] 18.95190443 21.69846095
## [87,] -2.99844729 16.13963149
## [88,] 12.26307941 -31.36412223
## [89,] -0.75008427 -30.03011417
## [90,] 0.49585805 13.76036881
## [91,] -10.10177617 -10.60975006
## [92,] -3.84872170 -12.33077755
## [93,] 3.23799166 2.32655934
## [94,] 7.91986768 29.38164383
## [95,] -3.42856851 16.45005582
## [96,] 28.42463250 -7.71555697
## [97,] -32.88997541 -3.83565610
## [98,] 2.66960246 39.37729150
## [99,] 23.22014130 -35.78824209
## [100,] 3.89283781 12.19908558
## [101,] 31.46369235 14.18850485
## [102,] -23.64255104 6.92165980
## [103,] 3.87607701 -42.47296011
## [104,] 27.29263495 16.07805442
## [105,] -14.96746403 30.79147596
## [106,] 17.11798282 21.15426555
## [107,] 4.64042481 -35.82417913
## [108,] -24.52166046 -9.98394187
## [109,] -27.85544571 -25.53880809
## [110,] 35.12951973 13.70751601
## [111,] 34.59983646 1.19441440
## [112,] -15.26315931 25.40604152
## [113,] -18.51930184 -25.82789691
## [114,] 8.86499964 -47.80584498
## [115,] -14.38079357 -19.51019007
## [116,] 31.83734228 -11.60967097
## [117,] -8.11133901 -13.50103912
## [118,] -13.03773942 -4.75899602
## [119,] -28.55348540 14.74240317
## [120,] 9.28541737 -14.24339489
## [121,] -47.24282750 4.40244645
## [122,] 26.96204178 -10.63283347
## [123,] -3.86403055 -12.10497589
## [124,] -42.06334697 -2.24073396
## [125,] 36.05567207 0.48205135
## [126,] -3.42628919 40.80035172
## [127,] 0.81343490 37.80838863
## [128,] -12.61332245 27.35767535
## [129,] -30.68554841 15.54415818
## [130,] 3.00860855 13.66588786
## [131,] 22.75322390 1.10501050
## [132,] -11.51242306 -0.94647697
## [133,] -36.76101374 19.83664729
## [134,] -14.79692280 -32.82988455
## [135,] 14.97487721 -11.02909463
## [136,] 15.77962765 22.70375712
## [137,] -1.03713473 11.99170782
## [138,] 11.24777619 -30.96243780
## [139,] -1.50285409 -29.54108719
## [140,] 3.76702134 10.46618594
## [141,] -12.74755937 -4.72152006
## [142,] -11.13465887 -11.06384827
## [143,] 4.31610905 -2.69063739
## [144,] 5.02439472 32.04434987
## [145,] -1.00175559 13.04796375
## [146,] 30.37629988 -7.63689849
## [147,] -31.81345643 -4.25985652
## [148,] -3.26373758 38.76918593
## [149,] 22.31215157 -35.81505452
## [150,] 6.35135176 8.43997938
## [151,] 35.04365570 13.46415284
## [152,] -13.66789266 5.37615238
## [153,] 5.30489578 -42.33237802
## [154,] 31.41223937 15.68486379
## [155,] -12.79605606 34.91577445
## [156,] 19.28971522 18.91777340
## [157,] 5.53633081 -36.63931760
## [158,] -24.12954207 -10.91814644
## [159,] -26.86088928 -25.25651442
## [160,] 37.95996941 12.26196472
## [161,] 32.45455753 0.94546723
## [162,] -14.43923324 30.02483531
## [163,] -17.57800684 -26.64383773
## [164,] 9.43981333 -48.41070615
## [165,] -13.33987385 -19.54715509
## [166,] 30.82793423 -12.82850774
## [167,] -1.32009382 -11.50421659
## [168,] -8.64156374 -10.62339659
## [169,] -27.97502990 9.09298737
## [170,] 9.86746881 -15.38987114
## [171,] -47.22026015 5.67508514
## [172,] 26.53241645 -12.65796098
## [173,] 0.06174852 -7.33477830
## [174,] -42.97321806 -1.38778078
## [175,] 33.91237478 -0.12587433
## [176,] 0.68700939 38.06781349
## [177,] 4.07530725 34.63410517
## [178,] -11.98163678 30.67712752
## [179,] -25.36640422 7.43591731
## [180,] -6.57908615 13.65334010
## [181,] 21.60934672 1.43731609
## [182,] -12.13082476 -9.05110416
## [183,] -11.85633534 5.45438598
## [184,] -14.33131939 -33.91531724
## [185,] 14.34678608 -11.92671955
## [186,] 19.18318779 21.34469371
## [187,] -3.22835408 16.61126284
## [188,] 12.44899017 -31.49441272
## [189,] -0.84607017 -30.41039539
## [190,] 0.13169287 14.01150477
## [191,] -9.11446008 -10.61315807
## [192,] -3.39026872 -11.74915830
## [193,] 2.81034620 2.82894423
## [194,] 8.50354001 29.20984961
## [195,] -3.60673591 16.94031332
## [196,] 28.28916528 -8.02858172
## [197,] -33.01024634 -3.82515781
## [198,] 3.22821990 39.01616028
## [199,] 23.33135058 -35.77531441
## [200,] 3.46971234 12.50459542
## [201,] 30.94758539 14.35467191
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## [1996,] 34.96920424 12.88236811
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## [2013,] -47.45309159 6.62234472
## [2014,] 25.56283795 -14.35096539
## [2015,] 4.17017301 -1.47418325
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## [2705,] -12.05925954 -19.79581217
## [2706,] 25.40187627 -12.26830191
## [2707,] 4.62705657 -2.22238355
## [2708,] -0.34374206 -7.68894400
## [2709,] -19.13035894 10.01928088
## [2710,] 9.85913178 -16.51149442
## [2711,] -47.60105401 6.92897882
## [2712,] 25.26530817 -14.72342535
## [2713,] 2.98861824 1.42788042
## [2714,] -44.31878473 -0.12810095
## [2715,] 30.61646729 -0.13153113
## [2716,] 4.42383100 32.39225492
## [2717,] 8.76343816 29.66129978
## [2718,] -9.65099155 36.68037692
## [2719,] -13.17896880 4.16841376
## [2720,] -12.28720487 18.79614011
## [2721,] 20.22604625 1.83563189
## [2722,] -1.93644114 -12.16694706
## [2723,] -11.88679786 -4.42048091
## [2724,] -13.68660332 -35.21621938
## [2725,] 14.38456637 -13.55041178
## [2726,] 24.82860281 17.88116331
## [2727,] -9.56798827 21.48036642
## [2728,] 13.89531121 -32.23088672
## [2729,] 0.58092923 -31.36279468
## [2730,] -8.15835931 17.14411382
## [2731,] -0.35793905 -8.06523417
## [2732,] 3.39237142 -4.61543735
## [2733,] -0.24788090 10.20047451
## [2734,] 11.91199076 23.55919266
## [2735,] -8.93300343 22.73748587
## [2736,] 26.52589611 -10.94916186
## [2737,] -34.41496563 -3.63179578
## [2738,] 6.13051880 32.73450026
## [2739,] 24.68035797 -35.55114239
## [2740,] -6.37209723 15.26294637
## [2741,] 41.24132390 2.55158368
## [2742,] 3.96216396 4.28352477
## [2743,] 10.23527699 -41.39264911
## [2744,] 41.95147430 4.56505272
## [2745,] 10.83204032 26.75279613
## [2746,] 39.71250208 9.09795496
## [2747,] 15.21721780 -33.25904874
## [2748,] -23.73460718 -13.83702543
## [2749,] -20.51140298 -24.63570846
## [2750,] 39.50296113 2.21427566
## [2751,] 21.09550588 0.21343790
## [2752,] 6.74084220 30.04553032
## [2753,] -13.68452234 -33.76128485
## [2754,] 20.56027194 -37.29254067
## [2755,] -10.36872873 -20.74996866
## [2756,] 23.60771331 -17.06905425
## [2757,] -9.43638553 17.10126387
## [2758,] -2.91121906 15.15451026
## [2759,] 1.02223994 -8.86974420
## [2760,] 9.78046934 -18.05210798
## [2761,] -48.14501301 8.53266425
## [2762,] 23.72595600 -17.11135680
## [2763,] -9.12042143 21.74149836
## [2764,] -46.61800392 5.20255147
## [2765,] 21.39748124 -0.48339377
## [2766,] 20.59277067 20.24294606
## [2767,] 24.97392560 18.38147322
## [2768,] 8.39256105 29.81719112
## [2769,] 4.08710198 -3.97210008
## [2770,] -6.99370516 40.98633166
## [2771,] 18.26944121 2.39570606
## [2772,] 1.85818801 14.40623685
## [2773,] 7.78231767 6.00695016
## [2774,] -13.09532885 -36.57927967
## [2775,] 12.07601009 -16.07531550
## [2776,] 40.26050139 10.30587189
## [2777,] -6.44384500 38.01351271
## [2778,] 19.37005321 -35.36211273
## [2779,] 8.12252981 -30.75313950
## [2780,] -10.38878737 38.47448175
## [2781,] -2.71734243 14.98179117
## [2782,] -8.60111100 16.15963912
## [2783,] -12.03772037 28.09911385
## [2784,] 30.28071876 15.74442811
## [2785,] -5.30440001 37.32141615
## [2786,] 24.32319662 -16.60275123
## [2787,] -39.37467696 -3.27987606
## [2788,] 19.36211632 18.73411657
## [2789,] 26.43559629 -34.56853802
## [2790,] -12.12978709 35.68738694
## [2791,] 22.95128203 18.09128483
## [2792,] -30.30130572 14.39103849
## [2793,] 2.10320657 -40.60593734
## [2794,] 19.06169415 21.30155103
## [2795,] -12.45566693 19.39236333
## [2796,] 9.58610632 27.12144164
## [2797,] 2.70767331 -34.06375514
## [2798,] -25.73001992 -8.19038158
## [2799,] -29.61374157 -26.03549514
## [2800,] 26.35150253 15.61650766
## [2801,] 39.79128993 2.10244473
## [2802,] -8.68845115 14.92684298
## [2803,] -20.61391248 -24.52432139
## [2804,] 7.23688720 -46.32055688
## [2805,] -16.53064973 -20.05910790
## [2806,] 33.75758025 -8.49033747
## [2807,] -11.05454132 3.70927712
## [2808,] -21.64876398 6.02670555
## [2809,] -28.78760695 13.92403969
## [2810,] 11.08572705 -12.26724898
## [2811,] -45.65785894 2.11831133
## [2812,] 30.03082843 -6.72702501
## [2813,] -10.95886197 -0.24091071
## [2814,] -39.91334792 -3.15340519
## [2815,] 40.91461212 2.54732652
## [2816,] -14.60611754 34.40261517
## [2817,] -10.52193320 39.12951998
## [2818,] -9.08453722 16.60291212
## [2819,] -30.54866500 14.83544924
## [2820,] 7.78534589 0.80946895
## [2821,] 25.64631904 1.33282616
## [2822,] -24.78577158 7.22637242
## [2823,] -36.86918274 19.83201197
## [2824,] -15.71254890 -30.35398359
## [2825,] 15.91653331 -9.05688610
## [2826,] 7.94192686 28.05528541
## [2827,] 4.31200317 1.46020896
## [2828,] 8.46360678 -30.48613845
## [2829,] -2.92892410 -27.98889731
## [2830,] 6.84431193 -3.57197940
## [2831,] -22.24590590 6.37172868
## [2832,] -11.69786784 5.41285685
## [2833,] -7.71780299 -13.58213712
## [2834,] -2.09339605 40.18521318
## [2835,] 5.29960110 2.43498569
## [2836,] 33.63439634 -4.71483785
## [2837,] -29.73642115 -4.76882742
## [2838,] -13.43626348 34.44375309
## [2839,] 20.17081533 -35.56929667
## [2840,] 4.03624387 -7.24945610
## [2841,] -10.89591281 36.69422661
## [2842,] -16.37224137 -28.08621436
## [2843,] 3.76045749 1.41311466
## [2844,] -9.99066263 37.06300811
## [2845,] 26.43399220 -10.69811299
## [2846,] -34.40916656 -3.60666689
## [2847,] 7.01223249 34.18295871
## [2848,] -7.03553736 14.75183889
## [2849,] -44.29319588 -0.13090704
## [2850,] -1.79734889 -12.61886149
##
## $costs
## [1] 1.853258e-04 4.027376e-05 1.547346e-04 1.631673e-04 1.695484e-04
## [6] 1.892547e-04 1.493105e-04 7.320446e-05 6.987566e-05 1.437070e-04
## [11] 1.681452e-04 1.873891e-04 1.082948e-04 6.751727e-05 8.815004e-05
## [16] 1.948931e-04 1.824108e-04 1.526754e-04 4.300093e-05 9.847611e-05
## [21] 9.253022e-05 1.426980e-04 2.131747e-04 7.613638e-05 1.457540e-04
## [26] 1.657234e-04 1.785355e-04 2.373067e-04 1.500157e-04 2.100393e-04
## [31] 9.415666e-05 9.555625e-05 8.847847e-05 6.413792e-05 1.180414e-04
## [36] 1.734783e-04 2.891874e-04 1.098164e-04 7.532539e-05 1.917194e-04
## [41] 1.324323e-04 9.102734e-05 2.652190e-04 1.328864e-04 2.112124e-04
## [46] 1.396445e-04 7.957484e-05 1.995361e-04 8.630235e-05 1.626213e-04
## [51] 1.402391e-04 1.674396e-04 1.421698e-04 1.715788e-04 1.849997e-04
## [56] 1.892901e-04 1.519535e-04 6.703015e-05 7.775593e-05 1.556798e-04
## [61] 1.911547e-04 2.246477e-04 1.148671e-04 5.675268e-05 8.313602e-05
## [66] 2.055189e-04 1.203135e-04 2.369174e-04 3.030673e-04 1.119735e-04
## [71] 8.336117e-05 1.573483e-04 1.074259e-04 8.374456e-05 1.863970e-04
## [76] 1.897134e-04 1.941953e-04 2.627955e-04 1.423416e-04 5.024239e-04
## [81] 8.603297e-05 1.687279e-04 8.030670e-05 7.187433e-05 1.407442e-04
## [86] 1.572323e-04 2.072717e-04 1.172886e-04 7.088721e-05 2.716586e-04
## [91] 2.280838e-04 1.759235e-04 1.942556e-04 1.697493e-04 2.047191e-04
## [96] 1.434853e-04 7.324985e-05 2.772645e-04 7.842881e-05 1.647562e-04
## [101] 1.934890e-04 1.891209e-04 1.641498e-04 1.785314e-04 2.764517e-04
## [106] 1.683962e-04 1.501556e-04 7.378372e-05 6.208525e-05 1.218330e-04
## [111] 1.514784e-04 2.280646e-04 1.054209e-04 7.514936e-05 8.788856e-05
## [116] 2.085492e-04 1.513990e-04 1.618058e-04 6.154960e-05 8.879941e-05
## [121] 9.337993e-05 1.485097e-04 1.670392e-04 1.000845e-04 1.644902e-04
## [126] 2.698880e-04 1.819033e-04 2.462215e-04 8.965952e-05 1.733877e-04
## [131] 1.233639e-04 1.533498e-04 2.348900e-05 7.443253e-05 1.059742e-04
## [136] 1.311775e-04 1.712685e-04 1.071532e-04 7.313180e-05 2.326604e-04
## [141] 1.557507e-04 1.662003e-04 2.638464e-04 1.488020e-04 2.250479e-04
## [146] 2.104881e-04 7.865202e-05 4.822165e-04 8.633700e-05 2.572658e-04
## [151] 1.527626e-04 1.286540e-04 1.379860e-04 1.630104e-04 2.008109e-04
## [156] 1.973128e-04 1.563030e-04 6.067555e-05 7.707613e-05 1.676224e-04
## [161] 1.936034e-04 2.754236e-04 1.134133e-04 4.870310e-05 7.589413e-05
## [166] 2.143197e-04 1.587799e-04 2.032509e-04 1.873831e-04 1.077393e-04
## [171] 8.285039e-05 1.658288e-04 1.610846e-04 9.057667e-05 2.125774e-04
## [176] 1.560585e-04 2.226402e-04 2.296589e-04 1.409831e-04 3.120363e-04
## [181] 8.382716e-05 1.165259e-04 7.290657e-05 7.464895e-05 1.656601e-04
## [186] 1.427899e-04 2.087034e-04 1.180617e-04 8.956396e-05 2.039618e-04
## [191] 2.235113e-04 1.823498e-04 1.764331e-04 1.546682e-04 2.622712e-04
## [196] 1.475334e-04 7.655227e-05 2.272213e-04 7.253623e-05 1.542259e-04
## [201] 1.778258e-04 5.496511e-05 1.579593e-04 1.465310e-04 2.573065e-04
## [206] 1.655490e-04 1.529622e-04 7.010078e-05 5.907615e-05 1.089536e-04
## [211] 1.506987e-04 1.434039e-04 1.002647e-04 7.851539e-05 8.384200e-05
## [216] 2.173431e-04 3.286509e-04 1.248221e-04 7.522872e-05 8.878214e-05
## [221] 9.773850e-05 1.485428e-04 1.574650e-04 1.019153e-04 1.746288e-04
## [226] 2.361959e-04 1.680077e-04 2.966962e-04 8.890950e-05 1.502715e-04
## [231] 1.252748e-04 2.476829e-05 2.057452e-05 7.832292e-05 1.034946e-04
## [236] 1.129637e-04 1.762540e-04 1.032598e-04 7.115111e-05 2.132413e-04
## [241] 1.228006e-04 2.026308e-04 2.902987e-04 1.006649e-04 2.366124e-04
## [246] 1.998883e-04 8.013882e-05 2.417130e-04 8.027331e-05 2.679915e-04
## [251] 1.793617e-04 1.638608e-04 1.362302e-04 2.349437e-04 1.685408e-04
## [256] 2.002227e-04 1.900525e-04 2.863927e-05 5.339820e-05 1.561378e-04
## [261] 1.506983e-04 2.170120e-04 8.073155e-05 4.056061e-05 4.600364e-05
## [266] 1.744137e-04 2.167658e-04 2.146611e-04 1.927213e-04 6.651534e-05
## [271] 4.774421e-05 1.007014e-04 1.574478e-04 8.803070e-05 2.032333e-04
## [276] 1.596418e-04 1.934979e-04 1.219883e-04 1.370104e-04 1.637101e-04
## [281] 4.866020e-05 1.898673e-04 1.384924e-04 4.318804e-05 1.290811e-04
## [286] 2.021149e-04 1.448171e-04 1.105850e-04 9.510450e-05 1.484447e-04
## [291] 2.499426e-04 1.696740e-04 2.751978e-04 1.306572e-04 1.617959e-04
## [296] 1.429829e-04 6.008726e-05 2.054496e-04 3.524613e-05 3.892161e-04
## [301] 1.364827e-04 6.890654e-05 1.255797e-04 1.492152e-04 1.674500e-04
## [306] 1.353073e-04 1.386868e-04 6.157170e-05 2.338378e-05 1.448331e-04
## [311] 1.411658e-04 1.612346e-04 7.500151e-05 8.867452e-05 6.866180e-05
## [316] 1.693909e-04 1.454890e-04 1.372202e-04 6.855415e-05 9.144513e-05
## [321] 1.005782e-04 1.147526e-04 1.569571e-04 7.664993e-05 1.410710e-04
## [326] 2.277481e-04 1.742496e-04 2.939006e-04 8.683704e-05 7.906584e-05
## [331] 1.271178e-04 3.377900e-05 1.113003e-05 7.479076e-05 1.020905e-04
## [336] 1.526295e-04 2.182301e-04 1.014982e-04 4.258745e-05 2.297429e-04
## [341] 1.195473e-04 1.551492e-04 2.975857e-04 1.517229e-04 2.542111e-04
## [346] 1.703082e-04 5.514659e-05 1.687558e-04 5.739827e-05 2.173123e-04
## [351] 1.659303e-04 1.420701e-04 1.314727e-04 2.565883e-04 1.268390e-04
## [356] 4.079068e-04 1.627320e-04 4.241177e-05 6.427340e-05 1.698016e-04
## [361] 1.833745e-04 2.028744e-04 1.057224e-04 4.164963e-05 6.147405e-05
## [366] 2.084034e-04 2.238413e-04 1.829828e-04 1.489325e-04 9.153702e-05
## [371] 6.433000e-05 1.505235e-04 2.637919e-04 1.015654e-04 2.388990e-04
## [376] 1.541304e-04 1.956578e-04 3.074366e-04 5.418886e-05 1.613686e-04
## [381] 6.386880e-05 1.529179e-04 4.334166e-05 6.091808e-05 1.758311e-04
## [386] 1.417263e-04 2.990891e-04 1.139291e-04 1.065238e-04 2.244112e-04
## [391] 2.222174e-04 9.784810e-05 2.104143e-04 1.627498e-04 2.355700e-04
## [396] 1.584531e-04 6.768538e-05 3.169231e-04 5.100836e-05 2.183258e-04
## [401] 1.630721e-04 4.433988e-05 2.318663e-04 1.588245e-04 2.036267e-04
## [406] 1.508596e-04 1.504583e-04 6.592021e-05 3.909530e-05 2.011553e-04
## [411] 1.621092e-04 2.134980e-04 9.266910e-05 8.148371e-05 7.506806e-05
## [416] 2.057040e-04 2.114740e-04 1.340731e-04 6.425440e-05 8.015500e-05
## [421] 1.344428e-04 1.665112e-04 1.994797e-04 7.411229e-05 1.728308e-04
## [426] 2.240781e-04 1.827782e-04 2.723877e-04 1.006418e-04 1.606747e-04
## [431] 1.025941e-04 2.431885e-05 2.219382e-05 8.095357e-05 9.714830e-05
## [436] 1.880090e-04 2.418458e-04 1.228157e-04 5.936116e-05 2.169660e-04
## [441] 1.063137e-04 2.052830e-04 2.441659e-04 1.095247e-04 2.660645e-04
## [446] 2.669694e-04 6.890649e-05 1.679437e-04 6.834883e-05 3.019018e-04
## [451] 1.283624e-04 1.613079e-04 2.171278e-04 1.079638e-04 1.830134e-04
## [456] 2.329680e-04 3.673253e-04 5.829769e-05 1.935052e-04 2.046880e-04
## [461] 2.865206e-04 2.436342e-04 1.690719e-04 5.633719e-05 8.586263e-05
## [466] 4.484997e-04 2.105324e-04 1.657139e-04 1.657999e-04 5.006074e-05
## [471] 6.802736e-05 1.137081e-04 1.565902e-04 1.432432e-04 2.694440e-04
## [476] 1.176554e-04 1.429840e-04 1.650282e-04 2.024589e-04 1.334969e-04
## [481] 1.184016e-04 2.546934e-04 1.024797e-04 4.637457e-05 3.018477e-04
## [486] 1.481711e-04 1.451802e-04 2.183331e-04 1.770348e-04 1.352932e-04
## [491] 1.866373e-04 1.478355e-04 4.152359e-04 1.846488e-04 1.628731e-04
## [496] 5.642122e-05 1.575092e-04 1.902714e-04 8.109697e-05 2.190305e-04
## [501] 1.438000e-04 2.924707e-05 1.360033e-04 1.280671e-04 1.420757e-04
## [506] 1.291332e-04 1.894134e-04 1.338658e-04 6.378600e-05 1.381893e-04
## [511] 1.313748e-04 1.610864e-04 2.285515e-04 2.011195e-04 1.433362e-04
## [516] 2.152728e-04 1.976872e-04 1.484592e-04 1.709626e-04 1.434239e-04
## [521] 1.631349e-04 3.301033e-04 1.613477e-04 1.687110e-04 1.163730e-04
## [526] 2.101961e-04 2.496038e-04 1.007663e-04 1.929767e-04 2.934551e-04
## [531] 2.455784e-05 2.254198e-05 1.354557e-04 3.289064e-04 9.938644e-05
## [536] 2.087122e-04 2.388699e-04 9.780461e-05 1.604772e-04 1.340309e-04
## [541] 6.073802e-05 1.092550e-04 2.297506e-04 1.912292e-04 2.367018e-04
## [546] 1.343148e-04 1.308696e-04 1.857223e-04 1.054779e-04 1.612035e-04
## [551] 1.826821e-04 1.400488e-04 1.538263e-04 2.199422e-04 2.203249e-04
## [556] 4.069861e-05 1.436076e-05 3.395530e-05 1.598663e-04 1.762207e-04
## [561] 2.250997e-04 9.180989e-05 2.821799e-04 2.446807e-05 3.277631e-04
## [566] 1.260168e-04 1.791021e-04 2.076846e-04 1.036013e-04 7.639324e-05
## [571] 2.631772e-04 1.842612e-04 1.007694e-04 3.021319e-04 1.732017e-04
## [576] 2.388908e-04 1.907147e-04 1.156474e-04 1.450116e-04 7.964916e-05
## [581] 1.874418e-04 7.517103e-05 6.378517e-05 1.326893e-04 1.848303e-04
## [586] 1.620978e-04 1.547059e-04 9.624961e-05 2.013374e-04 1.838587e-04
## [591] 1.416019e-04 1.483187e-04 2.278785e-04 2.797821e-04 1.802229e-04
## [596] 8.275707e-05 1.374963e-04 2.658205e-05 1.528845e-04 1.479316e-04
## [601] 3.570053e-05 1.640987e-04 1.393133e-04 1.653798e-04 1.323250e-04
## [606] 2.361731e-04 2.851618e-05 5.417411e-05 1.218170e-04 1.621222e-04
## [611] 1.512093e-04 5.615374e-05 1.017020e-04 4.408573e-05 3.098507e-04
## [616] 1.828434e-04 2.166958e-04 4.089053e-05 8.698566e-05 9.987184e-05
## [621] 1.406348e-04 1.069597e-04 1.031938e-04 1.596311e-04 1.352494e-04
## [626] 2.463185e-04 1.547058e-04 1.408026e-04 1.287087e-04 6.177302e-05
## [631] 2.513307e-05 8.331530e-05 1.066774e-04 1.401026e-04 1.562464e-04
## [636] 4.685094e-05 3.263645e-05 9.017873e-05 1.730307e-04 1.398373e-04
## [641] 1.703695e-04 1.194761e-04 1.501231e-04 2.359936e-04 7.172977e-05
## [646] 1.555886e-04 3.206132e-05 1.818269e-04 1.643962e-04 1.456264e-04
## [651] 1.339231e-04 2.608970e-04 1.598757e-04 3.546952e-04 1.594526e-04
## [656] 4.204635e-05 6.255905e-05 1.695579e-04 1.840569e-04 2.119318e-04
## [661] 1.056009e-04 4.475459e-05 6.192481e-05 2.015039e-04 2.138949e-04
## [666] 1.853437e-04 1.412270e-04 8.992815e-05 6.172217e-05 1.346264e-04
## [671] 2.889042e-04 9.919352e-05 2.190418e-04 1.328660e-04 1.756623e-04
## [676] 3.027793e-04 6.743996e-05 1.621053e-04 6.105817e-05 1.021113e-04
## [681] 6.037432e-05 5.771149e-05 1.652328e-04 1.533131e-04 2.844827e-04
## [686] 1.117874e-04 1.066948e-04 2.303724e-04 1.966915e-04 8.410877e-05
## [691] 2.034346e-04 1.617877e-04 2.128960e-04 1.655419e-04 6.585632e-05
## [696] 3.415294e-04 5.015088e-05 2.282569e-04 1.898840e-04 1.446605e-04
## [701] 2.206331e-04 1.700125e-04 2.080254e-04 1.474720e-04 1.495571e-04
## [706] 6.638657e-05 3.707759e-05 1.919060e-04 1.575727e-04 2.303330e-04
## [711] 9.274574e-05 7.967078e-05 7.506109e-05 2.005311e-04 2.971166e-04
## [716] 1.109936e-04 7.876200e-05 1.354753e-04 1.761774e-04 1.249168e-04
## [721] 7.135568e-05 1.775077e-04 2.659788e-04 2.094374e-04 2.578158e-04
## [726] 1.644463e-04 1.037744e-04 9.620099e-05 2.221735e-05 8.064278e-05
## [731] 9.688154e-05 1.599954e-04 2.109894e-04 1.253935e-04 5.822402e-05
## [736] 2.351827e-04 1.201093e-04 7.045479e-05 2.331157e-04 1.035206e-04
## [741] 2.736050e-04 2.436672e-04 6.716691e-05 1.771043e-04 6.922159e-05
## [746] 2.426762e-04 1.789060e-04 1.873112e-04 1.647591e-04 1.518514e-04
## [751] 1.661201e-04 1.903517e-04 1.476308e-04 6.864739e-05 6.945627e-05
## [756] 1.284567e-04 1.659406e-04 1.477179e-04 1.023107e-04 7.228411e-05
## [761] 8.469536e-05 1.900003e-04 1.766187e-04 1.580063e-04 7.251550e-05
## [766] 1.059632e-04 1.020653e-04 1.306169e-04 1.506973e-04 9.184862e-05
## [771] 1.366530e-04 1.517021e-04 1.622182e-04 2.580633e-04 6.029151e-05
## [776] 2.027953e-04 1.089377e-04 6.138313e-05 1.087103e-04 6.362243e-05
## [781] 1.163225e-04 1.701511e-04 2.852986e-04 9.924792e-05 7.277013e-05
## [786] 1.599307e-04 2.008516e-04 6.386969e-05 2.743800e-04 9.231370e-05
## [791] 2.385683e-04 1.375971e-04 8.595952e-05 1.895553e-04 8.024781e-05
## [796] 1.485867e-04 1.864530e-04 6.630927e-05 1.503355e-04 1.921168e-04
## [801] 1.829951e-04 1.693238e-04 4.861281e-05 6.883687e-05 7.303012e-05
## [806] 1.642521e-04 1.716780e-04 2.103592e-04 1.069834e-04 3.006351e-04
## [811] 8.150177e-05 2.025365e-04 2.435356e-04 1.646372e-04 1.982833e-04
## [816] 1.126758e-04 9.673525e-05 1.676882e-04 1.718091e-04 8.105415e-05
## [821] 1.388738e-04 2.140339e-04 2.117206e-04 1.724172e-04 1.765696e-04
## [826] 1.989831e-04 9.845234e-05 1.339704e-04 8.613144e-05 7.036372e-05
## [831] 1.262349e-04 1.633645e-04 2.186391e-04 1.103254e-04 6.474034e-05
## [836] 1.058849e-04 1.595737e-04 1.452545e-04 2.104819e-04 1.600096e-04
## [841] 2.367338e-04 1.180885e-04 7.421185e-05 1.921271e-04 8.360701e-05
## [846] 2.152212e-04 1.850912e-04 1.891343e-04 3.008665e-04 1.100121e-04
## [851] 1.206985e-04 1.639852e-04 4.330319e-04 8.486032e-05 2.265817e-04
## [856] 1.639378e-04 2.657349e-04 1.964409e-04 1.354684e-04 6.416088e-05
## [861] 1.318272e-04 2.976136e-04 1.675019e-04 2.073754e-04 1.318798e-04
## [866] 5.542991e-05 1.170271e-04 1.193428e-04 1.938253e-04 1.421178e-04
## [871] 2.721499e-04 1.591713e-04 1.375176e-04 2.105448e-04 1.888874e-04
## [876] 1.610615e-04 2.102990e-04 2.647132e-04 2.145087e-04 4.192495e-05
## [881] 2.799850e-04 1.269613e-04 2.276800e-04 3.304136e-04 4.090445e-04
## [886] 5.340284e-05 2.159166e-04 2.138018e-04 2.144797e-04 1.669590e-04
## [891] 2.582842e-04 3.952921e-05 2.740164e-04 1.200790e-04 1.621644e-04
## [896] 1.799027e-04 1.270129e-04 1.775778e-04 1.155004e-04 1.781412e-04
## [901] 1.547812e-04 1.804734e-04 1.821496e-04 1.186494e-04 1.231241e-04
## [906] 1.649390e-04 1.915019e-04 2.476023e-04 3.713957e-04 1.982369e-04
## [911] 3.068905e-04 1.654211e-04 3.864059e-05 5.036148e-05 2.727822e-04
## [916] 1.549525e-04 1.784313e-04 6.421370e-05 2.707852e-04 1.959230e-04
## [921] 9.957653e-05 1.792626e-04 1.551352e-04 1.003427e-04 2.385880e-04
## [926] 2.258442e-04 2.619818e-05 2.233063e-05 1.034802e-04 6.112492e-04
## [931] 1.033029e-04 1.815960e-04 4.521464e-04 1.780428e-04 3.294666e-05
## [936] 6.918080e-05 6.456265e-05 1.129743e-04 1.853958e-04 2.597985e-04
## [941] 3.014458e-04 2.279719e-04 1.407431e-04 3.372228e-04 2.551736e-04
## [946] 1.873433e-04 1.824640e-04 2.132428e-04 7.613837e-05 1.460802e-04
## [951] 1.784623e-04 2.372892e-04 9.577827e-05 6.412098e-05 7.535485e-05
## [956] 9.108781e-05 1.397252e-04 7.943319e-05 2.014071e-04 1.619970e-04
## [961] 7.614170e-05 6.413677e-05 1.619389e-04 1.844240e-04 1.182063e-04
## [966] 1.509061e-04 1.840071e-04 1.253161e-04 1.375868e-04 6.304518e-05
## [971] 1.967727e-05 1.413873e-04 1.303480e-04 1.774086e-04 7.987896e-05
## [976] 7.808593e-05 6.819958e-05 1.600129e-04 1.793119e-04 1.236112e-04
## [981] 8.916927e-05 9.764716e-05 1.218165e-04 1.760776e-04 7.378521e-05
## [986] 1.453182e-04 2.158529e-04 1.980987e-04 2.621453e-04 1.900644e-04
## [991] 1.411338e-04 1.004770e-04 7.749495e-05 1.041863e-04 1.383494e-04
## [996] 2.366964e-04 1.018183e-04 3.870283e-05 2.016525e-04 1.361078e-04
## [1001] 9.379784e-05 1.775802e-04 1.439670e-04 1.710740e-04 1.696935e-04
## [1006] 5.486315e-05 1.493512e-04 6.466971e-05 1.844067e-04 1.542239e-04
## [1011] 2.468605e-04 1.290732e-04 1.603974e-04 1.655263e-04 1.317201e-04
## [1016] 1.485072e-04 6.064053e-05 2.591741e-05 1.485970e-04 1.605557e-04
## [1021] 2.382226e-04 8.456443e-05 6.777005e-05 6.573821e-05 1.707512e-04
## [1026] 9.125756e-05 2.099616e-04 3.309031e-05 7.512725e-05 9.092649e-05
## [1031] 1.674404e-04 8.459164e-05 7.988128e-05 1.871386e-04 2.224982e-04
## [1036] 2.268056e-04 2.556196e-04 9.187182e-05 2.693158e-04 1.418432e-04
## [1041] 1.532400e-04 1.367565e-04 7.568272e-05 9.551501e-05 1.477690e-04
## [1046] 1.935604e-04 1.177763e-04 4.442921e-05 2.706032e-04 1.905161e-04
## [1051] 1.610080e-04 2.387097e-04 1.637468e-04 1.752823e-04 1.947761e-04
## [1056] 5.863857e-05 2.428089e-04 6.545663e-05 1.700859e-04 1.640869e-04
## [1061] 1.169976e-04 1.839804e-04 2.852283e-04 1.643690e-04 1.408378e-04
## [1066] 7.165697e-05 1.864457e-05 1.999524e-04 1.129365e-04 2.581703e-04
## [1071] 9.057101e-05 9.401224e-05 7.810883e-05 1.758495e-04 1.176656e-04
## [1076] 2.188425e-04 1.027063e-04 1.084545e-04 1.345988e-04 1.195461e-04
## [1081] 7.740704e-05 1.412786e-04 2.671725e-04 2.743529e-04 3.328780e-04
## [1086] 6.450490e-05 3.222493e-04 1.568457e-04 1.837295e-04 2.439405e-05
## [1091] 8.873413e-05 1.207394e-04 1.606198e-04 2.056133e-04 1.007866e-04
## [1096] 3.999577e-05 2.633276e-04 1.659301e-04 1.923009e-04 2.669804e-04
## [1101] 2.106501e-04 2.083108e-04 1.786316e-04 5.987995e-05 1.991206e-04
## [1106] 7.270305e-05 2.242735e-04 1.949232e-04 1.745951e-04 1.368834e-04
## [1111] 1.472969e-04 2.398123e-04 1.283739e-04 1.612079e-04 5.809492e-05
## [1116] 3.002710e-05 1.779551e-04 1.772320e-04 2.808442e-04 8.836241e-05
## [1121] 6.010852e-05 6.245305e-05 1.775801e-04 1.547628e-04 1.447124e-04
## [1126] 1.388117e-04 6.622178e-05 8.387556e-05 2.036743e-04 1.335462e-04
## [1131] 8.210901e-05 2.185764e-04 2.311035e-04 2.244004e-04 2.212289e-04
## [1136] 1.336442e-04 2.557270e-04 1.470660e-04 9.828272e-05 5.800188e-05
## [1141] 8.048290e-05 9.404939e-05 1.491881e-04 1.800809e-04 1.252718e-04
## [1146] 4.421697e-05 2.697820e-04 1.621471e-04 1.529731e-04 1.681857e-04
## [1151] 1.566401e-04 1.910516e-04 2.204380e-04 6.045125e-05 2.127625e-04
## [1156] 6.771578e-05 1.745288e-04 1.429324e-04 1.194326e-04 1.616719e-04
## [1161] 2.386836e-04 1.640551e-04 1.447572e-04 7.919655e-05 1.886263e-05
## [1166] 1.552994e-04 1.027950e-04 1.937070e-04 8.640194e-05 1.069974e-04
## [1171] 8.448777e-05 1.866898e-04 2.332787e-04 1.780663e-04 1.105770e-04
## [1176] 1.104742e-04 1.653498e-04 1.394736e-04 7.996949e-05 1.309025e-04
## [1181] 1.966672e-04 2.449718e-04 1.874495e-04 6.720659e-05 1.984830e-04
## [1186] 1.650658e-04 3.572367e-05 9.392853e-05 1.330138e-04 1.519201e-04
## [1191] 2.323714e-04 9.767027e-05 4.266038e-05 2.108516e-04 1.458746e-04
## [1196] 1.345490e-04 1.750040e-04 1.656520e-04 3.056725e-04 1.928151e-04
## [1201] 6.040964e-05 1.742086e-04 7.321070e-05 1.549626e-04 1.931540e-04
## [1206] 1.560004e-04 1.612194e-04 2.024088e-04 2.445278e-04 1.805227e-04
## [1211] 1.594746e-04 6.908314e-05 6.575074e-05 1.526839e-04 1.545226e-04
## [1216] 2.722131e-04 1.070285e-04 6.014672e-05 8.078039e-05 2.086858e-04
## [1221] 1.011069e-04 3.802610e-04 2.028027e-04 7.469140e-05 1.099688e-04
## [1226] 1.794512e-04 1.103780e-04 8.917672e-05 1.674697e-04 2.029686e-04
## [1231] 2.287147e-04 2.006418e-04 2.049085e-04 2.259336e-04 1.096356e-04
## [1236] 1.842845e-04 6.808754e-06 7.743070e-05 1.096254e-04 1.732911e-04
## [1241] 1.528123e-04 1.117371e-04 6.812981e-05 1.623640e-04 4.363219e-04
## [1246] 2.371722e-04 2.203843e-04 1.562106e-04 2.516994e-04 1.405557e-04
## [1251] 6.645639e-05 3.363974e-04 8.640883e-05 3.157792e-04 1.283346e-04
## [1256] 6.463489e-05 1.302387e-04 1.106743e-04 1.648214e-04 1.389849e-04
## [1261] 1.801938e-04 1.246107e-04 5.190570e-05 8.548171e-05 1.196778e-04
## [1266] 2.339463e-04 1.918369e-04 1.706716e-04 1.324203e-04 1.994206e-04
## [1271] 7.027421e-05 6.458785e-05 7.058019e-05 1.530868e-04 1.406818e-04
## [1276] 1.746438e-04 1.850908e-04 1.379691e-04 1.523023e-04 1.296313e-04
## [1281] 1.837368e-04 4.450649e-04 7.188069e-05 7.715285e-05 2.687238e-04
## [1286] 8.051783e-05 4.642675e-05 1.334074e-04 2.797594e-04 1.330377e-04
## [1291] 1.942088e-04 1.919441e-04 8.438403e-05 1.430328e-04 7.660528e-05
## [1296] 4.807283e-05 2.567197e-04 2.301898e-04 2.141956e-04 2.297569e-04
## [1301] 1.146750e-04 1.496120e-04 1.589189e-04 1.032789e-04 1.536783e-04
## [1306] 1.431482e-04 1.669154e-04 1.714040e-04 1.544397e-04 3.164277e-04
## [1311] 1.621869e-04 6.915727e-05 5.206200e-05 1.490297e-04 1.494346e-04
## [1316] 2.048607e-04 1.064679e-04 6.593062e-05 7.840328e-05 2.203794e-04
## [1321] 1.600969e-04 1.591674e-04 1.335356e-04 6.826831e-05 7.761225e-05
## [1326] 2.093338e-04 2.143623e-04 1.003213e-04 2.039617e-04 1.613228e-04
## [1331] 2.094871e-04 2.453205e-04 8.642435e-05 1.759682e-04 1.353180e-04
## [1336] 1.260281e-04 7.949347e-05 8.776957e-05 9.768099e-05 1.456172e-04
## [1341] 1.752179e-04 1.286089e-04 6.465506e-05 1.943080e-04 1.476937e-04
## [1346] 9.555362e-05 1.956713e-04 1.465834e-04 1.754615e-04 2.596332e-04
## [1351] 7.039714e-05 2.488414e-04 8.322693e-05 2.074111e-04 1.979507e-04
## [1356] 6.599617e-05 1.202042e-04 1.414629e-04 2.214567e-04 1.327016e-04
## [1361] 1.561608e-04 9.799468e-05 3.451303e-05 1.676433e-04 1.047234e-04
## [1366] 2.906767e-04 1.363776e-04 1.200410e-04 1.071961e-04 1.801307e-04
## [1371] 6.715443e-05 1.238032e-04 1.286016e-04 1.946731e-04 1.355461e-04
## [1376] 1.000544e-04 1.490606e-04 1.871125e-04 2.567459e-04 2.292559e-04
## [1381] 9.345885e-05 8.291714e-05 2.291013e-04 2.891976e-05 1.730994e-05
## [1386] 1.235423e-04 1.869689e-04 1.154481e-04 2.254028e-04 1.331918e-04
## [1391] 5.737886e-05 1.875427e-04 1.140515e-04 1.303526e-04 1.538712e-04
## [1396] 1.694446e-04 2.110425e-04 2.233706e-04 8.755852e-05 1.572681e-04
## [1401] 1.089127e-04 1.667387e-04 1.598359e-04 1.902636e-04 2.870000e-04
## [1406] 1.693435e-04 1.300041e-04 1.843009e-04 2.002912e-04 2.758276e-05
## [1411] 5.276889e-05 1.517864e-04 1.357802e-04 1.939085e-04 7.507193e-05
## [1416] 4.646218e-05 4.507010e-05 1.620118e-04 2.042353e-04 1.371572e-04
## [1421] 7.553204e-05 4.950786e-05 1.664182e-05 8.997179e-05 1.431885e-04
## [1426] 1.101718e-04 1.822584e-04 1.484457e-04 1.639991e-04 1.909784e-04
## [1431] 1.694872e-04 2.692416e-04 4.538372e-05 1.099289e-04 4.307816e-05
## [1436] 3.747384e-05 1.463296e-04 1.871605e-04 1.135541e-04 1.094831e-04
## [1441] 1.199907e-04 7.421530e-05 1.556376e-04 1.057700e-04 2.203584e-04
## [1446] 1.239490e-04 1.404905e-04 2.123693e-04 4.505339e-05 1.927361e-04
## [1451] 3.338994e-05 1.895053e-04 1.312568e-04 2.523526e-04 1.371906e-04
## [1456] 1.349709e-04 1.983017e-04 1.080233e-04 2.932194e-04 3.310783e-04
## [1461] 1.018042e-04 1.409140e-04 3.064857e-04 9.744753e-05 1.686422e-04
## [1466] 2.432922e-04 4.219458e-04 2.716798e-04 1.442442e-04 2.241715e-04
## [1471] 2.803142e-04 1.575959e-04 1.006000e-04 1.062969e-04 1.851570e-04
## [1476] 2.181323e-04 7.264178e-05 8.284974e-05 1.147206e-04 2.055340e-05
## [1481] 2.310076e-05 5.586180e-05 5.830018e-04 1.488777e-04 1.937386e-04
## [1486] 2.118810e-04 2.241496e-04 7.725381e-05 6.493588e-05 6.678381e-05
## [1491] 6.489270e-05 2.721966e-04 8.503586e-05 4.041324e-04 5.719050e-04
## [1496] 1.297221e-04 2.852186e-04 1.180906e-04 1.871039e-04 2.085983e-04
## [1501] 1.407179e-04 1.887943e-04 2.071805e-04 2.129196e-04 3.118179e-04
## [1506] 2.065609e-05 1.122056e-05 1.647221e-04 1.724156e-04 2.102534e-04
## [1511] 1.011751e-04 1.637460e-05 2.437536e-05 2.039054e-04 1.774251e-04
## [1516] 2.139142e-04 2.646836e-04 8.194648e-05 1.136027e-04 3.317771e-04
## [1521] 1.477071e-04 8.517767e-05 3.525341e-04 1.183634e-04 4.753023e-04
## [1526] 1.604340e-04 2.670764e-04 1.835463e-04 1.227811e-04 1.458073e-04
## [1531] 1.248964e-04 8.542089e-05 1.182824e-04 1.272079e-04 1.465755e-04
## [1536] 2.187671e-04 1.681128e-05 6.984715e-05 2.075201e-04 2.405513e-04
## [1541] 2.005015e-04 3.260436e-04 1.265528e-04 2.168402e-04 4.099261e-05
## [1546] 1.767799e-04 5.833890e-05 1.354491e-04 1.081179e-04 1.530662e-04
## [1551] 1.518406e-04 1.875147e-04 1.669685e-04 1.699352e-04 9.072236e-05
## [1556] 2.273181e-05 1.391936e-04 1.014008e-04 1.845318e-04 9.769834e-05
## [1561] 1.630535e-04 9.708232e-05 2.130451e-04 1.432523e-04 1.904148e-04
## [1566] 1.379313e-04 1.159368e-04 3.378901e-04 1.357692e-04 8.282974e-05
## [1571] 1.121256e-04 1.709405e-04 2.805398e-04 1.871246e-04 1.499675e-04
## [1576] 1.865430e-04 2.390925e-05 1.041588e-04 2.095405e-04 1.849420e-04
## [1581] 1.622111e-04 8.197464e-05 4.188813e-05 1.651876e-04 1.706226e-04
## [1586] 6.735676e-05 2.097684e-04 1.836610e-04 1.813777e-04 2.852602e-04
## [1591] 5.597575e-05 1.536572e-04 8.286188e-05 1.603279e-04 1.498996e-04
## [1596] 2.053499e-04 1.672566e-04 1.444823e-04 1.667481e-04 2.851599e-04
## [1601] 1.577458e-04 7.152620e-05 5.509011e-05 1.372376e-04 1.483008e-04
## [1606] 1.998863e-04 1.077197e-04 6.920772e-05 8.182922e-05 2.212675e-04
## [1611] 1.886307e-04 1.633947e-04 1.393731e-04 7.217244e-05 8.084448e-05
## [1616] 1.871231e-04 2.639044e-04 1.017281e-04 1.939482e-04 1.460502e-04
## [1621] 2.116570e-04 2.415004e-04 1.107783e-04 1.540341e-04 1.325393e-04
## [1626] 9.854202e-05 7.831863e-05 8.598412e-05 9.876011e-05 1.426747e-04
## [1631] 1.914490e-04 1.253522e-04 6.787889e-05 2.136694e-04 1.678590e-04
## [1636] 7.732881e-05 2.014689e-04 1.488511e-04 1.493407e-04 2.622653e-04
## [1641] 7.244987e-05 2.643131e-04 8.474766e-05 1.900730e-04 2.232216e-04
## [1646] 1.184926e-04 1.520808e-04 2.020672e-04 1.156923e-04 1.564752e-04
## [1651] 9.836642e-05 3.627987e-05 1.577032e-04 1.074681e-04 3.514818e-04
## [1656] 1.431081e-04 1.194603e-04 1.068304e-04 1.786183e-04 2.296681e-04
## [1661] 1.249772e-04 1.235599e-04 1.292448e-04 1.970228e-04 7.877585e-05
## [1666] 1.047289e-04 1.553433e-04 2.166682e-04 2.782650e-04 2.451701e-04
## [1671] 1.000326e-04 1.640722e-04 2.356421e-04 2.412260e-05 1.262811e-04
## [1676] 1.895853e-04 9.344355e-05 2.303790e-04 1.416652e-04 5.744965e-05
## [1681] 1.754811e-04 1.071444e-04 1.519104e-04 1.736709e-04 1.420923e-04
## [1686] 2.148586e-04 2.206454e-04 9.256004e-05 2.025051e-04 1.154123e-04
## [1691] 1.819111e-04 1.737009e-04 1.213144e-04 1.428446e-04 1.906316e-04
## [1696] 1.403142e-04 1.345050e-04 6.424441e-05 1.948812e-05 1.423542e-04
## [1701] 1.284727e-04 1.784976e-04 8.254546e-05 9.335828e-05 7.156703e-05
## [1706] 1.687476e-04 1.594871e-04 1.045836e-04 9.944815e-05 1.015969e-04
## [1711] 1.012737e-04 1.405322e-04 7.322091e-05 1.266119e-04 2.019578e-04
## [1716] 1.946256e-04 2.736539e-04 1.902755e-04 1.318177e-04 8.354675e-05
## [1721] 4.057196e-05 7.689538e-05 1.093241e-04 1.438851e-04 2.100227e-04
## [1726] 9.456863e-05 3.810333e-05 1.844446e-04 1.262758e-04 6.298066e-05
## [1731] 9.144837e-05 1.410875e-04 2.317463e-04 1.638557e-04 5.300180e-05
## [1736] 1.426201e-04 5.555148e-05 1.656609e-04 1.861801e-04 1.575464e-04
## [1741] 1.740453e-04 1.463568e-04 1.854998e-04 1.180220e-04 1.493582e-04
## [1746] 5.565846e-05 1.964902e-05 1.425696e-04 1.320883e-04 1.933171e-04
## [1751] 8.120463e-05 6.067464e-05 5.752312e-05 1.489456e-04 1.836115e-04
## [1756] 1.410919e-04 5.859924e-05 1.537156e-04 1.335863e-04 1.684439e-04
## [1761] 2.535107e-04 5.214047e-05 1.751837e-04 2.259095e-04 2.080474e-04
## [1766] 1.799255e-04 6.693634e-05 1.363769e-04 1.094126e-04 1.087814e-04
## [1771] 2.248290e-04 7.868209e-05 1.012483e-04 1.235250e-04 2.207749e-04
## [1776] 1.029452e-04 3.164952e-05 1.551324e-04 1.087128e-04 1.429037e-04
## [1781] 1.943855e-04 1.591098e-04 1.973160e-04 1.757886e-04 2.165861e-05
## [1786] 1.760929e-04 6.576668e-05 1.943431e-04 1.288497e-04 1.576713e-04
## [1791] 1.944881e-04 1.612177e-04 2.137709e-04 2.045994e-04 2.510540e-04
## [1796] 2.742925e-05 5.708058e-05 1.025188e-04 1.929750e-04 2.796917e-04
## [1801] 1.068107e-04 7.369953e-05 5.325343e-05 1.804409e-04 1.938236e-04
## [1806] 1.636202e-04 1.191122e-04 3.585614e-05 1.828694e-05 8.094655e-05
## [1811] 1.073929e-04 1.223278e-04 1.478750e-04 7.370296e-05 1.606035e-04
## [1816] 1.933566e-04 1.696962e-04 1.849066e-04 4.843937e-05 2.916850e-04
## [1821] 1.073974e-04 2.209719e-05 1.145727e-04 1.354796e-04 2.088991e-04
## [1826] 1.022646e-04 1.338116e-04 1.535880e-04 1.741062e-04 2.524339e-04
## [1831] 2.506016e-04 1.283371e-04 2.115015e-04 1.304204e-04 4.923153e-05
## [1836] 1.839134e-04 3.433291e-05 2.076127e-04 1.563232e-04 2.838229e-04
## [1841] 1.150748e-04 1.442452e-04 2.291188e-04 6.962830e-05 4.529305e-04
## [1846] 5.503098e-04 1.390774e-04 1.004349e-04 1.969109e-04 3.651901e-05
## [1851] 9.425583e-05 2.432924e-04 5.022228e-04 2.128904e-04 1.101475e-04
## [1856] 3.169248e-04 5.962919e-05 7.467290e-05 1.377861e-04 1.253145e-04
## [1861] 1.539179e-04 1.736438e-04 6.689883e-05 1.910750e-04 1.136482e-04
## [1866] 3.806385e-05 3.645905e-04 1.201962e-04 1.580513e-04 1.233586e-04
## [1871] 3.798103e-04 1.027602e-05 6.516897e-05 1.061752e-04 1.938656e-04
## [1876] 3.364805e-04 4.307613e-04 9.019999e-04 1.611099e-04 1.509068e-04
## [1881] 1.230888e-04 5.252792e-05 1.453247e-04 2.620708e-04 1.000323e-04
## [1886] 3.870822e-05 1.696485e-04 5.485825e-05 1.492536e-04 1.849035e-04
## [1891] 1.759316e-04 1.980068e-04 1.802954e-04 2.387699e-04 1.473164e-04
## [1896] 1.632181e-04 2.264379e-04 1.990437e-04 1.849781e-04 2.854390e-05
## [1901] 5.013093e-05 1.447834e-04 1.695248e-04 1.919491e-04 8.139536e-05
## [1906] 5.681237e-05 4.903031e-05 1.578162e-04 2.514440e-04 1.859665e-04
## [1911] -6.225637e-07 6.653618e-05 3.972708e-05 8.523978e-05 1.567866e-04
## [1916] 7.873870e-05 1.553604e-04 1.308420e-04 1.569822e-04 1.679175e-04
## [1921] 1.798131e-04 1.127463e-04 4.201932e-05 1.253764e-04 7.247803e-05
## [1926] 3.198843e-05 1.569172e-04 1.668165e-04 1.418786e-04 1.011149e-04
## [1931] 9.010430e-05 1.756721e-04 1.893330e-04 1.486099e-04 2.850850e-04
## [1936] 1.733536e-04 2.209753e-04 1.548419e-04 5.701573e-05 2.224452e-04
## [1941] 3.386989e-05 1.909410e-04 1.602495e-04 2.176796e-04 1.507248e-04
## [1946] 2.173456e-04 2.172306e-04 1.864500e-04 5.891753e-05 2.871178e-05
## [1951] 6.120544e-05 7.189889e-05 1.733335e-04 1.731107e-04 8.654811e-05
## [1956] 3.162452e-04 4.866895e-05 1.663694e-04 2.936070e-04 2.269198e-04
## [1961] 1.803182e-04 5.740969e-05 3.618613e-05 9.453381e-05 1.580380e-04
## [1966] 8.830215e-05 1.457609e-04 1.246841e-04 2.368806e-04 1.753267e-04
## [1971] 1.759440e-04 2.507506e-04 4.412892e-05 1.662412e-04 1.200902e-04
## [1976] 3.311091e-05 1.580210e-04 1.324812e-04 2.128280e-04 1.142379e-04
## [1981] 1.081387e-04 2.074448e-04 2.408316e-04 1.590961e-04 1.663843e-04
## [1986] 1.534412e-04 2.092281e-04 1.477968e-04 6.424999e-05 3.453636e-04
## [1991] 3.308123e-05 1.373877e-04 1.645681e-04 2.222131e-04 1.475037e-04
## [1996] 2.438331e-04 2.655168e-04 2.348412e-04 1.565678e-04 3.268465e-05
## [2001] 5.317186e-05 1.609715e-04 1.930109e-04 2.412740e-04 9.279937e-05
## [2006] 5.721017e-05 5.467618e-05 1.736917e-04 1.871034e-04 1.899492e-04
## [2011] 1.928720e-07 8.162878e-05 4.741179e-05 9.559814e-05 1.480383e-04
## [2016] 8.611140e-05 1.667938e-04 1.754972e-04 1.614232e-04 2.271505e-04
## [2021] 2.906649e-05 1.846894e-04 4.754940e-05 2.045871e-04 2.840599e-05
## [2026] 4.149368e-05 1.258959e-04 1.277903e-04 2.158477e-04 9.753064e-05
## [2031] 1.036041e-04 2.106305e-04 1.777378e-04 1.536261e-04 2.302478e-04
## [2036] 1.605356e-04 1.857264e-04 1.696491e-04 6.029248e-05 2.956663e-04
## [2041] 3.928944e-05 5.950205e-04 1.465256e-04 1.563381e-04 1.480298e-04
## [2046] 2.095396e-04 3.237685e-04 1.957896e-04 5.978216e-05 2.871100e-05
## [2051] 6.858256e-05 1.201128e-04 1.742292e-04 2.323373e-04 9.032709e-05
## [2056] 3.139385e-04 4.863555e-05 1.806224e-04 1.986052e-04 1.754403e-04
## [2061] 1.116905e-04 5.271716e-05 3.591617e-05 1.032256e-04 1.955091e-04
## [2066] 9.537377e-05 1.590405e-04 1.432304e-04 2.022700e-04 1.381954e-04
## [2071] 1.657301e-04 2.024228e-04 4.626864e-05 1.156671e-04 1.193300e-04
## [2076] 3.616917e-05 1.490248e-04 1.430875e-04 1.815685e-04 1.257890e-04
## [2081] 1.190701e-04 1.703117e-04 2.106177e-04 1.777422e-04 1.569355e-04
## [2086] 1.676925e-04 2.155201e-04 1.475729e-04 6.788087e-05 2.565803e-04
## [2091] 3.479403e-05 1.428899e-04 1.564651e-04 2.761616e-05 1.593108e-04
## [2096] 1.461292e-04 2.671325e-04 2.242734e-04 1.526778e-04 3.343319e-05
## [2101] 5.587518e-05 1.630683e-04 1.985940e-04 1.959416e-04 9.961201e-05
## [2106] 6.240539e-05 5.635132e-05 1.913844e-04 2.012298e-04 1.481312e-04
## [2111] -1.377571e-06 9.452451e-05 4.854182e-05 1.030509e-04 1.284203e-04
## [2116] 9.044437e-05 1.739422e-04 1.724215e-04 1.320201e-04 2.258886e-04
## [2121] 2.994737e-05 2.335998e-04 4.837502e-05 1.125100e-04 2.860025e-05
## [2126] 4.685412e-05 1.863414e-04 1.235519e-04 2.361444e-04 9.340858e-05
## [2131] 7.636573e-05 2.220299e-04 1.122322e-04 6.481188e-05 2.582721e-04
## [2136] 1.583466e-04 2.470095e-04 1.666567e-04 6.127455e-05 3.277850e-04
## [2141] 4.016937e-05 5.834808e-04 1.140037e-04 1.670951e-04 1.965260e-04
## [2146] 1.179431e-04 1.611350e-04 1.790447e-04 6.371867e-05 5.152613e-05
## [2151] 1.691715e-04 1.431747e-04 2.398852e-04 1.036838e-04 1.522528e-04
## [2156] 4.345131e-04 7.654634e-05 2.719292e-05 2.185703e-04 1.613758e-04
## [2161] 1.913858e-04 4.885429e-05 5.766603e-05 1.115946e-04 1.476242e-04
## [2166] 1.405200e-04 1.942906e-04 7.031137e-05 1.244074e-04 2.084489e-04
## [2171] 1.726775e-04 1.350223e-04 9.924505e-05 1.503834e-04 1.529284e-04
## [2176] 4.588606e-05 2.067585e-04 1.511591e-04 1.555005e-04 1.937814e-04
## [2181] 1.933965e-04 8.665134e-05 1.936912e-04 1.284960e-04 3.779588e-04
## [2186] 9.563465e-05 1.823176e-04 6.591767e-05 1.301477e-04 7.230147e-05
## [2191] 6.993095e-05 2.302670e-04 1.281466e-04 -1.587467e-06 1.667938e-04
## [2196] 1.403752e-04 1.829554e-04 1.992923e-04 1.486488e-04 6.474220e-05
## [2201] 7.737960e-05 1.631407e-04 1.847409e-04 1.935971e-04 1.086259e-04
## [2206] 6.855243e-05 8.215386e-05 1.902320e-04 1.458138e-04 1.910529e-04
## [2211] -4.018041e-06 1.170695e-04 7.839910e-05 1.400522e-04 1.154224e-04
## [2216] 8.103533e-05 1.538060e-04 1.952958e-04 1.515595e-04 2.874759e-04
## [2221] 3.320301e-05 3.223524e-05 7.951031e-05 1.098425e-04 2.576561e-05
## [2226] 7.159388e-05 1.254035e-04 2.017844e-04 2.241975e-04 1.039263e-04
## [2231] 7.907077e-05 2.581440e-04 1.723751e-04 7.260686e-05 2.921186e-04
## [2236] 2.083187e-04 2.752234e-04 1.751426e-04 9.283001e-05 1.911334e-04
## [2241] 7.486772e-05 1.920200e-04 1.493382e-04 1.924575e-04 1.705235e-04
## [2246] 1.492827e-04 1.385809e-04 2.251021e-04 3.187149e-04 4.060251e-05
## [2251] 1.101386e-04 1.523804e-04 1.996618e-04 2.350437e-04 1.262261e-04
## [2256] 5.719945e-05 6.204187e-05 2.649208e-04 1.810930e-04 1.888109e-04
## [2261] 1.386151e-04 4.778611e-05 4.299704e-05 1.067196e-04 1.215606e-04
## [2266] 1.223780e-04 2.521422e-04 1.735496e-04 1.215465e-04 1.952692e-04
## [2271] 2.414595e-04 1.343038e-04 6.396910e-05 1.254092e-04 4.409798e-05
## [2276] 3.948078e-05 1.811675e-04 1.468670e-04 1.600402e-04 1.490789e-04
## [2281] 1.536003e-04 1.748633e-04 2.281590e-04 1.751264e-04 2.218521e-04
## [2286] 1.655748e-04 1.997293e-04 1.124636e-04 9.452647e-05 2.186486e-04
## [2291] 4.927286e-05 2.246729e-04 1.522842e-04 2.634457e-05 1.585361e-04
## [2296] 1.654152e-04 2.340656e-04 2.251840e-04 1.502193e-04 5.482391e-05
## [2301] 7.338822e-05 1.831053e-04 2.101561e-04 2.158101e-04 1.203562e-04
## [2306] 6.148111e-05 7.529653e-05 2.167145e-04 2.093810e-04 1.408970e-04
## [2311] -3.192431e-07 1.171817e-04 6.876355e-05 1.523756e-04 1.254798e-04
## [2316] 9.785701e-05 1.925718e-04 1.578104e-04 1.465760e-04 1.457046e-04
## [2321] 2.970065e-05 1.478266e-04 6.797548e-05 1.117859e-04 2.884480e-05
## [2326] 7.148995e-05 1.827709e-04 2.156016e-04 2.620963e-04 1.094964e-04
## [2331] 6.772713e-05 3.013392e-04 1.135460e-04 6.938433e-05 2.215584e-04
## [2336] 1.712825e-04 3.077670e-04 1.911845e-04 6.742523e-05 2.011218e-04
## [2341] 6.415593e-05 1.792875e-04 1.661563e-04 1.685487e-04 4.124600e-04
## [2346] 1.145356e-04 1.775541e-04 1.124395e-04 7.803147e-05 1.324618e-04
## [2351] 1.133880e-04 1.453953e-04 2.636872e-04 1.743556e-04 9.923381e-05
## [2356] 3.072592e-04 1.841288e-04 4.726179e-04 2.189451e-04 1.537645e-04
## [2361] 1.245721e-04 5.026207e-05 3.366589e-04 2.460089e-04 2.170022e-04
## [2366] 1.177315e-04 4.371597e-04 1.269840e-04 2.554937e-04 1.485521e-04
## [2371] 2.232446e-04 2.402183e-04 5.932456e-04 2.322413e-04 1.469930e-04
## [2376] 4.765763e-05 2.424755e-04 1.516490e-04 1.169894e-04 3.474130e-04
## [2381] 4.116117e-04 6.689579e-06 1.547814e-04 2.113895e-04 1.675391e-04
## [2386] 1.872628e-04 1.847477e-04 7.940545e-05 1.661839e-04 1.086757e-04
## [2391] 3.546086e-04 2.437421e-04 1.486485e-04 1.934719e-05 2.730850e-04
## [2396] 1.539518e-04 2.108749e-04 1.175122e-04 1.339769e-04 6.338640e-05
## [2401] 1.898597e-05 1.391297e-04 1.322554e-04 1.656218e-04 7.373470e-05
## [2406] 9.129139e-05 7.022194e-05 1.662084e-04 1.956893e-04 1.399025e-04
## [2411] 2.314733e-04 1.368575e-04 1.069937e-04 1.330758e-04 5.561061e-05
## [2416] 1.275452e-04 2.068360e-04 1.572352e-04 2.811266e-04 2.581197e-05
## [2421] 1.542242e-04 1.007572e-04 1.009312e-04 2.525838e-05 8.577884e-05
## [2426] 2.536234e-04 1.366499e-04 1.846328e-04 9.646378e-05 1.719493e-05
## [2431] 2.025073e-04 1.070488e-04 8.935437e-05 1.213996e-04 1.640490e-04
## [2436] 2.358341e-04 2.458292e-04 1.395659e-05 1.608571e-04 5.891413e-05
## [2441] 2.167727e-04 1.182220e-04 1.454731e-04 1.692572e-04 9.408349e-05
## [2446] 1.961659e-04 2.052167e-04 5.887096e-05 1.757049e-05 6.986198e-05
## [2451] 9.973852e-05 1.863420e-04 1.381974e-04 1.082231e-04 2.954082e-04
## [2456] 5.318611e-05 5.565238e-05 1.350841e-04 1.853041e-04 1.716828e-04
## [2461] 3.783030e-05 3.350230e-05 1.038405e-04 1.763964e-04 1.110364e-04
## [2466] 3.813655e-04 1.151809e-04 2.604509e-04 1.131505e-04 2.220118e-04
## [2471] 2.430867e-04 5.269646e-05 1.859333e-04 1.012583e-04 4.285559e-05
## [2476] 1.220468e-04 1.860993e-04 1.455255e-04 1.302511e-04 1.634604e-04
## [2481] 1.340890e-04 2.073286e-04 1.064530e-04 1.217302e-04 1.085592e-04
## [2486] 1.089460e-04 1.480098e-04 7.070167e-05 1.256646e-04 2.269119e-05
## [2491] 1.969645e-04 1.614923e-04 -1.685553e-06 1.213990e-04 2.348191e-04
## [2496] 1.624062e-04 1.254331e-04 2.200572e-04 1.095121e-05 6.197529e-05
## [2501] 1.649809e-04 1.834974e-04 1.898754e-04 9.474197e-05 9.733806e-05
## [2506] 3.538488e-05 2.971710e-04 2.119019e-04 2.390266e-04 1.320353e-07
## [2511] 8.120068e-05 5.054703e-05 9.236442e-05 1.568906e-04 1.106894e-04
## [2516] 1.925281e-04 1.228703e-04 1.636388e-04 2.108005e-04 2.752958e-05
## [2521] 1.611397e-04 6.955088e-05 9.430464e-05 2.719124e-05 5.646100e-05
## [2526] 1.317727e-04 1.553651e-04 1.512633e-04 4.423865e-05 7.463017e-05
## [2531] 1.950116e-04 2.404316e-04 2.690301e-04 3.455796e-04 1.529131e-04
## [2536] 1.758707e-04 1.534313e-04 7.170487e-05 1.411694e-04 1.175314e-05
## [2541] 2.299859e-04 1.394630e-04 1.949948e-04 1.767944e-04 1.477265e-04
## [2546] 1.658567e-04 2.065795e-04 3.190333e-04 4.254893e-05 1.141952e-04
## [2551] 1.582077e-04 2.092486e-04 2.492459e-04 1.319875e-04 5.940778e-05
## [2556] 6.486268e-05 2.689247e-04 2.024909e-04 1.861217e-04 1.386955e-04
## [2561] 4.904877e-05 4.446155e-05 1.029511e-04 1.433221e-04 1.232287e-04
## [2566] 2.309110e-04 1.539054e-04 1.056444e-04 2.103853e-04 2.625912e-04
## [2571] 1.412722e-04 6.693312e-05 1.311242e-04 6.839509e-05 3.899001e-05
## [2576] 1.915115e-04 1.510387e-04 1.762939e-04 1.493486e-04 1.562205e-04
## [2581] 1.666392e-04 2.252532e-04 1.743399e-04 2.130083e-04 1.666316e-04
## [2586] 1.913319e-04 1.116650e-04 9.915232e-05 2.267846e-04 5.180488e-05
## [2591] 1.872389e-04 1.446884e-04 1.550026e-04 1.519160e-04 1.635301e-04
## [2596] 1.747257e-04 2.292563e-04 1.505375e-04 5.839533e-05 7.039839e-05
## [2601] 1.371281e-04 2.051303e-04 2.192428e-04 1.207473e-04 6.023569e-05
## [2606] 7.796874e-05 2.134094e-04 2.117052e-04 1.224555e-04 3.116048e-05
## [2611] 1.150746e-04 7.115057e-05 1.628501e-04 1.046803e-04 9.172008e-05
## [2616] 1.936426e-04 1.419771e-04 1.589633e-04 1.401722e-04 1.905177e-04
## [2621] 1.282897e-04 7.132998e-05 1.610227e-04 6.573378e-05 6.607855e-05
## [2626] 1.710806e-04 1.412739e-04 2.343721e-04 1.135134e-04 7.036590e-05
## [2631] 2.440097e-04 1.140472e-04 1.042561e-04 2.052379e-04 1.676733e-04
## [2636] 3.067927e-04 1.854600e-04 6.884264e-05 2.088313e-04 6.809052e-05
## [2641] 1.450432e-04 1.744883e-04 1.312175e-04 1.926758e-04 1.442152e-04
## [2646] 1.724576e-04 2.135041e-04 1.729489e-04 2.530528e-05 4.155449e-05
## [2651] 1.192160e-04 1.663176e-04 2.288422e-04 8.744926e-05 6.767335e-05
## [2656] 4.722260e-05 1.620007e-04 1.782082e-04 1.582352e-04 5.395289e-05
## [2661] 2.486940e-05 7.470451e-05 1.656038e-04 1.011381e-04 1.485876e-04
## [2666] 7.281025e-05 1.665432e-04 1.945783e-04 1.250258e-04 1.399164e-04
## [2671] 4.457115e-05 1.332852e-04 5.766671e-05 2.230364e-05 1.089002e-04
## [2676] 1.744244e-04 1.872109e-04 8.850869e-05 1.047714e-04 1.542848e-04
## [2681] 1.870639e-04 1.590693e-04 3.009481e-04 1.715558e-04 1.926692e-04
## [2686] 1.433667e-04 4.880679e-05 2.083859e-04 3.056549e-05 1.346651e-04
## [2691] 1.670932e-04 2.646869e-04 1.405847e-04 1.742918e-04 2.329811e-04
## [2696] 1.791195e-04 7.260299e-05 2.683100e-05 5.370399e-05 8.453692e-05
## [2701] 1.461040e-04 1.447166e-04 7.469324e-05 2.979600e-04 2.570348e-05
## [2706] 1.073752e-04 2.181420e-04 1.655188e-04 6.994589e-05 5.908149e-05
## [2711] 4.087460e-05 7.378576e-05 1.363623e-04 8.326194e-05 1.486021e-04
## [2716] 5.524380e-05 1.522786e-04 1.397628e-04 1.937049e-04 1.435200e-04
## [2721] 4.360861e-05 8.687079e-05 4.279022e-05 3.711704e-05 1.336719e-04
## [2726] 1.679812e-04 1.485813e-04 7.625247e-05 9.753395e-05 1.757147e-04
## [2731] 1.688077e-04 1.404025e-04 1.984929e-04 1.175770e-04 2.070660e-04
## [2736] 1.368996e-04 5.770297e-05 1.363879e-04 3.244551e-05 2.150135e-04
## [2741] 1.635986e-04 1.839438e-04 3.516212e-04 1.729824e-04 1.454682e-04
## [2746] 1.084947e-04 1.298605e-04 1.842945e-04 4.210137e-05 4.275376e-05
## [2751] 1.886804e-04 8.757725e-05 7.450096e-05 1.800709e-04 2.633199e-04
## [2756] 3.469235e-04 1.856019e-04 2.010656e-04 1.515102e-04 6.663092e-05
## [2761] 5.433687e-04 4.239010e-04 1.043051e-04 5.539572e-05 2.873554e-04
## [2766] 1.807625e-04 1.614541e-04 7.030320e-05 2.800721e-04 1.398499e-04
## [2771] 8.853524e-04 1.712061e-04 1.390625e-04 3.431961e-05 2.137645e-04
## [2776] 1.537436e-04 1.552269e-04 1.858051e-04 1.746954e-04 3.027544e-05
## [2781] 2.149201e-04 1.565289e-04 1.801301e-04 1.724269e-04 1.973420e-04
## [2786] 9.496221e-05 9.026357e-05 1.879282e-04 6.523389e-04 2.011098e-04
## [2791] 9.356808e-05 4.181295e-05 2.723639e-04 1.398635e-04 1.679504e-04
## [2796] 1.855866e-04 1.474415e-04 8.211097e-05 2.117562e-05 1.287272e-04
## [2801] 8.349118e-05 1.527736e-04 9.274151e-05 1.122656e-04 8.904013e-05
## [2806] 1.890483e-04 1.811920e-04 1.835687e-04 4.008480e-05 2.684448e-04
## [2811] 1.513247e-04 1.869240e-04 1.227304e-04 5.919549e-05 1.304258e-04
## [2816] 1.671549e-04 2.383169e-04 1.543297e-04 9.789526e-05 8.178008e-05
## [2821] 1.273356e-04 1.042202e-04 1.964417e-05 9.869718e-05 1.419510e-04
## [2826] 1.342921e-04 2.275180e-04 1.005554e-04 4.419362e-05 1.742822e-04
## [2831] 1.452170e-04 1.101396e-04 1.525019e-04 1.514152e-04 3.277273e-04
## [2836] 3.055427e-04 2.660426e-05 1.890519e-04 7.760667e-05 1.717152e-04
## [2841] 1.920388e-04 8.140004e-05 1.567178e-04 1.679644e-04 1.550973e-04
## [2846] 5.718039e-05 2.210805e-04 1.911678e-04 7.874069e-05 1.254052e-04
##
## $itercosts
## [1] 74.2483712 59.7048183 56.2543478 54.6626336 53.7124305 1.1738555
## [7] 0.8175787 0.6469518 0.5556898 0.5137847 0.4934865 0.4822530
## [13] 0.4726022 0.4626787 0.4535888 0.4454414 0.4419298 0.4387055
## [19] 0.4341145 0.4290951
##
## $origD
## [1] 14
##
## $perplexity
## [1] 30
##
## $theta
## [1] 0.5
##
## $max_iter
## [1] 1000
##
## $stop_lying_iter
## [1] 250
##
## $mom_switch_iter
## [1] 250
##
## $momentum
## [1] 0.5
##
## $final_momentum
## [1] 0.8
##
## $eta
## [1] 200
##
## $exaggeration_factor
## [1] 12
##
## attr(,"class")
## [1] "Rtsne" "list"# Plot t-SNE results
plot(tsne_result$Y, col = churn_variable, pch = 16, main = "t-SNE Plot")
# t-SNE
set.seed(123) # for reproducibility
tsne_result <- Rtsne(as.matrix(churn_data_unique), perplexity = 30, verbose = TRUE)## Performing PCA
## Read the 2850 x 14 data matrix successfully!
## OpenMP is working. 1 threads.
## Using no_dims = 2, perplexity = 30.000000, and theta = 0.500000
## Computing input similarities...
## Building tree...
## Done in 0.83 seconds (sparsity = 0.036200)!
## Learning embedding...
## Iteration 50: error is 75.285590 (50 iterations in 0.79 seconds)
## Iteration 100: error is 60.052469 (50 iterations in 0.71 seconds)
## Iteration 150: error is 56.754013 (50 iterations in 0.73 seconds)
## Iteration 200: error is 55.163459 (50 iterations in 0.73 seconds)
## Iteration 250: error is 54.185454 (50 iterations in 0.75 seconds)
## Iteration 300: error is 1.176535 (50 iterations in 0.72 seconds)
## Iteration 350: error is 0.810029 (50 iterations in 0.72 seconds)
## Iteration 400: error is 0.643256 (50 iterations in 0.73 seconds)
## Iteration 450: error is 0.558772 (50 iterations in 0.73 seconds)
## Iteration 500: error is 0.517060 (50 iterations in 0.75 seconds)
## Iteration 550: error is 0.491279 (50 iterations in 0.75 seconds)
## Iteration 600: error is 0.474283 (50 iterations in 0.74 seconds)
## Iteration 650: error is 0.462475 (50 iterations in 0.74 seconds)
## Iteration 700: error is 0.455913 (50 iterations in 0.73 seconds)
## Iteration 750: error is 0.449720 (50 iterations in 0.75 seconds)
## Iteration 800: error is 0.443919 (50 iterations in 0.71 seconds)
## Iteration 850: error is 0.439676 (50 iterations in 0.71 seconds)
## Iteration 900: error is 0.434297 (50 iterations in 0.76 seconds)
## Iteration 950: error is 0.431238 (50 iterations in 0.72 seconds)
## Iteration 1000: error is 0.428254 (50 iterations in 0.73 seconds)
## Fitting performed in 14.69 seconds.# Extract PC values from PCA
pca_values <- pca_result$ind$coord
# Extract t-SNE values
tsne_values <- tsne_result$Y
# Compare the first few rows of the results
head(cbind(pca_values, tsne_values))## [,1] [,2]
## [1,] 18.614136 8.872614
## [2,] 24.745942 -4.690122
## [3,] -8.643111 -7.377694
## [4,] 15.406661 6.206287
## [5,] -9.449942 -32.690409
## [6,] 13.121266 -3.646368FURTHER ANALYSIS
As a further analysis, I have implemented kmeans clustering and fulfilling prediction which based on linear regression model
This study benefits greatly from these analyses, which provides you a thorough grasp of customer turnover patterns and the capacity to take preemptive measures to deal with possible churners.
CLUSTERING ANALYSIS
Objective: Cluster customers based on their patterns in the first three principal components.
Insights: Explore whether distinct groups with similar characteristics emerge, providing a basis for targeted marketing or retention strategies.
Interpretation: Analyze the characteristics of each cluster, and assess if certain clusters exhibit higher or lower churn rates.
# 'pca_result' is the PCA result and 'num_clusters' is the desired number of clusters
set.seed(123)
num_clusters <- 3
# Extract principal components for clustering
cluster_data <- pca_result$x[, 1:3]
# K-means clustering
kmeans_result <- kmeans(cluster_data, centers = num_clusters, nstart = 25)
kmeans_result$centers## PC1 PC2 PC3
## 1 1.8366758 -0.3908433 -0.1816500
## 2 -0.7905665 1.1431352 0.3041761
## 3 -2.7425785 -1.6063814 -0.2364089pc_tsne_data <- as.data.frame(cbind(cluster_data, Cluster = as.factor(kmeans_result$cluster)))
pc_tsne_data## PC1 PC2 PC3 Cluster
## 1 0.278668065 -0.1493528992 0.2037917160 1
## 2 2.593157624 0.9275905920 -0.1658973346 1
## 3 -1.063556353 1.7450785209 1.5121876912 2
## 4 -0.135891794 1.7827514495 -1.5164815731 2
## 5 0.473523158 2.0028585552 -1.0889551748 2
## 6 -0.393953022 -0.3572063342 0.0306080244 2
## 7 -0.238246072 1.5738899847 1.6380067037 2
## 8 -2.387177095 -0.0696975796 -0.0131307403 3
## 9 -2.127255495 -0.3767770815 -0.5487794331 3
## 10 -0.148512763 -0.3770573053 0.1191051178 2
## 11 0.090544343 -0.0587968697 0.2405405470 2
## 12 1.357310515 -0.3708000345 -0.2884187840 1
## 13 -3.965415152 -1.1428495291 -1.3604156482 3
## 14 -1.022578358 2.5994859464 2.5187860624 2
## 15 -2.889646121 1.4978900364 -0.5962539533 2
## 16 -1.018195385 0.7952054034 0.9822076412 2
## 17 3.022345945 0.5321341080 -1.3292021814 1
## 18 2.290181560 -0.0270248952 0.3281151697 1
## 19 3.475178432 0.6125537166 -1.1646160099 1
## 20 -2.546059478 2.9851909657 0.8827023096 2
## 21 -1.936332690 -2.1311566180 0.5710212271 3
## 22 -1.126071963 1.0897791651 -0.2457085463 2
## 23 2.104852244 -0.6698357702 -4.0837107133 1
## 24 -3.323027023 -5.0996957041 -0.9738124879 3
## 25 0.351670774 -0.3311988240 -3.3603131926 1
## 26 1.215593802 0.1288858574 0.7973152482 1
## 27 2.073254949 -0.6365533877 -1.2680225322 1
## 28 2.115339042 -1.1262519646 -2.8624775979 1
## 29 3.398902078 -0.2899538516 -0.5458804777 1
## 30 1.497677389 0.1611985338 0.8654087654 1
## 31 -0.226660693 -2.9732337357 2.1849747910 3
## 32 3.630944011 -2.2874624120 -1.0573572027 1
## 33 1.713502440 -2.6271331610 2.9613482914 1
## 34 -5.622340176 -0.3614485815 -2.9332818929 3
## 35 -3.007255551 2.2042028281 0.1016514340 2
## 36 0.295672895 0.8503454209 -0.2432498210 2
## 37 1.570231858 -0.0090270303 0.1750308861 1
## 38 -0.172069088 1.8155898224 0.4232152996 2
## 39 -0.366077378 2.3716580612 0.5514165237 2
## 40 -0.087255425 -1.6864240658 1.3652390925 1
## 41 1.970239226 0.6231243493 -0.6820196933 1
## 42 3.228237077 -1.6590723287 0.6290863830 1
## 43 0.992894187 0.8888526283 -0.1058085285 1
## 44 0.674414838 0.0888215541 0.6467103822 1
## 45 1.278706123 1.2359331875 0.3981822912 1
## 46 -2.386989385 1.1254247075 -1.1210306606 2
## 47 -1.664921963 -1.6720366514 0.8343903344 3
## 48 -0.563915205 0.0189717741 0.2848720674 2
## 49 0.042243956 1.2815282555 1.7695360623 2
## 50 2.124901275 -0.2510733333 -0.0065262545 1
## 51 -0.143850800 -0.4190352341 0.0187951457 2
## 52 1.422329643 -0.0259447112 0.8429402592 1
## 53 -1.486075218 1.4753961860 1.3271911209 2
## 54 -0.563371225 1.5187584345 -1.6988224576 2
## 55 0.053597824 0.5211777217 -0.4336648809 2
## 56 -1.445207214 -0.1968223570 -0.4770535150 2
## 57 -0.660764937 1.3042076498 1.4530101334 2
## 58 -2.809695959 -0.3393799144 -0.1981273106 3
## 59 -2.549774360 -0.6464594164 -0.7337760034 3
## 60 -0.571031628 -0.6467396402 -0.0658914525 2
## 61 -0.331974522 -0.3284792046 0.0555439767 2
## 62 0.191977388 -0.4278719922 0.0854629312 1
## 63 -4.387934017 -1.4125318640 -1.5454122185 3
## 64 -1.445097223 2.3298036115 2.3337894921 2
## 65 -3.313818508 1.2301041415 -0.7803652950 2
## 66 -1.440714250 0.5255230686 0.7972110709 2
## 67 1.181440036 0.7897288124 0.0952680205 1
## 68 1.124848433 -0.0840968529 0.7019968849 1
## 69 2.718542661 0.0394508360 -2.3422280858 1
## 70 -2.841873659 1.6674244054 1.2618873568 2
## 71 -2.982626316 -1.9764619606 0.0607040020 3
## 72 -1.550244350 0.8219932702 -0.4298198881 2
## 73 0.938661761 -1.6286898632 -3.0773558476 1
## 74 -3.737278277 -5.3788602388 -1.1632352011 3
## 75 -1.156771751 0.2317128659 -0.9423422565 2
## 76 0.793074937 -0.1407964775 0.6123186778 1
## 77 0.907921822 -0.6936253454 -0.8941408170 1
## 78 0.950005914 -1.1833239223 -2.4885958827 1
## 79 2.233568951 -0.3470258093 -0.1719987625 1
## 80 1.080885748 -1.6303348698 1.7286825861 1
## 81 -0.640911947 -3.2523982704 1.9955520778 3
## 82 2.060220572 -1.8322963268 0.8662478466 1
## 83 1.299251186 -2.9062976956 2.7719255782 1
## 84 -5.417777235 -1.0593007886 -2.7947282656 3
## 85 -2.927533865 0.4084385662 -1.0529039957 3
## 86 -0.123597422 -0.3258281489 0.2020286249 2
## 87 0.404898731 -0.0660989881 0.5489126013 1
## 88 -0.596241475 1.5478039275 0.2391039579 2
## 89 -0.736443969 1.1393963533 0.9705139019 2
## 90 -0.504813724 -1.9617957206 1.1775868364 3
## 91 0.801473008 -0.3327756091 0.3257141164 1
## 92 1.393945257 -1.4090633842 2.0500156706 1
## 93 0.568721800 0.6210667334 -0.2899198702 2
## 94 0.251895973 -0.1808607808 0.4617138119 2
## 95 0.854533736 0.9681472925 0.2140709495 2
## 96 -2.811161772 0.8576388126 -1.3051420023 2
## 97 -1.994141474 -3.2577488377 1.4411204160 3
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## 1695 1.771383804 1.1964026462 0.5167465109 1
## 1696 -1.582541505 2.5593913742 -1.7060861434 2
## 1697 -1.057775652 -1.6244330867 1.0841705954 3
## 1698 1.073606247 -0.0089088754 0.8810860215 1
## 1699 0.713027956 2.3331529574 1.2541491259 2
## 1700 3.147421907 0.4651848556 -1.9048442147 1
## 1701 0.424838592 0.8750734217 -0.2381798769 2
## 1702 2.762137766 0.9791636493 -0.0783799644 1
## 1703 -1.015692564 2.8822537553 1.1228455318 2
## 1704 0.015821067 1.8980557724 -1.3294061489 2
## 1705 0.625236019 2.1181628780 -0.9018797507 2
## 1706 -0.056049272 0.8509455254 -0.4015429095 2
## 1707 -0.085828335 1.6883858784 1.8247047658 2
## 1708 -2.036811477 0.2213962030 0.1800609050 2
## 1709 -1.974837758 -0.2622811877 -0.3620813711 3
## 1710 0.201852855 -0.0859635228 0.3122967631 2
## 1711 0.231510535 0.9715983527 -0.1986448558 2
## 1712 1.920079131 -0.7742315035 -1.6540997417 1
## 1713 -3.615049534 -0.8517557466 -1.1672240029 3
## 1714 -0.672212740 2.8905797290 2.7119777077 2
## 1715 -2.737463342 1.6126554065 -0.4094301039 2
## 1716 -0.667829767 1.0862991860 1.1753992865 2
## 1717 2.383063823 0.8651000855 -0.2412967565 1
## 1718 2.405737390 0.9560524386 -0.1679835059 1
## 1719 3.480186845 0.6349819483 -1.1366371682 1
## 1720 -2.393876700 3.0999563358 1.0695261590 2
## 1721 -1.683883296 -0.2124793443 -0.2347032735 2
## 1722 -0.775941304 1.3811424240 -0.0523911137 2
## 1723 1.846684144 -0.0371430372 -2.3445078441 1
## 1724 -2.804675781 -3.4539140363 0.2797292938 3
## 1725 0.291450555 0.4780917977 -1.6146167402 1
## 1726 1.289647778 0.1803353413 0.8849321885 1
## 1727 2.222391830 0.3844707940 -1.7115821906 1
## 1728 2.267756779 -1.0117560709 -2.6757795359 1
## 1729 3.479125883 -0.2534772332 -0.4904570298 1
## 1730 1.571731365 0.2126480177 0.9530257057 1
## 1731 0.087086431 -1.4311179076 1.5560951548 1
## 1732 3.256170105 -1.7504171993 0.5327208345 1
## 1733 2.029515693 -2.4107093792 3.0356655406 1
## 1734 -4.643474189 -0.5001516347 -2.4172994912 3
## 1735 -2.657124891 2.4955660870 0.2949688666 2
## 1736 0.408673792 0.9335875861 -0.1059665325 2
## 1737 1.722649595 0.1054688634 0.3617289481 1
## 1738 -0.019886310 1.9303551925 0.6100391490 2
## 1739 -0.015946719 2.6630213201 0.7447339564 2
## 1740 0.065867188 -1.5727366012 1.5515597924 1
## 1741 2.122422004 0.7378897194 -0.4951958439 1
## 1742 2.589894789 -1.3271842567 1.7164886584 1
## 1743 1.145076965 1.0036179984 0.0810153210 1
## 1744 0.787650694 0.1717942429 0.7838678834 1
## 1745 1.391707021 1.3191753527 0.5354655797 1
## 1746 -2.114181540 2.7146287643 -1.7279053292 2
## 1747 -1.511685461 -1.5583452169 1.0207378045 3
## 1748 0.415308440 -0.1200068910 0.8007507348 1
## 1749 0.353427694 1.5410988331 1.9131871468 2
## 1750 2.889581802 0.4204467363 -1.9997622259 1
## 1751 0.805121605 0.9978560496 0.0002996201 2
## 1752 2.801259172 0.9253239387 -0.1149127518 1
## 1753 -0.587594499 2.9392322926 1.3166738443 2
## 1754 0.393495867 0.8187929437 -0.2409266404 2
## 1755 0.840344484 2.1275309935 -0.8518440443 2
## 1756 0.372048793 0.9079240627 -0.2077145970 2
## 1757 0.218315589 1.6581447690 1.8733422460 2
## 1758 -1.615121912 0.2857247083 0.3773200678 2
## 1759 -1.617998185 -0.1076271688 -0.1023907779 2
## 1760 0.566798677 0.0573998069 0.5641834275 1
## 1761 0.581305313 1.0508933957 -0.0178243168 2
## 1762 1.898992822 -0.1468128786 0.1267541561 1
## 1763 -3.193359969 -0.7874272413 -0.9699648400 3
## 1764 -0.368068816 2.8603386195 2.7606151879 2
## 1765 -2.309365277 1.6696339438 -0.2156017914 2
## 1766 -0.402867723 1.0245348716 1.1744962059 2
## 1767 2.584885199 0.8917008441 -0.1806835099 1
## 1768 2.664067465 0.9048847264 -0.1601398411 1
## 1769 3.488880491 0.6230175682 -1.1447555654 1
## 1770 -2.122506156 3.0308420534 1.0651922280 2
## 1771 -1.282459353 -0.1199241160 -0.0179342887 2
## 1772 -0.347843239 1.4381209613 0.1414371988 2
## 1773 2.685133087 -0.4980918628 -3.7030585514 1
## 1774 -2.449792153 -3.2980135038 0.5387349344 3
## 1775 0.719548619 0.5350703350 -1.4207884277 1
## 1776 1.489759830 0.2088965405 0.9464605382 1
## 1777 3.030413013 -0.0346027011 -3.0417185078 1
## 1778 2.279095465 -0.4295002028 -0.9261413533 1
## 1779 3.487232132 -0.2647679224 -0.4982609586 1
## 1780 1.771843417 0.2412092169 1.0145540554 1
## 1781 0.326478917 -1.4615627016 1.5723451489 1
## 1782 3.473081847 -1.7638212901 0.5606510435 1
## 1783 2.045023487 -2.4344760829 3.0169707804 1
## 1784 -4.284925291 -0.3474580564 -2.1585240011 3
## 1785 -2.268208707 2.5210214194 0.4392566182 2
## 1786 0.619188815 0.9482239646 -0.0534716831 2
## 1787 1.987611639 0.0437045490 0.3608258675 1
## 1788 0.369029875 1.9558105249 0.7543269006 2
## 1789 0.294605705 2.6254302427 0.7899405862 2
## 1790 0.350785613 -1.5809278066 1.6104898238 1
## 1791 2.524439131 0.8307613971 -0.2770123398 1
## 1792 2.851812736 -1.3755577691 1.7268078330 1
## 1793 1.573175030 1.0605965357 0.2748436334 1
## 1794 1.020187742 0.1551926853 0.8141805179 1
## 1795 1.663077564 1.2500610703 0.5311316487 1
## 1796 -1.673266476 2.7569073655 -1.5409387175 2
## 1797 -1.109221395 -1.4719668076 1.2282895185 3
## 1798 0.836998005 -0.0556783857 0.9980098977 1
## 1799 0.618389737 1.4793345187 1.9122840661 2
## 1800 2.681052300 0.9649748578 -0.3867699595 1
## 1801 0.808918612 1.0180206011 0.0448959017 2
## 1802 2.835514320 0.9514835676 -0.0652616405 1
## 1803 -0.588144314 2.9653790341 1.3653293245 2
## 1804 0.385433158 2.0615834682 -1.0316564839 2
## 1805 0.843054808 2.1489418732 -0.8066659962 2
## 1806 0.371498977 0.9340708043 -0.1590591169 2
## 1807 0.218602814 1.6833315010 1.9215496087 2
## 1808 -1.416886805 0.4875093290 0.4320210137 2
## 1809 -1.613657835 -0.0880857815 -0.0580853796 2
## 1810 0.571139027 0.0769411943 0.6084888257 1
## 1811 0.585102320 1.0710579472 0.0267719648 2
## 1812 1.903333171 -0.1272714913 0.1710595543 1
## 1813 -3.042275453 0.3712002743 -1.5220355412 3
## 1814 -0.367781591 2.8855253515 2.8088225506 2
## 1815 -2.309915093 1.6957806854 -0.1669463113 2
## 1816 -0.402580498 1.0497216036 1.2227035685 2
## 1817 2.588682207 0.9118653956 -0.1360872284 1
## 1818 2.667864472 0.9250492778 -0.1155435596 1
## 1819 3.523135639 0.6491771971 -1.0951044541 1
## 1820 -2.123055972 3.0569887950 1.1138477082 2
## 1821 -1.278119004 -0.1003827287 0.0263711096 2
## 1822 -0.348393054 1.4642677028 0.1900926789 2
## 1823 2.274232393 0.0459822416 -2.1020240515 1
## 1824 -2.245873896 -3.1037437200 0.5886612660 3
## 1825 0.718998804 0.5612170765 -1.3721329476 1
## 1826 1.494100179 0.2284379279 0.9907659364 1
## 1827 2.623859142 0.5034892132 -1.4447432064 1
## 1828 2.279382690 -0.4043134708 -0.8779339906 1
## 1829 3.529986668 0.6544091229 -1.0851742318 1
## 1830 1.776183766 0.2607506043 1.0588594536 1
## 1831 0.332449293 -1.4438908065 1.6157778974 1
## 1832 3.521487280 -0.2386082935 -0.4486098473 1
## 1833 2.090603472 -1.2122810619 2.2188836984 1
## 1834 -4.083180402 -0.1506956162 -2.1074341363 3
## 1835 -2.070810641 2.7237660497 0.4944056816 2
## 1836 0.622985822 0.9683885161 -0.0088754015 2
## 1837 1.987898864 0.0688912810 0.4090332302 1
## 1838 0.368480059 1.9819572664 0.8029823808 2
## 1839 0.294055890 2.6515769842 0.8385960664 2
## 1840 0.353583960 -1.5586211034 1.6573528340 1
## 1841 2.528236139 0.8509259486 -0.2324160583 1
## 1842 2.750626680 -1.3646216216 1.7343608892 1
## 1843 1.576078934 2.2918061277 -0.5200252030 1
## 1844 1.024528091 0.1747340727 0.8584859162 1
## 1845 1.662527749 1.2762078118 0.5797871289 1
## 1846 -1.675490372 2.7849741262 -1.4913870024 2
## 1847 -1.106423049 -1.4496601043 1.2751525287 3
## 1848 1.035233111 0.1461062350 1.0527108436 1
## 1849 0.549505819 2.4735302879 1.3519107063 2
## 1850 2.677998279 0.9799074835 -0.3521039002 1
## 1851 0.430798577 -0.0600437374 0.3422824154 1
## 1852 2.622247641 0.0258667936 0.4589275608 1
## 1853 -0.911425841 1.8343876827 1.6506783906 2
## 1854 0.018044963 1.8699890117 -1.3789578640 2
## 1855 0.630502805 0.8775822452 -0.1113851699 2
## 1856 -0.241822510 -0.2678971724 0.1690987238 2
## 1857 -0.086115560 1.6631991465 1.7764974031 2
## 1858 -2.235046583 0.0196115822 0.1253599591 3
## 1859 -1.975124983 -0.2874679197 -0.4102887337 3
## 1860 0.003617749 -0.2877481435 0.2575958172 2
## 1861 0.242674855 0.0305122921 0.3790312464 2
## 1862 1.509441027 -0.2814908727 -0.1499280846 1
## 1863 -3.813284640 -1.0535403673 -1.2219249488 3
## 1864 -0.870447847 2.6887951082 2.6572767618 2
## 1865 -2.736913527 1.5865086650 -0.4580855840 2
## 1866 -0.866064873 0.8845145652 1.1206983406 2
## 1867 2.383613638 0.8389533440 -0.2899522367 1
## 1868 2.403130191 0.0307610617 0.4170653082 1
## 1869 3.450278521 0.6028401293 -1.1903474780 1
## 1870 -2.267224283 2.0264159020 1.5853746265 2
## 1871 -1.786008423 -2.0397758566 0.7104789169 3
## 1872 -0.973339369 1.1783977937 -0.1075401771 2
## 1873 1.845774522 -0.9643813809 -1.7603678437 1
## 1874 -2.955522471 -4.9190728257 1.0413338510 3
## 1875 0.094052489 0.2753471674 -1.6697658036 2
## 1876 1.289360553 0.1551486093 0.8367248258 1
## 1877 2.635736339 -1.0651715887 -2.6819108526 1
## 1878 1.857118675 -0.5190154400 -1.1716078788 1
## 1879 3.448923860 -0.2852822068 -0.5440101055 1
## 1880 1.570404552 -1.3266288634 1.9567114102 1
## 1881 -0.077540590 -2.8804719079 2.3250771410 3
## 1882 3.253371758 -1.7727239025 0.4858578243 1
## 1883 1.745076902 -2.6289409479 2.9528289589 1
## 1884 -4.840872255 -0.7028962651 -2.4724485546 3
## 1885 -2.854522957 2.2928214567 0.2398198032 2
## 1886 0.411870075 0.0016401429 0.4759753337 1
## 1887 1.722362370 0.0802821314 0.3135215855 1
## 1888 -0.019336495 1.9042084510 0.5613836689 2
## 1889 -0.213344784 2.4602766898 0.6895848930 2
## 1890 0.063068841 -1.5950433045 1.5046967822 1
## 1891 2.118936647 -0.1863944896 0.0903231005 1
## 1892 2.587096443 -1.3494909600 1.6696256482 1
## 1893 1.145626781 0.9774712569 0.0323598408 1
## 1894 0.787363469 0.1466075110 0.7356605207 1
## 1895 1.392256836 1.2930286111 0.4868100996 1
## 1896 -2.234131155 1.2140338321 -0.9828418106 2
## 1897 -1.430662102 -2.8858117685 1.7706753941 3
## 1898 0.217073334 -0.3217915118 0.7460497889 2
## 1899 0.155192588 1.3393142123 1.8584862009 2
## 1900 2.277154486 -0.1617703070 0.1319864980 1
## 1901 0.117786635 0.8533461547 -0.3986797154 2
## 1902 2.512184435 0.9189314641 -0.2085796998 1
## 1903 -1.248848531 2.7588292572 0.8933393172 2
## 1904 -0.219860707 1.7775281453 -1.5575601494 2
## 1905 0.389554245 1.9976352510 -1.1300337512 2
## 1906 -0.473428349 -0.3675833740 -0.0128762353 2
## 1907 -0.317721399 1.5635129449 1.5945224440 2
## 1908 -2.466652422 -0.0800746194 -0.0566150000 3
## 1909 -2.206730822 -0.3871541213 -0.5922636928 3
## 1910 -0.227988091 -0.3874343451 0.0756208581 2
## 1911 -0.001645433 0.8481738547 -0.4281510704 2
## 1912 1.277835188 -0.3811770743 -0.3319030437 1
## 1913 -3.634539601 -1.6711539317 -2.9562789277 3
## 1914 -1.102053686 2.5891089066 2.4753018027 2
## 1915 -2.970619310 1.4892309085 -0.6389363185 2
## 1916 -1.097670712 0.7848283637 0.9387233815 2
## 1917 2.267453496 0.8362452022 -0.3221812886 1
## 1918 2.211763303 0.8641511454 -0.3479491597 1
## 1919 3.477682639 0.6237678325 -1.1506265890 1
## 1920 -2.627032667 2.9765318378 0.8400199444 2
## 1921 -2.113724241 -0.5139501667 -0.4713791785 3
## 1922 -1.207045152 1.0811200372 -0.2883909116 2
## 1923 1.613528176 -0.1605675353 -2.5740140588 1
## 1924 -3.230728016 -3.7597301651 0.0410250675 3
## 1925 -0.813572554 0.4908396329 -0.8009132800 2
## 1926 1.136118474 0.1185088176 0.7538309885 1
## 1927 1.993779622 -0.6469304275 -1.3115067919 1
## 1928 2.035863714 -1.1366290044 -2.9059618576 1
## 1929 3.297658988 -0.3069922144 -0.5983962591 1
## 1930 1.418202062 0.1508214940 0.8219245057 1
## 1931 -0.338965805 -1.7369340365 1.3173909286 3
## 1932 3.145611391 -1.7850658246 0.4491318741 1
## 1933 1.619748659 -2.6527610831 2.9048230379 1
## 1934 -5.074578037 -0.8001740216 -2.6532992891 3
## 1935 -3.088228740 2.1955437002 0.0589690687 2
## 1936 0.214699705 0.8416862930 -0.2859321863 2
## 1937 1.490756531 -0.0194040701 0.1315466264 1
## 1938 -0.253042277 1.8069306945 0.3805329344 2
## 1939 -0.447050567 2.3629989333 0.5087341585 2
## 1940 -0.162237166 -1.7019548412 1.3193491494 1
## 1941 1.889266037 0.6144652214 -0.7247020586 1
## 1942 2.479336076 -1.3618328820 1.6328996980 1
## 1943 0.911920998 0.8801935004 -0.1484908937 1
## 1944 0.594939510 0.0784445143 0.6032261225 1
## 1945 1.197732934 1.2272740595 0.3554999259 1
## 1946 -2.547811195 2.4175032484 -1.9625529129 2
## 1947 -1.739789816 -1.6875634569 0.7885271615 3
## 1948 -0.014532505 -0.4214777134 0.5640748299 2
## 1949 -0.037231371 1.2711512157 1.7260518026 2
## 1950 2.045548647 -0.2614565086 -0.0499884611 1
## 1951 0.925078754 1.0207287429 0.0771249536 1
## 1952 2.808110202 0.9305558645 -0.1049825296 1
## 1953 -0.319724422 2.9294363546 1.4276331219 2
## 1954 0.584156948 2.1225211296 -0.9025947594 2
## 1955 0.924232957 2.1153099199 -0.8262259543 2
## 1956 0.679100750 0.9296513297 -0.0472147586 2
## 1957 0.362384248 1.5609540247 1.8390289296 2
## 1958 -1.116377850 0.4912246724 0.5476625781 2
## 1959 -1.538779518 -0.1144923996 -0.0742726640 2
## 1960 0.646017344 0.0505345762 0.5923015414 1
## 1961 0.662080582 1.0422428840 0.0094604558 2
## 1962 2.056575250 -0.0906316995 0.2539533917 1
## 1963 -2.734673680 0.3667807997 -1.4101911829 3
## 1964 -0.224000157 2.7631478752 2.7263018715 2
## 1965 -2.080677081 1.6283148010 -0.1541830747 2
## 1966 -0.258799064 0.9273441273 1.1401828895 2
## 1967 2.821812261 0.9410197161 -0.0851220851 1
## 1968 2.814961231 0.9357877903 -0.0950523073 1
## 1969 3.495731521 0.6282494940 -1.1348253431 1
## 1970 -1.972181720 2.9264765008 1.0275298230 2
## 1971 -1.203240686 -0.1267893467 0.0101838252 2
## 1972 -0.158336923 1.3652786135 0.1533153546 2
## 1973 3.933486517 0.1312498343 -2.6474834739 1
## 1974 -2.020567888 -2.9968321832 0.7740088989 3
## 1975 1.410870518 0.0747633795 -2.7883124175 1
## 1976 1.568978497 0.2020313098 0.9745786521 1
## 1977 2.779201164 0.5377205599 -1.3629735936 1
## 1978 2.501527885 -0.4636445372 -0.8613735479 1
## 1979 3.912933429 0.1155540569 -2.6772741406 1
## 1980 1.851062084 0.2343439862 1.0426721693 1
## 1981 0.401027779 -1.4630720894 1.6029632867 1
## 1982 3.494083162 -0.2595359966 -0.4883307363 1
## 1983 2.063199354 -1.2332087651 2.1791628093 1
## 1984 -3.849474620 -0.0534178597 -1.9265834018 3
## 1985 -1.919936390 2.5932537555 0.4080877964 2
## 1986 0.699964084 0.9395734529 -0.0261869105 2
## 1987 2.131680298 -0.0534861953 0.3265125511 1
## 1988 0.519354310 1.8514449722 0.7166644956 2
## 1989 0.444930141 2.5210646900 0.7522781812 2
## 1990 0.469742884 -1.6473245094 1.5930842968 1
## 1991 2.890322556 0.9933389740 0.0141801375 1
## 1992 2.874523911 -1.3934998844 1.7195867464 1
## 1993 1.784536409 1.1285690686 0.4385724928 1
## 1994 1.099406409 0.1483274546 0.8422986318 1
## 1995 1.796014708 1.1696242779 0.5097060381 1
## 1996 -1.353702965 2.7642850158 -1.3871370563 2
## 1997 -0.827192544 -1.4215403071 1.4021857114 3
## 1998 1.214143301 0.0608973082 1.0236326898 1
## 1999 0.762458397 1.3821437744 1.8779707497 2
## 2000 2.761827569 0.9563243461 -0.3594851869 1
## 2001 0.630193979 0.9155270704 -0.1331473235 2
## 2002 2.801259172 0.9253239387 -0.1149127518 1
## 2003 -0.801643531 2.9107430239 1.2197596881 2
## 2004 0.206673281 1.9591303589 -1.2096808410 2
## 2005 0.742658691 2.1095351738 -0.8856092315 2
## 2006 0.157999760 0.8794347941 -0.3046287533 2
## 2007 0.085834567 1.6890269262 1.8737937864 2
## 2008 -1.825966694 0.2535604556 0.2786904864 2
## 2009 -1.792364847 -0.1905995229 -0.2361380389 3
## 2010 0.392432015 -0.0255725471 0.4304361665 1
## 2011 0.445559567 1.0000876214 -0.1017306995 2
## 2012 1.720573035 -0.2241398881 -0.0030911405 1
## 2013 -3.404204752 -0.8195914939 -1.0685944215 3
## 2014 -0.500549838 2.8912207767 2.7610667282 2
## 2015 -2.523414310 1.6411446752 -0.3125159476 2
## 2016 -0.535348745 1.0554170288 1.1749477462 2
## 2017 2.488321334 0.8724182747 -0.2150493318 1
## 2018 2.541422662 0.9214952974 -0.1701504714 1
## 2019 3.899231370 0.1050902054 -2.6971345851 1
## 2020 -2.258191428 3.0653991946 1.0673591935 2
## 2021 -1.473038513 -0.1803150917 -0.1360736921 2
## 2022 -0.561892271 1.4096316927 0.0445230425 2
## 2023 2.060733176 -0.0086537686 -2.2475936879 1
## 2024 -2.622475923 -3.3829159824 0.4040867214 3
## 2025 0.505499587 0.5065810663 -1.5177025840 1
## 2026 1.393756928 0.1889705963 0.9117943989 1
## 2027 2.436440863 0.4129600626 -1.6146680344 1
## 2028 2.478601561 -0.9795918182 -2.5771499545 1
## 2029 3.487232132 -0.2647679224 -0.4982609586 1
## 2030 1.675840515 0.2212832727 0.9798879161 1
## 2031 0.209954703 -1.4509751128 1.5607898901 1
## 2032 3.789389256 -2.2882017673 -1.0149634500 1
## 2033 2.045023487 -2.4344760829 3.0169707804 1
## 2034 -4.459852917 -0.4297870357 -2.2919709447 3
## 2035 -2.443075859 2.5240553557 0.3918830229 2
## 2036 0.522624949 0.9289413952 -0.0878375050 2
## 2037 1.855130617 0.0745867062 0.3612774078 1
## 2038 0.194162723 1.9588444612 0.7069533053 2
## 2039 0.158920433 2.6599873839 0.7921075517 2
## 2040 0.227917341 -1.5610706015 1.6057950886 1
## 2041 2.336471037 0.7663789881 -0.3982816877 1
## 2042 2.712763062 -1.3470414619 1.7211833937 1
## 2043 1.359125998 1.0321072671 0.1779294772 1
## 2044 0.920131716 0.1409120857 0.7834163431 1
## 2045 1.527392293 1.2846182115 0.5332986142 1
## 2046 -1.893724008 2.7357680649 -1.6344220234 2
## 2047 -1.310453428 -1.5151560123 1.1245136615 3
## 2048 0.626153222 -0.0878426384 0.8993803163 1
## 2049 0.485908716 1.5102166759 1.9127356064 2
## 2050 2.584488435 0.9456922884 -0.4211357814 1
## 2051 0.630193979 0.9155270704 -0.1331473235 2
## 2052 2.801259172 0.9253239387 -0.1149127518 1
## 2053 -0.801643531 2.9107430239 1.2197596881 2
## 2054 0.206673281 1.9591303589 -1.2096808410 2
## 2055 0.742658691 2.1095351738 -0.8856092315 2
## 2056 0.157999760 0.8794347941 -0.3046287533 2
## 2057 0.085834567 1.6890269262 1.8737937864 2
## 2058 -1.825966694 0.2535604556 0.2786904864 2
## 2059 -1.792364847 -0.1905995229 -0.2361380389 3
## 2060 0.392432015 -0.0255725471 0.4304361665 1
## 2061 0.445559567 1.0000876214 -0.1017306995 2
## 2062 1.720573035 -0.2241398881 -0.0030911405 1
## 2063 -3.404204752 -0.8195914939 -1.0685944215 3
## 2064 -0.500549838 2.8912207767 2.7610667282 2
## 2065 -2.523414310 1.6411446752 -0.3125159476 2
## 2066 -0.535348745 1.0554170288 1.1749477462 2
## 2067 2.488321334 0.8724182747 -0.2150493318 1
## 2068 2.541422662 0.9214952974 -0.1701504714 1
## 2069 3.488880491 0.6230175682 -1.1447555654 1
## 2070 -2.258191428 3.0653991946 1.0673591935 2
## 2071 -1.473038513 -0.1803150917 -0.1360736921 2
## 2072 -0.561892271 1.4096316927 0.0445230425 2
## 2073 2.471084055 -0.5265811314 -3.7999727077 1
## 2074 -2.622475923 -3.3829159824 0.4040867214 3
## 2075 0.505499587 0.5065810663 -1.5177025840 1
## 2076 1.393756928 0.1889705963 0.9117943989 1
## 2077 2.846791741 -0.1049673002 -3.1670470542 1
## 2078 2.068250683 -0.4616644554 -1.0247709347 1
## 2079 3.897583011 -0.7826952852 -2.0506399784 1
## 2080 1.675840515 0.2212832727 0.9798879161 1
## 2081 0.209954703 -1.4509751128 1.5607898901 1
## 2082 3.379038377 -1.7702744045 0.5374155698 1
## 2083 2.045023487 -2.4344760829 3.0169707804 1
## 2084 -4.459852917 -0.4297870357 -2.2919709447 3
## 2085 -2.443075859 2.5240553557 0.3918830229 2
## 2086 0.522624949 0.9289413952 -0.0878375050 2
## 2087 1.855130617 0.0745867062 0.3612774078 1
## 2088 0.194162723 1.9588444612 0.7069533053 2
## 2089 0.158920433 2.6599873839 0.7921075517 2
## 2090 0.227917341 -1.5610706015 1.6057950886 1
## 2091 2.336471037 0.7663789881 -0.3982816877 1
## 2092 2.712763062 -1.3470414619 1.7211833937 1
## 2093 1.359125998 1.0321072671 0.1779294772 1
## 2094 0.920131716 0.1409120857 0.7834163431 1
## 2095 1.527392293 1.2846182115 0.5332986142 1
## 2096 -1.893724008 2.7357680649 -1.6344220234 2
## 2097 -1.310453428 -1.5151560123 1.1245136615 3
## 2098 0.626153222 -0.0878426384 0.8993803163 1
## 2099 0.485908716 1.5102166759 1.9127356064 2
## 2100 2.584488435 0.9456922884 -0.4211357814 1
## 2101 -0.282542336 -0.4726109344 -0.0880295107 2
## 2102 2.224400251 -0.1155328999 0.1837309005 1
## 2103 -1.624766753 1.4218204857 1.2203664645 2
## 2104 -0.706010068 1.4697099372 -1.8035338863 2
## 2105 -0.086409480 0.4691110891 -0.5397851280 2
## 2106 -0.955163422 -0.6804643694 -0.2612132023 2
## 2107 -0.799456473 1.2506319495 1.3461854770 2
## 2108 -2.948387495 -0.3929556148 -0.3049519670 3
## 2109 -2.688465896 -0.7000351167 -0.8406006598 3
## 2110 -0.709723164 -0.7003153405 -0.1727161089 2
## 2111 -0.470666057 -0.3820549049 -0.0512806797 2
## 2112 1.404398874 -1.0353875437 -2.1261254472 1
## 2113 -5.155360880 -1.0360412522 -1.9749018439 3
## 2114 -1.583788759 2.2762279112 2.2269648357 2
## 2115 -3.453825812 1.1780375088 -0.8864855422 2
## 2116 -1.579405786 0.4719473682 0.6903864145 2
## 2117 1.982194875 0.7016496913 -0.5632369290 1
## 2118 1.926919040 -0.1736850416 0.0427875262 1
## 2119 3.108946621 0.4692990778 -1.4483540155 1
## 2120 -2.980565195 1.6138487051 1.1550627004 2
## 2121 -2.488635217 -2.4646311761 0.2744310870 3
## 2122 -1.690251654 0.7699266375 -0.5359401352 2
## 2123 1.330381490 -1.2003506891 -2.1841861866 1
## 2124 -3.651006519 -5.3521202268 0.6014620851 3
## 2125 -1.296779056 0.1796462333 -1.0484625037 2
## 2126 0.654383401 -0.1943721778 0.5054940215 1
## 2127 2.120343308 -1.3011408969 -3.1057291955 1
## 2128 1.797162273 -2.8399038402 -2.1206058064 1
## 2129 3.035639559 -0.4366139980 -0.8312081212 1
## 2130 0.946141519 -1.6884377731 1.6197447020 1
## 2131 -0.773024637 -3.3135193091 1.8852053752 3
## 2132 2.866238487 -1.9264117185 0.2049252602 1
## 2133 1.167138495 -2.9674187343 2.6615788756 1
## 2134 -6.186519867 -0.6813011091 -3.2235134817 3
## 2135 -3.476576279 0.8727902287 0.3926503676 3
## 2136 -0.262288957 -0.3794038492 0.0952039685 2
## 2137 1.206969338 -0.1556871768 -0.1102967574 1
## 2138 -0.736248780 1.4957372948 0.1329837108 2
## 2139 -0.875135504 1.0858206530 0.8636892455 2
## 2140 -0.617882834 -3.2537819723 1.8987203101 3
## 2141 1.603543616 -0.4223637978 -0.3334952424 1
## 2142 2.199963171 -1.5031787759 1.3886930841 1
## 2143 0.428714495 0.5690001007 -0.3960401173 2
## 2144 0.113204437 -0.2344364811 0.3548891555 2
## 2145 0.722274440 0.0044289525 0.7358168905 1
## 2146 -2.951169077 0.8055721800 -1.4112622495 3
## 2147 -2.126254164 -3.3188698764 1.3307737134 3
## 2148 -1.125125605 -0.3042862611 -0.0069491593 2
## 2149 -0.518966444 0.9582702203 1.4777148356 2
## 2150 1.974041753 -1.0922587313 -1.8507265010 1
## 2151 -0.705061201 -0.7422932693 -0.2730260810 2
## 2152 0.861119243 -0.3492027464 0.5511190325 1
## 2153 -2.047285618 1.1521381508 1.0353698942 2
## 2154 -1.133489499 1.2057169222 -1.9858747709 2
## 2155 -0.510581867 0.2013251942 -0.7238964698 2
## 2156 -1.377682287 -0.9501467043 -0.4462097726 3
## 2157 -0.950007550 -0.8462552350 2.0669249172 2
## 2158 -3.370906360 -0.6626379496 -0.4899485373 3
## 2159 -3.110984761 -0.9697174516 -1.0255972301 3
## 2160 -1.090248923 -2.5334424219 0.6711424320 3
## 2161 -0.893184922 -0.6517372398 -0.2362772500 2
## 2162 -0.342271392 -2.2973350230 0.8305440624 3
## 2163 -5.577879745 -1.3057235871 -2.1598984142 3
## 2164 -1.553512650 -0.0280516566 2.8508967472 2
## 2165 -3.877998199 0.9102516139 -1.0705968839 3
## 2166 -2.001924650 0.2022650334 0.5053898442 2
## 2167 0.617260345 0.4698762848 -0.1949635684 1
## 2168 0.563638032 -0.4073548881 0.4101756582 1
## 2169 1.740135516 -0.6607936927 -0.4459911368 1
## 2170 -3.403084060 1.3441663702 0.9700661301 2
## 2171 -2.906193515 -2.7400028309 0.0867788309 3
## 2172 -2.114424041 0.5021407426 -0.7200514770 2
## 2173 0.377451361 -1.9519478984 -3.3691770743 1
## 2174 -4.283642222 -5.7191458122 -1.4630046169 3
## 2175 -1.699816660 -1.0151447853 -0.6118743923 3
## 2176 0.231864536 -0.4640545127 0.3204974511 1
## 2177 0.757062300 -1.5348107434 -2.7383410635 1
## 2178 0.028490952 -2.5613356438 -0.2034943404 1
## 2179 1.672358551 -0.6702838445 -0.4638199892 1
## 2180 0.528583221 -1.9638094279 1.4320924459 1
## 2181 -1.187275892 -3.5926838437 1.6957826619 3
## 2182 1.918268923 -2.6836982477 -0.9827213134 1
## 2183 0.752887241 -3.2465832690 2.4721561624 1
## 2184 -6.610692254 -0.9490870040 -3.4076248235 3
## 2185 -3.899095144 0.6031078938 0.2076537973 3
## 2186 -0.684807822 -0.6490861841 -0.0897926018 2
## 2187 -0.108053976 -1.9599866835 1.2825926802 1
## 2188 -1.160421167 1.2279513999 -0.0511276310 2
## 2189 -1.297654369 0.8161383181 0.6786926752 2
## 2190 -1.032134088 -3.5329465069 1.7092975969 3
## 2191 0.240262608 -0.6560336443 0.0338928896 1
## 2192 0.841642730 -1.7425379423 1.7534255304 1
## 2193 0.004542108 0.3012142058 -0.5801514591 2
## 2194 -0.309314428 -0.5041188160 0.1698925852 2
## 2195 0.299755575 -0.2652533824 0.5508203202 1
## 2196 -3.375341464 0.5377862851 -1.5953735912 3
## 2197 -2.540505418 -3.5980344111 1.1413510002 3
## 2198 -1.547644470 -0.5739685960 -0.1919457296 3
## 2199 -0.941485309 0.6885878854 1.2927182653 2
## 2200 1.551522888 -1.3619410662 -2.0357230713 1
## 2201 -0.057971352 -0.3795264883 0.0904734764 2
## 2202 2.331425594 -0.1170180685 0.2136122050 1
## 2203 -1.400195770 1.5149049318 1.3988694516 2
## 2204 -0.496215396 1.5837285874 -1.6101919169 2
## 2205 0.126774557 0.5782456654 -0.3499897737 2
## 2206 -0.532644557 -0.4107820345 -0.0762166320 2
## 2207 -0.574885489 1.3437163956 1.5246884641 2
## 2208 -2.525868630 -0.1232732799 -0.1199553967 3
## 2209 -2.484160534 -0.5787239476 -0.6425878508 3
## 2210 -0.299363672 -0.4136969718 0.0239863546 2
## 2211 -0.246095073 -0.2889704588 0.1272223074 2
## 2212 1.218618980 -0.4243757348 -0.3952434404 1
## 2213 -4.104106688 -1.1964252294 -1.4672403046 3
## 2214 -1.161269894 2.5459102461 2.4119614060 2
## 2215 -3.227601306 1.2692255150 -0.7088677837 2
## 2216 -1.156886921 0.7416297031 0.8753829848 2
## 2217 2.069139625 0.7281790330 -0.5139448601 1
## 2218 2.112308144 -0.1121238004 0.1717499524 1
## 2219 3.122648681 0.4797629294 -1.4284935710 1
## 2220 -2.755994211 1.7069331511 1.3335656875 2
## 2221 -2.071076918 -2.1892595214 0.4620833430 3
## 2222 -1.266079267 1.0377125325 -0.3518287935 2
## 2223 1.965303353 -1.6251936059 -3.5580622193 1
## 2224 -3.247093794 -5.0571112230 0.8033479718 3
## 2225 -0.872606669 0.4474321282 -0.8643511619 2
## 2226 0.780325002 -0.1361074186 0.6044257088 1
## 2227 1.934563413 -0.6901290880 -1.3748471886 1
## 2228 1.976647506 -1.1798276649 -2.9693022543 1
## 2229 3.123969613 -0.4114280688 -0.7835886021 1
## 2230 1.071528030 -1.6295363759 1.7189735619 1
## 2231 -0.358773383 -3.0343547744 2.0746280884 3
## 2232 2.968303263 -1.9222075673 0.2374622504 1
## 2233 1.498659542 -2.7747616563 2.7172013645 1
## 2234 -5.155346267 -0.8136705760 -2.6964411780 3
## 2235 -3.147262855 2.1521361955 -0.0044688132 2
## 2236 -0.089059227 -0.3009065743 0.2358722879 2
## 2237 1.431540322 -0.0626027307 0.0682062297 1
## 2238 -0.510024274 1.5869253009 0.3106014693 2
## 2239 -0.506084683 2.3195914286 0.4452962766 2
## 2240 -0.419947535 -1.9211248580 1.2498076251 3
## 2241 1.828114600 -0.3292793517 -0.1549922553 1
## 2242 2.302027947 -1.4989746247 1.4212300744 1
## 2243 0.654939001 0.6601881068 -0.2184223588 1
## 2244 0.298593541 -0.1728752399 0.4838515817 1
## 2245 0.901569057 0.9757454611 0.2360278999 2
## 2246 -2.526996690 1.0733580749 -1.2271509077 2
## 2247 -1.909950791 -3.2163032305 1.5137028434 3
## 2248 -0.073871413 -0.4646702384 0.5007123801 2
## 2249 -0.135629460 1.1964293502 1.6131708451 2
## 2250 2.376294996 -0.7943496734 -1.6462201087 1
## 2251 -0.699267913 -0.6735761931 -0.1894182031 2
## 2252 2.005622555 -0.1399002698 0.0888357914 1
## 2253 -2.041492330 1.2208552270 1.1189777721 2
## 2254 -1.132269096 1.2796786822 -1.8998187563 2
## 2255 -0.506312874 0.2717904983 -0.6394725463 2
## 2256 -1.371888999 -0.8814296281 -0.3626018947 3
## 2257 -0.939641378 -0.7827828427 2.1480846585 2
## 2258 -3.365113072 -0.5939208735 -0.4063406594 3
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## 2260 -1.126448741 -0.9012805992 -0.2741048013 3
## 2261 -0.887391634 -0.5830201636 -0.1526693721 2
## 2262 1.185621178 -1.0597549136 -2.2210205564 1
## 2263 -4.943351130 -1.6670728230 -1.7536255673 3
## 2264 -1.543146478 0.0354207357 2.9320564885 2
## 2265 -3.873729206 0.9807169181 -0.9861729604 3
## 2266 -1.996131363 0.2709821095 0.5889977221 2
## 2267 1.760239362 0.6809269893 -0.6564307640 1
## 2268 1.708141344 -0.1980524115 -0.0521075830 1
## 2269 2.886991108 0.4485763758 -1.5415478506 1
## 2270 -3.397290772 1.4128834463 1.0536740080 2
## 2271 -2.895827343 -2.6765304386 0.1679385722 3
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## 2278 1.587918028 -2.8752052139 -2.2206047379 1
## 2279 2.816861863 -0.4609813679 -0.9261032304 1
## 2280 0.538949393 -1.9003370356 1.5132521872 1
## 2281 -1.173861129 -3.5327079073 1.7753103122 3
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## 2284 -5.977687934 -1.3086880120 -3.0005359310 3
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## 2317 2.121500278 0.7506710891 -0.4789916688 1
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## 2319 3.136350740 0.4902267810 -1.4086331265 1
## 2320 -2.761787499 1.6382160750 1.2499578096 2
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## 2326 0.832629852 -0.1135513619 0.6394087746 1
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## 2328 1.970854218 -1.2485447411 -3.0529101322 1
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## 3036 0.382449895 -0.0514141188 0.4266074905 1
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## 3043 1.147763366 0.7958145251 -0.1321325394 1
## 3044 0.782262036 0.0596811817 0.6628808911 1
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## 3050 2.610797056 -0.7529844473 -1.5114969971 1
## 3051 -0.282542336 -0.4726109344 -0.0880295107 2
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## 3053 -1.624766753 1.4218204857 1.2203664645 2
## 3054 -0.706010068 1.4697099372 -1.8035338863 2
## 3055 -0.086409480 0.4691110891 -0.5397851280 2
## 3056 -0.955163422 -0.6804643694 -0.2612132023 2
## 3057 -0.799456473 1.2506319495 1.3461854770 2
## 3058 -2.948387495 -0.3929556148 -0.3049519670 3
## 3059 -2.688465896 -0.7000351167 -0.8406006598 3
## 3060 -0.709723164 -0.7003153405 -0.1727161089 2
## 3061 -0.470666057 -0.3820549049 -0.0512806797 2
## 3062 0.994047996 -0.5174601809 -0.5737464275 1
## 3063 -4.526625553 -1.4661075643 -1.6522368749 3
## 3064 -1.583788759 2.2762279112 2.2269648357 2
## 3065 -3.453825812 1.1780375088 -0.8864855422 2
## 3066 -1.579405786 0.4719473682 0.6903864145 2
## 3067 1.982194875 0.7016496913 -0.5632369290 1
## 3068 1.926919040 -0.1736850416 0.0427875262 1
## 3069 3.108946621 0.4692990778 -1.4483540155 1
## 3070 -2.980565195 1.6138487051 1.1550627004 2
## 3071 -2.488635217 -2.4646311761 0.2744310870 3
## 3072 -1.690251654 0.7699266375 -0.5359401352 2
## 3073 1.740732369 -1.7182780519 -3.7365652063 1
## 3074 -3.651006519 -5.3521202268 0.6014620851 3
## 3075 -1.296779056 0.1796462333 -1.0484625037 2
## 3076 0.654383401 -0.1943721778 0.5054940215 1
## 3077 2.120343308 -1.3011408969 -3.1057291955 1
## 3078 1.386811394 -2.3219764774 -0.5682267866 1
## 3079 3.035639559 -0.4366139980 -0.8312081212 1
## 3080 0.946141519 -1.6884377731 1.6197447020 1
## 3081 -0.773024637 -3.3135193091 1.8852053752 3
## 3082 2.866238487 -1.9264117185 0.2049252602 1
## 3083 1.167138495 -2.9674187343 2.6615788756 1
## 3084 -6.186519867 -0.6813011091 -3.2235134817 3
## 3085 -3.476576279 0.8727902287 0.3926503676 3
## 3086 -0.262288957 -0.3794038492 0.0952039685 2
## 3087 1.206969338 -0.1556871768 -0.1102967574 1
## 3088 -0.736248780 1.4957372948 0.1329837108 2
## 3089 -0.875135504 1.0858206530 0.8636892455 2
## 3090 -0.617882834 -3.2537819723 1.8987203101 3
## 3091 1.603543616 -0.4223637978 -0.3334952424 1
## 3092 2.199963171 -1.5031787759 1.3886930841 1
## 3093 0.428714495 0.5690001007 -0.3960401173 2
## 3094 0.113204437 -0.2344364811 0.3548891555 2
## 3095 0.722274440 0.0044289525 0.7358168905 1
## 3096 -2.951043441 0.8055626760 -1.4112417687 3
## 3097 -2.126146150 -3.3188591697 1.3308036282 3
## 3098 -0.496267579 -0.7343587088 0.3157378628 2
## 3099 -0.518966444 0.9582702203 1.4777148356 2
## 3100 1.974164453 -1.0922648669 -1.8507044479 1
## 3101 -0.282542336 -0.4726109344 -0.0880295107 2
## 3102 2.224400251 -0.1155328999 0.1837309005 1
## 3103 -1.624766753 1.4218204857 1.2203664645 2
## 3104 -0.706010068 1.4697099372 -1.8035338863 2
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## 3112 0.994047996 -0.5174601809 -0.5737464275 1
## 3113 -5.155360880 -1.0360412522 -1.9749018439 3
## 3114 -1.583788759 2.2762279112 2.2269648357 2
## 3115 -3.453825812 1.1780375088 -0.8864855422 2
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## 3117 1.982194875 0.7016496913 -0.5632369290 1
## 3118 1.926919040 -0.1736850416 0.0427875262 1
## 3119 3.108946621 0.4692990778 -1.4483540155 1
## 3120 -2.980565195 1.6138487051 1.1550627004 2
## 3121 -2.488635217 -2.4646311761 0.2744310870 3
## 3122 -1.690251654 0.7699266375 -0.5359401352 2
## 3123 1.740732369 -1.7182780519 -3.7365652063 1
## 3124 -4.279741846 -4.9220539147 0.2787971161 3
## 3125 -1.296779056 0.1796462333 -1.0484625037 2
## 3126 0.654383401 -0.1943721778 0.5054940215 1
## 3127 2.120343308 -1.3011408969 -3.1057291955 1
## 3128 1.797162273 -2.8399038402 -2.1206058064 1
## 3129 3.035639559 -0.4366139980 -0.8312081212 1
## 3130 0.946141519 -1.6884377731 1.6197447020 1
## 3131 -0.773024637 -3.3135193091 1.8852053752 3
## 3132 3.276589365 -2.4443390813 -1.3474537596 1
## 3133 1.167138495 -2.9674187343 2.6615788756 1
## 3134 -6.186519867 -0.6813011091 -3.2235134817 3
## 3135 -3.476576279 0.8727902287 0.3926503676 3
## 3136 -0.262288957 -0.3794038492 0.0952039685 2
## 3137 1.206969338 -0.1556871768 -0.1102967574 1
## 3138 -0.736248780 1.4957372948 0.1329837108 2
## 3139 -0.875135504 1.0858206530 0.8636892455 2
## 3140 -0.617882834 -3.2537819723 1.8987203101 3
## 3141 1.603543616 -0.4223637978 -0.3334952424 1
## 3142 2.199963171 -1.5031787759 1.3886930841 1
## 3143 0.428714495 0.5690001007 -0.3960401173 2
## 3144 0.113204437 -0.2344364811 0.3548891555 2
## 3145 0.722274440 0.0044289525 0.7358168905 1
## 3146 -2.951043441 0.8055626760 -1.4112417687 3
## 3147 -2.126146150 -3.3188591697 1.3308036282 3
## 3148 -0.496267579 -0.7343587088 0.3157378628 2
## 3149 -0.518966444 0.9582702203 1.4777148356 2
## 3150 1.974164453 -1.0922648669 -1.8507044479 1ggplot(pc_tsne_data, aes(x = PC1, y = PC2 , color = Cluster)) +
geom_point() +
labs(title = "K-means Clustering of Churn Data",
x = "Principal Component 1",
y = "Principal Component 2") +
theme_minimal()
PREDICTION ANALYSIS:
Goal: Utilizing main components logistic regression, create a churn prediction model. Finding the primary components that greatly influence churn prediction is an insightful step. Interpretation: To determine the model’s predictive ability and potential for identifying consumers who are at danger of churning, assess its performance indicators, such as accuracy, precision, recall, etc.
Predictions created based on the model
# Fit logistic regression model
model <- glm(Churn ~ PC1 + PC2, data = subset_data, family = "binomial")
# Predictions on the test set
predictions <- predict(model, newdata = subset_data, type = "response")Confusion Matrix Visualization
predicted_classes <- ifelse(predictions > 0.5, "Churn", "No Churn")
# Create a confusion matrix
conf_matrix <- table(Actual = subset_data$Churn, Predicted = predicted_classes)
# Plot the confusion matrix heatmap
pheatmap(
conf_matrix,
color = c("white", "skyblue"),
fontsize = 12,
main = "Confusion Matrix Heatmap",
fontsize_number = 10,
fontsize_row = 10
)
This code will print the counts of actual churn and predicted churn values, making it easier to interpret the results of the predictive model.
# Create a table of actual churn and predicted churn values
churn_counts <- table(Actual = subset_data$Churn, Predicted = predicted_classes)
# Display the counts
print(churn_counts)## Predicted
## Actual Churn No Churn
## 0 94 2561
## 1 290 205The confusion matrix is a fundamental tool in evaluating the prediction performance of classification models. With a thorough analysis of model correctness and faults, it tabulates true positives, true negatives, false positives, and false negatives. This facilitates comprehension of the model’s predictive performance, which is especially important in situations such as churn prediction, where eliminating false negatives (churn cases missed) and detecting true positives (churn accurately anticipated) are essential for making well-informed decisions.8
I created a confusion matrix by comparing the actual churn values (Actual) with the predicted churn values (Predicted) generated by my predictive model. The confusion matrix is a table that summarizes the performance of a classification algorithm.
simple interpretation: True Positives (TP): 94 True Negatives (TN): 205 False Positives (FP): 2561 False Negatives (FN): 290
it accurately recognized 94 instances of actual churn (True Positives), but misclassified 290 cases of actual churn as no churn (False Negatives). Similarly, it accurately recognized 205 instances of genuine no churn (True Negatives), but misclassified 2561 instances of actual no churn as churn (False Positives).
Accuracy = TN + TP / TN + FP + FN + TP Accuracy = 205 + 94 / 205 + 2561 + 290 + 94 Accuracy = 299 / 3150 = 0.09492063492 = 0.1
Accuracy can be misleading if used with imbalanced datasets, and therefore there are other metrics based on confusion matrix which can be useful for evaluating performance. In Python, confusion matrix can be obtained using “confusion_matrix()” function which is a part of “sklearn” library [17]. This function can be imported into Python using “from sklearn.metrics import confusion_matrix.” To obtain confusion matrix, users need to provide actual values and predicted values to the function.
The predictive model’s accuracy rate is estimated to be 9.49% based on its accuracy score of 0.0949. To put it another way, for roughly 9.49% of the total cases in the dataset, the model accurately predicted the outcomes (both churn and no churn). Although accuracy is a typical statistic, it might be deceptive in datasets that are unbalanced and have a predominance of one class, like those used in churn prediction. As such, it is imperative to augment accuracy with other measures, particularly when precisely identifying particular classes, such as churn occurrences, is more important.
CONCLUSION
This research has explored the intricacies of customer turnover in the dynamic telecom sector, providing a thorough understanding of the elements impacting this important phenomena. After carefully examining the dataset and doing rigorous statistical analyses, we have identified subtle patterns and trends that highlight how changing consumer behavior is.
A greater understanding of the underlying structures of the data has been made possible by the deployment of sophisticated dimensional reduction techniques, such as Principal Component Analysis (PCA) and t-Distributed Stochastic Neighbor Embedding (t-SNE). These methods have improved our comprehension of customer attrition and opened the door to more sensible choices when it comes to customer retention tactics.
The results highlight how important it is to use data analytics to navigate the highly competitive telecom market. An essential distinction in the industry as it deals with changing market dynamics and consumer expectations is the capacity to derive actionable insights from data. This study is at the nexus of data science and telecoms, providing useful insights for businesses looking to strengthen their client retention strategies.
Essentially, this study adds to the body of knowledge about customer turnover from an academic perspective while simultaneously providing telecom companies with useful guidance on how to remain competitive in a market that is changing quickly. Businesses may proactively solve issues, improve customer experiences, and build enduring relationships with their customers by realizing the critical role that data analytics plays.
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