Interpreting PCA Results

Welcome to our exploration of Interpreting Principal Component Analysis (PCA) Results and its application in Machine Learning. Today, we will first generate a synthetic dataset with features inherently influenced by various built-in factors. Next, we will computationally implement PCA and explore variable interactions. We will then compare the performance of models trained using the original features and the principal components derived from PCA. Let's dive right in!

Incorporating PCA-reduced data into Machine Learning models can significantly enhance our model's efficiency and lessen the issue of overfitting. PCA aids in reducing dimensionality without losing much information. This feature becomes increasingly useful when we deal with real-life datasets that have numerous attributes or features.

Our first step is to create a synthetic dataset, which consists of several numeric features that naturally influence each other. The purpose of including these dependencies is to later determine if PCA can detect these implicit relationships among the features.

Note: In this example, the feature monthly_calls is generated as monthly_calls <- 100 + 2 * tenure + 0.5 * data_usage, which is a nearly exact linear combination of tenure and data_usage (with no added noise). This makes the dataset less realistic, but it helps illustrate how PCA can identify and handle such linear dependencies.

set.seed(42) # Set random seed for reproducibility

# Number of samples
n_samples <- 1000

# Generate features
tenure <- round(rnorm(n_samples, mean = 24, sd = 6))  # Average tenure of 24 months
monthly_charges <- rnorm(n_samples, mean = 70, sd = 12)  # Average monthly charge of $70
data_usage <- rnorm(n_samples, mean = 20, sd = 5)  # Average data usage of 20 GB
monthly_calls <- 100 + 2 * tenure + 0.5 * data_usage  # More calls with higher tenure and data usage (deterministic)
customer_satisfaction <- sample(1:10, n_samples, replace = TRUE)  # Satisfaction scores from 1 to 10

# Derived correlated features
total_charges <- monthly_charges * tenure
age_of_account <- tenure + rnorm(n_samples, mean = 0, sd = 1)  # Very similar to tenure

# Binary target variable 'Churn' - arbitrary function influenced by different factors
churn <- as.integer((tenure < 12) | (monthly_charges > 100) | (data_usage > 30) | (customer_satisfaction < 4))

Now, let's put our data into a data frame:

# Create Data Frame
df <- data.frame(
  Monthly_Charges = monthly_charges,
  Total_Charges = total_charges,
  Tenure = tenure,
  Data_Usage = data_usage,
  Monthly_Calls = monthly_calls,
  Age_of_Account = age_of_account,
  Customer_Satisfaction = customer_satisfaction,
  Churn = churn
)

# Separate data and target variable for Logistic Regression
data <- df
target <- data$Churn
data$Churn <- NULL

This portion of the code generates random variables to simulate typical customer usage data. This includes usage facts such as monthly_charges, monthly_calls, and data_usage, and a binary variable churn that is influenced by these features. All this data is assembled together in a data frame.