Computational Linear Algebra: PCA & k-Means Clustering

Project Overview

This project applies Principal Component Analysis (PCA) for dimensionality reduction and k-Means Clustering for market segmentation on a dataset containing psychological and demographic features.

The core objective is to analyze the underlying data structure, compare the impact of StandardScaler and MinMaxScaler preprocessing methods on the results, and identify meaningful clusters (segments) of individuals based on their personality traits, interests, and lifestyle features.

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Team

• MOHAMMADI, Arian (ID: 346278)
• KOSE, Mustafa Kerem (ID: 339018)

Academic Year: 2024/2025

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Data & Preprocessing

Feature Selection

The analysis utilized features from the following categories:

• Fixed: Personality, Health
• Personal: Finance
• Entertainment: Movies, Music

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Data Preparation Steps

1. Sampling: A random sample of 2/3 of the total rows was used.
2. Imputation:
   • Missing values were filled using the mode for categorical features.
   • Missing values were filled using the mean for numerical features.
3. Encoding: Categorical variables were converted using Label Encoding.
4. Scaling: Two datasets were prepared for comparative analysis:
   • Xstd_df: Scaled using StandardScaler (Mean ≈ 0, Variance ≈ 1).
   • Xmm_df: Scaled using MinMaxScaler (Range [0, 1]).

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Key Findings

PCA for Dimensionality Reduction

PCA was performed to retain components explaining at least 33% of the variance, capping the maximum number of components at 5.

Standard Scaled (Xstd_df)

• Components Used (m): 5
• Rationale: Chosen for optimal contribution, as equal variance makes the dataset highly suitable for PCA.

MinMax Scaled (Xmm_df)

• Components Used (m): 4
• Rationale: Achieves the target 33% variance with fewer components.

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k-Means Clustering Results

The optimal number of clusters (k = 4) was determined using the Silhouette Score for both datasets.

Scaling Method	k	Silhouette Score
MinMax Scaled	4	0.161
Standard Scaled	4	0.158

Observation: MinMax scaling produced slightly better separation.

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Cluster Interpretations (Centroid Analysis)

The four clusters were profiled by interpreting the Principal Components (PCs) that defined their cluster centroids:

1. High-Energy Social Adventurers

• Dominant Features:
  • High values in Social Dynamism (MinMax)
  • High Energy Lifestyle (Standard)
• Evaluation Notes:
  Strongly validated by External Evaluation (Age) — peak representation among younger individuals (19–21).

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2. Emotional & Romantic

• Dominant Features:
  • High Emotional Instability (MinMax)
  • High Emotional Reactivity (Standard)
• Evaluation Notes:
  Linked to emotional/sentimental characteristics, consistent across both scaling methods.

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3. Practical / Financial Focus

• Dominant Features:
  • High Consumerism/Materialism (MinMax)
  • High Strategic Thinking (Standard)
• Evaluation Notes:
  Describes individuals driven by financial concerns or a practical, disciplined approach to life.

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4. Values & Well-being

• Dominant Features:
  • High Cultural Action/Tradition (MinMax)
  • High Values and Well-being (Standard)
• Evaluation Notes:
  Individuals focused on personal growth, empathy, and cultural values.

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Dependencies

This analysis requires standard Python libraries for data manipulation and machine learning:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn.preprocessing import StandardScaler, MinMaxScaler
from sklearn.decomposition import PCA
from sklearn.cluster import KMeans
from sklearn.metrics import silhouette_score, silhouette_samples