🏃‍♂️ Human Activity Recognition (HAR) – Data Analysis

Multivariate Analysis • PCA • Factor Analysis • ICA • Sensor Data Modeling

This repository contains a full multivariate analysis of the Human Activity Recognition (HAR) dataset collected from smartphone sensors. The goal is to uncover latent structures, evaluate redundancy among features, and compare PCA, FA, and ICA in modeling human motion signals.

“The primary objective of this report is to analyse the latent underlying structures of this dataset using multivariate methods… and interpret the resulting latent dimensions in the context of human physical activity.”

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📌 Dataset Overview

The dataset is a simplified version of the UCI HAR dataset and includes:

• 30 volunteers performing 6 activities
  • Walking
  • Walking Upstairs
  • Walking Downstairs
  • Sitting
  • Standing
  • Laying
• 33 features total
  • 20 time‑domain features
  • 13 frequency‑domain features (via FFT)
• 10,299 observations
• No missing values

All features are normalized to [-1, 1], enabling fair comparison across sensors.

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🔍 Data Inspection

Outliers

• Outliers detected using z-score > 5 (very conservative).
• Fewer than 50 outliers per feature → < 1% of data.
• Outliers mainly appear in 3‑axial raw signals (e.g., tBodyAcc-mean()-Z).
• Magnitude and frequency-domain features show almost no outliers.

Interpretation:

• Directional axes (X/Y/Z) capture abrupt motion → more variability.
• Magnitude and FFT features are smoother → more stable.

“Retaining these outliers preserves the natural variability of human motion… important for meaningful latent structure analysis.”

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📊 Exploratory Analysis

Boxplots (Figure 1)

• All features normalized to [-1, 1], but variance differs substantially.
• Time-domain 3‑axial signals show larger variability.
• Magnitude and frequency-domain features are more stable.

Implications:

• Standardization is required before multivariate modeling.
• Variance differences suggest non-normality and heterogeneous structure.

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🔥 Correlation Structure (Figure 2)

Key patterns:

• Strong correlation block (r > 0.8):

  • 13 frequency-domain features
  • 5 magnitude time-domain features
    → Suggests a shared latent factor (overall movement intensity)

• Weak correlations between:

  • 3‑axial time-domain features
  • frequency/magnitude features
    → Suggests independent signal sources

Methodological implications:

• PCA is appropriate for dimensionality reduction.
• FA is suitable for modeling shared covariance among magnitude/FFT features.
• ICA is promising because independence exists across feature groups.
• CCA/MLR are not suitable due to lack of strong cross-block correlations.

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🧪 Normality Assessment

QQ Plots (Figure 3)

• Most features deviate from the normal line → non-normal distributions.

Mardia’s Test

• Skewness = 623.41, p = 0.0
• Kurtosis = 3244.07, p = 0.0
  → Strong rejection of multivariate normality.

Implications:

• FA assumptions partially violated.
• ICA is well justified (requires non-Gaussianity).
• PCA remains valid but sensitive to heavy tails.

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🧮 Principal Component Analysis (PCA)

PCA Findings (Figure 4)

• PC1 explains ~50% of total variance.
• PC1–PC6 explain ~75% of variance.
• PC1–PC7 explain ~80% of variance.

Component selection:

• Kaiser criterion → retain 6 PCs
• Eigenvalue ratios (>2) support similar range
• Scree plot elbow at PC1–PC2

Conclusion:
The 33-dimensional dataset can be reduced to 6–7 principal components with minimal information loss.

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🧱 Factor Analysis (FA)

Suitability

• KMO = 0.9291 → excellent factorability
• Bartlett test unstable numerically, but KMO strongly supports FA

Factor selection (Figure 5)

• Parallel analysis → 6 factors
• Likelihood ratio test → 6-factor model fits adequately (p > 0.05)

Interpretation

• FA captures shared structure among:
  • frequency-domain features
  • magnitude features
• 3‑axial time-domain features load weakly → partially independent

Conclusion:
FA reveals latent movement-intensity factors, but does not fully capture independent directional signals.

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🧬 Independent Component Analysis (ICA)

ICA is motivated by:

• strong non-normality
• weak correlations across feature groups
• multiple independent physical processes (gravity, body acceleration, jerk, gyro)

ICA extracts statistically independent components that likely correspond to:

• gravity-related motion
• body acceleration
• jerk signals
• rotational motion (gyroscope)

ICA complements PCA/FA by revealing independent signal sources rather than covariance-based structure.

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🧠 Method Comparison

Method	Captures	Strengths	Limitations
PCA	Max variance directions	Great for dimensionality reduction	Sensitive to heavy tails
FA	Shared covariance	Strong for magnitude/FFT block	Assumes MVN; weaker for 3‑axial signals
ICA	Independent sources	Ideal for non-Gaussian sensor data	Components may be unstable

Conclusion:
HAR data is best understood as a combination of shared movement-intensity factors and independent directional/rotational components.

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📁 Repository Structure

├── data/                 # HAR dataset (not included)
├── notebooks/            # EDA, PCA, FA, ICA analysis notebooks
├── src/                  # Modular Python scripts for analysis
├── figures/              # Exported plots (boxplots, heatmaps, PCA, FA, ICA)
└── README.md             # Project documentation